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MODELING, ANALYSIS, AND DESIGN OF MULTIRATE SYSTEMS By Mauro Cimino Bachelor of Engineering (“Laurea di primo livello in Ingegneria Informatica”) University of Rome “Tor Vergata” Rome, Italy 2003 Master of Engineering (“Laurea specialistiva in Ingegneria dell’Automazione”) University of Rome “Tor Vergata” Rome, Italy 2005 Submitted to the Faculty of the Graduate College of the Oklahoma State University in partial fulfillment of the requirements for the Degree of DOCTOR OF PHILOSOPHY December, 2010 COPYRIGHT By Mauro Cimino December, 2010 MODELING, ANALYSIS, AND DESIGN OF MULTIRATE SYSTEMS Dissertation Approved: Thesis Adviser, Dr. Prabhakar R. Pagilla Committee Member, Dr. Lawrence L. Hoberock Committee Member, Dr. Gary E. Young Committee Member, Dr. Martin Hagan Dean of the Graduate College ii ACKNOWLEDGMENTS I wish to thank my advisor, Dr. Prabhakar R. Pagilla, for his guidance and support towards the achievement of my doctoral degree, and for his friendship. I also wish to thank my committee members, Dr. Lawrence L. Hoberock, Dr. Gary Young, and Dr. Martin Hagan for their insightful suggestions and opinions. Un ringraziamento particolare va ai miei genitori e a mio fratello che mi hanno sempre sostenuto ed incoraggiato durante tutto il periodo di permanenza in Oklahoma. iii TABLE OF CONTENTS Chapter Page 1 Introduction 1 2 Preliminaries 6 2.1 Upsampling and Downsampling operations . . . . . . . . . . . . . . 7 2.1.1 Timedomain characteristics . . . . . . . . . . . . . . . . . . . 7 2.1.2 Transformdomain characteristics . . . . . . . . . . . . . . . . 9 2.2 LPTV systems and shiftinvariance property . . . . . . . . . . . . . . 11 2.3 Lifting technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4 Modified Ztransforms . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3 Design of LPTV controllers for inputstate model matching 19 3.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.2 Insights on the controller structure . . . . . . . . . . . . . . . . . . . 23 3.3 Multirate control design . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.3.1 Design of the gain matrices KL and LL . . . . . . . . . . . . . 27 3.3.2 Design of Cϕ and Dx to achieve ripplefree response . . . . . . 30 3.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4 Frequency domain modeling and parametrization of LTI Controllers 42 4.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.2 Modeling the multirate system . . . . . . . . . . . . . . . . . . . . . . 46 4.3 Closedloop stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 iv 4.4 Model matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.4.1 Model matching at the slow rate . . . . . . . . . . . . . . . . . 55 4.4.2 Model matching at the fast rate . . . . . . . . . . . . . . . . . 57 4.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 5 Ripplefree conditions in multirate systems using LTI controllers 63 5.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 5.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 5.3 Ripplefree conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.4 Digital higherorder filters . . . . . . . . . . . . . . . . . . . . . . . . 75 5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 6 Conclusions and future work 77 BIBLIOGRAPHY 79 v LIST OF FIGURES Figure Page 1.1 HDD logical organization [1]. . . . . . . . . . . . . . . . . . . . . . . 1 2.1 Upsampling and downsampling operations. . . . . . . . . . . . . . . 9 2.2 Example corresponding to the block diagram of Fig. 2.1B with N = 2. The digital signal yT (on the left) is first downsampled (center) and then upsampled (on the right). . . . . . . . . . . . . . . . . . . . . . 9 2.3 Examples of multirate systems. . . . . . . . . . . . . . . . . . . . . . 11 2.4 A particular example of a causal linear LPTV system. . . . . . . . . . 13 3.1 Control system structure . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.2 Closedloop system response to a step reference signal: the state variables of the multirate closedloop system are compared with the state variables of the desired slowrate LTI system. The signals x1, x2 are the velocity and position, respectively, of the double integrator continuoustime plant. The closedloop system output is y(t) = x2(t). . . . . . . 40 4.1 Multirate control scheme. . . . . . . . . . . . . . . . . . . . . . . . . 45 4.2 Multirate control scheme. . . . . . . . . . . . . . . . . . . . . . . . . 46 5.1 Multirate Control System. . . . . . . . . . . . . . . . . . . . . . . . . 64 5.2 Multirate Control System. . . . . . . . . . . . . . . . . . . . . . . . . 65 5.3 Plant output y(t). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.4 Controller output uT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 vi NOMENCLATURE Symbols and operators R Real number field N Natural number field N+ Set of all the nonnegative natural numbers t Time variable (t ∈ R) T Control update rate (or base rate) Ts Measurement update rate Z Transform operator LN Lifting operator Shift operator at the sampling time T d Unit delay at the sampling time T s Shift operator at the sampling time Ts ds Unit delay at the sampling time Ts ↑N Upsampling operator by a factor N ↓N Downsampling operator by a factor N j Imaginary unity (√−1) ⊤ Transpose operator ker Null space operator P(s) Laplace transform of the controlled plant H(s) Laplace transform of the zeroorder hold y(·) Plant output signal r(·) Reference signal LTI Linear, timeinvariant LPTV Linear, periodically timevarying HDD Harddisk drive FIR Finite impulse response vii CHAPTER 1 Introduction Multirate systems are widely used in process and manufacturing industries. In some applications the rate of update of the feedback measurement is slower than the controller update rate. One of these applications is the control of the HardDisk Drive (HDD) Read/Write (RW) head [2]. In HDD data are stored on the platters along thousands of concentric circular tracks. As shown in Fig. 1.1, each track is divided into sectors, which are usually the smallest addressable units containing the data stored on the harddisk drive. Each sector stores a certain amount of user data, and Figure 1.1: HDD logical organization [1]. reserves the remaining for control and management purposes of the drive. Those additional bytes are usually stored at the beginning of the sectors and include the sector ID information. The sector ID is useful for locating the sector on the disk when certain data have to be retrieved or stored. The angular position of the R/W magnetic head is obtained every time the R/W head passes through the part of the sector containing the sector ID information. Therefore, the angular position is obtained at different rates depending on the rotational speed of the platters. The control input 1 to the VoiceCoil Motor Actuator, which drives the arm containing the R/W head, is provided by the digital board installed on the back of the HDD. The multirate nature of the system is due to the fact that the discretetime control action provided by the digital board updates at a rate much faster than the rate of update of the R/W head position feedback. Other examples of multirate systems where the rate of update of the feedback measurement is slower than the update rate of the control signal include the octane rating control in the continuous catalytic reforming process [3] and the control of chemical concentrations in distillation columns [4]. Sometimes, these multirate systems are treated as singlerate ones, and classic design tools for singlerate systems are utilized to develop control strategies. However, neglecting the multirate nature of these systems may lead to unwanted results. Therefore, the development of strategies to model, analyze and control multirate systems is very important and a large amount of work has been done in literature to address these issues. In the late 1950’s the frequency decomposition and the switch decomposition techniques [5–9] were utilized in singlerate sampleddata systems to analyze the response of the continuoustime plant between samples. These two techniques were shown to be related in [10]. The switch decomposition was first used in [6] to model multirate timevarying systems as singlerate Linear TimeInvariant (LTI) ones. This in turn facilitated the use of existing singlerate techniques to design controllers for multirate systems. However, the main drawback of this technique was that the modeled system could be very difficult to analyze in some cases, based on the rates of operation of the samplers. In the statespace domain, a different technique, referred to as the lifting technique [11] (or as blocking technique in signal processing), was developed to analyze the intersample behavior of singlerate sampleddata systems. This technique, which was initially used in [12] also to model Linear Periodically TimeVarying (LPTV) systems as singlerate LTI ones, became commonly used in the multirate sys 2 tems literature for both modeling and design problems. Compared to the frequency and switch decomposition techniques, the lifting technique provides a more compact solution for modeling multirate systems as singlerate systems. Several control problems for multirate systems have been addressed in the last three decades using multirate control structures: optimal control [13–15], pole placement [16, 17], model matching [18, 19], robust control [18, 20–23], ripplefree output tracking and regulation [24–27], and controller parametrization [23, 28, 29]. In most of these multirate control problems the plant to be controlled is represented by a LTI system, and the controller is designed so that the control signal updates several times between two consecutive feedback measurements. This control strategy was shown to be much more successful than the classical LTI control strategy for most of the above mentioned control problems. Due to this reason and to the simplicity of designing these controllers with the use of the lifting technique, this multirate control strategy became very popular in the literature. In particular, the class of controllers which received most of the attention in the multirate control system area is the one of multirate LPTV controllers, characterized not just by having different input and output update rates but also by the periodically timevarying nature of their gains. The reason for the popularity of multirate LPTV controllers was mainly due to the existence of the lifting technique. In fact this technique allows one to rewrite multirate LPTV systems as singlerate LTI ones, hence facilitating LPTV controller design. The effectiveness of a multirate LPTV controller over a singlerate one was initially shown in [20] and [18]. In particular, the theory developed in [20] for singleinput singleoutput (SISO) bicausal plants, showed that LPTV controllers can be designed to remove blocking zeros, to obtain an arbitrarily large gain margin (in singlerate LTI control systems, instead, the gain margin is known to be limited by the unstable zeros and poles of the plant [30]), and to simultaneously stabilize any finite set of plants. In [18] it is also shown that poles and zeros of the closedloop transfer function can 3 be arbitrarily placed, resulting in the opportunity to meet any gain and phase margin specifications. However, a drawback derived from the use of LPTV controllers may sometimes be the presence of ripples in the output of the continuoustime plant. The rippleeffect is a common undesired phenomenon in sampleddata systems. In the case of singlerate systems it is well known to be mostly due to the presence of unobservable dynamics [9] generated through the choice of a critical sampling period [31]. The output of multirate systems, unlike that of singlerate systems, can be affected also by another form of ripples which is due to the timevarying nature of the closedloop system. In control systems where the measurement update rate is a multiple of the control update rate this undesired rippleeffect, unlike the one due to the selection of a critical sampling period, can be detected from the output of the system sampled at the control update rate, but is not visible from the measurements of the plant output (which is sampled at the slower measurement update rate). Even though several solutions to the steadystate ripple effect have been found in the context of output tracking and regulation problems [24–27], the transient response of the closedloop system may still be negatively affected by the use of LPTV controllers. Therefore, with the aim of reducing nonlinear behaviors in both transient and steadystate response, in this work a particular subclass of multirate LPTV controllers is considered (chapters 4 and 5). Each member of this class comprises an upsampling operator (which is a static timevarying operator) followed by a LTI system. The use of this controller structure (as opposed to the more general form of multirate LPTV controllers) may reduce undesired oscillatory behavior in the transient response of the continuoustime plant. One of the main contributions provided in chapter 4 consists of a procedure to parameterize the set of all the LTI controllers for which model matching is achieved with a desired digital LTI system in the two cases: (1) the desired LTI system updates at the control update rate, (2) the desired LTI system updates at 4 the measurement update rate. The analysis of the effects of these controllers on the steadystate response of the closedloop system is provided in chapter 5. In particular, conditions to achieve a steadystate ripplefree response are obtained, and a design procedure that satisfy the conditions is provided. These conditions, differently from [24–27], are obtained in the context of a model matching problem using the above mentioned controller structure. The model matching problem addressed in chapter 4 is the common inputoutput model matching problem, where the closedloop system output is required to match the output of a desired system when the same reference signal is provided to the two systems. The more complex problem of inputstate matching using LPTV controllers is addressed in chapter 3. Differently from the inputoutput model matching problem, the inputstate matching problem requires the state of the multirate closedloop system to match the state of a desired LTI system. To address this problem a more general form of multirate LPTV controllers is considered. In the same chapter conditions to achieve a steadystate ripplefree response to step reference signals are also provided. Preliminary mathematical tools, which are necessary for the formulation of the problems and the development of results in the above mentioned chapters, are given in chapter 2. A discussion about the topics of future research is given in chapter 6. 5 CHAPTER 2 Preliminaries The mathematical tools shown in this chapter will be utilized in the rest of this work to analyze and design multirate sampleddata systems. It is well known that the output of a digital LTI system in a singlerate framework can be analyzed using the tools shown in every digital control system book (see [32, 33]). However, for systems with inputs and outputs updating at different rates, other tools, such as upsamplers, downsamplers, lifting technique and modified Ztransform, can be needed for analysis and modeling. These tools and the properties needed for subsequent developments are discussed in this chapter. Some preliminary notations are introduced first. The control systems considered in this work involve digital signals and systems updating at two different rates, 1/T and 1/Ts, where Ts = NT, and N > 1 ∈ N+. Since Ts > T, Tsupdating and Tupdating signals (or systems) will be respectively referred to as slowrate or fastrate signals (respectively, systems). To distinguish between slowrate and fastrate signals (or system) the superscripts T and NT, respectively, will be utilized to refer to their rate of update. The unit delay operators d and ds will be utilized in the Ztransforms of fastupdating and slowupdating sequences, respectively, to distinguish between unit delays of T and NT seconds. Also, lower case and upper case letters will be used to refer to signal (or systems) expressed in timedomain and transform domain, respectively. Therefore, the Ztransforms of a fastupdating sequence yT [k] and a 6 slowupdating sequence yNT [k] are given by Y T (d) , Z{yT [k]} = X∞ k=0 yT [k]dk (2.1) Y NT (ds) , Z{yNT [k]} = X∞ k=0 yNT [k]dks (2.2) Wherever it is clear from the context, the arguments k (in the time domain) and d or ds (in the transform domain) will be omitted. 2.1 Upsampling and Downsampling operations Modeling a multirate system involves the use of upsampling and downsampling operations. In this section the discussion is limited to only the properties of these operations which will be useful for subsequent developments. A more extensive discussion of these operations is given in [34, 35]. 2.1.1 Timedomain characteristics Upsampling or downsampling a digital signal by a factor N respectively corresponds to increasing or decreasing by N times the rate of update of the original signal. Since only signals updating at the rates 1/NT and 1/T are considered, only upsampling and downsampling operations by a factor N are utilized in this work to switch between these two rates. Therefore, the bracket notations [·]T and [·]NT , where the superscript denotes the new rate of update of the signal, are utilized to refer to upsampled and downsampled signals (or systems), respectively. Nested notations are also utilized, when needed, to keep track of the sequence of upsampling and downsampling operations on signals. For instance, the notation [[·]NT ]T is utilized to denote signals that have been downsampled first and then upsampled; some properties of these nested notations are given in the following, after the definitions of upsampling 7 and downsampling operations. Let VT and VNT be respectively the spaces of Tupdating and NTupdating sequences. The upsampling and downsampling operators by a factor N, respectively denoted as ↑N and ↓N, are defined as ↑N: VNT → VT , ↓N: VT → VNT . Therefore, upsampling a sequence yNT by a factor N corresponds to increasing by N times the rate of update of the signal. This is done by inserting N −1 zerovalued samples between two consecutive samples of the original signal, that is, ↑N yNT , [yNT ]T [k] = yNT [k/N], if N divides k 0, otherwise. (2.3) Therefore, the upsampled sequence [yNT ]T , contains all the information needed to reconstruct yNT . Conversely, downsampling a sequence yT corresponds to reducing by N times the rate of update of the signal. This is done by discarding the last N −1 samples for every consecutive group of N samples of the initial sequence, that is, ↓N yT , [yT ]NT [k] = yT [kN], k = 0, 1, 2, . . . (2.4) This results in a loss of highfrequency information about the original sequence yT . Some of the main properties of these operations are discussed in [34, 35]. One property of the upsampling and downsampling operators is that ↓N↑N= I where I is the identity operator. This property, which can be verified by using (2.4) and (2.3), implies that [[yNT ]T ]NT = yNT , for any slowupdating sequence yNT . Other properties are shown through the two block diagrams in Fig. 2.1(A) and Fig. 2.1(B) where the blocks (↑N) and (↓N) respectively represent the upsampling and down 8 sampling operations by a factor N. The block scheme in Fig. 2.1(A) shows a portion (A) (B) NT y(t) yT N N N yNT [yT ]NT [yNT ]T [[yT ]NT ]T Figure 2.1: Upsampling and downsampling operations. of a dualrate control scheme, where the continuous signal y(t) is slowsampled at the rate 1/NT and then upsampled by a factor N. The signals [yNT ]T and [[yT ]NT ]T of Fig. 2.1(A) and (B), respectively, are equivalent [9]. This implies the equivalence of the signals yNT and [yT ]NT . Fig. 2.2 shows an example corresponding to the evolution of a signal yT in input to the block diagram of Fig. 2.1B, for the case N = 2. Figure 2.2: Example corresponding to the block diagram of Fig. 2.1B with N = 2. The digital signal yT (on the left) is first downsampled (center) and then upsampled (on the right). 2.1.2 Transformdomain characteristics In the Ztransform domain, [Y NT ]T and [Y T ]NT are utilized to denote the Ztransform of the upsampled sequence [yNT ]T and the downsampled sequence [yT ]NT , respectively. These transforms can be computed as follows [34, 35]: [Y NT ]T (d) = Y NT (dN) [Y T ]NT (ds) = 1 N NX−1 k=0 Y T (d1/N s e−2πjk/N) (2.5) 9 where j denotes the imaginary unit. The rules for nested upsampling and downsampling operations discussed previously for signals in timedomain also apply to signals in Ztransform domain, that is, [Y T ]NT = Y NT , [[Y NT ]T ]NT = Y NT . Considering that a transfer function is the impulse response of a LTI system, the same notations and properties considered so far for Ztransforms of sequences apply also to transfer functions of digital systems. The following two properties of upsampling and downsampling operators are useful for the convolutions between signals and systems: [Y NT 1 Y NT 2 ]T (d) = [Y NT 1 ]T (d)[Y NT 2 ]T (d) = Y NT 1 (dN)Y NT 2 (dN) (2.6) h Y T 3 Y NT 2 T iNT (ds) = [Y T 3 ]NT (ds)Y NT 2 (ds) (2.7) where Y NT 1 , Y NT 2 and Y T 3 denote the Ztransforms of signals and/or systems. These two properties can be obtained using the definitions (2.5). In particular, it is shown in (2.6) that the upsampling operation is linear with respect to the product, that is [Y NT 1 Y NT 2 ]T = [Y NT 1 ]T [Y NT 2 ]T . However, it is important to notice that this does not hold for the downsampling operation, that is, [ATBT ]NT 6= [AT ]NT [BT ]NT . (2.8) Properties (2.6) and (2.7) are useful to model the two block diagrams in Fig. 2.3. In Fig. 2.3(A) the output Y T , Z{yT } is related to the input UT , Z{uT } by the expression Y T = GTUT . However, since the downsampling operation is not linear with respect to the product, [Y T ]NT 6= [GT ]NT [UT ]NT . Therefore, [Y T ]NT = [GTUT ]NT . 10 (A) (B) uT GT (d) yT N N N [yT ]NT eNT CNT 1 (ds) CT 2 (d) uNT 1 [uNT 1 ]T uT2 [uT2 ]NT Figure 2.3: Examples of multirate systems. In Fig. 2.3(B) the signal [UT 2 ]NT , Z{[uT2 ]NT } can be obtained using (2.5) and (2.7) as follows: [UT 2 ]NT = h CT 2 UNT 1 T iNT = [CT 2 ]NTUNT 1 = [CT 2 ]NTCNT 1 ENT where UNT 1 , Z{uNT 1 } and ENT , Z{eNT }. 2.2 LPTV systems and shiftinvariance property In this section, the definitions of timeinvariant, periodically timevarying, and (m, n) shiftinvariant systems are given. These definitions will be useful to classify the multirate systems utilized in the next chapters. Let : VT → VT and s : VNT → VNT be the linear transformations respectively corresponding to the delay operators d and ds, that is, Z{ uT1 } = dUT 1 , Z{ suNT 2 } = dsUNT 2 where UT 1 and UNT 2 are the Ztransforms of the sequences uT1 and uNT 2 , respectively. Then it is well known that a Tupdating and a NTupdating causal linear system with impulse response cT and cNT , respectively, are timeinvariant if: pcT = cT p, ps cNT = cNT ps , p ∈ N+. 11 This definition does not apply to upsampling and downsampling operators. In fact, it follows from (2.3) and (2.4) that s ↓N =↓N N (2.9) N ↑N =↑N s (2.10) N ↑N↓N =↑N↓N N (2.11) i ↑N↓N 6=↑N↓N i, i = 1, . . . ,N − 1. (2.12) Equations (2.9) and (2.10) clearly show that the upsampling and downsampling operators are not timeinvariant. In fact, the upsampling operator has the inputoutput property that shifting the input by 1 sample results in shifting the output by N samples. For the downsampling operator, instead, a shifting of the input by N samples results in a shifting of the output by 1 sample. For these reasons, the upsampling and downsampling operators are usually referred to as (N, 1) and (1,N)shiftinvariant [36], respectively. The shiftinvariance property for multirate systems can be considered as a generalization of the timeinvariance property for singlerate systems. Equations (2.11) and (2.12) show that the system made of a downsampling operator followed by an upsampling operator is a particular example of a causal linear (N,N)shiftinvariant system, for which a shifting of the input by i samples results in a shifting of the output by i samples if and only if i = kN, k ∈ N. All (N,N) shiftinvariant systems are also called periodically timevarying systems, or Nperiodic systems [18]. A general statespace representation of all causal LPTV systems operating with sampling time T is given by ξT [kN + i + 1] = AiξT [kN + i] + BiuT [kN + i] yT [kN + i] = CiξT [kN + i] + DiuT [kN + i] (2.13) 12 where ξT , uT and yT are the state, input and output variables of the system, k ∈ N, i = 0, . . . ,N − 1, and the matrices Ai, Bi, Ci and Di vary as i goes from 0 to N − 1. A particular example of a LPTV system, which will be used in chapter 4 and 5 is the one shown in Fig. 2.4 where CT denotes the transfer function of a Tupdating, uT yT N N CT Figure 2.4: A particular example of a causal linear LPTV system. causal LTI system. If (AC,BC,CC,DC) is a statespace representation of CT , it is possible to show that the LPTV system in Fig. 2.4 has a statespace representation of the form (2.13) with (Ai,Bi,Ci,Di) = (AC,BC,CC,DC) if i = 0 (I, 0, 0, 0) if i = 1, . . . ,N − 1. 2.3 Lifting technique The lifting technique, which dates back to [11], has been utilized in the last three decades to convert multirate systems into singlerate timeinvariant systems [37]. The notion of the lifting operator and its application to multirate systems are explained in this section. Let VNT N be the set of slowupdating sequences with elements as N dimensional vectors. The lifting operator LN and the lifted version wNT L [k] of a sequence wT [k] ∈ VT are defined as LN : VT → VNT N : wT [k] → wNT L [k] 13 where wNT L [k] , wNT L,0 [k] wNT L,1 [k] ... wNT L,N−1[k] = wT [Nk] wT [Nk + 1] ... wT [Nk + (N − 1)] . (2.14) Since wNT L , LN(wT ) updates at the slow rate 1/NT, its Ztransform, denoted as WNT L , is given by: WNT L , Z{LN(wT )} = WNT L,0 ,WNT L,1 , · · · ,WNT L,N−1 ⊤ where the symbol ⊤ denotes the transpose operator and WNT L,i (ds) = P∞ k=0 wNT L,i [k]dks , i = 0, . . . ,N − 1. The lifting technique can be used to convert LPTV systems of the form (2.13) into equivalent LTI systems. This can be done as follows. Let (Ai,Bi,Ci,Di), i = 0, . . . ,N − 1, be the statespace representation of a SISO LPTV system with period N, input uT and output yT . Its lifted version corresponds to the LTI system, between the input uNT L , LN(uT ) and output yNT L , LN(yT ), with statespace representation 14 (AL,BL,CL,DL) given by AL , NY−1 i=0 Ai = AN−1 · · ·A1A0 BL , BL,0 · · · BL,N−1 , BL,i , QN−1 k=i+1 Ak Bi, i = 0, . . . ,N − 2 BN−1 i = N − 1 CL , C⊤ L,0 · · · C⊤ L,N−1 ⊤ , CL,i , C0, i = 0 Ci Qi−1 k=0 Ak, i = 1, . . . ,N − 1 DL , DL,(0,0), · · · ,DL,(0,N−1) ... ... DL,(N−1,0), · · · ,DL,(N−1,N−1) ,DL,(m,n) , Cm Qm−1 k=n+1 Ak Bn m > n − 1 CmBm−1 m = n − 1 Dm m = n 0 m < n In the transform domain, such a lifted system is represented by the N × N transfer matrix Y NT L = CLds (I − dsAL)−1 BL + DL UNT L (2.15) where Y NT L , Z{yNT L } and UNT L , Z{uNT L }. This shows that the lifted representation of a LPTV system is LTI, and that the lifted system output contains all the information needed to reconstruct the output of the original system. Notice that if also the original system was LTI, that is, (Am,Bm,Cm,Dm) = (An,Bn,Cn,Dn) for 15 m, n = 0, . . . ,N − 1, the lifted representation (AL,BL,CL,DL) would take the form AL , AN, BL , AN−1B AN−2B · · · AB B CL , C CA CA2 ... CAN−1 , DL , D 0 · · · 0 0 CB D · · · 0 0 CAB CB · · · 0 0 ... ... ... 0 CAN−2B CAN−3B · · · CB D and the (m, n)th entry (m, n = 0, . . . ,N − 1) of the corresponding N × N transfer matrix (2.15) would be given by CAm(d−1 s I − AN)−1AN−1−nB + CAm−n−1B, m > n CAm(d−1 s I − AN)−1AN−1−nB + D, m = n CAm(d−1 s I − AN)−1AN−1−nB, m < n. (2.16) It is important to notice that, even though the lifted system is obtained by augmenting the input and output sequences with their values at N consecutive time instants, the orders of a system and its lifted representation are the same. Moreover, since it is always possible to retrieve a system from its lifted representation, the lifting operation is invertible: L−1 N LN = I. The same procedure shown above to obtain a lifted representation of a system can be used also for LPTV systems with input and output updating at different rates, that is, for multirate LPTV systems. In particular, given a SISO LPTV system with period N1, and input and output updating at the sampling periods T1 = N2T and T2 = N3T, respectively, the corresponding lifted representation can be obtained by choosing N = lcm(N1,N2,N3) (where lcm stands for least common multiple, and N1, N2, N3 are positive integers), and by removing from CL and DL and from BL 16 the rows and columns, respectively, corresponding to time instants when the input and output are not available. For instance, consider the case in which N3 = 1 and N1 = N2 = N, that is, the system output updates N times between two consecutive updates of the input, and the matrices Ai, Bi, Ci, Di are periodic with period N. The lifted representation of such a multirate LPTV system can be obtained by removing the last N −1 block columns from BL and DL, and by leaving AL and CL unaltered. Notice that the lifted representation of a multirate LPTV system is a singlerate LTI system. 2.4 Modified Ztransforms The ith modified Ztransform [9] of a fastupdating sequence gT [k], denoted as ˜G NT {i} , is the Ztransform of the sequence {gT [kN + i]}k, that is, ˜G NT {i} (ds) , X∞ k=0 gT [kN + i]dks = Z gT [kN + i] , i = 0, . . . ,N − 1. (2.17) It is possible to show that the first N modified Ztransforms of a sequence gT can be utilized to generate its Ztransform, GT , through the following expression: GT (d) = ˜G NT {0} (dN) + d˜GNT {1} (dN) + · · · + dN−1 ˜G NT {N−1}(dN). (2.18) If gT is the impulse response of a causal LTI system with transfer function GT , it is possible to show that the modified Ztransforms of gT are related to the entries of the lifted system GNT L . This can be shown as follows. Let yNT L and uNT L be the lifted output and input, respectively, of the system GNT L . From the definition of lifted signals in (2.14) it can be easily shown that the entries of the Ndimensional vectors 17 Y NT L and UNT L are respectively the modified Ztransforms of yT and uT , that is, Y T = Y NT L,0 + dY NT L,1 + · · · + dN−1Y NT L,N−1 UT = UNT L,0 + dUNT L,1 + · · · + dN−1UNT L,N−1. If uT is an impulse, UNT L,i ≡ 0 for i = 1, . . . ,N − 1, and UNT L,0 ≡ 1. Therefore, Y T = GNT L,(0,0)UNT L,0 + dGNT L,(1,0)UNT L,0 + · · · + dN−1GNT L,(N−1,0)UNT L,0 = GNT L,(0,0) + dGNT L,(1,0) + · · · + dN−1GNT L,(N−1,0). Moreover, since uT is an impulse, gT = yT , and hence ˜G NT {i} = GNT L,(i,0), i = 0, . . . ,N − 1 (2.19) where GNT L,(i,0) is the ith entry in the first column of the lifted system GNT L . 18 CHAPTER 3 Design of LPTV controllers for inputstate model matching In this chapter a statespace approach to design a controller for multirate systems is proposed. The multirate system comprises a continuoustime plant whose output is measured at the slow rate, 1/Ts, and is provided to a digital periodically timevarying controller operating at the faster rate, 1/T. The first control design objectives is to ensure that the closedloop state vector matches the state vector of a desired system at the slow measurement rate 1/Ts. This control problem is referred to as inputstate matching. The second objective is to ensure that the continuoustime plant exhibits a ripplefree steadystate response to step reference signals. The model matching problem for multirate systems usually refers in literature to the common inputoutput model matching problem. This problem was initially solved in [19] with the use of LPTV controllers first by converting the multirate system into a lifted LTI one, and then by placing all the zeros and poles of the lifted system to match the dynamics of a desired singlerate LTI system. The opportunity to generate a control action between two consecutive feedback measurements was the key for arbitrary placement of both zeros and poles of the lifted LTI system. However, the use of LPTV controllers resulted in undesired steadystate intersample behavior (also referred to as ripple effect) of the closedloop system response. This undesired behavior was shown in [19] to be a sideeffect due to the placement of the zeros of the closedloop system. The problem of ripples in the steadystate output of multirate systems received a large attention in literature. One of the main results was obtained in [24], followed by [26, 27]. However, the main objective of these works is to solve 19 the output regulation and tracking problems in multirate systems, away from the context of model matching. Conditions for a ripplefree response in the context of an inputoutput model matching problem, using a particular class of LPTV controllers, are given in [38]. The inputstate matching problem considered in this chapter, differently from the classical inputoutput model matching problem, requires the state of the multirate system to match the state of a desired LTI system at the measurement sampling instants. This problem has been initially addressed in [2] for multirate systems with slowrate measurement updates and fastrate control action. However, the strategy proposed in [2] allows to consider only desired systems with no forcing input (that is, of the type ζNT [k + 1] = FζNT [k]). One of the contributions in this chapter is to provide necessary and sufficient conditions to achieve closedloop inputstate matching at the measurement update rate with a desired singlerate LTI system which includes the forcing input (that is, of the type ζNT [k + 1] = FζNT [k] + GrNT [k]). Moreover, conditions are also given to achieve a ripplefree step response of the closedloop system. In particular, it will be shown that these conditions depend not just on the ratio between the measurementrate and control updaterate, but also on the steadystate value of the state vector of the desired system to match. To solve the inputstate matching problem a particular set of causal multirate LPTV controllers is considered, and insights on the nature of these controllers is provided. Moreover, a systematic procedure to design such controllers is given, and an example is shown to corroborate it. The theory and the procedure developed in this chapter can be applied to any controllable LTI plant. The initial work of the author was published in [39] and [40], and the subsequent developments can be found in [41]. This chapter is organized as follows. Section 3.1 gives the problem formulation. Insight on the controller structure is provided in section 3.2. The solution to the considered control problem and the design procedure are given in section 3.3. An 20 example with simulations is provided in section 3.4, and conclusions are given in section 3.5. 3.1 Problem formulation Consider the continuoustime LTI system x˙ (t) = Acx(t) + Bcu(t) y(t) = Ccx(t) (3.1) where x(t) ∈ Rnx is the state vector, u(t) ∈ Rnu is the control input vector, and y(t) ∈ Rny is the output vector to be regulated to the constant reference signal rNT [k] ∈ Rny . The control system structure in Fig. 3.1 is considered. The digital controller C operates at the sampling period T, and the measurements of the plant state x(t) are available at the slow sampling period Ts = NT. C ZOH (sI − Ac)−1Bc Ts x Cc rNT uT xNT y Figure 3.1: Control system structure The following causal, digital multirate controller C is considered ϕT [kN + i + 1] = Kϕ,iϕT [kN] + Kx,ixNT [k] + LirNT [k] uT [kN + i] = CϕϕT [kN + i] + DxxNT [k] (3.2) where ϕT ∈ Rn' with nϕ ∈ N+, and i = 0, . . . ,N − 1. The control design objectives are stated in the following problem. Problem 3.1 Design the matrices Cϕ, Dx, Kϕ,i, Kx,i, Li, i = 0, . . . ,N − 1, of the controller (3.2) to achieve closedloop state matching, at the slow rate 1/NT, with the 21 desired singlerate system ζNT [k + 1] = FζNT [k] + GrNT [k] yNT [k] = HζNT [k] (3.3) where ζNT [k] ∈ Rnx+n', H , [Cc, 0ny×n'], and the matrices F, G are chosen such that the desired system (3.3) is stable and exhibits zero steadystate regulation error to a unit step input. Moreover, the matrices Cϕ, Dx, Kϕ,i, Kx,i, and Li have to be designed in order to achieve a ripplefree closedloop response to step reference signals. By “ripplefree” it is meant that the continuoustime regulation error, e(t) = y(t)−r∞ (where r∞ , limk→∞ rNT [k]), has to be zero at steadystate within two consecutive measurement update instants, that is, lim k→∞ Z (k+1)Ts kTs e⊤(t)e(t)dt = 0. (3.4) To ensure the noncriticality of the sampling period T [31] and the stabilizability of the periodic discretetime system formed by the cascade connection of the zeroorder hold, plant and slowrate sampler [18, 23], the following is assumed. Assumption 3.1 The sampling times involved in the multirate system are not critical, that is, λa − λb 6= j 2πk NT , ∀k ∈ N, k 6= 0 where λa and λb are any two distinct eigenvalues of the block diagonal matrix Ac 0nx×1 01×nx 01,1 formed by augmenting the matrix Ac of the continuoustime plant (3.1) with a zero 22 eigenvalue corresponding to the dynamics of the step reference signal rNT . This assumption can be readily satisfied with a proper selection of T and N. 3.2 Insights on the controller structure One may wonder as to why a controller of the form (3.2) is considered instead of one of the more common form ¯ϕT [kN + i + 1] = ¯K ϕ,i ¯ ϕT [kN + i] + ¯K x,ixNT [k] + ¯L irNT [k] ¯uT [kN + i] = Cϕ ¯ ϕT [kN + i] + DxxNT [k] (3.5) where the state ¯ ϕ on the righthand side of (3.5) depends on i. As shown below in Theorem 3.1, this is due to the fact that the set of controllers of the form (3.2) contains the set of all controllers of the form (3.5). Theorem 3.1 Consider the set of all causal LPTV controllers of the form (3.5), where the fastrate updates of the state variable ¯ ϕT influence the controller dynamics. The following two statements hold: 1. Every controller of the form (3.5) can always be written in the form (3.2). 2. A controller in the form (3.2) can be written in the form (3.5) if and only if K⊤ ϕ,i ∈ Span{K⊤ ϕ,i−1}, for i = 1, . . . ,N − 1. Proof. It is straightforward to see through the recursion of both the controllers (3.2) and (3.5) that the two forms are equivalent, that is, ¯ ϕT [kN +i+1] = ϕT [kN +i+1] 23 for every integer k and every i = 0, . . . ,N − 1, if and only if Kϕ,0 = ¯K ϕ,0, Kx,0 = ¯K x,0, L0 = ¯L 0 Kϕ,i = ¯K ϕ,iKϕ,i−1 i = 1, . . . ,N − 1 Kx,i = ¯K ϕ,iKx,i−1 + ¯K x,i i = 1, . . . ,N − 1 Li = ¯K ϕ,iLi−1 + ¯L i i = 1, . . . ,N − 1 (3.6) Statement 1) directly follows from the equations given in (3.6). From the second equation of (3.6) it is clear that it is not possible to rewrite controller (3.2) in the form (3.5) unless K⊤ ϕ,i ∈ Span{K⊤ ϕ,i−1}, for i = 1, . . . ,N −1. 3.3 Multirate control design Problem 3.1 can be divided into the two following subproblems: 1) model matching with the desired system (3.3) and 2) ripplefree steadystate response of the continuoustime plant. It will be shown in this section that the solution to the first control problem requires the design of only the controller periodically timevarying matrices Kx,i, Kϕ,i, and Li, i = 0, . . . ,N − 1. The solution to the second control problem depends on the choice of the remaining controller matrices Cϕ and Dx. To show this, it is necessary to rewrite the closedloop system in a different form. With this aim, consider the zeroorder hold equivalent [32] of the plant (3.1) at the sampling period T xT [kN + i + 1] = xT [kN + i] + uT [kN + i] yT [kN + i] = CcxT [kN + i] (3.7) where = eAcT , = Z T 0 eAcλBcdλ. (3.8) 24 The discretetime system (3.7) and the proposed timevarying controller (3.2) can be rewritten in the following compact form: ξT [kN + i + 1] = ¯ ξT [kN + i] + ¯ wT [kN + i] yNT [k] = HξT [kN] (3.9) where ξT [kN +i] , [(xT [kN +i])⊤, (ϕT [kN +i])⊤]⊤ is the extended state vector, and the matrices ¯ , ¯ are given by ¯ , s Cϕ 0n'×nx 0n'×n' , ¯ , 0nx×n' In'×n' , with s = + Dx, and wT [kN + i] can be thought of as a pseudocontrol action given by wT [kN + i] , KiξT [kN] + LirNT [k] (3.10) with Ki , [Kx,i,Kϕ,i]. Let wNT L be the lifted pseudocontrol signal obtained from wT as shown in section 2.3. The induced lifted version of the system (3.9) is given by: ξT [(k + 1)N] = ¯ LξT [kN] + ¯ LwNT L [k] yNT [k] = HξT [kN] (3.11) where ¯ L , ¯ N, ¯ L , ¯ N−1¯ , · · · , ¯ ¯ ,¯ , wNT L [k] , wT [kN] ... wT [kN + N − 1] . 25 Considering that the state signal ξT in (3.10) is available only at the measurement update rate, the lifted version of the pseudocontrol (3.10) takes the form wNT L [k] = KLξT [kN] + LLrNT [k] (3.12) where KL and LL are constant matrices given by KL , K0 K1 ... KN−1 = Kx,0 Kϕ,0 Kx,1 Kϕ,1 ... ... Kx,N−1 Kϕ,N−1 , LL , L0 L1 ... LN−1 . (3.13) Notice that the lifted singlerate LTI system formed by (3.11) and (3.12) describes the dynamics of the original LPTV closedloop system, formed by (3.2) and (3.7), at the sampling period Ts. In particular, the closedloop singlerate LTI system obtained by combining (3.11) and (3.12) can be rewritten as ξT [(k + 1)N] = (¯ L + ¯ LKL)ξT [kN] + ¯ LLLrNT [k] yNT [k] = HξT [kN]. (3.14) From the closedloop system (3.14) it is clear that, for some given matrices Cϕ and Dx, state matching is achieved at the slowrate 1/Ts with the desired system (3.3) if and only if the periodically timevarying matrices Kx,i, Kϕ,i, and Li, i = 0, . . . ,N −1, are selected such that KL and LL satisfy ¯ L + ¯ LKL = F (3.15) ¯ LLL = G. (3.16) 26 As a consequence, the ripplefree requirement, mainly relies on the remaining constant matrices Cϕ and Dx. 3.3.1 Design of the gain matrices KL and LL In this section, necessary and sufficient conditions are given for the existence of a solution (KL, LL) to (3.15) and (3.16) for any pair (F,G) characterizing the desired system (3.3). The following lemma gives a preliminary result needed to obtain those conditions. Lemma 3.1 (On the existence of a right inverse of ¯ L) The matrix ¯ L, defined for the system (3.11), has a right inverse ¯ + L for any integer nϕ if and only if 1. nx ≤ (N − 1) 2. the pair ( s, Cϕ) is controllable Proof. Considering that ¯ p, where p ∈ N+, is given by ¯ p = ps p−1 s Cϕ 0n'×nx 0n'×n' , ¯ L can be rewritten as ¯ L = ˜R 0nx×n' 0n'×n'(N−1) In' (3.17) where ˜R , N−2 s Cϕ · · · sCϕ Cϕ . (3.18) For ¯ + L to exist, ¯ L must have full row rank. By inspection this occurs if and only if 27 ˜R has full row rank, that is, nx. Sufficiency (⇐). Since N ≥ nx + 1 and the pair ( s, Cϕ) is controllable, ˜R is a full row rank matrix. Necessity (⇒) The necessity of conditions 1 and 2 is now proven by contradiction. Assume that the pair ( s, Cϕ) is not controllable but that condition 1 holds, that is, nx ≤ (N − 1). As a consequence of the CayleyHamilton theorem, the matrices ps , where p ≥ nx, can be expressed as a linear combination of the set of matrices q s, q = 1, . . . , nx − 1. Therefore ˜R does not have full row rank, that is, Rank(˜R ) < nx. Conversely, assume that the pair ( s, Cϕ) is controllable but that nx > (N − 1). Then, for any integer nϕ < nx/(N − 1) (notice that since nx > (N − 1), nϕ = 1 < nx/(N − 1)), the number of columns of ˜R is nϕ(N − 1) < nx. This contradicts the hypothesis that ¯ + L has a right inverse. The following theorem provides the solution of the state matching problem. Theorem 3.2 Under the assumption 3.1, there exists a controller of the form (3.2) such that the state of the closedloop multirate system matches the state of the desired singlerate system (3.3) at the rate 1/Ts, for any pair (F,G), if and only if a. nx ≤ (N − 1) b. the pair (Ac,Bc) is controllable c. Rank(Cϕ) = nu Proof. Sufficiency (⇐). Let conditions a, b and c hold. The existence of matrices KL and LL that satisfy (3.15) and (3.16) has to be shown. Let λ⋆ be an eigenvalue of s and let v⊤ 6= 0 be the corresponding left eigenvector, that is, v⊤ s = λ⋆v⊤. Since (Ac,Bc) is controllable, by assumption 3.1 ( , ) is also controllable. Since static state feedback does not affect the controllability of a system, the pair ( s, ) is controllable. Therefore, it has to be that v⊤ [ s − λ⋆I, ] 6= 0, that is, v⊤ 6= 0. Since 28 Cϕ is a full row rank matrix, then also v⊤Cϕ 6= 0, that is v⊤ [ s − λ⋆I, Cϕ] 6= 0. Therefore, the pair ( s, Cϕ) is controllable, and by lemma 3.1, the right inverse ¯ + L of ¯ L exists. In turn, the solution (KL,LL) to (3.15) and (3.16) is given by KL = ¯ + L(F − ¯ L) (3.19) LL = ¯ + LG (3.20) Necessity (⇒). Since there exists a solution (KL, LL) to (3.15) and (3.16), then [F − ¯ L,G] ∈ Span(¯ L). Since this has to hold for any pair (F,G), it must be that Rank(¯ L) = nx + nφ. In other words, ¯ L must have a right inverse. By lemma 3.1, this implies that nx ≤ N − 1 and that the pair ( s, Cϕ) is controllable. Let λ⋆ be an eigenvalue of s and let v⊤ 6= 0 be the corresponding left eigenvector, that is, v⊤ s = λ⋆v⊤. The proof will proceed by contradiction. Assume first that Rank(Cϕ) = nu but that the pair (Ac,Bc) is not controllable. Then ( s, ) is not controllable, that is, there exists an eigenvector of s such that v⊤ = 0. In turn, this implies v⊤Cϕ = 0, which is a contradiction since the pair ( s, Cϕ) is controllable. Conversely, assume that the pair (Ac,Bc) is controllable but that Rank(Cϕ) < nu. Since the pair (Ac,Bc) is controllable, also the pair ( s, ) is controllable. Therefore, v⊤ s = λ⋆v⊤ and v⊤ 6= 0 for every eigenvectoreigenvalue pair (v⊤, λ⋆) of s. However, since Rank(Cϕ) < nu there may exist an eigenvector v⊤ of s such that v⊤Cϕ = 0. However, this contradicts the fact that the pair ( s, Cϕ) is controllable. Remark 3.1 (On the values of N and nϕ) Notice that conditions a and c of Theorem 3.2 require the selection of N and nϕ, respectively, such that N ≥ nx + 1 and nϕ ≥ nu. 29 3.3.2 Design of Cϕ and Dx to achieve ripplefree response Theorem 3.2 provided conditions for the existence of a solution to the state matching problem for any desired system (3.3), and required the matrices KL and LL to be selected according to (3.19) and (3.20). In this section conditions are found on the controller matrices Cϕ and Dx such that the closedloop multirate system, obtained by designing KL and LL according to (3.19) and (3.20), also exhibits a ripplefree steadystate response to step reference signals. It will be shown at the end of this section that it is possible to design Cϕ and Dx to guarantee a ripplefree steadystate response only if the steadystate value of the desired system (3.3) satisfies some conditions dependent on the null space of the matrix [ − I, ]. Therefore, let us start by denoting with M the (nx + nϕ) × ny matrix characterizing the steadystate value of the desired system states, that is, ζNT [k] → Mr∞ as k → ∞ (3.21) and by referring to Ma and Mb respectively as the nx ×ny and the nϕ ×ny partitions of M such that M , [M⊤ a ,M⊤ b ]⊤. Moreover, let us denote with Ns , [S⊤ a , S⊤ b ]⊤ the matrix whose columns constitute a basis for the null space of [ − I, ], that is, − I Sa Sb = 0 (3.22) where the rowdimensions of Sa and Sb are nx and nu, respectively. The following lemma provides insights on the conditions on the matrix M that are necessary and sufficient in order to guarantee a ripplefree steadystate response to step reference signals. The results of this lemma are utilized later in Theorem 3.3 to derive necessary and sufficient conditions on the matrices Cϕ and Dx. 30 Lemma 3.2 Let the conditions of Theorem 3.2 be satisfied, and let KL and LL be given by (3.19) and (3.20), respectively. The closedloop system in Fig. 3.1 exhibits a ripplefree response to step reference signals if and only if Li = Mb − KiM, i = 0, . . . ,N − 1 (3.23) Proof. Sufficiency (⇐). As ζNT [k] → Mr∞, also ξT [kN] → Mr∞ by the state matching condition. Therefore, xNT [k] → Mar∞ and ϕT [kN] → Mbr∞ as k → ∞. Therefore, the controller dynamics (3.2) at steadystate can be rewritten in the following compact form: ϕT [kN + i + 1] = KiMr∞ + Lir∞ uT [kN + i] = CϕϕT [kN + i] + DxMar∞ (3.24) If (3.23) holds, then KiMr∞ +Lir∞ = Mbr∞. This implies that ϕT [kN +i] → Mbr∞ as ζNT [k] → Mr∞. As a result the controller output uT [kN +i] becomes constant as k → ∞, and therefore, the closedloop system exhibits a ripplefree response. Necessity (⇒). The closedloop system exhibits a ripplefree response to step reference signals and state matching at the slowrate with ζNT [k]. Therefore, since ζNT [k] tends to a constant as k goes to infinity, also ϕT [kN + i] tends to a constant. Because of the state matching condition, ϕT [kN +i] → Mbr∞ as k → ∞. As a result, from (3.24) KiMr∞ + Lir∞ = Mbr∞. The matrix ˜R defined in (3.18) will be utilized in the rest of this section to derive the subsequent results. Theorem 3.3 Let the conditions of Theorem 3.2 be satisfied, and let KL and LL be given by (3.19) and (3.20), respectively. Moreover, let ˜M b be the (N − 1)blocks matrix where each block is equal to Mb, that is, ˜M b , [M⊤ b , · · · ,M⊤ b ]⊤. The closed 31 loop system in Fig. 3.1 exhibits a ripplefree steadystate response to step reference signals if and only if there exist matrices Cϕ and Dx such that • ˜M b ∈ Span{˜R ⊤} • the columns of the matrix Ma DxMa + CϕMb (3.25) are contained in the range space of Ns Proof. By lemma 3.2 it is necessary and sufficient to show that LL + KLM = Mb,L where Mb,L , [ ˜M ⊤ b ,M⊤ b ]⊤. Using (3.19) and (3.20), and considering that (F −I)M + G = 0, the following hold: LL + KLM = ¯ + LG + ¯ + L(F − ¯ L)M = ¯ + L G + (F − I)M − (¯ L − I)M = −¯ + L(¯ L − I)M = −¯ + L ( Ns − I)Ma + N−1 s CϕMb −Mb . (3.26) Sufficiency (⇐). Since the columns of the matrix in (3.25) are in the range space of Ns, s − I Cϕ M = 0. Therefore, ( s − I)Ma = −CϕMb. Considering also that ( Ns − I) = ( N−1 s + · · · + 32 s + I)( s − I), (3.26) can be further expanded as LL + KLM = ˜R ⊤(˜R ˜R ⊤)−1 0 0 I ˜R 0 0 I Mb,L where the first matrix on the righthand side is the right inverse, ¯ + L, of the matrix ¯ L given in (3.17). Therefore, LL + KLM = ˜R ⊤(˜R ˜R ⊤)−1 ˜R 0 0 I Mb,L Let (U, V, ) be the singular value decomposition of ˜R , that is, ˜R = U 0 V ⊤ where U and V are nx×nx and nϕ(N −1)×nϕ(N −1) unitary matrices, respectively, and is a square, diagonal, nonsingular nx ×nx matrix (because Rank{˜R } = nx). It is possible to show that ˜R ⊤(˜R ˜R ⊤)−1 ˜R V = ⊤( ⊤)−1 0 0 0 = V I 0 0 0 (3.27) Therefore, (V, V, I) is the singular value decomposition of the symmetric and square matrix ˜R ⊤(˜R ˜R ⊤)−1 ˜R . Let V be partitioned as V = [V1, V2], where V1 is an nϕ(N − 1) × nx matrix. Then, the matrix ˜R ⊤(˜R ˜R ⊤)−1 ˜R acts as an identity operator for all the vectors in the range space of V1, that is, ˜R ⊤(˜R ˜R ⊤)−1 ˜R V1 = V1. Since also ˜R ⊤(˜R ˜R ⊤)−1˜R ˜R ⊤ = ˜R ⊤, the columns of ˜R ⊤ must be in the range space of V1. However, since Rank{V1} = Rank{˜R ⊤} = nx, Span{V1} ≡ Span{˜R ⊤}. Since by hypothesis 33 ˜M b ∈ Span{˜R ⊤}, (3.26) can be rewritten as LL + KLM = Mb,L Necessity (⇒). Because of the state matching at the slowrate 1/Ts, ξT [kN] → Mr∞ as k → ∞. Since ξT [kN] , [(xT [kN])⊤, (ϕT [kN])⊤]⊤, xNT [k] , xT [kN] → Mar∞ and ϕT [kN] → Mbr∞. Since the closedloop response is ripplefree at steadystate, xT [kN + i] → Mar∞ and ϕT [kN + i] → Mbr∞ for every i = 0, . . . ,N − 1 as k → ∞. Therefore, from (3.7), ( + Dx − I)Mar∞ + CϕMbr∞ = 0 or equivalently s − I Cϕ M = 0 The last equality can be rewritten as − I Ma DxMa +Mb = 0 which proves the second condition of the theorem. Also, since LL + KLM = Mb,L, it must be that ˜M b ∈ Span{˜R ⊤}. Notice that since the columns of the matrix in (3.25) have to be in the range space of Ns, it is necessary that Ma ∈ Span(Sa). This is the only condition of Theorem 3.3 not dependent on the selection of Cϕ and Dx. If P denotes the matrix such that Ma = SaP, the remaining conditions of Theorem 3.3 are satisfied if and only if there 34 exist matrices Mb, Cϕ, Dx and Q such that Mb = ( ps Cϕ)⊤Q, p = 0, . . . ,N − 2, (3.28) DxMa + CϕMb = SbP. (3.29) Equations (3.28) and (3.29) always admit a solution if the desired system is chosen so that Ma has full column rank and Mb = 0. In fact, in that case one can select Q = 0, Dx = (M⊤ a Ma)−1M⊤ a SbP and Cϕ to be any matrix. Therefore, if the pair (F,G) of the desired system (3.3) is chosen such that M = S⊤ a 0n'×ny ⊤ P, (3.30) where P is any matrix such that SaP has full column rank, then there exist matrices Cϕ and Dx for which the multirate closedloop system exhibits a ripplefree steadystate response to step reference signals. Notice that, since can be safely assumed to have full column rank, it is straightforward to verify through (3.22) that Sa has full column rank. Therefore, there always exist a P such that Ma , SaP which has full column rank. It remains to investigate how (F,G) can be selected in order for (3.30) to be satisfied. One way consists of selecting F as desired, and G given by G = −(F − I) SaP 0 (3.31) In fact, by applying the final value theorem to the Ztransform of the state variables 35 ζNT we have: lim z→1Z{ζNT } = lim z→1 (zI − F)−1G = SaP 0 where z corresponds to the unit advance operator in the Ztransform domain. Therefore, complete freedom is given to the choice of the F. It is important to notice that the strategy to select the pair (F,G) in order for the desired system to satisfy (3.30) is not unique. The following theorem summarizes the results obtained in this section. Theorem 3.4 Consider a stable desired digital LTI system of the form (3.3), and let (Ac,Bc,Cc) be a representation of the continuoustime plant to be controlled. Under assumption 3.1, it is possible to design the timevarying controller (3.2) for the continuoustime plant (3.1) to achieve closedloop state matching at the measurement update rate with the desired system (3.3) if and only if 1. the pair (Ac,Bc) is controllable 2. N ≥ nx + 1 The unique solution for the matrices Kϕ,i, Kx,i and Li of the controller (3.2) can be obtained by choosing any full row rank matrix Cϕ, and it is given by KL = ¯ + L(F − ¯ L) LL = ¯ + LG where KL and LL are uniquely related to Kϕ,i, Kx,i and Li through (3.13). With a controller designed in such a way, the closedloop system exhibits a steadystate ripplefree response to step reference signals if and only if there exist matrices Q and 36 P such that ( ps Cϕ)⊤Q = Mb p = 0, . . . ,N − 2 Ma = SaP DxMa + CϕMb = SbP Sufficient conditions for the existence of a solution to the latter equations are that Ma ∈ Span(Sa), Rank(Ma) = ny and Mb = 0. In such a case the matrix Dx can be selected as Dx = (M⊤ a Ma)−1M⊤ a SbP. (3.32) To complete this section a procedure to design the controller is provided. This procedure is based on the sufficient conditions of Theorem 3.4. Procedure  Step 1. Select N to satisfy assumption 3.1 and so that N ≥ nx+1 (see Theorem 3.2).  Step 2. Obtain the discrete equivalent representation, ( , ), of the continuoustime controllable plant (3.1), where is a full column rank matrix.  Step 3. Find a basis, Ns , [S⊤ a , S⊤ b ]⊤, for the null space of the matrix [ −I, ].  Step 4. Select any desired system of the form (3.3) where F is any matrix and G satisfies (3.31) (see Theorem 3.3), for any full column rank matrix P of appropriate dimension.  Step 5. Select Cϕ to be any full row rank matrix (according to Theorem 3.2) and Dx = (M⊤ a Ma)−1M⊤ a SbP. 37  Step 6. Construct the matrices ¯ L, ¯ L and compute the right inverse ¯ + L = ¯ ⊤L (¯ L¯ ⊤L )−1.  Step 7. After computing the matrices KL and LL according to (3.19) and (3.20), obtain the controller matrices Kx,i, Kϕ,i and Li as shown in (3.13). 3.4 Example In this section the results summarized in Theorem 3.4 will be applied to a doubleintegrator system with statespace representation Ac = 0 0 1 0 , Bc = 1 0 , Cc = 0 1 Since nx = 2 it is required to choose N ≥ 3. For simplicity, let us consider the case N = 3 and T = 1. The zeroorder hold equivalent of the plant operating at the fast rate 1/T is given by = 1 0 T 1 , = T 5T2 , Cc = 0 1 A basis for the null space of [ − I, ] is Ns = [0, 1, 0]⊤. Therefore, Sa = [0, 1]⊤, and Sb = 0. Let us choose the matrix F of the desired system (3.3) to be zerovalued, and let us select the matrix G according to (3.31). Hence, the desired system statespace representation is given: F = 0 0 0 0 0 0 0 0 0 , G = 0 1 0 As a result, Ma = [0, 1]⊤, Rank{Ma} = ny, Mb = 0. 38 Let Cϕ = 1. Therefore, the matrix Dx, computed through (3.32), is given by Dx = [0, 0]. Then, the matrices Kx,i, Kϕ,i and Li of the controller (3.2), designed according to (3.19) and (3.20), are given by Kx,0 = [−5/(2T),−1/(10T2)], Kϕ,0 = −2, L0 = 1/(10T2) Kx,1 = [3/(2T), 1/(10T2)], Kϕ,1 = 1, L1 = −1/(10T2) Kx,2 = [0, 0], Kϕ,2 = 0, L2 = 0 The step response of the closedloop multirate system comprising the continuoustime plant and the digital controller is compared with the response of the desired system in as shown in Fig. 3.2. From Fig. 3.2 it is clear that the state of the closedloop system matches at every measurement sampling instant the state of the desired system (3.3), and that the response of the closedloop system is ripplefree at steadystate. 3.5 Conclusions In this chapter the inputstate matching problem with a desired singlerate digital system was considered for the class of multirate systems with slow measurement update rate and fast controller update rate. In particular, given any desired singlerate LTI system operating at the measurement update rate, conditions and a controller design procedure were given for which the closedloop system state matches the state of the desired system at that measurement update rate. Moreover, it was shown that, if the input matrix of the desired system is properly selected, a ripplefree steadystate response of the closedloop system can be obtained. Despite the constraints on the input matrix of the desired system, the developed design procedure gives full freedom on the choice of the closedloop eigenstructure. This allows to potentially determine the shape of the state response of the closedloop system. It is important to notice that 39 0 3 6 9 12 15 −1 −0.5 0 0.5 1 x1(t) zNT 1 [k] 0 3 6 9 12 15 0 0.2 0.4 0.6 0.8 1 x2(t) zNT 2 [k] 0 3 6 9 12 15 −1 −0.5 0 0.5 1 xT 3[k] zNT 3 [k] Figure 3.2: Closedloop system response to a step reference signal: the state variables of the multirate closedloop system are compared with the state variables of the desired slowrate LTI system. The signals x1, x2 are the velocity and position, respectively, of the double integrator continuoustime plant. The closedloop system output is y(t) = x2(t). 40 even though the conditions stated in this chapter allow to avoid the presence of ripples in the steadystate response of the continuoustime plant, undesired oscillations may appear in the transient response due to the timevarying nature of the control system. These undesired oscillations may be due to the particular choice of the eigenstructure of the desired system. Therefore, future research will focus on finding conditions on the choice of the desired modal response to avoid possible ripples also in the closedloop system transient response. Moreover, the theory shown in chapter 3 was obtained assuming the availability of full state information. Therefore, the cases in which only partial information is available will be considered in the future. 41 CHAPTER 4 Frequency domain modeling and parametrization of LTI Controllers Several control strategies have been developed in the last three decades for multirate systems. In some of the cases the measurements from the plant are available only after a relatively long period of time compared to the operating rate of the digital controller. In those cases, the use of a control action updating at a faster rate than the measurement update rate was shown to increase some robustness indexes, such as gain and phase margins [18]. To keep the robustness advantages deriving from the use of a controller operating at a rate faster than the measurement update rate, while reducing the undesired transient intersample behavior of the closedloop system and guaranteeing a ripplefree steadystate response, only controllers formed by the cascade of an upsapling operator with factor N and a LTI system are considered in this chapter. These controllers, belonging to the class of causal (N, 1)shift invariant systems, receive the slowupdating measurements from the plant and generate a fastupdating control signal. With this control structure the main cause of undesired intersample behavior (both at transient and steadystate) can be isolated and directly addressed. In fact, as it will be shown in chapter 5, the main cause of undesired intersample behavior is the use of the (N, 1)shift invariant upsampling operators. Therefore, the undesired intersample behavior can be reduced by requiring the LTI system embedded in the control structure to filter appropriately the output of the upsamplers. This motivates the use of this particular class of (N, 1)shiftinvariant controllers. One of the main contributions of this chapter consists of a novel procedure to 42 parameterize the set of all the stabilizing (N, 1)shiftinvariant controllers with the above mentioned structure for which model matching is achieved with a desired digital LTI system in the two cases: (1) the desired LTI system updates at the control update rate, (2) the desired LTI system updates at the measurement update rate. Notice that, to the author’s best knowledge, the problem in the case (1) has never been considered before in literature. The problem of parametrizing controllers in the multirate systems framework was initially addressed in [28] and [29]. The results in those two works are the milestones for the parametrization of (m, n)shiftinvariant controllers in the multirate systems framework. In particular, a parametrization of all the stabilizing (m, n)shiftinvariant controllers for the particular set of (n,m)shiftinvariant plants comprising zeroorder holds, a continuoustime causal LTI system and sampling operators was proposed in [28]. The theory in [28] was then generalized in [29] to consider all the possible (n,m)shiftinvariant plants. In both [28] and [29] the parametrization theory results in the complete set of (m, n)shiftinvariant controllers for the corresponding class of plants. The parametrization procedure required first to design the controller for the lifted version of the plant, and then to apply the inverse lifting process to the controller designed for the lifted plant. Since in this chapter it is considered the case of plants whose output is measured at the rate 1/Ts and whose input updates at the faster rate 1/T, the parametrization is restricted to (N, 1)shiftinvariant controllers. In particular, only those (N, 1) shiftinvariant controllers made of a downsampling operator followed by a digital LTI system are considered for the reasons explained earlier. The solution to this problem cannot be obtained simply by using the theory developed in [28] and [29] for the case m = N and n = 1. This is due to the fact that the solution of the parametrization problem considered in [28] and [29], in the case m = N and n = 1, would be the entire set of (N, 1)shiftinvariant controllers, including those which 43 cannot be expressed with the series connection of a downsampling operator and a digital LTI system. Two other major differences between the works in [28] and [29] and the theory shown in this chapter are the following. (1) In the solution proposed in this chapter there is no need to consider any causality constraint on the controller designed for the lifted system. (2) The background objective of the parametrization in this chapter is not just the stability of the closedloop system but also the inputoutput model matching with a desired LTI system. The author’s preliminary results are shown in [42, 43]. This chapter is organized as follows. The problem formulation is given in section 4.1. Section 4.2 describes the frequency domain approach utilized to model the multirate system. Closedloop stability and the model matching problems, respectively, are discussed in sections 4.3 and 4.4. Section 4.5 shows an application of the developed theory. Conclusions are given in section 4.6. 4.1 Problem formulation Let P(s) be the Laplace transform of the continuoustime system x˙ (t) = Acx(t) + Bcu(t) y(t) = Ccx(t) + Dcu(t) (4.1) where x(t) ∈ Rnx is the state vector, u(t) ∈ R is the control signal, and y(t) ∈ R is the measured and controlled plant output to be regulated to the constant reference rNT [k] ∈ R. Let G(s) , P(s)H(s), where H(s) , (1−esT )/s is a the zeroorder hold operating at the control update rate 1/T. Consider the multirate system in Fig. 4.1, where the (N, 1)shiftinvariant feedback and feedforward controllers are made of an upsampling operator followed by the LTI digital systems CT 1 and CT 2 , respectively. 44 The control signal takes the form UT = CT 2 [RNT ]T −CT 1 [Y NT ]T , where UT , Z{uT }, +  N N NT rNT CT 1 (d) CT 2 (d) ZOH P(s) [rNT ]T uT y(t) yNT [yNT ]T Figure 4.1: Multirate control scheme. RNT , Z{rNT } and Y NT , Z{yNT }. Notice that in the block diagram of Fig. 4.1 the output of the plant is measured at the slow rate 1/NT, and the control signal uT updates N times between two consecutive measurement updates. The following assumption on the continuoustime plant (4.1) is necessary for the subsequent results. Assumption 4.1 The continuoustime system (4.1) is controllable and observable. Moreover, assumption 3.1 will be considered in this chapter to guarantee the controllability and reconstructibility of the periodic discretetime system formed by the cascade connection of the zeroorder hold, the plant G(s) and the slowrate sampler [18, 23]. In the following, since the design of the (N, 1)shiftinvariant controllers shown in Fig. 4.1 consists in the design of the LTI systems CT 1 and CT 2 , the problem of parameterizing such a set of (N, 1)shiftinvariant controllers will be more conveniently referred to as the problem of parameterizing the LTI controllers CT 1 and CT 2 . Hence, given the multirate system in Fig. 4.1, the goals are the following: 1. Modeling in the transform domain the closedloop multirate system both at the slowrate, 1/Ts, and at the fastrate, 1/T, using the tools introduced in chapter 2. The closedloop system must be expressed in a form suitable for the next objectives. 45 2. Parameterizing the set of stabilizing LTI controllers for the multirate system shown in Fig. 4.1. 3. Parameterizing the set of LTI controllers for which closedloop model matching is achieved at the slowrate with a desired stable slowrate LTI system. 4. Parameterizing the set of LTI controllers for which closedloop model matching is achieved at the fastrate with a desired stable fastrate LTI system. To achieve these goals, necessary and sufficient conditions to achieve stability and model matching will also be obtained. These conditions will be used to parameterize the corresponding sets of LTI controllers CT 1 and CT 2 . 4.2 Modeling the multirate system Let GT be the Ztransform, at the sampling period T, of the plant (4.1) preceded by the zeroorder hold. A digital control system corresponding to the sampleddata control system in Fig. 4.1 can be obtained first by replacing the slowrate sampler with a fastrate sampler followed by a downsampling operator, and then by replacing the cascade made by the zeroorder hold, the plant P(s) and the fastrate sampler with the digital system GT . The resulting digital system is shown in Fig. 4.2. The following +  N N N rNT CT 1 (d) CT 2 (d) GT (d) yT [rNT ]T uT yNT [yNT ]T Figure 4.2: Multirate control scheme. proposition can be proved by using properties (2.6) and (2.7) of the upsampling and downsampling operators. Proposition 4.1 The slowrate closedloop transfer function KNT of the system in Fig. 4.1, between the slowrate reference signal RNT and the measured plant output 46 Y NT , is given by KNT , Y NT RNT = GTCT 2 NT 1 + [GTCT 1 ]NT (4.2) The fastrate closedloop transfer function KT , between the upsampled reference signal [RNT ]T and the fastrate plant output Y T , Z{yT }, is given by KT , Y T [RNT ]T = GT CT 2 − CT 1 [[GTCT 2 ]NT ]T 1 + [[GTCT 1 ]NT ]T (4.3) Proof. The plant output Y T is given by Y T = GTCT 2 [RNT ]T − GTCT 1 [Y NT ]T (4.4) The slowrate version Y NT of the plant output can be obtained as follows: Y NT =[GTCT 2 [RNT ]T ]NT − [GTCT 1 [Y NT ]T ]NT =[GTCT 2 ]NTRNT − [GTCT 1 ]NTY NT Equation (4.2) is obtained by grouping Y NT . Equation (4.3) follows after the substitution of Y NT into (4.4). Remark 4.1 Notice that KNT can be obtained just by downsampling KT , that is, [KT ]NT = [GTCT 2 ]NT − GTCT 1 [[GTCT 2 ]NT ]T 1 + [[GTCT 1 ]NT ]T NT = [GTCT 2 ]NT − [GTCT 1 ]NT [GTCT 2 ]NT 1 + [GTCT 1 ]NT = [GTCT 2 ]NT 1 + [GTCT 1 ]NT = KNT However, it is not possible to obtain KT by upsampling KNT because the additional information contained in KT about the fastrate impulse response of the closedloop 47 system cannot be recovered with an upsampling operation. In fact, [KNT ]T = [[GTCT 2 ]NT ]T 1 + [[GTCT 1 ]NT ]T 6= KT As shown in (2.8), [GTCT i ]NT 6= [GT ]NT [CT i ]NT , i = 1, 2. Therefore, the classic control techniques developed for singlerate systems cannot be used to design CT 1 and CT 2 . To overcome this problem, the following proposition, using the procedure developed in [42, 44, 45], introduces an invertible transformation which allows to rewrite KNT and KT in a form more suitable for analysis and design. Proposition 4.2 The closedloop transfer functions (4.2) and (4.3) can be written in the following form: KNT (ds) = [GT T ]NT (ds) ˜ CNT 2 (ds) 1 + [GT T ]NT (ds) ˜ CNT 1 (ds) (4.5) KT (d) = GT (d) T (d) ˜ CNT 2 (dN) − GT (d) T (d) ˜ CNT 1 (dN)[GT T ]NT (dN) ˜ CNT 2 (dN) 1 + [GT T ]NT (dN) ˜ CNT 1 (dN) (4.6) where T , [1, d, · · · , d(N−1)], and ˜ CNT i is a column vector containing all the modified Ztransforms of Z−1{CT i }, that is, ˜ CNT i , ˜ CNT i{0} · · · ˜ CNT i{N−1} ⊤ , i = 1, 2. Proof. As shown in (2.18), the controller transfer functions CT i (d), i = 1, 2, can be rewritten as the weighted sum of its upsampled modified Ztransforms, that is, CT i (d) = NX−1 k=0 dk h ˜ CNT i{k} iT = T [ ˜ CNT i ]T = T (d) ˜ CNT i (dN) (4.7) 48 Therefore, the following equalities hold GT (d)CT i (d) = GT (d) T (d)˜C NT i (dN) [GTCT i ]NT = [GT T [ ˜ CNT i ]T ]NT = [GT T ]NT ˜ CNT i and the discretetime closedloop transfer functions (4.5) and (4.6) can be obtained by substituting the last two equalities into (4.2) and (4.3), respectively. Unlike the forms given in (4.2) and (4.3), the closedloop transfer functions (4.5) and (4.6) are more suitable for designing a controller because ˜ CNT i (dN) (i = 1, 2) and [GT T ]NT (dN) are separated in their denominators and numerators. Observe that the transformation from CT i (d) to ˜ CNT i (dN) is invertible. Therefore, a controller designed for the closedloop systems (4.5) or (4.6) can be converted back into an equivalent controller for the closedloop systems (4.2) or (4.3), respectively, by using (4.7). Consider now the following coprime factorizations of the transfer functions involved in the multirate system: GT T NT = BNT ANT , GT = NTG DTG , ˜ CNT 1 = Y NT 1 XNT , ˜ CNT 2 = Y NT 2 XNT (4.8) where ANT and XNT are proper and stable scalar transfer functions, BNT is a proper and stable 1×N transfer function, Y NT 1 and Y NT 2 are proper and stable N×1 transfer functions, DTG and NTGare polynomials. Notice that, as explained in remark 4.3, there always exists a stable leftcoprime factorization A(dN),B(dN) for [ TGT ]NT (dN). By using these factorizations, the closedloop transfer functions (4.5) and (4.6) can be 49 rewritten as KNT = BNT Y NT 2 FNT (4.9) KT (d) = NTG (d) DTG (d) T (d)(FNT (dN)I − Y NT 1 (dN)BNT (dN)) FNT (dN) Y NT 2 (dN) XNT (dN) (4.10) where I is the identity matrix of dimension N ×N, and FNT , ANTXNT +BNTY NT 1 . Remark 4.2 Due to the stability of the factorization (4.8) it is clear from (4.9) that the stability of KNT can be achieved if and only if the transfer function F is a unit in the space of the rational functions (that is, it is stable and proper and its inverse is also stable and proper). In other words, KNT can be regarded as the closedloop transfer function of a singlerate control system operating at the slow rate 1/NT, and hence, the stability conditions for singlerate systems discussed in [46] can be applied. However, it is not so straightforward to see if the same conditions apply to KT . In fact, as shown in equation (4.10), the stability of the closedloop multirate system is not achieved if the transmission zeros of the transfer function matrix FNT (dN)I − Y NT 1 (dN)BNT (dN) do not cancel the unstable zeros of the cascade DTG (d)XNT (dN). Remark 4.3 The statespace realization, ( ¯ A, ¯B , ¯C , ¯D ), of GT T NT (dN) is related to the system in (4.1) by A¯ = ANc , ¯B = [AN−1 c Bc, · · · ,AcBc,Bc] ¯ C = Cc, ¯D = [Dc, 0, · · · , 0]. By assumptions 3.1 and 4.1 the pair (A¯,C¯) is detectable [18, 23], and hence, there exists a matrix H such that the spectrum of ¯ A + H ¯C lies in the open unit circle. Therefore, a stable leftcoprime parametrization of GT T NT (dN) exists and it is 50 given by [47] BNT = Ccds(I − ds(ANc + HCc))−1(¯B + H ¯D ) + ¯D ANT = Ccds(I − ds(ANc + HCc))−1H + 1. (4.11) To facilitate the readability of the rest of this chapter, the notations NG, DG, , A, B, X, Y1, Y2 and F, will be utilized to mean NTG (d), DTG (d), T (d), ANT (dN), BNT (dN), XNT (dN), Y NT 1 (dN), Y NT 2 (dN) and FNT (dN), respectively, unless otherwise stated. 4.3 Closedloop stability In this section, necessary and sufficient conditions are given for the existence of LTI controllers CT 1 ,CT 2 which stabilize the closedloop multirate system KT in Fig. 4.1. The following lemma is useful to obtain subsequent results. Lemma 4.1 If F is a unit in the space of rational functions, all the unstable zeros of the cascade DGX are transmission zeros of the transfer function matrix (FI − Y1B). Proof. Since F is a unit in the space of rational functions in the variable dN, it follows that F 6= 0 and 1−BF−1Y1 6= 0. By contradiction assume that 1−BF−1Y1 = 0. Then, the following statements are equivalent: (AX + BY1)−1BY1 = 1, BY1 = AX + BY1, AX = 0. The latter equation, however, cannot be true because it would imply either one or both the transfer functions A and X to be identically null. Therefore 1 − BF−1Y1 6= 0, and by using the matrix inversion formula the transfer function matrix (FI − Y1B)−1 can be written as (FI − Y1B)−1 = F−1I + F−1Y1BF−1 1 − BF−1Y1 (4.12) 51 Considering that 1 − BF−1Y1 = AXF−1, equation (4.27) can be rewritten as (FI − Y1B)−1 = (AXI + Y1B)F−1 AX (4.13) To show that all the zeros of AX are poles of (FI − Y1B)−1, it is necessary and sufficient to prove that the transmission zeros of (AXI + Y1B)F−1 do not cancel those of AX. Let d0 be an unstable zero of AX. Since AX is a scalar transfer function, d0 is also a blocking zero of AX, that is, A(d0)X(d0)δ = 0, ∀δ 6= 0, δ ∈ R (4.14) By contradiction, assume that d0 is also an unstable transmission zero of (AXI + Y1B)F−1, that is, there exists a column vector 0 6= β ∈ RN such that (A(d0)X(d0)I + Y1(d0)B(d0))F−1(d0)β = 0. Since F is stable by definition, d0 cannot be a transmission zero of F−1. It follows that d0 is a transmission zero of Y1B. It is shown below that d0 is also a transmission zero of BY1. Let U1, V1, U2 and V2 be unimodular polynomial matrices, in the variable dN, such that the SmidthMcMillan decomposition of Y1 and B are given by Y1 = U1[ ⊤1 , 0]⊤V1 and B = U2[ 2, 0]V2, where 1 and 2 are stable transfer functions in the variable dN. Considering that the cascade connections Y1B and BY1 are given by Y1B = U1 1V1U2 2 0 0 0 V2, BY1 = U2 2V2U1 1V2 d0 is a transmission zero of Y1B if and only if it is also a zero of BY1, that is, B(d0)Y1(d0)δ = 0, ∀δ 6= 0, δ ∈ R (4.15) 52 By combining (4.14) and (4.15) it follows that (A(d0)X(d0) + B(d0)Y1(d0))δ = 0, ∀δ 6= 0, δ ∈ R, which means that d0 is also an unstable zero of F. This, however, contradicts the hypothesis that F is a unit. Therefore, there cannot be any cancelation between the unstable transmission zeros of (FI − Y1B)−1 and those of AX. In other words, all the unstable zeros of AX are poles of (FI − Y1B)−1. Considering that [GT T ]NT is obtained from GT T through (2.5), it is possible to show that A = DG where is a stable transfer function of the variable d. Therefore, equation (4.13) can be rewritten as (FI − Y1B)−1 = (AXI + Y1B) F−1 DGX (4.16) It follows that the unstable zeros of the cascade DGX are poles of (FI − Y1B)−1, and hence, the transmission zeros of (FI − Y1B). Remark 4.4 (On the transmission zeros of (FI − Y1B)) Notice that the zeros of A are symmetric with respect to the origin. Moreover, for every root γ of DG, A has N zeros with the same absolute value, γ, equally spaced around the origin of the complex plane. Therefore, for every unstable root of DG, has N − 1 unstable zeros with the same absolute value. This can be easily verified using (2.5). Therefore, the unstable transmission zeros of (FI − Y1B) are not just the unstable roots of DG but also the unstable zeros of the transfer function . The following theorem gives necessary and sufficient conditions for the existence of stabilizing LTI controllers. Theorem 4.2 will provide the parametrization of the set of all the feedback controllers CT 1 stabilizing the multirate system in Fig. 4.1. Theorem 4.1 The LTI controllers CT 1 , X−1Y1, CT 2 , X−1Y2 53 stabilize the closedloop multirate system KT if and only if F is a unit. Proof. (⇒) By hypothesis the closedloop system (4.10) is stable. By construction A, B, Y1, Y2 and X are proper and stable, which implies that also F is a proper stable transfer function. By inspection on the closedloop transfer function (4.10) it is straightforward to show that if F has an unstable zero, that zero is a pole of KT . Since KT is stable, F cannot have any unstable zero. Therefore, F is a unit in the space of rational functions in the variable dN. (⇐) By hypothesis F is a unit. Therefore the stability of KT is guaranteed by lemma 4.1. Theorem 4.2 All the stabilizing LTI feedback controllers CT 1 for the closedloop multirate system in Fig. 4.1 are of the form CT 1 = Y1 − A X + B (4.17) where is any proper stable N × 1 transfer function matrix in the variable dN, and the pair (X, Y1) satisfies the Bezout equation AX + BY1 = 1. Proof. (⇐) Let (X, Y1) be a particular solution of the Bezout equation. Then also ¯X , ¯Y1 , where ¯X , X +B and ¯Y1 , Y1 −A , is a solution of the Bezout equation, that is, A(X + B ) + B(Y1 − A ) = 1. Therefore, by theorem 4.1, every controller of the form (4.17) is a stabilizing controller. (⇒) Let (X, Y1) and ¯X , ¯Y1 be two different stable solutions of the Bezout equation. It has to be shown that there always exists a stable transfer function (dN), such that the solution ¯X , ¯Y1 can be written as ¯X = X +B , ¯Y1 = Y1−A . Since (X, Y ) and ¯X , ¯Y1 are two different solutions of the Bezout equation, A(X − ¯X ) = −B(Y1− ¯ Y1). If = (Y1− ¯ Y1)/A then ¯X = X+B and ¯Y1 = Y1−A . It remains to be shown that is stable. Since A and B are coprime, A is also a factor of (Y1 − ¯ Y1). Therefore, 54 the stability of (Y1 − ¯ Y1) proves the stability of . Moreover, since Y1, ¯ Y1 and A are functions of dN, then also is a function of dN. 4.4 Model matching In this section, necessary and sufficient conditions are given for the existence of LTI controllers CT 1 and CT 2 such that the dynamics of the slowrate closedloop system KNT and of the fastrate closedloop system KT match with the dynamics of a desired slowrate system KNT d and fastrate system KT d , respectively. 4.4.1 Model matching at the slow rate The closedloop transfer function of the multirate system in Fig. 4.1 between the slowrate reference signal RNT and the slowrate plant output Y NT is given in (4.5). By simple inspection it is possible to see that (4.5) represents the closedloop transfer function of a singlerate system where the plant, the feedback controller and the feedforward controller are given by the transfer matrices [GT T ]NT , ˜ CNT 1 and ˜ CNT 2 , respectively. Therefore, by using the theory developed in [48] for singlerate systems, and considering the factorizations (4.8), it is possible to conclude that the transfer function KNT is stable if and only if F is a unit. Moreover, as shown in [18] it is always possible to arbitrarily place both zeros and poles of KNT to achieve model matching (at the slowrate) with any desired transfer function if the couple (Ac,Bc) is controllable, assumption 3.1 holds, and N −1 ≥ nx. As an extension, the following theorem provides the parametrization of the set of all the stabilizing controllers for the multirate system in Fig. 4.1 such that model matching is achieved at the slowrate with a desired transfer function KNT d . Theorem 4.3 Let assumptions 3.1 and 4.1 be verified, and N − 1 ≥ nx. Consider the factorizations (4.8) for the transfer functions involved in the closedloop system 55 (4.5). Let KNT d be a desired stable transfer function. All the LTI controllers CT 1 and CT 2 such that model matching is achieved are of the form CT 1 = Y1 − A X + B , CT 2 = Y2 X + B (4.18) where is any proper stable N × 1 transfer function of the variable dN, and Y1, Y2, X satisfy the following equations AX + BY1 = 1 (4.19) BY2 = KNT d (4.20) Proof. (⇐) Let CT 1 and CT 2 as in (4.18). It has to be proved that closedloop model matching is achieved with the transfer function KNT d . By using (4.7), CT 1 and CT 2 can be rewritten as CT 1 (d) = (d)CNT 1 (dN) and CT 2 (d) = (d)CNT 2 (dN), where CNT 1 (dN) , Y1 − A X + B , CNT 2 (dN) , Y2 X + B Considering that F = 1 (from (4.19)), and by using equation (4.20), the result can be obtained by substitution of CNT 1 (dN) and CNT 2 (dN) in the multirate closedloop system (4.5). (⇒) Assume that the model matching is achieved through the LTI controller CT 1 = ¯X −1 ¯ Y1 and CT 2 = ¯X −1 ¯ Y2. It has to be proved that there exists a transfer function matrix such that CT 1 and CT 2 can be written in the form (4.18). Since the closedloop system is stable, by theorem 4.2 there exists a transfer function such that CT 1 is expressed in the form (4.17). Therefore ¯X = X + B and ¯ Y1 = Y1 − A , where the pair (X, Y1) is a particular solution of the Bezout equation (4.19). Since the pair (Ac,Bc) is controllable and N − 1 ≥ nx, B is right invertible [49]. Let Y2 be the right inverse of B. Since B is stable and due to the structure of B, all the 56 invariant zeros of B are stable. Therefore Y2 is also stable and can be selected so that (4.20) is satisfied. 4.4.2 Model matching at the fast rate The following theorem provides necessary and sufficient conditions for the existence of LTI controllers CT 1 and CT 2 such that the closedloop transfer function KT matches with a desired transfer function KT d . Theorem 4.4 Consider the factorizations (4.8) for the transfer functions involved in the closedloop system (4.6). Let KT d be a stable singlerate system operating at the sampling time T. It is possible to design the LTI controllers CT 1 = X−1Y1 and CT 2 = X−1Y2, for the multirate system shown in Fig. 4.1, to achieve model matching with the system KT d (d) if and only if (a) A and B are relatively right prime (b) S is a left divisor of KT d , where S is defined as S , NG DG FI − Y1B FX (4.21) (c) the unstable zeros NG(d)(d) are zeros of KT d (d), where (d) is a stable transfer function such that A(dN) = DG(d)(d). Proof. (⇒) By hypothesis the closedloop multirate system matches the stable system KT d . By theorem 4.1, the stability of the closedloop system implies that F is a unit. Therefore, (a) follows. Since Y2 is stable by definition and, by equation (4.10) KT = SY2 = KT d , (b) follows. The stability of Y2 guarantees that no illicit cancelation occurs between the poles of Y2 and the zeros of S. (c) follows from the fact that the unstable zeros of S correspond to the unstable zeros of NG, as shown in (4.16). 57 (⇐) (a),(b) and (c) are true by hypothesis. Then there exists a pair (X, Y1) such that F = 1, and therefore the closedloop system is stable by virtue of theorem 4.1. Further, since (b) and (c) hold, there exists a stable transfer function matrix Y2 such that SY2 = KT d . Therefore, by equation (4.10) the closedloop system KT equals KT d . It is clear from theorem 4.4 that the desired system KT d cannot be selected arbitrarily. In fact, due to condition (c) of theorem 4.4 the set of zeros of the desired system KT d must contain the unstable zeros of the plant transfer function GT plus, for every unstable pole of GT , the N − 1 unstable zeros of discussed in remark 4.4. Therefore, consider the following factorizations of the polynomial NG and of the transfer function : NG = Nin G Nout G , = inout (4.22) where Nin G and in are polynomials of the variable d containing all and only the roots of NG and of the numerator polynomial of , respectively, contained in the closed unit circle; Nout G and out are respectively a Hurwitz polynomial and a stable minimumphase transfer function satisfying (4.22). According to condition (c) of theorem 4.4, the desired fastrate closedloop transfer function KT d must take the form KT d (d) = Nin G (d)in(d) ¯K T d (d) (4.23) where ¯K T d is a stable and proper transfer function. The following Theorem provides a parametrization of the set of all the stabilizing LTI controllers CT 1 and CT 2 such that the fastrate closedloop transfer function KT matches with a transfer function KT d factorized in the form (4.23). Theorem 4.5 Consider the factorizations (4.8) and (4.22). Let KT d be a desired 58 stable transfer function in the form (4.23). All the LTI controllers CT 1 and CT 2 such that model matching is achieved with KT d can be written as CT 1 = ¯X −1 ¯ Y1, CT 2 = ¯X −1 ¯ Y2, (4.24) with ¯X , ¯ Y1 and ¯ Y2 given by ¯X = X + B ¯ Y1 = Y1 − A ¯ Y2 = (A(X + B ) I + (Y1 − A )B) ˜M NT (dN) (4.25) where is any proper stable N ×1 transfer function of the variable dN, ˜MNT (ds) is a N×1 vector containing the N modified Ztransforms of Z−1{ ¯K T d (d)/(Nout G (d)out(d))}, and X, Y1 satisfy the Bezout equation (4.19). Proof. (⇐) Let CT 1 and CT 2 be given by (4.24) and (4.25). It has to be proved that closedloop model matching is achieved between the fastrate closedloop transfer function KT and KT d . Let us start by showing that KT , in (4.10), can be rewritten as KT = NG (AXI + Y1B)−1 Y2 (4.26) In fact, using the matrix inversion formula, and considering that 1−BF−1Y1 = AX = DGX 6= 0 and F = 1, the transfer matrix (FI − Y1B)−1 can be written as (FI − Y1B)−1 = F−1I + F−1Y1BF−1 1 − BF−1Y1 = I + Y1B AX = AXI + Y1B AX = AXI + Y1B DGX (4.27) The closedloop transfer function (4.26) is obtained with the substitution of the inverse 59 of (4.27) into (4.10). By using the controllers CT 1 and CT 2 given in (4.24) and (4.25) (that is, by replacing X, Y1 and Y2 in (4.26) with ¯X , ¯ Y1, ¯Y2), after some simplifications, the transfer function KT becomes KT = NG ˜M NT (dN). where, by construction, (d) ˜M NT (dN) = ¯K T d (d)/(Nout G (d)out(d)). Hence, KT (d) = Nin G (d)in(d) ¯K T d (d) = KT (d). Considering that ¯ Y2 is stable by construction, model matching is achieved. (⇒) Assume that the model matching is achieved through the LTI controller CT 1 = ¯X −1 ¯ Y1 and CT 2 = ¯X −1 ¯ Y2. It has to be proved that there exists a transfer function matrix such that ¯X , ¯ Y1 and ¯ Y2 can be written in the form (4.25). The closedloop transfer function of the system can be written as KT = NG A ¯X I + ¯ Y1B −1 ¯ Y2 (4.28) Since KT is stable there always exists a transfer function such that ¯X = X + B and ¯ Y1 = Y1 − A , where the pair (X, Y1) is a particular solution of the Bezout equation (4.19), [44]. Therefore, equation (4.28) can be rewritten as KT = NG (A(X + B )I + (Y1 − A )B)−1 ¯ Y2. (4.29) Considering that (d) ˜M NT (dN) = ¯K T d (d)/(Nout G (d)out(d)), ¯ Y2 must take the form shown in (4.25). It remains to be shown that ¯ Y2 is proper. Since ¯K T d represents a causal system in its minimal form, the only way for ¯K T d (d)/(Nout G (d)out(d)) to be not proper is that the numerator of Nout G out contains a factor dk with k ≥ 1. However, this is not possible since all the roots of the numerator of NG inside the unit circle, including the roots in the origin corresponding to terms like dk, are included by 60 construction into Nin G in. Therefore, due also to the properness of A, B, X, Y1 and , it is possible to conclude that ¯Y2 is proper. 4.5 Example Consider the plant transfer function P(s) = 1/(s + 1). Let H(s) = (1 − e−sT )/s be the zeroorder hold operating at the sampling time T = 0.01s. Then GT (d) = (1 − h)d/(1 − hd), where h , e−T . Considering N = 2, it follows that GT T = (1−h)d 1−hd (1−h)d2 1−hd , GT T NT (dN) = (−1+h)hd2 −1+h2d2 (−1+h)d2 −1+h2d2 According to the factorizations given by (4.8), let A and B be defined respectively as A = −1 + h2d2 and B = [(−1 + h)d2h, (−1 + h)d2]. A simple solution to the Bezout equation is given by X = −1 and Y1 = [1/(−1 + h), h]⊤. Therefore, all the stabilizing LTI controllers for the multirate system are of the form CT 1 (d) = 1 −1+h + hd − (−1 + h2d2)(ω1 + ω2d) −1 + ω1(−1 + h)d2h + ω2(−1 + h)d2 (4.30) where ω1(d2) and ω2(d2) are the entries of the stable and proper transfer function matrix = [ω1, ω2]. A necessary condition to achieve model matching is that the set of zeros of the desired closedloop transfer function contains the unstable zeros of NG, where = −(1 + hd) is such that A = DG. Notice that in the example DG does not have any unstable root, and therefore A and do not have any unstable zeros. Therefore, the only requirement on KT d is that KT d = d ¯K T d , where ¯K T d is any stable and proper transfer function. Model matching is achieved by selecting ¯ Y2 as shown in (4.25). 61 4.6 Conclusions In this chapter the problem of parameterizing the set of LTI controllers for the class of dualrate systems with slow measurement update rate and fast controller update rate was considered. The controller structure considered in this chapter is different from the one considered in chapter 3. In particular, these controllers are made of an upsampling operator and an LTI system. Upsampling operators are utilized to convert the update rate of the controller inputs into the controller update rate. A downsampling operator following the plant discretized at the fast control update rate was utilized to model the original system as one with fastupdating input and slowupdating output. The use of the invertible transformation (4.7) and of upsampling and downsampling operators, was shown to be beneficial to write the closedloop transfer function of the multirate system in Fig. 4.1 in the more familiar form of a singlerate closedloop system. A procedure was also given to parametrize the set of controllers CT 1 and CT 2 for which model matching is achieved in the two cases of a fastupdating and a slowupdating desired LTI system KT d and KNT d , respectively. In such parametrizations the free parameter can be chosen as any stable transfer function matrix to span the entire parametrized set. Due to the popularity and efficacy of PID controllers, a topic of future research may be the one of restricting the parametrization of controllers for multirate systems to only and all the PID controllers for which certain performance criteria can be met. In turn this problem can be extended to the one parametrizing the set of fixed structure controllers for multirate systems. Moreover, future research will focus also on the parametrization of the set of LTI controllers for which certain robustness criteria, in addition to stability, are met. 62 CHAPTER 5 Ripplefree conditions in multirate systems using LTI controllers In this chapter it is studied the effect of the (N, 1)shiftinvariant controllers used in chapter 4 on the steadystate response of the closedloop system. In particular, conditions and a design procedure are given to avoid the problem of ripples at steadystate. In the context of a output regulation and tracking problem, necessary and sufficient conditions were given in [26, 27] to achieve a ripplefree response to sinusoidalexponential reference signals in the case that (m, n)shiftinvariant controllers are utilized. In particular, a continuoustime internal model of the reference signals has to be included in the forward path of the control system. For constant reference signals however, the presence of a continuoustime zeroorder hold in cascade with the plant may not be sufficient to guarantee a ripplefree steadystate response, and two possible solutions can be utilized: either the controller is augmented with a fastrate digital integrator [27], or the controller coefficients are forced to converge at steadystate [25]. The use of timevarying controllers, however, may significantly worsen the intersample behavior of the transient response of the closedloop system, but no work has been done to directly address this problem. If the (N, 1)shiftinvariant controllers shown in chapter 4 are utilized, the main cause of ripples in the steadystate output can be isolated. In fact it will be shown that ripples may exist in the steadystate response of the closedloop system in Fig. 4.1 mainly because of the presence of the timevarying upsampling operators. Therefore, ripples can be avoided at steadystate, and the undesired nonlinear behavior may be 63 reduced during the transient, simply with the use of particular LTI filters connected to upsampling operators. Necessary and sufficient conditions on the structure of these filters are obtained in this chapter. The author’s initial work is shown in [38, 50]. This chapter is organized as follows. The problem formulation is given in section 5.1. Before deriving the main results of this chapter, two numerical examples that clearly motivate the need for subsequent analysis are given in section 5.2. Section 5.3 contains the main results of this chapter. Some particular solutions to the addressed problem are shown in section 5.4. Conclusions are given in section 5.5. 5.1 Problem Formulation Denote with P(s) the Laplace transform of a continuoustime LTI SISO plant to be controlled, and with y(t) its output to be regulated to the reference signal rNT . The plant output is measured at the slowrate 1/NT, where N is a positive integer, and the control signal uT updates at the faster rate 1/T. The considered multirate system is shown in Fig. 5.1. This control system is similar to the one shown in Fig. 4.1, for the + + N N NT rNT CT 1 (d) CT 2 (d) ZOH P(s) uT y(t) yNT FT (d) FT (d) Figure 5.1: Multirate Control System. exception of the additional LTI systems FT placed between the upsampling operators and the controllers CT i , i = 1, 2. Therefore, the digital control action provided by the controller takes the following form UT = CT 1 FT [Y NT ]T + CT 2 FT [RNT ]T 64 where UT , Z{uT }, Y NT , Z{yNT } and RNT , Z{rNT }. It is possible to show that the slowrate samplers in Fig. 5.1 can be modeled with a fastrate sampler followed by a downsampling operator. This alternative structure, shown in Fig. 5.2, is more suitable for modeling and analysis. + + N N N T rNT CT 1 (d) CT 2 (d) ZOH P(s) uT y(t) yNT FT (d) FT (d) Figure 5.2: Multirate Control System. The following assumption, together with assumption 3.1, is considered. Assumption 5.1 Given a filter FT , there exist stable controllers CT 1 and CT 2 that stabilize the slowrate reference to output transfer function Y NT = 1 − GTCT 1 FT NT −1 GTCT 2 FT NT RNT (5.1) and such that the closedloop multirate system exhibits zero steadystate error at the measurement update rate in response to a step reference signal, that is, lim k→∞ e(kNT) = 0 where e(t) , y(t) − r(t). The stability condition in assumption 5.1 can be met by designing the controller CT 1 as discussed in chapter 4, and by ensuring that CT 2 is stable. The condition, in assumption 5.1, about the zero steadystate error of the closedloop system at the measurement update rate, is satisfied if the forward path of the block diagram in Fig. 5.1 contains a continuoustime internal model of the step reference signal [24]. 65 Therefore, due to the presence of the zeroorder hold H(s), the controllers CT 1 and CT 2 can be designed to satisfy such a condition. Since the achievement of zero steadystate error at the measurement update rate is not sufficient to guarantee a ripplefree response to a step reference signal (because the error e(t) may be different from zero between two consecutive measurement sampling instants), it is desirable to obtain necessary and sufficient conditions on the LTI filter FT to achieve a steadystate ripplefree closedloop response, that is, lim k→∞ Z (k+1)NT kNT e⊤(τ )e(τ )dτ = 0. 5.2 Example The effect of the selection of an inappropriate filter FT on the continuoustime closedloop output y(t) is discussed in this section. In particular, two different examples are considered, each one obtained with the use of a different filter FT , to show that ripples can occur in the system output even though the systems CT 1 and CT 2 are LTI. The main purpose of this section is to motivate the need for the analysis developed subsequently in this chapter. Consider the control scheme in Fig. 5.2 where the plant transfer function is given by the doubleintegrator P(s) = 1/s2 and the controller sampling period is T = 0.1 sec. The plant output, y(t), and the step reference signal, r(t), are sampled at the rate 1/NT, where N = 3. Let KNT be the closedloop transfer function between the measured output Y NT and the slowrate reference signal RNT , that is, KNT , Y NT /RNT . Using the procedure shown in chapter 4, let the controllers CT 1 and CT 2 be designed in order to guarantee that KNT matches with the desired transfer function KNT d given by KNT d , 1 12 d(1 + 0.2d) 1 − 1.8d + 0.9d2 (5.2) 66 The cases (a) in which FT = 1 and (b) in which FT = 1 + d + d2 are considered for this example. In the case (a) the stable controllers CT 1 and CT 2 are given by CT 1 = 3 50 (−9 + 7d), CT 2 = 1 0.3 − 0.54d3 + 0.27d6 In the case (b), instead, the stable controllers CT 1 and CT 2 are given by CT 1 = −22 + 18d 0.36 + 0.09d3 , CT 2 = 10 − 8d + 2d2 3.6 − 5.58d3 + 1.62d6 + 0.81d9 Notice that with the designed controllers and the selected sampling period T, assumptions 3.1 and 5.1 are satisfied in both the cases (a) and (b). The closedloop system behavior to a unit step reference signal in the cases (a) and (b) is shown in Fig. 5.3 and Fig. 5.4. In particular, Fig. 5.3 compares the closedloop continuoustime response y(t) with the response of the desired digital system KNT d , and Fig. 5.4 shows the behavior of the control signal uT . As shown in Fig. 5.3, despite the continuoustime output y(t) equals in both the cases the output of the desired system at the measurement update rate (that is, at the time instants t = kNT, k = 0, 1, 2, . . .), an undesirable highfrequency oscillatory behavior characterizes the continuoustime response of the system in the case (a). This oscillatory behavior, which at steadystate is referred to as rippleeffect, does not appear in the continuoustime output of case (b) where a different filter FT has been utilized. Similarly, Fig. 5.4 shows the presence of persistent oscillations in the control signal uT in the case (a), and no oscillations in the case (b) where uT converges to a constant value. This fundamental difference between the step responses in the cases (a) and (b) shows that an inappropriate choice of the filter FT can cause a rippleeffect at steadystate as well as undesirable high frequency oscillations in the transient. Therefore, a characterization of the set of filters FT which help to prevent the rippleeffect is needed. 67 0 5 10 15 20 0 0.5 1 1.5 2 Time (sec) FT=1 Desired system response FT=1+d+d2 Figure 5.3: Plant output y(t). 0 5 10 15 20 −150 −100 −50 0 50 100 150 Time (sec) FT=1+d+d2 FT=1 Figure 5.4: Controller output uT . Remark 5.1 (On the stability of the fastrate closedloop system) Due to the presence of persisting oscillations at steadystate, it appears from Fig. 5.3 for the case (a) that the closedloop system is stable at the measurement updaterate but only marginally stable at the control updaterate. However, it is shown in chapter 4 that the fastrate closedloop system KT , Y T /[RNT ]T in Fig. 5.1 is stable if and only if the slowrate closedloop system KNT , Y NT /RNT is stable. Therefore, since KNT is stable (by assumption 5.1) also KT is stable. In particular, for the case (a) the stable 68 fastrate closedloop transfer function is given by KT = 12d + 36d2 + 60d3 + 39d4 + 8d5 + 12d6 + 7d7 720 − 1296d3 + 648d6 Therefore the sustained oscillations at steadystate are not due to the closedloop system instability, but due to the timevarying nature of the upsampling operators. 5.3 Ripplefree conditions It is shown in this section that the ripplefree behavior of the closedloop system response depends only on the choice of the digital filter FT . Moreover, necessary and sufficient conditions are given on the coefficients of the filter FT to guarantee a ripplefree closedloop response to step reference signals. Some preliminary results are given first to facilitate the proof of the main theorems. Proposition 5.1 (From [25]) Let GT be the transfer function of a LTI system, and let GNT L be its lifted version, whose elements are given in (2.16). Then the row sum of all the rows of GNT L (1) is equal, that is, XN k=1 GNT L,(m,k)(1) = XN k=1 GNT L,(n,k)(1), ∀m, n = 0 . . . ,N − 1 (5.3) The following result is a consequence of proposition 5.1. Proposition 5.2 Let ANT and BNT be two realvalued N ×1 transfer matrices, such that BNT = DNTANT , where DNT is a N × N transfer matrix. The following statements hold: 1. If the Ndimensional vector ANT (1) has all equal entries and DNT satisfies property (5.3), BNT (1) = DNT (1)ANT (1) is also a vector with all equal entries. 2. If ANT (1) and BNT (1) are each Ndimensional vectors with all equal entries, 69 DNT satisfies property (5.3). Proof. Part 1 is proved by noticing that BNT (1) = αANT (1), where α is the sum of all the elements in one row of DNT (1). Part 2 can be proved as follows. Since all the entries ANT k (1) (k = 0, . . . ,N − 1) of ANT (1) are equal, NX−1 k=0 DNT (i,k)(1)ANT k (1) = NX−1 k=0 DNT (i,k)(1) ! ¯a, i = 0, . . . ,N − 1 where ¯a = ANT k (1) (k = 0, . . . ,N − 1), and DNT (i,k) is the (i, k)th entry of DNT . Since all the entries of BNT (1) are equal, DNT (1) must satisfy property (5.3). The following result extends property (5.3) to the series connection of LTI systems. Proposition 5.3 Let DNT and ¯D NT be two N × N transfer matrices. The series connection DNT ¯D NT satisfies property (5.3) if and only if DNT and ¯D NT individually satisfy the same property. Proof. Let X ⊂ RN be the set of all the realvalued Ndimensional vectors xi (i = 1, 2, . . .) with not all the entries equal. Moreover, let X− be the complement of X in the space of all the realvalued Ndimensional vectors RN, and let x− i (i = 1, 2, . . .) denote the ith element of X−. (⇐). If DNT and ¯D NT satisfy property (5.3), DNT (1)x−i ∈ X− and ¯D NT (1)x−i ∈ X−, for every i ∈ N. Therefore, DNT (1)¯D NT (1)x− i ∈ X− which implies that every row sum of DNT (1)¯D NT (1) is equal. (⇒). By contradiction, assume that ¯D NT does not satisfy property (5.3) and that DNT does satisfy it. Therefore, there exists a vector x− i such that ¯D NT (1)x− i /∈ X− 70 and DNT (1)¯D NT (1)x− i /∈ X−. However, the last statement contradicts the hypothesis that the series connection DNT ¯D NT satisfies property (5.3). A similar argument can be made in the case that DNT does not satisfy property (5.3) and that ¯D NT does satisfy it. The following two theorems give the main results of this chapter. Theorem 5.1 Under the assumptions 3.1 and 5.1, the system in Fig. 5.1 exhibits a steadystate ripplefree response to step reference signals if and only if the first column of the lifted filter, FNT L (obtained as shown in (2.16)), has all equal entries for ds = 1, that is, FNT L (1) = α ⋆ · · · ⋆ ... ... ... α ⋆ · · · ⋆ (5.4) where α is any real scalar number. Proof. (⇒). As discussed in section 2.1, the upsampling operators insert N −1 zerovalued samples between two consecutive samples of the original signal. This behavior of the upsampling operators can be taken into account by assuming that the lifted filter matrix FNT L is identically zero except for the first column, that is, FNT L = FNT L,(1,1) 0 · · · 0 ... ... ... FNT L,(N,1) 0 · · · 0 (5.5) By using (5.5) as the lifted matrix of the filter FT , it is possible to show that the lifted plant input, UNT L , is related to the lifted reference signal RNT L , Z{LN(rT )}, 71 by the expression I − CNT 1,L FNT L GNT L UNT L = CNT 2,L FNT L RNT L (5.6) where CNT 1,L and CNT 2,L are the lifted versions of the controllers CNT 1 and CNT 2 , respectively. The stability of the closedloop system (5.1) ensures that the steadystate closedloop response yNT and the control action uNT are constant. Therefore, to achieve a ripplefree response to step reference signals, it remains to be shown that also uT is constant at steadystate. Moreover, in order to feed the feedback loop with a signal constant at steadystate, it is necessary that the output of the controller CT 2 also reaches a steadystate value. In other word it is necessary that all the entries of the vector uNT L , as well as all the entries of the inverse Ztransform of the right hand side of (5.6), have the same steadystate value. By applying the final value theorem to equation (5.6), and by proposition 5.2, this implies that the N ×N matrix S , I − CNT 1,L FNT L GNT L has to satisfy property (5.3). In turn, by proposition 5.3, this requires that the matrix FNT L GNT L satisfies the same property. Considering that FNT L (1)GNT L (1) is of the following form FNT L (1)GNT L (1) = FNT L,(0,0)(1)GNT L,(0,0)(1) · · · FNT L,(0,0)(1)GNT L,(0,N−1)(1) ... ... FNT L,(N−1,0)(1)GNT L,(0,0)(1) · · · FNT L,(N−1,0)(1)GNT L,(0,N−1)(1) , in order for the matrix FNT L GNT L to satisfy property (5.3), it must be that FNT L,(i,0)(1) = α, i = 0, . . . ,N − 1, where α can be any real number. (⇐). If FNT L (1) is in the form (5.4), the matrix S , (I − CNT 1,L FNT L GNT L ) satisfies property 5.3. Moreover, considering that the vector rNT L , LN(rT ) has all equal entries, at steadystate the right hand side of (5.6) is also a vector with all equal 72 entries. Therefore, by proposition 5.2, the lifted plant input UNT L (1) has all equal entries, and this is sufficient to guarantee that the plant output is ripplefree at steadystate. The following theorem gives closedform conditions on finite impulse response filters such that their lifted version is in the form (5.4) at steadystate. Theorem 5.2 Let the filter FT be a finite impulse response system of the form FT , a0 + a1d + a2d2 + · · · + andn (5.7) The lifted system FNT L (dN) of FT is in the form (5.4) for dN = 1 if and only if Xp k=0 aNk+i = α, i = 0, . . . ,N − 1 (5.8) where p = ⌊(n − i)/N⌋ is the largest integer not greater than (n − i)/N. Proof. As shown in (2.19), the first column of FNT L contains the modified Ztransforms of FT . In particular, from the definition of modified Ztransform (2.17), the first N modified Ztransforms of FT are given by ˜ FNT 0 = a0 + aNd + a2Nd2 + · · · + a⌊n/N⌋Nd⌊n/N⌋ ˜ FNT 1 = a1 + aN+1d + · · · + a⌊(n−1)/N⌋N+1d⌊(n−1)/N⌋ ... = ... ˜ FNT N−1 = aN−1 + a2N−1d + · · · + a⌊(n−N+1)/N⌋N+N−1d⌊(n−N+1)/N⌋ In general, the ith modified Ztransform is given by ˜ FNT i (d) = Xp k=0 aNk+idk (5.9) 73 where p = ⌊(n − i)/N⌋. Therefore, in order for FNT L (1) to be of the form (5.4), it is necessary and sufficient that (5.8) is satisfied. Theorem 5.2 gives necessary and sufficient conditions for the filter FT in Fig. 5.1 to guarantee a ripplefree closedloop response to step reference signals, in the case that FT is a FIR filter. The conditions are expressed as constraints on the sum of groups of coefficients in FT . Remark 5.2 The result obtained in theorem 5.2 is based on the assumption that FT is a FIR filter. However, if it is desired to use additional poles in the filter FT , while preserving the ripplefree behavior of the closedloop system, the added poles must not cancel the zeros introduced by the FIR part of the filter. Similarly, additional zeros can be used as long as they are not at the origin (to avoid cancelation with the poles at the origin of the FIR part of the filter). Remark 5.3 The results shown in this work are also valid in the case that the feedforward and the feedback filters FT in Fig. 5.1 are not the same. However, to achieve a ripplefree closedloop response it is important that both the filters satisfy condition (5.4). Remark 5.4 The idea behind the separation of the controllers into the two parts, CT i (for i = 1, 2) and FT is to facilitate the design process. In fact, by keeping FT and CT i separated one can first select the appropriate FT to achieve ripplefree response (and possibly also to filter the system output), and then design the remaining part of the controller, CT i , to meet other performance criteria. However, FT and CT i can be joined to form one transfer function for implementation purposes. 74 5.4 Digital higherorder filters A simple example of a FIR filter that satisfies condition (5.8) is the digital ZOH (ZOH) : FT = 1 + d + d2 + · · · + dN−1 = 1 − dN 1 − d (5.10) In fact, by replacing n with N − 1 in (5.9), it can be shown that ˜ FNT i = ai = 1, i = 0, . . . ,N − 1. The digital ZOH filter in (5.10) serves to hold the output signal of the upsampling operator between two consecutive slow measurement updates. However, the filter in (5.10) is not the only solution. In fact some other examples include higher order holds (details on how to obtain a representation for higher order holds are given in [51]). For instance, in the cases of the first order hold (FOH) and the triangular hold (TH), the corresponding modified Ztransforms are given by: (FOH) : ˜ FNT i (d) = N + i N − i N d ⇒ ˜ FNT i (1) = 1 (TH) : ˜ FNT i (d) = i N + N − i N d ⇒ ˜ FNT i (1) = 1 The use of any one of these filters guarantees a ripplefree closedloop response to step reference signals. 5.5 Conclusions For the class of controllers considered in chapter 4, a good choice of the sampling period and the stability of the closedloop system are necessary but not sufficient to achieve a ripplefree response at steadystate. In fact, one of the main causes of ripples in the response of such a multirate system, where the output of the plant is measured at a slow rate and LTI controllers are utilized to provide a faster control action, is the manner in which the slow measurements are fed into the controller. The inclusion of digital prefilters, in cascade with the controller transfer function, 75 was shown in this chapter to be useful in avoiding ripples in the step response of the closedloop system. Necessary and sufficient conditions on the coefficients of such filters were also provided to characterize the entire set of filters for which a ripplefree steadystate response can be achieved. 76 CHAPTER 6 Conclusions and future work In this work several control problems were considered for the class of multirate systems with slow measurement update rate and fast controller update rate. Such control problems can be grouped as follows: 1) inputstate matching with a desired singlerate digital system, 2) parametrization of LTI controllers for dualrate systems, and 3) ripplefree response of multirate systems with upsampling and downsampling operators. The solution to the second problem was facilitated by the use of the upsampling and downsampling operators as modeling tools. However, the timevarying nature of these two operators was shown to be the main cause of ripples in the steadystate response of the multirate system. A method to avoid ripples in the step response with the use of FIR filters in cascade with the upsampling operators was given in chapter 5. Future research that is relevant to the material covered is given at the end of each chapter. The first use of upsampling and downsampling operators can be found in the signal processing area [34]. In particular, with the use of these operators, the information contained in any signal can be converted to a pyramidal set of information where each layer of the pyramid characterizes the signal properties at a particular scale. If, for instance, this multiscale signal characterization is applied to the output signal of the plant, one may potentially design a controller to achieve a desired closedloop performance at each particular scale. The main disadvantage of this multiscale theory, that was originally named as subband coding, is the difficulty in the choice of the scale factors, which correspond to the factors of the upsampling and down 77 sampling operators [52]. This is due to the fact that the frequency domain properties of a signal are usually timevarying. Therefore, a way to express the scale factors as functions of time would be needed. To overcome this problem, and provide a better timefrequency localization of the signal properties, a new multiscale theory based on the use of wavelets was recently adopted [53]. The use of wavelets to brake down the signal information into a finite discrete set of useful and meaningful characteristics at different scales resulted being very successful. If one wants to apply this multiscale theory to the plant output signal (in a control system), a filter may be utilized to generate the scaled information to be passed on to the controller. This procedure, however, would result in delaying the feedback measurement, which is highly undesired. One way to overcome this issue would be to model the plant also by using the waveletbased multiscale theory. Few attempts have been made to model LTI systems using this theory [54–56], but the results obtained so far in the literature are complicated to analyze and to use (due to the complexity of the original wavelets). 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Title  Modeling, Analysis, and Design of Multirate Systems 
Date  20101201 
Author  Cimino, Mauro 
Department  Mechanical Engineering 
Document Type  
Full Text Type  Open Access 
Abstract  Multirate systems refer to the class of control systems where digital signals and systems updating at different rates are utilized. These systems have been the subject of extensive research for many decades. This work focuses on the class of multirate systems where the controller update rate is faster than the measurement update rate. This class of systems can be found in several applications of electromechanical and chemical processes. The classical model matching problem that is studied in the literature requires the inputoutput properties of the closedloop multirate system to match those of a desired singlerate linear timeinvariant system. The model matching problem from an inputstate viewpoint is considered in this work: Given a desired linear timeinvariant system, find conditions and provide a controller design procedure to achieve matching between the closedloop system and the desired system state variables at the measurement update rate. The advantage of solving this inputstate model matching problem derives from the opportunity to select a desired behavior for the closedloop system state variables. Necessary and sufficient conditions are given to solve this problem and, at the same time, to avoid the presence of ripples in the steadystate output of the continuoustime plant. In addition, a procedure is provided to design linear timevarying controllers for this control problem. The second part of this work focuses on the design of linear timeinvariant controllers for multirate systems. A new frequencydomain approach to model multirate systems is developed. This approach facilitates the design and the parametrization of linear timeinvariant controllers. The new modeling approach is based on the use of upsampling and downsampling operators. The developed parametrization procedure is utilized to parametrize the set of stabilizing linear timeinvariant controllers for which model matching is achieved at the fast control update rate with a desired singlerate system. Even though the use of upsampling and downsampling operators facilitates the parametrization of linear timeinvariant controllers, this may cause ripples in the closedloop system response. Therefore, necessary and sufficient conditions are also derived to guarantee ripplefree response to step reference signals. The conditions are given in terms of the constraints on the coefficients of finite impulse response filters included at the input of the feedback and feedforward controllers. 
Note  Dissertation 
Rights  © Oklahoma Agricultural and Mechanical Board of Regents 
Transcript  MODELING, ANALYSIS, AND DESIGN OF MULTIRATE SYSTEMS By Mauro Cimino Bachelor of Engineering (“Laurea di primo livello in Ingegneria Informatica”) University of Rome “Tor Vergata” Rome, Italy 2003 Master of Engineering (“Laurea specialistiva in Ingegneria dell’Automazione”) University of Rome “Tor Vergata” Rome, Italy 2005 Submitted to the Faculty of the Graduate College of the Oklahoma State University in partial fulfillment of the requirements for the Degree of DOCTOR OF PHILOSOPHY December, 2010 COPYRIGHT By Mauro Cimino December, 2010 MODELING, ANALYSIS, AND DESIGN OF MULTIRATE SYSTEMS Dissertation Approved: Thesis Adviser, Dr. Prabhakar R. Pagilla Committee Member, Dr. Lawrence L. Hoberock Committee Member, Dr. Gary E. Young Committee Member, Dr. Martin Hagan Dean of the Graduate College ii ACKNOWLEDGMENTS I wish to thank my advisor, Dr. Prabhakar R. Pagilla, for his guidance and support towards the achievement of my doctoral degree, and for his friendship. I also wish to thank my committee members, Dr. Lawrence L. Hoberock, Dr. Gary Young, and Dr. Martin Hagan for their insightful suggestions and opinions. Un ringraziamento particolare va ai miei genitori e a mio fratello che mi hanno sempre sostenuto ed incoraggiato durante tutto il periodo di permanenza in Oklahoma. iii TABLE OF CONTENTS Chapter Page 1 Introduction 1 2 Preliminaries 6 2.1 Upsampling and Downsampling operations . . . . . . . . . . . . . . 7 2.1.1 Timedomain characteristics . . . . . . . . . . . . . . . . . . . 7 2.1.2 Transformdomain characteristics . . . . . . . . . . . . . . . . 9 2.2 LPTV systems and shiftinvariance property . . . . . . . . . . . . . . 11 2.3 Lifting technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4 Modified Ztransforms . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3 Design of LPTV controllers for inputstate model matching 19 3.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.2 Insights on the controller structure . . . . . . . . . . . . . . . . . . . 23 3.3 Multirate control design . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.3.1 Design of the gain matrices KL and LL . . . . . . . . . . . . . 27 3.3.2 Design of Cϕ and Dx to achieve ripplefree response . . . . . . 30 3.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4 Frequency domain modeling and parametrization of LTI Controllers 42 4.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.2 Modeling the multirate system . . . . . . . . . . . . . . . . . . . . . . 46 4.3 Closedloop stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 iv 4.4 Model matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.4.1 Model matching at the slow rate . . . . . . . . . . . . . . . . . 55 4.4.2 Model matching at the fast rate . . . . . . . . . . . . . . . . . 57 4.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 5 Ripplefree conditions in multirate systems using LTI controllers 63 5.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 5.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 5.3 Ripplefree conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.4 Digital higherorder filters . . . . . . . . . . . . . . . . . . . . . . . . 75 5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 6 Conclusions and future work 77 BIBLIOGRAPHY 79 v LIST OF FIGURES Figure Page 1.1 HDD logical organization [1]. . . . . . . . . . . . . . . . . . . . . . . 1 2.1 Upsampling and downsampling operations. . . . . . . . . . . . . . . 9 2.2 Example corresponding to the block diagram of Fig. 2.1B with N = 2. The digital signal yT (on the left) is first downsampled (center) and then upsampled (on the right). . . . . . . . . . . . . . . . . . . . . . 9 2.3 Examples of multirate systems. . . . . . . . . . . . . . . . . . . . . . 11 2.4 A particular example of a causal linear LPTV system. . . . . . . . . . 13 3.1 Control system structure . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.2 Closedloop system response to a step reference signal: the state variables of the multirate closedloop system are compared with the state variables of the desired slowrate LTI system. The signals x1, x2 are the velocity and position, respectively, of the double integrator continuoustime plant. The closedloop system output is y(t) = x2(t). . . . . . . 40 4.1 Multirate control scheme. . . . . . . . . . . . . . . . . . . . . . . . . 45 4.2 Multirate control scheme. . . . . . . . . . . . . . . . . . . . . . . . . 46 5.1 Multirate Control System. . . . . . . . . . . . . . . . . . . . . . . . . 64 5.2 Multirate Control System. . . . . . . . . . . . . . . . . . . . . . . . . 65 5.3 Plant output y(t). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.4 Controller output uT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 vi NOMENCLATURE Symbols and operators R Real number field N Natural number field N+ Set of all the nonnegative natural numbers t Time variable (t ∈ R) T Control update rate (or base rate) Ts Measurement update rate Z Transform operator LN Lifting operator Shift operator at the sampling time T d Unit delay at the sampling time T s Shift operator at the sampling time Ts ds Unit delay at the sampling time Ts ↑N Upsampling operator by a factor N ↓N Downsampling operator by a factor N j Imaginary unity (√−1) ⊤ Transpose operator ker Null space operator P(s) Laplace transform of the controlled plant H(s) Laplace transform of the zeroorder hold y(·) Plant output signal r(·) Reference signal LTI Linear, timeinvariant LPTV Linear, periodically timevarying HDD Harddisk drive FIR Finite impulse response vii CHAPTER 1 Introduction Multirate systems are widely used in process and manufacturing industries. In some applications the rate of update of the feedback measurement is slower than the controller update rate. One of these applications is the control of the HardDisk Drive (HDD) Read/Write (RW) head [2]. In HDD data are stored on the platters along thousands of concentric circular tracks. As shown in Fig. 1.1, each track is divided into sectors, which are usually the smallest addressable units containing the data stored on the harddisk drive. Each sector stores a certain amount of user data, and Figure 1.1: HDD logical organization [1]. reserves the remaining for control and management purposes of the drive. Those additional bytes are usually stored at the beginning of the sectors and include the sector ID information. The sector ID is useful for locating the sector on the disk when certain data have to be retrieved or stored. The angular position of the R/W magnetic head is obtained every time the R/W head passes through the part of the sector containing the sector ID information. Therefore, the angular position is obtained at different rates depending on the rotational speed of the platters. The control input 1 to the VoiceCoil Motor Actuator, which drives the arm containing the R/W head, is provided by the digital board installed on the back of the HDD. The multirate nature of the system is due to the fact that the discretetime control action provided by the digital board updates at a rate much faster than the rate of update of the R/W head position feedback. Other examples of multirate systems where the rate of update of the feedback measurement is slower than the update rate of the control signal include the octane rating control in the continuous catalytic reforming process [3] and the control of chemical concentrations in distillation columns [4]. Sometimes, these multirate systems are treated as singlerate ones, and classic design tools for singlerate systems are utilized to develop control strategies. However, neglecting the multirate nature of these systems may lead to unwanted results. Therefore, the development of strategies to model, analyze and control multirate systems is very important and a large amount of work has been done in literature to address these issues. In the late 1950’s the frequency decomposition and the switch decomposition techniques [5–9] were utilized in singlerate sampleddata systems to analyze the response of the continuoustime plant between samples. These two techniques were shown to be related in [10]. The switch decomposition was first used in [6] to model multirate timevarying systems as singlerate Linear TimeInvariant (LTI) ones. This in turn facilitated the use of existing singlerate techniques to design controllers for multirate systems. However, the main drawback of this technique was that the modeled system could be very difficult to analyze in some cases, based on the rates of operation of the samplers. In the statespace domain, a different technique, referred to as the lifting technique [11] (or as blocking technique in signal processing), was developed to analyze the intersample behavior of singlerate sampleddata systems. This technique, which was initially used in [12] also to model Linear Periodically TimeVarying (LPTV) systems as singlerate LTI ones, became commonly used in the multirate sys 2 tems literature for both modeling and design problems. Compared to the frequency and switch decomposition techniques, the lifting technique provides a more compact solution for modeling multirate systems as singlerate systems. Several control problems for multirate systems have been addressed in the last three decades using multirate control structures: optimal control [13–15], pole placement [16, 17], model matching [18, 19], robust control [18, 20–23], ripplefree output tracking and regulation [24–27], and controller parametrization [23, 28, 29]. In most of these multirate control problems the plant to be controlled is represented by a LTI system, and the controller is designed so that the control signal updates several times between two consecutive feedback measurements. This control strategy was shown to be much more successful than the classical LTI control strategy for most of the above mentioned control problems. Due to this reason and to the simplicity of designing these controllers with the use of the lifting technique, this multirate control strategy became very popular in the literature. In particular, the class of controllers which received most of the attention in the multirate control system area is the one of multirate LPTV controllers, characterized not just by having different input and output update rates but also by the periodically timevarying nature of their gains. The reason for the popularity of multirate LPTV controllers was mainly due to the existence of the lifting technique. In fact this technique allows one to rewrite multirate LPTV systems as singlerate LTI ones, hence facilitating LPTV controller design. The effectiveness of a multirate LPTV controller over a singlerate one was initially shown in [20] and [18]. In particular, the theory developed in [20] for singleinput singleoutput (SISO) bicausal plants, showed that LPTV controllers can be designed to remove blocking zeros, to obtain an arbitrarily large gain margin (in singlerate LTI control systems, instead, the gain margin is known to be limited by the unstable zeros and poles of the plant [30]), and to simultaneously stabilize any finite set of plants. In [18] it is also shown that poles and zeros of the closedloop transfer function can 3 be arbitrarily placed, resulting in the opportunity to meet any gain and phase margin specifications. However, a drawback derived from the use of LPTV controllers may sometimes be the presence of ripples in the output of the continuoustime plant. The rippleeffect is a common undesired phenomenon in sampleddata systems. In the case of singlerate systems it is well known to be mostly due to the presence of unobservable dynamics [9] generated through the choice of a critical sampling period [31]. The output of multirate systems, unlike that of singlerate systems, can be affected also by another form of ripples which is due to the timevarying nature of the closedloop system. In control systems where the measurement update rate is a multiple of the control update rate this undesired rippleeffect, unlike the one due to the selection of a critical sampling period, can be detected from the output of the system sampled at the control update rate, but is not visible from the measurements of the plant output (which is sampled at the slower measurement update rate). Even though several solutions to the steadystate ripple effect have been found in the context of output tracking and regulation problems [24–27], the transient response of the closedloop system may still be negatively affected by the use of LPTV controllers. Therefore, with the aim of reducing nonlinear behaviors in both transient and steadystate response, in this work a particular subclass of multirate LPTV controllers is considered (chapters 4 and 5). Each member of this class comprises an upsampling operator (which is a static timevarying operator) followed by a LTI system. The use of this controller structure (as opposed to the more general form of multirate LPTV controllers) may reduce undesired oscillatory behavior in the transient response of the continuoustime plant. One of the main contributions provided in chapter 4 consists of a procedure to parameterize the set of all the LTI controllers for which model matching is achieved with a desired digital LTI system in the two cases: (1) the desired LTI system updates at the control update rate, (2) the desired LTI system updates at 4 the measurement update rate. The analysis of the effects of these controllers on the steadystate response of the closedloop system is provided in chapter 5. In particular, conditions to achieve a steadystate ripplefree response are obtained, and a design procedure that satisfy the conditions is provided. These conditions, differently from [24–27], are obtained in the context of a model matching problem using the above mentioned controller structure. The model matching problem addressed in chapter 4 is the common inputoutput model matching problem, where the closedloop system output is required to match the output of a desired system when the same reference signal is provided to the two systems. The more complex problem of inputstate matching using LPTV controllers is addressed in chapter 3. Differently from the inputoutput model matching problem, the inputstate matching problem requires the state of the multirate closedloop system to match the state of a desired LTI system. To address this problem a more general form of multirate LPTV controllers is considered. In the same chapter conditions to achieve a steadystate ripplefree response to step reference signals are also provided. Preliminary mathematical tools, which are necessary for the formulation of the problems and the development of results in the above mentioned chapters, are given in chapter 2. A discussion about the topics of future research is given in chapter 6. 5 CHAPTER 2 Preliminaries The mathematical tools shown in this chapter will be utilized in the rest of this work to analyze and design multirate sampleddata systems. It is well known that the output of a digital LTI system in a singlerate framework can be analyzed using the tools shown in every digital control system book (see [32, 33]). However, for systems with inputs and outputs updating at different rates, other tools, such as upsamplers, downsamplers, lifting technique and modified Ztransform, can be needed for analysis and modeling. These tools and the properties needed for subsequent developments are discussed in this chapter. Some preliminary notations are introduced first. The control systems considered in this work involve digital signals and systems updating at two different rates, 1/T and 1/Ts, where Ts = NT, and N > 1 ∈ N+. Since Ts > T, Tsupdating and Tupdating signals (or systems) will be respectively referred to as slowrate or fastrate signals (respectively, systems). To distinguish between slowrate and fastrate signals (or system) the superscripts T and NT, respectively, will be utilized to refer to their rate of update. The unit delay operators d and ds will be utilized in the Ztransforms of fastupdating and slowupdating sequences, respectively, to distinguish between unit delays of T and NT seconds. Also, lower case and upper case letters will be used to refer to signal (or systems) expressed in timedomain and transform domain, respectively. Therefore, the Ztransforms of a fastupdating sequence yT [k] and a 6 slowupdating sequence yNT [k] are given by Y T (d) , Z{yT [k]} = X∞ k=0 yT [k]dk (2.1) Y NT (ds) , Z{yNT [k]} = X∞ k=0 yNT [k]dks (2.2) Wherever it is clear from the context, the arguments k (in the time domain) and d or ds (in the transform domain) will be omitted. 2.1 Upsampling and Downsampling operations Modeling a multirate system involves the use of upsampling and downsampling operations. In this section the discussion is limited to only the properties of these operations which will be useful for subsequent developments. A more extensive discussion of these operations is given in [34, 35]. 2.1.1 Timedomain characteristics Upsampling or downsampling a digital signal by a factor N respectively corresponds to increasing or decreasing by N times the rate of update of the original signal. Since only signals updating at the rates 1/NT and 1/T are considered, only upsampling and downsampling operations by a factor N are utilized in this work to switch between these two rates. Therefore, the bracket notations [·]T and [·]NT , where the superscript denotes the new rate of update of the signal, are utilized to refer to upsampled and downsampled signals (or systems), respectively. Nested notations are also utilized, when needed, to keep track of the sequence of upsampling and downsampling operations on signals. For instance, the notation [[·]NT ]T is utilized to denote signals that have been downsampled first and then upsampled; some properties of these nested notations are given in the following, after the definitions of upsampling 7 and downsampling operations. Let VT and VNT be respectively the spaces of Tupdating and NTupdating sequences. The upsampling and downsampling operators by a factor N, respectively denoted as ↑N and ↓N, are defined as ↑N: VNT → VT , ↓N: VT → VNT . Therefore, upsampling a sequence yNT by a factor N corresponds to increasing by N times the rate of update of the signal. This is done by inserting N −1 zerovalued samples between two consecutive samples of the original signal, that is, ↑N yNT , [yNT ]T [k] = yNT [k/N], if N divides k 0, otherwise. (2.3) Therefore, the upsampled sequence [yNT ]T , contains all the information needed to reconstruct yNT . Conversely, downsampling a sequence yT corresponds to reducing by N times the rate of update of the signal. This is done by discarding the last N −1 samples for every consecutive group of N samples of the initial sequence, that is, ↓N yT , [yT ]NT [k] = yT [kN], k = 0, 1, 2, . . . (2.4) This results in a loss of highfrequency information about the original sequence yT . Some of the main properties of these operations are discussed in [34, 35]. One property of the upsampling and downsampling operators is that ↓N↑N= I where I is the identity operator. This property, which can be verified by using (2.4) and (2.3), implies that [[yNT ]T ]NT = yNT , for any slowupdating sequence yNT . Other properties are shown through the two block diagrams in Fig. 2.1(A) and Fig. 2.1(B) where the blocks (↑N) and (↓N) respectively represent the upsampling and down 8 sampling operations by a factor N. The block scheme in Fig. 2.1(A) shows a portion (A) (B) NT y(t) yT N N N yNT [yT ]NT [yNT ]T [[yT ]NT ]T Figure 2.1: Upsampling and downsampling operations. of a dualrate control scheme, where the continuous signal y(t) is slowsampled at the rate 1/NT and then upsampled by a factor N. The signals [yNT ]T and [[yT ]NT ]T of Fig. 2.1(A) and (B), respectively, are equivalent [9]. This implies the equivalence of the signals yNT and [yT ]NT . Fig. 2.2 shows an example corresponding to the evolution of a signal yT in input to the block diagram of Fig. 2.1B, for the case N = 2. Figure 2.2: Example corresponding to the block diagram of Fig. 2.1B with N = 2. The digital signal yT (on the left) is first downsampled (center) and then upsampled (on the right). 2.1.2 Transformdomain characteristics In the Ztransform domain, [Y NT ]T and [Y T ]NT are utilized to denote the Ztransform of the upsampled sequence [yNT ]T and the downsampled sequence [yT ]NT , respectively. These transforms can be computed as follows [34, 35]: [Y NT ]T (d) = Y NT (dN) [Y T ]NT (ds) = 1 N NX−1 k=0 Y T (d1/N s e−2πjk/N) (2.5) 9 where j denotes the imaginary unit. The rules for nested upsampling and downsampling operations discussed previously for signals in timedomain also apply to signals in Ztransform domain, that is, [Y T ]NT = Y NT , [[Y NT ]T ]NT = Y NT . Considering that a transfer function is the impulse response of a LTI system, the same notations and properties considered so far for Ztransforms of sequences apply also to transfer functions of digital systems. The following two properties of upsampling and downsampling operators are useful for the convolutions between signals and systems: [Y NT 1 Y NT 2 ]T (d) = [Y NT 1 ]T (d)[Y NT 2 ]T (d) = Y NT 1 (dN)Y NT 2 (dN) (2.6) h Y T 3 Y NT 2 T iNT (ds) = [Y T 3 ]NT (ds)Y NT 2 (ds) (2.7) where Y NT 1 , Y NT 2 and Y T 3 denote the Ztransforms of signals and/or systems. These two properties can be obtained using the definitions (2.5). In particular, it is shown in (2.6) that the upsampling operation is linear with respect to the product, that is [Y NT 1 Y NT 2 ]T = [Y NT 1 ]T [Y NT 2 ]T . However, it is important to notice that this does not hold for the downsampling operation, that is, [ATBT ]NT 6= [AT ]NT [BT ]NT . (2.8) Properties (2.6) and (2.7) are useful to model the two block diagrams in Fig. 2.3. In Fig. 2.3(A) the output Y T , Z{yT } is related to the input UT , Z{uT } by the expression Y T = GTUT . However, since the downsampling operation is not linear with respect to the product, [Y T ]NT 6= [GT ]NT [UT ]NT . Therefore, [Y T ]NT = [GTUT ]NT . 10 (A) (B) uT GT (d) yT N N N [yT ]NT eNT CNT 1 (ds) CT 2 (d) uNT 1 [uNT 1 ]T uT2 [uT2 ]NT Figure 2.3: Examples of multirate systems. In Fig. 2.3(B) the signal [UT 2 ]NT , Z{[uT2 ]NT } can be obtained using (2.5) and (2.7) as follows: [UT 2 ]NT = h CT 2 UNT 1 T iNT = [CT 2 ]NTUNT 1 = [CT 2 ]NTCNT 1 ENT where UNT 1 , Z{uNT 1 } and ENT , Z{eNT }. 2.2 LPTV systems and shiftinvariance property In this section, the definitions of timeinvariant, periodically timevarying, and (m, n) shiftinvariant systems are given. These definitions will be useful to classify the multirate systems utilized in the next chapters. Let : VT → VT and s : VNT → VNT be the linear transformations respectively corresponding to the delay operators d and ds, that is, Z{ uT1 } = dUT 1 , Z{ suNT 2 } = dsUNT 2 where UT 1 and UNT 2 are the Ztransforms of the sequences uT1 and uNT 2 , respectively. Then it is well known that a Tupdating and a NTupdating causal linear system with impulse response cT and cNT , respectively, are timeinvariant if: pcT = cT p, ps cNT = cNT ps , p ∈ N+. 11 This definition does not apply to upsampling and downsampling operators. In fact, it follows from (2.3) and (2.4) that s ↓N =↓N N (2.9) N ↑N =↑N s (2.10) N ↑N↓N =↑N↓N N (2.11) i ↑N↓N 6=↑N↓N i, i = 1, . . . ,N − 1. (2.12) Equations (2.9) and (2.10) clearly show that the upsampling and downsampling operators are not timeinvariant. In fact, the upsampling operator has the inputoutput property that shifting the input by 1 sample results in shifting the output by N samples. For the downsampling operator, instead, a shifting of the input by N samples results in a shifting of the output by 1 sample. For these reasons, the upsampling and downsampling operators are usually referred to as (N, 1) and (1,N)shiftinvariant [36], respectively. The shiftinvariance property for multirate systems can be considered as a generalization of the timeinvariance property for singlerate systems. Equations (2.11) and (2.12) show that the system made of a downsampling operator followed by an upsampling operator is a particular example of a causal linear (N,N)shiftinvariant system, for which a shifting of the input by i samples results in a shifting of the output by i samples if and only if i = kN, k ∈ N. All (N,N) shiftinvariant systems are also called periodically timevarying systems, or Nperiodic systems [18]. A general statespace representation of all causal LPTV systems operating with sampling time T is given by ξT [kN + i + 1] = AiξT [kN + i] + BiuT [kN + i] yT [kN + i] = CiξT [kN + i] + DiuT [kN + i] (2.13) 12 where ξT , uT and yT are the state, input and output variables of the system, k ∈ N, i = 0, . . . ,N − 1, and the matrices Ai, Bi, Ci and Di vary as i goes from 0 to N − 1. A particular example of a LPTV system, which will be used in chapter 4 and 5 is the one shown in Fig. 2.4 where CT denotes the transfer function of a Tupdating, uT yT N N CT Figure 2.4: A particular example of a causal linear LPTV system. causal LTI system. If (AC,BC,CC,DC) is a statespace representation of CT , it is possible to show that the LPTV system in Fig. 2.4 has a statespace representation of the form (2.13) with (Ai,Bi,Ci,Di) = (AC,BC,CC,DC) if i = 0 (I, 0, 0, 0) if i = 1, . . . ,N − 1. 2.3 Lifting technique The lifting technique, which dates back to [11], has been utilized in the last three decades to convert multirate systems into singlerate timeinvariant systems [37]. The notion of the lifting operator and its application to multirate systems are explained in this section. Let VNT N be the set of slowupdating sequences with elements as N dimensional vectors. The lifting operator LN and the lifted version wNT L [k] of a sequence wT [k] ∈ VT are defined as LN : VT → VNT N : wT [k] → wNT L [k] 13 where wNT L [k] , wNT L,0 [k] wNT L,1 [k] ... wNT L,N−1[k] = wT [Nk] wT [Nk + 1] ... wT [Nk + (N − 1)] . (2.14) Since wNT L , LN(wT ) updates at the slow rate 1/NT, its Ztransform, denoted as WNT L , is given by: WNT L , Z{LN(wT )} = WNT L,0 ,WNT L,1 , · · · ,WNT L,N−1 ⊤ where the symbol ⊤ denotes the transpose operator and WNT L,i (ds) = P∞ k=0 wNT L,i [k]dks , i = 0, . . . ,N − 1. The lifting technique can be used to convert LPTV systems of the form (2.13) into equivalent LTI systems. This can be done as follows. Let (Ai,Bi,Ci,Di), i = 0, . . . ,N − 1, be the statespace representation of a SISO LPTV system with period N, input uT and output yT . Its lifted version corresponds to the LTI system, between the input uNT L , LN(uT ) and output yNT L , LN(yT ), with statespace representation 14 (AL,BL,CL,DL) given by AL , NY−1 i=0 Ai = AN−1 · · ·A1A0 BL , BL,0 · · · BL,N−1 , BL,i , QN−1 k=i+1 Ak Bi, i = 0, . . . ,N − 2 BN−1 i = N − 1 CL , C⊤ L,0 · · · C⊤ L,N−1 ⊤ , CL,i , C0, i = 0 Ci Qi−1 k=0 Ak, i = 1, . . . ,N − 1 DL , DL,(0,0), · · · ,DL,(0,N−1) ... ... DL,(N−1,0), · · · ,DL,(N−1,N−1) ,DL,(m,n) , Cm Qm−1 k=n+1 Ak Bn m > n − 1 CmBm−1 m = n − 1 Dm m = n 0 m < n In the transform domain, such a lifted system is represented by the N × N transfer matrix Y NT L = CLds (I − dsAL)−1 BL + DL UNT L (2.15) where Y NT L , Z{yNT L } and UNT L , Z{uNT L }. This shows that the lifted representation of a LPTV system is LTI, and that the lifted system output contains all the information needed to reconstruct the output of the original system. Notice that if also the original system was LTI, that is, (Am,Bm,Cm,Dm) = (An,Bn,Cn,Dn) for 15 m, n = 0, . . . ,N − 1, the lifted representation (AL,BL,CL,DL) would take the form AL , AN, BL , AN−1B AN−2B · · · AB B CL , C CA CA2 ... CAN−1 , DL , D 0 · · · 0 0 CB D · · · 0 0 CAB CB · · · 0 0 ... ... ... 0 CAN−2B CAN−3B · · · CB D and the (m, n)th entry (m, n = 0, . . . ,N − 1) of the corresponding N × N transfer matrix (2.15) would be given by CAm(d−1 s I − AN)−1AN−1−nB + CAm−n−1B, m > n CAm(d−1 s I − AN)−1AN−1−nB + D, m = n CAm(d−1 s I − AN)−1AN−1−nB, m < n. (2.16) It is important to notice that, even though the lifted system is obtained by augmenting the input and output sequences with their values at N consecutive time instants, the orders of a system and its lifted representation are the same. Moreover, since it is always possible to retrieve a system from its lifted representation, the lifting operation is invertible: L−1 N LN = I. The same procedure shown above to obtain a lifted representation of a system can be used also for LPTV systems with input and output updating at different rates, that is, for multirate LPTV systems. In particular, given a SISO LPTV system with period N1, and input and output updating at the sampling periods T1 = N2T and T2 = N3T, respectively, the corresponding lifted representation can be obtained by choosing N = lcm(N1,N2,N3) (where lcm stands for least common multiple, and N1, N2, N3 are positive integers), and by removing from CL and DL and from BL 16 the rows and columns, respectively, corresponding to time instants when the input and output are not available. For instance, consider the case in which N3 = 1 and N1 = N2 = N, that is, the system output updates N times between two consecutive updates of the input, and the matrices Ai, Bi, Ci, Di are periodic with period N. The lifted representation of such a multirate LPTV system can be obtained by removing the last N −1 block columns from BL and DL, and by leaving AL and CL unaltered. Notice that the lifted representation of a multirate LPTV system is a singlerate LTI system. 2.4 Modified Ztransforms The ith modified Ztransform [9] of a fastupdating sequence gT [k], denoted as ˜G NT {i} , is the Ztransform of the sequence {gT [kN + i]}k, that is, ˜G NT {i} (ds) , X∞ k=0 gT [kN + i]dks = Z gT [kN + i] , i = 0, . . . ,N − 1. (2.17) It is possible to show that the first N modified Ztransforms of a sequence gT can be utilized to generate its Ztransform, GT , through the following expression: GT (d) = ˜G NT {0} (dN) + d˜GNT {1} (dN) + · · · + dN−1 ˜G NT {N−1}(dN). (2.18) If gT is the impulse response of a causal LTI system with transfer function GT , it is possible to show that the modified Ztransforms of gT are related to the entries of the lifted system GNT L . This can be shown as follows. Let yNT L and uNT L be the lifted output and input, respectively, of the system GNT L . From the definition of lifted signals in (2.14) it can be easily shown that the entries of the Ndimensional vectors 17 Y NT L and UNT L are respectively the modified Ztransforms of yT and uT , that is, Y T = Y NT L,0 + dY NT L,1 + · · · + dN−1Y NT L,N−1 UT = UNT L,0 + dUNT L,1 + · · · + dN−1UNT L,N−1. If uT is an impulse, UNT L,i ≡ 0 for i = 1, . . . ,N − 1, and UNT L,0 ≡ 1. Therefore, Y T = GNT L,(0,0)UNT L,0 + dGNT L,(1,0)UNT L,0 + · · · + dN−1GNT L,(N−1,0)UNT L,0 = GNT L,(0,0) + dGNT L,(1,0) + · · · + dN−1GNT L,(N−1,0). Moreover, since uT is an impulse, gT = yT , and hence ˜G NT {i} = GNT L,(i,0), i = 0, . . . ,N − 1 (2.19) where GNT L,(i,0) is the ith entry in the first column of the lifted system GNT L . 18 CHAPTER 3 Design of LPTV controllers for inputstate model matching In this chapter a statespace approach to design a controller for multirate systems is proposed. The multirate system comprises a continuoustime plant whose output is measured at the slow rate, 1/Ts, and is provided to a digital periodically timevarying controller operating at the faster rate, 1/T. The first control design objectives is to ensure that the closedloop state vector matches the state vector of a desired system at the slow measurement rate 1/Ts. This control problem is referred to as inputstate matching. The second objective is to ensure that the continuoustime plant exhibits a ripplefree steadystate response to step reference signals. The model matching problem for multirate systems usually refers in literature to the common inputoutput model matching problem. This problem was initially solved in [19] with the use of LPTV controllers first by converting the multirate system into a lifted LTI one, and then by placing all the zeros and poles of the lifted system to match the dynamics of a desired singlerate LTI system. The opportunity to generate a control action between two consecutive feedback measurements was the key for arbitrary placement of both zeros and poles of the lifted LTI system. However, the use of LPTV controllers resulted in undesired steadystate intersample behavior (also referred to as ripple effect) of the closedloop system response. This undesired behavior was shown in [19] to be a sideeffect due to the placement of the zeros of the closedloop system. The problem of ripples in the steadystate output of multirate systems received a large attention in literature. One of the main results was obtained in [24], followed by [26, 27]. However, the main objective of these works is to solve 19 the output regulation and tracking problems in multirate systems, away from the context of model matching. Conditions for a ripplefree response in the context of an inputoutput model matching problem, using a particular class of LPTV controllers, are given in [38]. The inputstate matching problem considered in this chapter, differently from the classical inputoutput model matching problem, requires the state of the multirate system to match the state of a desired LTI system at the measurement sampling instants. This problem has been initially addressed in [2] for multirate systems with slowrate measurement updates and fastrate control action. However, the strategy proposed in [2] allows to consider only desired systems with no forcing input (that is, of the type ζNT [k + 1] = FζNT [k]). One of the contributions in this chapter is to provide necessary and sufficient conditions to achieve closedloop inputstate matching at the measurement update rate with a desired singlerate LTI system which includes the forcing input (that is, of the type ζNT [k + 1] = FζNT [k] + GrNT [k]). Moreover, conditions are also given to achieve a ripplefree step response of the closedloop system. In particular, it will be shown that these conditions depend not just on the ratio between the measurementrate and control updaterate, but also on the steadystate value of the state vector of the desired system to match. To solve the inputstate matching problem a particular set of causal multirate LPTV controllers is considered, and insights on the nature of these controllers is provided. Moreover, a systematic procedure to design such controllers is given, and an example is shown to corroborate it. The theory and the procedure developed in this chapter can be applied to any controllable LTI plant. The initial work of the author was published in [39] and [40], and the subsequent developments can be found in [41]. This chapter is organized as follows. Section 3.1 gives the problem formulation. Insight on the controller structure is provided in section 3.2. The solution to the considered control problem and the design procedure are given in section 3.3. An 20 example with simulations is provided in section 3.4, and conclusions are given in section 3.5. 3.1 Problem formulation Consider the continuoustime LTI system x˙ (t) = Acx(t) + Bcu(t) y(t) = Ccx(t) (3.1) where x(t) ∈ Rnx is the state vector, u(t) ∈ Rnu is the control input vector, and y(t) ∈ Rny is the output vector to be regulated to the constant reference signal rNT [k] ∈ Rny . The control system structure in Fig. 3.1 is considered. The digital controller C operates at the sampling period T, and the measurements of the plant state x(t) are available at the slow sampling period Ts = NT. C ZOH (sI − Ac)−1Bc Ts x Cc rNT uT xNT y Figure 3.1: Control system structure The following causal, digital multirate controller C is considered ϕT [kN + i + 1] = Kϕ,iϕT [kN] + Kx,ixNT [k] + LirNT [k] uT [kN + i] = CϕϕT [kN + i] + DxxNT [k] (3.2) where ϕT ∈ Rn' with nϕ ∈ N+, and i = 0, . . . ,N − 1. The control design objectives are stated in the following problem. Problem 3.1 Design the matrices Cϕ, Dx, Kϕ,i, Kx,i, Li, i = 0, . . . ,N − 1, of the controller (3.2) to achieve closedloop state matching, at the slow rate 1/NT, with the 21 desired singlerate system ζNT [k + 1] = FζNT [k] + GrNT [k] yNT [k] = HζNT [k] (3.3) where ζNT [k] ∈ Rnx+n', H , [Cc, 0ny×n'], and the matrices F, G are chosen such that the desired system (3.3) is stable and exhibits zero steadystate regulation error to a unit step input. Moreover, the matrices Cϕ, Dx, Kϕ,i, Kx,i, and Li have to be designed in order to achieve a ripplefree closedloop response to step reference signals. By “ripplefree” it is meant that the continuoustime regulation error, e(t) = y(t)−r∞ (where r∞ , limk→∞ rNT [k]), has to be zero at steadystate within two consecutive measurement update instants, that is, lim k→∞ Z (k+1)Ts kTs e⊤(t)e(t)dt = 0. (3.4) To ensure the noncriticality of the sampling period T [31] and the stabilizability of the periodic discretetime system formed by the cascade connection of the zeroorder hold, plant and slowrate sampler [18, 23], the following is assumed. Assumption 3.1 The sampling times involved in the multirate system are not critical, that is, λa − λb 6= j 2πk NT , ∀k ∈ N, k 6= 0 where λa and λb are any two distinct eigenvalues of the block diagonal matrix Ac 0nx×1 01×nx 01,1 formed by augmenting the matrix Ac of the continuoustime plant (3.1) with a zero 22 eigenvalue corresponding to the dynamics of the step reference signal rNT . This assumption can be readily satisfied with a proper selection of T and N. 3.2 Insights on the controller structure One may wonder as to why a controller of the form (3.2) is considered instead of one of the more common form ¯ϕT [kN + i + 1] = ¯K ϕ,i ¯ ϕT [kN + i] + ¯K x,ixNT [k] + ¯L irNT [k] ¯uT [kN + i] = Cϕ ¯ ϕT [kN + i] + DxxNT [k] (3.5) where the state ¯ ϕ on the righthand side of (3.5) depends on i. As shown below in Theorem 3.1, this is due to the fact that the set of controllers of the form (3.2) contains the set of all controllers of the form (3.5). Theorem 3.1 Consider the set of all causal LPTV controllers of the form (3.5), where the fastrate updates of the state variable ¯ ϕT influence the controller dynamics. The following two statements hold: 1. Every controller of the form (3.5) can always be written in the form (3.2). 2. A controller in the form (3.2) can be written in the form (3.5) if and only if K⊤ ϕ,i ∈ Span{K⊤ ϕ,i−1}, for i = 1, . . . ,N − 1. Proof. It is straightforward to see through the recursion of both the controllers (3.2) and (3.5) that the two forms are equivalent, that is, ¯ ϕT [kN +i+1] = ϕT [kN +i+1] 23 for every integer k and every i = 0, . . . ,N − 1, if and only if Kϕ,0 = ¯K ϕ,0, Kx,0 = ¯K x,0, L0 = ¯L 0 Kϕ,i = ¯K ϕ,iKϕ,i−1 i = 1, . . . ,N − 1 Kx,i = ¯K ϕ,iKx,i−1 + ¯K x,i i = 1, . . . ,N − 1 Li = ¯K ϕ,iLi−1 + ¯L i i = 1, . . . ,N − 1 (3.6) Statement 1) directly follows from the equations given in (3.6). From the second equation of (3.6) it is clear that it is not possible to rewrite controller (3.2) in the form (3.5) unless K⊤ ϕ,i ∈ Span{K⊤ ϕ,i−1}, for i = 1, . . . ,N −1. 3.3 Multirate control design Problem 3.1 can be divided into the two following subproblems: 1) model matching with the desired system (3.3) and 2) ripplefree steadystate response of the continuoustime plant. It will be shown in this section that the solution to the first control problem requires the design of only the controller periodically timevarying matrices Kx,i, Kϕ,i, and Li, i = 0, . . . ,N − 1. The solution to the second control problem depends on the choice of the remaining controller matrices Cϕ and Dx. To show this, it is necessary to rewrite the closedloop system in a different form. With this aim, consider the zeroorder hold equivalent [32] of the plant (3.1) at the sampling period T xT [kN + i + 1] = xT [kN + i] + uT [kN + i] yT [kN + i] = CcxT [kN + i] (3.7) where = eAcT , = Z T 0 eAcλBcdλ. (3.8) 24 The discretetime system (3.7) and the proposed timevarying controller (3.2) can be rewritten in the following compact form: ξT [kN + i + 1] = ¯ ξT [kN + i] + ¯ wT [kN + i] yNT [k] = HξT [kN] (3.9) where ξT [kN +i] , [(xT [kN +i])⊤, (ϕT [kN +i])⊤]⊤ is the extended state vector, and the matrices ¯ , ¯ are given by ¯ , s Cϕ 0n'×nx 0n'×n' , ¯ , 0nx×n' In'×n' , with s = + Dx, and wT [kN + i] can be thought of as a pseudocontrol action given by wT [kN + i] , KiξT [kN] + LirNT [k] (3.10) with Ki , [Kx,i,Kϕ,i]. Let wNT L be the lifted pseudocontrol signal obtained from wT as shown in section 2.3. The induced lifted version of the system (3.9) is given by: ξT [(k + 1)N] = ¯ LξT [kN] + ¯ LwNT L [k] yNT [k] = HξT [kN] (3.11) where ¯ L , ¯ N, ¯ L , ¯ N−1¯ , · · · , ¯ ¯ ,¯ , wNT L [k] , wT [kN] ... wT [kN + N − 1] . 25 Considering that the state signal ξT in (3.10) is available only at the measurement update rate, the lifted version of the pseudocontrol (3.10) takes the form wNT L [k] = KLξT [kN] + LLrNT [k] (3.12) where KL and LL are constant matrices given by KL , K0 K1 ... KN−1 = Kx,0 Kϕ,0 Kx,1 Kϕ,1 ... ... Kx,N−1 Kϕ,N−1 , LL , L0 L1 ... LN−1 . (3.13) Notice that the lifted singlerate LTI system formed by (3.11) and (3.12) describes the dynamics of the original LPTV closedloop system, formed by (3.2) and (3.7), at the sampling period Ts. In particular, the closedloop singlerate LTI system obtained by combining (3.11) and (3.12) can be rewritten as ξT [(k + 1)N] = (¯ L + ¯ LKL)ξT [kN] + ¯ LLLrNT [k] yNT [k] = HξT [kN]. (3.14) From the closedloop system (3.14) it is clear that, for some given matrices Cϕ and Dx, state matching is achieved at the slowrate 1/Ts with the desired system (3.3) if and only if the periodically timevarying matrices Kx,i, Kϕ,i, and Li, i = 0, . . . ,N −1, are selected such that KL and LL satisfy ¯ L + ¯ LKL = F (3.15) ¯ LLL = G. (3.16) 26 As a consequence, the ripplefree requirement, mainly relies on the remaining constant matrices Cϕ and Dx. 3.3.1 Design of the gain matrices KL and LL In this section, necessary and sufficient conditions are given for the existence of a solution (KL, LL) to (3.15) and (3.16) for any pair (F,G) characterizing the desired system (3.3). The following lemma gives a preliminary result needed to obtain those conditions. Lemma 3.1 (On the existence of a right inverse of ¯ L) The matrix ¯ L, defined for the system (3.11), has a right inverse ¯ + L for any integer nϕ if and only if 1. nx ≤ (N − 1) 2. the pair ( s, Cϕ) is controllable Proof. Considering that ¯ p, where p ∈ N+, is given by ¯ p = ps p−1 s Cϕ 0n'×nx 0n'×n' , ¯ L can be rewritten as ¯ L = ˜R 0nx×n' 0n'×n'(N−1) In' (3.17) where ˜R , N−2 s Cϕ · · · sCϕ Cϕ . (3.18) For ¯ + L to exist, ¯ L must have full row rank. By inspection this occurs if and only if 27 ˜R has full row rank, that is, nx. Sufficiency (⇐). Since N ≥ nx + 1 and the pair ( s, Cϕ) is controllable, ˜R is a full row rank matrix. Necessity (⇒) The necessity of conditions 1 and 2 is now proven by contradiction. Assume that the pair ( s, Cϕ) is not controllable but that condition 1 holds, that is, nx ≤ (N − 1). As a consequence of the CayleyHamilton theorem, the matrices ps , where p ≥ nx, can be expressed as a linear combination of the set of matrices q s, q = 1, . . . , nx − 1. Therefore ˜R does not have full row rank, that is, Rank(˜R ) < nx. Conversely, assume that the pair ( s, Cϕ) is controllable but that nx > (N − 1). Then, for any integer nϕ < nx/(N − 1) (notice that since nx > (N − 1), nϕ = 1 < nx/(N − 1)), the number of columns of ˜R is nϕ(N − 1) < nx. This contradicts the hypothesis that ¯ + L has a right inverse. The following theorem provides the solution of the state matching problem. Theorem 3.2 Under the assumption 3.1, there exists a controller of the form (3.2) such that the state of the closedloop multirate system matches the state of the desired singlerate system (3.3) at the rate 1/Ts, for any pair (F,G), if and only if a. nx ≤ (N − 1) b. the pair (Ac,Bc) is controllable c. Rank(Cϕ) = nu Proof. Sufficiency (⇐). Let conditions a, b and c hold. The existence of matrices KL and LL that satisfy (3.15) and (3.16) has to be shown. Let λ⋆ be an eigenvalue of s and let v⊤ 6= 0 be the corresponding left eigenvector, that is, v⊤ s = λ⋆v⊤. Since (Ac,Bc) is controllable, by assumption 3.1 ( , ) is also controllable. Since static state feedback does not affect the controllability of a system, the pair ( s, ) is controllable. Therefore, it has to be that v⊤ [ s − λ⋆I, ] 6= 0, that is, v⊤ 6= 0. Since 28 Cϕ is a full row rank matrix, then also v⊤Cϕ 6= 0, that is v⊤ [ s − λ⋆I, Cϕ] 6= 0. Therefore, the pair ( s, Cϕ) is controllable, and by lemma 3.1, the right inverse ¯ + L of ¯ L exists. In turn, the solution (KL,LL) to (3.15) and (3.16) is given by KL = ¯ + L(F − ¯ L) (3.19) LL = ¯ + LG (3.20) Necessity (⇒). Since there exists a solution (KL, LL) to (3.15) and (3.16), then [F − ¯ L,G] ∈ Span(¯ L). Since this has to hold for any pair (F,G), it must be that Rank(¯ L) = nx + nφ. In other words, ¯ L must have a right inverse. By lemma 3.1, this implies that nx ≤ N − 1 and that the pair ( s, Cϕ) is controllable. Let λ⋆ be an eigenvalue of s and let v⊤ 6= 0 be the corresponding left eigenvector, that is, v⊤ s = λ⋆v⊤. The proof will proceed by contradiction. Assume first that Rank(Cϕ) = nu but that the pair (Ac,Bc) is not controllable. Then ( s, ) is not controllable, that is, there exists an eigenvector of s such that v⊤ = 0. In turn, this implies v⊤Cϕ = 0, which is a contradiction since the pair ( s, Cϕ) is controllable. Conversely, assume that the pair (Ac,Bc) is controllable but that Rank(Cϕ) < nu. Since the pair (Ac,Bc) is controllable, also the pair ( s, ) is controllable. Therefore, v⊤ s = λ⋆v⊤ and v⊤ 6= 0 for every eigenvectoreigenvalue pair (v⊤, λ⋆) of s. However, since Rank(Cϕ) < nu there may exist an eigenvector v⊤ of s such that v⊤Cϕ = 0. However, this contradicts the fact that the pair ( s, Cϕ) is controllable. Remark 3.1 (On the values of N and nϕ) Notice that conditions a and c of Theorem 3.2 require the selection of N and nϕ, respectively, such that N ≥ nx + 1 and nϕ ≥ nu. 29 3.3.2 Design of Cϕ and Dx to achieve ripplefree response Theorem 3.2 provided conditions for the existence of a solution to the state matching problem for any desired system (3.3), and required the matrices KL and LL to be selected according to (3.19) and (3.20). In this section conditions are found on the controller matrices Cϕ and Dx such that the closedloop multirate system, obtained by designing KL and LL according to (3.19) and (3.20), also exhibits a ripplefree steadystate response to step reference signals. It will be shown at the end of this section that it is possible to design Cϕ and Dx to guarantee a ripplefree steadystate response only if the steadystate value of the desired system (3.3) satisfies some conditions dependent on the null space of the matrix [ − I, ]. Therefore, let us start by denoting with M the (nx + nϕ) × ny matrix characterizing the steadystate value of the desired system states, that is, ζNT [k] → Mr∞ as k → ∞ (3.21) and by referring to Ma and Mb respectively as the nx ×ny and the nϕ ×ny partitions of M such that M , [M⊤ a ,M⊤ b ]⊤. Moreover, let us denote with Ns , [S⊤ a , S⊤ b ]⊤ the matrix whose columns constitute a basis for the null space of [ − I, ], that is, − I Sa Sb = 0 (3.22) where the rowdimensions of Sa and Sb are nx and nu, respectively. The following lemma provides insights on the conditions on the matrix M that are necessary and sufficient in order to guarantee a ripplefree steadystate response to step reference signals. The results of this lemma are utilized later in Theorem 3.3 to derive necessary and sufficient conditions on the matrices Cϕ and Dx. 30 Lemma 3.2 Let the conditions of Theorem 3.2 be satisfied, and let KL and LL be given by (3.19) and (3.20), respectively. The closedloop system in Fig. 3.1 exhibits a ripplefree response to step reference signals if and only if Li = Mb − KiM, i = 0, . . . ,N − 1 (3.23) Proof. Sufficiency (⇐). As ζNT [k] → Mr∞, also ξT [kN] → Mr∞ by the state matching condition. Therefore, xNT [k] → Mar∞ and ϕT [kN] → Mbr∞ as k → ∞. Therefore, the controller dynamics (3.2) at steadystate can be rewritten in the following compact form: ϕT [kN + i + 1] = KiMr∞ + Lir∞ uT [kN + i] = CϕϕT [kN + i] + DxMar∞ (3.24) If (3.23) holds, then KiMr∞ +Lir∞ = Mbr∞. This implies that ϕT [kN +i] → Mbr∞ as ζNT [k] → Mr∞. As a result the controller output uT [kN +i] becomes constant as k → ∞, and therefore, the closedloop system exhibits a ripplefree response. Necessity (⇒). The closedloop system exhibits a ripplefree response to step reference signals and state matching at the slowrate with ζNT [k]. Therefore, since ζNT [k] tends to a constant as k goes to infinity, also ϕT [kN + i] tends to a constant. Because of the state matching condition, ϕT [kN +i] → Mbr∞ as k → ∞. As a result, from (3.24) KiMr∞ + Lir∞ = Mbr∞. The matrix ˜R defined in (3.18) will be utilized in the rest of this section to derive the subsequent results. Theorem 3.3 Let the conditions of Theorem 3.2 be satisfied, and let KL and LL be given by (3.19) and (3.20), respectively. Moreover, let ˜M b be the (N − 1)blocks matrix where each block is equal to Mb, that is, ˜M b , [M⊤ b , · · · ,M⊤ b ]⊤. The closed 31 loop system in Fig. 3.1 exhibits a ripplefree steadystate response to step reference signals if and only if there exist matrices Cϕ and Dx such that • ˜M b ∈ Span{˜R ⊤} • the columns of the matrix Ma DxMa + CϕMb (3.25) are contained in the range space of Ns Proof. By lemma 3.2 it is necessary and sufficient to show that LL + KLM = Mb,L where Mb,L , [ ˜M ⊤ b ,M⊤ b ]⊤. Using (3.19) and (3.20), and considering that (F −I)M + G = 0, the following hold: LL + KLM = ¯ + LG + ¯ + L(F − ¯ L)M = ¯ + L G + (F − I)M − (¯ L − I)M = −¯ + L(¯ L − I)M = −¯ + L ( Ns − I)Ma + N−1 s CϕMb −Mb . (3.26) Sufficiency (⇐). Since the columns of the matrix in (3.25) are in the range space of Ns, s − I Cϕ M = 0. Therefore, ( s − I)Ma = −CϕMb. Considering also that ( Ns − I) = ( N−1 s + · · · + 32 s + I)( s − I), (3.26) can be further expanded as LL + KLM = ˜R ⊤(˜R ˜R ⊤)−1 0 0 I ˜R 0 0 I Mb,L where the first matrix on the righthand side is the right inverse, ¯ + L, of the matrix ¯ L given in (3.17). Therefore, LL + KLM = ˜R ⊤(˜R ˜R ⊤)−1 ˜R 0 0 I Mb,L Let (U, V, ) be the singular value decomposition of ˜R , that is, ˜R = U 0 V ⊤ where U and V are nx×nx and nϕ(N −1)×nϕ(N −1) unitary matrices, respectively, and is a square, diagonal, nonsingular nx ×nx matrix (because Rank{˜R } = nx). It is possible to show that ˜R ⊤(˜R ˜R ⊤)−1 ˜R V = ⊤( ⊤)−1 0 0 0 = V I 0 0 0 (3.27) Therefore, (V, V, I) is the singular value decomposition of the symmetric and square matrix ˜R ⊤(˜R ˜R ⊤)−1 ˜R . Let V be partitioned as V = [V1, V2], where V1 is an nϕ(N − 1) × nx matrix. Then, the matrix ˜R ⊤(˜R ˜R ⊤)−1 ˜R acts as an identity operator for all the vectors in the range space of V1, that is, ˜R ⊤(˜R ˜R ⊤)−1 ˜R V1 = V1. Since also ˜R ⊤(˜R ˜R ⊤)−1˜R ˜R ⊤ = ˜R ⊤, the columns of ˜R ⊤ must be in the range space of V1. However, since Rank{V1} = Rank{˜R ⊤} = nx, Span{V1} ≡ Span{˜R ⊤}. Since by hypothesis 33 ˜M b ∈ Span{˜R ⊤}, (3.26) can be rewritten as LL + KLM = Mb,L Necessity (⇒). Because of the state matching at the slowrate 1/Ts, ξT [kN] → Mr∞ as k → ∞. Since ξT [kN] , [(xT [kN])⊤, (ϕT [kN])⊤]⊤, xNT [k] , xT [kN] → Mar∞ and ϕT [kN] → Mbr∞. Since the closedloop response is ripplefree at steadystate, xT [kN + i] → Mar∞ and ϕT [kN + i] → Mbr∞ for every i = 0, . . . ,N − 1 as k → ∞. Therefore, from (3.7), ( + Dx − I)Mar∞ + CϕMbr∞ = 0 or equivalently s − I Cϕ M = 0 The last equality can be rewritten as − I Ma DxMa +Mb = 0 which proves the second condition of the theorem. Also, since LL + KLM = Mb,L, it must be that ˜M b ∈ Span{˜R ⊤}. Notice that since the columns of the matrix in (3.25) have to be in the range space of Ns, it is necessary that Ma ∈ Span(Sa). This is the only condition of Theorem 3.3 not dependent on the selection of Cϕ and Dx. If P denotes the matrix such that Ma = SaP, the remaining conditions of Theorem 3.3 are satisfied if and only if there 34 exist matrices Mb, Cϕ, Dx and Q such that Mb = ( ps Cϕ)⊤Q, p = 0, . . . ,N − 2, (3.28) DxMa + CϕMb = SbP. (3.29) Equations (3.28) and (3.29) always admit a solution if the desired system is chosen so that Ma has full column rank and Mb = 0. In fact, in that case one can select Q = 0, Dx = (M⊤ a Ma)−1M⊤ a SbP and Cϕ to be any matrix. Therefore, if the pair (F,G) of the desired system (3.3) is chosen such that M = S⊤ a 0n'×ny ⊤ P, (3.30) where P is any matrix such that SaP has full column rank, then there exist matrices Cϕ and Dx for which the multirate closedloop system exhibits a ripplefree steadystate response to step reference signals. Notice that, since can be safely assumed to have full column rank, it is straightforward to verify through (3.22) that Sa has full column rank. Therefore, there always exist a P such that Ma , SaP which has full column rank. It remains to investigate how (F,G) can be selected in order for (3.30) to be satisfied. One way consists of selecting F as desired, and G given by G = −(F − I) SaP 0 (3.31) In fact, by applying the final value theorem to the Ztransform of the state variables 35 ζNT we have: lim z→1Z{ζNT } = lim z→1 (zI − F)−1G = SaP 0 where z corresponds to the unit advance operator in the Ztransform domain. Therefore, complete freedom is given to the choice of the F. It is important to notice that the strategy to select the pair (F,G) in order for the desired system to satisfy (3.30) is not unique. The following theorem summarizes the results obtained in this section. Theorem 3.4 Consider a stable desired digital LTI system of the form (3.3), and let (Ac,Bc,Cc) be a representation of the continuoustime plant to be controlled. Under assumption 3.1, it is possible to design the timevarying controller (3.2) for the continuoustime plant (3.1) to achieve closedloop state matching at the measurement update rate with the desired system (3.3) if and only if 1. the pair (Ac,Bc) is controllable 2. N ≥ nx + 1 The unique solution for the matrices Kϕ,i, Kx,i and Li of the controller (3.2) can be obtained by choosing any full row rank matrix Cϕ, and it is given by KL = ¯ + L(F − ¯ L) LL = ¯ + LG where KL and LL are uniquely related to Kϕ,i, Kx,i and Li through (3.13). With a controller designed in such a way, the closedloop system exhibits a steadystate ripplefree response to step reference signals if and only if there exist matrices Q and 36 P such that ( ps Cϕ)⊤Q = Mb p = 0, . . . ,N − 2 Ma = SaP DxMa + CϕMb = SbP Sufficient conditions for the existence of a solution to the latter equations are that Ma ∈ Span(Sa), Rank(Ma) = ny and Mb = 0. In such a case the matrix Dx can be selected as Dx = (M⊤ a Ma)−1M⊤ a SbP. (3.32) To complete this section a procedure to design the controller is provided. This procedure is based on the sufficient conditions of Theorem 3.4. Procedure  Step 1. Select N to satisfy assumption 3.1 and so that N ≥ nx+1 (see Theorem 3.2).  Step 2. Obtain the discrete equivalent representation, ( , ), of the continuoustime controllable plant (3.1), where is a full column rank matrix.  Step 3. Find a basis, Ns , [S⊤ a , S⊤ b ]⊤, for the null space of the matrix [ −I, ].  Step 4. Select any desired system of the form (3.3) where F is any matrix and G satisfies (3.31) (see Theorem 3.3), for any full column rank matrix P of appropriate dimension.  Step 5. Select Cϕ to be any full row rank matrix (according to Theorem 3.2) and Dx = (M⊤ a Ma)−1M⊤ a SbP. 37  Step 6. Construct the matrices ¯ L, ¯ L and compute the right inverse ¯ + L = ¯ ⊤L (¯ L¯ ⊤L )−1.  Step 7. After computing the matrices KL and LL according to (3.19) and (3.20), obtain the controller matrices Kx,i, Kϕ,i and Li as shown in (3.13). 3.4 Example In this section the results summarized in Theorem 3.4 will be applied to a doubleintegrator system with statespace representation Ac = 0 0 1 0 , Bc = 1 0 , Cc = 0 1 Since nx = 2 it is required to choose N ≥ 3. For simplicity, let us consider the case N = 3 and T = 1. The zeroorder hold equivalent of the plant operating at the fast rate 1/T is given by = 1 0 T 1 , = T 5T2 , Cc = 0 1 A basis for the null space of [ − I, ] is Ns = [0, 1, 0]⊤. Therefore, Sa = [0, 1]⊤, and Sb = 0. Let us choose the matrix F of the desired system (3.3) to be zerovalued, and let us select the matrix G according to (3.31). Hence, the desired system statespace representation is given: F = 0 0 0 0 0 0 0 0 0 , G = 0 1 0 As a result, Ma = [0, 1]⊤, Rank{Ma} = ny, Mb = 0. 38 Let Cϕ = 1. Therefore, the matrix Dx, computed through (3.32), is given by Dx = [0, 0]. Then, the matrices Kx,i, Kϕ,i and Li of the controller (3.2), designed according to (3.19) and (3.20), are given by Kx,0 = [−5/(2T),−1/(10T2)], Kϕ,0 = −2, L0 = 1/(10T2) Kx,1 = [3/(2T), 1/(10T2)], Kϕ,1 = 1, L1 = −1/(10T2) Kx,2 = [0, 0], Kϕ,2 = 0, L2 = 0 The step response of the closedloop multirate system comprising the continuoustime plant and the digital controller is compared with the response of the desired system in as shown in Fig. 3.2. From Fig. 3.2 it is clear that the state of the closedloop system matches at every measurement sampling instant the state of the desired system (3.3), and that the response of the closedloop system is ripplefree at steadystate. 3.5 Conclusions In this chapter the inputstate matching problem with a desired singlerate digital system was considered for the class of multirate systems with slow measurement update rate and fast controller update rate. In particular, given any desired singlerate LTI system operating at the measurement update rate, conditions and a controller design procedure were given for which the closedloop system state matches the state of the desired system at that measurement update rate. Moreover, it was shown that, if the input matrix of the desired system is properly selected, a ripplefree steadystate response of the closedloop system can be obtained. Despite the constraints on the input matrix of the desired system, the developed design procedure gives full freedom on the choice of the closedloop eigenstructure. This allows to potentially determine the shape of the state response of the closedloop system. It is important to notice that 39 0 3 6 9 12 15 −1 −0.5 0 0.5 1 x1(t) zNT 1 [k] 0 3 6 9 12 15 0 0.2 0.4 0.6 0.8 1 x2(t) zNT 2 [k] 0 3 6 9 12 15 −1 −0.5 0 0.5 1 xT 3[k] zNT 3 [k] Figure 3.2: Closedloop system response to a step reference signal: the state variables of the multirate closedloop system are compared with the state variables of the desired slowrate LTI system. The signals x1, x2 are the velocity and position, respectively, of the double integrator continuoustime plant. The closedloop system output is y(t) = x2(t). 40 even though the conditions stated in this chapter allow to avoid the presence of ripples in the steadystate response of the continuoustime plant, undesired oscillations may appear in the transient response due to the timevarying nature of the control system. These undesired oscillations may be due to the particular choice of the eigenstructure of the desired system. Therefore, future research will focus on finding conditions on the choice of the desired modal response to avoid possible ripples also in the closedloop system transient response. Moreover, the theory shown in chapter 3 was obtained assuming the availability of full state information. Therefore, the cases in which only partial information is available will be considered in the future. 41 CHAPTER 4 Frequency domain modeling and parametrization of LTI Controllers Several control strategies have been developed in the last three decades for multirate systems. In some of the cases the measurements from the plant are available only after a relatively long period of time compared to the operating rate of the digital controller. In those cases, the use of a control action updating at a faster rate than the measurement update rate was shown to increase some robustness indexes, such as gain and phase margins [18]. To keep the robustness advantages deriving from the use of a controller operating at a rate faster than the measurement update rate, while reducing the undesired transient intersample behavior of the closedloop system and guaranteeing a ripplefree steadystate response, only controllers formed by the cascade of an upsapling operator with factor N and a LTI system are considered in this chapter. These controllers, belonging to the class of causal (N, 1)shift invariant systems, receive the slowupdating measurements from the plant and generate a fastupdating control signal. With this control structure the main cause of undesired intersample behavior (both at transient and steadystate) can be isolated and directly addressed. In fact, as it will be shown in chapter 5, the main cause of undesired intersample behavior is the use of the (N, 1)shift invariant upsampling operators. Therefore, the undesired intersample behavior can be reduced by requiring the LTI system embedded in the control structure to filter appropriately the output of the upsamplers. This motivates the use of this particular class of (N, 1)shiftinvariant controllers. One of the main contributions of this chapter consists of a novel procedure to 42 parameterize the set of all the stabilizing (N, 1)shiftinvariant controllers with the above mentioned structure for which model matching is achieved with a desired digital LTI system in the two cases: (1) the desired LTI system updates at the control update rate, (2) the desired LTI system updates at the measurement update rate. Notice that, to the author’s best knowledge, the problem in the case (1) has never been considered before in literature. The problem of parametrizing controllers in the multirate systems framework was initially addressed in [28] and [29]. The results in those two works are the milestones for the parametrization of (m, n)shiftinvariant controllers in the multirate systems framework. In particular, a parametrization of all the stabilizing (m, n)shiftinvariant controllers for the particular set of (n,m)shiftinvariant plants comprising zeroorder holds, a continuoustime causal LTI system and sampling operators was proposed in [28]. The theory in [28] was then generalized in [29] to consider all the possible (n,m)shiftinvariant plants. In both [28] and [29] the parametrization theory results in the complete set of (m, n)shiftinvariant controllers for the corresponding class of plants. The parametrization procedure required first to design the controller for the lifted version of the plant, and then to apply the inverse lifting process to the controller designed for the lifted plant. Since in this chapter it is considered the case of plants whose output is measured at the rate 1/Ts and whose input updates at the faster rate 1/T, the parametrization is restricted to (N, 1)shiftinvariant controllers. In particular, only those (N, 1) shiftinvariant controllers made of a downsampling operator followed by a digital LTI system are considered for the reasons explained earlier. The solution to this problem cannot be obtained simply by using the theory developed in [28] and [29] for the case m = N and n = 1. This is due to the fact that the solution of the parametrization problem considered in [28] and [29], in the case m = N and n = 1, would be the entire set of (N, 1)shiftinvariant controllers, including those which 43 cannot be expressed with the series connection of a downsampling operator and a digital LTI system. Two other major differences between the works in [28] and [29] and the theory shown in this chapter are the following. (1) In the solution proposed in this chapter there is no need to consider any causality constraint on the controller designed for the lifted system. (2) The background objective of the parametrization in this chapter is not just the stability of the closedloop system but also the inputoutput model matching with a desired LTI system. The author’s preliminary results are shown in [42, 43]. This chapter is organized as follows. The problem formulation is given in section 4.1. Section 4.2 describes the frequency domain approach utilized to model the multirate system. Closedloop stability and the model matching problems, respectively, are discussed in sections 4.3 and 4.4. Section 4.5 shows an application of the developed theory. Conclusions are given in section 4.6. 4.1 Problem formulation Let P(s) be the Laplace transform of the continuoustime system x˙ (t) = Acx(t) + Bcu(t) y(t) = Ccx(t) + Dcu(t) (4.1) where x(t) ∈ Rnx is the state vector, u(t) ∈ R is the control signal, and y(t) ∈ R is the measured and controlled plant output to be regulated to the constant reference rNT [k] ∈ R. Let G(s) , P(s)H(s), where H(s) , (1−esT )/s is a the zeroorder hold operating at the control update rate 1/T. Consider the multirate system in Fig. 4.1, where the (N, 1)shiftinvariant feedback and feedforward controllers are made of an upsampling operator followed by the LTI digital systems CT 1 and CT 2 , respectively. 44 The control signal takes the form UT = CT 2 [RNT ]T −CT 1 [Y NT ]T , where UT , Z{uT }, +  N N NT rNT CT 1 (d) CT 2 (d) ZOH P(s) [rNT ]T uT y(t) yNT [yNT ]T Figure 4.1: Multirate control scheme. RNT , Z{rNT } and Y NT , Z{yNT }. Notice that in the block diagram of Fig. 4.1 the output of the plant is measured at the slow rate 1/NT, and the control signal uT updates N times between two consecutive measurement updates. The following assumption on the continuoustime plant (4.1) is necessary for the subsequent results. Assumption 4.1 The continuoustime system (4.1) is controllable and observable. Moreover, assumption 3.1 will be considered in this chapter to guarantee the controllability and reconstructibility of the periodic discretetime system formed by the cascade connection of the zeroorder hold, the plant G(s) and the slowrate sampler [18, 23]. In the following, since the design of the (N, 1)shiftinvariant controllers shown in Fig. 4.1 consists in the design of the LTI systems CT 1 and CT 2 , the problem of parameterizing such a set of (N, 1)shiftinvariant controllers will be more conveniently referred to as the problem of parameterizing the LTI controllers CT 1 and CT 2 . Hence, given the multirate system in Fig. 4.1, the goals are the following: 1. Modeling in the transform domain the closedloop multirate system both at the slowrate, 1/Ts, and at the fastrate, 1/T, using the tools introduced in chapter 2. The closedloop system must be expressed in a form suitable for the next objectives. 45 2. Parameterizing the set of stabilizing LTI controllers for the multirate system shown in Fig. 4.1. 3. Parameterizing the set of LTI controllers for which closedloop model matching is achieved at the slowrate with a desired stable slowrate LTI system. 4. Parameterizing the set of LTI controllers for which closedloop model matching is achieved at the fastrate with a desired stable fastrate LTI system. To achieve these goals, necessary and sufficient conditions to achieve stability and model matching will also be obtained. These conditions will be used to parameterize the corresponding sets of LTI controllers CT 1 and CT 2 . 4.2 Modeling the multirate system Let GT be the Ztransform, at the sampling period T, of the plant (4.1) preceded by the zeroorder hold. A digital control system corresponding to the sampleddata control system in Fig. 4.1 can be obtained first by replacing the slowrate sampler with a fastrate sampler followed by a downsampling operator, and then by replacing the cascade made by the zeroorder hold, the plant P(s) and the fastrate sampler with the digital system GT . The resulting digital system is shown in Fig. 4.2. The following +  N N N rNT CT 1 (d) CT 2 (d) GT (d) yT [rNT ]T uT yNT [yNT ]T Figure 4.2: Multirate control scheme. proposition can be proved by using properties (2.6) and (2.7) of the upsampling and downsampling operators. Proposition 4.1 The slowrate closedloop transfer function KNT of the system in Fig. 4.1, between the slowrate reference signal RNT and the measured plant output 46 Y NT , is given by KNT , Y NT RNT = GTCT 2 NT 1 + [GTCT 1 ]NT (4.2) The fastrate closedloop transfer function KT , between the upsampled reference signal [RNT ]T and the fastrate plant output Y T , Z{yT }, is given by KT , Y T [RNT ]T = GT CT 2 − CT 1 [[GTCT 2 ]NT ]T 1 + [[GTCT 1 ]NT ]T (4.3) Proof. The plant output Y T is given by Y T = GTCT 2 [RNT ]T − GTCT 1 [Y NT ]T (4.4) The slowrate version Y NT of the plant output can be obtained as follows: Y NT =[GTCT 2 [RNT ]T ]NT − [GTCT 1 [Y NT ]T ]NT =[GTCT 2 ]NTRNT − [GTCT 1 ]NTY NT Equation (4.2) is obtained by grouping Y NT . Equation (4.3) follows after the substitution of Y NT into (4.4). Remark 4.1 Notice that KNT can be obtained just by downsampling KT , that is, [KT ]NT = [GTCT 2 ]NT − GTCT 1 [[GTCT 2 ]NT ]T 1 + [[GTCT 1 ]NT ]T NT = [GTCT 2 ]NT − [GTCT 1 ]NT [GTCT 2 ]NT 1 + [GTCT 1 ]NT = [GTCT 2 ]NT 1 + [GTCT 1 ]NT = KNT However, it is not possible to obtain KT by upsampling KNT because the additional information contained in KT about the fastrate impulse response of the closedloop 47 system cannot be recovered with an upsampling operation. In fact, [KNT ]T = [[GTCT 2 ]NT ]T 1 + [[GTCT 1 ]NT ]T 6= KT As shown in (2.8), [GTCT i ]NT 6= [GT ]NT [CT i ]NT , i = 1, 2. Therefore, the classic control techniques developed for singlerate systems cannot be used to design CT 1 and CT 2 . To overcome this problem, the following proposition, using the procedure developed in [42, 44, 45], introduces an invertible transformation which allows to rewrite KNT and KT in a form more suitable for analysis and design. Proposition 4.2 The closedloop transfer functions (4.2) and (4.3) can be written in the following form: KNT (ds) = [GT T ]NT (ds) ˜ CNT 2 (ds) 1 + [GT T ]NT (ds) ˜ CNT 1 (ds) (4.5) KT (d) = GT (d) T (d) ˜ CNT 2 (dN) − GT (d) T (d) ˜ CNT 1 (dN)[GT T ]NT (dN) ˜ CNT 2 (dN) 1 + [GT T ]NT (dN) ˜ CNT 1 (dN) (4.6) where T , [1, d, · · · , d(N−1)], and ˜ CNT i is a column vector containing all the modified Ztransforms of Z−1{CT i }, that is, ˜ CNT i , ˜ CNT i{0} · · · ˜ CNT i{N−1} ⊤ , i = 1, 2. Proof. As shown in (2.18), the controller transfer functions CT i (d), i = 1, 2, can be rewritten as the weighted sum of its upsampled modified Ztransforms, that is, CT i (d) = NX−1 k=0 dk h ˜ CNT i{k} iT = T [ ˜ CNT i ]T = T (d) ˜ CNT i (dN) (4.7) 48 Therefore, the following equalities hold GT (d)CT i (d) = GT (d) T (d)˜C NT i (dN) [GTCT i ]NT = [GT T [ ˜ CNT i ]T ]NT = [GT T ]NT ˜ CNT i and the discretetime closedloop transfer functions (4.5) and (4.6) can be obtained by substituting the last two equalities into (4.2) and (4.3), respectively. Unlike the forms given in (4.2) and (4.3), the closedloop transfer functions (4.5) and (4.6) are more suitable for designing a controller because ˜ CNT i (dN) (i = 1, 2) and [GT T ]NT (dN) are separated in their denominators and numerators. Observe that the transformation from CT i (d) to ˜ CNT i (dN) is invertible. Therefore, a controller designed for the closedloop systems (4.5) or (4.6) can be converted back into an equivalent controller for the closedloop systems (4.2) or (4.3), respectively, by using (4.7). Consider now the following coprime factorizations of the transfer functions involved in the multirate system: GT T NT = BNT ANT , GT = NTG DTG , ˜ CNT 1 = Y NT 1 XNT , ˜ CNT 2 = Y NT 2 XNT (4.8) where ANT and XNT are proper and stable scalar transfer functions, BNT is a proper and stable 1×N transfer function, Y NT 1 and Y NT 2 are proper and stable N×1 transfer functions, DTG and NTGare polynomials. Notice that, as explained in remark 4.3, there always exists a stable leftcoprime factorization A(dN),B(dN) for [ TGT ]NT (dN). By using these factorizations, the closedloop transfer functions (4.5) and (4.6) can be 49 rewritten as KNT = BNT Y NT 2 FNT (4.9) KT (d) = NTG (d) DTG (d) T (d)(FNT (dN)I − Y NT 1 (dN)BNT (dN)) FNT (dN) Y NT 2 (dN) XNT (dN) (4.10) where I is the identity matrix of dimension N ×N, and FNT , ANTXNT +BNTY NT 1 . Remark 4.2 Due to the stability of the factorization (4.8) it is clear from (4.9) that the stability of KNT can be achieved if and only if the transfer function F is a unit in the space of the rational functions (that is, it is stable and proper and its inverse is also stable and proper). In other words, KNT can be regarded as the closedloop transfer function of a singlerate control system operating at the slow rate 1/NT, and hence, the stability conditions for singlerate systems discussed in [46] can be applied. However, it is not so straightforward to see if the same conditions apply to KT . In fact, as shown in equation (4.10), the stability of the closedloop multirate system is not achieved if the transmission zeros of the transfer function matrix FNT (dN)I − Y NT 1 (dN)BNT (dN) do not cancel the unstable zeros of the cascade DTG (d)XNT (dN). Remark 4.3 The statespace realization, ( ¯ A, ¯B , ¯C , ¯D ), of GT T NT (dN) is related to the system in (4.1) by A¯ = ANc , ¯B = [AN−1 c Bc, · · · ,AcBc,Bc] ¯ C = Cc, ¯D = [Dc, 0, · · · , 0]. By assumptions 3.1 and 4.1 the pair (A¯,C¯) is detectable [18, 23], and hence, there exists a matrix H such that the spectrum of ¯ A + H ¯C lies in the open unit circle. Therefore, a stable leftcoprime parametrization of GT T NT (dN) exists and it is 50 given by [47] BNT = Ccds(I − ds(ANc + HCc))−1(¯B + H ¯D ) + ¯D ANT = Ccds(I − ds(ANc + HCc))−1H + 1. (4.11) To facilitate the readability of the rest of this chapter, the notations NG, DG, , A, B, X, Y1, Y2 and F, will be utilized to mean NTG (d), DTG (d), T (d), ANT (dN), BNT (dN), XNT (dN), Y NT 1 (dN), Y NT 2 (dN) and FNT (dN), respectively, unless otherwise stated. 4.3 Closedloop stability In this section, necessary and sufficient conditions are given for the existence of LTI controllers CT 1 ,CT 2 which stabilize the closedloop multirate system KT in Fig. 4.1. The following lemma is useful to obtain subsequent results. Lemma 4.1 If F is a unit in the space of rational functions, all the unstable zeros of the cascade DGX are transmission zeros of the transfer function matrix (FI − Y1B). Proof. Since F is a unit in the space of rational functions in the variable dN, it follows that F 6= 0 and 1−BF−1Y1 6= 0. By contradiction assume that 1−BF−1Y1 = 0. Then, the following statements are equivalent: (AX + BY1)−1BY1 = 1, BY1 = AX + BY1, AX = 0. The latter equation, however, cannot be true because it would imply either one or both the transfer functions A and X to be identically null. Therefore 1 − BF−1Y1 6= 0, and by using the matrix inversion formula the transfer function matrix (FI − Y1B)−1 can be written as (FI − Y1B)−1 = F−1I + F−1Y1BF−1 1 − BF−1Y1 (4.12) 51 Considering that 1 − BF−1Y1 = AXF−1, equation (4.27) can be rewritten as (FI − Y1B)−1 = (AXI + Y1B)F−1 AX (4.13) To show that all the zeros of AX are poles of (FI − Y1B)−1, it is necessary and sufficient to prove that the transmission zeros of (AXI + Y1B)F−1 do not cancel those of AX. Let d0 be an unstable zero of AX. Since AX is a scalar transfer function, d0 is also a blocking zero of AX, that is, A(d0)X(d0)δ = 0, ∀δ 6= 0, δ ∈ R (4.14) By contradiction, assume that d0 is also an unstable transmission zero of (AXI + Y1B)F−1, that is, there exists a column vector 0 6= β ∈ RN such that (A(d0)X(d0)I + Y1(d0)B(d0))F−1(d0)β = 0. Since F is stable by definition, d0 cannot be a transmission zero of F−1. It follows that d0 is a transmission zero of Y1B. It is shown below that d0 is also a transmission zero of BY1. Let U1, V1, U2 and V2 be unimodular polynomial matrices, in the variable dN, such that the SmidthMcMillan decomposition of Y1 and B are given by Y1 = U1[ ⊤1 , 0]⊤V1 and B = U2[ 2, 0]V2, where 1 and 2 are stable transfer functions in the variable dN. Considering that the cascade connections Y1B and BY1 are given by Y1B = U1 1V1U2 2 0 0 0 V2, BY1 = U2 2V2U1 1V2 d0 is a transmission zero of Y1B if and only if it is also a zero of BY1, that is, B(d0)Y1(d0)δ = 0, ∀δ 6= 0, δ ∈ R (4.15) 52 By combining (4.14) and (4.15) it follows that (A(d0)X(d0) + B(d0)Y1(d0))δ = 0, ∀δ 6= 0, δ ∈ R, which means that d0 is also an unstable zero of F. This, however, contradicts the hypothesis that F is a unit. Therefore, there cannot be any cancelation between the unstable transmission zeros of (FI − Y1B)−1 and those of AX. In other words, all the unstable zeros of AX are poles of (FI − Y1B)−1. Considering that [GT T ]NT is obtained from GT T through (2.5), it is possible to show that A = DG where is a stable transfer function of the variable d. Therefore, equation (4.13) can be rewritten as (FI − Y1B)−1 = (AXI + Y1B) F−1 DGX (4.16) It follows that the unstable zeros of the cascade DGX are poles of (FI − Y1B)−1, and hence, the transmission zeros of (FI − Y1B). Remark 4.4 (On the transmission zeros of (FI − Y1B)) Notice that the zeros of A are symmetric with respect to the origin. Moreover, for every root γ of DG, A has N zeros with the same absolute value, γ, equally spaced around the origin of the complex plane. Therefore, for every unstable root of DG, has N − 1 unstable zeros with the same absolute value. This can be easily verified using (2.5). Therefore, the unstable transmission zeros of (FI − Y1B) are not just the unstable roots of DG but also the unstable zeros of the transfer function . The following theorem gives necessary and sufficient conditions for the existence of stabilizing LTI controllers. Theorem 4.2 will provide the parametrization of the set of all the feedback controllers CT 1 stabilizing the multirate system in Fig. 4.1. Theorem 4.1 The LTI controllers CT 1 , X−1Y1, CT 2 , X−1Y2 53 stabilize the closedloop multirate system KT if and only if F is a unit. Proof. (⇒) By hypothesis the closedloop system (4.10) is stable. By construction A, B, Y1, Y2 and X are proper and stable, which implies that also F is a proper stable transfer function. By inspection on the closedloop transfer function (4.10) it is straightforward to show that if F has an unstable zero, that zero is a pole of KT . Since KT is stable, F cannot have any unstable zero. Therefore, F is a unit in the space of rational functions in the variable dN. (⇐) By hypothesis F is a unit. Therefore the stability of KT is guaranteed by lemma 4.1. Theorem 4.2 All the stabilizing LTI feedback controllers CT 1 for the closedloop multirate system in Fig. 4.1 are of the form CT 1 = Y1 − A X + B (4.17) where is any proper stable N × 1 transfer function matrix in the variable dN, and the pair (X, Y1) satisfies the Bezout equation AX + BY1 = 1. Proof. (⇐) Let (X, Y1) be a particular solution of the Bezout equation. Then also ¯X , ¯Y1 , where ¯X , X +B and ¯Y1 , Y1 −A , is a solution of the Bezout equation, that is, A(X + B ) + B(Y1 − A ) = 1. Therefore, by theorem 4.1, every controller of the form (4.17) is a stabilizing controller. (⇒) Let (X, Y1) and ¯X , ¯Y1 be two different stable solutions of the Bezout equation. It has to be shown that there always exists a stable transfer function (dN), such that the solution ¯X , ¯Y1 can be written as ¯X = X +B , ¯Y1 = Y1−A . Since (X, Y ) and ¯X , ¯Y1 are two different solutions of the Bezout equation, A(X − ¯X ) = −B(Y1− ¯ Y1). If = (Y1− ¯ Y1)/A then ¯X = X+B and ¯Y1 = Y1−A . It remains to be shown that is stable. Since A and B are coprime, A is also a factor of (Y1 − ¯ Y1). Therefore, 54 the stability of (Y1 − ¯ Y1) proves the stability of . Moreover, since Y1, ¯ Y1 and A are functions of dN, then also is a function of dN. 4.4 Model matching In this section, necessary and sufficient conditions are given for the existence of LTI controllers CT 1 and CT 2 such that the dynamics of the slowrate closedloop system KNT and of the fastrate closedloop system KT match with the dynamics of a desired slowrate system KNT d and fastrate system KT d , respectively. 4.4.1 Model matching at the slow rate The closedloop transfer function of the multirate system in Fig. 4.1 between the slowrate reference signal RNT and the slowrate plant output Y NT is given in (4.5). By simple inspection it is possible to see that (4.5) represents the closedloop transfer function of a singlerate system where the plant, the feedback controller and the feedforward controller are given by the transfer matrices [GT T ]NT , ˜ CNT 1 and ˜ CNT 2 , respectively. Therefore, by using the theory developed in [48] for singlerate systems, and considering the factorizations (4.8), it is possible to conclude that the transfer function KNT is stable if and only if F is a unit. Moreover, as shown in [18] it is always possible to arbitrarily place both zeros and poles of KNT to achieve model matching (at the slowrate) with any desired transfer function if the couple (Ac,Bc) is controllable, assumption 3.1 holds, and N −1 ≥ nx. As an extension, the following theorem provides the parametrization of the set of all the stabilizing controllers for the multirate system in Fig. 4.1 such that model matching is achieved at the slowrate with a desired transfer function KNT d . Theorem 4.3 Let assumptions 3.1 and 4.1 be verified, and N − 1 ≥ nx. Consider the factorizations (4.8) for the transfer functions involved in the closedloop system 55 (4.5). Let KNT d be a desired stable transfer function. All the LTI controllers CT 1 and CT 2 such that model matching is achieved are of the form CT 1 = Y1 − A X + B , CT 2 = Y2 X + B (4.18) where is any proper stable N × 1 transfer function of the variable dN, and Y1, Y2, X satisfy the following equations AX + BY1 = 1 (4.19) BY2 = KNT d (4.20) Proof. (⇐) Let CT 1 and CT 2 as in (4.18). It has to be proved that closedloop model matching is achieved with the transfer function KNT d . By using (4.7), CT 1 and CT 2 can be rewritten as CT 1 (d) = (d)CNT 1 (dN) and CT 2 (d) = (d)CNT 2 (dN), where CNT 1 (dN) , Y1 − A X + B , CNT 2 (dN) , Y2 X + B Considering that F = 1 (from (4.19)), and by using equation (4.20), the result can be obtained by substitution of CNT 1 (dN) and CNT 2 (dN) in the multirate closedloop system (4.5). (⇒) Assume that the model matching is achieved through the LTI controller CT 1 = ¯X −1 ¯ Y1 and CT 2 = ¯X −1 ¯ Y2. It has to be proved that there exists a transfer function matrix such that CT 1 and CT 2 can be written in the form (4.18). Since the closedloop system is stable, by theorem 4.2 there exists a transfer function such that CT 1 is expressed in the form (4.17). Therefore ¯X = X + B and ¯ Y1 = Y1 − A , where the pair (X, Y1) is a particular solution of the Bezout equation (4.19). Since the pair (Ac,Bc) is controllable and N − 1 ≥ nx, B is right invertible [49]. Let Y2 be the right inverse of B. Since B is stable and due to the structure of B, all the 56 invariant zeros of B are stable. Therefore Y2 is also stable and can be selected so that (4.20) is satisfied. 4.4.2 Model matching at the fast rate The following theorem provides necessary and sufficient conditions for the existence of LTI controllers CT 1 and CT 2 such that the closedloop transfer function KT matches with a desired transfer function KT d . Theorem 4.4 Consider the factorizations (4.8) for the transfer functions involved in the closedloop system (4.6). Let KT d be a stable singlerate system operating at the sampling time T. It is possible to design the LTI controllers CT 1 = X−1Y1 and CT 2 = X−1Y2, for the multirate system shown in Fig. 4.1, to achieve model matching with the system KT d (d) if and only if (a) A and B are relatively right prime (b) S is a left divisor of KT d , where S is defined as S , NG DG FI − Y1B FX (4.21) (c) the unstable zeros NG(d)(d) are zeros of KT d (d), where (d) is a stable transfer function such that A(dN) = DG(d)(d). Proof. (⇒) By hypothesis the closedloop multirate system matches the stable system KT d . By theorem 4.1, the stability of the closedloop system implies that F is a unit. Therefore, (a) follows. Since Y2 is stable by definition and, by equation (4.10) KT = SY2 = KT d , (b) follows. The stability of Y2 guarantees that no illicit cancelation occurs between the poles of Y2 and the zeros of S. (c) follows from the fact that the unstable zeros of S correspond to the unstable zeros of NG, as shown in (4.16). 57 (⇐) (a),(b) and (c) are true by hypothesis. Then there exists a pair (X, Y1) such that F = 1, and therefore the closedloop system is stable by virtue of theorem 4.1. Further, since (b) and (c) hold, there exists a stable transfer function matrix Y2 such that SY2 = KT d . Therefore, by equation (4.10) the closedloop system KT equals KT d . It is clear from theorem 4.4 that the desired system KT d cannot be selected arbitrarily. In fact, due to condition (c) of theorem 4.4 the set of zeros of the desired system KT d must contain the unstable zeros of the plant transfer function GT plus, for every unstable pole of GT , the N − 1 unstable zeros of discussed in remark 4.4. Therefore, consider the following factorizations of the polynomial NG and of the transfer function : NG = Nin G Nout G , = inout (4.22) where Nin G and in are polynomials of the variable d containing all and only the roots of NG and of the numerator polynomial of , respectively, contained in the closed unit circle; Nout G and out are respectively a Hurwitz polynomial and a stable minimumphase transfer function satisfying (4.22). According to condition (c) of theorem 4.4, the desired fastrate closedloop transfer function KT d must take the form KT d (d) = Nin G (d)in(d) ¯K T d (d) (4.23) where ¯K T d is a stable and proper transfer function. The following Theorem provides a parametrization of the set of all the stabilizing LTI controllers CT 1 and CT 2 such that the fastrate closedloop transfer function KT matches with a transfer function KT d factorized in the form (4.23). Theorem 4.5 Consider the factorizations (4.8) and (4.22). Let KT d be a desired 58 stable transfer function in the form (4.23). All the LTI controllers CT 1 and CT 2 such that model matching is achieved with KT d can be written as CT 1 = ¯X −1 ¯ Y1, CT 2 = ¯X −1 ¯ Y2, (4.24) with ¯X , ¯ Y1 and ¯ Y2 given by ¯X = X + B ¯ Y1 = Y1 − A ¯ Y2 = (A(X + B ) I + (Y1 − A )B) ˜M NT (dN) (4.25) where is any proper stable N ×1 transfer function of the variable dN, ˜MNT (ds) is a N×1 vector containing the N modified Ztransforms of Z−1{ ¯K T d (d)/(Nout G (d)out(d))}, and X, Y1 satisfy the Bezout equation (4.19). Proof. (⇐) Let CT 1 and CT 2 be given by (4.24) and (4.25). It has to be proved that closedloop model matching is achieved between the fastrate closedloop transfer function KT and KT d . Let us start by showing that KT , in (4.10), can be rewritten as KT = NG (AXI + Y1B)−1 Y2 (4.26) In fact, using the matrix inversion formula, and considering that 1−BF−1Y1 = AX = DGX 6= 0 and F = 1, the transfer matrix (FI − Y1B)−1 can be written as (FI − Y1B)−1 = F−1I + F−1Y1BF−1 1 − BF−1Y1 = I + Y1B AX = AXI + Y1B AX = AXI + Y1B DGX (4.27) The closedloop transfer function (4.26) is obtained with the substitution of the inverse 59 of (4.27) into (4.10). By using the controllers CT 1 and CT 2 given in (4.24) and (4.25) (that is, by replacing X, Y1 and Y2 in (4.26) with ¯X , ¯ Y1, ¯Y2), after some simplifications, the transfer function KT becomes KT = NG ˜M NT (dN). where, by construction, (d) ˜M NT (dN) = ¯K T d (d)/(Nout G (d)out(d)). Hence, KT (d) = Nin G (d)in(d) ¯K T d (d) = KT (d). Considering that ¯ Y2 is stable by construction, model matching is achieved. (⇒) Assume that the model matching is achieved through the LTI controller CT 1 = ¯X −1 ¯ Y1 and CT 2 = ¯X −1 ¯ Y2. It has to be proved that there exists a transfer function matrix such that ¯X , ¯ Y1 and ¯ Y2 can be written in the form (4.25). The closedloop transfer function of the system can be written as KT = NG A ¯X I + ¯ Y1B −1 ¯ Y2 (4.28) Since KT is stable there always exists a transfer function such that ¯X = X + B and ¯ Y1 = Y1 − A , where the pair (X, Y1) is a particular solution of the Bezout equation (4.19), [44]. Therefore, equation (4.28) can be rewritten as KT = NG (A(X + B )I + (Y1 − A )B)−1 ¯ Y2. (4.29) Considering that (d) ˜M NT (dN) = ¯K T d (d)/(Nout G (d)out(d)), ¯ Y2 must take the form shown in (4.25). It remains to be shown that ¯ Y2 is proper. Since ¯K T d represents a causal system in its minimal form, the only way for ¯K T d (d)/(Nout G (d)out(d)) to be not proper is that the numerator of Nout G out contains a factor dk with k ≥ 1. However, this is not possible since all the roots of the numerator of NG inside the unit circle, including the roots in the origin corresponding to terms like dk, are included by 60 construction into Nin G in. Therefore, due also to the properness of A, B, X, Y1 and , it is possible to conclude that ¯Y2 is proper. 4.5 Example Consider the plant transfer function P(s) = 1/(s + 1). Let H(s) = (1 − e−sT )/s be the zeroorder hold operating at the sampling time T = 0.01s. Then GT (d) = (1 − h)d/(1 − hd), where h , e−T . Considering N = 2, it follows that GT T = (1−h)d 1−hd (1−h)d2 1−hd , GT T NT (dN) = (−1+h)hd2 −1+h2d2 (−1+h)d2 −1+h2d2 According to the factorizations given by (4.8), let A and B be defined respectively as A = −1 + h2d2 and B = [(−1 + h)d2h, (−1 + h)d2]. A simple solution to the Bezout equation is given by X = −1 and Y1 = [1/(−1 + h), h]⊤. Therefore, all the stabilizing LTI controllers for the multirate system are of the form CT 1 (d) = 1 −1+h + hd − (−1 + h2d2)(ω1 + ω2d) −1 + ω1(−1 + h)d2h + ω2(−1 + h)d2 (4.30) where ω1(d2) and ω2(d2) are the entries of the stable and proper transfer function matrix = [ω1, ω2]. A necessary condition to achieve model matching is that the set of zeros of the desired closedloop transfer function contains the unstable zeros of NG, where = −(1 + hd) is such that A = DG. Notice that in the example DG does not have any unstable root, and therefore A and do not have any unstable zeros. Therefore, the only requirement on KT d is that KT d = d ¯K T d , where ¯K T d is any stable and proper transfer function. Model matching is achieved by selecting ¯ Y2 as shown in (4.25). 61 4.6 Conclusions In this chapter the problem of parameterizing the set of LTI controllers for the class of dualrate systems with slow measurement update rate and fast controller update rate was considered. The controller structure considered in this chapter is different from the one considered in chapter 3. In particular, these controllers are made of an upsampling operator and an LTI system. Upsampling operators are utilized to convert the update rate of the controller inputs into the controller update rate. A downsampling operator following the plant discretized at the fast control update rate was utilized to model the original system as one with fastupdating input and slowupdating output. The use of the invertible transformation (4.7) and of upsampling and downsampling operators, was shown to be beneficial to write the closedloop transfer function of the multirate system in Fig. 4.1 in the more familiar form of a singlerate closedloop system. A procedure was also given to parametrize the set of controllers CT 1 and CT 2 for which model matching is achieved in the two cases of a fastupdating and a slowupdating desired LTI system KT d and KNT d , respectively. In such parametrizations the free parameter can be chosen as any stable transfer function matrix to span the entire parametrized set. Due to the popularity and efficacy of PID controllers, a topic of future research may be the one of restricting the parametrization of controllers for multirate systems to only and all the PID controllers for which certain performance criteria can be met. In turn this problem can be extended to the one parametrizing the set of fixed structure controllers for multirate systems. Moreover, future research will focus also on the parametrization of the set of LTI controllers for which certain robustness criteria, in addition to stability, are met. 62 CHAPTER 5 Ripplefree conditions in multirate systems using LTI controllers In this chapter it is studied the effect of the (N, 1)shiftinvariant controllers used in chapter 4 on the steadystate response of the closedloop system. In particular, conditions and a design procedure are given to avoid the problem of ripples at steadystate. In the context of a output regulation and tracking problem, necessary and sufficient conditions were given in [26, 27] to achieve a ripplefree response to sinusoidalexponential reference signals in the case that (m, n)shiftinvariant controllers are utilized. In particular, a continuoustime internal model of the reference signals has to be included in the forward path of the control system. For constant reference signals however, the presence of a continuoustime zeroorder hold in cascade with the plant may not be sufficient to guarantee a ripplefree steadystate response, and two possible solutions can be utilized: either the controller is augmented with a fastrate digital integrator [27], or the controller coefficients are forced to converge at steadystate [25]. The use of timevarying controllers, however, may significantly worsen the intersample behavior of the transient response of the closedloop system, but no work has been done to directly address this problem. If the (N, 1)shiftinvariant controllers shown in chapter 4 are utilized, the main cause of ripples in the steadystate output can be isolated. In fact it will be shown that ripples may exist in the steadystate response of the closedloop system in Fig. 4.1 mainly because of the presence of the timevarying upsampling operators. Therefore, ripples can be avoided at steadystate, and the undesired nonlinear behavior may be 63 reduced during the transient, simply with the use of particular LTI filters connected to upsampling operators. Necessary and sufficient conditions on the structure of these filters are obtained in this chapter. The author’s initial work is shown in [38, 50]. This chapter is organized as follows. The problem formulation is given in section 5.1. Before deriving the main results of this chapter, two numerical examples that clearly motivate the need for subsequent analysis are given in section 5.2. Section 5.3 contains the main results of this chapter. Some particular solutions to the addressed problem are shown in section 5.4. Conclusions are given in section 5.5. 5.1 Problem Formulation Denote with P(s) the Laplace transform of a continuoustime LTI SISO plant to be controlled, and with y(t) its output to be regulated to the reference signal rNT . The plant output is measured at the slowrate 1/NT, where N is a positive integer, and the control signal uT updates at the faster rate 1/T. The considered multirate system is shown in Fig. 5.1. This control system is similar to the one shown in Fig. 4.1, for the + + N N NT rNT CT 1 (d) CT 2 (d) ZOH P(s) uT y(t) yNT FT (d) FT (d) Figure 5.1: Multirate Control System. exception of the additional LTI systems FT placed between the upsampling operators and the controllers CT i , i = 1, 2. Therefore, the digital control action provided by the controller takes the following form UT = CT 1 FT [Y NT ]T + CT 2 FT [RNT ]T 64 where UT , Z{uT }, Y NT , Z{yNT } and RNT , Z{rNT }. It is possible to show that the slowrate samplers in Fig. 5.1 can be modeled with a fastrate sampler followed by a downsampling operator. This alternative structure, shown in Fig. 5.2, is more suitable for modeling and analysis. + + N N N T rNT CT 1 (d) CT 2 (d) ZOH P(s) uT y(t) yNT FT (d) FT (d) Figure 5.2: Multirate Control System. The following assumption, together with assumption 3.1, is considered. Assumption 5.1 Given a filter FT , there exist stable controllers CT 1 and CT 2 that stabilize the slowrate reference to output transfer function Y NT = 1 − GTCT 1 FT NT −1 GTCT 2 FT NT RNT (5.1) and such that the closedloop multirate system exhibits zero steadystate error at the measurement update rate in response to a step reference signal, that is, lim k→∞ e(kNT) = 0 where e(t) , y(t) − r(t). The stability condition in assumption 5.1 can be met by designing the controller CT 1 as discussed in chapter 4, and by ensuring that CT 2 is stable. The condition, in assumption 5.1, about the zero steadystate error of the closedloop system at the measurement update rate, is satisfied if the forward path of the block diagram in Fig. 5.1 contains a continuoustime internal model of the step reference signal [24]. 65 Therefore, due to the presence of the zeroorder hold H(s), the controllers CT 1 and CT 2 can be designed to satisfy such a condition. Since the achievement of zero steadystate error at the measurement update rate is not sufficient to guarantee a ripplefree response to a step reference signal (because the error e(t) may be different from zero between two consecutive measurement sampling instants), it is desirable to obtain necessary and sufficient conditions on the LTI filter FT to achieve a steadystate ripplefree closedloop response, that is, lim k→∞ Z (k+1)NT kNT e⊤(τ )e(τ )dτ = 0. 5.2 Example The effect of the selection of an inappropriate filter FT on the continuoustime closedloop output y(t) is discussed in this section. In particular, two different examples are considered, each one obtained with the use of a different filter FT , to show that ripples can occur in the system output even though the systems CT 1 and CT 2 are LTI. The main purpose of this section is to motivate the need for the analysis developed subsequently in this chapter. Consider the control scheme in Fig. 5.2 where the plant transfer function is given by the doubleintegrator P(s) = 1/s2 and the controller sampling period is T = 0.1 sec. The plant output, y(t), and the step reference signal, r(t), are sampled at the rate 1/NT, where N = 3. Let KNT be the closedloop transfer function between the measured output Y NT and the slowrate reference signal RNT , that is, KNT , Y NT /RNT . Using the procedure shown in chapter 4, let the controllers CT 1 and CT 2 be designed in order to guarantee that KNT matches with the desired transfer function KNT d given by KNT d , 1 12 d(1 + 0.2d) 1 − 1.8d + 0.9d2 (5.2) 66 The cases (a) in which FT = 1 and (b) in which FT = 1 + d + d2 are considered for this example. In the case (a) the stable controllers CT 1 and CT 2 are given by CT 1 = 3 50 (−9 + 7d), CT 2 = 1 0.3 − 0.54d3 + 0.27d6 In the case (b), instead, the stable controllers CT 1 and CT 2 are given by CT 1 = −22 + 18d 0.36 + 0.09d3 , CT 2 = 10 − 8d + 2d2 3.6 − 5.58d3 + 1.62d6 + 0.81d9 Notice that with the designed controllers and the selected sampling period T, assumptions 3.1 and 5.1 are satisfied in both the cases (a) and (b). The closedloop system behavior to a unit step reference signal in the cases (a) and (b) is shown in Fig. 5.3 and Fig. 5.4. In particular, Fig. 5.3 compares the closedloop continuoustime response y(t) with the response of the desired digital system KNT d , and Fig. 5.4 shows the behavior of the control signal uT . As shown in Fig. 5.3, despite the continuoustime output y(t) equals in both the cases the output of the desired system at the measurement update rate (that is, at the time instants t = kNT, k = 0, 1, 2, . . .), an undesirable highfrequency oscillatory behavior characterizes the continuoustime response of the system in the case (a). This oscillatory behavior, which at steadystate is referred to as rippleeffect, does not appear in the continuoustime output of case (b) where a different filter FT has been utilized. Similarly, Fig. 5.4 shows the presence of persistent oscillations in the control signal uT in the case (a), and no oscillations in the case (b) where uT converges to a constant value. This fundamental difference between the step responses in the cases (a) and (b) shows that an inappropriate choice of the filter FT can cause a rippleeffect at steadystate as well as undesirable high frequency oscillations in the transient. Therefore, a characterization of the set of filters FT which help to prevent the rippleeffect is needed. 67 0 5 10 15 20 0 0.5 1 1.5 2 Time (sec) FT=1 Desired system response FT=1+d+d2 Figure 5.3: Plant output y(t). 0 5 10 15 20 −150 −100 −50 0 50 100 150 Time (sec) FT=1+d+d2 FT=1 Figure 5.4: Controller output uT . Remark 5.1 (On the stability of the fastrate closedloop system) Due to the presence of persisting oscillations at steadystate, it appears from Fig. 5.3 for the case (a) that the closedloop system is stable at the measurement updaterate but only marginally stable at the control updaterate. However, it is shown in chapter 4 that the fastrate closedloop system KT , Y T /[RNT ]T in Fig. 5.1 is stable if and only if the slowrate closedloop system KNT , Y NT /RNT is stable. Therefore, since KNT is stable (by assumption 5.1) also KT is stable. In particular, for the case (a) the stable 68 fastrate closedloop transfer function is given by KT = 12d + 36d2 + 60d3 + 39d4 + 8d5 + 12d6 + 7d7 720 − 1296d3 + 648d6 Therefore the sustained oscillations at steadystate are not due to the closedloop system instability, but due to the timevarying nature of the upsampling operators. 5.3 Ripplefree conditions It is shown in this section that the ripplefree behavior of the closedloop system response depends only on the choice of the digital filter FT . Moreover, necessary and sufficient conditions are given on the coefficients of the filter FT to guarantee a ripplefree closedloop response to step reference signals. Some preliminary results are given first to facilitate the proof of the main theorems. Proposition 5.1 (From [25]) Let GT be the transfer function of a LTI system, and let GNT L be its lifted version, whose elements are given in (2.16). Then the row sum of all the rows of GNT L (1) is equal, that is, XN k=1 GNT L,(m,k)(1) = XN k=1 GNT L,(n,k)(1), ∀m, n = 0 . . . ,N − 1 (5.3) The following result is a consequence of proposition 5.1. Proposition 5.2 Let ANT and BNT be two realvalued N ×1 transfer matrices, such that BNT = DNTANT , where DNT is a N × N transfer matrix. The following statements hold: 1. If the Ndimensional vector ANT (1) has all equal entries and DNT satisfies property (5.3), BNT (1) = DNT (1)ANT (1) is also a vector with all equal entries. 2. If ANT (1) and BNT (1) are each Ndimensional vectors with all equal entries, 69 DNT satisfies property (5.3). Proof. Part 1 is proved by noticing that BNT (1) = αANT (1), where α is the sum of all the elements in one row of DNT (1). Part 2 can be proved as follows. Since all the entries ANT k (1) (k = 0, . . . ,N − 1) of ANT (1) are equal, NX−1 k=0 DNT (i,k)(1)ANT k (1) = NX−1 k=0 DNT (i,k)(1) ! ¯a, i = 0, . . . ,N − 1 where ¯a = ANT k (1) (k = 0, . . . ,N − 1), and DNT (i,k) is the (i, k)th entry of DNT . Since all the entries of BNT (1) are equal, DNT (1) must satisfy property (5.3). The following result extends property (5.3) to the series connection of LTI systems. Proposition 5.3 Let DNT and ¯D NT be two N × N transfer matrices. The series connection DNT ¯D NT satisfies property (5.3) if and only if DNT and ¯D NT individually satisfy the same property. Proof. Let X ⊂ RN be the set of all the realvalued Ndimensional vectors xi (i = 1, 2, . . .) with not all the entries equal. Moreover, let X− be the complement of X in the space of all the realvalued Ndimensional vectors RN, and let x− i (i = 1, 2, . . .) denote the ith element of X−. (⇐). If DNT and ¯D NT satisfy property (5.3), DNT (1)x−i ∈ X− and ¯D NT (1)x−i ∈ X−, for every i ∈ N. Therefore, DNT (1)¯D NT (1)x− i ∈ X− which implies that every row sum of DNT (1)¯D NT (1) is equal. (⇒). By contradiction, assume that ¯D NT does not satisfy property (5.3) and that DNT does satisfy it. Therefore, there exists a vector x− i such that ¯D NT (1)x− i /∈ X− 70 and DNT (1)¯D NT (1)x− i /∈ X−. However, the last statement contradicts the hypothesis that the series connection DNT ¯D NT satisfies property (5.3). A similar argument can be made in the case that DNT does not satisfy property (5.3) and that ¯D NT does satisfy it. The following two theorems give the main results of this chapter. Theorem 5.1 Under the assumptions 3.1 and 5.1, the system in Fig. 5.1 exhibits a steadystate ripplefree response to step reference signals if and only if the first column of the lifted filter, FNT L (obtained as shown in (2.16)), has all equal entries for ds = 1, that is, FNT L (1) = α ⋆ · · · ⋆ ... ... ... α ⋆ · · · ⋆ (5.4) where α is any real scalar number. Proof. (⇒). As discussed in section 2.1, the upsampling operators insert N −1 zerovalued samples between two consecutive samples of the original signal. This behavior of the upsampling operators can be taken into account by assuming that the lifted filter matrix FNT L is identically zero except for the first column, that is, FNT L = FNT L,(1,1) 0 · · · 0 ... ... ... FNT L,(N,1) 0 · · · 0 (5.5) By using (5.5) as the lifted matrix of the filter FT , it is possible to show that the lifted plant input, UNT L , is related to the lifted reference signal RNT L , Z{LN(rT )}, 71 by the expression I − CNT 1,L FNT L GNT L UNT L = CNT 2,L FNT L RNT L (5.6) where CNT 1,L and CNT 2,L are the lifted versions of the controllers CNT 1 and CNT 2 , respectively. The stability of the closedloop system (5.1) ensures that the steadystate closedloop response yNT and the control action uNT are constant. Therefore, to achieve a ripplefree response to step reference signals, it remains to be shown that also uT is constant at steadystate. Moreover, in order to feed the feedback loop with a signal constant at steadystate, it is necessary that the output of the controller CT 2 also reaches a steadystate value. In other word it is necessary that all the entries of the vector uNT L , as well as all the entries of the inverse Ztransform of the right hand side of (5.6), have the same steadystate value. By applying the final value theorem to equation (5.6), and by proposition 5.2, this implies that the N ×N matrix S , I − CNT 1,L FNT L GNT L has to satisfy property (5.3). In turn, by proposition 5.3, this requires that the matrix FNT L GNT L satisfies the same property. Considering that FNT L (1)GNT L (1) is of the following form FNT L (1)GNT L (1) = FNT L,(0,0)(1)GNT L,(0,0)(1) · · · FNT L,(0,0)(1)GNT L,(0,N−1)(1) ... ... FNT L,(N−1,0)(1)GNT L,(0,0)(1) · · · FNT L,(N−1,0)(1)GNT L,(0,N−1)(1) , in order for the matrix FNT L GNT L to satisfy property (5.3), it must be that FNT L,(i,0)(1) = α, i = 0, . . . ,N − 1, where α can be any real number. (⇐). If FNT L (1) is in the form (5.4), the matrix S , (I − CNT 1,L FNT L GNT L ) satisfies property 5.3. Moreover, considering that the vector rNT L , LN(rT ) has all equal entries, at steadystate the right hand side of (5.6) is also a vector with all equal 72 entries. Therefore, by proposition 5.2, the lifted plant input UNT L (1) has all equal entries, and this is sufficient to guarantee that the plant output is ripplefree at steadystate. The following theorem gives closedform conditions on finite impulse response filters such that their lifted version is in the form (5.4) at steadystate. Theorem 5.2 Let the filter FT be a finite impulse response system of the form FT , a0 + a1d + a2d2 + · · · + andn (5.7) The lifted system FNT L (dN) of FT is in the form (5.4) for dN = 1 if and only if Xp k=0 aNk+i = α, i = 0, . . . ,N − 1 (5.8) where p = ⌊(n − i)/N⌋ is the largest integer not greater than (n − i)/N. Proof. As shown in (2.19), the first column of FNT L contains the modified Ztransforms of FT . In particular, from the definition of modified Ztransform (2.17), the first N modified Ztransforms of FT are given by ˜ FNT 0 = a0 + aNd + a2Nd2 + · · · + a⌊n/N⌋Nd⌊n/N⌋ ˜ FNT 1 = a1 + aN+1d + · · · + a⌊(n−1)/N⌋N+1d⌊(n−1)/N⌋ ... = ... ˜ FNT N−1 = aN−1 + a2N−1d + · · · + a⌊(n−N+1)/N⌋N+N−1d⌊(n−N+1)/N⌋ In general, the ith modified Ztransform is given by ˜ FNT i (d) = Xp k=0 aNk+idk (5.9) 73 where p = ⌊(n − i)/N⌋. Therefore, in order for FNT L (1) to be of the form (5.4), it is necessary and sufficient that (5.8) is satisfied. Theorem 5.2 gives necessary and sufficient conditions for the filter FT in Fig. 5.1 to guarantee a ripplefree closedloop response to step reference signals, in the case that FT is a FIR filter. The conditions are expressed as constraints on the sum of groups of coefficients in FT . Remark 5.2 The result obtained in theorem 5.2 is based on the assumption that FT is a FIR filter. However, if it is desired to use additional poles in the filter FT , while preserving the ripplefree behavior of the closedloop system, the added poles must not cancel the zeros introduced by the FIR part of the filter. Similarly, additional zeros can be used as long as they are not at the origin (to avoid cancelation with the poles at the origin of the FIR part of the filter). Remark 5.3 The results shown in this work are also valid in the case that the feedforward and the feedback filters FT in Fig. 5.1 are not the same. However, to achieve a ripplefree closedloop response it is important that both the filters satisfy condition (5.4). Remark 5.4 The idea behind the separation of the controllers into the two parts, CT i (for i = 1, 2) and FT is to facilitate the design process. In fact, by keeping FT and CT i separated one can first select the appropriate FT to achieve ripplefree response (and possibly also to filter the system output), and then design the remaining part of the controller, CT i , to meet other performance criteria. However, FT and CT i can be joined to form one transfer function for implementation purposes. 74 5.4 Digital higherorder filters A simple example of a FIR filter that satisfies condition (5.8) is the digital ZOH (ZOH) : FT = 1 + d + d2 + · · · + dN−1 = 1 − dN 1 − d (5.10) In fact, by replacing n with N − 1 in (5.9), it can be shown that ˜ FNT i = ai = 1, i = 0, . . . ,N − 1. The digital ZOH filter in (5.10) serves to hold the output signal of the upsampling operator between two consecutive slow measurement updates. However, the filter in (5.10) is not the only solution. In fact some other examples include higher order holds (details on how to obtain a representation for higher order holds are given in [51]). For instance, in the cases of the first order hold (FOH) and the triangular hold (TH), the corresponding modified Ztransforms are given by: (FOH) : ˜ FNT i (d) = N + i N − i N d ⇒ ˜ FNT i (1) = 1 (TH) : ˜ FNT i (d) = i N + N − i N d ⇒ ˜ FNT i (1) = 1 The use of any one of these filters guarantees a ripplefree closedloop response to step reference signals. 5.5 Conclusions For the class of controllers considered in chapter 4, a good choice of the sampling period and the stability of the closedloop system are necessary but not sufficient to achieve a ripplefree response at steadystate. In fact, one of the main causes of ripples in the response of such a multirate system, where the output of the plant is measured at a slow rate and LTI controllers are utilized to provide a faster control action, is the manner in which the slow measurements are fed into the controller. The inclusion of digital prefilters, in cascade with the controller transfer function, 75 was shown in this chapter to be useful in avoiding ripples in the step response of the closedloop system. Necessary and sufficient conditions on the coefficients of such filters were also provided to characterize the entire set of filters for which a ripplefree steadystate response can be achieved. 76 CHAPTER 6 Conclusions and future work In this work several control problems were considered for the class of multirate systems with slow measurement update rate and fast controller update rate. Such control problems can be grouped as follows: 1) inputstate matching with a desired singlerate digital system, 2) parametrization of LTI controllers for dualrate systems, and 3) ripplefree response of multirate systems with upsampling and downsampling operators. The solution to the second problem was facilitated by the use of the upsampling and downsampling operators as modeling tools. However, the timevarying nature of these two operators was shown to be the main cause of ripples in the steadystate response of the multirate system. A method to avoid ripples in the step response with the use of FIR filters in cascade with the upsampling operators was given in chapter 5. Future research that is relevant to the material covered is given at the end of each chapter. The first use of upsampling and downsampling operators can be found in the signal processing area [34]. In particular, with the use of these operators, the information contained in any signal can be converted to a pyramidal set of information where each layer of the pyramid characterizes the signal properties at a particular scale. If, for instance, this multiscale signal characterization is applied to the output signal of the plant, one may potentially design a controller to achieve a desired closedloop performance at each particular scale. The main disadvantage of this multiscale theory, that was originally named as subband coding, is the difficulty in the choice of the scale factors, which correspond to the factors of the upsampling and down 77 sampling operators [52]. This is due to the fact that the frequency domain properties of a signal are usually timevarying. Therefore, a way to express the scale factors as functions of time would be needed. To overcome this problem, and provide a better timefrequency localization of the signal properties, a new multiscale theory based on the use of wavelets was recently adopted [53]. The use of wavelets to brake down the signal information into a finite discrete set of useful and meaningful characteristics at different scales resulted being very successful. If one wants to apply this multiscale theory to the plant output signal (in a control system), a filter may be utilized to generate the scaled information to be passed on to the controller. This procedure, however, would result in delaying the feedback measurement, which is highly undesired. One way to overcome this issue would be to model the plant also by using the waveletbased multiscale theory. Few attempts have been made to model LTI systems using this theory [54–56], but the results obtained so far in the literature are complicated to analyze and to use (due to the complexity of the original wavelets). 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