Robust and perfect tracking of discrete-time systems

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1 Abstract Automatica (0) } This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Kenko Uchida under the direction of Editor Tamer Basar. * Corresponding author. Tel.: #--; fax: #--. address: bmchen@nus.edu.sg (B. M. Chen). Brief Paper Robust perfect tracking of discrete-time systems Ben M. Chen*, Zongli Lin, Kexiu Liu Department of Electrical & Computer Engineering, The National University of Singapore, Singapore, Singapore Department of Electrical & Computer Engineering, The University of Virginia, Charlottesville, VA, USA Received July 00; revised April 0; received in "nal form July 0 We study in this paper the problem of robust perfect tracking for discrete-time linear multivariable systems. By robust perfect tracking, we mean the ability of a controller to track a given reference signal with arbitrarily small settling time in the face of external disturbances initial conditions. A set of necessary su$cient conditions under which the proposed problem is solvable is obtained, under these conditions, constructive algorithms are given that yield the required solutions. As is general to discrete-time systems, the solvability conditions to the above problem are quite restrictive. To relax these conditions, we propose an almost perfect tracking scheme, which is capable of tracking references precisely after certain initial steps. 0 Elsevier Science Ltd. All rights reserved. Keywords: Robust control; Tracking; Disturbance rejection; Discrete-time systems; Control Systems. Introduction to the problem The problem of robust perfect tracking (RPT) is to design a controller such that the resulting closed-loop system is asymptotically stable the controlled output perfectly tracks a given reference signal in the presence of any initial conditions external disturbances. By robust perfect tracking, we mean the ability of a controller to track a given reference signal with arbitrarily small settling time in the face of external disturbances initial conditions. The subject of continuous-time systems has been fully examined in a recent work by Liu, Chen, Lin (0). The main objective of this paper is to study such a problem for discrete-time systems. To be more speci"c, we present in this paper the robust perfect tracking problem for the following discrete-time system, Σ : x(k#)"ax#bu#ew, x(0)"x, y"c x#d w, h"c x#d u#d w, 000-/0/$ - see front matter 0 Elsevier Science Ltd. All rights reserved. PII: S ( 0 ) () where x is the state, u is the control input, w q is the external disturbance, y p is the measurement output, h l is the output to be controlled. We also assume that the pair (A, B) is stabilizable the pair (A, C ) is detectable. For future references, we de"ne Σ Σ to be the subsystems characterized by the matrix quadruples (A, B, C, D ) (A, E, C, D ), respectively. Given the external disturbance wl, p[,r], any reference signal vector r l, the RPT problem for the discrete-time system () is to "nd a parameterized dynamic measurement feedback control law of the following form: v(k#)"a (ε)v#b (ε)y#g(ε)r, u"c (ε)v#d (ε)y#h(ε)r, such that when () is applied to (): (i) There exists an εh'0 such that the resulting closedloop system with r"0 w"0 is asymptotically stable for all ε(0, εh]; (ii) Let h(k, ε) be the closed-loop controlled output response let e(k, ε) be the resulting tracking error, i.e., e(k, ε):"h(k, ε)!r. Then, for any x, e P0asεP0. In this paper, we will derive a set of necessary su$cient conditions under which the proposed robust ()

2 B. M. Chen et al. / Automatica (0) } perfect tracking problem has a solution, under these conditions, develop algorithms for the construction of feedback laws that solve the proposed problem. It turns out that the solvability conditions for the above proposed RPT problem are quite restrictive compared to its counterpart in the continuous-time case (see Liu et al., 0). In particular, the discrete-time problem has a quite restrictive constraint on the in"nite zero structure of the given system, while the in"nite zero structure can be arbitrary in its continuous-time counterpart. To relax these conditions, we will introduce a modi"ed problem, which can be solved for a much larger class of discretetime systems. This modi"ed formulation will yield internally stabilizing control laws that are capable of tracking reference signals with some delays. If we know the reference signal a few steps ahead, the modi"ed tracking control scheme will then track the reference precisely after certain steps, provided that the given plant satis"es a new set of more relaxed conditions. We note that this is a unique feature of the discrete-time problem. It is impossible to achieve such a performance for continuoustime systems. Throughout this paper, the following notation will be used: X denotes the pseudo inverse of X; U, are, respectively, the open unit disc, the unit circle the set of complex numbers outside the unit circle on the complex plane; Ker(X) is the kernel of X; Im(X) is the image of X; Given a linear system Σ H characterized by a matrix quadruple (A, B, C, D), we de"ne (see e.g., Chen, 00): (i) V U (Σ H ) is the maximal subspace of for which there exists an F such that V U is (A#BF)-invariant is contained in Ker(C#DF), the eigenvalues of (A#BF) V U are contained in U. (ii) S U (Σ H ) is the minimal subspace of for which there exists a K such that S U is (A#KC)-invariant contains Im(B#KD), the eigenvalues of the map which is induced by (A#KC) on the factor space /S U are contained in U.. Solvability conditions solutions The following theorem gives a set of necessary su$cient conditions under which the proposed RPT problem is solvable for the given plant (). We will show the su$ciency of these conditions by explicitly constructing the required control laws. Theorem. Consider the given system () with its external disturbance wl, p[,r], its initial condition x(0)"x. Then, for any reference signal r, the proposed robust perfect problem is solvable by the control law of () if only if the following conditions are satisxed: () (A, B) is stabilizable (A, C ) is detectable; () D #D SD "0, where S"!(D D )D D D (D D ); () Σ is right invertible of minimum phase with no inxnite zeros; () Ker(C #D SC ) MC Im(D ) :"ζ C ζim(d ). Proof. We "rst show that Conditions } in the theorem are necessary. Let us consider the case when r,0. Then, the proposed robust perfect tracking problem reduces to the perfect regulation problem. We can reformulate the perfect regulation problem for the given system () as the well studied almost disturbance decoupling problem for the following system: x(k#)"ax#bu#[e I]w, x(0)"0, y"c x#[d 0]w, () h"c x#d u#[d 0]w. For easy reference, we let ΣI be characterized (A,[EI], C,[D 0]). Following the results of the discrete-time almost disturbance decoupling problem (see Chen, 00), we can show that if the almost disturbance decoupling problem for the above system is solvable, we have: (i) (A, B) is stabilizable (A, C ) is detectable; (ii)d #D SD "0,where S"!(D D )D D D (D D ); (iii) Im([E#BSD I])LV U (Σ )#B Ker(D ); (iv) Ker(C #D SC )MS U (ΣI )C Im(D ); "nally (v) S U (ΣI )LV U (Σ ). Next, it is easy to see that S U (ΣI )" hence Condition (v) implies that V U (Σ )", or equivalently, Σ is right invertible with no in"nite zeros no invariant zeros in. Furthermore, Condition (iv) reduces to the condition Ker(C #D SC )MC Im(D ). Thus, it remains to be shown that if the proposed RPT problem is solvable, the subsystem Σ must be of minimum phase. In what follows, we proceed to show such a fact: First, we note that the second condition, i.e., D #D SD "0, implies that if we apply a pre-output feedback law, u"sy, to System (), the resulting new system will have a zero direct feed-through term from w to h. Hence, without loss of any generality, we hereafter assume that D "0 throughout the rest of the paper. Next, we show that if the robust perfect tracking problem is solvable for general nonzero reference r, Σ must be of minimum phase, i.e., Σ cannot have any invariant zeros on the unit circle. In fact, this condition must hold even for the case when w"0 x "0, i.e., for the robust perfect tracking of the following system: x(k#)"ax#bu, y"c x, e"c x#d u!r"h!r. Now, if we treat r as an external disturbance, then the above problem is again equivalent to the well-known () 0

3 B. M. Chen et al. / Automatica (0) } almost disturbance decoupling problem with measurement feedback with internal stability for the following system: x(k#)"ax#bu, y " C x r, e"c x#d u!r. Without loss of generality, we assume that the quadruple (A, B, C, D ) has been transformed into the form of the special coordinate basis of Sannuti Saberi () (see also Chen, 00), i.e., we have x"x x x, h"h, r"r, e"e "h!r, () u" u u x (k#)"a x #B h, () x (k#)"a x #B h, () x (k#)"a x #B h #B [E x () #E x ]#B u, () e "C x #C x #C x #u!r, () where λ(a ) λ(a ) are, respectively, the invariant zeros of (A, B, C, D ) in, (A, B ) is controllable. In order to bring the subsystem from u to e into the stard form of the special coordinate basis, we need to change h in ()}() to e #r, i.e., x (k#)"a x #B e #B r, () x (k#)"a x #B e #B r, () x (k#)"a x #B e # B [E x #E x ] #B u #B r. () It is easy to see that the subsystem from the controlled input, i.e., (u u ), to the error output, i.e., e, is now in the stard form of the special coordinate basis. It then follows from the result of Chen (00) (i.e., Proposition..) that if the almost disturbance decoupling problem with measurement feedback with internal stability for system () is solvable, there must exist a nonzero vector ξ such that ξ(λi!a )"0 ξb "0, which implies that the matrix pair (A, B ) is not completely controllable. Following the property of the special coordinate basis (see Chen, 00), the uncontrollability of (A, B ) implies the unstabilizability of the pair (A, B), which is obviously a contradiction. Hence, x must be nonexistent Σ must be of minimum phase. This proves the necessary part. We note that for the case when D "0, the direct feedthrough term D must be a zero matrix as well, the last condition, i.e., Condition (iv), of Theorem reduces to Ker(C )MKer(C ). We will show the su$ciency of those conditions in Theorem by explicitly constructing controllers which solve the proposed robust perfect tracking problem under Conditions } of Theorem. This will be done in the next subsections. It turns out that the control laws, which solve the robust perfect tracking for the given plant () under the solvability of Theorem, need not be parameterized by any tuning parameter. Thus, () can be replaced by v(k#)"a v#b y#gr, () u"c v#d y#hr furthermore, the resulting tracking error e can be made identically zero for all k*0... Solutions to state feedback case When all states of the plant are measured for feedback, i.e., C "I D "0, the problem can be solved by a static control law. In fact, the conditions in Theorem are reduced to: (i) (A, B) is stabilizable; (ii) D "0; (iii) Σ is right invertible of minimum phase with no in"nite zeros. We construct in this subsection a state feedback control law, u"fx#hr () which solves the robust perfect tracking (RPT) problem for () under the above conditions. Step S.: This step is to transform the subsystem from u to h of the given system () into the special coordinate basis, i.e., to "nd nonsingular state, input output transformations Γ, Γ Γ to put it into the structural form (see Chen, 00, for details), i.e., Γ(A!B C )Γ " A 0 B E A, () 0 ΓBΓ "Γ[B B ]Γ " B B B, () ΓD Γ "[I 0], () ΓC Γ "ΓC Γ "[C C ], () where λ(a) are the invariant zeros of (A, B, C, D ) (A, B ) is controllable. Step S.: Choose an appropriate F such that A "A!B F is stable. 0

4 B. M. Chen et al. / Automatica (0) } Step S.: Finally, we let F"!Γ C E C F Γ H"Γ 0 I Γ. () Theorem. Consider the discrete-time system () with any external disturbance w any initial condition x(0). Assume that all its states are measured for feedback. If D "0 Σ is stabilizable, right invertible of minimum phase with no inxnite zeros, then, for any reference signal r, the proposed robust perfect tracking is solved by the control law of () with F H as given in (). Proof. It follows from some simple calculations... Solutions to measurement feedback case Without loss of generality, we assume throughout this subsection that matrix D "0. It turns out that, for discrete-time systems, the full order observer based control law is not capable of achieving the robust perfect tracking performance, because there is a delay of one step in the observer itself. Thus, we will focus on the construction of a reduced order measurement feedback control law to solve the RPT problem. For simplicity of presentation, we assume that matrices C D have already been transformed into the following forms: C " 0 C I 0 D " D 0, () where D is of full row rank. We "rst partition the following system: x(k#)"ax#bu#[e I ]w, y"c x#[d 0]w () in conformity with the structures of C D in (), i.e., δ(x ) δ(x ) " A y y " 0 A # B A A x x B u# E C I 0 x I 0 w, () E 0 I x # D w, where δ(x )"x (k#) δ(x )"x (k#). Obviously, y "x is directly available hence need not be estimated. Next, let Σ be characterized by (A, E, C, D ) " A,[E 0 I ], C A, D 0 0 E I 0. () It is straightforward to verify that Σ is right invertible with no "nite in"nite zeros. Moreover, (A, C )is detectable if only if (A, C ) is detectable. We are ready to present the following algorithm. Step R.: For the given system (), we again assume that all the state variables of () are measurable then follow Steps S.}S. of the algorithm of the previous subsection to construct gain matrices F H. We also partition F in conformity with x x of () as follows: F"[F F ]. () Step R.: Let K be an appropriate dimensional constant matrix such that the eigenvalues of A #K C "A #[K K ] C () A are all in. This can be done because (A, C ) is detectable. Step R.: Let G"(B #K B )H, G "[!K, A #K A!(A #K C )K ], () A "A #B F #K C #K B F, B "G #(B #K B )[0, F!F K ], C "F, D "[0, F!F K ]. () Step R.: Finally, we obtain the following reduced order measurement feedback control law: v(k#)"a v#b y#gr, u"c v#d y#hr. () Theorem. Consider the given system () with any external disturbance w any initial condition x(0). If Conditions } of Theorem are satisxed, then, for any reference signal r, the proposed RPT is solved by the reduced order measurement feedback control laws of (). Proof. Due to space limitation, the proof is omitted (see Chen, 00 for details). The su$ciency of Theorem is obvious now in view of the result of Theorem. The proof of Theorem is thus completed. 0

5 B. M. Chen et al. / Automatica (0) }. An almost perfect tracking problem As has been seen in the previous section, the solvability conditions for the robust perfect tracking problem are generally too strong. We introduce now a modi"ed problem, the almost perfect tracking problem, which can be solved for a much larger class of discrete-time systems with any in"nite zero structure. This modi"ed formulation will yield internally stabilizing control laws that are capable of tracking reference signal r with some delays. If we know the reference signal a few steps ahead, the modi"ed tracking control scheme will then track the reference precisely after certain steps. For simplicity, we consider in this section the discretetime system () without external disturbances, i.e., x(0)"x, Σ :x(k#)"ax#bu, y"c x, h"c x#d u. () Let us "rst consider the case that the reference r l to be tracked is a known vector sequence, which implies that r(k#d), 0)d)κ d, is known for some integer κ d *0. This is a quite reasonable assumption in most practical situations when one wants to track references such as step functions, ramp functions sinusoidal functions. We will later deal with the case when r(k#d), d'0, is unknown. We are ready to formally de"ne the almost perfect tracking problem. Given the discrete-time system () with initial condition x(0)"x the reference r with r(k#d), 0)d)κ d, being known for a nonnegative integer κ d, the (κ d,κ ) almost perfect tracking problem, where κ is another nonnegative integer, is to "nd a dynamic measurement feedback control law of the following form: v(k#)"a v#b y #G r##g d r(k#κ d), u"c v#d y #H r##h d r(k#κ d) () such that when () is applied to (), () the resulting closed-loop system is internally stable; () for any initial condition x, the resulting tracking error e,0 orh,r, for all k*κ. Theorem. Consider the discrete-time plant () with x(0)"x, with (i) (A, B) being stabilizable (A, C ) being observable; (ii) Σ being right invertible of minimum phase. Let the inxnite zero structure of Σ be given as S (Σ )"q,, q, with q ))q, let the controllability index of (A, C ) be C"k,, k p, with k ))k p. Then, the (κ d,κ ) almost perfect tracking problem is solvable for any reference with κ d "q κ "q #k p!. Proof. We prove this theorem by explicitly constructing the required control law. Let us "rst construct the special coordinate basis of Σ. It follows from Sannuti Saberi () (see also Chen, 00) that there exist nonsingular state, output input transformations Γ, Γ Γ, which will take Σ into the stard format of the special coordinate basis, i.e., x"γ x, h"γ hi, u"γ u, r"γ r, () x "x x x, x "x x, r "r r, hi" h h, x "x x, u "u u u "u u u, h "h h, r " r r, δ(x )"A x #B h # h, () δ(x )"A x #B h # h #B E x #B u, () h "C x #C x #C x #u, u () for each i",, m, x δ(x )"A x # h # h #B u #E x #E x # E x, () h "C x "x, h "C x, () where δ()"(k#), the triple (A, B, C ) has the special structure as given in (..) of Chen (00). Furthermore, has the following special format: "[ 0 0] with its last row always being identically zero. Next, we partition, i",, m, as follows: ", " () 0

6 B. M. Chen et al. / Automatica (0) } de"ne a new controlled output h h (k#q )! hi " ]hi(k#q!j) [ h (k#q )! [ ]hi(k#q () Then, it is straightforward to verify that y can be expressed as hi "C[ x #D[ u () with C C[ " C E E C E () 0 0 D[ " I 0 I 0, where E E"E E "E E "E E E E E,,. () Let AI"ΓAΓ BI"ΓBΓ, let Σ[ be characterized by (AI, BI, C[, D[ ). It is simple to show that the auxiliary system Σ[ is right invertible of minimum phase with no in"nite zeros. We "rst assume that C "I follow Steps S.}S. of the previous section to obtain a state feedback control law u "FIx #HI r, () where " r r r (k#q )! [ r (k#q )! [ ]r (k#q!j) ]r (k#q () which has the following properties: (i) AI#BIFI is asymptotically stable, (ii) the resulting hi "r. This implies that the actual controlled output h is capable of precisely tracking the given reference r after q steps. Rewriting (), we obtain u"γ u "Γ [FIx #HI r ] "Γ [FIx #HI r #HI r (k#) ##HI r (k#q )] () for some,,,. Let F"Γ FIΓ, H "Γ HI Γ, for j"0,,, m. We have u"fx#h r#h r(k#) ##H r(k#q ). () Next, we proceed to construct a reduced order measurement feedback controller. We follow Steps R.}R. of the previous section to obtain matrices B, B, F, F gain matrices A, B, C D as given in (). Note that the pair (A, C ) is observable (A, C ) has a controllability index k,, k p, it is easy to show that (A, C ) is also observable the controllability index of (A, C ) is given by k!,, k p!. It follows from Theorem.. of Chen (00) that there exists a gain matrix K such that A #K C has all its eigenvalues at the origin (A #K C ) p,0. () We thus choose such a K in constructing gain matrices A, B, C D. The reduced order measurement feedback law is then given by v(k#)"a v#b y# G r(k#j), u"c v#d y# H r(k#j), () where G "(B #K B )H, i"0,,, m. Let x "x!v#k x. It is straightforward to verify that the closed-loop system comprising the given system () the reduced order measurement feedback control law of () can be rewritten as follows: x (k#)"(a #K C )x, () x(k#)"(a#bf)x!bf x # BH r(k#j), () h"(c #D F)x!D F x # D H r(k#j). () Thus, it is easy to see that the closed-loop system is asymptotically stable as A#BF A #K C have eigenvalues inside the unit circle. Clearly, for any initial condition, () implies that x "0 for all k*k p!. Hence, for k*k p!, () () reduce to x(k#)"(a#bf)x# BH r(k#j), h"(c #D F)x# D H r(k#j) () 0

7 B. M. Chen et al. / Automatica (0) } which are precisely the same as the closed-loop dynamics under the state feedback law. If we treat x(k p!) as a new initial condition to (), it will take another q steps for h to precisely track the reference r. Thus, we have h"r for all k*q #k p!. Hence, the (κ d,κ ) almost perfect tracking problem is solved by the control law () with κ d "q κ "q #k p!. Remark. Consider the given plant () which has all properties as stated in Theorem. Then, the (κ d,κ ) almost perfect tracking problem is solvable by a full order measurement feedback controller of the form () with κ d "q κ "q #k p. It can be shown that the required solution is given by v(k#)"a v# BH r(k#j)!ky, u"fv# H r(k#j), () where A "A#BF#KC, K being chosen such that (A#KC ) p "0. Remark. For simplicity, we consider Σ to be a single output system with a relative degree q. Clearly, if the reference r(k#d) is unknown for all d'0, then the full order output feedback controller () with r(k#) being replaced by r will be capable of tracking the reference with a delay of q steps after q #k p initial steps. Similarly, under the same situation, the reduced order output feedback controller () with r(k#) being replaced by r will track the reference with a delay of q steps after q #k p! initial steps.. Conclusions We have studied the problem of robust perfect tracking for discrete-time linear time-invariant multivariable systems. A set of necessary su$cient conditions under which the proposed problem is solvable is obtained, under these conditions, constructive algorithms are given that yield required solutions. We have also proposed in this paper an almost perfect tracking scheme, which can track references precisely after certain initial steps. References Chen, B. M. (00). Robust H control. New York: Springer. Liu, K., Chen, B. M., & Lin, Z. (0). On the problem of robust perfect tracking for linear systems with external disturbances. International Journal of Control,, }. Sannuti, P., & Saberi, A. (). A special coordinate basis of multivariable linear systems*finite in"nite zero structure, squaring down decoupling. International Journal of Control,, }. Ben M. Chen, born on November,, in Fuqing, Fujian, China, received his B.S. degree in mathematics computer science from Amoy University, Xiamen, China, in, M.S. degree in electrical engineering from Gonzaga University, Spokane, Washington, in, Ph.D. degree in electrical computer engineering from Washington State University, Pullman, Washington, in. He was a software engineer from to in South-China Computer Corporation, China, was an assistant professor from to in the Electrical Engineering Department of the State University of New York at Stony Brook. Since, he has been with the Department of Electrical Computer Engineering, National University of Singapore, where he is currently an associate professor. His current research interests are in linear control system theory, control applications, development of web-based virtual laboratories network security systems. He is an author or co-author of "ve monographs, Hard Disk Drive ServoSystems(London: Springer, 0), Robust H Control (London: Springer, 00), H Control Its Applications (London: Springer, ), H Optimal Control (London: Prentice Hall, ), Loop Transfer Recovery: Analysis Design (London: Springer, ), one textbook, Basic Circuit Analysis (Singapore: Prentice Hall, st Ed., ; nd Ed., ). He was an associate editor in } on the Conference Editorial Board of IEEE Control Systems Society. He currently serves as an associate editor of IEEE Transactions on Automatic Control. Zongli Lin was born in Fuqing, Fujian, China on February,. He received his B.S. degree in mathematics computer science from Amoy University, Xiamen, China, in, his Master of Engineering degree in automatic control from Chinese Academy of Space Technology, Beijing, China, in, his Ph.D. degree in electrical computer engineering from Washington State University, Pullman, Washington, in May. From July to July, Dr. Lin worked as a control engineer at Chinese Academy of Space Technology. In January, he joined the Department of Applied Mathematics Statistics, State University of New York at Stony Brook as a visiting assistant professor, where he became an assistant professor in September. Since July, he has been with the Department of Electrical Computer Engineering at University of Virginia, where he is currently an associate professor. His current research interests include nonlinear control, robust control, control of systems with saturating actuators. In these areas he has published several papers. He is also the author of the book, Low Gain Feedback (London:Springer, ) a co-author (with Tingshu Hu) of the recent book, Control Systems with Actuator Saturation: Analysis Design, (Boston:BirkhaK user, 0). A senior member of IEEE, Dr. Lin was an associate editor on the Conference Editorial Board of the IEEE Control Systems Society currently serves as an Associate Editor of IEEE Transactions on Automatic Control. He is also a member of the IEEE Control Systems Society's Technical Committee on Nonlinear Systems Control heads its Working Group on Control with Constraints. He is the recipient of a US O$ce of Naval Research Young Investigator Award. Kexiu Liu received his B.S. degree from Harbin Institute of Technology, China, in, M.S. degree from Shanghai Jiaotong University, China, in. He is now a Ph.D. cidate at the National University of Singapore. He joined Seagate Technology in December 00. His current research "elds include optimal control, robust control, robust perfect tracking hard disk control design. 0