Abstract and Alied Analysis Volume 203 Article ID 97546 5 ages htt://dxdoiorg/055/203/97546 Research Article Controllability of Linear Discrete-Time Systems with Both Delayed States and Delayed Inuts Hong Shi Guangming Xie 23 and Wenguang Luo 4 Deartment of Mathematics and Physics Beijing Institute of Petrochemical Technology Beijing 0267 China 2 State Key Laboratory for Turbulence and Comlex Systems College of Engineering Peking University Beijing 0087 China 3 School of Electrical and Electronics Engineering East China Jiaotong University Nanchang 33003 China 4 School of Electric and Information Engineering Guangxi University of Science and Technology Liuzhou 545006 China Corresondence should be addressed to Guangming Xie; xiegming@kueducn Received 29 November 202; Acceted 3 January 203 Academic Editor: Valery Y Glizer Coyright 203 Hong Shi et al This is an oen access article distributed under the Creative Commons Attribution License which ermits unrestricted use distribution and reroduction in any medium rovided the original work is roerly cited The controllability issues for discrete-time linear systems with delay in state and control are addressed By introducing a new concet the controllability realization index (CRI) the characteristic of controllability is revealed An easily testable necessary and sufficient condition for the controllability of discrete-time linear systems with state and control delay is established Introduction The concet of controllability first given by Kalman in the 960s [] lays a fundamental role in the modern control theory and has close connections with ole assignment structure decomosition quadratic control and so forth [2 3] The various asects of the controllability of linear systems with delay were considered by several authors [4 ] The discrete cases have been considered by Klamka [7] Watanabe [8]and Phat [9] but the mathematical conditions given for investigating the controllability are not suitable for real verification and alication In our recent aer [2] a new concet called controllability realization index (CRI) is roosed which is crucial in determining the controllability of such kind of discrete systems with delays In that aer it is roved that the value of CRI is finite for discrete systems with delays and a general CRI value for lanar discrete systems with delays is given Thusthejudgingconditionofcontrollabilityforlanarcase is established In this aer we will extend our result to the more general case namely discrete systems with any order with time delays both in state and in control This aer is organized as follows In Section 2 some basic definitions and reliminary results are resented Section 3 is the main results An easily testable necessary and sufficient condition for the controllability of discretetime linear systems with state and control delay is established A numerical examle is given in Section 4 Finally the conclusion is rovided in Section 5 2 Problem Formulation and Preliminaries In this aer we consider the discrete-time case the system model is described as follows: x (k) A i x (k i) q B j u(k j) () where x(k) R n is the state u(k) R s is the inut A i R n n and B j R n s are constant matrices and the ositive integers i j are the lengths of the stes of time delays The initial states x( )x( )x(0)and the initial inut u( q) u( q )u( )are given arbitrarily The controllability discussed here refers to the unconstrained controllability or comletely controllability Definition (controllability) The system () is said to be (comletely) controllable if for any initial inut u( q) u( q )u( )anyinitialstatex( )x( )x(0)and
2 Abstract and Alied Analysis any terminal state x f there exist a ositive integer k and inut u(0)u()such that x(k) x f Definition 2 (controllability realization index CRI) For the system () if there exists a ositive integer K such that for any initial inut u( q)u( q)u( )initialstate x( )x( )x(0)andanyterminalstatex f there exists an inut u(0)u(k ) such that x(k) x f then one calls K the controllability realization index (CRI) of the system () Obviously if exists such K is not unique so the one calls the smallest K among them the minimum controllability realization index (MinCRI) Denote by N thenonnegativeintegersetthematrices A A N R n n are said to be linearly deendent on R n n if there exist scalars c c N R not all are zero such that N i c ia i 0Inthefollowingstatementsan{A A N } will be used to denote the sace constructed by the linear combinations of matrices {A A N } 3 Main Results 3 Delay in State In this section we first investigate the controllability of the systems only with delay in state x (k) A i x (k i) Bu(k) (2) Now we introduce a matrix sequence {G k } k0 Rn n as follows: I { A G k i G i { A i G i { if k0 if k if k2 Lemma 3 The general solution of the system (2) is given by x (k) Ψ(kx( )x(0)) k (3) G k i Bu (i) k N where Ψ(kx( )x(0))is the art of the solution with zero inut Proof See Aendix A Lemma 4 The matrix sequence {G k } k0 given by (3) satisfies (4) san {G 0 G k }san {G 0 G n() } (5) Proof See Aendix B Theorem 5 The system (2) is controllable if and only if rank [G 0 B G n() B] n Proof (Necessity) If the system is controllable then we know that san {G 0 G k } R n ThusbyLemma 4wehave rank [G 0 B G n() B] n (Sufficiency) By Lemma3 we have x(n())ψ(n() x( )x(0)) n() G n() i Bu (i) Since rank [G 0 B G n() B] n foranyinitialstate x( )x(0) and any terminal state x(n( )) wecan select aroriate inuts u(0)u(n() )such that the equation x(n( )) x f wherex(n( )) is given by the above equation and x f is arbitrary Thus the system is controllable Corollary 6 n( ) is a CRI of the system (2) Proof It is directly followed from Theorem 5 Remark 7 This work has imroved the result in [2] When thesystemissecondorderthatisn2werove2 2 to be a CRI value which differs from the CRI value 24 in [2] The difference lies in that the CRI of a system is not unique For ractical alications obviously the less the better 32 Delays in Both State and Inut Now we investigate the controllability of the system () We only consider the case when qforthecaseq the discussion is similar (Without loss of generality we assume that > q andlet B j 0 jqq2 then we come back to the q case) Lemma 8 The solution of the system () can be exressed as x (k) Φ(kx( )x(0) u( )u( )) where k H k i u (i) k N G 0 B 0 k { G H k k i B i { G k i B i { if k0 if k if k; and Φ(kx( )x(0)u( )u( ))is the art of the solution corresonding only to the initial state and initial inut Proof The roof is similar to that of Lemma 3 Lemma 9 The matrix sequence {H k } k0 given by (8) satisfies san {H 0 H k }san {H 0 H n() } (9) (6) (7) (8)
Abstract and Alied Analysis 3 Proof See Aendix C Theorem 0 The system () is controllable if and only if rank [H 0 H n() ]n Proof The roof is similar to that of Theorem 5 Corollary n( ) is a CRI of the system () Proof It is directly followed from Theorem 0 Remark 2 This corollary rovides a comlete and verifiable method to testify the controllability of a general discretetime system with delay in state or in control or both Aroximately the comutation work for each H k is O(n 3 ) andtheentiretestingworktakeso( 2 n 4 ) 4 Examle In this section we resent a numeric examle Aendices A Proof of Lemma 3 Proof Now we rove that (4)holds For k0wehave x () For kwehave x (2) A i x ( i) Bu(0) Ψ(0x( )x(0))g 0 Bu (0) i A i x ( i) Bu() A i x ( i) A 0 x () Bu() (A) Examle 3 Consider the system () withq2n 3 and 0 0 0 A 0 [ 0 0 0] A [ 0 0] A 2 [ 0 ] [ 0 0 0] 0 0 B 0 B [ ] B 2 [ ] (0) By simle calculation we get G 0 [ 0 0] G [ 0 0 0] G 2 [ 0 0] [ 0 0 0] 2 0 0 3 0 0 G 3 [ 0 ] G 4 [ 0 0] H 0 H H 2 2 4 H 3 [ ] H 4 [ ] [ ] Thus by Theorem 0 the system is controllable 5 Conclusion () In this aer we have investigated the controllability of discrete-time linear systems with time delays Necessary and sufficient conditions have been established for discrete-time linear systems with state delay or both state and control delays The roosed conditions are suitable for real verification and can be efficiently comuted i A i x ( i) A 0 Ψ(0x( )x(0)) A 0 G 0 Bu (0) Bu() Ψ(x( )x(0))g Bu (0) G 0 Bu () (A2) For k wehave x (k) ik ik A i x (k i) Bu(k) A i x (k i) A i x (k i) ( ik A i x (k i) Bu(k) A i (Ψ(x( )x(0)) A i x (k i) G j Bu (j)) G 0 Bu (k) A i Ψ(x( )x(0))) A i ( G j Bu (j)) G 0 Bu (k)
4 Abstract and Alied Analysis Ψ(kx( )x(0)) G 0 Bu (k) Ψ(kx( )x(0)) For k>wehave x (k) ( A i x (k i) Bu(k) k j A i (Ψ(x( )x(0)) A i G j Bu (j) G k j Bu (j) G 0 Bu (k) G j Bu (j)) G 0 Bu (k) A i Ψ(x( )x(0))) A i ( Ψ(kx( )x(0)) j jk G j Bu (j)) G 0 Bu (k) A i G j Bu (j) G 0 Bu (k) A i G j Bu (j) G 0 Bu (k) Ψ(kx( )x(0)) B Proof of Lemma 4 (A3) G k j Bu (j) G 0 Bu (k) Proof We introduce a new matrix sequence {W k } k R n() n given by (A4) W k G [ ] (B) [ G k ] It is easy to verify that W k AW k k > (B2) G k where A 0 A A I 0 0 A [ (B3) ] [ 0 I 0 ] By the well-known Hamilton-Caylay Theorem for any k n( ) wehave It follows that Hence we have W k san {W W n() } G k san {G 0 G n() } san {G 0 G k }san {G 0 G n() } C Proof of Lemma 9 (B4) (B5) (B6) Proof Consider the matrix sequence {W k } k Rn() n given by (B) it is easy to verify that H k W T k B 0 By the roof of Lemma 4wehave This imlies that [ ] k (C) [ B ] W k AW k k (C2) san {H H k }san {H H n() } (C3) Hence we have san {H 0 H k }san {H 0 H n() } (C4) Acknowledgments The authors would like to thank the Associate Editor Professor Valery Y Glizer and the anonymous reviewer for their valuable comments and suggestions which significantly contributed to imroving the quality of the ublication This work is suorted by the National Natural Science Foundation (NNSF) of China (60774089 0972003) the Foundation Grant of Guangxi Key Laboratory of Automobile Comonents and Vehicle Technology (3-A-03-0) and the Oening Project of Guangxi Key Laboratory of Automobile Comonents and Vehicle Technology (202KFZD03) It is alsosuortedbythebeijingcityboardofeducationscience and Technology Program: modeling analysis and control of swarming behavior of multile dynamic agent systems
Abstract and Alied Analysis 5 References [] R E Kalman On the general theory of control systems in Proceedings of the st IFAC Congressvol48 492Moscow Russia 960 [2] T Kailath Linear Systems Prentice-Hall Englewood Cliffs NJ USA 980 [3] E D Sontag Mathematical Control Theory: Deterministic Finite Dimensional Systemsvol6SringerNewYorkNYUSA990 [4] H W Sorenson Controllability and observability of linear stochastic time-discrete control systems in Advances in Control Systems vol 6 95 58 Academic Press New York NY USA 968 [5] J Klamka Controllability of delayed dynamical systems in Proceedings of 4th IFAC congressvol4485 490 [6] M Fliess Some new interretations of controllability and their ractical imlications Annual Reviews in Control vol23 97 206 999 [7] J Klamka Relative and absolute controllability of discrete systems with delays in control International Control vol26no65 74977 [8] K Watanabe Further study of sectral controllability of systems with multile commensurate delays in state variables International Controlvol39no3497 505984 [9] V N Phat Controllability of discrete-time systems with multile delays on controls and states International Controlvol49no5645 654989 [0] V Y Glizer Euclidean sace controllability of singularly erturbed linear systems with state delay Systems & Control Lettersvol43no38 9200 [] J Klamka Absolute controllability of linear systems with timevariable delays in control International Control vol 26no57 63977 [2] H Shi G Xie and W Luo Controllability analysis of linear discrete-time systems with time-delay in state Abstract and Alied Analysis vol 202 Article ID 490903 ages 202
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