Necessary and Sufficient Stability Condition of Discrete State Delay Systems

Transcription

1 nternational Journal ecessary of Control, and Automation, Sufficient and Stability Systems, Condition vol, of no Discrete 4, pp State , Delay December Systems ecessary and Sufficient Stability Condition of Discrete State Delay Systems Young Soo Suh, Young Shick Ro, Hee Jun Kang, and Hong Hee Lee Abstract: A new method to solve a Lyapunov equation for a discrete delay system is proposed Using this method, a Lyapunov equation can be solved from a simple linear equation and -th power of a constant matrix, where is the state delay Combining a Lyapunov equation and frequency domain stability, a new stability condition is proposed for a discrete state delay system whose state delay is not exactly known but only known to lie in a certain interval Keywords: Time delay systems, stability, Lyapunov function, discrete systems TRODUCTO State delays are frequently encountered in control problems of many physical systems n particular, continuous state delay systems have been received a lot of attentions and many stability results have been proposed (see [-3] and its references) On the other hand, there has been less attention to the following discrete state delay system x( k + ) = A x( k) + A x( k ), () where x R n is a state The reason of less attention is not surprising since system () can be transformed into an equivalent non-delayed system ntroducing an augmented new state z( k) as follows: xk ( ) ( ) n xk R n zk ( ), R xk ( ) we can obtain an equivalent non-delayed system where z( k + ) = A z( k ), () Manuscript received July 7, 004; revised October 6, 004; accepted October 9, 004 Recommended by Editorial Board member Seung-Hi Lee under the direction of Editor-in- Chief Myung Jin Chung This work was supported by 004 Research Fund of University of Ulsan Young Soo Suh, Young Shick Ro, Hee Jun Kang, and Hong Hee Lee are with Department of Electrical Engineering, University of Ulsan, San 9, Muger-dong, am-gu, Ulsan , Korea ( s: suh@ieeeorg, ysro@uouulsan ackr, {hjkang, hhlee}@mailulsanackr) A A A ow stability of () can be investigated by simply checking stability of an ordinary non-delayed system () However, there are two important cases that stability check of the equivalent non-delayed system is not adequate for stability check of () The first case is when the state delay is very large, and the second case is the state delay is not exactly known but only known to lie in a certain interval f is very large, then the A matrix of () is very large n ( + ) n ( + ) (note A R ) Hence stability check of A is numerically demanding and sometimes an unstable task f the state delay is only known to lie in a certain interval (for example, [0, max ] ), then stability should be checked for each [0, max ], which is also a numerically demanding task, in particular for large max To cope with these two cases, we propose a new non-conservative stability condition of () by carefully investigating a Lyapunov equation for () n Section, it is shown that a solution of a Lyapunov equation for () can be transformed into a simple linear equation, where the only term depending on is -th power of a constant matrix Hence, even for large, the computation is simple n Section 3, combining the constant matrix with frequency domain interpretations, we propose a stability condition for (), where the state delay is only known to lie in a certain interval n Section 4, a numerical example is given to illustrate the results of this paper n Section 5, conclusion is provided

2 50 Young Soo Suh, Young Shick Ro, Hee Jun Kang, and Hong Hee Lee The works which are most related to ours are [4-9] n [4-7], delay independent stability conditions are considered The conditions are, however, conservative when the state delay is known to lie in a certain interval n [8], a robust stability problem is considered for a exactly known delay case n [9], a delay dependent stability condition is derived based on Razumikhin-type theorem n [0,], similar delay dependent stability condition is used to derive a controller All these delay dependent stability conditions are sufficient conditions, while in this paper the necessary and sufficient condition is proposed otation is standard For a matrix M C n n given by m m n M =, mn mnn csm is defined by n n nn n csm [ m m m m ]' C Symbol denotes the kronecker product LYAPUOV FUCTO The Lyapunov function for () is defined by V( z( k)) z( k)' Pz( k ), n ( + ) n ( + ) where symmetric matrix P R is partitioned compatible to the partition of z( k ) and labeled as follows: P00 P0( ) P0() 0( ) (, ) (,) P P P P = P0( ) P(, ) P (,) t is standard from the following V( zk ( + )) V( zk ( )) = zk ( )'( A ' PA Pzk ) ( ) that the system () is stable if and only if there exists P = P ' > 0 satisfying A ' PA P < 0 From the special structure of z( k ) ( z( k ) is stable if and only if x( k ) is stable), the condition A ' PA P < 0 can be modified as in the following lemma Lemma : System () is stable if and only if there exists P = P ' > 0 with P 00 > 0 satisfying Q 0 A ' PA P + = (3) n n for some Q= Q' R > 0 Proof: ote that (3) is not a standard Lyapunov form Standard Lyapunov theorem [] states that () is stable if and only if there exists P = P ' > 0 satisfying A ' PA P + Q = 0 (4) ( ) ( ) for some + + = n n Q Q' R > 0 We show that P = P ' 0 with P 00 > 0 satisfying (3) exists if and only if P = P ' > 0 satisfying (4) exists To do this, we construct P from P Let ˆ n ( + ) n ( + ) P R be defined by 0 ε Pˆ( ε ) ( ) ε ε > 0 where all unspecified entries are 0 Then, A ' ˆ P( ε) A P( ε) ε ε = Let then, n ( + ) n ( + ) P R be defined by, ε P( ε) = P+ Pˆ ( ε ), (5) A ' ( ) P ε A P( ε) Q+ ε = (6) ((4) (3) part) f P = P ' 0 with P 00 > 0 satisfying (3) exists, then P is positive definite and satisfies (6) f we choose ε > 0 such that Q+ ε <0, then (4) is satisfied ((4) (3) part) Suppose P = P ' > 0 satisfying (6) with Q+ ε < 0 and ε > 0 exists Then by

3 ecessary and Sufficient Stability Condition of Discrete State Delay Systems 503 partitioning P as in (5), we can show that P satisfies (3) Furthemore, since P > 0, P 00 ((,) element of P ) is positive definite Remaining to show is P 0 Since P > 0, for any n ( + ) x 0 R, the following is satisfied : x ' P ( ε) x = x ' Px + x ' Pˆ ( ε) x = xpx ' + f( x ) ε > 0, (7) where f( x ) R 0 Since (7) is satisfied for all ε satisfying 0 < ε Q, ε independent x ' Px term should be greater or equal to 0 Thus P 0 Matrix P has 3 variables: P 00, P0() i = P0 ()' i and P(, i j) = P ( j,)' i The following lemma simplifies the expression of P using one variable Xi () Lemma : The solution P satisfying (3) is given by P00 = X(0), P0() i = X() i A, A' Xi ( j)' A,0 j i, P(, i j) = A' X( j i) A,0 i j, where X( k),0 k is given by A ' X(0) A + A X( )' A + A ' X( ) A + A' X(0) A X(0) + Q= 0, X(0) = X(0)', X( k + ) = A0' X( k) + AX( k)', 0 k 0 0 (8) (9) Proof: Straightforward calculation of (3) yields (8) and (9) Since the matrix difference equation (third equation) in (9) is not in an easy form to solve, the matrix difference equation is transformed into a kind of two point boundary value problem in the next lemma Throughout the paper, A 0 is assumed to be nonsingular: we note that most discrete systems have nonsingular A 0 matrices Lemma 3: The matrix difference equation in (9) is equivalent to the following: csx ( k + ) csx ( k ) = A B csx ( k), A BA A ( BB) csx ( k) (0) where A ( A 0 '), B ( A ') T, and Row vector n n T [ T T T ], T R () Tl, l n is defined by T( i ) n j e( j ) n i, i, j n, + + n where el R, l n is a row vector whose l - th element is and all other elements are 0 Proof: ote the following properties of column string cs( ABC) = ( C ' A) csb, () csa' = TcsA, n n where ABC,, R Using these properties, we obtain from the forward difference equation in (9): l csx ( k + ) = AcsX ( k) + BcsX ( k ) (3) Thus from the above equation, we obtain the backward equation (note that A is nonsingular if and only A is nonsingular) 0 ' csx ( k) = A csx ( k + ) A BcsX ( k ) Letting k k in the above equation, we obtain csx ( k ) = A BAcsX ( k) + (4) A ( BB) csx( k) Combining (3) and (4), we obtain (0) For later reference, we define two matrices H (see (0)) and J : A B 0 H, J A BA A ( BB) 0 (5) ow return to the problem of solving (9) for some Q The matrix Lyapunov equation (first equation) and the matrix difference equation (third equation) are coupled in (9) To solve the Lyapunov equation, it is necessary to obtain an X (0) and X( ) pair satisfying the matrix difference equation, or equivalently (0) The constraint imposed on any X (0) and X( ) pair satisfying (0) can be stated using the boundary condition: csx ( ) csx (0) = (0) H ( ) csx csx,

4 504 Young Soo Suh, Young Shick Ro, Hee Jun Kang, and Hong Hee Lee and the above equation can be expressed using J (see (5)) as follows: csx (0) ( JH ) = 0 ( ) (6) csx Does an X (0) and X( ) pair satisfying (6) always exist? For example, if dim ull( JH ) = 0, then only [( csx (0))'( csx ( ))']' = 0 can be the solution to (6) The next lemma shows that dim ull( JH ) = n and thus there exists a nontrivial X (0) and X( ) pair satisfying (6) Lemma 4: The following is satisfied dim ull( JH ) = n (7) The proof of Lemma 4 needs the following lemma Lemma 5: f z is an eigenvalue of H, then z is also an eigenvalue of H Proof: f z is an eigenvalue of H, there exists a [ xy ' ']' 0 satisfying x x H = z y y The upper block of the above equation is Ax+ By = zx (8) The lower block is A BAx + A ( BB) y = zy (9) nserting (8) into (9), we obtain ow consider z y = Bx+ Ay (0) x Bx + Ay H = y A BAy + A ( BB) y From (0), the upper block of the above equation is z y Using (8) and (0), the lower block is given by A BAy + A ( BB) y = z x Hence we obtain x x H = z y y () From the proof of Lemma 5, we note that eigenvalues and eigenvectors of H (for simplicity, all eigenvalues of H are assumed to be simple and nonzero) can be expressed as X Y X Y Σ H =, () Y X Y X Σ Σ C n n where diagonal elements are eigenvalues of H Proof of Lemma 4: First we will show that is a diagonal matrix whose dim ull( JH ) n (3) Let v is defined by X Y v α, Y X Σ n where α C, then v ull( JH ) from () Hence we have dim ull( JH ) X Y dim Range( ) = n Y X Σ Similarly, we can show that dim ull( JH ) n (4) From (3) and (4), we can conclude that only two eigenvalues ± and thus dim ull( JH ) = n dimull( JH ) n JH has From (3), we obtain(7) From (7), the singular value decomposition of ( JH ) is given by Σ 0 * ( JH ) = U 0 0 V, where U, V are unitary matrices, and Σ R n n is a diagonal matrix whose diagonal elements are nonzero singular values of ( JH ) Thus the constraint on an X (0) and X( ) pair satisfying (0) is given by

5 ecessary and Sufficient Stability Condition of Discrete State Delay Systems 505 where csx (0) [ R R] = 0 ( ), (5) csx [ R R ] Σ [ 0] V Using (5), the coupled equations (9) can be reduced to a simple linear equation Lemma 6: An X (0) and X( ) pair satisfying (9) can be computed by the following equation: ( A0' A0') + ( A' A') R ( A0' A') T + ( A' A0') csx (0) R csx ( ) csq =, 0 * (6) where R and R are from (5) Proof: The upper equation of (6) is from the first equation of (9) and the lower equation is from (5) Remark : Once X (0) and X( ) are obtained, Xi (), i can be computed easily from (0) For example, X () and X( ) are computed by csx () csx (0) = ( ) H ( ) csx csx Hence the Lyapunov equation (3) can be solved from a simple linear equation (6) and up to -th power of the constant matrix H 3 STABLTY CODTO n this section, we show that eigenvalues and eigenvectors of H are closely connected with frequency domain stability of () Based on this observation, we propose a new stability condition which ensure stability of () for all [0, max ] System () is stable if and only if the characteristic equation + det( z A z A ) = 0 (7) has all its roots inside the unit circle Since det M = det M ', () is stable if and only if the characteristic equation + det( z A ' z A ') = 0 (8) has all its roots inside the unit circle We will provide a simple way to check whether all roots of (8) lie inside the unit circle Define r+ r f (,) zr det( z A' z A'), W() r z C f(,) z r = 0, W (, r ε) z C z z < ε, zˆ W( r) B { } { ˆ (9) n the next lemma, we show that if r is changed to r + r and r is small, each element of W( r + r ) is near W( r ) Lemma 7: For any ε > 0, there is δ > 0 such that for every z W( r + r ), the following is satisfied: z W B (, r ε ) whenever r < δ (30) Proof: ote that f ( zr, ) can be written as follows: (,) = n () kr f zr a zz k k= 0 for some polynomials ak ( z ) and f ( zr, ) is an analytic function of z Thus f ( zr, ) is a continuous function of z f we define then mr (, ε ) inf f(,) zr, (3) z C W B ( r, ε ) z W B (, r ε ) whenever f ( zr, ) < mrε (, ) (3) ow suppose z W( r + r ) Since f ( zr, ) is an analytic function of r, for any mrε (, ) > 0, there is δ > 0 such that f ( zr, + r) f( zr, ) < mrε (, ) whenever r < δ (33) Since f( z, r + r ) = 0, we have f ( zr, ) < mrε (, ) whenever r < δ From (3), this implies (30) Lemma 7 states that small change of r guarantees small change of the location of W( r ) From this observation, we obtain the following lemma Lemma 8: f () is stable for = 0 and W( r ) does not have an element on the unit circle for all real number r [0, ], then () is stable for max [0, max ] Proof: Let ρ ( r) be defined by ρ () r sup z z W( r) }

6 506 Young Soo Suh, Young Shick Ro, Hee Jun Kang, and Hong Hee Lee From Lemma 7, for each ε > 0, there is δ > 0 such that ρ( r + r) < ρ( r) + ε whenever r < δ Thus if ρ ( r) < 0 and z W( r ) is not on the unit circle for 0 r max, ρ ( r) < for 0 r max This means that all roots of (8) lies inside the unit circle for 0 max The next lemma shows that unit circle element of W() r can be checked from eigenvalues of H Lemma 9: f W( r ) has a unit circle element, then the root is an eigenvalue of H Proof: f f( z, r ) = 0 has a root z = e, w R on the unit circle, there is a nonzero x C n satisfying ( e A ' A ' e ) x= 0 (34) r Let α C n be defined by Let α α = = xe α n n r v C be defined by u v =, u where u is defined by αx u α nx (35) ote v 0 from the construction We will prove this lemma by showing that ( e H) v = 0 From the definition of H, we obtain p ( e H) v q ( e A) u Bu = A BAu + ( e A ( BB)) u We will show that p = q = 0 The i -th block of p is given by i = ( 0') αi ' ααi αi r p e A x A e r = ( e A ' A ' e ) x= 0 (36) The lower block vector q is given by q = A BAu + ( e A ( BB)) u = A { BAu+ ( e A + BB) u} Using Au = e u Bu (from the fact p = 0 obtain: ), we q = A { e Bu BBu + ( e A + BB) u} = A e { Bu+ ( A e ) u} = A e { p} = 0 The last equation is from (36) Thus we have proved ( e H) v = 0 Using Lemma (9), we can compute max such that () is stable for all [0, max ] i Theorem : Let e, wi R 0 be a unit circle eigenvalue of H and v i be the corresponding eigenvector Let r R 0 be defined by i βk m(ln( )) / wi, wi 0, ri γ k 0, wi = 0, (37) where βk is k -th element of v i, γ k is ( n + k) -th element of v i k n can be chosen arbitrarily as long as k -th element of v i is nonzero (i) f () is stable for = 0 and max is the greatest integer not larger than min r i, then () is stable for all [0, max ] (ii) f () is stable for = 0 and H does not have a unit circle eigenvalue, then () is stable for all 0 Proof: Suppose k = in (37) From (35), we obtain β = ii r e γ and r = r i is a unit circle root of f( z, r ) = 0 From Lemma 9, W( r ) does not have an element on the unit circle for r min r i Thus (8) does not have a root on the unit circle for min r i Hence from Lemma 8, () is stable for all [0, max ] The proof of (ii) is immediate from Lemma 8 and Lemma 9 Remark : The stability condition (ii) is called delay independent stability condition [5,6]

7 ecessary and Sufficient Stability Condition of Discrete State Delay Systems UMERCAL EXAMPLE Consider the following system xk ( + ) = ( ) 0 07 xk x( k ) 0 04 (38) The system is stable for = 0 Eigenvalues of H are given by {099, 030, 0686, 097 ± j 0346, 4650, 3305, 3456} ote that there exists a unit circle root j = e, w 0, and w = 0368 The corresponding eigenvector v is given by v = [ 00784, 09, 0687, 0655, 0089,0988, 0099, 0678]' From (37), we obtain r = 0483, and thus max = 0 Hence we can conclude that (38) is stable for [0,0] n fact, by checking A, we can verify that (38) is stable for 0 and unstable = For comparison, the stability is also checked with the stability condition in [9]: the system is determined to be stable for [0,7] Also, to check delay independent stability condition, stability of (38) is checked with 0 0 A = α 0 04 t turns out that the system is delay independently stable (ie, H matrix does not have unit circle eigenvalues) if 0 α 0834 For comparison, the delay independent stability is also checked with the stability condition in [4]: the system is determined to be delay independently stable if 0 α 0834 To investigate how two delay independent stability conditions are related is an interesting further research issue 4 COCLUSO The main contribution of this paper is as follows First, we have proposed an easy method to solve a Lyapunov equation for a state delay system Using the proposed method, we can easily solve a Lyapunov equation even for large ; thus the proposed method can be useful when a Lyapunov equation for large should be solved, for example gramian computation in the balanced model reduction of a large delay state delay system Secondly, we have proposed a new stability condition based on the relationship between frequency domain stability and a constant matrix that appears in a Lyapunov equation The proposed stability condition ensures stability of a discrete state delay system whose state delay is not exactly known but known to lie in a certain interval REFERECES [] S iculescu, E Verriest, L Dugard, and J M Dion, Stability of linear systems with delayed state: A guided tour, Proc of FAC nternational Workshop on Linear Time Delay Systems, pp 37-44, 998 [] M Ekanayake, K Premaratne, C Douligeris, and P Bauer, Stability of discrete-time systems with time-varying delays, Proc of American Control Conference, pp , 00 [3] D L Debeljkovic and M Aleksandric, Lyapunov and non-lyapunov stability of linear discrete time delay systems, Proc of American Control Conference, pp , 003 [4] E Verriest and A F vanov, Robust stability of delay-difference equations, Proc of the 34th Conference on Decision and Control, pp , 995 [5] J W Wu and K S Hong, Delay-independent exponential stability criteria for time-varying discrete delay systems, EEE Trans Automat Contr, vol 38, no 4, pp 8-84, 994 [6] H Trinh and M Aldeen, D-stability analysis of discrete-delay perturbed systems, nt J Contr, vol 6, no, pp , 995 [7] S H Song and J K Kim, H control of discrete-time linear systems with norm-bounded uncertainties and time delay in state, Automatica, vol 34, no, pp 37-39, 998 [8] H Trinh and M Aldeen, Robust stability of singularly perturbed discrete-delay systems, EEE Trans Automat Contr, vol 40, no 9, pp 60-63, 995 [9] B Lee and J G Lee, Delay-dependent stability criteria for discrete-time delay systems, Proc of American Control Conference, pp 39-30, 999 [0] K T Kim, S H Cho, K H Bang, and H B Park, H control for discrete-time linear systems with time-varying delays in state, Proc of 7th Annual Conference of the ndustrial Electronics Society, pp 707-7, 00 [] W H Chen, Z H Guan, and X Lu, Delaydependent guaranteed cost control for uncertain discrete-time systems with delay, EE Proceedings, vol 50, no 4, pp 4-46, 003

8 508 Young Soo Suh, Young Shick Ro, Hee Jun Kang, and Hong Hee Lee [] M Hahn, Theory and Application of Lyapunov's Direct Method, J: Prentice Hall, 963 Young Soo Suh received the BS and MS degrees in Department of Control and nstrumentation from Seoul ational University, in 990 and 99, respectively He received the PhD degree in Mathematical Engineering from the Tokyo university in 997 He is an associate professor of Department of Electrical Engineering, University of Ulsan His research interests include time delay systems and networked control systems Young Shick Ro received the BS, MS, and PhD degrees in Electrical Engineering from Yonsei University, in 98, 983, and 987, respectively He is a professor of Department of Electrical engineering, University of Ulsan His research interests include robotics and machine vision Hee Jun Kang received the BS and MS degrees in Mechanical engineering from Seoul ational University, in 985 and 987, respectively He received the PhD degree in Mechanical Engineering from university of Texas at Austin in 99 He is a professor of Department of Electrical Engineering, University of Ulsan His research interests include robot control & applications, haptics and visual servoing industrial network Hong Hee Lee received the BS, MS, and PhD degrees in Electrical Engineering from Seoul ational University in 980, 98, and 990, respectively He is a professor of Department of Electrical Engineering, University of Ulsan His research interests include power electronics and