Observer-based stabilization of switching linear systems

Transcription

1 Available online at wwwsciencedirectcom Automatica 9 (2) wwwelseviercom/locate/automatica Brief paper Observer-based stabilization of switching linear systems ZG Li a, CY Wen b, YC Soh b a Laboratories for Information Technology, Agency for Science, Technology and Research, 2 Heng Mui Keng Terrace, Singapore 96, Singapore b School of Electrical and Electronic Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 69798, Singapore Received 9 August 998 received in revised form 29 August 22 accepted October 22 Abstract In this paper, we present a deep pole assignment method to study the observer-based stabilization of switching linear systems where the dynamics of each mode are known a priori but the switching times of modes are arbitrary The design can be used for both nite and innite switched linear systems We emphasize our paper on the case where the switchings of the observer and controller do not coincide with those of the system? 22 Elsevier Science Ltd All rights reserved Keywords: Switching linear system Observer Controller Pole assignment Introduction A switching linear system is one where the system dynamics is linear, but time varying, and switches among a nite number of modes (Fu & Barmish, 986 Hilhorst, Amerongen, Lohnberg, & Tulleken, 994 Ezzine & Haddad, 989) In order to obtain fast and accurate responses of these processes, appropriate controllers and observers should be designed and stored for the dierent modes At any moment, the controller and the observer corresponding to the right mode should be used These kinds of systems is called switching linear systems and it can be used to model synchronously switching linear systems, networks with periodically varying switches and systems subject to failures (Ezzine & Haddad, 989) The stability analysis and stabilization of switching systems have been studied by a number of researchers Michel and Hu (998) have considered the stability of switched systems via a comparison theory, which is also very useful for the stability analysis of other discontinuous systems Ezzine and Haddad (989) studied the problems of controllability, observability and stability of periodic switched This paper was not presented at any IFAC meeting This paper was recommended for publication in revised form by Associate Editor Hassan Khalil under the direction of Editor Tamer Basar Corresponding author Tel: fax: addresses: ezgli@lita-staredusg (ZG Li), ecywen@ntuedusg (CY Wen), eycsoh@ntuedusg (YC Soh) linear systems Wicks, Peleties, and De-carlo (998) designed a switching law for the stabilization of a class of switched systems, which is an NP-hard problem (Blondel & Tsitsiklis, 997) Fu and Barmish (986) have studied the stabilization of the nite switching linear systems successfully However, the method in Fu and Barmish (986) cannot be used to consider the stabilization of innite switching linear systems It should also be noted that some basic problems have been outlined in Liberzon and Morse (999) However, the observer-based stabilization of switching linear systems with innite switching times has not been considered yet, especially in the case where the switchings of the observer and controller do not coincide exactly with those of the system In this paper, we overcome the potential problem by introducing a deep pole assignment method The pole assignment method is used to develop an observer and a controller for each frozen system These observers and controllers form a switching observer and a switching controller for the whole switching control systems where the number of switchings involved can be innite or nite, and these observers and controllers are stored Because the frozen system cannot be known at any moment, we need to identify the switching instances of the system and the next frozen system This can be done by checking the observer error Once the next frozen system is known, the controller and the observer should be switched to the ones corresponding to the next frozen system It is well known that the poles of a controllable (or observable) frozen system can be assigned to any 5-98//$ - see front matter? 22 Elsevier Science Ltd All rights reserved PII: S5-98(2)267-4

2 58 ZG Li et al / Automatica 9(2) given values (Chen, 984) If the real parts of poles are negative enough, then the Lyapunov function V (X (t))= X (t) 2 will decrease along each frozen system with any given decay rate even in the case where there are overlaps between the switchings of the system and the switchings of the controllers and the observers This ensures that V (X (t)) decreases along the whole system X (t) =A(m(t))X (t) with any given decay rate Thus the system X (t)=a(m(t))x (t) is stable in the sense of Lyapunov We require that the interval between any two successive switchings has a lower bound T as in Narendra and Balakrishnan (99) However, our T is only required to be known and it can be arbitrarily small The controller uses the estimate of the state by the observer for feedback rather than the state of the system The controller stabilizes the system in the sense of Lyapunov This is very important because with Lyapunov stability, we can get a handle on the types of overshoot behavior Further, the switchings of the controller and the observer do not need to coincide exactly with the switchings of the system The result of this paper shows that for any desired decay rate, the closed-loop switching system would yield a stable system with the required decay rate The rest of the paper is organized as follows In the following section, problem formulation is given The main result is derived in Section In Section 4, a numerical example is given to illustrate the application of the main results Concluding remarks are given in Section 5 2 Problem formulation In this paper, we shall design an observer-based controller for the stabilization of the following switching single input single output linear system: X (t)=a(m(t))x (t)+b(m(t))u(t) () y(t)=c(m(t))x (t) where X (t) is the system state vector of dimension r, u(t)is the control input, y(t) is the output, and m(t) is the mode index which is a piecewise constant taking values in the nite index set M = 2:::n} We shall refer the ith mode of the system as continuous variable dynamic system i (CVDS i) The switching laws are dened by one of the following two methods: (I) The mode will automatically switch from mode i to mode j, if the duration time of mode i is i (Ezzine & Haddad, 989) (II) The mode will switch from mode i to mode j, ifthe state of mode i is in a given switching conditional set S ij (X (t)), which is of the following form (Pettersson & Lennartson, 996): S ij (X (t)) = X (t) h ij (X (t))=}: (2) We let tk s denote the kth switching instance of the system Further, we make the following assumption about () Assumption (A(i)c(i)) is observable () (A(i)b(i)) is controllable (4) T = inf k ts k+ tk} s (5) where T is known but can be arbitrarily small We propose the following r-dimensional observer for mode i: X (t)=a(i)x (t)+b(i)u(t)+ L(i)(y(t) y (t)) X (t )=X y (t)=c(i)x (t): (6) Based on the estimate X (t), we develop a controller of the form u(t)= K(i)X (t): (7) The requirement of the design is that the closed-loop switching linear system () satises a prescribed decay rate Thus, the observer-based stabilization problem can be formulated as follows: Consider the switching system () with switching law (I) or (II), and satises Assumption Given any decay rate, design an r-dimensional observer of the form (6), a feedback controller of the form (7) and the switching laws of the controller and the observer, such that the closed-loop system satises lim t e t X (t) =: The solution to this problem is composed of two steps: Step : Design an observer and a controller for each subsystem Step 2: Dene a switching law for these observers and controllers Generally, the controller corresponding to the active subsystem should be used However, we cannot know the initial subsystem and the subsequent subsystems in advance Thus, we need to impose some small delay on the switchings of the controllers so that we can identify the initial subsystems and the subsequent active subsystems Then, the controller corresponding to the active subsystem is switched into action Note that there will be overshoot in the interval before the right observer and controller are activated Thus, we need to present a method to constrain the overshoot and to ensure sucient decay of the overshoot during the interval when the observer and controller corresponding to the active subsystem are used This is possible because the poles of each observable and controllable subsystem can be assigned arbitrarily The whole process will be shown in detail in the next section

3 ZG Li et al / Automatica 9(2) Main results In this section, we shall rst present some preliminaries to illustrate the process of pole assignment for the observer and the controller for each subsystem Lemma (Wilkinson, 965) For any given degenerate rate and ::: r with i j when i j, let + + r Q( ::: r )= ( + ) 2 ( + r ) 2 ( + ) r ( + r ) r f(s r)=(s ) (s r ) = s r + d r s r + + d s + d E( ::: r )= d d d 2 d r D( ::: r )= : + r Then Q ( ::: r )E( ::: r )Q( ::: r ) = D( ::: r ): (8) Lemma 2 Let g i (s)= si A(i) =s r +a r (i)s r + +a (i)s+a (i) A r T (i)b(i) P(i)= A(i)b(i) b(i) a r (i) a (i) a 2 (i) a (i) a r (i) a r (i) (i)= a (i) a 2 (i) a (i) a r (i) c(i)a r (i) : c(i)a(i) c(i) If (A(i)b(i)) is controllable and (A(i)c(i)) is observable, then there exists K(i) and L(i) given by K(i)=[d c a (i):::d c r a r (i)]p (i) (9) L(i)= (i)[d o a (i):::d o r a r (i)] () such that Ã(i)=P(i)Q( c ::: c r)d( c ::: c r) Q ( c ::: c r)p (i) () Â(i)= (i)(q T ) ( o ::: o r )D( o ::: o r ) Q T ( ::: o r o )(i) (2) where Ã(i) =A(i) b(i)k(i) Â(i) =A(i) L(i)c(i): Proof It is straightforward by using Lemma and the pole assignment method (Chen, 984) Lemma Suppose that (c(i)a(i)) is observable, then for any L R r,(c(i)a(i) Lc(i)) is observable For any desired decay rate and observer state X (t), we denote e(t)=x (t) X (t) e t e 2t ( ::: r t)= e rt (i ::: c rt) c = eã(i)t = P(i)Q(i c ::: c rt)(i c ::: c rt) () Q (i c ::: c r)p (i) 2 (i o ::: o r t)=eâ(i)t = (i)(q T ) (i o ::: o r t)(i o ::: o r t) Q T (i o ::: o r t)(i) (4) T

4 52 ZG Li et al / Automatica 9(2) (i c ::: c r o ::: o r t 2 t ) = t2 t eã(i)(t2 t) b(i) [d o a (i):::d o r a r (i)]p (i)eâ(i)(t t) dt: (5) Then, we have [ ] [ ][ ] X (t) A(i) b(i)k(i) b(i)k(i) X (t) = ė(t) A(i) L(i)c(i) e(t) [ ] X (t) ỹ(t)=y(t) y (t)=[ c(i)] : (6) e(t) Suppose that the mode of the system is i within the interval [tj sts j+ ], using Lemmas and 2, we have e(t s j+)= 2 (i o ::: o r t s j+ t s j)e (ts j+ ts j ) e(t s j) X (t s j+)= (i c ::: c rt s j+ t s j)e (ts j+ ts j ) X (t s j) Thus, we have + (i c ::: c r o ::: o r t s j+ t s j) e (ts j+ ts j ) e(t s j): e(t s j+) 6 =2 max( T 2 2 )e (ts j+ ts j ) e(t s j) (7) X (t s j+) 6 =2 max( T )e (ts j+ ts j ) X (t s j) + =2 max( T )e (ts j+ ts j ) e(t s j) : (8) From the above derivations, we know that the key problem is how to choose j c (j = 2:::r) and o j (j = 2:::r) such that for any given decay rate, all t T and all i = 2:::n, we have max ( T (i ::: c rt) c (i ::: c rt)) c max ( T 2 (i o ::: o r t) 2 (i o ::: o r t)) max ( T (i ::: c r c ::: o r o t) (i ::: c r c ::: o r o t)) where max (A) denotes the maximum eigenvalue of matrix A, and 4 can be chosen to give some exibility in determining the switching time of the observer and the controller This problem will be solved by using a deep pole assignment method through the following three lemmas The proofs are omitted due to the space limitation Lemma 4 Given any desired decay rate and t, there exist c j (i t) (j = 2:::r) and o j (i t) (j = 2:::r), such that max ( T (i c ::: c rt) (i c ::: c rt)) max ( T 2 (i o ::: o r t) 2 (i o ::: o r t)) max ( T (i c ::: c r o ::: o r t) (i c ::: c r o ::: o r t)) : Lemma 5 Given any, there exist ˆc j (i ) (j = 2:::r) and ˆ o j (i ) (j = 2:::r), such that max ( T (i ˆ c ::: ˆ c rt) (i ˆ c ::: ˆ c rt)) max ( T 2 (i ˆ o ::: ˆ o r t) 2 (i ˆ o ::: ˆ o r t)) max ( T (i ˆ c ::: ˆ c r ˆ o ::: ˆ o r t) (i ˆ c ::: ˆ c r ˆ o ::: ˆ o r t)) for t [T i ] i 2:::n}, where i is a nite positive number that bounds the switching interval of CVDS i Lemma 6 Given any desired decay rate, there exist c j (j = 2:::r) and o j (j = 2:::r), such that max ( T (i c ::: c rt) (i c ::: c rt)) max ( T 2 (i o ::: o r t) 2 (i o ::: o r t)) max ( T (i c ::: c r o ::: o r t) (i c ::: c r o ::: o r t)) for t T i 2:::n} Based on the above ve lemmas, we can design the subcontroller and subobserver for each subsystem The initial states for the rst subobserver are arbitrary while the initial states for the subsequent subobserver are the same as the end states of the previous subobserver Meanwhile, we also need to identify the active subsystems, the switching instances of the system such that the appropriate switchings of the controller and the observer can be well dened

5 ZG Li et al / Automatica 9(2) To identify the mode of the system, we note that for any given A(î) L(i)K(i)b(i)c(î), there exists a constant ti 6 T= where, such that e (A(î) L(i)c(î))t i i î 6 n (9) e 2t i e (A(î) b(î)k(i))t i i î 6 n (2) e 2t i t i e (A(î) b(î)k(i))(t i t) b(î)k(i)e (A(î) L(i)c(î))t dt 2 e 2t i i î 6 n: (2) Remark Note that (9) (2) hold when t i =, the right sides of (9) (2) are continuous functions of t i It follows that there exists a t i 6 T= such that (9) (2) hold Suppose that the mode of the system is within [tj s ts j ] Since [c( )A( ) L(k)c( )] is observable for any mode k M, e(t) and X (t) can be obtained by using the value of ỹ() in(6) when is in the interval [tj sts j + t ], and it is of the following form: [ t+t e(t)= e (A(lj) L(k)c()) T (t +t ) c T ( ) t c( )e (A(lj) L(k)c())(t +t ) d ] t+t e (A(lj) L(k)c()) T (t+t ) c T ( )ỹ()d t X (t)=e(t)+x (t): Then, we can compute ŷ()=c(k)x () [t s j ts j + t î ] for a certain k Ifŷ()=y() holds for all [t s j ts j + t ], then the mode of the system is k Otherwise, we need to check other mode In this way, we can identify the active mode of the system Next, we need to identify the switching instance of the system as follows: t s = t tj s = sup t tj X c (t tl j ) 2 t 6 2 j e2(t t t ) X (t ) 2 }: Finally, we need to dene the switching instances of the observers and the controllers Suppose that tj o and tj c are respectively the jth switching instance of the observer and the controller, we choose tj c = to j and it is dened according to the following ve cases: Case : (j), 2 (j) and (j) 2 (j) t o j (j) 2 (j): (22) Case 2: (j), 2 (j) and (j) 2 (j)= tj o = inf t}: (2) t (j) 2(j) Case : (j) and 2 (j)= tj o (j): (24) Case 4: (j)= and 2 tj o 2 (j): (25) Case 5: (j)= and 2 (j)= tj o = tj s + 2 T (26) where ( (j)= t t tj o + 2 ) T t t s j 6 2 T t ) e(t ) 2 6 e(t tl j ) 2 2 j e2(t t 6 } 4 2 j e2(t t j t) e(t ) 2 (27) ( 2 (j)= t t tj o + 2 ) T t t s j 6 2 T t ) ( X (t ) 2 +(2j ) e(t ) 2 ) 2 j e2(t t 6 X (t t j ) j e2(t t } + X (t ) 2 ) t ) ((2j ) e(t ) 2 (28) with t o = tc = t + T=, and e(t ) and X (t ) are obtained by the following equations: [ t+( T=) e(t )= T (t +( T=) t) c T (l ) t e (A(l) L(l )c(l )) c(l )e (A(l) L(l )c(l ))(t +( T=) t) dt ] t+( T=) e (A(l) L(l )c(l )) T (t +( T=) t) c T (l )y(t)dt t X (t )=e(t )+X (t ) where l is the initial mode of the system Remark 2 In the sets (27) and (28), the key idea is to restrict X (tj o t ) 2 6 ( 4 =2 j )e 2(to j t t ) ((2j ) e(t ) 2 + X (t ) 2 ) and e(tj o t j ) 2 6 ( 4 =2 j )e 2(to j t j t) e(t ) 2 } when tj o t j tj s With the above switchings for the controller and observer, we have the following result

6 522 ZG Li et al / Automatica 9(2) Theorem (Observer-based switching control theorem) Consider the switching system () with switching laws (I) or (II) Suppose that the system satises Assumption, then for any given decay rate, there exist an r dimensional observer of the form (6) and a feedback controller of the form (7) for mode i with the switching instances of the controller and the observer dened in (22), such that lim t e t X (t) =: Proof For each subsystem i and any given decay rate Using Lemmas 4 6, we can select c :::c r, o :::o r Based on these eigenvalues, an r dimensional observer of the form (6) and a feedback controller of the form (7) can be designed by using Lemmas 2 The design of the switchings of the controller and the observer is given in (22) Suppose that the switching instances of the system are t s ts 2 :::ts g and the switching instances of the controller are t c tc 2 :::tc g, and t t c g Note that for nite switching systems, g is nite and g for innite switching systems Further, we suppose that the mode of the system is l i within the interval [t s j ts j ] It can be easily proved by the induction method that (a) For any j, we have t c j t j = t o j t j t s j: (29) (b) For any j, e(t o j t j ) o 2 j e2(t j t j t) e(t ) 2 () holds for both (j)= and (j) (c) For any j, X (t o j t j ) o 2 j e2(t j t j t) ((2j ) e(t ) 2 + X (t ) 2 ) () holds for both 2 (j)= and 2 (j) We shall now prove that for any t s g+ t to g, we have e(t) 2 6 M e 2(t to g ) e(t o g) 2 X (t) 2 6 M e 2(t to g ) ( e(t o g) 2 + X (t o g) 2 ): Using Lemmas 4 6, we have e(t) 2 = e T (t o g)eât (l g)(t t o g ) eâ(lg)(t to g ) e(t o g) = e 2(t to g ) e T (t o g) T 2 (l g o ::: o r t t o g) 2 (l g o ::: o r t t o g)e(t o g) 6 max ( T 2 (l g o ::: o r t t o g) 2 (l g o ::: o r t t o g))e 2(t to g ) e(t o g) 2 6 M e 2(t to g ) e(t o g) 2 : (2) Similarly, X (t) 2 6 M e 2(t to g ) ( X (tg) o 2 + e(tg) o 2 ) where if the bound of the interval is known, then M = max max max j (i t)} 6j6 6i6n 6t6t i else M = max max max j(i t)} 6j6 6i6n 6t6 where (i t)= max ( T (i ::: c rt) c (i ::: c rt)) c 2 (i t)= max ( T 2 (i o ::: o r t) 2 (i o ::: o r t)) (i t)= max ( T (i c ::: c r o ::: o r t) (i ::: c r c ::: o r o t)): Finally, we shall prove that the required result holds Let M 2 = max max e (A(i)+b(i)K(j))t 2 e 2T max 6t6 T i j e (A(i)+ L(j)c(i))t 2 e 2T t e (A(i)+b(i)K(j))(t ) b(i)k(j)e (A(i)+ L(j)c(i)) d e 2T } : It follows that when tg+ o t ts g+, we have e(t) 2 6 M 2 e 2(ts g+ ) e(tg+) s 2 () X (t) 2 6 M 2 e 2(ts g+ ) ( e(tg+) s 2 + X (tg+) s 2 ): (4) Using inequalities () and (2), and when tg+ s t to g, we have e(t) 2 6 M e 2(t to g ) e(tg) o 2 6 2M e 2(t ts g +t g ) e(t s g t g ) 2 4M 2 g e2(t t) e(t ) 2 : Using inequality (), and when tg+ o t ts g+, we have e(t) 2 6 M 2 e 2(t ts g+ ) e(tg+) s 2 Thus e(t) 2 6 max M 2 2 g+ e2(t t) e(t ) 2 : 4 M M } g e2(t t) e(t ) 2 : Similarly, we have X (t) 2 max 4 M M } g e 2(t t) ((2g +) e(t ) 2 + X (t ) 2 ): 2

7 ZG Li et al / Automatica 9(2) Note that g =t and lim max 4 M M } 2 g 4 2 g ((2g +) e(t ) 2 + X (t ) 2 )=: It follows that lim t e t X (t) =: 4 Numerical example Example Consider a switching linear system which is composed of two modes: Mode : [ ] [ ] A() = b() = c()=[ ]: Mode 2: [ ] [ ] A(2) = b(2) = c(2)=[ ]: e t x 2 (t) t (second) Fig 2 The responses e t X 2 (t) of the system 5 Due to space limitation, we only consider switching law I The durations for modes and 2 are and 4 s, respectively The initial state of the system is X () = [ 2] T, the initial state of the observer is X ()=[5 5] T and the initial mode for the system is mode We choose the desired decay rate as = Let the controllers be K() = [25 ] and K(2) = [ 25] and the observers be [ ] [ ] 25 L() = and L(2) = : 25 The time responses of e t X (t) and control input of the switching control system are given in Figs Clearly, the states converge exponentially to zero e t x (t) u (t) Conclusion t (second) Fig The control input u(t) of the system In this paper, we have used the pole assignment method to design an observer-based controller for both the nite and innite switching linear systems The switching of the controller and the observer do not need to coincide exactly with the switchings of the system We have reconstructed the state of the system and the error between the state and the estimate of the system by the output of the system in some interval The state and the error have been used to determine the switching instances of the controller and the observer It has been shown that the proposed method can achieve any prescribed decay rate t (second) Fig The responses e t X (t) of the system References Blondel, V, & Tsitsiklis, J N (997) NP-hardness of some linear control design problems SIAM Journal on Control Optimization, 5, Chen, C T (984) Linear system theory and design New York: Rinehart and Winston

8 524 ZG Li et al / Automatica 9(2) Ezzine, J, & Haddad, A H (989) Controllability and observability of hybrid systems International Journal of Control, 49, Fu, M, & Barmish, B R (986) Adaptive stabilization of linear systems via switching control IEEE Transactions on Automatic Control,, 97 Hilhorst, R A, Amerongen, J V, Lohnberg, P, & Tulleken, H J A F (994) A supervisor control of mode-switch processes Automatica,, 9 Liberzon, D, & Morse, A S (999) Basic problems in stability and design of switched systems IEEE Control Systems Magazines, 9, 59 7 Michel, A N, & Hu, B (998) A comparison theory for stability analysis of discontinuous dynamic systems Part I: Results involving stability preserving mappings In Proceedings of the 7th IEEE CDC (pp 65 64) Narendra, K S, & Balakrishnan, J (99) Improving transient response of adaptive control systems using multiple models and switching In Proceedings of the 2nd IEEE CDC (pp 67 72) Pettersson, S, & Lennartson, B (996) Stability and robustness for hybrid systems In Proceedings of the 5th IEEE CDC (pp 22 27) Wicks, M A, Peleties, P, & De-Carlo, R A (998) Switched controller synthesis for quadratic stabilization of a pair of unstable linear systems European Journal on Control, 4, 4 47 Wilkinson, J H (965) The algebraic eigenvalue problem Oxford: Clarendon Press Zhengguo LI received the B Sci degree and the M Eng Degree from Northeastern University in 992 and 995, respectively, and received the PhD degree from Nanyang Technological University in 2 His research interests include hybrid systems, video processing and chaotic secure communication Currently, he is working at the agency for science, technology and research Changyun Wen received his BEng from Xian Jiaotong University in 98 and PhD from the University of Newcastle in Australia From August 989 August 99, he was a Postdoctoral Fellow at the University of Adelaide, Australia Since August 99, he has been with the School of Electrical and Electronic Engineering at Nanyang Technological University where he is currently an Associate Professor His major research interests are in the areas of adaptive control, iterative learning control, robust control, signal processing and their applications to ATM congestion control He is an Associate Editor of IEEE Transaction on Automatic Control Yeng Chai Soh received the BEng (Hons I) degree in electrical and electronic engineering from the University of Canterbury, New Zealand, in 98, and the PhD degree in electrical engineering from the University of Newcastle, Australia, in 987 From 986 to 987, he was a research assistant in the Department of Electrical and Computer Engineering, University of Newcastle He joined the Nanyang Technological University, Singapore, in 987 where he is currently a professor in the School of Electrical and Electronic Engineering Since 995, he has been the Head of the Control and Instrumentation Division His current research interests are in the areas of robust system theory and applications signal processing, estimation and ltering model reduction and hybrid systems