Title Necessary an sufficient conitions for analysis an synthesis of markov jump linear systems with incomplete transition escriptions Author(s) Zhang, L; Lam, J Citation Ieee Transactions On Automatic Control, 200, v. 55 n. 7, p. 695-70 Issue Date 200 URL http://hl.hanle.net/0722/3946 Rights IEEE Transactions on Automatic Control. Copyright IEEE.; 200 IEEE. Personal use of this material is permitte. However, permission to reprint/republish this material for avertising or promotional purposes or for creating new collective works for resale or reistribution to servers or lists, or to reuse any copyrighte component of this work in other works must be obtaine from the IEEE.; This work is license uner a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
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Necessary an Sufficient Conitions for Analysis an Synthesis of Markov Jump Linear Systems With Incomplete Transition Descriptions Lixian Zhang an James Lam Abstract This technical note is concerne with exploring a new approach for the analysis an synthesis for Markov jump linear systems with incomplete transition escriptions. In the stuy, not all the elements of the transition rate matrices (TRMs) in continuous-time omain, or transition probability matrices (TPMs) in iscrete-time omain are assume to be known. By fully consiering the properties of the TRMs an TPMs, an the convexity of the uncertain omains, necessary an sufficient criteria of stability an stabilization are obtaine in both continuous an iscrete time. Numerical examples are use to illustrate the results. Inex Terms Markov jump linear systems, stability, stabilization. I. INTRODUCTION Markov jump linear system (MJLS) is a class of multi-moal systems in which the transitions among ifferent moes are governe by a Markov chain. The stuies of these systems are motivate by the powerful moeling capability of Markov chains in practical applications, an many useful results have been obtaine, see [] [] for instance. The concepts of semi-markov chain, hien Markov chain, time-nonhomogeneous Markov chain, have also been importe to the control community in recent years an have promote many applications of MJLSs [2] [4]. However, in most of the stuies, complete knowlege of the moe transitions is require as a prerequisite for analysis an synthesis of MJLSs. This means that the transition probabilities (TPs) of the unerlying Markov chain are assume to be completely known. However, in practice, incomplete TPs are often encountere especially if aequate samples of the transitions are costly or time-consuming to obtain. Examples with such ifficulties can be foun in many fiels, such as communication systems with elay variations an packet losses, biochemical systems with iverse changes of environmental conitions, temperature, humiity. To relax the assumption that all the TPs are known, a new concept for MJLSs with partially unknown TPs is propose [5] an a series of stuies have been carrie out [6] [8]. The propose systems are therefore more general, by which much more complex switching phenomena can be moele. Meanwhile, as two extreme cases, the so-calle switche systems uner arbitrary switching [9], [20] an the conventional Markov jump systems are covere in the framework. However, although the works lai a conceptual founation for analysis 008-9286/$26.00 200 IEEE Manuscript receive December 0, 2009; revise February 03, 200 an March 2, 200 First publishe April 0, 200; current version publishe July 08, 200. This work was supporte in part by RGC HU 737/09E, National Natural Science Founation of China (6090400/F03007), Fun of Ministry of Eucation of China (200923022007), China Postoctoral Science Founation (20090460069), Outstaning Youth Science Fun of China (60825303), 973 Project (2009CB320600) in China, Major program of National Natural Science Founation of China (9096005/A0202), Heilongjiang Postoctoral Science Founation, China, an Overseas Talents Founation of Harbin Institute of Technology. Recommene by Associate Eitor P. Shi. L. X. Zhang is with the Space Control an Inertial Technology Research Center, Harbin Institute of Technology, Harbin, Heilongjiang 5000, China (e-mail: lixianzhang@hit.eu.cn). J. Lam is with the Department of Mechanical Engineering, University of Hong ong, Hong ong, China (e-mail: james.lam@hku.hk). Digital Object Ientifier 0.09/TAC.200.2046607
696 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 55, NO. 7, JULY 200 an synthesis of MJLSs, the approach evelope still has room for improvement in terms of conservatism. In fact, the properties of both the transition rate matrix (TRM) in continuous-time omain an the transition probability matrix (TPM) in iscrete-time omain have not been fully use. In this technical note, a new approach will be explore for the analysis an synthesis of MJLSs with incomplete escription of their transitions. Using the properties that the sum of each row is zero in a TRM or one in a TPM, together with the convexity of the uncertain omains, necessary an sufficient conitions for the stability analysis an stabilization synthesis problems are first erive for both continuous-time an iscrete-time cases. The conservatism in the previous stuies is eliminate by consiering the fact that the unknown elements of each row in TRM or TPM form a polytope. Moreover, for the continuous-time case, the ifficulty that the unknown elements contain iagonal elements is also overcome by introucing a lower boun of the iagonal element without aitional conservatism. The rest of the technical note is organize as follows. We formulate the consiere systems in Section II. Section III is evote to the issue of stability an stabilization for the system in both continuous-time an iscrete-time cases. Numerical examples are provie to emonstrate the theoretical finings. The technical note is conclue in Section IV. Notation: The notation use in this technical note is stanar. The superscript T stans for matrix transposition, n enotes the n imensional Eucliean space; + represents the sets of positive integers, respectively. For the notation (; F; P), represents the sample space, F is the -algebra of subsets of the sample space an P is the probability measure on F. E[] stans for the mathematical expectation. In aition, in symmetric block matrices or long matrix expressions, we use 3 as an ellipsis for the terms that are introuce by symmetry an iagfm ;M 2 ;...;M N g stans for a block-iagonal matrix constitute by M ;M 2 ;...;M N. The notation P > 0( 0) means P is real symmetric positive (semi-positive) efinite, an M i is aopte to enote M for brevity. I an 0 represent respectively, ientity matrix an zero matrix. Matrices, if their imensions are not explicitly state, are assume to be compatible for algebraic operations. II. PRELIMINARIES Given the probability space (; F; P) an consier the following continuous-time an iscrete-time MJLS, respectively: an _x(t) =A(r t )x(t)+b(r t )u(t) () x(k +)=A(r k )x(k)+b(r k )u(k) (2) where x(t) 2 n (respectively, x(k)) is the state vector an u(t) 2 l (respectively, u(k)) is the control input. The stochastic process fr t ;t 0g (respectively, the Markov chain fr k ;k 0g), taking values in a finite set I = f;...;ng, governs the switching among the ifferent system moes. In the continuous-time MJLS, fr t ;t 0g is a continuous-time, iscrete-state homogeneous Markov process an has the following moe transition probabilities: Pr(r t+h = jjr t = i) = ijh + o(h); if j 6= i + iih + o(h); if j = i where h>0, lim h!0 (o(h)=h) =0an ij 0 (i; j 2I, j 6= i) enotes the switching rate from moe i at time t to moe j at time t + h, an ii = 0 j=;j6=i ij for all i 2I. Hence, the transition rate matrix (TRM) in the Markov process is given by 3 = 2 N 2 22 2N... N N2 NN For the iscrete-time case, the process fr k ;k 0g is escribe by a iscrete-time homogeneous Markov chain, which takes values in finite set I with moe transition probabilities Pr(r k+ = jjr k = i) = ij N where ij 0; 8i; j 2I, an j= ij =. Likewise, the transition probability matrix (TPM) is given by 5 = 2 N 2 22 2N... N N2 NN The set I contains N moes of system () (or system (2)) an for r t = i 2I(respectively, r k = i), the system matrices of the i th moe are enote by A i, B i, C i, D i, E i, F i, which are real an known. The transition rates or probabilities escribe above are consiere to be partially available, that is, some elements in matrix 3 or 5 are unknown. Take system () or system (2) with 4 operation moes for example, the TRM 3 or TPM 5 may be written as ^ 2 ^ 3 4 ^ ^ ^ 2 22 23 24 ^ 3 32 ^ ; 33 34 ^ ^ 4 42 43 44 ^ 2 3 ^ 4 ^ 2 ^ 22 ^ 23 24 3 ^ 32 33 ^ 34 ^ 4 ^ 42 43 44 where each unknown element is labele with a hat ^. For convenience, 8i 2I, we enote I In aition, if I = fj : ij (or ij) is knowng ; I U = fj : ij (or ij) is unknowng (3) 6= ;, I is further escribe as I = f ; 2 ;...; m g ; m i 2f; 2;...;N 0 2g (4) where s 2 +, s 2f; 2;...;m ig, represents the inex of the s th known element in the i th row of matrix 3 or 5. Also, throughout the technical note, we enote = ij ; = : : ij : In the continuous-time case, when ^ ii is unknown, it is necessary to provie a lower boun for it an we have. Remark : The case m i = N 0 ; 8i 2 I, is exclue in (4), which means if we have only one unknown element, one can naturally calculate it from the known elements in each row an the TRM or TPM property. For MJLSs, the following stability efinition will be use [], [2]. Definition : System () (respectively, (2)) is sai to be stochastically stable if for u(t) 0 (respectively, u(k) 0) an every initial conition x 0 2 n an r 0 2I, the following hols: E kx(t)k 2 jx 0 ;r 0 < respectively; E 0 k=0 kx(k)k 2 jx 0;r 0 < :
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 55, NO. 7, JULY 200 697 III. STABILITY AND STABILIZATION In this section, we will erive the stochastic stability criteria for system () an system (2) when the transition rates or probabilities are partially unknown, an to esign a state-feeback stabilizing controller such that the close-loop system is stochastically stable. The moe-epenent controller consiere here has the form u(t) =(r t )x(t) (respectively; u(k) =(r k )x(k)) (5) where i (8r t = i 2 I,orr k = i 2 I) are the controller gains to be etermine. First, we provie the following preliminary stability results for MJLSs with completely known TRM or TPM. Lemma ([]): System () (with u(t) 0) is stochastically stable if an only if there exists a set of positive-efinite matrices P i, i 2I, satisfying A T i P i + P ia i + P < 0 (6) where P = ij P j. Lemma 2 ([2]): System (2) (with u(k) 0) is stochastically stable if an only if there exists a set of positive-efinite matrices P i, i 2I, satisfying where P = ijp j. A T i P A i 0 P i < 0 (7) A. Continuous-Time Case Let us first give the stability result for the unforce system () (with u(t) 0). The following theorem presents a necessary an sufficient conition on the stochastic stability of the consiere system with partially unknown transition rates. Theorem : Consier unforce system () with partially unknown transition rates. The corresponing system is stochastically stable if an only if there exists a set of matrices P i > 0, i 2I, such that, 8i 2I U ; A T i P i + P ia i + P U ; where P unknown iagonal element. P j < 0; if i 2I (8) + Pi Pj Pj < 0; if i 2I U (9) = ijp j an is a given lower boun for the Proof: We shall separate the proof into two cases, i 2I an, an bear in min that system () is stochastically stable if i 2 I U an only if (6) hols. ) Case : i 2I. It shoul be first note that in this case one has nee to consier < 0 here since in the i th row of the TRM are known. Now we rewrite the left-han sie of (6) as T 2 i = A i P i + P ia i + P + = 0. We only =0means the elements P j where the elements ; U, are unknown. Since we have 0 ( = 0 ) an ( = 0 )=, we know that P j 2 i = A T i P i + P ia i + P i 0 Pj : Therefore, for 0, 2i < 0 is equivalent to AT i P i + P i A i + P i 0 P j < 0; U, which implies that, in the presence of unknown elements, the system stability is ensure if an only if (8) hols. 2) Case 2: i 2I U. 2 i =. Also, we only consier ^ ii < here since if ^ ii =, then the ith row of the TRM is completely known. Now the left-han sie of the stability conition in (6) can be rewritten as In this case, ^ ii is unknown, 2 i = + ^ ii P i + = A T i P i + P ia i + P + 0^ ii 0 0 an ^ ii + ^ iip i ;j6=i 0^ ii 0 ;j6=i P j: P j Likewise, since we have 0 ( = 0 ^ ii 0 ) an ( = 0 ^ ii 0 ;j6=i )=, we know that ;j6=i 0^ ii 0 +^ iip i 0 ^ iip j 0 Pj which means that 2 i < 0 is equivalent to U, j 6= i As ^ ii is lower boune by,wehave + ^ ii P i 0 ^ ii P j 0 P j < 0: (0) ^ ii < which implies that ^ ii may take any value between [ ; 0 + ] for some <0arbitrarily small. Then ^ ii can be further written as a convex combination ^ ii = + +(0) where takes value arbitrarily in [0, ]. Thus, (0) hols if an only if U, j 6= i A T i P i +P i A i +P 0 P i+ P j P j +(P i 0P j ) < 0 () an + P i 0 P j 0 P j < 0 (2) simultaneously hol. Since is arbitrarily small, () hols if an only if A T i P i + P ia i + P Pi < 0 which is the case in (2) when j = i; U. Hence (0) is equivalent to (9).
698 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 55, NO. 7, JULY 200 Therefore, in the presence of unknown elements in the TRM, one can reaily conclue that the system is stable if an only if (8) an (9) hol for i 2I an i 2I U, respectively. Remark 2: The stability criterion evelope in Theorem is less conservative than the one obtaine in [5]. More specifically, in Theorem of [5], if i 2I, the conitions are which, since + < 0 A T i P i + P i A i + P j 0 < 0, ensure + A T i P i +P ia i + P 0 AT i P i +P ia i +P j < 0 which is (8). Also, if i 2I U, the criteria in Theorem of [5] are + < 0 A T i P i + P i A i + P j 0; U ; j = i A T i P i + P i A i + P j 0; U ; j 6= i: In this case, since < 0 an + A T i P i +P i A i +P which guarantees + > 0,wehave + A T i P i +P i A i +P i + (3) 0 A T i P i +P ia i +P j < 0 P i 0 P j 0 P j < 0; : U Therefore, the conitions (8), (9) are less conservative than (3). Note that the obtaine conitions are without loss of generality since the lower boun,,of^ ii is allowe to be arbitrarily negative. Now let us consier the stabilization problem of system () in the presence of unknown elements in the TRM. The following theorem presents a necessary an sufficient criterion for the existence of a moeepenent stabilizing controller of the form in (5). Theorem 2: Consier system () with partially unknown transition rates. If there exist matrices X i > 0 an Y i ; 8i 2Isuch that 3 i + ii X i T X i 3 0X 0 < 0; 3 3 0X j U ; 3 i + Xi T if i 2I (4) Xi 3 0X 0 < 0; 3 3 0X j U ; where 3 i = Ai X i + X i A T i X T if i 2I U (5) + B i Y i + Y T i B T i an =iag X ;...;X ; = i X i ;...; i X i (6) an 8s 2f; 2;...;m i g, s is escribe in (4), s 6= i, then there exists a moe-epenent stabilizing controller of the form in (5) such that the close-loop system is stochastically stable. Moreover, if the LMIs in (4), (5) have a solution, an amissible controller gain is given by i = Y ix 0 i : (7) Proof: Consier system () with the control input (5) an replace A i by A i +B i i in (8), (9), respectively. Then, if i 2I, performing a congruence transformation to (8) by P 0 i, we can obtain (A i + B i i)p 0 i + P 0 i (A i + B i i) T + P 0 i P P 0 i 0 P 0 i P jp 0 i < 0: (8) Setting X i = P 0 i, Y i = i X i an consiering (6), by Schur complement, one can obtain that (8) is equivalent to (4). In a similar way, if i 2I U, (5) can be worke out from (9). Meanwhile, ue to Y i = ix i, the esire controller gain is given by (7). Remark 3: It is note from (5) that if the iagonal elements in the TRM contain unknown ones, the system stability, the existence of the amissible controller an the controller gains solution will be epenent on. This epenency, therefore, will reuce the conservatism -inepenent results obtaine in [5]. existe in the previous B. Discrete-Time Case The following theorem presents a necessary an sufficient conition on the stochastic stability of the unforce system (2) with partially unknown transition probabilities. Theorem 3: Consier the unforce system (2) with partially unknown transition probabilities. The corresponing system is stochastically stable if an only if there exists a set of matrices P i > 0, i 2I such that A T i where P P + = ijp j. Pj Ai 0 Pi < 0; 8j 2I (9) Proof: It shoul be first note that in the iscrete-time case, an we exclue =here since it means that all the elements in the i th row are known. Now the left-han sie of stability conition (7) in Lemma 2 can be rewritten as 9 i = A T i P i + = A T i P i + 0 P j A i 0 P i 0 P j A i 0 P i where the elements, j 2I U, are unknown. Since 0 (= 0 ) ; 8j 2I U an (^ ij= 0 )=, we know that 9 i = 0 A T i P + Pj Ai 0 Pi Therefore, for 0 0, 9i < 0 is equivalent to AT i (P + ( 0 )Pj)Ai 0 Pi < 0; 8j 2 I U, which implies that, in the presence of unknown elements, the system stability is ensure if an only if (9) hols. Remark 4: Analogous to Remark 2 for the continuous-time case, the necessary an sufficient criterion evelope in Theorem 3 is also less conservative when compare with Theorem 3 in [5], where the stability conitions are given by A T i P A i 0 P i < 0 A T i P j A i 0 P i < 0; U :
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 55, NO. 7, JULY 200 699 The inequalities yiel A T i P A i 0 P i + A T i P j A i 0P i <0; U which is (9). Therefore, combine with Remark, it is seen that the approach aopte in Theorems an 3 in this technical note, which uses the TRM or TPM property (the sum of all the elements in each row is zero or one), gives the necessary an sufficient criteria an are less conservative than the existing results. Now consier the system (2) with control input u(k), the following theorem presents a conition for the existence of a moe-epenent stabilizing controller with the form in (5). Theorem 4: Consier system (2) with partially unknown transition probabilities. If there exist matrices X i > 0 an Y i ; 8i 2Isuch that where L X 0X i L T (AiXi + BiYi) 3 0X = p i I;...; i I; 0 I T < 0 (20) (2) =iag X ;...;X ;X j ; j 2I U (22) an 8s 2 f; 2;...;m ig, s is escribe in (4), then there exists a moe-epenent stabilizing controller of the form in (5) such that the close-loop system is stochastically stable. Moreover, if the LMIs in (20) have a solution, an amissible controller gain is given by (7). Proof: First of all, by Theorem 3, we know that system (2) is stochastically stable with partially unknown transition probabilities if the inequality (9) hols. By Schur complement, (9) is equivalent to 0P i 3 3 3 3 3 p i P A i 0P 3 3 3 3 p i P A i 0 0P 3 3 3...... 3 3 p i P A i 0 0 0P 3 0 P ja i 0 0 0 0P j <0: (23) Now, consier the system with the control input (5) an replace A i by A i + B i i in (23). Setting X i = 0 P i, performing a congruence transformation to (23) by iag[x i ; X ] an applying the change of variable Y i = ix i, we can reaily obtain (20). Therefore, if (20) hols, (9) will be satisfie in Theorem 3, that is, the unerlying system is stochastically stable. Meanwhile, ue to Y i = ix i, the esire controller gain is given by (7). Remark 5: In contrast with the continuous-time case, the iscretetime case is relatively simpler since all the elements in the TPM are nonnegative an we nee not istinguish the cases of iagonal elements known or unknown. Remark 6: It is note that an interesting conclusion can be irectly rawn from Theorem an Theorem 3. That is, when all the elements in the TRM or TPM are unknown, the unerlying systems are subject to switchings without known statistics. This leas to the so-calle eterministic switche systems uner arbitrary switchings (see [9], [20] for continuous-time an iscrete-time case, respectively). We can therefore obtain the necessary an sufficient stability criterion of such switche systems in continuous-time an iscrete-time cases, respectively. More specifically, in the iscrete-time case, we have the stability conition is A T i P ja i 0P i < 0; 8i2j 2I2I, which is reuce from (9) when all the elements in the TPM are unknown. Likewise, for the TABLE I CONTROLLERS FOR TRM (25) continuous-time case, if all the elements in the TRM are unknown, the conitions leas to (9) only an it reuces to A T i P i + P i A i + (P i 0 P j ) < 0: (24) Since can be arbitrarily negative, inequality (24) requires P i = P which leas to the conition P j A T i P + PA i < 0; 8i 2 j 2I2I: C. Numerical Examples The valiity an the reuction of conservatism of the results obtaine above are verifie by the following numerical examples. Example : Consier MJLS () with three operation moes an the following system matrices: A = 00:50 00:75 ; A 2 = 00:20 0: A 3 = 0 ; B = 5 0 ; B 2 = 02 0 ; B 3 = 02 : Assume the TRM is given by TMR = 02:4 00:33 0:4 Moe 3 4 0:3 ^2 ^3 2 0:7 0:2 0:5 3 ^3 ^32 00:5 ; (25) where ; 8i 2 j 2I2I U enote the unknown elements. The purpose of this example is to verify the reuce conservatism of the obtaine results in the continuous-time case. First, one can check that the open loop system is unstable by both Theorem in the technical note an Theorem in [5]. Then, base on Theorem 2 in the technical note, we obtain the controller gains for the system as shown in Table I. However, it is verifie that the stabilization criterion evelope previously cannot yiel a feasible solution of the controller, which shows that the evelope approach in the technical note is less conservative. Notice that in Example, all the iagonal elements of TRM (25) are known. Now we further provie another example with unknown iagonal elements in the TRM to illustrate the epenency of controller of the corresponing unknown iagonal element. Example 2: Consier MJLS () with four operation moes an the following system matrices: esign on the lower boun 05 07:5 A = 0 0 A 3 = 02 0 0 ; A 4 = ; A 2 = 2:4 03:3 0 4 0 02:3 0 0 ; B = 0 ; B2 = 0 0 ; B3 = 02 ; ; B4 = 0 :
700 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 55, NO. 7, JULY 200 TABLE II CONTROLLERS FOR TRM (26) TABLE III CONTROLLERS FOR TPM (27) The comparison of Theorem 4 in the technical note with Theorem 4 in [5] is summarize in Table III, where the reuction of conservatism of the new criterion is emonstrate. The TRM is given by TMR= Moe 2 3 4 0:3 0:2 ^3 ^4 2 ^2 ^22 0:5 0:5 3 0: ^32 02:5 ^34 4 0:4 0:2 0:6 0:2 (26) In the 2n row of TRM (26), the iagonal element ^ 22 is unknown, we assign its lower boun (2) a priori with ifferent values. It can be checke that the open-loop system is unstable base on Theorem in [5], or Theorem in this technical note for any (2) 2 (0; 0]. Then, by Theorem 2 in [5] an Theorem 2 in the technical note with ifferent (2), we obtain the controller gains as shown in Table II. It is seen from Table II that the obtaine controller gains are epenent on (2). By applying the bisection metho with the conitions in Theorem 2, one can further obtain the minimal value of (2), below which the stabilizing controller will not exist (here we get (2) = 02:2758 by some stanar numerical software). It is also worth mentioning here that, for some systems, one may obtain that the controller solution is inepenent on the boun of iagonal elements, as the system in Example of [5] shows that the controller exists espite that ^ 22 is unknown an has no given lower boun. Example 3: Consier MJLS (2) with four operation moes an the following system matrices: A = 0:25 2:5 02:5 0:5 00:25 A 3 = 2:5 03:0 ; A 2 = ; A 4 = 0:25 00:83 2:5 03:5 :5 00:56 2:5 02:75 ; B = 2 ; B 2 = 0 ; B 3 = ; B 4 = 0:8 0 : Moreover, the TPM is given by TMR = Moe 4 3 4 0:3 0:2 0: 0:4 2 ^ 2 0:2 0:3 ^ 24 3 ^ 3 ^ 32 00:5 0:5 4 0:2 0:2 0: 0:5 ; (27) IV. CONCLUSION In this technical note, we have revisite the analysis an synthesis problems of Markov jump linear system with incomplete transition escriptions. Necessary an sufficient criteria are obtaine for MJLSs in both continuous-time omain an iscrete-time omain by fully exploiting the properties of the transition rates matrix an the transition probabilities matrix. The conservatism of the approach evelope previously, which only leas to sufficient conitions for the system, is reuce by the newly evelope approach. Numerical examples have verifie the theoretical results given in the technical note. It is expecte that the approach can be further use for other analysis an synthesis issues such as H analysis, H synthesis an other applications such as Markov jumping neural networks, e.g., [2] with incomplete transition escriptions therein. ACNOWLEDGMENT The authors woul like to thank the associate eitor an the reviewers for their helpful comments an suggestions which have helpe improve the presentation of the technical note. REFERENCES [] E.. Boukas, Stochastic Switching Systems: Analysis an Design. Berlin, Germany: Birkhauser, 2005. [2] O. L. V. Costa, M. D. Fragoso, an R. P. Marques, Discrete-Time Markovian Jump Linear Systems. Lonon, U..: Springer-Verlag, 2005. [3] Y. G. Fang an. A. Loparo, Stabilization of continuous-timie jump linear systems, IEEE Trans. Autom. Control, vol. 47, no. 0, pp. 590 603, Oct. 2002. [4] X. Feng,. A. Loparo, Y. Ji, an H. J. Chizeck, Stochastic stability properties of jump linear systems, IEEE Trans. Autom. Control, vol. 37, no., pp. 38 52, Jan. 992. [5] G. Nakura, Stochastic optimal tracking with preview by state feeback for linear iscrete-time Markovian jump systems, Int. J. Innovative Comp, Inform. Control, vol. 6, no., pp. 5 27, 200. [6] P. Shi, E.. Boukas, an R.. Agarwal, Control of Markovian jump iscrete-time systems with norm boune uncertainty an unknown elay, IEEE Trans. Autom. Control, vol. 44, no., pp. 239 244, Nov. 999. [7] P. Shi, E.. Boukas, an R.. Agarwal, alman filtering for continuous-time uncertain systems with Markovian jumping parameters, IEEE Trans. Autom. Control, vol. 44, no. 8, pp. 592 597, Aug. 999. [8] Z. Wang, Y. Liu, L. Yu, an X. Liu, Exponential stability of elaye recurrent neural networks with Markovian jumping parameters, Phys. Lett. A, vol. 356, no. 4, pp. 346 352, 2006. [9] Z. Wang, H. Qiao, an. Burnham, On stabilization of bilinear uncertain time-elay stochastic systems with Markovian jumping parameters, IEEE Trans. Autom. Control, vol. 47, no. 4, pp. 640 646, Apr. 2002.
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 55, NO. 7, JULY 200 70 [0] S. Xu an T. Chen, Robust H control for uncertain iscrete-time stochastic bilinear systems with Markovian switching, Int. J. Robust Nonlin. Control, vol. 5, no. 5, pp. 20 27, 2005. [] Q. Ding an M. Zhong, On esigning H fault etection filter for Markovian jump linear systems with polytopic uncertainties, Int. J. Innovative Comp. Inform. Control, vol. 6, no. 3, pp. 995 004, 200. [2] B. Bercu, F. Dufour, an G. Yin, Almost sure stabilization for feeback controls of regime-switching linear systems with a hien Markov chain, IEEE Trans. Autom. Control, vol. 54, no. 9, pp. 24 225, Sep. 2009. [3] C. Schwartz an A. H. Haa, Control of jump linear systems having semi-markov sojourn times, in Proc. IEEE 42n Conf. Decision Control, Ma, HI, 2003, pp. 2804 2805. [4] L. X. Zhang, H estimation for iscrete-time piecewise homogeneous Markov jump linear systems, Automatica, vol. 45, no., pp. 24 225, 2009. [5] L. X. Zhang an E.. Boukas, Stability an stabilization of Markovian jump linear systems with partly unknown transition probability, Automatica, vol. 45, no. 2, pp. 463 468, 2009. [6] L. X. Zhang an E.. Boukas, H control for iscrete-time Markovian jump linear systems with partly unknown transition probabilities, Int. J. Robust Nonlin. Control, vol. 9, no. 5, pp. 868 883, 2009. [7] L. X. Zhang an E.. Boukas, H control of a class of extene Markov jump linear systems, IET Control Theory Appl., vol. 3, no. 7, pp. 834 842, 2009. [8] L. X. Zhang, E.. Boukas, an J. Lam, Analysis an synthesis of Markov jump linear systems with time-varying elays an partially known transition probabilities, IEEE Trans. Autom. Control, vol. 53, no. 0, pp. 2458 2464, Sep. 2008. [9] J. Daafouz, P. Rieinger, an C. Iung, Stability analysis an control synthesis for switche systems: A switche Lyapunov function approach, IEEE Trans. Autom. Control, vol. 47, no., pp. 883 887, Nov. 2002. [20] D. Liberzon, Switching in Systems an Control. Berlin, Germany: Birkhauser, 2003. [2] Y. Liu, Z. Wang, J. Liang, an X. Liu, Stability an synchronization of iscrete-time Markovian jumping neural networks with mixe moeepenent time-elays, IEEE Trans. Neural Networks, vol. 20, no. 7, pp. 02 6, Jul. 2009. Topological Obstructions to Submanifol Stabilization Abol-Reza Mansouri, Member, IEEE Abstract We consier the problem of local asymptotic feeback stabilization via a continuously ifferentiable feeback law of a control system _x = (x u) efine in Eucliean space (with being continuously ifferentiable) to a compact, connecte, oriente -imensional submanifol of with coimension strictly larger than one. We obtain necessary conitions on the topology of for such a stabilizing feeback law to exist. This extens the work one in [6], where only the coimension one case was treate. We also briefly iscuss the case where the control is only assume continuous. Inex Terms Euler-Poincare characteristic, homology groups, submanifol stabilization. I. INTRODUCTION Consier the following moification of Brockett s non-holonomic integrator [], introuce in [6]: In 3 (with canonical coorinate functions x; y; z), we efine (I) _x = u; _y = v; _z =(yu 0 xv)e z where u; v are the control functions. The control function f is given x here, with x = y an u = u,by v z (x; u) 7! f ((x; u)) = u v (yu 0 xv)e z ; an is continuously ifferentiable. It is clear that f is not onto any neighborhoo of the origin in 3 ; inee, no point on the z-axis of 3 other than the origin is in the range of f. It follows [] that there exists no continuously ifferentiable feeback law that can stabilize this system to the origin. Consier now the problem of asymptotically stabilizing this control system to a submanifol of 3 homeomorphic to the unit sphere S 2 of 3 ; efining 6 f; = (x; u) 2 3 2 2 jf (x; u) 6= 0 we easily have that 6 f; = 3 2 ( 2 nf0g); hence, we obtain H 2 (6 f;; ) ' H 2 ( 2 nf0g; ) = 0, where H k (; ) enotes the k th singular homology group with coefficients in, an ' enotes a group isomorphism (see e.g., Chapter 4 of [7]). On the other han, the Euler-Poincaré characteristic (S 2 ) of S 2 is non-zero (see Chapter 4 of [7]). It follows therefore from Theorem 4 of [6] that there exists no continuously ifferentiable feeback law stabilizing the above control system to S 2. Consier now the problem of asymptotically stabilizing this control system to the unit circle in the xy-plane, efine by S 2 \fz = 0g. As note in [6], this stabilization is achievable, an Manuscript receive January 28, 2009; revise September 09, 2009. First publishe April 0, 200; current version publishe July 08, 200. This work was supporte in part by the Natural Sciences an Engineering Research Council of Canaa. Recommene by Associate Eitor D. Liberzon. The author is with the Department of Mathematics an Statistics, Queen s University, ingston, ON 7L 3N6, Canaa (e-mail: mansouri@mast.queensu. ca). Digital Object Ientifier 0.09/TAC.200.2046922 008-9286/$26.00 200 IEEE