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Finite-time H∞ control for singular Markovian jump systems with partly unknown transition rates
Applied Mathematical Modelling xxx (2015) xxx–xxx
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Finite-time H1 control for singular Markovian jump systems with partly unknown transition rates Li Li a,b, Qingling Zhang a,c,⇑ a
Institute of Systems Science, Northeastern University, Shenyang, Liaoning Province 110819, PR China College of Mathematics and Physics, Bohai University, Jinzhou, Liaoning Province 121013, PR China c The State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang, Liaoning Province 110819, PR China b
a r t i c l e
i n f o
Article history: Received 23 July 2013 Received in revised form 22 December 2014 Accepted 17 April 2015 Available online xxxx Keywords: Singular Markovian jump systems Singular stochastic finite-time stability Singular stochastic finite-time boundedness Partly unknown transition rates Linear matrix inequalities (LMIs)
a b s t r a c t This paper investigates the problem of finite-time H1 control for a class of singular Markovian jump systems with partly unknown transition rates. Firstly, sufficient conditions on singular stochastic finite-time boundedness of singular Markovian jump systems with partly unknown transition rates are obtained. Secondly, the results are extended to singular stochastic H1 finite-time boundedness of singular Markovian jump systems with partly unknown transition rates. Then state feedback controllers are designed to ensure the singular stochastic finite-time boundedness and the singular stochastic H1 finite-time boundedness of the underlying closed singular Markovian jump systems in the forms of strict LMIs. Finally, numerical examples are given to illustrate the validity of the proposed methods. Ó 2015 Published by Elsevier Inc.
1. Introduction Singular systems, which are also referred to as generalized systems, descriptor systems, implicit systems, or differential– algebraic systems, have attracted many researchers due to the fact that singular systems have extensive applications in many practical systems such as electrical circuits, power systems, networks and other systems [1,2]. In recent years, many research topics on singular systems have been extensively studied such as stability and stabilization [3,4], H1 control problem [5,6], and so forth. A lot of attention has been paid to the study of Markovian jump systems (MJSs), which are a special kind of hybrid systems, because they have the advantage of better representing physical systems with random changes in their structures and parameters, and have successful applications in economic systems, manufacturing systems and communication systems [7–9]. Many important issues have been studied for this kind of systems, such as stability analysis, stabilization, H2 and H1 control [10–13]. When singular systems experience abrupt changes in their structures, it is natural to model them as singular Markovian jump systems (SMJSs) [14–16]. On the other hand, in most of the studies, complete knowledge of the mode transition rates is required as a prerequisite for analysis and synthesis of MJSs. Therefore, rather than having a large complexity to measure or estimate all the transition rates, it is significant and necessary to further study more general jump systems with partly unknown transition rates. Many attracting problems have been investigated and solved for these systems, such as stability analysis, stabilization [17,18] and H1 filter [19–21]. In many practical applications, one may be interested in not only the stability of the system but also the ⇑ Corresponding author at: Institute of Systems Science, Northeastern University, Shenyang, Liaoning Province 110819, PR China. E-mail address: [email protected] (Q. Zhang). http://dx.doi.org/10.1016/j.apm.2015.04.044 0307-904X/Ó 2015 Published by Elsevier Inc.
Please cite this article in press as: L. Li, Q. Zhang, Finite-time H1 control for singular Markovian jump systems with partly unknown transition rates, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.04.044
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L. Li, Q. Zhang / Applied Mathematical Modelling xxx (2015) xxx–xxx
behavior of the system over a fixed short time. For instance, large values of the state are not acceptable in the presence of saturations [22,23]. Compared with the classical Lyapunov asymptotical stability, in order to deal with these transient performances of control dynamic systems, finite-time stability or short-time stability was introduced. Some appealing results were obtained to ensure finite-time stability, finite-time boundedness and finite-time stabilization of various systems including linear systems, nonlinear systems, and stochastic systems [24–28]. To the best of our knowledge, the problem of singular stochastic finite-time H1 control for singular Markovian jump systems (SMJSs) with partly unknown transition rates has not been fully investigated, which motivates the main purpose of our study. In this paper, we are concerned with the problem of finite-time H1 control for SMJSs with partly unknown transition rates. And SMJSs with completely known transition rates are viewed as special cases of the ones tackled here. Firstly, we give the concepts of singular stochastic finite-time boundedness (SSFTB) and singular stochastic H1 finite-time boundedness (SSH1 FTB) of SMJSs. The main contribute lies in tractable sufficient conditions obtained to guarantee SSFTB and SSH1 FTB of SMJSs with partly unknown transition rates. Then, a finite-time H1 state feedback controller is designed in the form of LMIs, which ensures SSH1 FTB of the closed-loop system. Numerical examples are provided to demonstrate the effectiveness of the main results. Notation: Throughout this paper, the notations used are fairly standard, for real symmetric matrices A and B, the notation A P B (respectively, A > B) means that the matrix A B is positive semi-definite (respectively, positive definite). AT represents the transpose of a matrix A, and A1 represents the inverse of a matrix A. kmax B (kmin B) is the maximum (respectively, minimum) eigenvalue of a matrix B. diagf g stands for a block-diagonal matrix. I is the unit matrix with appropriate dimensions, and in a matrix, the term of symmetry is stated by the asterisk ‘⁄’. Let Rn stands for the n-dimensional Euclidean space, Rnm is the set of all real matrices, and k k stands for the Euclidean norm of vectors. Efg denotes the mathematics expectation of the stochastic process or vector. Ln2 ½0; 1Þ stands for the space of n-dimensional square integrable functions on ½0; 1Þ. 2. Basic definitions and lemmas Consider a class of SMJSs as follows:
_ ¼ Aðr t ÞxðtÞ þ Bðr t ÞuðtÞ þ Bw ðr t ÞwðtÞ ExðtÞ zðtÞ ¼ Cðr t ÞxðtÞ þ Dðr t ÞuðtÞ þ Dw ðr t ÞwðtÞ
ð1Þ
;
where xðtÞ 2 Rn is the state vector, uðtÞ 2 Rm is the control input, zðtÞ 2 Rp is the controlled output, E 2 Rnn with rankE ¼ r 6 n; wðtÞ 2 Rq is the disturbance which belongs to Lq2 ½0; 1Þ, and satisfies
Z
T
2
wT ðtÞwðtÞdt 6 d ;
d P 0;
ð2Þ
0
Aðrt Þ; Bðrt Þ; Bw ðrt Þ; Cðr t Þ; Dðr t Þ and Dw ðr t Þ are real known matrices with appropriate dimensions. frt ; t P 0g is a continuous-time Markov process with right continuous trajectories taking values in a finite set given by S ¼ f1; 2; . . . ; Ng with the transition rate matrix (TRM) P , fpij g given by
Prfr tþh ¼ jjr t ¼ ig ¼
pij h þ oðhÞ; i – j ; 1 þ pii h þ oðhÞ; i ¼ j
where h > 0; limh!0 oðhÞ ¼ 0 and pij P 0, for j – i, is transition rate from mode i to j at time t þ h, which satisfies h P pii ¼ Nj¼1;j–i pij . The transition rates of the jumping process may be considered to be partly accessible in this paper. For instance, the TRM of system (1) may be expressed as
3 p11 p^ 12 p^ 1N 6p ^ 22 p2N 7 7 6 21 p 2
P¼6 6 ..
4 . p^ N1
.. .
..
.. 7 7; . 5
.
pN2 pNN
where each unknown element is labeled with a hat ‘ ˆ ’. For notational clarity, 8i 2 S, the set U ðiÞ denotes U ðiÞ ¼ U k [ U uk with ðiÞ
ðiÞ Uk
, fjjpij is known for i 2 ðiÞ Uk
¼
ðiÞ kj
ðiÞ Sg; U uk
ðiÞ ðiÞ ðiÞ fk1 ; k2 ; . . . ; kmi g;
, fjjpij is unknown for i 2 Sg. Moreover, if
ðiÞ Uk
– /, it is further described as
mi 2 f1; 2; . . . ; N 2g;
ðiÞ kj
ðiÞ
ð3Þ ðiÞ Uk
where 2 Zþ ; 1 6 6 N; j ¼ 1; 2; . . . ; mi represents the jth known element of the set P ðiÞ pk , j2UðiÞ pij , and when p^ ii is unknown, it is necessary to provide a lower bound pðiÞ d for it.
in the TRM P. We denote
k
Consider a state feedback controller
uðtÞ ¼ Kðr t ÞxðtÞ;
ð4Þ
Please cite this article in press as: L. Li, Q. Zhang, Finite-time H1 control for singular Markovian jump systems with partly unknown transition rates, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.04.044
L. Li, Q. Zhang / Applied Mathematical Modelling xxx (2015) xxx–xxx
3
where Kðr t Þ is the controller gain to be determined. The SMJS (1) with the controller (4) can be written in the form of the control system as follows:
(
t ÞxðtÞ þ Bw ðr t ÞwðtÞ _ ExðtÞ ¼ Aðr ; zðtÞ ¼ Cðr t ÞxðtÞ þ Dw ðr t ÞwðtÞ
ð5Þ
t Þ ¼ Aðr t Þ þ Bðr t ÞKðrt Þ and Cðr t Þ ¼ Cðr t Þ þ Dðr t ÞKðrt Þ. where Aðr Definition 1 [29] regularity and nonimpulse. (i) The continuous-time SMJS (5) is said to be regular in the time interval ½0; T, if the characteristic polynomial t ÞÞ is not identically zero for all t 2 ½0; T. detðsE Aðr t ÞÞÞ ¼ rankE for all (ii) The continuous-time SMJS (5) is said to be impulse free in the time interval ½0; T, if degðdetðsE Aðr t 2 ½0; T.
Definition 2 ([28,30] singular stochastic finite-time stability (SSFTS)). The closed-loop SMJS (5) with wðtÞ ¼ 0 is said to be SSFTS with respect to ðc1 ; c2 ; T; Rrt Þ, with c1 < c2 and Rrt > 0, if the system is regular and impulse free in the time interval ½0; T and satisfies
EfxT ð0ÞET Rrt Exð0Þg 6 c21 ) EfxT ðtÞET Rrt ExðtÞg < c22 :
ð6Þ
Definition 3 ([28,30] singular stochastic finite-time boundedness (SSFTB)). The closed-loop system (5) which satisfies (2) is said to be SSFTB with respect to ðc1 ; c2 ; T; Rrt ; dÞ, with c1 < c2 and Rrt > 0, if the system is regular and impulse free in the time interval ½0; T and the condition (6) holds. t ÞÞ ensure the existence and uniqueness of impulse free solution of the Remark 1. The regularity and nonimpulse of ðE; Aðr system (5) in the time interval ½0; T. Definition 4 ([28,30] singular stochastic H1 finite-time boundedness (SSH1 FTB)). The closed-loop SMJS (5) is said to be SSH1 FTB with respect to ðc1 ; c2 ; T; Rrt ; c; dÞ, if the closed-loop SMJS (5) is SSFTB with respect to ðc1 ; c2 ; T; Rrt ; dÞ, and under the zero initial condition the controlled output zðtÞ satisfies
E
Z
T
zT ðtÞzðtÞdt
< c2
0
Z
T
wT ðtÞwðtÞdt
ð7Þ
0
for any nonzero wðtÞ which satisfies (2), where c is a prescribed positive scalar. For notational simplicity, in the sequel, for each possible r t ¼ i 2 S, we will denote Aðr t Þ by Ai ; Aw ðr t Þ by Aw;i , and so on. Lemma 1 [31]. For given matrices E; X > 0; Y, if ET X þ Y KT is nonsingular, then there exist matrices S > 0; L such that 1
ES þ LHT ¼ ðET X þ Y KT Þ
, where X; S 2 Rnn ; Y; L 2 RnðnrÞ , and K; H 2 RnðnrÞ are any matrices with full column rank
T
satisfying E K ¼ 0; EH ¼ 0. 3. Main results In this section, we start by developing results that can be used to design a state feedback controller that ensures the SMJS (5) with partly unknown transition rates is SSFTB. Theorem 1. The closed-loop SMJS (5) with partly unknown transition rates is SSFTB with respect to ðc1 ; c2 ; T; Ri ; dÞ, if there exist a scalar s P 0, matrices P i > 0; Q 1i > 0; Q 2i > 0, and Si ; i 2 S satisfying:
X1ij < 0;
ðiÞ 8j 2 U uk ;
if i 2 U k ;
X2ij < 0;
ðiÞ 8j 2 U uk ;
if i 2 U uk ;
1
1
ET X i ¼ ET R2i Q 1i R2i E;
ðiÞ
ð8Þ
ðiÞ
ð9Þ ð10Þ
Please cite this article in press as: L. Li, Q. Zhang, Finite-time H1 control for singular Markovian jump systems with partly unknown transition rates, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.04.044
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L. Li, Q. Zhang / Applied Mathematical Modelling xxx (2015) xxx–xxx 2
maxfkmax ðQ 1i Þgc21 þ maxfkmax ðQ 2i Þgd < minfkmin ðQ 1i Þgc22 esT ; i2S
i2S
ðiÞ
where P k ,
P j2U
ðiÞ k
ð11Þ
i2S
^ ii in the TRM P, and R 2 RnðnrÞ is any pij Pj ; pðiÞ is a given lower bound for the unknown diagonal element p d
T matrix with full column satisfying ET R ¼ 0; X i ¼ ET P i þ Si RT ,
"
# ðiÞ ðiÞ T T Xi þ XTA P k pk P j sPi E X Ti Bw;i A i i þE i ; Q 2i
"
# ðiÞ ðiÞ ðiÞ ðiÞ T T Xi þ XTA P k þ pd P i pd Pj pk Pj sP i E X Ti Bw;i A i i þE i : Q 2i
1 ij
X ¼
2 ij
X ¼
Proof. Firstly, we prove the SMJS (5) with partly unknown transition rates is regular and impulse free in the time interval ½0; T . By Schur complement Lemma, (8) and (9), we have
0 T T B ðiÞ T Xi þ XT A A i i i þ ðpii sÞE P i E < E @pk P j
T Xi þ XT A A i i i þ
1
X ðiÞ l2U k ;l–i
ðiÞ ðiÞ pil Pl C AE 6 0; 8j 2 U uk ; if i 2 U k ;
ð12Þ
ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ T T pðiÞ P k þ pd Pj þ pk Pj E 6 0; 8j 2 U uk ; if i 2 U uk : d s E Pi E < E
ð13Þ
I 0 Since rankE ¼ r 6 n, there exist nonsingular matrices G and H such that GEH ¼ r , then R can be rewritten as 0 0 0 i H ¼ Ai1 Ai2 ; GT Pi G1 ¼ Pi1 Pi2 ; 2 RðnrÞðnrÞ is any nonsingular matrix. Denote GA R ¼ GT , where U U Pi3 Pi4 Ai3 Ai4 Si1 ðiÞ ðiÞ T T , for every i 2 S. If i 2 U k ; 8j 2 U uk , pre- and post-multiplying (12) by H and H respectively, and if H Si ¼ Si2 ðiÞ
ðiÞ
i 2 U uk ; 8j 2 U uk , pre- and post-multiplying (13) by HT and H respectively, in these two cases we all can obtain that for every T U i4 < 0, which implies A i4 is nonsingular. Thus for every i 2 S; det sE A i is not identity zero and ST þ Si2 U TA i 2 S; A i4 i2 deg det sE Ai ¼ rankE. According to Definition 1, the system (5) with partly unknown transition rates is regular and impulse free in the time interval ½0; T . Let us consider the Lyapunov functional candidate as V ðxðtÞ; iÞ ¼ xT ðtÞET P i Exðt Þ for the system (5). Let L be the weak infinitesimal generator along the solution of the system (5), then, for each i 2 S, we have
2 3 N ! X T T N TXi þ XTA i þ X p E P E X B A ij j w;i 6 7 i i LV ðxðt Þ; iÞ ¼ 2xT ðt Þ ET Pi þ Si RT Ex_ ðtÞ þ xT ðt Þ pij ET Pj E xðtÞ ¼ gT ðtÞ4 i 5gðt Þ; j¼1 j¼1
0
ð14Þ
where gðtÞ ¼ xT ðt Þ
wT ðt Þ
T
.
Using the similar method in [32], we prove that (8) and (9) imply that
2
T T 6 A X i þ X i Ai þ Xi ¼ 4 i
j¼1
pij ET Pj E sET Pi E X Ti Bw;i 7
Case 1 i 2
ðiÞ Uk
3
N X
5 < 0:
ð15Þ
Q 2i ðiÞ
ðiÞ
ðiÞ
It should be noted that in this case one has pk 6 0, We only need to consider pk < 0. Here since pk ¼ 0 ðiÞ
means that all the elements in the row of the TRM P are known. So in this case, we have pk > 0, since we also have P ^ ij = pðiÞ ^ ij = pðiÞ 6 1 and j2UðiÞ p ¼ 1, from (8) we obtain that 06 p k k uk
Xi ¼
X
p^ ij = pðiÞ X1ij < 0: k
ðiÞ j2U uk
ðiÞ
So if i 2 U k (8) implies (15).
Please cite this article in press as: L. Li, Q. Zhang, Finite-time H1 control for singular Markovian jump systems with partly unknown transition rates, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.04.044
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L. Li, Q. Zhang / Applied Mathematical Modelling xxx (2015) xxx–xxx
ðiÞ ðiÞ ^ ii is lower bounded by pðiÞ ^ ii < pðiÞ ^ ii may take any Case 2 i 2 U uk In this case, as p , we have pd 6 p which implies that p d k h i ðiÞ ðiÞ ^ ii can be further written as a convex combination value between pd ; pk þ e for some e < 0 arbitrarily small. Then p ðiÞ p^ ii ¼ apðiÞ þ ae þ ð1 aÞpd , where a takes value arbitrarily in ½0; 1. And from (9), we can obtain that k
"
# ðiÞ ðiÞ T TXi þ XT A P k pk Pi sPi E X Ti Bw;i A i i þE i
< 0:
ð16Þ
Q 2i
Furthermore we have that
"
ðiÞ ðiÞ ðiÞ ðiÞ T T Xi þ XT A P k pk Pi þ pk Pj pk Pj þ e Pi Pj sPi E A i i þE i
X Ti Bw;i
Q 2i
# < 0:
ð17Þ
Then by (9) and (17) using the nature of the convex combination and Schur complement Lemma, we can obtain that
8j 2 U ðiÞ ; j–i uk " 3 ij
X ¼
# ðiÞ T T Xi þ XTA ^ ii Pi p ^ ii P j pðiÞ Pk þ p A X Ti Bw;i i i þE i k P j sP i E
< 0:
ð18Þ
Q 2i
P ^ ij = p ^ ii pðjÞ 6 1 and ¼ 1, from (18), we know that Since we have 0 6 p p^ ij = p^ ii pðiÞ ðiÞ k k j2U uk ;j–i P ðiÞ ðiÞ 3 ^ ij = p ^ ii pk Xi ¼ j2U ðiÞ ;j–i p Xij < 0. So if i 2 U uk , (9) implies (15) holds. Therefore if (8) and (9) hold, we conclude that uk
(15) holds. From (14) and (15), we have
LV ðxðtÞ; iÞ < sV ðxðtÞ; iÞ þ wT ðt ÞQ 2i wðtÞ:
ð19Þ
Further, (19) can be rewritten as
L½est VðxðtÞ; iÞ < est wT ðtÞQ 2i wðtÞ:
ð20Þ
Integrating (20) from 0 to t, we can obtain
est E fV ðxðtÞ; iÞg < E fV ðxð0Þ; r 0 Þg þ
Z
t
E esh wT ðhÞQ 2i wðhÞ dh:
ð21Þ
0
Noting that
s P 0; t 2 ½0; T and the condition (10), we have
Z t n o E xT ðt ÞET Pi ExðtÞ ¼ E fV ðxðt Þ; iÞg < est E fV ðxð0Þ; r0 Þg þ est esh wT ðhÞQ 2i wðhÞdh 0 2 6 est maxfkmax ðQ 1i Þgc21 þ maxfkmax ðQ 2i Þgd : i2S
i2S
Taking into account that
n o n o n o 1 1 E xT ðt ÞET Pi ExðtÞ ¼ E xT ðt ÞET R2i Q 1i R2i ExðtÞ P minfkmin ðQ 1i ÞgE xT ðt ÞET Ri Exðt Þ : i2S
We obtain 2
maxi2S fkmax ðQ 1i Þgc21 þ maxfkmax ðQ 2i Þgd n o n o n o i2S T T sT E xT ðt ÞET Ri Exðt Þ 6 max kmax Q 1 ð t ÞE P Ex ð t Þ : E x < e i 1i i2S minfkmin ðQ 1i Þg i2S
n o Therefore, it follows that the condition (11) implies E xT ðtÞET Ri ExðtÞ < c22 for all t 2 ½0; T . This completes the proof of the theorem. h Corollary 1. The closed-loop SMJS (5) with wðt Þ ¼ 0 and partly unknown transition rates is SSFTS with respect to ðc1 ; c2 ; T; Rrt Þ, if there exist a scalar s P 0, matrices P i > 0; Q 1i > 0 and Si ; i 2 S satisfying (10) and
ðiÞ ðiÞ T TXi þ XT A P k pk Pj sPi E < 0; A i i þE i
ðiÞ 8j 2 U uk ;
ðiÞ ðiÞ ðiÞ ðiÞ T TXi þ XT A P k þ pd Pi pd Pj pk Pj sPi E < 0; A i i þE i maxfkmax ðQ 1i Þgc21 < minfkmin ðQ 1i Þgc22 esT ; i2S
i2S
ðiÞ
if i 2 U k ; ðiÞ 8j 2 U uk ;
ð22Þ ðiÞ
if i 2 U uk ;
ð23Þ ð24Þ
where the notation of X i is the same as in Theorem 1. Please cite this article in press as: L. Li, Q. Zhang, Finite-time H1 control for singular Markovian jump systems with partly unknown transition rates, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.04.044
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L. Li, Q. Zhang / Applied Mathematical Modelling xxx (2015) xxx–xxx
ðiÞ ðiÞ ^ ii < pðiÞ Remark 2. In Theorem 1, we only consider p and pd < pk , otherwise we can get that all the elements of the ith k row in the TRM P are completely known. At the same time if mi ¼ N 1 in (3), this means that we have only one unknown element. We also can calculate it from the known elements, so that is all the elements of the ith row in the TRM P are completely known. In above cases, by the proof of Theorem 1 we can use (15) instead of (8) or (9). Therefore if mi ¼ N 1 or mi ¼ N in (3) for every i 2 S the underlying system is the one with completely known transition rates. By Theorem 1, Corollary 1 and Remark 2, we have the following corollaries:
Corollary 2. The closed-loop SMJS (5) with completely known transition rates is SSFTB with respect to ðc1 ; c2 ; T; Rrt ; dÞ if there exist a scalar s P 0, matrices Pi > 0; Q 1i > 0; Q 2i > 0 and Si ; i 2 S satisfying (10), (11) and (15). Corollary 3. The closed-loop SMJS (5) with wðt Þ ¼ 0 and completely known transition rates is SSFTS with respect to ðc1 ; c2 ; T; Rrt Þ, if there exist a scalar s P 0, matrices Pi > 0; Q 1i > 0 and Si ; i 2 S satisfying (10), (24) and
T Xi þ XT A A i i þ i
N X
pij ET Pj E sET Pi E < 0;
ð25Þ
j¼1
where the notation of X i is the same as in Theorem 1. Theorem 2. Given T > 0 and wðt Þ satisfying (2), the closed-loop SMJS (5) with partly unknown transition rates controlled by a state feedback controller (4) with K i ¼ Li Y 1 is SSFTB with respect to ðc1 ; c2 ; T; Rrt ; dÞ if there exist scalars i i > 0; Wi > 0; Q 2i > 0; Li and matrices P Si ; i 2 S, such that
2
1 6 Ui
6 D1ij ¼ 6 4 2
2
6 Ui 6 D2ij ¼ 6 4
Bw;i
if i 2 U k ;
3 Ir 0 7 7 7 < 0; 0 5 2 J ij
ðiÞ 8j 2 U uk ;
if i 2 U uk ;
2
g1 In < Ri G1 6 4 1 2
½Ir 0GEY i G
T
Ir
2
d g2 esT c22 c1
ð26Þ
ðiÞ
ð27Þ
0
0
3 0 7 T 12 5G Ri < In ;
ð28Þ
Wi
0 < Q 2i < g2 Iq ; "
ðiÞ
Ni2 HT
Q 2i
ðiÞ 8j 2 U uk ;
Ni1 HT
Q 2i
Bw;i
3 Ir 0 7 7 7 < 0; 0 5 1 J ij
s P 0; g1 > 0; g2 > 0,
ð29Þ c1 g1
# < 0;
ð30Þ
where
qffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffi T qffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffi T 1 pikðiÞ Y i ; . . . ; pikðiÞ Y Ti ; pikðiÞ Y Ti ; . . . ; pikðiÞm Y Ti ; pðiÞ k Y i , J ij ¼ diag 1 l1 lþ1 i n o ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ T Ir NkðiÞ ; . . . ; NkðiÞ ; NkðiÞ ; . . . ; NkðiÞ ; Nj ; kðiÞ ; p 2 k1 ; . . . kl1 ; klþ1 ; . . . kmi ; j , U2i ¼ pd s Y Ti ET þ Y Ti ATi þ l ¼ i; Np ¼ ½Ir 0GEY p G m 0 1 l1 lþ1 i qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffi T qffiffiffiffiffiffiffiffiffiffi I ðiÞ T 2 N i2 ¼ pikðiÞ Y i ; . . . ; pikðiÞm Y Ti ; pðiÞ ; Nj ; Nq ¼ ½Ir 0GEY q GT r ; q 2 LTi BTi þ Ai Y i þ Bi Li , d pk Y i ; J ij ¼ diag NkðiÞ ; . . . ; NkðiÞ mi 0 1 1 i n o ðiÞ ðiÞ i þ T T; R 2 RnðnrÞ is any matrix with full column rank and satisfies ER ¼ 0; G; H are nonsingular k1 ; . . . ; kmi ; j ; Y i ¼ EP Si R I 0 matrices that make GEH ¼ r . 0 0
U1i ¼ ðpii sÞY Ti ET þ Y Ti ATi þ LTi BTi þ Ai Y i þ Bi Li ; Ni1 ¼
Proof. We firstly prove that conditions (26) and (27) imply conditions (8) and (9). From (26) and (27), we can get that Y i is i þ T T , so nonsingular, and Y i ¼ EP Si R
i ET P 0: Y Ti ET ¼ EY i ¼ EP
ð31Þ
Please cite this article in press as: L. Li, Q. Zhang, Finite-time H1 control for singular Markovian jump systems with partly unknown transition rates, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.04.044
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Y i11 Y i12 Denoting H1 Y i GT ¼ , from (31), it is easy to obtain that Y i12 ¼ 0, and Y i11 is symmetric, then we have Y i21 Y i22 Y i11 0 and Y i22 are nonsingular. Therefore we conclude that , so we have Y i11 H1 Y i GT ¼ Y i21 Y i22 I Y 1 0 i11 GT Y 1 and ½ Ir 0 GEY j GT r ¼ Y j11 is nonsingular. Using Lemma 1, there exists 1 i H ¼ 0 Y 1 Y 1 i22 Y i21 Y i11 i22 T 1 T T nðnr Þ is a matrix with full column rank and satisfies ET R ¼ 0. So we have X i , Y i ¼ E Pi þ Si R , where P i > 0 and R 2 R " # 1 1 Ir T Ir 1 T Ir 1 1 T Y j11 0 1 T Y j11 ½ Ir 0 H ¼ H H ½ Ir 0 GEY j G ½ Ir 0 H ¼ H ¼ ET Y 1 H1 ¼ HT HT ET GT GT Y 1 j H H j ¼ E Xj: 0 0 0 0 0 T
ð32Þ
Now pre- and post-multiply (26) by diagfX Ti ; In ; Ir ; . . . ; Ir g and its transposition respectively, pre- and post-multiply (27) by |fflfflfflfflffl{zfflfflfflfflffl} mi
diagfX Ti ; In ; Ir ; . . . ; Ir g and its transposition respectively. And using Schur complement Lemma, we can obtain that (26) and |fflfflfflfflffl{zfflfflfflfflffl} mi þ1
(27) imply that (8) and (9) hold. By (32), letting
2
Q 1i ¼
1 6 Ri 2 G T 4
½Ir 0GEY i GT
1 Ir 0
we have
2 1 2
1 2
T
T T
1 2
ð33Þ
W1 i
0
T
3 0 7 12 5GRi ;
12
T6
E Ri Q 1i Ri E ¼ H H E Ri Ri G 4
½Ir 0GEY i G
T
1 Ir 0
0
3
" 1 0 7 12 12 1 T Y i11 5GRi Ri EHH ¼ H 0 1
Wi
0 0
# H1 ¼ ET X i :
Therefore we obtain that (10) holds. By (28) and (29), we have that In < Q 1i < g1 In , so maxi2S fkmax ðQ 1i Þg < 1
1
g1
; mini2S fkmin ðQ 1i Þg > 1; maxi2S fkmax ðQ 2i Þg < g2 . By (30) we can obtain that (11) holds. Therefore if (26)–(30) hold, the
closed-loop system (5) with partly unknown transition rates is SSFTB with respect to ðc1 ; c2 ; T; Rrt ; dÞ via the state feedback h controller (4) with K i ¼ Li Y 1 i . This completes the proof of the theorem. Corollary 4. Given T > 0 and wðtÞ ¼ 0, the closed-loop SMJS (5) with partly unknown transition rates controlled by a state feedback controller (4) with K i ¼ Li Y 1 is SSFTS with respect to ðc1 ; c2 ; T; Rrt Þ, if there exist scalars i i > 0; Wi > 0; Li and P Si ; i 2 S , such that (28) holds and
2
T 1 6 Ui Ni1 H
4
ðiÞ 8j 2 U uk ;
if i 2 U k ;
3 Ir 0 7 5 < 0;
ðiÞ 8j 2 U uk ;
if i 2 U uk ;
ðiÞ
ð34Þ
ðiÞ
ð35Þ
J 1ij
2
T 2 6 Ui Ni2 H
4
J 2ij
"
3 Ir 0 7 5 < 0;
s P 0; g1 > 0, matrices
esT c22
c1
c1
g1
# < 0;
ð36Þ
where the notations of Y i ; U1i ; U2i ; N i1 ; N i2 ; J 1ij and J 2ij are the same as in Theorem 2. Corollary 5. Given T > 0 and wðtÞ satisfying (2), the closed-loop SMJS (5) with completely known transition rates controlled by a state feedback controller (4) with K i ¼ Li Y 1 is SSFTB with respect to ðc1 ; c2 ; T; Rrt ; dÞ, if there exist scalars i i > 0; Wi > 0; Q 2i > 0; Li and matrices P Si ; i 2 S such that (28)–(30) and
2 6 Ui 6 6 4
Bw;i Q i2
3 Ir 0 7 7 7 < 0; 5 0
s P 0; g1 > 0; g2 > 0,
Ni HT
ð37Þ
J i
Please cite this article in press as: L. Li, Q. Zhang, Finite-time H1 control for singular Markovian jump systems with partly unknown transition rates, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.04.044
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L. Li, Q. Zhang / Applied Mathematical Modelling xxx (2015) xxx–xxx
where Ui ¼ ðpii sÞY Ti ET þ Y Ti ATi þ LTi BTi þ Ai Y i þ Bi Li , N i ¼ J i ¼ diagfN1 ; . . . ; Ni1 ; Niþ1 ; . . . ; NN g; Nj ¼ ½Ir 0GEY j GT
hpffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi i pffiffiffiffiffiffiffiffiffiffi pi1 Y Ti ; . . . ; pii1 Y Ti ; piiþ1 Y Ti ; . . . ; piN Y Ti , Ir , and the other notations are the same as in Theorem 2. 0
Corollary 6. Given T > 0 and wðt Þ ¼ 0, the closed-loop SMJS (5) with completely known transition rates controlled by a state feedback controller (4) with K i ¼ Li Y 1 is SSFTS with respect to ðc1 ; c2 ; T; Rrt Þ, if there exist scalars i i > 0; Wi > 0; Li and P Si ; i 2 S such that (28), (36) and
2
ÞY Ti ET
6 ðpii s 4
þ
Y Ti ATi
þ
LTi BTi
þ Ai Y i þ Bi Li
3 Ir Ni H 7 0 5 < 0; J i
s P 0; g1 > 0, matrices
T
ð38Þ
where the notations of Y i ; N i and J i are the same as in Corollary 5. Then the result of Theorems 1 and 2 will be extended to SSH1 FTB of this system. Theorem 3. Given T > 0 and wðtÞ satisfying (2), the closed-loop SMJS (5) with partly unknown transition rates is SSH1 FTB with respect to ðc1 ; c2 ; T; Rrt ; c; dÞ, if there exist scalars s P 0; c > 0, matrices P i > 0; Q 1i > 0 and Si ; i 2 S satisfying (10) and
K1ij < 0;
8j 2 U ðiÞ uk ;
if i 2 U k ;
K2ij < 0;
8j 2 U ðiÞ uk ;
if i 2 U uk ;
ðiÞ
ð39Þ
ðiÞ
ð40Þ
2
maxfkmax ðQ 1i Þgc21 þ c2 d < minfkmin ðQ 1i Þgc22 esT ; i2S
ð41Þ
i2S
where
2
K ¼4 1 ij
2
K2ij ¼ 4
ðiÞ ðiÞ T T Xi þ XT A TC P k pk Pj sPi E þ C A i i þE i i i
T Dw;i X Ti Bw;i þ C i
c2 I þ DTw;i Dw;i
3 5;
ðiÞ ðiÞ ðiÞ ðiÞ T TXi þ XTA T P k þ pd Pi pd Pj pk P j sPi E A i i þ Ci Ci þ E i
T Dw;i X Ti Bw;i þ C i
c2 I þ DTw;i Dw;i
3 5;
the other notations are the same as in Theorem 1. Proof. It is clearly seen that
2 4
ðiÞ ðiÞ T T Xi þ XT A TC P k pk Pj sPi E þ C A i i þE i i i
T Dw;i X Ti Bw;i þ C i
3
5 c2 I þ DTw;i Dw;i 2 3 " # T
TXi þ XTA i þ ET P ðiÞ pðiÞ Pj sPi E X T Bw;i C A i i i k k i 5þ ¼4 i C DTw;i c2 I
Dw;i < 0:
ð42Þ
Let c2 I , Q 2i , this together with (42) implies (8). In the same way we can get that (40) implies (9). So based on (10), (39)–(41), the SMJS (5) with partly unknown transition rates is SSFTB with respect to ðc1 ; c2 ; T; Rrt ; dÞ. In addition, for the SMJS (5), choose the same Lyapunov function as in Theorem 1, we have
! N X T T T T LV ðxðt Þ; iÞ ¼ 2x ðtÞ E Pi þ Si R Ex_ ðt Þ þ x ðt Þ pij E Pj E xðtÞ T
T Xi þ XTA ¼ xT ð t Þ A i i i þ
N X
!
j¼1
pij ET Pj E xðtÞ þ xT ðtÞX Ti Bw;i wðtÞ þ wT ðtÞBTw;i X i xðtÞ:
j¼1
According to (39) and (40), using the similar method in the proof of Theorem 1, we derive that
LV ðxðt Þ; iÞ 6 sV ðxðt Þ; iÞ þ c2 wT ðt ÞwðtÞ zT ðtÞzðt Þ
ð43Þ
L½est VðxðtÞ; iÞ 6 est ½c2 wT ðtÞwðtÞ zT ðtÞzðtÞ:
ð44Þ
and
Integrating (44) from 0 to t, under zero initial condition, we can obtain Please cite this article in press as: L. Li, Q. Zhang, Finite-time H1 control for singular Markovian jump systems with partly unknown transition rates, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.04.044
L. Li, Q. Zhang / Applied Mathematical Modelling xxx (2015) xxx–xxx
est E fV ðxðtÞ; iÞg 6 E
Z
t
9
esh c2 wT ðhÞwðhÞ zT ðhÞzðhÞ dh :
0
So E
nR t
o R t esh zT ðhÞzðhÞdh 6 0 esh c2 wT ðhÞwðhÞdh, 0
which further implies that
E
Z
T
zT ðtÞzðt Þdt
6 c2 esT
Z
0
T
wT ðtÞwðt Þdt:
0
¼ Therefore (7) holds with c
pffiffiffiffiffiffiffi esT c. This completes the proof of the theorem. h
Corollary 7. Given T > 0 and wðt Þ satisfying (2), the closed-loop SMJS (5) with completely known transition rates is SSH1 FTB with respect to ðc1 ; c2 ; T; Rrt ; c; dÞ, if there exist scalars s P 0; c > 0, matrices Pi > 0; Q 1i > 0; Si ; i 2 S satisfying (10), (41) and
2
T Xi þ X A 6A i i þ 6 i 4 T
N X j¼1
3
pij ET Pj E sET Pi E þ C Ti C i X Ti Bw;i þ C Ti Dw;i 7 2
c I þ
DTw;i Dw;i
7 < 0: 5
ð45Þ
In the following, a feedback controller is designed such that the SMJSs considered are SSH1 FTB. Theorem 4. Given T > 0 and wðtÞ satisfying (2), the closed-loop SMJS (5) with partly unknown transition rates controlled by a state feedback controller (4) with K i ¼ Li Y 1 is SSH1 FTB with respect to ðc1 ; c2 ; T; Rrt ; c; dÞ, if there exist scalars i s P 0; c > 0; g1 > 0, matrices Pi > 0; Wi > 0; Li and Si ; i 2 S, such that (28) and
2
1
6 Ui 6 6 H1ij ¼ 6 6 6 4 2
2 6 Ui
6 6
H2ij ¼ 6 6
6 4
"
2
Bw;i
Y Ti C Ti þ LTi DTi
c2 I
DTw;i
I
Bw;i
Y Ti C Ti þ LTi DTi
c2 I
DTw;i
I
d c2 esT c22
c1
c1
g1
3 Ir 0 7 7 7 7 < 0; 0 7 7 0 5 1 J ij
8j 2 U ðiÞ uk ;
if i 2 U k ;
3 Ir 0 7 7 7 7 < 0; 0 7 7 0 5 2 J ij
8j 2 U ðiÞ uk ;
if i 2 U uk ;
Ni1 HT
ðiÞ
ð46Þ
ðiÞ
ð47Þ
Ni2 HT
# < 0;
ð48Þ
where the notations of U1i ; N i1 ; J 1ij ; U2i ; N i2 ; J 2ij and H are the same as in Theorem 2. The proof of Theorem 4 is similar to the proof of Theorem 2 and thus it is omitted. Corollary 8. Given T > 0 and wðtÞ satisfying (2), the closed-loop SMJS (5) with completely known transition rates controlled by a state feedback controller (4) with K i ¼ Li Y 1 is SSH1 FTB with respect to ðc1 ; c2 ; T; Rrt ; c; dÞ, if there exist scalars i
s P 0; c > 0; g1 > 0, matrices Pi > 0; Wi > 0; Li and Si ; i 2 S, such that (28), (48) and 2
6 Ui 6 6 6 6 6 4
Bw;i
Y Ti C Ti þ LTi DTi
c2 I
DTw;i
I
3 Ir 0 7 7 7 7 < 0; 0 7 7 5 0
Ni HT
ð49Þ
J i
where the notations of Ui ; N i ; J i and H are the same as in Corollary 5. Remark 3. It is clear that Corollary 5, Corollary 6 and Corollary 8 provide sufficient conditions for the solvability of finite-time H1 control problem for SMJSs with completely known transition rates. Similar problems were studied in Theorem 2 and Theorem 4 in [30], but a set of matrices HðiÞ (i 2 S) were needed such that
Please cite this article in press as: L. Li, Q. Zhang, Finite-time H1 control for singular Markovian jump systems with partly unknown transition rates, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.04.044
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L. Li, Q. Zhang / Applied Mathematical Modelling xxx (2015) xxx–xxx
0 6 EP T ðiÞ ¼ PðiÞET ¼ ENXðiÞNT ET 6 HðiÞ; which increased the conservatism and complexity of the design procedure. In this paper, ET X i , ET Y 1 is expressed in a suiti able form (32), furthermore by this expression, Q 1i can be expressed properly in the form (33). So there is less conservatism in Corollaries 5, 6, 8 of this paper. And the results here are in the forms of strict LMIs when the parameter s P 0 is given but the results in [30] are not in the forms of strict LMIs under the same condition.
4. Numerical examples In this section, two examples are given to demonstrate the usability of the proposed results. Example 1. Consider a two-mode SMJS in [30] without uncertainties: 0:2 0:8 1:5 1 0:2 Mode 1: A1 ¼ ; B1 ¼ ; Bw;1 ¼ ; 0:1 2 3 0:5 0:1 0:2 2 1:2 1 1 ; B2 ¼ ; Bw;2 ¼ . Mode 2: A2 ¼ 0:3 1 4 0:5 2 And d ¼ 1. In addition, the transition rate matrix and singular matrix are given by, respectively, 1 0 1:2 1:2 P¼ ; E¼ . Then, we choose R1 ¼ R2 ¼ I2 ; T ¼ 2; c1 ¼ 1. When the parameter s changes between 0 0 1 1 ½1:5; 4:5, in Fig. 1, line 1 shows the local optimal bound of c2 by using Theorem 2 of [30], line 2 shows the local optimal bound of c2 by using Corollary 5 of this paper. By comparing the optimal bound of c2 in the specified range of the parameter s we can conclude the speed of achieving finite-time stabilization of these two methods. The results show that the method we obtain in Corollary 5 is better than the relative theorem in [30]. And we can see that when s ¼ 1:5, the value of c2 cannot been obtained by using Theorem 2 of [30], but the value of c2 can been solved by using Corollary 5 of this paper. This further proves that the method we obtain in this paper has less conservatism. Example 2. Consider the following three-mode SMJS with parameters: 0:5 0:75 0:5 0:01 Mode 1: A1 ¼ ; B1 ¼ ; Bw;1 ¼ ; C 1 ¼ ½ 1 1 ; D1 ¼ 0; Dw;1 ¼ 0:01; 1 2 1 0 3:4 2 2 0:05 ; B2 ¼ ; Bw;2 ¼ ; C 2 ¼ ½ 1 1 ; D2 ¼ 0; Dw;2 ¼ 0:02; Mode 2: A2 ¼ 1 3 0 0 0:2 1 1 0:03 ; B3 ¼ ; Bw;3 ¼ ; C 3 ¼ ½ 1 1 ; D3 ¼ 0; Dw;3 ¼ 0:03. Mode 3: A3 ¼ 1 0:5 3 0 p ffiffiffi 1 0 ; s ¼ 0:1; T ¼ 8; c1 ¼ 1; c2 ¼ 6 and d ¼ 1. The two cases of the transition rate matrices are described by: E¼ 0 0 2 3 2 3 ^ 12 p ^ 13 1:3 0:3 1 1:3 p 4 5 4 ^ 21 p ^ 22 1:1 5. Case1: P ¼ 0:9 2 1:1 ; Case 2: P ¼ p 0:2 0:3 0:5 0:2 0:3 0:5
120 line1 line2 100
c2
80
60
40
20
0 1.5
2
2.5
τ
3
3.5
4
4.5
Fig. 1. Comparison of c2 .
Please cite this article in press as: L. Li, Q. Zhang, Finite-time H1 control for singular Markovian jump systems with partly unknown transition rates, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.04.044
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L. Li, Q. Zhang / Applied Mathematical Modelling xxx (2015) xxx–xxx
x 10
7
Markov Chain
Open−loop state response
4 3.5 3 2.5
x1(t) x2(t)
4 3 2 1 0
2
4 Time t(sec)
6
8
2 1.5 1 0.5 0 −0.5
0
1
2
3
4
5
6
7
8
Time t(sec)
Fig. 2. State response of the open-loop system with completely known transition rates.
Markov Chain
Closed−loop state response
1.2
1
0.8
x1(t) x2(t)
4 3 2 1 0
2
0.6
4 Time t(sec)
6
8
0.4
0.2
0
−0.2
0
1
2
3
4
5
6
7
8
Time t(sec)
Fig. 3. State response of the closed-loop system with completely known transition rates.
x 10
2
Open−loop state response
7
Markov Chain
2.5
1.5
x1(t) x2(t)
4 3 2 1 0
2
4 Time t(sec)
6
8
1
0.5
0
−0.5
−1
0
1
2
3
4
5
6
7
8
Time t(sec)
Fig. 4. State response of the open-loop system with partly unknown transition rates.
Please cite this article in press as: L. Li, Q. Zhang, Finite-time H1 control for singular Markovian jump systems with partly unknown transition rates, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.04.044
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L. Li, Q. Zhang / Applied Mathematical Modelling xxx (2015) xxx–xxx
Markov Chain
1.2
Closed−loop state response
1
0.8
x1(t) x2(t)
4 3 2 1 0
2
4 Time t(sec)
6
8
0.6
0.4
0.2
0
−0.2
0
1
2
3
4
5
6
7
8
Time t(sec)
Fig. 5. State response of the closed-loop system with partly unknown transition rates.
Case1: By solving (28), (48) and (49) in Corollary 8, the controller gains are solved as: K 1 ¼ ½ 11:7947 5:6280 ; K 2 ¼ ½ 7:1390 0:0082 ; K 3 ¼ ½ 16:7359 7:7947 ; c ¼ 1:2813. To demonstrate the effectiveness of Corollary 8, assuming the initial condition xð0Þ ¼ ½0:7; 8:52T , Figs. 2 and 3 show state responses of the open-loop system and the closed-loop system controlled by (4) with completely known transition rates respectively. Case2: By solving (28) and (46)–(48) in Theorem 4, the controller gains are solved as: K 1 ¼ ½ 12:6218 5:5635 ; K 2 ¼ ½ 6:7600 0:0766 ; K 3 ¼ ½ 15:4519 7:2895 ; c ¼ 1:3652. To demonstrate the effectiveness of Theorem 4, assuming the initial condition xð0Þ ¼ ½0:7; 8:52T , Figs. 4 and 5 show state responses of the open-loop system and the closed-loop system controlled by (4) with partly unknown transition rates respectively. The disturbance rejection c we obtain in case 1 is smaller than in case 2. This is reasonable since the former uses more information on the transition rates.
5. Conclusions In this paper, we deal with the problem of finite-time H1 control for a class of singular Markovian jump systems with partly unknown transition rates. The considered systems are more general than the systems with completely known transition rates, which can be viewed as special cases of the ones tackled here. A number of sufficient conditions have been developed to ensure singular stochastic finite-time boundedness and singular stochastic H1 finite-time boundedness of singular Markovian jump systems. We have designed a state feedback controller which guarantees singular stochastic H1 finite-time boundedness of the closed-loop system. For computational convenience, all the conditions are given in the form of strict LMIs. Finally, numerical examples have been provided to demonstrate the effectiveness of the main results. Based on singular stochastic finite-time stability studied in this paper, we will study stochastic finite-time stability which has more extensive application for singular Markovian jump systems in the future. On the other hand, in view of its importance, singular Markovian jump systems with partly unknown transition rates will be further studied. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.apm. 2015.04.044. References [1] L. Dai, Singular Control Systems, Springer, Berlin, 1989. [2] F. Lewis, A survey of linear singular systems, Circ. Syst. Signal Process. 22 (1) (1986) 3–36. [3] S. Xu, J. Lam, Robust stability and stabilization of discrete singular systems: an equivalent characterization, IEEE Trans. Autom. Control 49 (4) (2004) 568–574. [4] Y. Xia, E. Boukas, P. Shi, J. Zhang, Stability and stabilization of continuous-time singular hybrid systems, Automatica 45 (6) (2009) 1504–1509. [5] Y. Xia, P. Shi, G. Liu, D. Rees, Robust mixed H1 =H2 state-feedback control for continuous-time descriptor systems with parameter uncertainties, Circ. Syst. Signal Process. 24 (4) (2005) 431–443.
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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Finite-time H1 control for singular Markovian jump systems with partly unknown transition rates Li Li a,b, Qingling Zhang a,c,⇑ a
Institute of Systems Science, Northeastern University, Shenyang, Liaoning Province 110819, PR China College of Mathematics and Physics, Bohai University, Jinzhou, Liaoning Province 121013, PR China c The State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang, Liaoning Province 110819, PR China b
a r t i c l e
i n f o
Article history: Received 23 July 2013 Received in revised form 22 December 2014 Accepted 17 April 2015 Available online xxxx Keywords: Singular Markovian jump systems Singular stochastic finite-time stability Singular stochastic finite-time boundedness Partly unknown transition rates Linear matrix inequalities (LMIs)
a b s t r a c t This paper investigates the problem of finite-time H1 control for a class of singular Markovian jump systems with partly unknown transition rates. Firstly, sufficient conditions on singular stochastic finite-time boundedness of singular Markovian jump systems with partly unknown transition rates are obtained. Secondly, the results are extended to singular stochastic H1 finite-time boundedness of singular Markovian jump systems with partly unknown transition rates. Then state feedback controllers are designed to ensure the singular stochastic finite-time boundedness and the singular stochastic H1 finite-time boundedness of the underlying closed singular Markovian jump systems in the forms of strict LMIs. Finally, numerical examples are given to illustrate the validity of the proposed methods. Ó 2015 Published by Elsevier Inc.
1. Introduction Singular systems, which are also referred to as generalized systems, descriptor systems, implicit systems, or differential– algebraic systems, have attracted many researchers due to the fact that singular systems have extensive applications in many practical systems such as electrical circuits, power systems, networks and other systems [1,2]. In recent years, many research topics on singular systems have been extensively studied such as stability and stabilization [3,4], H1 control problem [5,6], and so forth. A lot of attention has been paid to the study of Markovian jump systems (MJSs), which are a special kind of hybrid systems, because they have the advantage of better representing physical systems with random changes in their structures and parameters, and have successful applications in economic systems, manufacturing systems and communication systems [7–9]. Many important issues have been studied for this kind of systems, such as stability analysis, stabilization, H2 and H1 control [10–13]. When singular systems experience abrupt changes in their structures, it is natural to model them as singular Markovian jump systems (SMJSs) [14–16]. On the other hand, in most of the studies, complete knowledge of the mode transition rates is required as a prerequisite for analysis and synthesis of MJSs. Therefore, rather than having a large complexity to measure or estimate all the transition rates, it is significant and necessary to further study more general jump systems with partly unknown transition rates. Many attracting problems have been investigated and solved for these systems, such as stability analysis, stabilization [17,18] and H1 filter [19–21]. In many practical applications, one may be interested in not only the stability of the system but also the ⇑ Corresponding author at: Institute of Systems Science, Northeastern University, Shenyang, Liaoning Province 110819, PR China. E-mail address: [email protected] (Q. Zhang). http://dx.doi.org/10.1016/j.apm.2015.04.044 0307-904X/Ó 2015 Published by Elsevier Inc.
Please cite this article in press as: L. Li, Q. Zhang, Finite-time H1 control for singular Markovian jump systems with partly unknown transition rates, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.04.044
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behavior of the system over a fixed short time. For instance, large values of the state are not acceptable in the presence of saturations [22,23]. Compared with the classical Lyapunov asymptotical stability, in order to deal with these transient performances of control dynamic systems, finite-time stability or short-time stability was introduced. Some appealing results were obtained to ensure finite-time stability, finite-time boundedness and finite-time stabilization of various systems including linear systems, nonlinear systems, and stochastic systems [24–28]. To the best of our knowledge, the problem of singular stochastic finite-time H1 control for singular Markovian jump systems (SMJSs) with partly unknown transition rates has not been fully investigated, which motivates the main purpose of our study. In this paper, we are concerned with the problem of finite-time H1 control for SMJSs with partly unknown transition rates. And SMJSs with completely known transition rates are viewed as special cases of the ones tackled here. Firstly, we give the concepts of singular stochastic finite-time boundedness (SSFTB) and singular stochastic H1 finite-time boundedness (SSH1 FTB) of SMJSs. The main contribute lies in tractable sufficient conditions obtained to guarantee SSFTB and SSH1 FTB of SMJSs with partly unknown transition rates. Then, a finite-time H1 state feedback controller is designed in the form of LMIs, which ensures SSH1 FTB of the closed-loop system. Numerical examples are provided to demonstrate the effectiveness of the main results. Notation: Throughout this paper, the notations used are fairly standard, for real symmetric matrices A and B, the notation A P B (respectively, A > B) means that the matrix A B is positive semi-definite (respectively, positive definite). AT represents the transpose of a matrix A, and A1 represents the inverse of a matrix A. kmax B (kmin B) is the maximum (respectively, minimum) eigenvalue of a matrix B. diagf g stands for a block-diagonal matrix. I is the unit matrix with appropriate dimensions, and in a matrix, the term of symmetry is stated by the asterisk ‘⁄’. Let Rn stands for the n-dimensional Euclidean space, Rnm is the set of all real matrices, and k k stands for the Euclidean norm of vectors. Efg denotes the mathematics expectation of the stochastic process or vector. Ln2 ½0; 1Þ stands for the space of n-dimensional square integrable functions on ½0; 1Þ. 2. Basic definitions and lemmas Consider a class of SMJSs as follows:
_ ¼ Aðr t ÞxðtÞ þ Bðr t ÞuðtÞ þ Bw ðr t ÞwðtÞ ExðtÞ zðtÞ ¼ Cðr t ÞxðtÞ þ Dðr t ÞuðtÞ þ Dw ðr t ÞwðtÞ
ð1Þ
;
where xðtÞ 2 Rn is the state vector, uðtÞ 2 Rm is the control input, zðtÞ 2 Rp is the controlled output, E 2 Rnn with rankE ¼ r 6 n; wðtÞ 2 Rq is the disturbance which belongs to Lq2 ½0; 1Þ, and satisfies
Z
T
2
wT ðtÞwðtÞdt 6 d ;
d P 0;
ð2Þ
0
Aðrt Þ; Bðrt Þ; Bw ðrt Þ; Cðr t Þ; Dðr t Þ and Dw ðr t Þ are real known matrices with appropriate dimensions. frt ; t P 0g is a continuous-time Markov process with right continuous trajectories taking values in a finite set given by S ¼ f1; 2; . . . ; Ng with the transition rate matrix (TRM) P , fpij g given by
Prfr tþh ¼ jjr t ¼ ig ¼
pij h þ oðhÞ; i – j ; 1 þ pii h þ oðhÞ; i ¼ j
where h > 0; limh!0 oðhÞ ¼ 0 and pij P 0, for j – i, is transition rate from mode i to j at time t þ h, which satisfies h P pii ¼ Nj¼1;j–i pij . The transition rates of the jumping process may be considered to be partly accessible in this paper. For instance, the TRM of system (1) may be expressed as
3 p11 p^ 12 p^ 1N 6p ^ 22 p2N 7 7 6 21 p 2
P¼6 6 ..
4 . p^ N1
.. .
..
.. 7 7; . 5
.
pN2 pNN
where each unknown element is labeled with a hat ‘ ˆ ’. For notational clarity, 8i 2 S, the set U ðiÞ denotes U ðiÞ ¼ U k [ U uk with ðiÞ
ðiÞ Uk
, fjjpij is known for i 2 ðiÞ Uk
¼
ðiÞ kj
ðiÞ Sg; U uk
ðiÞ ðiÞ ðiÞ fk1 ; k2 ; . . . ; kmi g;
, fjjpij is unknown for i 2 Sg. Moreover, if
ðiÞ Uk
– /, it is further described as
mi 2 f1; 2; . . . ; N 2g;
ðiÞ kj
ðiÞ
ð3Þ ðiÞ Uk
where 2 Zþ ; 1 6 6 N; j ¼ 1; 2; . . . ; mi represents the jth known element of the set P ðiÞ pk , j2UðiÞ pij , and when p^ ii is unknown, it is necessary to provide a lower bound pðiÞ d for it.
in the TRM P. We denote
k
Consider a state feedback controller
uðtÞ ¼ Kðr t ÞxðtÞ;
ð4Þ
Please cite this article in press as: L. Li, Q. Zhang, Finite-time H1 control for singular Markovian jump systems with partly unknown transition rates, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.04.044
L. Li, Q. Zhang / Applied Mathematical Modelling xxx (2015) xxx–xxx
3
where Kðr t Þ is the controller gain to be determined. The SMJS (1) with the controller (4) can be written in the form of the control system as follows:
(
t ÞxðtÞ þ Bw ðr t ÞwðtÞ _ ExðtÞ ¼ Aðr ; zðtÞ ¼ Cðr t ÞxðtÞ þ Dw ðr t ÞwðtÞ
ð5Þ
t Þ ¼ Aðr t Þ þ Bðr t ÞKðrt Þ and Cðr t Þ ¼ Cðr t Þ þ Dðr t ÞKðrt Þ. where Aðr Definition 1 [29] regularity and nonimpulse. (i) The continuous-time SMJS (5) is said to be regular in the time interval ½0; T, if the characteristic polynomial t ÞÞ is not identically zero for all t 2 ½0; T. detðsE Aðr t ÞÞÞ ¼ rankE for all (ii) The continuous-time SMJS (5) is said to be impulse free in the time interval ½0; T, if degðdetðsE Aðr t 2 ½0; T.
Definition 2 ([28,30] singular stochastic finite-time stability (SSFTS)). The closed-loop SMJS (5) with wðtÞ ¼ 0 is said to be SSFTS with respect to ðc1 ; c2 ; T; Rrt Þ, with c1 < c2 and Rrt > 0, if the system is regular and impulse free in the time interval ½0; T and satisfies
EfxT ð0ÞET Rrt Exð0Þg 6 c21 ) EfxT ðtÞET Rrt ExðtÞg < c22 :
ð6Þ
Definition 3 ([28,30] singular stochastic finite-time boundedness (SSFTB)). The closed-loop system (5) which satisfies (2) is said to be SSFTB with respect to ðc1 ; c2 ; T; Rrt ; dÞ, with c1 < c2 and Rrt > 0, if the system is regular and impulse free in the time interval ½0; T and the condition (6) holds. t ÞÞ ensure the existence and uniqueness of impulse free solution of the Remark 1. The regularity and nonimpulse of ðE; Aðr system (5) in the time interval ½0; T. Definition 4 ([28,30] singular stochastic H1 finite-time boundedness (SSH1 FTB)). The closed-loop SMJS (5) is said to be SSH1 FTB with respect to ðc1 ; c2 ; T; Rrt ; c; dÞ, if the closed-loop SMJS (5) is SSFTB with respect to ðc1 ; c2 ; T; Rrt ; dÞ, and under the zero initial condition the controlled output zðtÞ satisfies
E
Z
T
zT ðtÞzðtÞdt
< c2
0
Z
T
wT ðtÞwðtÞdt
ð7Þ
0
for any nonzero wðtÞ which satisfies (2), where c is a prescribed positive scalar. For notational simplicity, in the sequel, for each possible r t ¼ i 2 S, we will denote Aðr t Þ by Ai ; Aw ðr t Þ by Aw;i , and so on. Lemma 1 [31]. For given matrices E; X > 0; Y, if ET X þ Y KT is nonsingular, then there exist matrices S > 0; L such that 1
ES þ LHT ¼ ðET X þ Y KT Þ
, where X; S 2 Rnn ; Y; L 2 RnðnrÞ , and K; H 2 RnðnrÞ are any matrices with full column rank
T
satisfying E K ¼ 0; EH ¼ 0. 3. Main results In this section, we start by developing results that can be used to design a state feedback controller that ensures the SMJS (5) with partly unknown transition rates is SSFTB. Theorem 1. The closed-loop SMJS (5) with partly unknown transition rates is SSFTB with respect to ðc1 ; c2 ; T; Ri ; dÞ, if there exist a scalar s P 0, matrices P i > 0; Q 1i > 0; Q 2i > 0, and Si ; i 2 S satisfying:
X1ij < 0;
ðiÞ 8j 2 U uk ;
if i 2 U k ;
X2ij < 0;
ðiÞ 8j 2 U uk ;
if i 2 U uk ;
1
1
ET X i ¼ ET R2i Q 1i R2i E;
ðiÞ
ð8Þ
ðiÞ
ð9Þ ð10Þ
Please cite this article in press as: L. Li, Q. Zhang, Finite-time H1 control for singular Markovian jump systems with partly unknown transition rates, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.04.044
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maxfkmax ðQ 1i Þgc21 þ maxfkmax ðQ 2i Þgd < minfkmin ðQ 1i Þgc22 esT ; i2S
i2S
ðiÞ
where P k ,
P j2U
ðiÞ k
ð11Þ
i2S
^ ii in the TRM P, and R 2 RnðnrÞ is any pij Pj ; pðiÞ is a given lower bound for the unknown diagonal element p d
T matrix with full column satisfying ET R ¼ 0; X i ¼ ET P i þ Si RT ,
"
# ðiÞ ðiÞ T T Xi þ XTA P k pk P j sPi E X Ti Bw;i A i i þE i ; Q 2i
"
# ðiÞ ðiÞ ðiÞ ðiÞ T T Xi þ XTA P k þ pd P i pd Pj pk Pj sP i E X Ti Bw;i A i i þE i : Q 2i
1 ij
X ¼
2 ij
X ¼
Proof. Firstly, we prove the SMJS (5) with partly unknown transition rates is regular and impulse free in the time interval ½0; T . By Schur complement Lemma, (8) and (9), we have
0 T T B ðiÞ T Xi þ XT A A i i i þ ðpii sÞE P i E < E @pk P j
T Xi þ XT A A i i i þ
1
X ðiÞ l2U k ;l–i
ðiÞ ðiÞ pil Pl C AE 6 0; 8j 2 U uk ; if i 2 U k ;
ð12Þ
ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ T T pðiÞ P k þ pd Pj þ pk Pj E 6 0; 8j 2 U uk ; if i 2 U uk : d s E Pi E < E
ð13Þ
I 0 Since rankE ¼ r 6 n, there exist nonsingular matrices G and H such that GEH ¼ r , then R can be rewritten as 0 0 0 i H ¼ Ai1 Ai2 ; GT Pi G1 ¼ Pi1 Pi2 ; 2 RðnrÞðnrÞ is any nonsingular matrix. Denote GA R ¼ GT , where U U Pi3 Pi4 Ai3 Ai4 Si1 ðiÞ ðiÞ T T , for every i 2 S. If i 2 U k ; 8j 2 U uk , pre- and post-multiplying (12) by H and H respectively, and if H Si ¼ Si2 ðiÞ
ðiÞ
i 2 U uk ; 8j 2 U uk , pre- and post-multiplying (13) by HT and H respectively, in these two cases we all can obtain that for every T U i4 < 0, which implies A i4 is nonsingular. Thus for every i 2 S; det sE A i is not identity zero and ST þ Si2 U TA i 2 S; A i4 i2 deg det sE Ai ¼ rankE. According to Definition 1, the system (5) with partly unknown transition rates is regular and impulse free in the time interval ½0; T . Let us consider the Lyapunov functional candidate as V ðxðtÞ; iÞ ¼ xT ðtÞET P i Exðt Þ for the system (5). Let L be the weak infinitesimal generator along the solution of the system (5), then, for each i 2 S, we have
2 3 N ! X T T N TXi þ XTA i þ X p E P E X B A ij j w;i 6 7 i i LV ðxðt Þ; iÞ ¼ 2xT ðt Þ ET Pi þ Si RT Ex_ ðtÞ þ xT ðt Þ pij ET Pj E xðtÞ ¼ gT ðtÞ4 i 5gðt Þ; j¼1 j¼1
0
ð14Þ
where gðtÞ ¼ xT ðt Þ
wT ðt Þ
T
.
Using the similar method in [32], we prove that (8) and (9) imply that
2
T T 6 A X i þ X i Ai þ Xi ¼ 4 i
j¼1
pij ET Pj E sET Pi E X Ti Bw;i 7
Case 1 i 2
ðiÞ Uk
3
N X
5 < 0:
ð15Þ
Q 2i ðiÞ
ðiÞ
ðiÞ
It should be noted that in this case one has pk 6 0, We only need to consider pk < 0. Here since pk ¼ 0 ðiÞ
means that all the elements in the row of the TRM P are known. So in this case, we have pk > 0, since we also have P ^ ij = pðiÞ ^ ij = pðiÞ 6 1 and j2UðiÞ p ¼ 1, from (8) we obtain that 06 p k k uk
Xi ¼
X
p^ ij = pðiÞ X1ij < 0: k
ðiÞ j2U uk
ðiÞ
So if i 2 U k (8) implies (15).
Please cite this article in press as: L. Li, Q. Zhang, Finite-time H1 control for singular Markovian jump systems with partly unknown transition rates, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.04.044
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ðiÞ ðiÞ ^ ii is lower bounded by pðiÞ ^ ii < pðiÞ ^ ii may take any Case 2 i 2 U uk In this case, as p , we have pd 6 p which implies that p d k h i ðiÞ ðiÞ ^ ii can be further written as a convex combination value between pd ; pk þ e for some e < 0 arbitrarily small. Then p ðiÞ p^ ii ¼ apðiÞ þ ae þ ð1 aÞpd , where a takes value arbitrarily in ½0; 1. And from (9), we can obtain that k
"
# ðiÞ ðiÞ T TXi þ XT A P k pk Pi sPi E X Ti Bw;i A i i þE i
< 0:
ð16Þ
Q 2i
Furthermore we have that
"
ðiÞ ðiÞ ðiÞ ðiÞ T T Xi þ XT A P k pk Pi þ pk Pj pk Pj þ e Pi Pj sPi E A i i þE i
X Ti Bw;i
Q 2i
# < 0:
ð17Þ
Then by (9) and (17) using the nature of the convex combination and Schur complement Lemma, we can obtain that
8j 2 U ðiÞ ; j–i uk " 3 ij
X ¼
# ðiÞ T T Xi þ XTA ^ ii Pi p ^ ii P j pðiÞ Pk þ p A X Ti Bw;i i i þE i k P j sP i E
< 0:
ð18Þ
Q 2i
P ^ ij = p ^ ii pðjÞ 6 1 and ¼ 1, from (18), we know that Since we have 0 6 p p^ ij = p^ ii pðiÞ ðiÞ k k j2U uk ;j–i P ðiÞ ðiÞ 3 ^ ij = p ^ ii pk Xi ¼ j2U ðiÞ ;j–i p Xij < 0. So if i 2 U uk , (9) implies (15) holds. Therefore if (8) and (9) hold, we conclude that uk
(15) holds. From (14) and (15), we have
LV ðxðtÞ; iÞ < sV ðxðtÞ; iÞ þ wT ðt ÞQ 2i wðtÞ:
ð19Þ
Further, (19) can be rewritten as
L½est VðxðtÞ; iÞ < est wT ðtÞQ 2i wðtÞ:
ð20Þ
Integrating (20) from 0 to t, we can obtain
est E fV ðxðtÞ; iÞg < E fV ðxð0Þ; r 0 Þg þ
Z
t
E esh wT ðhÞQ 2i wðhÞ dh:
ð21Þ
0
Noting that
s P 0; t 2 ½0; T and the condition (10), we have
Z t n o E xT ðt ÞET Pi ExðtÞ ¼ E fV ðxðt Þ; iÞg < est E fV ðxð0Þ; r0 Þg þ est esh wT ðhÞQ 2i wðhÞdh 0 2 6 est maxfkmax ðQ 1i Þgc21 þ maxfkmax ðQ 2i Þgd : i2S
i2S
Taking into account that
n o n o n o 1 1 E xT ðt ÞET Pi ExðtÞ ¼ E xT ðt ÞET R2i Q 1i R2i ExðtÞ P minfkmin ðQ 1i ÞgE xT ðt ÞET Ri Exðt Þ : i2S
We obtain 2
maxi2S fkmax ðQ 1i Þgc21 þ maxfkmax ðQ 2i Þgd n o n o n o i2S T T sT E xT ðt ÞET Ri Exðt Þ 6 max kmax Q 1 ð t ÞE P Ex ð t Þ : E x < e i 1i i2S minfkmin ðQ 1i Þg i2S
n o Therefore, it follows that the condition (11) implies E xT ðtÞET Ri ExðtÞ < c22 for all t 2 ½0; T . This completes the proof of the theorem. h Corollary 1. The closed-loop SMJS (5) with wðt Þ ¼ 0 and partly unknown transition rates is SSFTS with respect to ðc1 ; c2 ; T; Rrt Þ, if there exist a scalar s P 0, matrices P i > 0; Q 1i > 0 and Si ; i 2 S satisfying (10) and
ðiÞ ðiÞ T TXi þ XT A P k pk Pj sPi E < 0; A i i þE i
ðiÞ 8j 2 U uk ;
ðiÞ ðiÞ ðiÞ ðiÞ T TXi þ XT A P k þ pd Pi pd Pj pk Pj sPi E < 0; A i i þE i maxfkmax ðQ 1i Þgc21 < minfkmin ðQ 1i Þgc22 esT ; i2S
i2S
ðiÞ
if i 2 U k ; ðiÞ 8j 2 U uk ;
ð22Þ ðiÞ
if i 2 U uk ;
ð23Þ ð24Þ
where the notation of X i is the same as in Theorem 1. Please cite this article in press as: L. Li, Q. Zhang, Finite-time H1 control for singular Markovian jump systems with partly unknown transition rates, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.04.044
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ðiÞ ðiÞ ^ ii < pðiÞ Remark 2. In Theorem 1, we only consider p and pd < pk , otherwise we can get that all the elements of the ith k row in the TRM P are completely known. At the same time if mi ¼ N 1 in (3), this means that we have only one unknown element. We also can calculate it from the known elements, so that is all the elements of the ith row in the TRM P are completely known. In above cases, by the proof of Theorem 1 we can use (15) instead of (8) or (9). Therefore if mi ¼ N 1 or mi ¼ N in (3) for every i 2 S the underlying system is the one with completely known transition rates. By Theorem 1, Corollary 1 and Remark 2, we have the following corollaries:
Corollary 2. The closed-loop SMJS (5) with completely known transition rates is SSFTB with respect to ðc1 ; c2 ; T; Rrt ; dÞ if there exist a scalar s P 0, matrices Pi > 0; Q 1i > 0; Q 2i > 0 and Si ; i 2 S satisfying (10), (11) and (15). Corollary 3. The closed-loop SMJS (5) with wðt Þ ¼ 0 and completely known transition rates is SSFTS with respect to ðc1 ; c2 ; T; Rrt Þ, if there exist a scalar s P 0, matrices Pi > 0; Q 1i > 0 and Si ; i 2 S satisfying (10), (24) and
T Xi þ XT A A i i þ i
N X
pij ET Pj E sET Pi E < 0;
ð25Þ
j¼1
where the notation of X i is the same as in Theorem 1. Theorem 2. Given T > 0 and wðt Þ satisfying (2), the closed-loop SMJS (5) with partly unknown transition rates controlled by a state feedback controller (4) with K i ¼ Li Y 1 is SSFTB with respect to ðc1 ; c2 ; T; Rrt ; dÞ if there exist scalars i i > 0; Wi > 0; Q 2i > 0; Li and matrices P Si ; i 2 S, such that
2
1 6 Ui
6 D1ij ¼ 6 4 2
2
6 Ui 6 D2ij ¼ 6 4
Bw;i
if i 2 U k ;
3 Ir 0 7 7 7 < 0; 0 5 2 J ij
ðiÞ 8j 2 U uk ;
if i 2 U uk ;
2
g1 In < Ri G1 6 4 1 2
½Ir 0GEY i G
T
Ir
2
d g2 esT c22 c1
ð26Þ
ðiÞ
ð27Þ
0
0
3 0 7 T 12 5G Ri < In ;
ð28Þ
Wi
0 < Q 2i < g2 Iq ; "
ðiÞ
Ni2 HT
Q 2i
ðiÞ 8j 2 U uk ;
Ni1 HT
Q 2i
Bw;i
3 Ir 0 7 7 7 < 0; 0 5 1 J ij
s P 0; g1 > 0; g2 > 0,
ð29Þ c1 g1
# < 0;
ð30Þ
where
qffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffi T qffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffi T 1 pikðiÞ Y i ; . . . ; pikðiÞ Y Ti ; pikðiÞ Y Ti ; . . . ; pikðiÞm Y Ti ; pðiÞ k Y i , J ij ¼ diag 1 l1 lþ1 i n o ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ T Ir NkðiÞ ; . . . ; NkðiÞ ; NkðiÞ ; . . . ; NkðiÞ ; Nj ; kðiÞ ; p 2 k1 ; . . . kl1 ; klþ1 ; . . . kmi ; j , U2i ¼ pd s Y Ti ET þ Y Ti ATi þ l ¼ i; Np ¼ ½Ir 0GEY p G m 0 1 l1 lþ1 i qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffi T qffiffiffiffiffiffiffiffiffiffi I ðiÞ T 2 N i2 ¼ pikðiÞ Y i ; . . . ; pikðiÞm Y Ti ; pðiÞ ; Nj ; Nq ¼ ½Ir 0GEY q GT r ; q 2 LTi BTi þ Ai Y i þ Bi Li , d pk Y i ; J ij ¼ diag NkðiÞ ; . . . ; NkðiÞ mi 0 1 1 i n o ðiÞ ðiÞ i þ T T; R 2 RnðnrÞ is any matrix with full column rank and satisfies ER ¼ 0; G; H are nonsingular k1 ; . . . ; kmi ; j ; Y i ¼ EP Si R I 0 matrices that make GEH ¼ r . 0 0
U1i ¼ ðpii sÞY Ti ET þ Y Ti ATi þ LTi BTi þ Ai Y i þ Bi Li ; Ni1 ¼
Proof. We firstly prove that conditions (26) and (27) imply conditions (8) and (9). From (26) and (27), we can get that Y i is i þ T T , so nonsingular, and Y i ¼ EP Si R
i ET P 0: Y Ti ET ¼ EY i ¼ EP
ð31Þ
Please cite this article in press as: L. Li, Q. Zhang, Finite-time H1 control for singular Markovian jump systems with partly unknown transition rates, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.04.044
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Y i11 Y i12 Denoting H1 Y i GT ¼ , from (31), it is easy to obtain that Y i12 ¼ 0, and Y i11 is symmetric, then we have Y i21 Y i22 Y i11 0 and Y i22 are nonsingular. Therefore we conclude that , so we have Y i11 H1 Y i GT ¼ Y i21 Y i22 I Y 1 0 i11 GT Y 1 and ½ Ir 0 GEY j GT r ¼ Y j11 is nonsingular. Using Lemma 1, there exists 1 i H ¼ 0 Y 1 Y 1 i22 Y i21 Y i11 i22 T 1 T T nðnr Þ is a matrix with full column rank and satisfies ET R ¼ 0. So we have X i , Y i ¼ E Pi þ Si R , where P i > 0 and R 2 R " # 1 1 Ir T Ir 1 T Ir 1 1 T Y j11 0 1 T Y j11 ½ Ir 0 H ¼ H H ½ Ir 0 GEY j G ½ Ir 0 H ¼ H ¼ ET Y 1 H1 ¼ HT HT ET GT GT Y 1 j H H j ¼ E Xj: 0 0 0 0 0 T
ð32Þ
Now pre- and post-multiply (26) by diagfX Ti ; In ; Ir ; . . . ; Ir g and its transposition respectively, pre- and post-multiply (27) by |fflfflfflfflffl{zfflfflfflfflffl} mi
diagfX Ti ; In ; Ir ; . . . ; Ir g and its transposition respectively. And using Schur complement Lemma, we can obtain that (26) and |fflfflfflfflffl{zfflfflfflfflffl} mi þ1
(27) imply that (8) and (9) hold. By (32), letting
2
Q 1i ¼
1 6 Ri 2 G T 4
½Ir 0GEY i GT
1 Ir 0
we have
2 1 2
1 2
T
T T
1 2
ð33Þ
W1 i
0
T
3 0 7 12 5GRi ;
12
T6
E Ri Q 1i Ri E ¼ H H E Ri Ri G 4
½Ir 0GEY i G
T
1 Ir 0
0
3
" 1 0 7 12 12 1 T Y i11 5GRi Ri EHH ¼ H 0 1
Wi
0 0
# H1 ¼ ET X i :
Therefore we obtain that (10) holds. By (28) and (29), we have that In < Q 1i < g1 In , so maxi2S fkmax ðQ 1i Þg < 1
1
g1
; mini2S fkmin ðQ 1i Þg > 1; maxi2S fkmax ðQ 2i Þg < g2 . By (30) we can obtain that (11) holds. Therefore if (26)–(30) hold, the
closed-loop system (5) with partly unknown transition rates is SSFTB with respect to ðc1 ; c2 ; T; Rrt ; dÞ via the state feedback h controller (4) with K i ¼ Li Y 1 i . This completes the proof of the theorem. Corollary 4. Given T > 0 and wðtÞ ¼ 0, the closed-loop SMJS (5) with partly unknown transition rates controlled by a state feedback controller (4) with K i ¼ Li Y 1 is SSFTS with respect to ðc1 ; c2 ; T; Rrt Þ, if there exist scalars i i > 0; Wi > 0; Li and P Si ; i 2 S , such that (28) holds and
2
T 1 6 Ui Ni1 H
4
ðiÞ 8j 2 U uk ;
if i 2 U k ;
3 Ir 0 7 5 < 0;
ðiÞ 8j 2 U uk ;
if i 2 U uk ;
ðiÞ
ð34Þ
ðiÞ
ð35Þ
J 1ij
2
T 2 6 Ui Ni2 H
4
J 2ij
"
3 Ir 0 7 5 < 0;
s P 0; g1 > 0, matrices
esT c22
c1
c1
g1
# < 0;
ð36Þ
where the notations of Y i ; U1i ; U2i ; N i1 ; N i2 ; J 1ij and J 2ij are the same as in Theorem 2. Corollary 5. Given T > 0 and wðtÞ satisfying (2), the closed-loop SMJS (5) with completely known transition rates controlled by a state feedback controller (4) with K i ¼ Li Y 1 is SSFTB with respect to ðc1 ; c2 ; T; Rrt ; dÞ, if there exist scalars i i > 0; Wi > 0; Q 2i > 0; Li and matrices P Si ; i 2 S such that (28)–(30) and
2 6 Ui 6 6 4
Bw;i Q i2
3 Ir 0 7 7 7 < 0; 5 0
s P 0; g1 > 0; g2 > 0,
Ni HT
ð37Þ
J i
Please cite this article in press as: L. Li, Q. Zhang, Finite-time H1 control for singular Markovian jump systems with partly unknown transition rates, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.04.044
8
L. Li, Q. Zhang / Applied Mathematical Modelling xxx (2015) xxx–xxx
where Ui ¼ ðpii sÞY Ti ET þ Y Ti ATi þ LTi BTi þ Ai Y i þ Bi Li , N i ¼ J i ¼ diagfN1 ; . . . ; Ni1 ; Niþ1 ; . . . ; NN g; Nj ¼ ½Ir 0GEY j GT
hpffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi i pffiffiffiffiffiffiffiffiffiffi pi1 Y Ti ; . . . ; pii1 Y Ti ; piiþ1 Y Ti ; . . . ; piN Y Ti , Ir , and the other notations are the same as in Theorem 2. 0
Corollary 6. Given T > 0 and wðt Þ ¼ 0, the closed-loop SMJS (5) with completely known transition rates controlled by a state feedback controller (4) with K i ¼ Li Y 1 is SSFTS with respect to ðc1 ; c2 ; T; Rrt Þ, if there exist scalars i i > 0; Wi > 0; Li and P Si ; i 2 S such that (28), (36) and
2
ÞY Ti ET
6 ðpii s 4
þ
Y Ti ATi
þ
LTi BTi
þ Ai Y i þ Bi Li
3 Ir Ni H 7 0 5 < 0; J i
s P 0; g1 > 0, matrices
T
ð38Þ
where the notations of Y i ; N i and J i are the same as in Corollary 5. Then the result of Theorems 1 and 2 will be extended to SSH1 FTB of this system. Theorem 3. Given T > 0 and wðtÞ satisfying (2), the closed-loop SMJS (5) with partly unknown transition rates is SSH1 FTB with respect to ðc1 ; c2 ; T; Rrt ; c; dÞ, if there exist scalars s P 0; c > 0, matrices P i > 0; Q 1i > 0 and Si ; i 2 S satisfying (10) and
K1ij < 0;
8j 2 U ðiÞ uk ;
if i 2 U k ;
K2ij < 0;
8j 2 U ðiÞ uk ;
if i 2 U uk ;
ðiÞ
ð39Þ
ðiÞ
ð40Þ
2
maxfkmax ðQ 1i Þgc21 þ c2 d < minfkmin ðQ 1i Þgc22 esT ; i2S
ð41Þ
i2S
where
2
K ¼4 1 ij
2
K2ij ¼ 4
ðiÞ ðiÞ T T Xi þ XT A TC P k pk Pj sPi E þ C A i i þE i i i
T Dw;i X Ti Bw;i þ C i
c2 I þ DTw;i Dw;i
3 5;
ðiÞ ðiÞ ðiÞ ðiÞ T TXi þ XTA T P k þ pd Pi pd Pj pk P j sPi E A i i þ Ci Ci þ E i
T Dw;i X Ti Bw;i þ C i
c2 I þ DTw;i Dw;i
3 5;
the other notations are the same as in Theorem 1. Proof. It is clearly seen that
2 4
ðiÞ ðiÞ T T Xi þ XT A TC P k pk Pj sPi E þ C A i i þE i i i
T Dw;i X Ti Bw;i þ C i
3
5 c2 I þ DTw;i Dw;i 2 3 " # T
TXi þ XTA i þ ET P ðiÞ pðiÞ Pj sPi E X T Bw;i C A i i i k k i 5þ ¼4 i C DTw;i c2 I
Dw;i < 0:
ð42Þ
Let c2 I , Q 2i , this together with (42) implies (8). In the same way we can get that (40) implies (9). So based on (10), (39)–(41), the SMJS (5) with partly unknown transition rates is SSFTB with respect to ðc1 ; c2 ; T; Rrt ; dÞ. In addition, for the SMJS (5), choose the same Lyapunov function as in Theorem 1, we have
! N X T T T T LV ðxðt Þ; iÞ ¼ 2x ðtÞ E Pi þ Si R Ex_ ðt Þ þ x ðt Þ pij E Pj E xðtÞ T
T Xi þ XTA ¼ xT ð t Þ A i i i þ
N X
!
j¼1
pij ET Pj E xðtÞ þ xT ðtÞX Ti Bw;i wðtÞ þ wT ðtÞBTw;i X i xðtÞ:
j¼1
According to (39) and (40), using the similar method in the proof of Theorem 1, we derive that
LV ðxðt Þ; iÞ 6 sV ðxðt Þ; iÞ þ c2 wT ðt ÞwðtÞ zT ðtÞzðt Þ
ð43Þ
L½est VðxðtÞ; iÞ 6 est ½c2 wT ðtÞwðtÞ zT ðtÞzðtÞ:
ð44Þ
and
Integrating (44) from 0 to t, under zero initial condition, we can obtain Please cite this article in press as: L. Li, Q. Zhang, Finite-time H1 control for singular Markovian jump systems with partly unknown transition rates, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.04.044
L. Li, Q. Zhang / Applied Mathematical Modelling xxx (2015) xxx–xxx
est E fV ðxðtÞ; iÞg 6 E
Z
t
9
esh c2 wT ðhÞwðhÞ zT ðhÞzðhÞ dh :
0
So E
nR t
o R t esh zT ðhÞzðhÞdh 6 0 esh c2 wT ðhÞwðhÞdh, 0
which further implies that
E
Z
T
zT ðtÞzðt Þdt
6 c2 esT
Z
0
T
wT ðtÞwðt Þdt:
0
¼ Therefore (7) holds with c
pffiffiffiffiffiffiffi esT c. This completes the proof of the theorem. h
Corollary 7. Given T > 0 and wðt Þ satisfying (2), the closed-loop SMJS (5) with completely known transition rates is SSH1 FTB with respect to ðc1 ; c2 ; T; Rrt ; c; dÞ, if there exist scalars s P 0; c > 0, matrices Pi > 0; Q 1i > 0; Si ; i 2 S satisfying (10), (41) and
2
T Xi þ X A 6A i i þ 6 i 4 T
N X j¼1
3
pij ET Pj E sET Pi E þ C Ti C i X Ti Bw;i þ C Ti Dw;i 7 2
c I þ
DTw;i Dw;i
7 < 0: 5
ð45Þ
In the following, a feedback controller is designed such that the SMJSs considered are SSH1 FTB. Theorem 4. Given T > 0 and wðtÞ satisfying (2), the closed-loop SMJS (5) with partly unknown transition rates controlled by a state feedback controller (4) with K i ¼ Li Y 1 is SSH1 FTB with respect to ðc1 ; c2 ; T; Rrt ; c; dÞ, if there exist scalars i s P 0; c > 0; g1 > 0, matrices Pi > 0; Wi > 0; Li and Si ; i 2 S, such that (28) and
2
1
6 Ui 6 6 H1ij ¼ 6 6 6 4 2
2 6 Ui
6 6
H2ij ¼ 6 6
6 4
"
2
Bw;i
Y Ti C Ti þ LTi DTi
c2 I
DTw;i
I
Bw;i
Y Ti C Ti þ LTi DTi
c2 I
DTw;i
I
d c2 esT c22
c1
c1
g1
3 Ir 0 7 7 7 7 < 0; 0 7 7 0 5 1 J ij
8j 2 U ðiÞ uk ;
if i 2 U k ;
3 Ir 0 7 7 7 7 < 0; 0 7 7 0 5 2 J ij
8j 2 U ðiÞ uk ;
if i 2 U uk ;
Ni1 HT
ðiÞ
ð46Þ
ðiÞ
ð47Þ
Ni2 HT
# < 0;
ð48Þ
where the notations of U1i ; N i1 ; J 1ij ; U2i ; N i2 ; J 2ij and H are the same as in Theorem 2. The proof of Theorem 4 is similar to the proof of Theorem 2 and thus it is omitted. Corollary 8. Given T > 0 and wðtÞ satisfying (2), the closed-loop SMJS (5) with completely known transition rates controlled by a state feedback controller (4) with K i ¼ Li Y 1 is SSH1 FTB with respect to ðc1 ; c2 ; T; Rrt ; c; dÞ, if there exist scalars i
s P 0; c > 0; g1 > 0, matrices Pi > 0; Wi > 0; Li and Si ; i 2 S, such that (28), (48) and 2
6 Ui 6 6 6 6 6 4
Bw;i
Y Ti C Ti þ LTi DTi
c2 I
DTw;i
I
3 Ir 0 7 7 7 7 < 0; 0 7 7 5 0
Ni HT
ð49Þ
J i
where the notations of Ui ; N i ; J i and H are the same as in Corollary 5. Remark 3. It is clear that Corollary 5, Corollary 6 and Corollary 8 provide sufficient conditions for the solvability of finite-time H1 control problem for SMJSs with completely known transition rates. Similar problems were studied in Theorem 2 and Theorem 4 in [30], but a set of matrices HðiÞ (i 2 S) were needed such that
Please cite this article in press as: L. Li, Q. Zhang, Finite-time H1 control for singular Markovian jump systems with partly unknown transition rates, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.04.044
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L. Li, Q. Zhang / Applied Mathematical Modelling xxx (2015) xxx–xxx
0 6 EP T ðiÞ ¼ PðiÞET ¼ ENXðiÞNT ET 6 HðiÞ; which increased the conservatism and complexity of the design procedure. In this paper, ET X i , ET Y 1 is expressed in a suiti able form (32), furthermore by this expression, Q 1i can be expressed properly in the form (33). So there is less conservatism in Corollaries 5, 6, 8 of this paper. And the results here are in the forms of strict LMIs when the parameter s P 0 is given but the results in [30] are not in the forms of strict LMIs under the same condition.
4. Numerical examples In this section, two examples are given to demonstrate the usability of the proposed results. Example 1. Consider a two-mode SMJS in [30] without uncertainties: 0:2 0:8 1:5 1 0:2 Mode 1: A1 ¼ ; B1 ¼ ; Bw;1 ¼ ; 0:1 2 3 0:5 0:1 0:2 2 1:2 1 1 ; B2 ¼ ; Bw;2 ¼ . Mode 2: A2 ¼ 0:3 1 4 0:5 2 And d ¼ 1. In addition, the transition rate matrix and singular matrix are given by, respectively, 1 0 1:2 1:2 P¼ ; E¼ . Then, we choose R1 ¼ R2 ¼ I2 ; T ¼ 2; c1 ¼ 1. When the parameter s changes between 0 0 1 1 ½1:5; 4:5, in Fig. 1, line 1 shows the local optimal bound of c2 by using Theorem 2 of [30], line 2 shows the local optimal bound of c2 by using Corollary 5 of this paper. By comparing the optimal bound of c2 in the specified range of the parameter s we can conclude the speed of achieving finite-time stabilization of these two methods. The results show that the method we obtain in Corollary 5 is better than the relative theorem in [30]. And we can see that when s ¼ 1:5, the value of c2 cannot been obtained by using Theorem 2 of [30], but the value of c2 can been solved by using Corollary 5 of this paper. This further proves that the method we obtain in this paper has less conservatism. Example 2. Consider the following three-mode SMJS with parameters: 0:5 0:75 0:5 0:01 Mode 1: A1 ¼ ; B1 ¼ ; Bw;1 ¼ ; C 1 ¼ ½ 1 1 ; D1 ¼ 0; Dw;1 ¼ 0:01; 1 2 1 0 3:4 2 2 0:05 ; B2 ¼ ; Bw;2 ¼ ; C 2 ¼ ½ 1 1 ; D2 ¼ 0; Dw;2 ¼ 0:02; Mode 2: A2 ¼ 1 3 0 0 0:2 1 1 0:03 ; B3 ¼ ; Bw;3 ¼ ; C 3 ¼ ½ 1 1 ; D3 ¼ 0; Dw;3 ¼ 0:03. Mode 3: A3 ¼ 1 0:5 3 0 p ffiffiffi 1 0 ; s ¼ 0:1; T ¼ 8; c1 ¼ 1; c2 ¼ 6 and d ¼ 1. The two cases of the transition rate matrices are described by: E¼ 0 0 2 3 2 3 ^ 12 p ^ 13 1:3 0:3 1 1:3 p 4 5 4 ^ 21 p ^ 22 1:1 5. Case1: P ¼ 0:9 2 1:1 ; Case 2: P ¼ p 0:2 0:3 0:5 0:2 0:3 0:5
120 line1 line2 100
c2
80
60
40
20
0 1.5
2
2.5
τ
3
3.5
4
4.5
Fig. 1. Comparison of c2 .
Please cite this article in press as: L. Li, Q. Zhang, Finite-time H1 control for singular Markovian jump systems with partly unknown transition rates, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.04.044
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L. Li, Q. Zhang / Applied Mathematical Modelling xxx (2015) xxx–xxx
x 10
7
Markov Chain
Open−loop state response
4 3.5 3 2.5
x1(t) x2(t)
4 3 2 1 0
2
4 Time t(sec)
6
8
2 1.5 1 0.5 0 −0.5
0
1
2
3
4
5
6
7
8
Time t(sec)
Fig. 2. State response of the open-loop system with completely known transition rates.
Markov Chain
Closed−loop state response
1.2
1
0.8
x1(t) x2(t)
4 3 2 1 0
2
0.6
4 Time t(sec)
6
8
0.4
0.2
0
−0.2
0
1
2
3
4
5
6
7
8
Time t(sec)
Fig. 3. State response of the closed-loop system with completely known transition rates.
x 10
2
Open−loop state response
7
Markov Chain
2.5
1.5
x1(t) x2(t)
4 3 2 1 0
2
4 Time t(sec)
6
8
1
0.5
0
−0.5
−1
0
1
2
3
4
5
6
7
8
Time t(sec)
Fig. 4. State response of the open-loop system with partly unknown transition rates.
Please cite this article in press as: L. Li, Q. Zhang, Finite-time H1 control for singular Markovian jump systems with partly unknown transition rates, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.04.044
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L. Li, Q. Zhang / Applied Mathematical Modelling xxx (2015) xxx–xxx
Markov Chain
1.2
Closed−loop state response
1
0.8
x1(t) x2(t)
4 3 2 1 0
2
4 Time t(sec)
6
8
0.6
0.4
0.2
0
−0.2
0
1
2
3
4
5
6
7
8
Time t(sec)
Fig. 5. State response of the closed-loop system with partly unknown transition rates.
Case1: By solving (28), (48) and (49) in Corollary 8, the controller gains are solved as: K 1 ¼ ½ 11:7947 5:6280 ; K 2 ¼ ½ 7:1390 0:0082 ; K 3 ¼ ½ 16:7359 7:7947 ; c ¼ 1:2813. To demonstrate the effectiveness of Corollary 8, assuming the initial condition xð0Þ ¼ ½0:7; 8:52T , Figs. 2 and 3 show state responses of the open-loop system and the closed-loop system controlled by (4) with completely known transition rates respectively. Case2: By solving (28) and (46)–(48) in Theorem 4, the controller gains are solved as: K 1 ¼ ½ 12:6218 5:5635 ; K 2 ¼ ½ 6:7600 0:0766 ; K 3 ¼ ½ 15:4519 7:2895 ; c ¼ 1:3652. To demonstrate the effectiveness of Theorem 4, assuming the initial condition xð0Þ ¼ ½0:7; 8:52T , Figs. 4 and 5 show state responses of the open-loop system and the closed-loop system controlled by (4) with partly unknown transition rates respectively. The disturbance rejection c we obtain in case 1 is smaller than in case 2. This is reasonable since the former uses more information on the transition rates.
5. Conclusions In this paper, we deal with the problem of finite-time H1 control for a class of singular Markovian jump systems with partly unknown transition rates. The considered systems are more general than the systems with completely known transition rates, which can be viewed as special cases of the ones tackled here. A number of sufficient conditions have been developed to ensure singular stochastic finite-time boundedness and singular stochastic H1 finite-time boundedness of singular Markovian jump systems. We have designed a state feedback controller which guarantees singular stochastic H1 finite-time boundedness of the closed-loop system. For computational convenience, all the conditions are given in the form of strict LMIs. Finally, numerical examples have been provided to demonstrate the effectiveness of the main results. Based on singular stochastic finite-time stability studied in this paper, we will study stochastic finite-time stability which has more extensive application for singular Markovian jump systems in the future. On the other hand, in view of its importance, singular Markovian jump systems with partly unknown transition rates will be further studied. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.apm. 2015.04.044. References [1] L. Dai, Singular Control Systems, Springer, Berlin, 1989. [2] F. Lewis, A survey of linear singular systems, Circ. Syst. Signal Process. 22 (1) (1986) 3–36. [3] S. Xu, J. Lam, Robust stability and stabilization of discrete singular systems: an equivalent characterization, IEEE Trans. Autom. Control 49 (4) (2004) 568–574. [4] Y. Xia, E. Boukas, P. Shi, J. Zhang, Stability and stabilization of continuous-time singular hybrid systems, Automatica 45 (6) (2009) 1504–1509. [5] Y. Xia, P. Shi, G. Liu, D. Rees, Robust mixed H1 =H2 state-feedback control for continuous-time descriptor systems with parameter uncertainties, Circ. Syst. Signal Process. 24 (4) (2005) 431–443.
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