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Less conservative stabilization conditions for Markovian jump systems with partly unknown transition probabilities
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Journal of the Franklin Institute 351 (2014) 3042–3052 www.elsevier.com/locate/jfranklin
Short communication
Less conservative stabilization conditions for Markovian jump systems with partly unknown transition probabilities Sung Hyun Kimn School of Electrical Engineering, University of Ulsan, Daehak-ro 93, Nam-Gu, Ulsan 680-749, Republic of Korea Received 11 June 2013; received in revised form 1 October 2013; accepted 27 December 2013 Available online 14 January 2014
Abstract This paper is devoted to the robust stabilization problem for a class of MJLSs with mode transition probabilities taken to be partly known, unknown, and unknown but with known bounds. To meet the needs of less conservative stabilization conditions, this paper proposes a valuable approach that (1) can express the conditions in terms of matrix inequalities with homogeneous polynomial dependence on partly unknown transition probabilities and (2) can include all possible slack variables related to transition probability constraints in the relaxation process coupled with deriving a finite set of linear matrix inequalities (LMIs). Finally, two numerical examples are reported to illustrate the effectiveness of the derived stabilization conditions. & 2014 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction Over the past several years, Markovian jump linear systems (MJLSs) [1] have received considerable attention among control researchers because such systems are suitable to represent a class of dynamic systems subject to random abrupt variations in their structures. Thus, tremendous effort has been devoted to theoretical research on MJLSs with mode transitions governed by a Markov stochastic process (see, for example, [2–5,15,19–21]). In addition, the MJLS model has been extensively applied in many practical applications, such as networked n
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0016-0032/$32.00 & 2014 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfranklin.2013.12.023
S. Hyun Kim / Journal of the Franklin Institute 351 (2014) 3042–3052
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control systems (NCSs) [16,17,26] and actuator saturation [18]. A recent and interesting extension is the consideration of MJLSs with uncertain or partly unknown transition probabilities [22], which has emerged as a topic of significant interest because it is practically difficult to obtain the exact values of transition probabilities. In spite of this trend, most studies have focused on the case in which the mode transition probabilities are known exactly a priori, which may actually be a major source of system performance degradation or even instability in the presence of incomplete knowledge of transition probabilities (see [10,14] and references therein). Motivated by the above concerns, there has recently been significant progress on the analysis and synthesis of MJLSs with incomplete knowledge of probabilities, which can be classified into two categories: relative to uncertain transition probabilities and unknown transition probabilities. Specifically, the former has focused on shedding some light on the robustness problem of the norm-bounded or polytopic uncertainties presumed in the transition probability [8,9,11,12,17]. The latter has made considerable efforts to address MJLSs with partly unknown transition probabilities, which is very useful in the case in which there is no bound information for imprecise probabilities [13,14]. However, despite the fact that, with the development of computer technologies, the derivation of a set of less conservative stability analysis and stabilization conditions has become a key issue, little progress has been made toward reducing the conservatism that stems from incomplete knowledge of mode transition probabilities, which is the motivation behind this study. This paper considers a general framework for solving the stabilization problem for a class of MJLSs with transition probabilities taken to be partly known, unknown, and unknown but with known bounds. That is, MJLSs with uncertain or completely unknown transition probabilities correspond to two extreme cases of the considered systems. The distinguishing feature of the proposed approach is that it makes an impact on meeting the needs of less conservative stability analysis and stabilization conditions as an alternative to the aforementioned open problem. In this light, the main contributions of this paper can be highlighted as follows. First, we can express the stability analysis and stabilization conditions in terms of matrix inequalities with homogeneous polynomial dependence on partly unknown mode transition probabilities by defining three additional sets with respect to the measurability of the transition probabilities. Second, we can include all possible slack variables related to transition probability constraints in the relaxation process coupled with deriving a finite set of linear matrix inequalities (LMIs), which allows the convexity of the derived matrix inequalities to be sufficiently exploited. Finally, two numerical examples are given to illustrate the effectiveness of the derived stabilization conditions. Notation: Throughout this paper, standard notion will be adopted. The notations X Z Y and X4Y indicate that X Y is positive semi-definite and positive definite, respectively. In symmetric block matrices, ðnÞ is used as an ellipsis for terms that are induced by symmetry. For any square matrix Q, HeðQÞ ¼ Q þ QT , where QT denotes the transpose of Q. The notation diagðA; BÞ indicates a diagonal matrix with diagonal entries A and B; Nþ r ¼ f1; 2; …; rg; and colðv1 ; v2 ; …; vn Þ ¼ ½vT1 vT2 …vTn T for scalars or vectors vi. For real submatrices Si and Sij with appropriate dimensions, ½Si i A fa1 ;…;an g ¼ ½Sa1 …San and 2
Sa1 a1
6 ½Sij i;j A fa1 ;…;an g ¼ 4 ⋮ San a1
⋯
Sa1 an
⋱ ⋯
San an
3
⋮ 7 5:
The notation ei denotes the block entry matrix such that ½Sij i;j A Nþr ¼ ∑ri ¼ 1 ∑rj ¼ 1 eTi Sij ej .
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S. Hyun Kim / Journal of the Franklin Institute 351 (2014) 3042–3052
2. System description and preliminaries Consider the following MJLSs defined on a complete probability space ðΩ; F ; PÞ: ( x_ ðtÞ ¼ AðrðtÞÞxðtÞ þ BðrðtÞÞuðtÞ þ EðrðtÞÞpðtÞ; xð0Þ ¼ x0 ; qðtÞ ¼ GðrðtÞÞxðtÞ þ HðrðtÞÞuðtÞ;
ð1Þ
where xðtÞA Rnx and uðtÞA Rnu denote the state and the control input, respectively; x0 denotes the initial state; and pðtÞA Rnp and qðtÞA Rnq describe the structured feedback uncertainty such that pðtÞ ¼ ΔðtÞqðtÞ and ΔT ðtÞΔðtÞ r I. Here, frðtÞ; t Z 0g denotes a continuous-time Markov process on the probability space that takes the values in a finite state space Nþ s and has the mode transition probabilities ( π ij δt þ oðδtÞ if ja i; Prðrðt þ δtÞ ¼ jjrðtÞ ¼ iÞ ¼ ð2Þ 1 þ π ii δt þ oðδtÞ if j ¼ i; where δt40, limδt-0 ðoðδtÞ=δtÞ ¼ 0, and πij denotes the transition rate from mode i to mode j at time t þ δt. Furthermore, the mode transition rate matrix Π ¼ ½π ij i;j A Nþs belongs to ( s 1 if ja i SΠ 9 ½π ij i;j A Nþs j0 ¼ ∑ π ij ; 0r μij π ij ; μij ¼ ; 8 i; jA Nþ : s 1 otherwise j¼1 Next, to simplify the notation, we set Ai ¼ AðrðtÞ ¼ iÞ, Bi ¼ BðrðtÞ ¼ iÞ, E i ¼ EðrðtÞ ¼ iÞ, Gi ¼ GðrðtÞ ¼ iÞ, and H i ¼ HðrðtÞ ¼ iÞ, where the initial operation mode rð0Þ ¼ r 0 . For later convenience, we also define three additional sets with respect to the measurability of πij for i; jA Nþ s : þ Dþ i 9 fjjπ ij is known; 8 jA Ns g;
Di 9 fjjπ ij is unknown; 8j A Nþ s g;
D i 9 fjjπ ij is unknown but bounded as αij r π ij rβij ; 8 jA Nþ s g:
þ þ The following properties are then satisfied: Dþ i [ Di ¼ Ns , D i DDi , for all iA Ns . In addition, if the transition rate can be fully measured, then Di ¼ D i ¼ ∅.
Remark 1. In this paper, the transition rate πij is set to be partly known, that is, jA fDþ i ; Di jD i a ∅g. Lemma 1 (de Farias et al. [6]). The nominal system of Eq. (1) with zero input is said to be mean square stable if there exists a set of Pi for all iA Nþ s such that 04HeðPi Ai Þþ ∑sj ¼ 1 π ij Pj ; 8i A Nþ . s Lemma 2. The system of Eq. (1) with zero input is said to be robustly mean square stable if there exists a set of Pi, for all i A Nþ s , such that 2 3 s T π ij Pj þ Gi Gi Pi E i 7 6 HeðPi Ai Þ þ j ∑ ¼1 ð3Þ 044 5; 8 iA Nþ s : ðnÞ I Proof. From Lemma 1, it follows that 04xT ðtÞðHeðPi Ai Þ þ ∑sj ¼ 1 π ij Pj Þ x(t), which is in line with 04HeðxT ðtÞPi x_ ðtÞÞ þ xT ðtÞð∑sj ¼ 1 π ij Pj ÞxðtÞ. That is, in the sense that x_ ðtÞ ¼ Ai xðtÞ þ Ei pðtÞ,
S. Hyun Kim / Journal of the Franklin Institute 351 (2014) 3042–3052
the stability criterion becomes s
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!
04xT ðtÞ HeðPi Ai Þ þ ∑ π ij Pj xðtÞ þ 2xT ðtÞPi E i pðtÞ:
ð4Þ
j¼1
Furthermore, from pðtÞ ¼ ΔðtÞqðtÞ and ΔT ðtÞΔðtÞ r I, we can get qT ðtÞqðtÞZ pT ðtÞpðtÞ, i.e., ðGi xðtÞÞT ðGi xðtÞÞ Z pT ðtÞpðtÞ. Hence, by S-procedure, combining (4) and qT ðtÞqðtÞZ pT ðtÞpðtÞ yields 04xT ðtÞðHeðPi Ai Þ þ ∑sj ¼ 1 π ij Pj ÞxðtÞ þ 2xT ðtÞPi E i pðtÞ þ ðGi xðtÞÞT ðGi xðtÞÞ pT ðtÞpðtÞ, that is, 2 3 s " # " #T xðtÞ 6 HeðPi Ai Þ þ ∑ π ij Pj þ GTi Gi Pi E i 7 xðtÞ j¼1 04 : ð5Þ 4 5 pðtÞ pðtÞ ðnÞ I Therefore, we can see that Eq. (5) is assured by Eq. (3).
□
3. Main results Here, to facilitate the derivation, we define Π i ¼ ½π ij TjA Di and π i ¼ ∑j A Dþi π ij . 3.1. Stability analysis This section is devoted to addressing the robust stability analysis problem of MJLSs subject to þ Dþ i a Ns for any i in the absence of control input. Theorem 3. Suppose that there exist Hi, Sij, Xij, Y ij A Rðnx þnp Þðnx þnp Þ and symmetric matrices Pi A Rnx nx such that for all iA Nþ s , " # L0 ½Lj j A Di 04 ; ð6Þ ðnÞ ½Ljℓ j;ℓ A Di 0oHeðY ij Þ;
8 jA Di ;
0oHeðX ij Þ;
8 jA D i ;
ð7Þ
ð8Þ
where L0 ¼ M0 þ S 0 þ X 0 , Lj ¼ Mj þ S j þ X j þ Y j , Ljℓ ¼ S jℓ þ X jℓ in which ! T T M0 ¼ e1 HeðPi Ai Þ þ ∑ π ij Pj þ Gi Gi e1 þ He eT1 Pi Ei e2 þ eT2 ð I Þe2 ;
1 Pj e1 ; 2 S jℓ jℓ4j ¼ Sij þ Siℓ ;
Mj ¼ eT1
j A Dþ i
S 0 ¼ π i HeðH i Þ;
S j ¼ H i þ π i Sij ;
S jℓ jℓoj ¼ STij þ STiℓ ;
S jj ¼ He Sij ;
(
ðαij þ βij ÞX ij if jA D i Y j ¼ μij Y ij ; X 0 ¼ ∑ αij βij HeðX ij Þ; X j ¼ 0 otherwise jAD ( i 1 if ja i Heð X ij Þ if jA D i X jℓ jj a ℓ ¼ 0; X jj ¼ : ; μij ¼ 1 otherwise 0 otherwise Then, the unforced system of Eq. (1) is robustly mean square stable.
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Proof. The stability criterion in Eq. (3) can be rewritten as ! 04eT1 HeðPi Ai Þ þ ∑ π ij Pj þ ∑ π ij Pj þ GTi Gi e1 j A Di
j A Dþ i
8 iA Nþ s ;
þHeðeT1 Pi Ei e2 Þ þ eT2 ð IÞe2 ; which is equivalent to " #T " M0 I 04 ðnÞ Πi I
½Mj j A Di 0
#"
I
ð9Þ # 8 iA Nþ s :
;
Πi I
ð10Þ
Furthermore, from the constraint that Π A SΠ , we can obtain 0 r ∑ μij π ij HeðY ij Þ; j A Di
0 ¼ He
0oHeðY ij Þ;
8 jA Di ;
! π i I þ ∑ π ij I j A Di
iA Nþ s ;
!! H i þ ∑ π ij Sij j A Di
;
8i A Nþ s ;
which leads to Eq. (7), " #T " # #" 0 ½Y j j A Di I I 0r ; ðnÞ 0 Πi I Πi I "
I 0¼ Πi I
#T "
S0 ½S j j A Di ðnÞ ½S jℓ j;ℓ A Di
#"
ð11Þ
# I : Πi I
ð12Þ
In addition, for the transition rate such that π ij A D i , the following inequalities are satisfied: 0 r ∑ ðπ ij αij Þðπ ij βij ÞHeðX ij Þ; j A Di
0oHeðX ij Þ;
which leads to Eq. (8), #" " #T " # X0 ½X j j A Di I I 0r : ðnÞ ½X jℓ j;ℓ A Di Πi I Πi I
8 jA D i ;
iA Nþ s ;
ð13Þ
As a result, the so-called S-procedure [7] enables Eq. (10) subject to Eqs. (11)–(13) to be expressed as " #T " # #" L0 ½Lj j A Di I I 04 ; 8 iA Nþ s ; ðnÞ ½Ljℓ j;ℓ A Di Πi I Πi I which holds if Eq. (6) holds.
□
Corollary 4. Suppose that there exist matrices Hi, Sij, Xij, Y ij A Rnx nx and symmetric matrices Pi A Rnx nx such that Eqs. (6), (8), (7) for all iA Nþ π ij Pj and s , where M0 ¼ HeðPi Ai Þ þ ∑j A Dþ i Mj ¼ 12 Pj . Then, the nominal system of Eq. (1) with zero input is mean square stable. Proof. By Lemma 1, the mean square stability criterion is given by 04HeðPi Ai Þþ ∑j A Dþi π ij Pj þ ∑j A Di π ij Pj . The remaining relaxation process is analogous to the proof of Theorem 3. □
S. Hyun Kim / Journal of the Franklin Institute 351 (2014) 3042–3052
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3.2. Control synthesis This section is devoted to addressing the robust stabilization problem of MJLSs subject to þ Dþ i a Ns for any i. To this end, let us consider the state-feedback control law 8 iA Nþ s ;
uðtÞ ¼ F i xðtÞ;
ð14Þ
where F i ð ¼ FðrðtÞÞÞ denotes the controller gain to be designed. Then, the closed-loop system under Eq. (14) is given by ( x_ ðtÞ ¼ A~ i xðtÞ þ Ei pðtÞ; xð0Þ ¼ x0 ; ð15Þ ~ i xðtÞ; qðtÞ ¼ G ~ i ¼ Gi þ H i F i . Thus, in light of Eq. (15), applying the Schur where A~ i ¼ Ai þ Bi F i and G complement to Eq. (3) yields 2 3 0 I G~ i s 6 7 6 7 ð16Þ 046 ðnÞ HeðPi A~ i Þ þ ∑ π ij Pj Pi Ei 7: 4 5 j¼1 0 ðnÞ I Theorem 5. Suppose that there exist matrices F i ARnu nx , H~ i , S~ ij , X~ ij , Y~ ij A Rðnq þnx þnp Þðnq þnx þnp Þ and symmetric matrices P i and Qij A Rnx nx such that for all iA Nþ s , 2 3 ½L~ j j A Di L~ 0 5; 044 ð17Þ ðnÞ ½L~ jℓ j;ℓ A Di " 0r
Qij
Pi
#
ðnÞ P j
;
8 jA Nþ s =fig;
0oHeðY~ ij Þ;
8 jA Di ;
0oHeðX~ ij Þ;
8j A D i ;
ð18Þ ð19Þ
ð20Þ
~ 0 þ S~ 0 þ X~ 0 , L~ j ¼ M ~ j þ S~ j þ X~ j þ Y~ j , and L~ jℓ ¼ S~ jℓ þ X~ jℓ in which where L~ 0 ¼ M ~ 0 ¼ eT ð IÞe1 þ eT ðGi P i þ H i F i Þe2 þ eT HeðAi P i þ Bi F i Þ M 1 1 2 þ ∑ ðcij π ij Qij þ ð1 cij Þπ ij P i ÞÞe2 þ HeðeT2 E i e3 Þ þ eT3 ð IÞe3 ; j A Dþ i
1 T 1 ~ cij Qij þ 1 cij P i e2 ; S~ 0 ¼ π i He H~ i ; S~ j ¼ H~ i þ π i S~ ij ; M j ¼ e2 2 2 T T S~ jj ¼ HeðS~ ij Þ; S~ jℓ jℓ4j ¼ S~ ij þ S~ iℓ ; S~ jℓ jℓoj ¼ S~ ij þ S~ iℓ ; Y~ j ¼ μij Y~ ij ; X~ 0 ¼ ∑ αij βij HeðX~ ij Þ; X~ jℓ jj a ℓ ¼ 0;
( X~ j ¼
j A Di
(
ðαij þ βij ÞX~ ij
if jA D i
0
otherwise
;
X~ jj ¼
Heð X~ ij Þ
if jA D i
0
otherwise
;
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S. Hyun Kim / Journal of the Franklin Institute 351 (2014) 3042–3052
μij ¼
1
if ja i
1
otherwise
;
cij ¼
1
if j a i
0
otherwise
:
Then, the closed-loop system in Eq. (15) is robustly mean square stable. Proof. Let us define P i ¼ diagðI; P i ; IÞ, where P i ¼ Pi 1 . Then, pre- and post-multiplying Eq. (16) by P i becomes 2 3 0 I Gi P i þ H i F i s 6 7 6 7 046 ðnÞ HeðAi P i þ Bi F i Þ þ ∑ π ij P i Pj P i E i 7: ð21Þ j¼1 4 5 0 ðnÞ I Here, note that the (2,2)-block matrix of Eq. (21) is equal to HeðAi P i þ Bi F i Þþ ∑sj ¼ 1 ðcij π ij P i Pj P i þ ð1 cij Þπ ij P i Þ. Thus, by Eq. (18), condition (21) can be converted into 04eT1 ð IÞe1 þ eT1 ðGi P i þ H i F i Þe2 þ eT2 ðHeðAi P i þ Bi F i Þ þ ∑sj ¼ 1 ðcij π ij Qij þ ð1 cij Þπ ij P i ÞÞe2 þ HeðeT2 E i e3 Þ þ eT3 ð IÞe3 , which is equivalent to 04eT1 ð IÞe1 þ eT1 ðGi P i þ H i F i Þe2 þ eT2 ðHeðAi P i þ Bi F i Þ þ ∑j A Dþi ðcij π ij Qij þ ð1 cij Þπ ij P i Þ þ ∑j A Di ðcij π ij Qij þ ð1 cij Þπ ij P i ÞÞe2 þ HeðeT2 E i e3 Þ þ eT3 ð IÞe3 , which can be reduced to " #T " # #" ~ j ~ 0 ½M I I M j A Di 04 ð22Þ ; 8i A Nþ s : Πi I Πi I ðnÞ 0 Furthermore, as in the proof of Theorem 1, the constraint that Π A SΠ yields Eq. (19), " #T " # #" I I 0 ½Y~ j j A Di 0r ; Πi I Πi I ðnÞ 0 3" #T 2 ~ # ½S~ j j A Di S0 I 4 5 : 0¼ Πi I Πi I ðnÞ ½S~ jℓ j;ℓ A Di "
I
ð23Þ
ð24Þ
In addition, for the transition rate such that π ij A D i , the following inequalities are derived: (20), 3" " #T 2 ~ # ½X~ j j A Di X0 I I 4 5 0r : ð25Þ Πi I Πi I ðnÞ ½X~ jℓ j;ℓ A D i
Finally, by the S-procedure, combining Eqs. (22)–(25) yields 3" " #T 2 ~ # ½L~ j j A Di L0 I I 4 5 04 ; 8 iA Nþ s ; Πi I Πi I ðnÞ ½L~ jℓ j;ℓ A Di which holds if Eq. (17) holds.
□
Corollary 6. Suppose that there exist matrices F i A Rnu nx , H~ i , S~ ij , X~ ij , Y~ ij A Rnx nx ; and symmetric matrices P i and Qij A Rnx nx such that Eqs. (17)–(20) for all i A N þ s , where ~ 0 ¼ HeðAi P i þ Bi F i Þ þ ∑j A Dþ ðcij π ij Qij þ ð1 cij Þπ ij P i Þ and M ~ j ¼ 1 cij Qij þ 1 1 cij P i . M 2 2 i Then, the closed-loop system in Eq. (15) is mean square stable in the absence of uncertainties.
S. Hyun Kim / Journal of the Franklin Institute 351 (2014) 3042–3052
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Proof. By Eqs. (18) and (21), the mean square stability criterion for the nominal system of Eq. (15) is given by 04HeðAi P i þ Bi F i Þ þ ∑j A Dþi ðcij π ij Qij þ ð1 cij Þπ ij P i Þ þ ∑j A Di ðcij π ij Qij þ ð1 cij Þ π ij P i Þ. The remaining relaxation process is analogous to the proof of Theorem 5. □ Remark 2. The number of scalar variables involved in Theorem 5 and Corollary 6 is given by 4n2c þ nx ðnx þ 1Þ þ nx nu , where nc ¼ nq þ nx þ np in Theorem 5 and nc ¼ nx in Corollary 6. It should be noted that the computation cost of the proposed methods can be more demanding compared with other methods.
Remark 3. For Dþ i , Di , and D i , we can consider some special cases: (i) If Di a ∅ but ~ ~ ~ ~ D i ¼ ∅, then X ij ¼ 0. (ii) If Di ¼ ∅, then it can be assigned that H i ¼ 0, S ij ¼ 0, X ij ¼ 0, and Y~ ij ¼ 0. (iii) If Di ¼ Nþ s for all i, then the system is can be modeled as a switched linear system. (iv) If the number of elements in Dþ i is greater than or equal to s 1 for all i, then the system can be modeled as a system with completely known transition rates.
Remark 4. As reported in [23–25], the relaxation process for unknown modes under some given conditions plays an important role in enhancing the interactions among mode-dependent local conditions for stability and stabilization. For this reason, this paper sheds some light on exploring a useful method that involves the relaxation process with respect to partly unknown transition probabilities, coupled with deriving less conservative stabilization conditions. Remark 5. To set the values of αij and βij, we can consider a simple way originated from the property Π A SΠ . For instance, if π ii ¼ κ for any i, where κ40, then αij and βij can be set as 0 and κ, respectively. Remark 6. Since the use of slack variables requires more computation cost compared with other methods, there may be the need to balance the tradeoffs between the computational cost and the performance enhancement. 4. Numerical examples The effectiveness of the proposed result is verified through the following numerical examples.
4.1. Stabilization problem This example is provided to explore the applicability of Corollary 6 to the case where system uncertainties are not considered in the stabilization problem for MJLSs. To this end, let us consider the following MJLS, given in [14]: A1 ¼
15:0 10:0
B1 ¼
1:0 0:0
;
7:5 ; 10:0
B2 ¼
A2 ¼
0:0 1:0
2:4 10:0
3:3 ; 14:0
;
B3 ¼
1:0 2:0
A3 ¼
;
2:0 1:0 ; 10:0 10:0
B4 ¼
1:0 1:0
A4 ¼
10:0 10:0
2:3 ; 11:0
;
ð26Þ
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S. Hyun Kim / Journal of the Franklin Institute 351 (2014) 3042–3052
Table 1 Control gains for each α22 (Corollary 6). Methods
α22
F1
F2
F3
F4
[13] [14] Corollary 6 Corollary 6
2.5 2.5 2.5 4.0
– – [ 12.6031 40.0092] [ 14.2500 42.6077]
– – [ 23.1399 31.0898] [ 27.3739 38.6813]
– – [33.7007 33.0134] [37.5902 36.6212]
– – [26.6994 16.6430] [28.6931 17.4218]
Table 2 Control gains for each γ (Theorem 5). γ
F1
F2
F3
F4
0.05 0.11 0.12
[ 34.175 67.225] [ 42.916 71.230] –
[ 66.490 56.903] [ 76.034 53.307] –
[82.535 75.531] [126.087 107.098] –
[44.388 13.801] [68.928 7.748] –
2
1:3 0:2 6 π 21 π 22 6 Π¼6 4 0:1 π 32 0:4 0:2
π 13 0:5 2:5 0:6
3 π 14 0:5 7 7 7; π 34 5 1:2
Dþ 1 ¼ f1; 2g;
D1
Dþ 2 ¼ f3; 4g; D2
Dþ 3 ¼ f1; 3g; D3
þ Dþ 4 ¼ N4 ;
¼ f3; 4g; ¼ f1; 2g; ¼ f2; 4g; D4 ¼ ∅; Π 1 ¼ ½π 13 π 14 ; Π 2 ¼ ½π 21 π 22 ; Π 3 ¼ ½π 32 π 34 ; Π 4 ¼ 0; π 1 ¼ 1:1; π 2 ¼ 1:0; π 3 ¼ 2:4; π 4 ¼ 0:
First, the stability analysis of the unforced system of Eq. (26) can be carried out by Corollary 4, as in [13,14]. Next, to design the controller for Eq. (26), we find the solution to a set of LMIs in Corollary 6. Table 1 shows the control gains derived from the solution for α22 ¼ 2:5; 4:0, where other values of αij and βij are selected according to the way described in Remark 5. In addition, Table 1 reveals that the stabilization conditions of [13,14] do not provide feasible solutions for α22 ¼ 2:5, whereas the minimum allowable value of α22 achieved by Corollary 6 is less than 2.5. In this light, we can see that the proposed stabilization condition in Corollary 6 is less conservative in comparison with other methods. 4.2. Robust stabilization problem This example is provided to explore the applicability of Theorem 5 to the robust stabilization problem for MJLSs. For simulation, we choose α22 ¼ 2:5 and then add the following uncertain system matrices to Eq. (26): E i ¼ ½0:5 0:2T ;
Gi ¼ ½0:1 0:2;
H i ¼ γ;
8 iA Nþ 4:
ð27Þ
Then, from the solution to a set of LMIs in Theorem 5, we can obtain the control gains for each γ, which are listed in Table 2. As shown in Table 2, the maximum allowable value of γ achieved by Theorem 5 is given as 0.11. Furthermore, Fig. 1 shows the used mode evolution and the behavior of the state response corresponding to the obtained control gains, where it is assumed that x0 ¼ ½1:0 0:5T and r 0 ¼ 2. From Fig. 1, we can see that the listed control gains in Table 2 are available to stabilize Eqs. (26) and (27) in the sense that the state converges to zero as time increases. As a result, we can verify that the set of LMIs from Theorem 5 is very valuable for þ designing a robust controller for MJLSs subject to Dþ i a Ns for any i.
S. Hyun Kim / Journal of the Franklin Institute 351 (2014) 3042–3052 2
3051
γ = 0.05 γ = 0.11
1.5
x2
1
mode
0.5 0
4 3 2 1 0
−0.5 0
0.2
1
2
3
4
0.4
6 5 time
0.6
7
8
9
0.8
10
1
x1
Fig. 1. Mode evolution and state trajectory on the x1–x2 plane.
5. Concluding remarks In this paper, the robust stabilization problem for a class of continuous-time MJLSs has been investigated with consideration of mode transition probabilities, which are taken to be partly known, unknown, and unknown but with known bounds. To do so, a matrix inequality convexification approach is exploited to obtain a finite set of linear matrix inequalities (LMIs) from the stability analysis and stabilization conditions with homogeneous polynomial dependence on partly unknown transition probabilities, which has features that allow one to incorporate all possible slack variables involved in the transition probability constraints into the relaxation process. Acknowledgments This work was supported by 2014 Research Funds of Hyundai Heavy Industries for University of Ulsan.
References [1] X. Mao, Exponential stability of stochastic delay interval systems with Markovian switching, IEEE Trans. Autom. Control 47 (2002) 1604–1612. [2] I. Matei, J.S. Baras, A linear distributed filter inspired by the Markovian jump linear system filtering problem, Automatica 48 (2012) 1924–1928. [3] J. Dong, G.-H. Yang, Robust H2 control of continuous-time Markov jump linear systems, Automatica 44 (2008) 1431–1436. [4] H. Zhao, Q. Chen, S. Xu, H1 guaranteed cost control for uncertain Markovian jump systems with mode-dependent distributed delays and input delays, J. Frank. Inst. 346 (2009) 945–957. [5] H. Li, Q. Zhou, B. Chen, H. Liu, Parameter-dependent robust stability for uncertain Markovian jump systems with time delay, J. Frank. Inst. 348 (2011) 738–748. [6] D.P. de Farias, J.C. Geromel, J.B.R. do Val, O.L.V. Costa, Output feedback control of Markov jump linear systems in continuous-time, IEEE Trans. Autom. Control 45 (2000) 944–949. [7] S. Boyd, L.E. Chauoi, E. Feron, V. Balakrishan, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, PA, 1994.
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[8] E.K. Boukas, H1 control of discrete-time Markov jump systems with bounded transition probabilities, Optim. Control Appl. Methods 30 (2009) 477–494. [9] M. Karan, P. Shi, C. Kaya, Transition probability bounds for the stochastic stability robustness of continuous- and discrete-time Markovian jump linear systems, Automatica 42 (2006) 2159–2168. [10] M. Shen, G.-H. Yang, H2 state feedback controller design for continuous Markov jump linear systems with partly known information, Int. J. Syst. Sci. 43 (2012) 786–796. [11] J. Xiong, J. Lam, H. Gao, D.W.C. Ho, On robust stabilization of Markovian jump systems with uncertain switching probabilities, Automatica 41 (2005) 897–903. [12] J. Xiong, J. Lam, Robust H2 control of Markovian jump systems with uncertain switching probabilities, Int. J. Syst. Sci. 40 (2009) 255–265. [13] L. Zhang, E.-K. Boukas, Stability and stabilization of Markovian jump linear systems with partly unknown transition probabilities, Automatica 45 (2009) 463–468. [14] L. Zhang, J. Lam, Necessary and sufficient conditions for analysis and synthesis of Markov jump linear systems with incomplete transition descriptions, IEEE Trans. Autom. Control 55 (2010) 1695–1701. [15] S. He, F. Liu, Robust stabilization of stochastic Markovian jumping systems via proportional-integral control, Signal Process. 91 (2011) 2478–2486. [16] L. Xie, L. Xie, Stability analysis of networked sampled-data linear systems with Markovian packet losses, IEEE Trans. Autom. Control 54 (2009) 1375–1381. [17] S.H. Kim, P.-G. Park, Networked-based robust H1 control design using multiple levels of network traffic, Automatica 45 (2009) 764–770. [18] Y. Wang, C. Wang, Z. Zuo, Controller synthesis for Markovian jump systems with incomplete knowledge of transition probabilities and actuator saturation, J. Frank. Inst. 348 (2011) 2417–2429. [19] L. Xiong, J. Tian, X. Liu, Stability analysis for neutral Markovian jump systems with partially unknown transition probabilities, J. Frank. Inst. 349 (2012) 2193–2214. [20] E.K. Boukas, Z.K. Liu, Robust H1 control of discrete-time Markovian jump linear systems with mode-dependent time-delays, IEEE Trans. Autom. Control 46 (2001) 1918–1924. [21] O.L.V. Costa, R.P. Marques, Mixed H2 =H1 control of discrete-time Markovian jump linear systems, IEEE Trans. Autom. Control 43 (1998) 95–100. [22] E. Tian, D. Yue, G. Wei, Robust control for Markovian jump systems with partially known transition probabilities and nonlinearities, J. Frank. Inst. 350 (2013) 2069–2083. [23] P. Park, J.W. Ko, C. Jeong, Reciprocally convex approach to stability of systems with time-varying delays, Automatica 47 (2011) 235–238. [24] C.-H. Fang, Y.-S. Liu, S.-W. Kau, L. Hong, C.-H. Lee, A new LMI-based approach to relaxed quadratic stabilization of T–S fuzzy control systems, IEEE Trans. Fuzzy Syst. 14 (2006) 386–397. [25] A. Sala, C. Arino, Relaxed stability and performance conditions for Takagi–Sugeno fuzzy systems with knowledge on membership function overlap, IEEE Trans. Syst. Man. Cybern. B, Cybern. 37 (2007) 727–732. [26] R. Yang, P. Shi, G.-P. Liu, H. Gao, Network-based feedback control for systems with mixed delays based on quantization and dropout compensation, Automatica 47 (2011) 2805–2809.
Journal of the Franklin Institute 351 (2014) 3042–3052 www.elsevier.com/locate/jfranklin
Short communication
Less conservative stabilization conditions for Markovian jump systems with partly unknown transition probabilities Sung Hyun Kimn School of Electrical Engineering, University of Ulsan, Daehak-ro 93, Nam-Gu, Ulsan 680-749, Republic of Korea Received 11 June 2013; received in revised form 1 October 2013; accepted 27 December 2013 Available online 14 January 2014
Abstract This paper is devoted to the robust stabilization problem for a class of MJLSs with mode transition probabilities taken to be partly known, unknown, and unknown but with known bounds. To meet the needs of less conservative stabilization conditions, this paper proposes a valuable approach that (1) can express the conditions in terms of matrix inequalities with homogeneous polynomial dependence on partly unknown transition probabilities and (2) can include all possible slack variables related to transition probability constraints in the relaxation process coupled with deriving a finite set of linear matrix inequalities (LMIs). Finally, two numerical examples are reported to illustrate the effectiveness of the derived stabilization conditions. & 2014 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction Over the past several years, Markovian jump linear systems (MJLSs) [1] have received considerable attention among control researchers because such systems are suitable to represent a class of dynamic systems subject to random abrupt variations in their structures. Thus, tremendous effort has been devoted to theoretical research on MJLSs with mode transitions governed by a Markov stochastic process (see, for example, [2–5,15,19–21]). In addition, the MJLS model has been extensively applied in many practical applications, such as networked n
Tel.: þ82 52 259 2715; fax: þ82 52 259 1687. E-mail address: [email protected]
0016-0032/$32.00 & 2014 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfranklin.2013.12.023
S. Hyun Kim / Journal of the Franklin Institute 351 (2014) 3042–3052
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control systems (NCSs) [16,17,26] and actuator saturation [18]. A recent and interesting extension is the consideration of MJLSs with uncertain or partly unknown transition probabilities [22], which has emerged as a topic of significant interest because it is practically difficult to obtain the exact values of transition probabilities. In spite of this trend, most studies have focused on the case in which the mode transition probabilities are known exactly a priori, which may actually be a major source of system performance degradation or even instability in the presence of incomplete knowledge of transition probabilities (see [10,14] and references therein). Motivated by the above concerns, there has recently been significant progress on the analysis and synthesis of MJLSs with incomplete knowledge of probabilities, which can be classified into two categories: relative to uncertain transition probabilities and unknown transition probabilities. Specifically, the former has focused on shedding some light on the robustness problem of the norm-bounded or polytopic uncertainties presumed in the transition probability [8,9,11,12,17]. The latter has made considerable efforts to address MJLSs with partly unknown transition probabilities, which is very useful in the case in which there is no bound information for imprecise probabilities [13,14]. However, despite the fact that, with the development of computer technologies, the derivation of a set of less conservative stability analysis and stabilization conditions has become a key issue, little progress has been made toward reducing the conservatism that stems from incomplete knowledge of mode transition probabilities, which is the motivation behind this study. This paper considers a general framework for solving the stabilization problem for a class of MJLSs with transition probabilities taken to be partly known, unknown, and unknown but with known bounds. That is, MJLSs with uncertain or completely unknown transition probabilities correspond to two extreme cases of the considered systems. The distinguishing feature of the proposed approach is that it makes an impact on meeting the needs of less conservative stability analysis and stabilization conditions as an alternative to the aforementioned open problem. In this light, the main contributions of this paper can be highlighted as follows. First, we can express the stability analysis and stabilization conditions in terms of matrix inequalities with homogeneous polynomial dependence on partly unknown mode transition probabilities by defining three additional sets with respect to the measurability of the transition probabilities. Second, we can include all possible slack variables related to transition probability constraints in the relaxation process coupled with deriving a finite set of linear matrix inequalities (LMIs), which allows the convexity of the derived matrix inequalities to be sufficiently exploited. Finally, two numerical examples are given to illustrate the effectiveness of the derived stabilization conditions. Notation: Throughout this paper, standard notion will be adopted. The notations X Z Y and X4Y indicate that X Y is positive semi-definite and positive definite, respectively. In symmetric block matrices, ðnÞ is used as an ellipsis for terms that are induced by symmetry. For any square matrix Q, HeðQÞ ¼ Q þ QT , where QT denotes the transpose of Q. The notation diagðA; BÞ indicates a diagonal matrix with diagonal entries A and B; Nþ r ¼ f1; 2; …; rg; and colðv1 ; v2 ; …; vn Þ ¼ ½vT1 vT2 …vTn T for scalars or vectors vi. For real submatrices Si and Sij with appropriate dimensions, ½Si i A fa1 ;…;an g ¼ ½Sa1 …San and 2
Sa1 a1
6 ½Sij i;j A fa1 ;…;an g ¼ 4 ⋮ San a1
⋯
Sa1 an
⋱ ⋯
San an
3
⋮ 7 5:
The notation ei denotes the block entry matrix such that ½Sij i;j A Nþr ¼ ∑ri ¼ 1 ∑rj ¼ 1 eTi Sij ej .
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2. System description and preliminaries Consider the following MJLSs defined on a complete probability space ðΩ; F ; PÞ: ( x_ ðtÞ ¼ AðrðtÞÞxðtÞ þ BðrðtÞÞuðtÞ þ EðrðtÞÞpðtÞ; xð0Þ ¼ x0 ; qðtÞ ¼ GðrðtÞÞxðtÞ þ HðrðtÞÞuðtÞ;
ð1Þ
where xðtÞA Rnx and uðtÞA Rnu denote the state and the control input, respectively; x0 denotes the initial state; and pðtÞA Rnp and qðtÞA Rnq describe the structured feedback uncertainty such that pðtÞ ¼ ΔðtÞqðtÞ and ΔT ðtÞΔðtÞ r I. Here, frðtÞ; t Z 0g denotes a continuous-time Markov process on the probability space that takes the values in a finite state space Nþ s and has the mode transition probabilities ( π ij δt þ oðδtÞ if ja i; Prðrðt þ δtÞ ¼ jjrðtÞ ¼ iÞ ¼ ð2Þ 1 þ π ii δt þ oðδtÞ if j ¼ i; where δt40, limδt-0 ðoðδtÞ=δtÞ ¼ 0, and πij denotes the transition rate from mode i to mode j at time t þ δt. Furthermore, the mode transition rate matrix Π ¼ ½π ij i;j A Nþs belongs to ( s 1 if ja i SΠ 9 ½π ij i;j A Nþs j0 ¼ ∑ π ij ; 0r μij π ij ; μij ¼ ; 8 i; jA Nþ : s 1 otherwise j¼1 Next, to simplify the notation, we set Ai ¼ AðrðtÞ ¼ iÞ, Bi ¼ BðrðtÞ ¼ iÞ, E i ¼ EðrðtÞ ¼ iÞ, Gi ¼ GðrðtÞ ¼ iÞ, and H i ¼ HðrðtÞ ¼ iÞ, where the initial operation mode rð0Þ ¼ r 0 . For later convenience, we also define three additional sets with respect to the measurability of πij for i; jA Nþ s : þ Dþ i 9 fjjπ ij is known; 8 jA Ns g;
Di 9 fjjπ ij is unknown; 8j A Nþ s g;
D i 9 fjjπ ij is unknown but bounded as αij r π ij rβij ; 8 jA Nþ s g:
þ þ The following properties are then satisfied: Dþ i [ Di ¼ Ns , D i DDi , for all iA Ns . In addition, if the transition rate can be fully measured, then Di ¼ D i ¼ ∅.
Remark 1. In this paper, the transition rate πij is set to be partly known, that is, jA fDþ i ; Di jD i a ∅g. Lemma 1 (de Farias et al. [6]). The nominal system of Eq. (1) with zero input is said to be mean square stable if there exists a set of Pi for all iA Nþ s such that 04HeðPi Ai Þþ ∑sj ¼ 1 π ij Pj ; 8i A Nþ . s Lemma 2. The system of Eq. (1) with zero input is said to be robustly mean square stable if there exists a set of Pi, for all i A Nþ s , such that 2 3 s T π ij Pj þ Gi Gi Pi E i 7 6 HeðPi Ai Þ þ j ∑ ¼1 ð3Þ 044 5; 8 iA Nþ s : ðnÞ I Proof. From Lemma 1, it follows that 04xT ðtÞðHeðPi Ai Þ þ ∑sj ¼ 1 π ij Pj Þ x(t), which is in line with 04HeðxT ðtÞPi x_ ðtÞÞ þ xT ðtÞð∑sj ¼ 1 π ij Pj ÞxðtÞ. That is, in the sense that x_ ðtÞ ¼ Ai xðtÞ þ Ei pðtÞ,
S. Hyun Kim / Journal of the Franklin Institute 351 (2014) 3042–3052
the stability criterion becomes s
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!
04xT ðtÞ HeðPi Ai Þ þ ∑ π ij Pj xðtÞ þ 2xT ðtÞPi E i pðtÞ:
ð4Þ
j¼1
Furthermore, from pðtÞ ¼ ΔðtÞqðtÞ and ΔT ðtÞΔðtÞ r I, we can get qT ðtÞqðtÞZ pT ðtÞpðtÞ, i.e., ðGi xðtÞÞT ðGi xðtÞÞ Z pT ðtÞpðtÞ. Hence, by S-procedure, combining (4) and qT ðtÞqðtÞZ pT ðtÞpðtÞ yields 04xT ðtÞðHeðPi Ai Þ þ ∑sj ¼ 1 π ij Pj ÞxðtÞ þ 2xT ðtÞPi E i pðtÞ þ ðGi xðtÞÞT ðGi xðtÞÞ pT ðtÞpðtÞ, that is, 2 3 s " # " #T xðtÞ 6 HeðPi Ai Þ þ ∑ π ij Pj þ GTi Gi Pi E i 7 xðtÞ j¼1 04 : ð5Þ 4 5 pðtÞ pðtÞ ðnÞ I Therefore, we can see that Eq. (5) is assured by Eq. (3).
□
3. Main results Here, to facilitate the derivation, we define Π i ¼ ½π ij TjA Di and π i ¼ ∑j A Dþi π ij . 3.1. Stability analysis This section is devoted to addressing the robust stability analysis problem of MJLSs subject to þ Dþ i a Ns for any i in the absence of control input. Theorem 3. Suppose that there exist Hi, Sij, Xij, Y ij A Rðnx þnp Þðnx þnp Þ and symmetric matrices Pi A Rnx nx such that for all iA Nþ s , " # L0 ½Lj j A Di 04 ; ð6Þ ðnÞ ½Ljℓ j;ℓ A Di 0oHeðY ij Þ;
8 jA Di ;
0oHeðX ij Þ;
8 jA D i ;
ð7Þ
ð8Þ
where L0 ¼ M0 þ S 0 þ X 0 , Lj ¼ Mj þ S j þ X j þ Y j , Ljℓ ¼ S jℓ þ X jℓ in which ! T T M0 ¼ e1 HeðPi Ai Þ þ ∑ π ij Pj þ Gi Gi e1 þ He eT1 Pi Ei e2 þ eT2 ð I Þe2 ;
1 Pj e1 ; 2 S jℓ jℓ4j ¼ Sij þ Siℓ ;
Mj ¼ eT1
j A Dþ i
S 0 ¼ π i HeðH i Þ;
S j ¼ H i þ π i Sij ;
S jℓ jℓoj ¼ STij þ STiℓ ;
S jj ¼ He Sij ;
(
ðαij þ βij ÞX ij if jA D i Y j ¼ μij Y ij ; X 0 ¼ ∑ αij βij HeðX ij Þ; X j ¼ 0 otherwise jAD ( i 1 if ja i Heð X ij Þ if jA D i X jℓ jj a ℓ ¼ 0; X jj ¼ : ; μij ¼ 1 otherwise 0 otherwise Then, the unforced system of Eq. (1) is robustly mean square stable.
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S. Hyun Kim / Journal of the Franklin Institute 351 (2014) 3042–3052
Proof. The stability criterion in Eq. (3) can be rewritten as ! 04eT1 HeðPi Ai Þ þ ∑ π ij Pj þ ∑ π ij Pj þ GTi Gi e1 j A Di
j A Dþ i
8 iA Nþ s ;
þHeðeT1 Pi Ei e2 Þ þ eT2 ð IÞe2 ; which is equivalent to " #T " M0 I 04 ðnÞ Πi I
½Mj j A Di 0
#"
I
ð9Þ # 8 iA Nþ s :
;
Πi I
ð10Þ
Furthermore, from the constraint that Π A SΠ , we can obtain 0 r ∑ μij π ij HeðY ij Þ; j A Di
0 ¼ He
0oHeðY ij Þ;
8 jA Di ;
! π i I þ ∑ π ij I j A Di
iA Nþ s ;
!! H i þ ∑ π ij Sij j A Di
;
8i A Nþ s ;
which leads to Eq. (7), " #T " # #" 0 ½Y j j A Di I I 0r ; ðnÞ 0 Πi I Πi I "
I 0¼ Πi I
#T "
S0 ½S j j A Di ðnÞ ½S jℓ j;ℓ A Di
#"
ð11Þ
# I : Πi I
ð12Þ
In addition, for the transition rate such that π ij A D i , the following inequalities are satisfied: 0 r ∑ ðπ ij αij Þðπ ij βij ÞHeðX ij Þ; j A Di
0oHeðX ij Þ;
which leads to Eq. (8), #" " #T " # X0 ½X j j A Di I I 0r : ðnÞ ½X jℓ j;ℓ A Di Πi I Πi I
8 jA D i ;
iA Nþ s ;
ð13Þ
As a result, the so-called S-procedure [7] enables Eq. (10) subject to Eqs. (11)–(13) to be expressed as " #T " # #" L0 ½Lj j A Di I I 04 ; 8 iA Nþ s ; ðnÞ ½Ljℓ j;ℓ A Di Πi I Πi I which holds if Eq. (6) holds.
□
Corollary 4. Suppose that there exist matrices Hi, Sij, Xij, Y ij A Rnx nx and symmetric matrices Pi A Rnx nx such that Eqs. (6), (8), (7) for all iA Nþ π ij Pj and s , where M0 ¼ HeðPi Ai Þ þ ∑j A Dþ i Mj ¼ 12 Pj . Then, the nominal system of Eq. (1) with zero input is mean square stable. Proof. By Lemma 1, the mean square stability criterion is given by 04HeðPi Ai Þþ ∑j A Dþi π ij Pj þ ∑j A Di π ij Pj . The remaining relaxation process is analogous to the proof of Theorem 3. □
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3.2. Control synthesis This section is devoted to addressing the robust stabilization problem of MJLSs subject to þ Dþ i a Ns for any i. To this end, let us consider the state-feedback control law 8 iA Nþ s ;
uðtÞ ¼ F i xðtÞ;
ð14Þ
where F i ð ¼ FðrðtÞÞÞ denotes the controller gain to be designed. Then, the closed-loop system under Eq. (14) is given by ( x_ ðtÞ ¼ A~ i xðtÞ þ Ei pðtÞ; xð0Þ ¼ x0 ; ð15Þ ~ i xðtÞ; qðtÞ ¼ G ~ i ¼ Gi þ H i F i . Thus, in light of Eq. (15), applying the Schur where A~ i ¼ Ai þ Bi F i and G complement to Eq. (3) yields 2 3 0 I G~ i s 6 7 6 7 ð16Þ 046 ðnÞ HeðPi A~ i Þ þ ∑ π ij Pj Pi Ei 7: 4 5 j¼1 0 ðnÞ I Theorem 5. Suppose that there exist matrices F i ARnu nx , H~ i , S~ ij , X~ ij , Y~ ij A Rðnq þnx þnp Þðnq þnx þnp Þ and symmetric matrices P i and Qij A Rnx nx such that for all iA Nþ s , 2 3 ½L~ j j A Di L~ 0 5; 044 ð17Þ ðnÞ ½L~ jℓ j;ℓ A Di " 0r
Qij
Pi
#
ðnÞ P j
;
8 jA Nþ s =fig;
0oHeðY~ ij Þ;
8 jA Di ;
0oHeðX~ ij Þ;
8j A D i ;
ð18Þ ð19Þ
ð20Þ
~ 0 þ S~ 0 þ X~ 0 , L~ j ¼ M ~ j þ S~ j þ X~ j þ Y~ j , and L~ jℓ ¼ S~ jℓ þ X~ jℓ in which where L~ 0 ¼ M ~ 0 ¼ eT ð IÞe1 þ eT ðGi P i þ H i F i Þe2 þ eT HeðAi P i þ Bi F i Þ M 1 1 2 þ ∑ ðcij π ij Qij þ ð1 cij Þπ ij P i ÞÞe2 þ HeðeT2 E i e3 Þ þ eT3 ð IÞe3 ; j A Dþ i
1 T 1 ~ cij Qij þ 1 cij P i e2 ; S~ 0 ¼ π i He H~ i ; S~ j ¼ H~ i þ π i S~ ij ; M j ¼ e2 2 2 T T S~ jj ¼ HeðS~ ij Þ; S~ jℓ jℓ4j ¼ S~ ij þ S~ iℓ ; S~ jℓ jℓoj ¼ S~ ij þ S~ iℓ ; Y~ j ¼ μij Y~ ij ; X~ 0 ¼ ∑ αij βij HeðX~ ij Þ; X~ jℓ jj a ℓ ¼ 0;
( X~ j ¼
j A Di
(
ðαij þ βij ÞX~ ij
if jA D i
0
otherwise
;
X~ jj ¼
Heð X~ ij Þ
if jA D i
0
otherwise
;
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S. Hyun Kim / Journal of the Franklin Institute 351 (2014) 3042–3052
μij ¼
1
if ja i
1
otherwise
;
cij ¼
1
if j a i
0
otherwise
:
Then, the closed-loop system in Eq. (15) is robustly mean square stable. Proof. Let us define P i ¼ diagðI; P i ; IÞ, where P i ¼ Pi 1 . Then, pre- and post-multiplying Eq. (16) by P i becomes 2 3 0 I Gi P i þ H i F i s 6 7 6 7 046 ðnÞ HeðAi P i þ Bi F i Þ þ ∑ π ij P i Pj P i E i 7: ð21Þ j¼1 4 5 0 ðnÞ I Here, note that the (2,2)-block matrix of Eq. (21) is equal to HeðAi P i þ Bi F i Þþ ∑sj ¼ 1 ðcij π ij P i Pj P i þ ð1 cij Þπ ij P i Þ. Thus, by Eq. (18), condition (21) can be converted into 04eT1 ð IÞe1 þ eT1 ðGi P i þ H i F i Þe2 þ eT2 ðHeðAi P i þ Bi F i Þ þ ∑sj ¼ 1 ðcij π ij Qij þ ð1 cij Þπ ij P i ÞÞe2 þ HeðeT2 E i e3 Þ þ eT3 ð IÞe3 , which is equivalent to 04eT1 ð IÞe1 þ eT1 ðGi P i þ H i F i Þe2 þ eT2 ðHeðAi P i þ Bi F i Þ þ ∑j A Dþi ðcij π ij Qij þ ð1 cij Þπ ij P i Þ þ ∑j A Di ðcij π ij Qij þ ð1 cij Þπ ij P i ÞÞe2 þ HeðeT2 E i e3 Þ þ eT3 ð IÞe3 , which can be reduced to " #T " # #" ~ j ~ 0 ½M I I M j A Di 04 ð22Þ ; 8i A Nþ s : Πi I Πi I ðnÞ 0 Furthermore, as in the proof of Theorem 1, the constraint that Π A SΠ yields Eq. (19), " #T " # #" I I 0 ½Y~ j j A Di 0r ; Πi I Πi I ðnÞ 0 3" #T 2 ~ # ½S~ j j A Di S0 I 4 5 : 0¼ Πi I Πi I ðnÞ ½S~ jℓ j;ℓ A Di "
I
ð23Þ
ð24Þ
In addition, for the transition rate such that π ij A D i , the following inequalities are derived: (20), 3" " #T 2 ~ # ½X~ j j A Di X0 I I 4 5 0r : ð25Þ Πi I Πi I ðnÞ ½X~ jℓ j;ℓ A D i
Finally, by the S-procedure, combining Eqs. (22)–(25) yields 3" " #T 2 ~ # ½L~ j j A Di L0 I I 4 5 04 ; 8 iA Nþ s ; Πi I Πi I ðnÞ ½L~ jℓ j;ℓ A Di which holds if Eq. (17) holds.
□
Corollary 6. Suppose that there exist matrices F i A Rnu nx , H~ i , S~ ij , X~ ij , Y~ ij A Rnx nx ; and symmetric matrices P i and Qij A Rnx nx such that Eqs. (17)–(20) for all i A N þ s , where ~ 0 ¼ HeðAi P i þ Bi F i Þ þ ∑j A Dþ ðcij π ij Qij þ ð1 cij Þπ ij P i Þ and M ~ j ¼ 1 cij Qij þ 1 1 cij P i . M 2 2 i Then, the closed-loop system in Eq. (15) is mean square stable in the absence of uncertainties.
S. Hyun Kim / Journal of the Franklin Institute 351 (2014) 3042–3052
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Proof. By Eqs. (18) and (21), the mean square stability criterion for the nominal system of Eq. (15) is given by 04HeðAi P i þ Bi F i Þ þ ∑j A Dþi ðcij π ij Qij þ ð1 cij Þπ ij P i Þ þ ∑j A Di ðcij π ij Qij þ ð1 cij Þ π ij P i Þ. The remaining relaxation process is analogous to the proof of Theorem 5. □ Remark 2. The number of scalar variables involved in Theorem 5 and Corollary 6 is given by 4n2c þ nx ðnx þ 1Þ þ nx nu , where nc ¼ nq þ nx þ np in Theorem 5 and nc ¼ nx in Corollary 6. It should be noted that the computation cost of the proposed methods can be more demanding compared with other methods.
Remark 3. For Dþ i , Di , and D i , we can consider some special cases: (i) If Di a ∅ but ~ ~ ~ ~ D i ¼ ∅, then X ij ¼ 0. (ii) If Di ¼ ∅, then it can be assigned that H i ¼ 0, S ij ¼ 0, X ij ¼ 0, and Y~ ij ¼ 0. (iii) If Di ¼ Nþ s for all i, then the system is can be modeled as a switched linear system. (iv) If the number of elements in Dþ i is greater than or equal to s 1 for all i, then the system can be modeled as a system with completely known transition rates.
Remark 4. As reported in [23–25], the relaxation process for unknown modes under some given conditions plays an important role in enhancing the interactions among mode-dependent local conditions for stability and stabilization. For this reason, this paper sheds some light on exploring a useful method that involves the relaxation process with respect to partly unknown transition probabilities, coupled with deriving less conservative stabilization conditions. Remark 5. To set the values of αij and βij, we can consider a simple way originated from the property Π A SΠ . For instance, if π ii ¼ κ for any i, where κ40, then αij and βij can be set as 0 and κ, respectively. Remark 6. Since the use of slack variables requires more computation cost compared with other methods, there may be the need to balance the tradeoffs between the computational cost and the performance enhancement. 4. Numerical examples The effectiveness of the proposed result is verified through the following numerical examples.
4.1. Stabilization problem This example is provided to explore the applicability of Corollary 6 to the case where system uncertainties are not considered in the stabilization problem for MJLSs. To this end, let us consider the following MJLS, given in [14]: A1 ¼
15:0 10:0
B1 ¼
1:0 0:0
;
7:5 ; 10:0
B2 ¼
A2 ¼
0:0 1:0
2:4 10:0
3:3 ; 14:0
;
B3 ¼
1:0 2:0
A3 ¼
;
2:0 1:0 ; 10:0 10:0
B4 ¼
1:0 1:0
A4 ¼
10:0 10:0
2:3 ; 11:0
;
ð26Þ
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S. Hyun Kim / Journal of the Franklin Institute 351 (2014) 3042–3052
Table 1 Control gains for each α22 (Corollary 6). Methods
α22
F1
F2
F3
F4
[13] [14] Corollary 6 Corollary 6
2.5 2.5 2.5 4.0
– – [ 12.6031 40.0092] [ 14.2500 42.6077]
– – [ 23.1399 31.0898] [ 27.3739 38.6813]
– – [33.7007 33.0134] [37.5902 36.6212]
– – [26.6994 16.6430] [28.6931 17.4218]
Table 2 Control gains for each γ (Theorem 5). γ
F1
F2
F3
F4
0.05 0.11 0.12
[ 34.175 67.225] [ 42.916 71.230] –
[ 66.490 56.903] [ 76.034 53.307] –
[82.535 75.531] [126.087 107.098] –
[44.388 13.801] [68.928 7.748] –
2
1:3 0:2 6 π 21 π 22 6 Π¼6 4 0:1 π 32 0:4 0:2
π 13 0:5 2:5 0:6
3 π 14 0:5 7 7 7; π 34 5 1:2
Dþ 1 ¼ f1; 2g;
D1
Dþ 2 ¼ f3; 4g; D2
Dþ 3 ¼ f1; 3g; D3
þ Dþ 4 ¼ N4 ;
¼ f3; 4g; ¼ f1; 2g; ¼ f2; 4g; D4 ¼ ∅; Π 1 ¼ ½π 13 π 14 ; Π 2 ¼ ½π 21 π 22 ; Π 3 ¼ ½π 32 π 34 ; Π 4 ¼ 0; π 1 ¼ 1:1; π 2 ¼ 1:0; π 3 ¼ 2:4; π 4 ¼ 0:
First, the stability analysis of the unforced system of Eq. (26) can be carried out by Corollary 4, as in [13,14]. Next, to design the controller for Eq. (26), we find the solution to a set of LMIs in Corollary 6. Table 1 shows the control gains derived from the solution for α22 ¼ 2:5; 4:0, where other values of αij and βij are selected according to the way described in Remark 5. In addition, Table 1 reveals that the stabilization conditions of [13,14] do not provide feasible solutions for α22 ¼ 2:5, whereas the minimum allowable value of α22 achieved by Corollary 6 is less than 2.5. In this light, we can see that the proposed stabilization condition in Corollary 6 is less conservative in comparison with other methods. 4.2. Robust stabilization problem This example is provided to explore the applicability of Theorem 5 to the robust stabilization problem for MJLSs. For simulation, we choose α22 ¼ 2:5 and then add the following uncertain system matrices to Eq. (26): E i ¼ ½0:5 0:2T ;
Gi ¼ ½0:1 0:2;
H i ¼ γ;
8 iA Nþ 4:
ð27Þ
Then, from the solution to a set of LMIs in Theorem 5, we can obtain the control gains for each γ, which are listed in Table 2. As shown in Table 2, the maximum allowable value of γ achieved by Theorem 5 is given as 0.11. Furthermore, Fig. 1 shows the used mode evolution and the behavior of the state response corresponding to the obtained control gains, where it is assumed that x0 ¼ ½1:0 0:5T and r 0 ¼ 2. From Fig. 1, we can see that the listed control gains in Table 2 are available to stabilize Eqs. (26) and (27) in the sense that the state converges to zero as time increases. As a result, we can verify that the set of LMIs from Theorem 5 is very valuable for þ designing a robust controller for MJLSs subject to Dþ i a Ns for any i.
S. Hyun Kim / Journal of the Franklin Institute 351 (2014) 3042–3052 2
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γ = 0.05 γ = 0.11
1.5
x2
1
mode
0.5 0
4 3 2 1 0
−0.5 0
0.2
1
2
3
4
0.4
6 5 time
0.6
7
8
9
0.8
10
1
x1
Fig. 1. Mode evolution and state trajectory on the x1–x2 plane.
5. Concluding remarks In this paper, the robust stabilization problem for a class of continuous-time MJLSs has been investigated with consideration of mode transition probabilities, which are taken to be partly known, unknown, and unknown but with known bounds. To do so, a matrix inequality convexification approach is exploited to obtain a finite set of linear matrix inequalities (LMIs) from the stability analysis and stabilization conditions with homogeneous polynomial dependence on partly unknown transition probabilities, which has features that allow one to incorporate all possible slack variables involved in the transition probability constraints into the relaxation process. Acknowledgments This work was supported by 2014 Research Funds of Hyundai Heavy Industries for University of Ulsan.
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