Parameter-Dependent Lyapunov Function Approach to Robust ...
Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 2009
WeB03.3
Parameter-dependent Lyapunov function approach to robust stability analysis for discrete-time descriptor polytopic systems Xiangyu Gao1,2 , Guang-Ren Duan1,∗ , Xian Zhang2 Abstract— This paper investigates robust stability of discretetime descriptor polytopic systems (DTDPSs for short). The concept of affine generalized quadratic stability, which has less conservatism than generalized quadratic stability, of DTDPSs is proposed. It not only investigates affine generalized quadratic stability of DTDPSs implies robust stability, but also presents criterions in terms of linear matrix inequalities to test the (affine) generalized quadratic stability based on time-varying parameter-dependent Lyapunov function. Finally, a numerical example presents the effectiveness of the proposed method.
I. INTRODUCTION The Lyapunov theory has been one of the most appealing methods for investigating robust stability of linear polytopic systems (LPSs for short) during the last decade. In particular, some results are based on the concept of quadratic stability (see, e.g. [1]–[3]). Unfortunately, these results may be very conservative due to the use of a common Lyapunov function for the entire uncertainty set. In order to reduce the conservatism, several kinds of parameterdependent Lyapunov functions have been proposed in literature to investigate robust stability of LPSs. For example, Lyapunov functions with linear dependence (see [4]–[9] and references therein), Lyapunov functions with polynomial dependence [10], [11], homogeneous polynomially parameterdependent Lyapunov function of arbitrary degree [12], [13], homogeneous polynomially parameter-dependent quadratic Lyapunov functions [14], [15], Lyapunov functions which are parameter-dependent in negative as well as positive power series of parameters [16]. Besides, Lavaei and Aghdam [17] deal with the robust stability of discrete-time linear time-invariant systems with parametric uncertainties belonging to a semialgebraic set, which can solve wider problem than the ones given in [13] and [14] for discrete-time systems. By introducing extra variables and using additional LMIs, Leite and Peres [18] proposed a sufficient condition for robust D-stability of LPSs, which contains the condition for robust D-stability from [9] as a particular case and encompasses the robust stability results of [6] (discrete-time systems) and [7] This work was partially supported by the Major Program of National Natural Science Foundation of China under Grant No. 60710002 and Program for Changjiang Scholars and Innovative Research Team in University 1 Center for Control Theory and Guidance Technology, Harbin Institute of Technology, Harbin 150001, China. [email protected],
[email protected] 2
School of Harbin 150080,
Mathematical Science, Heilongjiang University, China. [email protected],
[email protected] ∗
Corresponding author.
978-1-4244-3872-3/09/$25.00 ©2009 IEEE
(continuous- time systems) when these special cases of Dstability are investigated. In addition, some robust stability results of linear parameter-varying systems with parametric uncertainties are obtained [19]–[21]. On the other hand, descriptor systems have been extensively studied during the past four decades due to their wide applications in circuits, economic, large scale systems, and other areas [22]. Many notions and results in statespace systems have been extended to descriptor systems [23], since the latter can present a much wider class of systems than state-space systems can. However, few results on robust stability analysis for descriptor polytopic systems have been reported so far [24]–[26]. So the study of such problems is of both practical and theoretical importance. In this paper, we will study robust stability of discretetime descriptor polytopic systems (DTDPSs for short) by constructing Lyapunov functions with linear dependence, which is a generalization of the results in [4]. The concept of affine generalized quadratic stability of DTDPSs, which has less conservatism than generalized quadratic stability, is proposed in Section II. Based on some preliminary results introduced in Section III, it is shown in Section IV that affine generalized quadratic stability of DTDPSs implies robust stability, and thereby, sufficient conditions for (affine) generalized quadratic stability of DTDPSs are investigated in terms of linear matrix inequalities (LMIs for short). At last, the effectiveness of the proposed method is demonstrated by a numerical example in Section V. It should be pointed out that the robust stability problem for descriptor polytopic systems is much complicated than that for LPSs because it require to consider not only stability and robustness, but also regularity and impulse-free for continuous-time (causality for discrete-time) descriptor systems, simultaneously. The notations occurred in this paper are as follows: Let < be the real number set, and Z + the nonnegative integer set. Denote by I[k1 , k2 ] the set {k1 , k1 + 1, · · · , k2 } for k1 , k2 ∈ Z + with k1 < k2 . Let ΛN := {(λ1 , · · · , λN ) :
N X
λi = 1, λi ≥ 0, ∀i ∈ I[1, N ]}.
i=1
For λ = (λ1 , · · · , λN ) ∈ ΛN and a group of matrices N P M1 , M2 , · · · , MN , we denote by M (λ) the matrix λ i Mi . i=1
For real symmetric matrices P and Q, P > Q(P < Q) means that the matrix P − Q is positive (negative) definite, and P ≥ Q(P ≤ Q) means that the matrix P − Q is
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WeB03.3 positive (negative) semi-definite. Denote by k · k2 either the Euclidean vector norm or its induced matrix 2-norm. The transpose, inverse, determinant, adjoint matrix and the minimum eigenvalue of the matrix A are represented by AT , A−1 , detA, adjA and λmin (A) respectively. Let A−T = (A−1 )T . II. PROBLEM FORMULATION AND DEFINITIONS Consider the DTDPS Exk+1 = A(αk )xk , k ∈ Z + ,
(1)
where xk is the state variable, E ∈ 0, ∀ i ∈ I[1, N ]. R Qi Lemma 2: [27] Let H ∈ Rn×n is a constant matrix such that H + H T < 0. Then H is µinvertible .¶ M1 M2 Lemma 3: [28] Let M = ∈ 0 for any αk ∈ ΛN . Using (7) we have X For the sake of simplicity, A˜i (αk ) will be abbreviated as A˜i below for i = 1, 2, 3, 4. From (7b),(8) and (9),we have µ ¶ X1 X2 < 0, (10) X2T X3 where ˜ 1 (αk+1 )A˜1 + A˜T X ˜ T (αk+1 )A˜1 − X ˜ 1 (αk ) X1 = A˜T1 X 3 2 ˜ 2 (αk+1 )A˜3 + A˜T X ˜ 3 (αk+1 )A˜3 , +A˜T1 X 3 ˜ 1 (αk+1 )A˜2 + A˜T X ˜ T (αk+1 )A˜2 X2 = A˜T1 X 3 2 T ˜ T ˜ ˜ ˜ ˜ +A1 X2 (αk+1 )A4 + A3 X3 (αk+1 )A˜4 , ˜ 1 (αk+1 )A˜2 + A˜T X ˜ T (αk+1 )A˜2 X3 = A˜T2 X 4 2 T T ˜ 2 (αk+1 )A˜4 + A˜ X ˜ 3 (αk+1 )A˜4 . +A˜ X 2
T
˜ 1 (αk+1 )A˜ − X ˜ 1 (αk ) < 0. A˜T X
Let λ = min {λi (αk )} and λ = max {λi (αk )}. Then 1≤i≤r
1≤i≤r
0 < λ ≤ λi (αk ) ≤ λ, ∀ αk ∈ ΛN , i = 1, 2, · · · , N. ˜ 1 (αk ) ≤ λIr , ∀ αk ∈ ΛN . Consequently, 0 < λIr ≤ X Similarly, from (12), there exists a positive number λ such that ˜ 1 (αk+1 )A˜ − X ˜ 1 (αk ) ≤ −λI. A˜T X
0 < λkyk k22 ≤ V (yk , k) ≤ λkyk k22 and ∆V (yk , k)
˜ 2T (αk+1 )A˜2 + A˜T2 X ˜ 2 (αk+1 )A˜4 + A˜T4 X ˜ 3 (αk+1 )A˜4 < 0. A˜T4 X ˜ 3 (αk+1 )A˜4 . We obtain ˜ 2 (αk+1 )A˜4 + 1 A˜T X Let H = A˜T2 X 2 4 from Lemma 2 that H is an invertible matrix, so is A˜4 . Hence the DTDPS (1) is causal and regular for all αk ∈ ΛN . Next, we show that the DTDPS (1) is stable. Let xk = Q0 [ykT zkT ]T , where yk ∈ 0 such that kA˜3 k2 ≤ ξ. Thus, it follows from (11b) that ˜ kzk k2 = k − A˜−1 4 A3 yk k2 −1 ˜ ≤ kA4 k2 kA˜3 k2 kyk k2 δ3 ξ ≤ kyk k2 . δ1
M1 =
So the system (1) is asymptotically stable for all αk ∈ ΛN . Since the parameter αk is not known a priori, the condition (7) is not numerically vertifiable. Due to the particular structure X(αk ) = α ˜ 1 (k)X1 + · · · + α ˜ N (k)XN , αk ∈ ΛN ,
(13)
i.e., that X(αk ) is affine in the parameters, the following sufficient condition can be obtained. Theorem 2: Given the vertex set {Aj , j ∈ I[1, N ]} and the bounds of the parameter increments 0 ≤ ρi ≤ 1, i ∈ I[1, N ]. The DTDPS (1) is AGQS if there exist matrices
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WeB03.3 R, Y ∈ 0, Qi > 0, i ∈ I[1, N ] such that
Qi RT 0
E T (Pi − Qi )E ≥ 0, ∀ i ∈ I[1, N ], R 0 Ai ATi Y T > 0, ∀ i ∈ I[1, N ], Y Ai Pi
(14a) (14b)
where Qi = Qi −
N P j=1
This, together with (17), implies that Q(αk+1 ) R 0 RT > 0, A(αk ) AT (αk )Y T T 0 Y A(αk ) Y + Y − P (αk+1 ) and hence µ A(αk ) − RT Q(αk+1 )−1 R Y A(αk ) It follows from
ρj Qj ,
Ai = E T (Pi − Qi )E + ATi R + RT Ai , N P Pi = Y T + Y − Pi − ρj Pj .
(Y − P (αk+1 ))P −1 (αk+1 )(Y − P (αk+1 ))T ≥ 0
(15) that
Y P −1 (αk+1 )Y T ≥ Y T + Y − P (αk+1 ).
j=1
Proof: From Lemma 1, we have that the inequality (14b) is equivalent to the Q(αk ) R 0 RT A(αk ) AT (αk )Y T > 0, ∀αk ∈ ΛN , (16) 0 Y A(αk ) P(αk ) where Q(αk )
= Q(αk ) −
A(αk )
=
P(αk )
N P j=1
ρj Qj ,
E T (P (αk ) − Q(αk ))E + RT A(αk ) +AT (αk )R, N P = Y T + Y − P (αk ) − ρj Pj .
Because of [Q(αk+1 )A(αk )−R]T Q−1 (αk+1 )[Q(αk+1 )A(αk )−R] ≥ 0, we derive that
N P i=1
α ˜ i (k) = 1 that
AT (αk )Q(αk+1 )A(αk ) ≥ AT (αk )R + RT A(αk ) − RT Q−1 (αk+1 )R = A(αk ) − RT Q−1 (αk+1 )R −E T (P (αk ) − Q(αk ))E.
Y T + Y − P (αk+1 ) N N N P P P = α ˜ i (k)(Y T + Y ) − α ˜ i (k)Pi − ∆˜ αi (k)Pi = =
i=1 N P i=1
i=1
α ˜ i (k)(Y
T
+ Y − Pi ) −
α ˜ i (k)(Y T + Y − Pi ) −
≥ Y T + Y − P (αk ) −
N P j=1
NP −1 j=1 NP −1 j=1
Applying Schur complement to (22) again, we obtain that
i=1
∆˜ αj (Pj − PN )
AT (αk )X(αk+1 )A(αk ) − E T X(αk )E < 0, ∀ αk ∈ ΛN .
∆˜ αj Pj + ∆˜ αN PN
ρj Pj
= P(αk ) > 0.
(17)
and Q(αk+1 )
This, together with (20), implies that µ ¶ A(αk ) − RT Q(αk+1 )−1 R AT (αk )Y T > 0. Y A(αk ) Y P −1 (αk+1 )Y T (21) Since detY 6= 0, the inequality (21) is equivalent to the following inequality ¶ µ A(αk ) − RT Q−1 (αk+1 )R AT (αk ) > 0. (22) A(αk ) P −1 (αk+1 )
j=1
It follows from (3), (4), (5) and
i=1 N P
YT
¶ AT (αk )Y T > 0. + Y − P (αk+1 ) (20)
= ≥
N P i=1 N P i=1
α ˜ i (k)Qi + α ˜ i (k)Qi −
= Q(αk ) −
N P i=1
N P i=1 N P i=1
∆˜ αi Qi ρi Qi
(18)
ρi Qi
= Q(k) > 0.
where X(αk ) = P (αk ) − Q(αk ). Since (14a) implies that E T X(αk )E ≥ 0, ∀ αk ∈ ΛN , it follows from Definition 3 and Theorem 1 that DTDPS (1) is AGQS. Remark 2: Although the slack variable approach of [31] is used in theory 2, the paper considers discrete time-varying descriptor polytopic systems, and X(αk ) of definition 3 is not positive definite but symmetric, so it is more difficult to deal with the problem than linear system. If Pi − Qi = X (a constant matrix) for all i ∈ I[1, N ], then a result on GQS can be deduced as follows. Corollary 1: Given the vertex set {Aj , j ∈ I[1, N ]} and the bounds of the parameter increments 0 ≤ ρi ≤ 1, i ∈ I[1, N ]. If there exist matrices R, X, Y ∈ 0, i ∈ I[1, N ] such that
Using Schur complement, we have from (16) and (18) that Q(αk+1 ) R 0 RT A(αk ) AT (αk )Y T > 0. (19) 0 Y A(αk ) P(αk )
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Xi RT 0
E T XE ≥ 0, ∀ i ∈ I[1, N ], R 0 Ai ATi Y T > 0, ∀i ∈ I[1, N ], Y Ai Pi
(23a) (23b)
WeB03.3
where Xi = Pi − X −
N X
−0.4 A1 = −0.08 0.5 1 A2 = −0.906 0
ρj (Pj − X),
j=1
Ai = E T XE + ATi R + RT Ai , Pi = Y T + Y − Pi −
N X
ρj Pj ,
j=1
then DTDPS (1) is GQS. If parameter αk ∈ ΛN of system (1) is constant (i.e., αk = α0 for any k ∈ Z + , which implies that bounds of parameter increment ρi = 0 for all i ∈ I[1, N ]), then the next two corollaries can be immediately obtained from Theorem 2 and Corollary 1, respectively. This offers sufficient conditions under which the system Exk+1 = A(α0 )xk
(24)
is AGQS and GQS, respectively. Corollary 2: Given the vertex set {Aj , j ∈ I[1, N ]} and the bounds of the parameter increments 0 ≤ ρi ≤ 1, i ∈ I[1, N ]. If there exist matrices R, Y ∈ 0, ∀i ∈ I[1, N ], Ai ATi Y T T Y Ai Y + Y − Pi
(25a) (25b)
where Ai = E T (Pi −Qi )E +ATi R +RT Ai , then the system (24) is AGQS. Corollary 3: Given the vertex set {Aj , j ∈ I[1, N ]} and the bounds of the parameter increments 0 ≤ ρi ≤ 1, i ∈ I[1, N ]. If there exist matrices R, X, Y ∈ 0, i ∈ I[1, N ] such that E T XE ≥ 0, ∀ i ∈ I[1, N ], (26a) Pi − X R 0 RT > 0, ∀i ∈ I[1, N ], Ai ATi Y T T 0 Y Ai Y + Y − Pi (26b) where Ai = E T XE + ATi R + RT Ai , then the system (24) is GQS. Remark 3: The results analogous to that of corollary 1 and 3 can be obtained in the literature [4] for linear system, since the paper considers time-varying descriptor systems, the results of corollary 1 and 3 have never been seen in the literatures which we have found.
0.5 0.3 −1 0 , 0 1 1 0.1 −1 0 . 0.2 1
Assume that ρ1 = 0.1 and ρ2 = 0.08. Using the LMI Control Toolbox in MATLAB, we obtain solutions of LMIs in (14) as follows: −0.1106 −0.0461 0.3773 R = −0.1674 −0.2574 0.1606 , 0.2697 −0.1751 1.5078 2.9311 2.2789 −0.0819 5.7271 −0.4469 , Y = 3.0400 −0.1039 −0.2922 1.4624 2.5059 2.7603 −0.1589 6.1193 −0.5031 , P1 = 2.7603 −0.1589 −0.5031 0.8926 3.7941 3.6328 −0.0872 5.1798 −0.2328 , P2 = 3.6328 −0.0872 −0.2328 0.8008 0.9537 0.8220 0.4714 Q1 = 0.8220 1.0119 0.1542 , 0.4714 0.1542 4.3302 1.0307 1.0038 0.2068 Q2 = 1.0038 1.5623 0.0933 . 0.2068 0.0933 4.1798 Therefore, by Theorem 2, we can conclude that the considered DTDPS (1) is AGQS. On the other hand, for the considered system, there is no solutions to the inequality (23) by LMI Toolbox in MATLAB, and hence the considered system is not GQS from Corollary 1. This shows that AGQS has less conservatism than GQS again. VI. CONCLUSIONS The robust stability analysis for DTDPS (1) is done by using the concept of affine generalized quadratic stability proposed here, which has less conservatism than generalized quadratic stability. LMI conditions for (affine) generalized quadratic stability have been given for this class of systems by using Lyapunov theory. The applicability of the proposed method is illustrated by an example.
V. NUMERICAL EXAMPLES
R EFERENCES
In this section, we provide an example to demonstrate the applicability of the proposed method. Consider the DTDPS (1) with 1 0 0 E = 0 1 0 , 0 0 0
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[17] J. Lavaei and A. G. Aghdam, Robust stability of LTI systems over semialgebraic sets using sum-of-squares matrix polynomials, IEEE Trans. Automat. Control, vol. 53, no. 1, pp. 417–423, 2008. [18] V. J. S. Leite and P. L. D. Peres, An improved LMI condition for robust D-stability of uncertain polytopic systems, IEEE Trans. Autom. Control, vol. 48, no. 3, pp. 500-504, 2003. [19] W. J. Mao, Robust stabilization of uncertain time-varying discrete systems and comments on ”an improved approach for constrained robust model predictive control,” Automatica, vol. 39, 1109-1112, 2003. [20] J. C. Geromel, Ratrizio Colameri, Robust stability of time varying polytopic systems, Systems Control Letters, vol. 55, 2006. [21] F. Blanchini, Stefano Miani, Carlo savorgnan, Stability results for linear parameter varying and switching systems, Automatica, vol. 43, 1817-1823, 2007. [22] F. L. Lewis, A survey of linear singular systems, Circ., Syst. Sign. Process., vol. 5, no. 1, pp. 3-36, 1986. [23] L. Dai, Singular Control systems, Springer, Berlin, 1989. [24] A. Bouali, M. Yagoubi and P. Chevrel, H2 gain scheduling control for rational LPV systems using the descriptor framework, Proc. 47th IEEE Confer. Decision and Control, pp. 3878–3883, 2008. [25] I. Masubuchi, J. Kato, M. Saeki and A. Ohara, Gain-scheduled controller design based on descriptor representation of LPV system: application to flight vehicle control, Proc. 43rd IEEE Confer. Decision and Control, pp. 815–820, 2004. [26] A. Bouali, M. Yagoubi and P. Chevrel, H2 gain scheduled observer dased controllers for rational LPV systems, Proc. 10th Intl. Confer. on Control, Automation, Robotics and Vision, pp. 1811–1816, 2008. [27] S. Xu, J. Lam, Robust Control and Filtering of Singular Systems, Lecture Notes in Control and Information Sciences, Springer, 2006. [28] S. Y. Xu and C. W. Yang, An algebraic approach to the robust stability analysis and robust stabilization of uncertain singular systems, Inter. J. Syst. Sci., vol. 31, no. 1, pp. 55-61, 2000. [29] C. G. Cao, Linear Algebra, Inner Mongolia Science and Technology Press (in Chinese), Inner Mongolia, 1999. [30] X. L. Zhang, Stability analysis of nonlinear time-varying discretetime systems (in Chinese), Master dissertation, East China Normal University, 2008. [31] M. C. de Oliveiraa, J. Bernussoub, J. C. Geromela, A new discretetime robust stability condition, Systems Control Lett., vol. 36, pp. 135-141, 1999.
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Parameter-dependent Lyapunov function approach to robust stability analysis for discrete-time descriptor polytopic systems Xiangyu Gao1,2 , Guang-Ren Duan1,∗ , Xian Zhang2 Abstract— This paper investigates robust stability of discretetime descriptor polytopic systems (DTDPSs for short). The concept of affine generalized quadratic stability, which has less conservatism than generalized quadratic stability, of DTDPSs is proposed. It not only investigates affine generalized quadratic stability of DTDPSs implies robust stability, but also presents criterions in terms of linear matrix inequalities to test the (affine) generalized quadratic stability based on time-varying parameter-dependent Lyapunov function. Finally, a numerical example presents the effectiveness of the proposed method.
I. INTRODUCTION The Lyapunov theory has been one of the most appealing methods for investigating robust stability of linear polytopic systems (LPSs for short) during the last decade. In particular, some results are based on the concept of quadratic stability (see, e.g. [1]–[3]). Unfortunately, these results may be very conservative due to the use of a common Lyapunov function for the entire uncertainty set. In order to reduce the conservatism, several kinds of parameterdependent Lyapunov functions have been proposed in literature to investigate robust stability of LPSs. For example, Lyapunov functions with linear dependence (see [4]–[9] and references therein), Lyapunov functions with polynomial dependence [10], [11], homogeneous polynomially parameterdependent Lyapunov function of arbitrary degree [12], [13], homogeneous polynomially parameter-dependent quadratic Lyapunov functions [14], [15], Lyapunov functions which are parameter-dependent in negative as well as positive power series of parameters [16]. Besides, Lavaei and Aghdam [17] deal with the robust stability of discrete-time linear time-invariant systems with parametric uncertainties belonging to a semialgebraic set, which can solve wider problem than the ones given in [13] and [14] for discrete-time systems. By introducing extra variables and using additional LMIs, Leite and Peres [18] proposed a sufficient condition for robust D-stability of LPSs, which contains the condition for robust D-stability from [9] as a particular case and encompasses the robust stability results of [6] (discrete-time systems) and [7] This work was partially supported by the Major Program of National Natural Science Foundation of China under Grant No. 60710002 and Program for Changjiang Scholars and Innovative Research Team in University 1 Center for Control Theory and Guidance Technology, Harbin Institute of Technology, Harbin 150001, China. [email protected],
[email protected] 2
School of Harbin 150080,
Mathematical Science, Heilongjiang University, China. [email protected],
[email protected] ∗
Corresponding author.
978-1-4244-3872-3/09/$25.00 ©2009 IEEE
(continuous- time systems) when these special cases of Dstability are investigated. In addition, some robust stability results of linear parameter-varying systems with parametric uncertainties are obtained [19]–[21]. On the other hand, descriptor systems have been extensively studied during the past four decades due to their wide applications in circuits, economic, large scale systems, and other areas [22]. Many notions and results in statespace systems have been extended to descriptor systems [23], since the latter can present a much wider class of systems than state-space systems can. However, few results on robust stability analysis for descriptor polytopic systems have been reported so far [24]–[26]. So the study of such problems is of both practical and theoretical importance. In this paper, we will study robust stability of discretetime descriptor polytopic systems (DTDPSs for short) by constructing Lyapunov functions with linear dependence, which is a generalization of the results in [4]. The concept of affine generalized quadratic stability of DTDPSs, which has less conservatism than generalized quadratic stability, is proposed in Section II. Based on some preliminary results introduced in Section III, it is shown in Section IV that affine generalized quadratic stability of DTDPSs implies robust stability, and thereby, sufficient conditions for (affine) generalized quadratic stability of DTDPSs are investigated in terms of linear matrix inequalities (LMIs for short). At last, the effectiveness of the proposed method is demonstrated by a numerical example in Section V. It should be pointed out that the robust stability problem for descriptor polytopic systems is much complicated than that for LPSs because it require to consider not only stability and robustness, but also regularity and impulse-free for continuous-time (causality for discrete-time) descriptor systems, simultaneously. The notations occurred in this paper are as follows: Let < be the real number set, and Z + the nonnegative integer set. Denote by I[k1 , k2 ] the set {k1 , k1 + 1, · · · , k2 } for k1 , k2 ∈ Z + with k1 < k2 . Let ΛN := {(λ1 , · · · , λN ) :
N X
λi = 1, λi ≥ 0, ∀i ∈ I[1, N ]}.
i=1
For λ = (λ1 , · · · , λN ) ∈ ΛN and a group of matrices N P M1 , M2 , · · · , MN , we denote by M (λ) the matrix λ i Mi . i=1
For real symmetric matrices P and Q, P > Q(P < Q) means that the matrix P − Q is positive (negative) definite, and P ≥ Q(P ≤ Q) means that the matrix P − Q is
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WeB03.3 positive (negative) semi-definite. Denote by k · k2 either the Euclidean vector norm or its induced matrix 2-norm. The transpose, inverse, determinant, adjoint matrix and the minimum eigenvalue of the matrix A are represented by AT , A−1 , detA, adjA and λmin (A) respectively. Let A−T = (A−1 )T . II. PROBLEM FORMULATION AND DEFINITIONS Consider the DTDPS Exk+1 = A(αk )xk , k ∈ Z + ,
(1)
where xk is the state variable, E ∈ 0, ∀ i ∈ I[1, N ]. R Qi Lemma 2: [27] Let H ∈ Rn×n is a constant matrix such that H + H T < 0. Then H is µinvertible .¶ M1 M2 Lemma 3: [28] Let M = ∈ 0 for any αk ∈ ΛN . Using (7) we have X For the sake of simplicity, A˜i (αk ) will be abbreviated as A˜i below for i = 1, 2, 3, 4. From (7b),(8) and (9),we have µ ¶ X1 X2 < 0, (10) X2T X3 where ˜ 1 (αk+1 )A˜1 + A˜T X ˜ T (αk+1 )A˜1 − X ˜ 1 (αk ) X1 = A˜T1 X 3 2 ˜ 2 (αk+1 )A˜3 + A˜T X ˜ 3 (αk+1 )A˜3 , +A˜T1 X 3 ˜ 1 (αk+1 )A˜2 + A˜T X ˜ T (αk+1 )A˜2 X2 = A˜T1 X 3 2 T ˜ T ˜ ˜ ˜ ˜ +A1 X2 (αk+1 )A4 + A3 X3 (αk+1 )A˜4 , ˜ 1 (αk+1 )A˜2 + A˜T X ˜ T (αk+1 )A˜2 X3 = A˜T2 X 4 2 T T ˜ 2 (αk+1 )A˜4 + A˜ X ˜ 3 (αk+1 )A˜4 . +A˜ X 2
T
˜ 1 (αk+1 )A˜ − X ˜ 1 (αk ) < 0. A˜T X
Let λ = min {λi (αk )} and λ = max {λi (αk )}. Then 1≤i≤r
1≤i≤r
0 < λ ≤ λi (αk ) ≤ λ, ∀ αk ∈ ΛN , i = 1, 2, · · · , N. ˜ 1 (αk ) ≤ λIr , ∀ αk ∈ ΛN . Consequently, 0 < λIr ≤ X Similarly, from (12), there exists a positive number λ such that ˜ 1 (αk+1 )A˜ − X ˜ 1 (αk ) ≤ −λI. A˜T X
0 < λkyk k22 ≤ V (yk , k) ≤ λkyk k22 and ∆V (yk , k)
˜ 2T (αk+1 )A˜2 + A˜T2 X ˜ 2 (αk+1 )A˜4 + A˜T4 X ˜ 3 (αk+1 )A˜4 < 0. A˜T4 X ˜ 3 (αk+1 )A˜4 . We obtain ˜ 2 (αk+1 )A˜4 + 1 A˜T X Let H = A˜T2 X 2 4 from Lemma 2 that H is an invertible matrix, so is A˜4 . Hence the DTDPS (1) is causal and regular for all αk ∈ ΛN . Next, we show that the DTDPS (1) is stable. Let xk = Q0 [ykT zkT ]T , where yk ∈ 0 such that kA˜3 k2 ≤ ξ. Thus, it follows from (11b) that ˜ kzk k2 = k − A˜−1 4 A3 yk k2 −1 ˜ ≤ kA4 k2 kA˜3 k2 kyk k2 δ3 ξ ≤ kyk k2 . δ1
M1 =
So the system (1) is asymptotically stable for all αk ∈ ΛN . Since the parameter αk is not known a priori, the condition (7) is not numerically vertifiable. Due to the particular structure X(αk ) = α ˜ 1 (k)X1 + · · · + α ˜ N (k)XN , αk ∈ ΛN ,
(13)
i.e., that X(αk ) is affine in the parameters, the following sufficient condition can be obtained. Theorem 2: Given the vertex set {Aj , j ∈ I[1, N ]} and the bounds of the parameter increments 0 ≤ ρi ≤ 1, i ∈ I[1, N ]. The DTDPS (1) is AGQS if there exist matrices
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WeB03.3 R, Y ∈ 0, Qi > 0, i ∈ I[1, N ] such that
Qi RT 0
E T (Pi − Qi )E ≥ 0, ∀ i ∈ I[1, N ], R 0 Ai ATi Y T > 0, ∀ i ∈ I[1, N ], Y Ai Pi
(14a) (14b)
where Qi = Qi −
N P j=1
This, together with (17), implies that Q(αk+1 ) R 0 RT > 0, A(αk ) AT (αk )Y T T 0 Y A(αk ) Y + Y − P (αk+1 ) and hence µ A(αk ) − RT Q(αk+1 )−1 R Y A(αk ) It follows from
ρj Qj ,
Ai = E T (Pi − Qi )E + ATi R + RT Ai , N P Pi = Y T + Y − Pi − ρj Pj .
(Y − P (αk+1 ))P −1 (αk+1 )(Y − P (αk+1 ))T ≥ 0
(15) that
Y P −1 (αk+1 )Y T ≥ Y T + Y − P (αk+1 ).
j=1
Proof: From Lemma 1, we have that the inequality (14b) is equivalent to the Q(αk ) R 0 RT A(αk ) AT (αk )Y T > 0, ∀αk ∈ ΛN , (16) 0 Y A(αk ) P(αk ) where Q(αk )
= Q(αk ) −
A(αk )
=
P(αk )
N P j=1
ρj Qj ,
E T (P (αk ) − Q(αk ))E + RT A(αk ) +AT (αk )R, N P = Y T + Y − P (αk ) − ρj Pj .
Because of [Q(αk+1 )A(αk )−R]T Q−1 (αk+1 )[Q(αk+1 )A(αk )−R] ≥ 0, we derive that
N P i=1
α ˜ i (k) = 1 that
AT (αk )Q(αk+1 )A(αk ) ≥ AT (αk )R + RT A(αk ) − RT Q−1 (αk+1 )R = A(αk ) − RT Q−1 (αk+1 )R −E T (P (αk ) − Q(αk ))E.
Y T + Y − P (αk+1 ) N N N P P P = α ˜ i (k)(Y T + Y ) − α ˜ i (k)Pi − ∆˜ αi (k)Pi = =
i=1 N P i=1
i=1
α ˜ i (k)(Y
T
+ Y − Pi ) −
α ˜ i (k)(Y T + Y − Pi ) −
≥ Y T + Y − P (αk ) −
N P j=1
NP −1 j=1 NP −1 j=1
Applying Schur complement to (22) again, we obtain that
i=1
∆˜ αj (Pj − PN )
AT (αk )X(αk+1 )A(αk ) − E T X(αk )E < 0, ∀ αk ∈ ΛN .
∆˜ αj Pj + ∆˜ αN PN
ρj Pj
= P(αk ) > 0.
(17)
and Q(αk+1 )
This, together with (20), implies that µ ¶ A(αk ) − RT Q(αk+1 )−1 R AT (αk )Y T > 0. Y A(αk ) Y P −1 (αk+1 )Y T (21) Since detY 6= 0, the inequality (21) is equivalent to the following inequality ¶ µ A(αk ) − RT Q−1 (αk+1 )R AT (αk ) > 0. (22) A(αk ) P −1 (αk+1 )
j=1
It follows from (3), (4), (5) and
i=1 N P
YT
¶ AT (αk )Y T > 0. + Y − P (αk+1 ) (20)
= ≥
N P i=1 N P i=1
α ˜ i (k)Qi + α ˜ i (k)Qi −
= Q(αk ) −
N P i=1
N P i=1 N P i=1
∆˜ αi Qi ρi Qi
(18)
ρi Qi
= Q(k) > 0.
where X(αk ) = P (αk ) − Q(αk ). Since (14a) implies that E T X(αk )E ≥ 0, ∀ αk ∈ ΛN , it follows from Definition 3 and Theorem 1 that DTDPS (1) is AGQS. Remark 2: Although the slack variable approach of [31] is used in theory 2, the paper considers discrete time-varying descriptor polytopic systems, and X(αk ) of definition 3 is not positive definite but symmetric, so it is more difficult to deal with the problem than linear system. If Pi − Qi = X (a constant matrix) for all i ∈ I[1, N ], then a result on GQS can be deduced as follows. Corollary 1: Given the vertex set {Aj , j ∈ I[1, N ]} and the bounds of the parameter increments 0 ≤ ρi ≤ 1, i ∈ I[1, N ]. If there exist matrices R, X, Y ∈ 0, i ∈ I[1, N ] such that
Using Schur complement, we have from (16) and (18) that Q(αk+1 ) R 0 RT A(αk ) AT (αk )Y T > 0. (19) 0 Y A(αk ) P(αk )
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Xi RT 0
E T XE ≥ 0, ∀ i ∈ I[1, N ], R 0 Ai ATi Y T > 0, ∀i ∈ I[1, N ], Y Ai Pi
(23a) (23b)
WeB03.3
where Xi = Pi − X −
N X
−0.4 A1 = −0.08 0.5 1 A2 = −0.906 0
ρj (Pj − X),
j=1
Ai = E T XE + ATi R + RT Ai , Pi = Y T + Y − Pi −
N X
ρj Pj ,
j=1
then DTDPS (1) is GQS. If parameter αk ∈ ΛN of system (1) is constant (i.e., αk = α0 for any k ∈ Z + , which implies that bounds of parameter increment ρi = 0 for all i ∈ I[1, N ]), then the next two corollaries can be immediately obtained from Theorem 2 and Corollary 1, respectively. This offers sufficient conditions under which the system Exk+1 = A(α0 )xk
(24)
is AGQS and GQS, respectively. Corollary 2: Given the vertex set {Aj , j ∈ I[1, N ]} and the bounds of the parameter increments 0 ≤ ρi ≤ 1, i ∈ I[1, N ]. If there exist matrices R, Y ∈ 0, ∀i ∈ I[1, N ], Ai ATi Y T T Y Ai Y + Y − Pi
(25a) (25b)
where Ai = E T (Pi −Qi )E +ATi R +RT Ai , then the system (24) is AGQS. Corollary 3: Given the vertex set {Aj , j ∈ I[1, N ]} and the bounds of the parameter increments 0 ≤ ρi ≤ 1, i ∈ I[1, N ]. If there exist matrices R, X, Y ∈ 0, i ∈ I[1, N ] such that E T XE ≥ 0, ∀ i ∈ I[1, N ], (26a) Pi − X R 0 RT > 0, ∀i ∈ I[1, N ], Ai ATi Y T T 0 Y Ai Y + Y − Pi (26b) where Ai = E T XE + ATi R + RT Ai , then the system (24) is GQS. Remark 3: The results analogous to that of corollary 1 and 3 can be obtained in the literature [4] for linear system, since the paper considers time-varying descriptor systems, the results of corollary 1 and 3 have never been seen in the literatures which we have found.
0.5 0.3 −1 0 , 0 1 1 0.1 −1 0 . 0.2 1
Assume that ρ1 = 0.1 and ρ2 = 0.08. Using the LMI Control Toolbox in MATLAB, we obtain solutions of LMIs in (14) as follows: −0.1106 −0.0461 0.3773 R = −0.1674 −0.2574 0.1606 , 0.2697 −0.1751 1.5078 2.9311 2.2789 −0.0819 5.7271 −0.4469 , Y = 3.0400 −0.1039 −0.2922 1.4624 2.5059 2.7603 −0.1589 6.1193 −0.5031 , P1 = 2.7603 −0.1589 −0.5031 0.8926 3.7941 3.6328 −0.0872 5.1798 −0.2328 , P2 = 3.6328 −0.0872 −0.2328 0.8008 0.9537 0.8220 0.4714 Q1 = 0.8220 1.0119 0.1542 , 0.4714 0.1542 4.3302 1.0307 1.0038 0.2068 Q2 = 1.0038 1.5623 0.0933 . 0.2068 0.0933 4.1798 Therefore, by Theorem 2, we can conclude that the considered DTDPS (1) is AGQS. On the other hand, for the considered system, there is no solutions to the inequality (23) by LMI Toolbox in MATLAB, and hence the considered system is not GQS from Corollary 1. This shows that AGQS has less conservatism than GQS again. VI. CONCLUSIONS The robust stability analysis for DTDPS (1) is done by using the concept of affine generalized quadratic stability proposed here, which has less conservatism than generalized quadratic stability. LMI conditions for (affine) generalized quadratic stability have been given for this class of systems by using Lyapunov theory. The applicability of the proposed method is illustrated by an example.
V. NUMERICAL EXAMPLES
R EFERENCES
In this section, we provide an example to demonstrate the applicability of the proposed method. Consider the DTDPS (1) with 1 0 0 E = 0 1 0 , 0 0 0
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