Improved Stability Criteria for Lurie Type Systems with Time-varying Delay

Transcription

1 Vol. 37, No. 5 ACTA ATOMATICA SINICA May, 011 Improved Stability Criteria for Lurie Type Systems with Time-varying Delay RAMAKRISHNAN Krishnan 1 RAY Goshaidas 1 Abstract In this technical note, we present a new stability analysis procedure for ascertaining the delay-dependent stability of a class of Lurie systems with time-varying delay and sector-bounded nonlinearity using Lyapunov-Krasovskii LK) functional approach. The proposed analysis, owing to the candidate LK functional and tighter bounding of its time-derivative, yields less conservative absolute and robust stability criteria for nominal and uncertain systems respectively. The effectiveness of the proposed criteria over some of the recently reported results is demonstrated using a numerical example. Key words Lurie systems, absolute stability, robust stability, Lyapunov-Krasovskii functional, linear matrix inequality DOI /SP.J The problem of absolute stability of Lurie systems has received considerable attention in control community, and many valuable results, such as Popov criterion, circle criterion, and Kalman-Yakubovih-Popov lemma have been reported in the past 13. Many typical nonlinear systems, such as Chua s circuit and the Lorenz system, can be classified into this type 4. Recently, reported results on master-slave synchronization for Lurie systems using time-delayed feedback control 57 has rekindled the interest on stability studies of the system, and has given a fresh impetus to the problem. As time-delay is often encountered in physical systems like communication systems, aircraft stabilization, nuclear reactor, process systems, population dynamics, etc., and is a source of poor performance and instability, the problem of absolute and robust stability of Lurie systems with time-delay attains considerable significance and interest in control literature. Depending upon whether or not the stability criteria for Lurie system contain the time-delay information, the criteria can be classified respectively into delay-dependent stability criteria and delay-independent stability criteria. In general, the delay-dependent criteria are less conservative than the delay-independent ones if delay size is very small. Hence, delay-dependent stability studies for Lurie systems with constant 810 and time-varying delay 1113 have been receiving increasing attention of the control community in recent years. In this paper, we research the problem of delaydependent stability of Lurie system with time-varying delay and sector-bounded nonlinearity using Lyapunov- Krasovskii LK) functional approach. Subsequently, absolute and robust stability criteria are derived respectively for nominal and uncertain Lurie systems in terms of linear matrix inequalities LMIs). To make the proposed criteria less conservative than the recently reported results 1113, a candidate LK functional is used in the delay-dependent stability analysis, and the cross-terms that emerge from the time-derivative of the functional are bounded tightly without neglecting any useful terms using minimal number of slack matrix variables. The proposed analysis, eventu- Manuscript received October 19, 010; accepted December 7, Department of Electrical Engineering, Indian Institute of Technology, Kharagpur, West Bengal 7130, India ally, culminates into a stability condition in convex LMI framework, and is solved non-conservatively at boundary conditions using standard numerical packages 14. Finally, a numerical example is employed to demonstrate the effectiveness of the proposed criteria. Notations. R n denotes the n-dimensional Euclidian space, R n m is the set of n m real matrices, I and 0 represents the identity matrix and null matrix of appropriate dimensions; the superscript T stands for the matrix transposition; X > 0 respectively X 0), for X R n m means that the matrix is real symmetric positive definite respectively, positive semi definite); R and Z denote the set of real numbers and integers; The represents the symmetric elements in a symmetric matrix. 1 System description and problem statement Consider a class of Lurie-type system with time-varying delay given by with ẋt) = A + Bxt ht)) + Dwt) zt) = M + Nxt ht)) wt) = ϕt, zt)) 1) = φt), t h, 0 ) R n, wt) R p, and zt) R q are the state, input, and output vectors of the system, respectively; A, B, D, M, and N are constant matrices of appropriate dimensions and the initial condition φt) is a continuous vector valued function; ϕt, zt)) R p is a nonlinear function that is piecewise continuous in t, and globally Lipschitz in zt); ϕt, 0) = 0, and satisfies the following sector condition t 0, zt) R p : ϕt, zt)) K 1zt) T ϕt, zt)) K zt) 0 3) K 1 and K are real constant matrices of appropriate dimensions, and K = K K 1 is a symmetric positivedefinite matrix. In other words, the nonlinear function ϕt, zt)) is said to belong to the sector K 1, K. On the other hand, if the nonlinear function ϕt, zt)) belongs to the sector 0, K, then, we have the following sector condition t 0, zt) R p : ϕ T t, zt))ϕt, zt)) Kzt) 0 4) The time-varying delay ht) is a continuous-time function satisfying the following conditions: 0 ht) h, ḣt) h d, t 0 5) In addition to absolute stability criterion, in this paper, robust stability criterion is also presented for the following class of uncertain Lurie systems: ẋt) = A + At)) + B + Bt))xt ht))+ D + Dt))wt) 6) zt) = M + Nxt ht)) wt) = ϕt, zt)) the time-varying uncertainties are of the form: At) Bt) Dt) = GF t) Ea E b E d G, E a, E b, and E d are known constant matrices of appropriate dimensions, and F t) is an unknown real timevarying matrix satisfying 7)

2 640 ACTA ATOMATICA SINICA Vol. 37 F T t)f t) I, t 8) The following lemmas are indispensable in deriving the proposed stability criterion, and they are stated below: Lemma For any constant matrix W R n n, a scalar γ > 0, and vector function ẋ : γ, 0 R n such that the integration tγ ẋt s)w ẋs)ds is well defined, then γ tγ δ γt) = ẋ T s)w ẋs)ds δ T γ t)π W δ γt) 9) xt γ) W W, Π W = W Lemma. Suppose r 1 rt) r, r ) : R + or Z +) R + or Z +). Then, for any R = R T > 0, free matrices T and Y, the following integral inequality holds: r1 tr ẋ T s)rẋs)ds δ T t)r rt))t R 1 T T + rt) r 1)Y R 1 Y T + Y Y + T T + Y Y + T T T δt) 10) δt) = x T t r 1) x T t rt)) x T t r ) T, T = T1 T T T T3 T T, and Y = Y1 T Y T Y3 T T. Proof. For any vectors z, y, and symmetric, positive definite matrix X, the following inequality holds: z T y z T X 1 z + y T Xy Substituting z = T T δt), y = rt) tr ẋs)ds, X = R, r rt) and subsequently using Jenson integral inequality, we get δ T t)t rt) tr δ T t)t R 1 T T δt) + ẋs)ds r rt)) rt) tr ẋ T s)rẋs)ds which, by Newton-Leibniz formula, is expressed as follows: rt) tr ẋ T s)rẋs)ds δ T t) r rt))t R 1 T T + 0 T T + 0 T T T δt) Similarly, we can deduce the following inequality as well: r1 ẋ T s)rẋs)ds δ T t) rt) r 1)Y R 1 Y T + trt) Y Y 0 + Y Y 0 T δt) Summation of the last two equations completes the proof of the Lemma. Lemma Suppose γ 1 γt) γ, γ ) : R + or Z +) R + or Z +). Then, for any constant matrices Ξ 1, Ξ, and Ω, the inequality Ω + γt) γ 1)Ξ 1 + γ γt))ξ holds if and only if the following boundary conditions hold: Ω + γ γ 1)Ξ 1 Ω + γ γ 1)Ξ Lemma Given matrices Q = Q T, H, E, and R = R T of appropriate dimensions Q + HF E + E T F T H T for all F satisfying F T F R holds, if and only if there exists some scalar, ɛ > 0 such that Q + ɛhh T + ɛ 1 E T RE The proposed absolute stability criterion The proposed absolute stability criterion for the system 1) satisfying 5) is stated below for ϕt, zt)) 0, K: Theorem 1. The system 1) satisfying 5) with ϕt, zt)) 0, K is absolutely stable for a given value of h and h d, if there exist real symmetric positive definite matrices P, Q, Z j, j = 1, ; matrices Q 11, Q 1, Q and slack matrices T i, Y i, M i, and N i, i = 1,, 3 of appropriate dimensions such that the following LMIs hold: Q11 Q 1 > 0 Q Π + Π 1 + Π T 1 Ā T h Ta 0 h Z1 Π + Π 1 + Π T 1 Ā T h Ya 0 h Z1 Φ + Φ 1 + Φ T 1 Ā T h Ma 0 h Z Φ + Φ 1 + Φ T 1 Ā T h Na 0 h Z 11) 1) 13) 14) Π and Φ are given on the top of the next page, and Π 1 = Y a Y a + T a T a 0 0 Y a = Y T 1 Y T Y T T T a = T T 1 T T T T T Φ 1 = 0 N a + M a N a M a 0 M a = 0 M T 1 M T M T 3 0 T N a = 0 N T 1 N T N T 3 0 T Ā = A B 0 0 D = h Z1 + Z)

3 No. 5 RAMAKRISHNAN Krishnan and RAY Goshaidas: Improved Stability Criteria for 641 Π = Φ = A T P + P A + Q + Q 11 P B Q 1 0 P D M T K T 1 h D)Q 0 0 N T K T Q Q 11 h Z Q1 + h Z 0 Q h Z 0 I A T P + P A + Q + Q 11 h Z1 P B Q1 + h Z1 0 P D M T K T 1 h D)Q 0 0 N T K T Q Q 11 Z1 h Q1 0 Q 0 I Γ = A T P + P A + Q + Q 11 P B Q 1 0 P D M T K T 1 h D)Q 0 0 N T K T Q Q 11 Q 1 0 Q 0 I Proof. Consider the following LK functional candidate: following inequality: V x t, t) = x T t)p + t h 0 h t+θ h t h xs) xs h ) t+θ tht) x T s)qxs)ds + T Q11 Q 1 Q ẋ T s)z 1ẋs)dsdθ + xs) xs h ) ds + ẋ T s)z ẋs)dsdθ 15) V x t, t) x T t)p A + Bxt ht)) + Dwt)) + δ T t)υδt) + x T t)q 1 h d )x T t ht))qxt ht)) + ẋ T t)ẋt) h th t h ẋ T s)z 1ẋs)ds ẋ T s)z ẋs)ds w T t)wt) w T t)km + Nxt ht)) 17) The time-derivative of the LK functional 15) along the trajectory of 1) is given by δt) = V x t, t) = x T t)p A + Bxt ht)) + Dwt)) + δ T t)υδt) + x T t)q 1 ḣt))xt t ht))qxt ht)) + ẋ T t)ẋt) h th xt h ) xt h) t h ẋ T s)z 1ẋs)ds ẋ T s)z ẋs)ds 16), Υ = Q11 Q1 0 Q Q 11 Q 1 Q Now, using the bound on the delay-derivative, and the sector condition defined in 4), we transform 16) into the By defining an augmented vector ζt) = x T t) x T t ht)) x T t h ) xt t h) w T t) T, we can express 17) as follows: V x t, t) ζ T t)γ + ĀTĀ)ζt) t h h th ẋ T s)z 1ẋs)ds ẋ T s)z ẋs)ds 18) Γ is given on the top of this page. Now, when 0 ht) h/, the cross-terms t h ẋ T s)z 1ẋs)ds and h th ẋt s)z ẋs)ds are dealt using Lemmas 1 and respectively as follows: t h ) ẋ T s)z 1ẋs)ds δ1 T h t) ht) T Z 1 1 T T + ht)y Z 1 1 Y T + Y Y + T T + Y Y + T T T δ 1t) 19)

4 64 ACTA ATOMATICA SINICA Vol. 37 and h th xt h) ẋ T s)z ẋs)ds xt h T ) δ 1t) = h Z h Z xt h ) 0) h Z xt h) x T t) x T t ht)) x T t h ) T T = T T 1 T T T T 3 Y = Y1 T Y T Y3 T In a similar manner, when h/ ht) h, the crossterms t h ẋ T s)z 1ẋs)ds and h th ẋt s)z ẋs)ds are dealt using Lemmas 1 and respectively as follows: t h ẋ T s)z 1ẋs)ds xt h ) T T T h Z1 h Z1 h Z1 xt h ) 1) and h ẋ T s)z ẋs)ds δ T t) h ht))mz 1 M T + th ht) h ) NZ 1 N T + N + M N M + T N + M N M δ t) ) δ t) = x T t ht)) x T t h ) xt t h) M = M T 1 M T M T 3 N = N T 1 N T N T 3 Hence, for 0 ht) h/, we use the relationships given in 19) and 0) in 18) and obtain the following inequality: T T V x t, t) ζ T h t) Ω 1 + ht) ht)y az 1 1 Ya T ) T az 1 1 T T a + T ζt) 3) similarly, for h/ ht) h, we use 1) and ) in 18) and obtain the following inequality condition: V x t, t) ζ T t) Ω + h ht))m az 1 Ma T + ht) h ) N az 1 Na T ζt) 4) Ω 1 = Π + Π 1 + Π T 1 + ĀT Ā and Ω = Φ + Φ1 + ΦT 1 + Ā T Ā. By Lyapunov-Krasovskii stability theorem18, the system 1) is absolutely stable, if V x t, t) λ for some λ > 0, if, for 0 ht) h/, ) h Ω 1 + ht) T az 1 1 Ta T + ht)y az 1 1 Ya T 5) and for h/ ht) h, Ω + h ht))m az 1 M T a + ht) h ) N az 1 Na T 6) By applying Lemma 3 and Schur complement 19 successively to 5) and 6), we deduce the LMIs stated in Theorem 1. 3 Robust stability criterion The proposed robust stability criterion for the uncertain system 6) satisfying the time-varying delay 5) is stated below for ϕt, zt)) 0, K: Theorem. The uncertain system 6) satisfying 5) with ϕt, zt)) 0, K is robustly absolutely stable for a given value of h and h d, if there exist real symmetric positive definite matrices P, Q, Z j, j = 1, ; matrices Q 11, Q 1, Q and slack matrices T i, Y i, M i and N i, i = 1,, 3 of appropriate dimensions, and scalars µ i > 0, i = 1 to 4 such that the following LMIs hold: Q11 Q 1 > 0 Q Π + Π 1 + Π T 1 Ā T P G µ1 Ē T h Ta G 0 0 µ 1I 0 0 µ 1I 0 h Z1 7) Π + Π 1 + Π T 1 Ā T P G µ Ē T h Ya G 0 0 µ I 0 0 µ I 0 h Z1 8) Φ + Φ 1 + Φ T 1 Ā T P G µ3 Ē T h Ma G 0 0 µ 3I 0 0 µ 3I 0 h Z 9) Φ + Φ 1 + Φ T 1 Ā T P G µ4 Ē T h Na G 0 0 µ 4I 0 0 µ 4I 0 h Z P = P T T Ē = E a E b 0 0 E d 30)

5 No. 5 RAMAKRISHNAN Krishnan and RAY Goshaidas: Improved Stability Criteria for 643 Proof. Replace the system matrices A, B, and D in LMIs 11) 14) with A + At)), B + Bt)), and D + Dt)), respectively. Then, from the definition of norm-bounded uncertainties 7), we obtain the following inequalities: Ω 1 + P GF t)ē) + P GF t)ē) T + h T b Z 1 1 T T b Ω 1 + P GF t)ē) + P GF t)ē) T + h Y b Z 1 1 Y T b Ω + P GF t)ē) + P GF t)ē) T + h M bz 1 M T b Ω + P GF t)ē) + P GF t)ē) T + h N b Z 1 N T b 31) 3) 33) 34) P = P T T T, Ē = Ē 0, T b = Ta T 0 T, and Y b = Ya T 0 T. Now, there exists µ i > 0, i = 1,, 3, 4 such that by Lemma 4, if 31) 34) hold, the following inequalities hold: Ω 1 + µ 1 1 P G) P G) T + µ 1Ē T Ē + h T b Z 1 1 T T b 35) Ω 1 + µ 1 P G) P G) T + µ Ē T Ē + h Y b Z 1 1 Y T b 36) Ω + µ 1 3 P G) P G) T + µ 3Ē T Ē + h M bz 1 M T b 37) Ω + µ 1 4 P G) P G) T + µ 4Ē T Ē + h N b Z 1 N T b 38) Schur complement to 35) 38) completes the proof. Remark 1. For ϕt, zt)) K 1, K, by using loop transformation, we can obtain the absolute and robust stability criteria by replacing A, B and K in Theorems 1 and with A DK 1M), B DK 1N), and K K 1), respectively. Remark. The reason for less conservativeness of the proposed stability criteria over the recently reported results 111 is attributed to the candidate LK functional used in the delay-dependent stability analysis. In 111, the LK functional is formulated by considering the variation of the delay ht) in the entire delay range, i.e., ht) 0, h; as in this paper, we have considered the variation of the delay in two equal segments of the delay-range viz. ht) 0, h/ and ht) h/, h, and have suitably embedded the segment information into the LK functional. The analysis, subsequently, paves way to stability criteria that yield less conservative stability criteria compared to those of 111. Remark 3. Though the delay-dependent stability analysis presented in 13 also splits the delay-interval 0, h into two equal sub-intervals and defines different energy functions on each interval, the following over-bounding: h h ) h ẋ T s)z ẋs)ds th ht) h ) h ẋ T s)z ẋs)ds tht) neglecting the terms h ht)) ht) ẋ T s)z th ẋs)ds, ht) h ) ht) ẋ T s)z th ẋs)ds, and h ht)) h tht) ẋt s)z ẋs)ds in the stability analysis introduces conservatism in the ensuing stability criteria. On the other hand, in the proposed stability analysis, we have reduced the conservatism of the stability criteria by using a candidate LK functional 0, and have bounded the timederivative of the functional tightly without neglecting any useful terms) using minimal number of slack matrix variables that are not redundant. This, in turn, yields less conservative stability criteria compared to those of 13. Remark 4. In situations there is no restriction on the derivative of the time-varying delay, the stability criteria can be deduced readily by letting Q = 0 in the Theorems 1 and. 4 A numerical example The effectiveness of the proposed stability criteria over 111 and 13 is demonstrated on a numerical example in this section. Example 1. Consider the nominal Lurie system given in 1) with following parameters: A = D = , B = 1 1, M = N = , K 1 = 0., K = 0.5 The maximum allowable delay bounds MADB) provided by the proposed stability criterion for Example 1 is listed in Table 1 for different values of h d. Owing to the use of a candidate LK functional 15) and tighter bounding conditions on the cross-terms, the proposed absolute stability criterion yields less conservative delay bounds than the existing results Consider the uncertain Lurie system which has the following parameters in addition to those furnished in Example 1: E a = E b = , E d = , G = The maximum delay bounds provided by the proposed robust stability criterion of Theorem for the uncertain system is listed in Table for different values of h d. For the uncertain case, the proposed robust stability criterion also yields less conservative delay bounds than those of the existing results. Table 1 Absolute stability: maximum allowable delay bound h for given h d Method h d > 1 11 h h a h Theorem 1 h

6 644 ACTA ATOMATICA SINICA Vol. 37 Table Robust stability: maximum allowable delay bound h for given h d Method h d > 1 11 h h a h Theorem h a These are the true results of Corollaries 1 and of 13. Remark 5. By solving the LMIs presented in Corollaries 1 and of 13, we have found that the delay bounds claimed in the paper are not true. In fact, the actual delay bounds provided by the corollaries are more conservative than what is actually claimed. These results are presented in Tables 1 and. 5 Conclusion In this note, we have proposed less conservative absolute and robust stability criteria for a class of Lurie systems with time-varying delay and sector-bounded nonlinearity. The delay-dependent stability analysis that yields the stability criteria uses a candidate LK functional, and the timederivative of the functional is bounded tightly without neglecting any useful terms using minimal number of slack matrix variables. The proposed analysis, subsequently, yields a stability criterion in convex LMI framework and is solved non-conservatively at boundary conditions using standard numerical packages. The effectiveness of the proposed criteria over recently reported results is demonstrated on a standard numerical example. As master-slave synchronization of Lurie systems with time-delayed feedback control is an important practical application, the future research is focused on delay-dependent stabilization of Lurie systems with time-varying delay. References 1 Popov V M. Hyperstability of Control Systems. New York: Springer, 1973 Khalil H K. Nonlinear Systems. pper Saddle River: Prentice-Hall, Liao X X. Absolute Stability of Nonlinear Control Systems. Beijing: Science Press, Wang J Z, Duan Z S, Yang Y, Huang L. Analysis and Control of Nonlinear Systems with Stationary Sets: Time-Domain and Frequency-Domain Methods. Singapore: World Scientific Publishing Company, Yalcin M E, Suykens J A K, Vandewalle J. Master slave synchronization of Lur e systems with time-delay. International Journal of Bifurcation and Chaos, 001, 116): Liao X X, Chen G R. Chaos synchronization of general Lur e systems via time-delay feedback control. International Journal of Bifurcation and Chaos, 003, 131): Cao J, Li H X, Ho D W C. Synchronization criteria of Lur e systems with time-delay feedback control. Chaos, Solitons and Fractals, 005, 34): He Y, Wu M. Absolute stability for multiple delay general Lur e control systems with multiple nonlinearities. Journal of Computational and Applied Mathematics, 003, 159): Wu M, Feng Z H, He Y. Improved delay-dependent absolute stability of Lur e systems with time-delay. International Journal of Control, Automation and Systems, 009, 76): He Y, Wu M, She J H, Liu G P. Robust stability for delay Lur e control systems with multiple nonlinearities. Journal of Computational and Applied Mathematics, 005, 176): Han Q L, Yue D. Absolute stability of Lur e systems with time-varying delay. IET Control Theory and Applications, 007, 13): Wu M, Feng Z Y, He Y, She J H. Improved delay-dependent absolute stability and robust stability for a class of nonlinear systems with time-varying delay. International Journal of Robust and Nonlinear Control, 010, 06): Gao Jin-Feng, Pan Hai-Peng, Ji Xiao-Fu. A new delaydependent absolute stability criterion for Lurie systems with time-varying delay. Acta Automatica Sinica, 010, 366): Gahinet P, Nemirovskii A, Laub A J, Chilali M. LMI control toolbox for use with MATLAB.. S.: Mathworks Inc, Han Q L. Absolute stablity of time-delayed systems with sector-bounded nonlinearity. Automatica, 005, 411): Yue D, Tian E G, Zhang Y J. A piecewise analysis method to stability analysis of linear continuous/discrete systems with time-varying delay. International Journal of Robust Nonlinear Control, 009, 1913): Wu M, He Y, She J H, Liu G P. Delay-dependent criteria for robust stability of time-varying delay systems. Automatica, 004, 408): Gu K, Kharitonov V L, Chen J. Stability of Time-delay Systems. Boston: Birkhauseräuser, Boyd S, Ghaoui L E, Feron E, Balakrishnan V. Linear Matrix Inequalities in System and Control Theory. Philadelphia: SIAM, Kwon O M, Park J H, Lee S M. An improved delaydependent criterion for asymptotic stability of uncertain dynamic systems with time-varying delays. Journal of Optimization Theory and Applications, 010, 145): RAMAKRISHNAN Krishnan Received his M. E. degree in control systems engineering from PSG College of Technology, India in He is currently a full-time Ph. D. scholar in the Department of Electrical Engineering, Indian Institute of Technology, Kharagpur, West Bengal, India. His research interest covers interval time-delay systems and LMI optimization. Corresponding author of this paper. ramkieeiit@gmail.com RAY Goshaidas Received his Ph. D. degree in control systems from Indian Institute of Technology in 198. He is currently a full professor in the Department of Electrical Engineering, Indian Institute of Technology, Kharagpur, West Bengal, India. His research interest covers de-centralized control, time-delay systems, robust control and optimal control. gray@ee.iitkgp.ernet.in