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Absolute stability of time-delay systems with sector-bounded nonlinearity
Automatica 41 (2005) 2171 – 2176 www.elsevier.com/locate/automatica
Technical communique
Absolute stability of time-delay systems with sector-bounded nonlinearity夡 Qing-Long Han∗ Faculty of Informatics and Communication, Central Queensland University, Rockhampton, Qld. 4702, Australia Received 29 December 2004; received in revised form 17 May 2005; accepted 14 August 2005 Available online 3 October 2005
Abstract This paper deals with the problem of absolute stability of time-delay systems with sector-bounded nonlinearity. Some new delay-dependent stability criteria are obtained and formulated in the form of linear matrix inequalities (LMIs). Neither model transformation nor bounding technique for cross terms is involved through derivation of the stability criteria. Numerical examples show that the results obtained in this paper improve the estimate of the stability limit over some existing result. 䉷 2005 Elsevier Ltd. All rights reserved. Keywords: Nonlinear systems; Time-delay; Sector condition; Uncertainty; Absolute stability; Linear matrix inequality (LMI)
1. Introduction Time-delays are frequently encountered in many fields of science and engineering, including communication network, manufacturing systems, biology, economy and other areas (Gu, Kharitonov, & Chen, 2003; Hale & Verduyn Lunel, 1993). During the last two decades, the problem of stability of linear timedelay systems has been the subject of considerable research efforts. Many significant results have been reported in the literature. For the recent progress, the reader is referred to Gu et al. (2003) and the references therein. Since the introduction of absolute stability by Lur’e (1957), the problem of absolute stability of a class of nonlinear control systems with a fixed matrix in the linear part of the system and one or multiple uncertain nonlinearities satisfying the sector constraints has been extensively studied (Aizerman & Gantmacher, 1964; Khalil, 1996; Liao, 1993; Lur’e, 1957; Popov, 1973; Yakubovich, Leonov, & Gelig, 2004). From the view of modern robustness theory, absolute stability can be considered as the robust global asymptotic stability with perturbations of nonlinearities from a given class of systems. 夡 This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Vladimir O. Nikiforov under the direction of Editor Paul Van den Hof. ∗ Tel.: +61 7 4930 9270; fax: +61 7 4930 9729. E-mail address: [email protected]
0005-1098/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2005.08.005
Due to time-delay occurring in practical systems, the problem of absolute stability for systems with delay has been also studied (Bliman, 2001; Gan & Ge, 2001; He & Wu, 2003; Li, 1963; Liao, 1993; Popov & Halanay, 1962; Somolines, 1977). However, the results mentioned above are delay-independent. When the time-delay is small, these results are often overly conservative; especially, they are not applicable to closed-loop systems which are open-loop unstable and are stabilized using delayed inputs, due to either delayed measurements or delayed actuator action in the input channels. Recently, Yu, Han, Yu, and Gao (2003) has studied the delay-dependent absolute stability for uncertain time-delay systems. Model transformation and bounding technique for cross terms are employed to derive some sufficient conditions. As is well known to the area of time-delay systems (Gu et al., 2003), model transformation sometimes will induce additional dynamics. Although a tighter bounding for cross terms can reduce the conservatism. However, there is no obvious way to obtain a much tighter bounding for cross terms. To sum up, in order to derive a much less conservative stability condition, we are in a position to avoid using both model transformation and bounding technique for cross terms, which motivates the present study. In this paper, we will deal with the problem of absolute stability for a class of time-delay systems which can be represented as a feedback connection of a linear dynamical system and a nonlinearity satisfying a sector constraint. Some new delaydependent absolute stability criteria will be derived without
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Q.-L. Han / Automatica 41 (2005) 2171 – 2176
employing any model transformation and bounding technique for cross terms. We will also give some numerical examples to show the effectiveness of the criteria. Notation. Rn denotes the n-dimensional Euclidean space. Rm×n is the set of all m × n real matrices. For symmetric matrices P and Q, the notation P > Q (P Q) means that matrix P − Q is positive definite (positive semi-definite). I is an identity matrix of appropriate dimensions. For an arbitrary matrix W and two symmetric matrices P and Q, the symmetric term in a symmetric matrix is denoted by *, i.e.
P ∗
W Q
=
P WT
W Q
the following integration is well defined, then
0
x˙ T (t + )W x(t ˙ + ) d −W W x(t) T T (x (t) x (t − )) . W −W x(t − )
−
−
(4)
Proof. Use Lemma 1 in Gu (2000) to obtain
0
−
x˙ T (t + )W x(t ˙ + ) d
.
T
0 −
W
x(t ˙ + ) d
0 −
x(t ˙ + ) d
= [x(t) − x(t − )] W [x(t) − x(t − )]. T
2. Problem statement
Re-arranging some terms yields (4).
Consider the following system: ˙ = Ax(t) + Bx(t − h) + Dw(t), x(t) z(t) = Mx(t) + N x(t − h), w(t) = −(t, z(t)),
3. Criterion with general sector condition (1)
with
We first consider the case when the nonlinear function (t, z(t)) belongs to a sector [0, K], i.e. (t, z(t)) satisfies T (t, z(t))[(t, z(t)) − Kz(t)] 0.
x() = () ∀ ∈ [−h, 0], Rn ,
(2) Rm
Rm
where x(t) ∈ w(t) ∈ and z(t) ∈ are the state vector, input vector and output vector of the system, respectively; h > 0 is the constant delay; (.) is a continuous vector valued initial function, A ∈ Rn×n , B ∈ Rn×n , D ∈ Rn×m , M ∈ Rm×n and N ∈ Rm×n are constant matrices; (t, z(t)) : [0, ∞) × Rm → Rm is a memory, possibly time-varying, nonlinear vector valued function which is piecewise continuous in t, globally Lipchitz in z(t), (t, 0) = 0, and satisfies the following sector condition for ∀t 0, ∀z(t) ∈ Rm : [(t, z(t)) − K1 z(t)]T [(t, z(t)) − K2 z(t)] 0,
(3)
where K1 and K2 are constant real matrices of appropriate dimensions and K = K1 − K2 is a symmetric positive definite matrix. It is customary that such a nonlinear function (t, z(t)) is said to belong to a sector [K1 , K2 ] (Khalil, 1996). We first introduce the following definition. Definition 1. The system described by (1)–(2) is said to be absolutely stable in the sector [K1 , K2 ] if the system is globally uniformly asymptotically stable for any nonlinear function (t, z(t)) satisfying (3). In this paper, we will attempt to formulate some practically computable criteria to check the absolute stability of the system described by (1)–(2). The following lemma is useful in deriving the criteria. Lemma 2. For any constant matrix W ∈ Rn×n , W = W T > 0, scalar > 0, and vector function x˙ : [−, 0] → Rn such that
(5)
We have the following result. Proposition 3. For given scalar h > 0, the system described by (1)–(2) with nonlinear connection function satisfying (5) is absolutely stable if there exist a scalar 0, real matrices P > 0, Q > 0, and R > 0 such that
(1, 1) ∗ ∗ ∗
PB + R −Q − R ∗ ∗
P D − M T K T −N T K T −2I ∗
hAT R hB T R < 0, hD T R −R
(6)
where (1, 1) AT P + P A + Q − R. Proof. Choose a Lyapunov–Krasovskii functional candidate as t V (t, xt ) = x T (t)P x(t) + x T ()Qx() d t−h t (h − t + )x˙ T ()(hR)x() ˙ d, + t−h
(7)
where P > 0, Q > 0, and R > 0. Taking the derivative of V (t, xt ) with respect to t along the trajectory of (1) yields V˙ (t, xt ) = x T (t)(AT P + P A + Q)x(t) + 2x T (t)P Bx(t − h) + 2x T (t)P Dw(t) − x T (t − h)Qx(t − h) + x˙ T (t)(h2 R)x(t) ˙ t − x˙ T ()(hR)x() ˙ d. t−h
Q.-L. Han / Automatica 41 (2005) 2171 – 2176
Use Lemma 2 to obtain t − x˙ T ()(hR)x() ˙ d t−h −R T T (x (t) x (t − h)) R
R −R
In view of Schur complement (see Fact 16), < 0 if there exist real matrices P > 0, Q > 0, and R > 0 and S > 0, and a scalar 0 such that (13).
x(t) . x(t − h)
Remark 4. Yu et al. (2003) considered the absolute stability of system (1) for the case of N = 0. The idea in Yu et al. (2003) was to first transform the system to a system with a distributed delay, then to apply an inequality to some cross terms. However, from the proof process of Proposition 3, one can clearly see that neither model transformation nor bounding technique for cross terms is involved. By Fact 17, one can also see that using the S-procedure in the proof process of Proposition 3 does not induce any conservatism due the case of p = 1. Therefore, the stability criterion is expected to be less conservative.
Noting that (1), the following holds:
T A T 2 T ˙ = q (t) B T (h2 R)(A B D)q(t), x˙ (t)(h R)x(t) DT where q T (t) = (x T (t) x T (t − h) w T (t)). Then we have V˙ (t, xt )q T (t)q(t), where
=
11 ∗ ∗
12 22 ∗
13 23 33
2173
Remark 5. From (13), one can see that > 0 is implied and ˜ = −1 Q, can be absorbed by other variables using P˜ = −1 P , Q −1 ˜ and R = R. Therefore, in the following, we use > 0 instead of 0.
,
with 11 AT P + P A + Q − R + AT (h2 R)A, 12 P B + R + AT (h2 R)B, 13 P D + AT (h2 R)D, 22 − Q − R + B T (h2 R)B, 23 B T (h2 R)D, 33 D T (h2 R)D. A sufficient condition for absolute stability of the system described by (1)–(2) is that there exist real matrices P > 0, Q > 0, and R > 0 such that V˙ (t, xt )q T (t)q(t) < 0,
Remark 6. Proposition 3 provides a delay-dependent absolute stability criterion for the system described by (1)–(2). If we set R = 0, then the Lyapunov–Krasovskii functional (7) ret duces to V˜ (t, xt ) = x T (t)P x(t) + t−h x T ()Qx() d. Similar to the proof of Proposition 3, using V˜ (t, xt ) we can obtain a delay-independent absolute stability condition for system (1)–(2). More specifically, we can conclude that for given scalar h > 0, the system described by (1)–(2) with nonlinear connection function satisfying (5) is absolutely stable if there exist a scalar > 0, real matrices P > 0 and Q > 0 such that
(8)
for all q(t) = 0 satisfying (5). From (1) and (5), we have w T (t)w(t) + w T (t)K[Mx(t) + N x(t − h)]0.
(9)
Then using the S-procedure in Yakubovich (1971) (see Fact 17), one can see that (8) is implied by the existence of a scalar 0 such that q T (t)q(t) − 2w T (t)w(t) − 2w T (t)K × [Mx(t) + N x(t − h)] < 0,
(10)
for all q(t) = 0. Rewrite (10) as q T (t)q(t) < 0, where
=
11 ∗ ∗
12 22 ∗
(11) 13 23 33
ij = ij (i, j = 1, 2), 23 = 23 − N T K T , 33 = 33 − 2I.
P B P D − M T K T −Q −N T K T ∗ −2I
< 0.
(12)
Remark 7. By Proposition 3, we can easily obtain a criterion for the special case of M = 0. It can be stated as that for given scalar h > 0, the system described by (1)–(2) with M = 0 and nonlinear connection function satisfying (5) is absolutely stable if there exist a scalar > 0, real matrices P > 0, Q > 0, and R > 0 such that
(1, 1) P B + R −Q − R ∗ ∗ ∗ ∗ ∗
PD −N T K T −2I ∗
hAT R T hB R < 0, hD T R −R
(13)
with 13 = 13 − M T K T ,
AT P + P A + Q ∗ ∗
,
where (1, 1) AT P + P A + Q − R. For the nonlinearity (t, z(t)) satisfying the more general sector condition (3), by applying an idea known as loop transformation (Khalil, 1996), we can conclude that the absolute stability of system (1)–(2) in the sector [K1 , K2 ] is equivalent
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Q.-L. Han / Automatica 41 (2005) 2171 – 2176
(3) and F (t) satisfying (17) is robustly absolutely stable if there exist scalars > 0, > 0, real matrices P > 0, Q > 0, and R > 0 such that
to that of the following system: x(t) ˙ = (A − DK 1 M)x(t) + (B − DK 1 N ) ×x(t − h) + D w(t), ˜ z(t) = Mx(t) + Nx(t − h), w(t) ˜ = −(t, ˜ z(t)),
(14)
˜ z(t)) satisfies for in the sector [0, K2 − K1 ], i.e. (t, ∀t 0, ∀z(t) ∈ Rm ,
(1, 1) ∗ ∗ ∗ ∗ ∗
(1, 2) −Q − R ∗ ∗ ∗ ∗
(1, 3) (2, 3) −2I ∗ ∗ ∗
PL 0 0 − I ∗ ∗
(1, 5) (2, 5) hD T R hLT R −R ∗
EaT EbT 0 < 0, (18) 0 0 − I
˜ T (t, z(t))[(t, ˜ z(t)) − (K2 − K1 )z(t)] 0, where by Proposition 3 we have the following result.
(1, 1) (A − DK 1 M)T P + P (A − DK 1 M) + Q − R,
Corollary 8. For given scalar h > 0, the system described by (1)–(2) with nonlinear connection function satisfying (3) is absolutely stable if there exist a scalar > 0, real matrices P > 0, Q > 0, and R > 0 such that
(1, 1) ∗ ∗ ∗
(1, 2) −Q − R ∗ ∗
(1, 3) (1, 4) (2, 3) (2, 4) < 0, −2I hD T R ∗ −R
(1, 2) P (B − DK 1 N ) + R, (1, 3) P D − M T (K2 − K1 )T , (1, 5) h(A − DK 1 M)T R, (2, 3) − N T (K2 − K1 )T , (2, 5) h(B − DK 1 N )T R.
(15) Remark 10. As a by-product, if we do not consider the nonlinearity, by Corollary 9 a delay-dependent stability criterion is immediately followed for the system.
where (1, 1) (A − DK 1 M)T P + P (A − DK 1 M) + Q − R,
4. Criterion with decentralized sector condition
(1, 2) P (B − DK 1 N ) + R,
Throughout this section, we assume that the nonlinearity (t, z(t)) is decentralized, i.e.
(1, 3) P D − M T (K2 − K1 )T , (1, 4) h(A − DK 1 M)T R,
T (t, z(t)) = (1 (t, z1 (t)), 2 (t, z2 (t)), . . . , m (t, zm (t))),
(2, 3) − N T (K2 − K1 )T , (2, 4) h(B − DK 1 N )T R. If there exist uncertainties in system’s matrices A and B, we can have different results depending on the uncertainty type. For the polytopic uncertainty, (15) needs to be satisfied for all the vertices. For norm-bounded uncertainty, system (1) becomes x(t) ˙ = [A + LF (t)Ea ]x(t) + [B + LF (t)Eb ] ×x(t − h) + Dw(t), z(t) = Mx(t) + N x(t − h), w(t) = −(t, z(t)),
(16)
where L, Ea , and Eb are known real constant matrices of appropriate dimensions, and F (t) is an unknown continuous timevarying matrix function satisfying F T (t)F (t)I .
(17)
(19)
and each component of (t, z(t)) satisfies the section condition
i zi2 (t) zi (t)i (t, zi (t)) i zi2 (t)
∀t 0.
(20)
For a given trajectory x(t) of system (1), define the time-varying gains by i (t, zi (t)) = ki (t)zi (t).
(21)
For different trajectories of the system, the time-varying gains defined in (21) are different. But, irrespective of the particular trajectories traced, it is always true that
i ki (t) i ,
(22)
Using the routine method of handling norm-bounded uncertainty (Han, 2002) by Proposition 3 we can obtain a more general result.
because i (t, zi (t)) satisfies (20). With this observation and defining K(t) = diag{k1 (t), k2 (t), . . . , km (t)}, the problem of absolute stability of system (1) with the (t, z(t)) satisfying (20) reduces to that of robust stability of the following system:
Corollary 9. For given scalar h > 0, the system described by (16), (2) with nonlinear connection function satisfying
x(t) ˙ = [A − DK(t)M]x(t) + [B − DK(t)N ]x(t − h),
(23)
Q.-L. Han / Automatica 41 (2005) 2171 – 2176
for all time-varying gains satisfying (21). The gains k1 (t) to km (t) take values in the convex hypercube H = K ∈ Rm | i ki i .
(24)
We denote by k (1) , k (2) , . . . , k (2 Corresponding to each vertex, let A(i) A − DK (i) M,
m)
B (i) B − DK (i) N.
Proposition 11. For given scalar h > 0, the system described by (1)–(2) with nonlinear connection function satisfying (20) is absolutely stable if there exist real matrices P > 0, Q > 0, and R > 0 such that (1, 1) P B (i) + R ∗ −Q − R ∗ ∗
hA(i)T R hB (i)T R −R
Table 1 Maximum allowed time-delay hmax using criteria in Yu et al. (2003) and this paper
Yu et al. (2003) Corollary 9
0.00 2.3295 2.4859
0.05 2.0703 2.2396
0.10 1.8481 2.0243
0.15 1.6537 1.8363
, the 2m vertices of H.
Similar to the proof of Proposition 3, we have the following result.
2175
< 0,
(25)
for all A(i) , B (i) (i = 1, 2, . . . , 2m ), where (1, 1) A(i)T P + P A(i) + Q − R. Remark 12. One can also consider the effect of uncertainties in system’s matrices A and B on the stability of the system. By Proposition 11 one can easily conclude the corresponding stability criteria for uncertainty of different type. Remark 13. It is interesting to note that h appears linearly in (13), (15), (18) and (25). Thus, a generalized eigenvalue problem as defined in Boyd, El Ghaoui, Feron, and Balakrishnan (1994) can be formulated to solve the minimum acceptable 1/ h and therefore the maximum hmax to maintain robust absolute stability as judged by these conditions. 5. Numerical examples Example 14. Consider the system described by (16)–(17) with −2 0 −1 0 −0.2 , B= , D= , −0.3 0 −0.9 −1 −1 M = ( 0.6 0.8 ) , N= ( 0 0 ) , K1 =0.2, K2 = 0.5,
0 1 0 L= , 0, Ea = Eb = . 0
0 1
Example 15. Consider the system described by (1), (19), (20) with −2.0 −1.0 0.5 1.0 A= , B= , 0.5 0.2 −0.1 −0.8 −0.5 0 0.4 0 D= , M= , 0 0.2 0 0.5 0.2 0 0.1 0 0.4 0 , K1 = N= , K2 = . 0 0.3 0 0.2 0 0.5 Since N = 0, no conclusion can be made using criteria in Yu et al. (2003). However, one can easily formulate a criterion along the idea in Yu et al. (2003) for the case of N = 0. By the new criterion, the numerical result is hmax = 1.6417. Applying Corollary 8, the maximum allowed time-delay is computed as hmax = 2.0239. Using Proposition 11 we have hmax = 2.0263. For this example, Proposition 11 can provide a slightly better result than Corollary 8. It is again to show that the criteria in this paper are less conservative than the corresponding criterion in Yu et al. (2003). 6. Conclusion The problem of absolute stability of a class of time-delay systems with sector-bounded nonlinearity has been addressed. Delay-dependent stability criteria with the general and decentralized sector conditions have been proposed. For the case of the decentralized sector condition, problem of absolute stability has been reduced to that of robust stability of linear timedelay systems. In order to reduce the conservatism of the criteria, we have avoided using model transformation and bounding technique for cross terms, which are widely used in deriving delay-dependent stability criteria for linear time-delay systems. Numerical examples have shown improvements over some existing result.
A=
Using the criterion in Yu et al. (2003) and Corollary 9 in this paper, Table 1 lists the maximum allowed time-delay hmax for robust absolute stability. It is clear to see that for this example the criterion in this paper can provide a less conservative result than that in Yu et al. (2003).
Acknowledgements The author would like to thank the Editor, Professor Paul M.J. Van den Hof, the Associate Editor and the anonymous reviewers for their constructive comments and suggestions to improve the quality of the paper. The research work was partially supported by Central Queensland University for the 2004 Research Advancement Awards Scheme Project “Analysis and Synthesis of Networked Control Systems” and the Strategic Research Project “Delay Effects: Analysis, Synthesis and Applications” (2003–2006).
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Q.-L. Han / Automatica 41 (2005) 2171 – 2176
Appendix A. Schur complement and S-procedure Schur complement is widely employed when a nonlinear matrix inequality is transformed into an linear matrix inequality (LMI) form. Fact 16. For matrices Q = QT , S and R = R T of appropriate dimensions, the inequality Q S >0 ST R is equivalent to the following two inequalities: R > 0, Q − SR −1 S T > 0. S-procedure (Yakubovich, 1971) plays an important role in absolute stability and robust stability theory. There are a number of variations, one of which is used in this paper can be stated as follows. Fact 17. Let Fi = FiT ∈ Rn×n , i = 0, 1, 2, . . . , p. Then the following statement is true T F0 > 0,
for all = 0 satisfying T Fi 0,
(A.1)
if there exist real scalars i 0, i = 1, 2, . . . , p such that F0 −
p
i Fi > 0.
(A.2)
i=1
For p = 1, these two statements are equivalent. References Aizerman, M. A., & Gantmacher, F. R. (1964). Absolute stability of regulator systems. San Francisco, CA: Holden-Day.
Bliman, P.-A. (2001). Lyapunov–Krasovskii functionals and frequency domain: delay-independent absolute stability criteria for delay systems. International Journal of Robust and Nonlinear Control, 11, 771–788. Boyd, S., El Ghaoui, L., Feron, E., & Balakrishnan, V. (1994). Linear matrix inequalities in systems and control theory. Philadelphia, PA: SIAM. Gan, Z. X., & Ge, W. G. (2001). Lyapunov functional for multiple delay general Lur’e control systems with multiple non-linearities. Journal of Mathematics Analysis and Applications, 259, 596–608. Gu, K. (2000). An integral inequality in the stability problem of time-delay systems. Proceedings of the 39th IEEE conference on decision and control (pp. 2805–2810). Sydney, Australia. Gu, K., Kharitonov, V. L., & Chen, J. (2003). Stability of time-delay systems. Boston: Birkhäuser. Hale, J. K., & Verduyn Lunel, S. M. (1993). Introduction to functional differential equations. New York: Springer. Han, Q.-L. (2002). Robust stability of uncertain delay-differential systems of neutral type. Automatica, 38, 719–723. He, Y., & Wu, M. (2003). Absolute stability for multiple delay general Lur’e control systems with multiple nonlinearities. Journal of Computational and Applied Mathematics, 159, 241–248. Khalil, H. K. (1996). Nonlinear systems. Upper Saddle River, NJ: PrenticeHall. Li, X.-J. (1963). On the absolute stability of systems with time lags. Chinese Mathematics, 4, 609–626. Liao, X. X. (1993). Absolute stability of nonlinear control systems. Beijing: Science Press. Lur’e, A. I. (1957). Some nonlinear problems in the theory of automatic control. London: H.M. Stationery Office. Popov, V. M. (1973). Hyperstability of control systems. New York, NY: Springer. Popov, V. M., & Halanay, A. (1962). About stability of non-linear controlled systems with delay. Automation and Remote Control, 23, 849–851. Somolines, A. (1977). Stability of Lurie type functional equations. Journal of Differential Equations, 26, 191–199. Yakubovich, V.A. (1971). S-procedure in nonlinear control theory. Vestnik Leningradskogo Universiteta, Ser. Matematika, Series 1, 13, 62–77. Yakubovich, V. A., Leonov, G. A., & Gelig, A. Kh. (2004). Stability of stationary sets in control systems with discontinuous nonlinearities. Singapore: World Scientific. Yu, L., Han, Q.-L., Yu, S., & Gao, J. (2003). Delay-dependent conditions for robust absolute stability of uncertain time-delay systems. Proceedings of the 42nd IEEE conference on decision and control (pp. 6033–6037). Maui, Hawaii USA.
Technical communique
Absolute stability of time-delay systems with sector-bounded nonlinearity夡 Qing-Long Han∗ Faculty of Informatics and Communication, Central Queensland University, Rockhampton, Qld. 4702, Australia Received 29 December 2004; received in revised form 17 May 2005; accepted 14 August 2005 Available online 3 October 2005
Abstract This paper deals with the problem of absolute stability of time-delay systems with sector-bounded nonlinearity. Some new delay-dependent stability criteria are obtained and formulated in the form of linear matrix inequalities (LMIs). Neither model transformation nor bounding technique for cross terms is involved through derivation of the stability criteria. Numerical examples show that the results obtained in this paper improve the estimate of the stability limit over some existing result. 䉷 2005 Elsevier Ltd. All rights reserved. Keywords: Nonlinear systems; Time-delay; Sector condition; Uncertainty; Absolute stability; Linear matrix inequality (LMI)
1. Introduction Time-delays are frequently encountered in many fields of science and engineering, including communication network, manufacturing systems, biology, economy and other areas (Gu, Kharitonov, & Chen, 2003; Hale & Verduyn Lunel, 1993). During the last two decades, the problem of stability of linear timedelay systems has been the subject of considerable research efforts. Many significant results have been reported in the literature. For the recent progress, the reader is referred to Gu et al. (2003) and the references therein. Since the introduction of absolute stability by Lur’e (1957), the problem of absolute stability of a class of nonlinear control systems with a fixed matrix in the linear part of the system and one or multiple uncertain nonlinearities satisfying the sector constraints has been extensively studied (Aizerman & Gantmacher, 1964; Khalil, 1996; Liao, 1993; Lur’e, 1957; Popov, 1973; Yakubovich, Leonov, & Gelig, 2004). From the view of modern robustness theory, absolute stability can be considered as the robust global asymptotic stability with perturbations of nonlinearities from a given class of systems. 夡 This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Vladimir O. Nikiforov under the direction of Editor Paul Van den Hof. ∗ Tel.: +61 7 4930 9270; fax: +61 7 4930 9729. E-mail address: [email protected]
0005-1098/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2005.08.005
Due to time-delay occurring in practical systems, the problem of absolute stability for systems with delay has been also studied (Bliman, 2001; Gan & Ge, 2001; He & Wu, 2003; Li, 1963; Liao, 1993; Popov & Halanay, 1962; Somolines, 1977). However, the results mentioned above are delay-independent. When the time-delay is small, these results are often overly conservative; especially, they are not applicable to closed-loop systems which are open-loop unstable and are stabilized using delayed inputs, due to either delayed measurements or delayed actuator action in the input channels. Recently, Yu, Han, Yu, and Gao (2003) has studied the delay-dependent absolute stability for uncertain time-delay systems. Model transformation and bounding technique for cross terms are employed to derive some sufficient conditions. As is well known to the area of time-delay systems (Gu et al., 2003), model transformation sometimes will induce additional dynamics. Although a tighter bounding for cross terms can reduce the conservatism. However, there is no obvious way to obtain a much tighter bounding for cross terms. To sum up, in order to derive a much less conservative stability condition, we are in a position to avoid using both model transformation and bounding technique for cross terms, which motivates the present study. In this paper, we will deal with the problem of absolute stability for a class of time-delay systems which can be represented as a feedback connection of a linear dynamical system and a nonlinearity satisfying a sector constraint. Some new delaydependent absolute stability criteria will be derived without
2172
Q.-L. Han / Automatica 41 (2005) 2171 – 2176
employing any model transformation and bounding technique for cross terms. We will also give some numerical examples to show the effectiveness of the criteria. Notation. Rn denotes the n-dimensional Euclidean space. Rm×n is the set of all m × n real matrices. For symmetric matrices P and Q, the notation P > Q (P Q) means that matrix P − Q is positive definite (positive semi-definite). I is an identity matrix of appropriate dimensions. For an arbitrary matrix W and two symmetric matrices P and Q, the symmetric term in a symmetric matrix is denoted by *, i.e.
P ∗
W Q
=
P WT
W Q
the following integration is well defined, then
0
x˙ T (t + )W x(t ˙ + ) d −W W x(t) T T (x (t) x (t − )) . W −W x(t − )
−
−
(4)
Proof. Use Lemma 1 in Gu (2000) to obtain
0
−
x˙ T (t + )W x(t ˙ + ) d
.
T
0 −
W
x(t ˙ + ) d
0 −
x(t ˙ + ) d
= [x(t) − x(t − )] W [x(t) − x(t − )]. T
2. Problem statement
Re-arranging some terms yields (4).
Consider the following system: ˙ = Ax(t) + Bx(t − h) + Dw(t), x(t) z(t) = Mx(t) + N x(t − h), w(t) = −(t, z(t)),
3. Criterion with general sector condition (1)
with
We first consider the case when the nonlinear function (t, z(t)) belongs to a sector [0, K], i.e. (t, z(t)) satisfies T (t, z(t))[(t, z(t)) − Kz(t)] 0.
x() = () ∀ ∈ [−h, 0], Rn ,
(2) Rm
Rm
where x(t) ∈ w(t) ∈ and z(t) ∈ are the state vector, input vector and output vector of the system, respectively; h > 0 is the constant delay; (.) is a continuous vector valued initial function, A ∈ Rn×n , B ∈ Rn×n , D ∈ Rn×m , M ∈ Rm×n and N ∈ Rm×n are constant matrices; (t, z(t)) : [0, ∞) × Rm → Rm is a memory, possibly time-varying, nonlinear vector valued function which is piecewise continuous in t, globally Lipchitz in z(t), (t, 0) = 0, and satisfies the following sector condition for ∀t 0, ∀z(t) ∈ Rm : [(t, z(t)) − K1 z(t)]T [(t, z(t)) − K2 z(t)] 0,
(3)
where K1 and K2 are constant real matrices of appropriate dimensions and K = K1 − K2 is a symmetric positive definite matrix. It is customary that such a nonlinear function (t, z(t)) is said to belong to a sector [K1 , K2 ] (Khalil, 1996). We first introduce the following definition. Definition 1. The system described by (1)–(2) is said to be absolutely stable in the sector [K1 , K2 ] if the system is globally uniformly asymptotically stable for any nonlinear function (t, z(t)) satisfying (3). In this paper, we will attempt to formulate some practically computable criteria to check the absolute stability of the system described by (1)–(2). The following lemma is useful in deriving the criteria. Lemma 2. For any constant matrix W ∈ Rn×n , W = W T > 0, scalar > 0, and vector function x˙ : [−, 0] → Rn such that
(5)
We have the following result. Proposition 3. For given scalar h > 0, the system described by (1)–(2) with nonlinear connection function satisfying (5) is absolutely stable if there exist a scalar 0, real matrices P > 0, Q > 0, and R > 0 such that
(1, 1) ∗ ∗ ∗
PB + R −Q − R ∗ ∗
P D − M T K T −N T K T −2I ∗
hAT R hB T R < 0, hD T R −R
(6)
where (1, 1) AT P + P A + Q − R. Proof. Choose a Lyapunov–Krasovskii functional candidate as t V (t, xt ) = x T (t)P x(t) + x T ()Qx() d t−h t (h − t + )x˙ T ()(hR)x() ˙ d, + t−h
(7)
where P > 0, Q > 0, and R > 0. Taking the derivative of V (t, xt ) with respect to t along the trajectory of (1) yields V˙ (t, xt ) = x T (t)(AT P + P A + Q)x(t) + 2x T (t)P Bx(t − h) + 2x T (t)P Dw(t) − x T (t − h)Qx(t − h) + x˙ T (t)(h2 R)x(t) ˙ t − x˙ T ()(hR)x() ˙ d. t−h
Q.-L. Han / Automatica 41 (2005) 2171 – 2176
Use Lemma 2 to obtain t − x˙ T ()(hR)x() ˙ d t−h −R T T (x (t) x (t − h)) R
R −R
In view of Schur complement (see Fact 16), < 0 if there exist real matrices P > 0, Q > 0, and R > 0 and S > 0, and a scalar 0 such that (13).
x(t) . x(t − h)
Remark 4. Yu et al. (2003) considered the absolute stability of system (1) for the case of N = 0. The idea in Yu et al. (2003) was to first transform the system to a system with a distributed delay, then to apply an inequality to some cross terms. However, from the proof process of Proposition 3, one can clearly see that neither model transformation nor bounding technique for cross terms is involved. By Fact 17, one can also see that using the S-procedure in the proof process of Proposition 3 does not induce any conservatism due the case of p = 1. Therefore, the stability criterion is expected to be less conservative.
Noting that (1), the following holds:
T A T 2 T ˙ = q (t) B T (h2 R)(A B D)q(t), x˙ (t)(h R)x(t) DT where q T (t) = (x T (t) x T (t − h) w T (t)). Then we have V˙ (t, xt )q T (t)q(t), where
=
11 ∗ ∗
12 22 ∗
13 23 33
2173
Remark 5. From (13), one can see that > 0 is implied and ˜ = −1 Q, can be absorbed by other variables using P˜ = −1 P , Q −1 ˜ and R = R. Therefore, in the following, we use > 0 instead of 0.
,
with 11 AT P + P A + Q − R + AT (h2 R)A, 12 P B + R + AT (h2 R)B, 13 P D + AT (h2 R)D, 22 − Q − R + B T (h2 R)B, 23 B T (h2 R)D, 33 D T (h2 R)D. A sufficient condition for absolute stability of the system described by (1)–(2) is that there exist real matrices P > 0, Q > 0, and R > 0 such that V˙ (t, xt )q T (t)q(t) < 0,
Remark 6. Proposition 3 provides a delay-dependent absolute stability criterion for the system described by (1)–(2). If we set R = 0, then the Lyapunov–Krasovskii functional (7) ret duces to V˜ (t, xt ) = x T (t)P x(t) + t−h x T ()Qx() d. Similar to the proof of Proposition 3, using V˜ (t, xt ) we can obtain a delay-independent absolute stability condition for system (1)–(2). More specifically, we can conclude that for given scalar h > 0, the system described by (1)–(2) with nonlinear connection function satisfying (5) is absolutely stable if there exist a scalar > 0, real matrices P > 0 and Q > 0 such that
(8)
for all q(t) = 0 satisfying (5). From (1) and (5), we have w T (t)w(t) + w T (t)K[Mx(t) + N x(t − h)]0.
(9)
Then using the S-procedure in Yakubovich (1971) (see Fact 17), one can see that (8) is implied by the existence of a scalar 0 such that q T (t)q(t) − 2w T (t)w(t) − 2w T (t)K × [Mx(t) + N x(t − h)] < 0,
(10)
for all q(t) = 0. Rewrite (10) as q T (t)q(t) < 0, where
=
11 ∗ ∗
12 22 ∗
(11) 13 23 33
ij = ij (i, j = 1, 2), 23 = 23 − N T K T , 33 = 33 − 2I.
P B P D − M T K T −Q −N T K T ∗ −2I
< 0.
(12)
Remark 7. By Proposition 3, we can easily obtain a criterion for the special case of M = 0. It can be stated as that for given scalar h > 0, the system described by (1)–(2) with M = 0 and nonlinear connection function satisfying (5) is absolutely stable if there exist a scalar > 0, real matrices P > 0, Q > 0, and R > 0 such that
(1, 1) P B + R −Q − R ∗ ∗ ∗ ∗ ∗
PD −N T K T −2I ∗
hAT R T hB R < 0, hD T R −R
(13)
with 13 = 13 − M T K T ,
AT P + P A + Q ∗ ∗
,
where (1, 1) AT P + P A + Q − R. For the nonlinearity (t, z(t)) satisfying the more general sector condition (3), by applying an idea known as loop transformation (Khalil, 1996), we can conclude that the absolute stability of system (1)–(2) in the sector [K1 , K2 ] is equivalent
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Q.-L. Han / Automatica 41 (2005) 2171 – 2176
(3) and F (t) satisfying (17) is robustly absolutely stable if there exist scalars > 0, > 0, real matrices P > 0, Q > 0, and R > 0 such that
to that of the following system: x(t) ˙ = (A − DK 1 M)x(t) + (B − DK 1 N ) ×x(t − h) + D w(t), ˜ z(t) = Mx(t) + Nx(t − h), w(t) ˜ = −(t, ˜ z(t)),
(14)
˜ z(t)) satisfies for in the sector [0, K2 − K1 ], i.e. (t, ∀t 0, ∀z(t) ∈ Rm ,
(1, 1) ∗ ∗ ∗ ∗ ∗
(1, 2) −Q − R ∗ ∗ ∗ ∗
(1, 3) (2, 3) −2I ∗ ∗ ∗
PL 0 0 − I ∗ ∗
(1, 5) (2, 5) hD T R hLT R −R ∗
EaT EbT 0 < 0, (18) 0 0 − I
˜ T (t, z(t))[(t, ˜ z(t)) − (K2 − K1 )z(t)] 0, where by Proposition 3 we have the following result.
(1, 1) (A − DK 1 M)T P + P (A − DK 1 M) + Q − R,
Corollary 8. For given scalar h > 0, the system described by (1)–(2) with nonlinear connection function satisfying (3) is absolutely stable if there exist a scalar > 0, real matrices P > 0, Q > 0, and R > 0 such that
(1, 1) ∗ ∗ ∗
(1, 2) −Q − R ∗ ∗
(1, 3) (1, 4) (2, 3) (2, 4) < 0, −2I hD T R ∗ −R
(1, 2) P (B − DK 1 N ) + R, (1, 3) P D − M T (K2 − K1 )T , (1, 5) h(A − DK 1 M)T R, (2, 3) − N T (K2 − K1 )T , (2, 5) h(B − DK 1 N )T R.
(15) Remark 10. As a by-product, if we do not consider the nonlinearity, by Corollary 9 a delay-dependent stability criterion is immediately followed for the system.
where (1, 1) (A − DK 1 M)T P + P (A − DK 1 M) + Q − R,
4. Criterion with decentralized sector condition
(1, 2) P (B − DK 1 N ) + R,
Throughout this section, we assume that the nonlinearity (t, z(t)) is decentralized, i.e.
(1, 3) P D − M T (K2 − K1 )T , (1, 4) h(A − DK 1 M)T R,
T (t, z(t)) = (1 (t, z1 (t)), 2 (t, z2 (t)), . . . , m (t, zm (t))),
(2, 3) − N T (K2 − K1 )T , (2, 4) h(B − DK 1 N )T R. If there exist uncertainties in system’s matrices A and B, we can have different results depending on the uncertainty type. For the polytopic uncertainty, (15) needs to be satisfied for all the vertices. For norm-bounded uncertainty, system (1) becomes x(t) ˙ = [A + LF (t)Ea ]x(t) + [B + LF (t)Eb ] ×x(t − h) + Dw(t), z(t) = Mx(t) + N x(t − h), w(t) = −(t, z(t)),
(16)
where L, Ea , and Eb are known real constant matrices of appropriate dimensions, and F (t) is an unknown continuous timevarying matrix function satisfying F T (t)F (t)I .
(17)
(19)
and each component of (t, z(t)) satisfies the section condition
i zi2 (t) zi (t)i (t, zi (t)) i zi2 (t)
∀t 0.
(20)
For a given trajectory x(t) of system (1), define the time-varying gains by i (t, zi (t)) = ki (t)zi (t).
(21)
For different trajectories of the system, the time-varying gains defined in (21) are different. But, irrespective of the particular trajectories traced, it is always true that
i ki (t) i ,
(22)
Using the routine method of handling norm-bounded uncertainty (Han, 2002) by Proposition 3 we can obtain a more general result.
because i (t, zi (t)) satisfies (20). With this observation and defining K(t) = diag{k1 (t), k2 (t), . . . , km (t)}, the problem of absolute stability of system (1) with the (t, z(t)) satisfying (20) reduces to that of robust stability of the following system:
Corollary 9. For given scalar h > 0, the system described by (16), (2) with nonlinear connection function satisfying
x(t) ˙ = [A − DK(t)M]x(t) + [B − DK(t)N ]x(t − h),
(23)
Q.-L. Han / Automatica 41 (2005) 2171 – 2176
for all time-varying gains satisfying (21). The gains k1 (t) to km (t) take values in the convex hypercube H = K ∈ Rm | i ki i .
(24)
We denote by k (1) , k (2) , . . . , k (2 Corresponding to each vertex, let A(i) A − DK (i) M,
m)
B (i) B − DK (i) N.
Proposition 11. For given scalar h > 0, the system described by (1)–(2) with nonlinear connection function satisfying (20) is absolutely stable if there exist real matrices P > 0, Q > 0, and R > 0 such that (1, 1) P B (i) + R ∗ −Q − R ∗ ∗
hA(i)T R hB (i)T R −R
Table 1 Maximum allowed time-delay hmax using criteria in Yu et al. (2003) and this paper
Yu et al. (2003) Corollary 9
0.00 2.3295 2.4859
0.05 2.0703 2.2396
0.10 1.8481 2.0243
0.15 1.6537 1.8363
, the 2m vertices of H.
Similar to the proof of Proposition 3, we have the following result.
2175
< 0,
(25)
for all A(i) , B (i) (i = 1, 2, . . . , 2m ), where (1, 1) A(i)T P + P A(i) + Q − R. Remark 12. One can also consider the effect of uncertainties in system’s matrices A and B on the stability of the system. By Proposition 11 one can easily conclude the corresponding stability criteria for uncertainty of different type. Remark 13. It is interesting to note that h appears linearly in (13), (15), (18) and (25). Thus, a generalized eigenvalue problem as defined in Boyd, El Ghaoui, Feron, and Balakrishnan (1994) can be formulated to solve the minimum acceptable 1/ h and therefore the maximum hmax to maintain robust absolute stability as judged by these conditions. 5. Numerical examples Example 14. Consider the system described by (16)–(17) with −2 0 −1 0 −0.2 , B= , D= , −0.3 0 −0.9 −1 −1 M = ( 0.6 0.8 ) , N= ( 0 0 ) , K1 =0.2, K2 = 0.5,
0 1 0 L= , 0, Ea = Eb = . 0
0 1
Example 15. Consider the system described by (1), (19), (20) with −2.0 −1.0 0.5 1.0 A= , B= , 0.5 0.2 −0.1 −0.8 −0.5 0 0.4 0 D= , M= , 0 0.2 0 0.5 0.2 0 0.1 0 0.4 0 , K1 = N= , K2 = . 0 0.3 0 0.2 0 0.5 Since N = 0, no conclusion can be made using criteria in Yu et al. (2003). However, one can easily formulate a criterion along the idea in Yu et al. (2003) for the case of N = 0. By the new criterion, the numerical result is hmax = 1.6417. Applying Corollary 8, the maximum allowed time-delay is computed as hmax = 2.0239. Using Proposition 11 we have hmax = 2.0263. For this example, Proposition 11 can provide a slightly better result than Corollary 8. It is again to show that the criteria in this paper are less conservative than the corresponding criterion in Yu et al. (2003). 6. Conclusion The problem of absolute stability of a class of time-delay systems with sector-bounded nonlinearity has been addressed. Delay-dependent stability criteria with the general and decentralized sector conditions have been proposed. For the case of the decentralized sector condition, problem of absolute stability has been reduced to that of robust stability of linear timedelay systems. In order to reduce the conservatism of the criteria, we have avoided using model transformation and bounding technique for cross terms, which are widely used in deriving delay-dependent stability criteria for linear time-delay systems. Numerical examples have shown improvements over some existing result.
A=
Using the criterion in Yu et al. (2003) and Corollary 9 in this paper, Table 1 lists the maximum allowed time-delay hmax for robust absolute stability. It is clear to see that for this example the criterion in this paper can provide a less conservative result than that in Yu et al. (2003).
Acknowledgements The author would like to thank the Editor, Professor Paul M.J. Van den Hof, the Associate Editor and the anonymous reviewers for their constructive comments and suggestions to improve the quality of the paper. The research work was partially supported by Central Queensland University for the 2004 Research Advancement Awards Scheme Project “Analysis and Synthesis of Networked Control Systems” and the Strategic Research Project “Delay Effects: Analysis, Synthesis and Applications” (2003–2006).
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Q.-L. Han / Automatica 41 (2005) 2171 – 2176
Appendix A. Schur complement and S-procedure Schur complement is widely employed when a nonlinear matrix inequality is transformed into an linear matrix inequality (LMI) form. Fact 16. For matrices Q = QT , S and R = R T of appropriate dimensions, the inequality Q S >0 ST R is equivalent to the following two inequalities: R > 0, Q − SR −1 S T > 0. S-procedure (Yakubovich, 1971) plays an important role in absolute stability and robust stability theory. There are a number of variations, one of which is used in this paper can be stated as follows. Fact 17. Let Fi = FiT ∈ Rn×n , i = 0, 1, 2, . . . , p. Then the following statement is true T F0 > 0,
for all = 0 satisfying T Fi 0,
(A.1)
if there exist real scalars i 0, i = 1, 2, . . . , p such that F0 −
p
i Fi > 0.
(A.2)
i=1
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