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An LMI approach to persistent bounded disturbance rejection for a class of nonlinear impulsive systems
Nonlinear Analysis: Hybrid Systems 1 (2007) 297–305 www.elsevier.com/locate/nahs
An LMI approach to persistent bounded disturbance rejection for a class of nonlinear impulsive systemsI Fei Hao a,∗ , Long Wang b a The Seventh Research Division, Beijing University of Aeronautics and Astronautics, Beijing, 100083, PR China b Center for Systems and Control, Department of Mechanics and Engineering Science, Peking University, Beijing 100871, PR China
Received 14 August 2005; accepted 1 March 2006
Abstract The problem of persistent bounded disturbance rejection for a class of nonlinear (Lipschitz-like) impulsive systems is considered through invariant set analysis using the Lyapunov function method. Conditions on a robust attractor for this class of systems are given in terms of linear matrix inequalities (LMIs), which ensure simultaneously internal stability and desired L 1 -performance. The obtained results are only dependent on the Lipschitz-like constant matrices without regard to the nonlinear forms. Based on the results, a simple approach to the design of a robust controller is presented. Finally, a numerical example is worked out to illustrate the efficiency of the theoretical results. c 2006 Elsevier Ltd. All rights reserved.
Keywords: Nonlinear impulsive systems; Disturbance rejection; LMI; L 1 -performance
1. Introduction There exist extensively impulsive phenomena in many areas of science and technology, such as neural networks, communication, rhythm in medicine and biology, optimal control in economics and so on. Moreover, many systems in those areas contain nonlinear terms in general. Impulsive differential equations, that is, differential equations involving impulsive effects, appear as a natural description of these phenomena of several real world problems. The theory of impulsive differential systems is still developing up to now. There have been many works devoted to the qualitative analysis of impulsive systems (see, e.g., [1] and the references therein). Recently, problems concerning control of impulsive systems have also attracted increasing attention [2–5]. Particularly, the controllability and the design problem of linear impulsive control systems have been addressed in [4] and [5]. On the other hand, it is not only theoretically interesting but also practically important to consider the effects of persistent bounded disturbances on a dynamical system. The problem of persistent bounded disturbance rejection for linear systems without impulsive effects was first formulated in [6] and has been extensively studied in recent years (see, e.g., [7,8] and the references therein). Very recently, we have investigated the problem of persistent bounded disturbance rejection control of linear I This work was supported by the National Natural Science Foundation of China under grants (No. 60504018, 60304014, and 10372002), and in part by the innovation foundation from the School of Science, Beihang University. ∗ Corresponding author. Tel.: +86 10 89183613. E-mail address: [email protected] (F. Hao).
c 2006 Elsevier Ltd. All rights reserved. 1751-570X/$ - see front matter doi:10.1016/j.nahs.2006.03.006
298
F. Hao, L. Wang / Nonlinear Analysis: Hybrid Systems 1 (2007) 297–305
impulsive systems in [9]. Furthermore, this problem for linear impulsive systems with parameter uncertainties has been studied. Sufficient conditions for the solvability of the (robust) problems were derived in terms of LMIs (see [9–11]). It is worth noting that when parameter uncertainties as well as nonlinearities appeared in a model, the H∞ problem for continuous systems (see [12]) and the corresponding discrete system (see [13]) were considered by Xie and De Souza. However, there have been few studies devoted to the problem of persistent bounded disturbance rejection for nonlinear impulsive systems. This brief intends to provide some basic results on the L 1 -control of persistently disturbed nonlinear impulsive systems through invariant set analysis using the Lyapunov function method. For a class of nonlinear impulsive systems (i.e., a linear part plus a nonlinear term with Lipschitz-like type functions, see Definition 1), we will establish conditions on a robust ellipsoidal attractor for the class of nonlinear systems in terms of linear matrix inequalities (LMIs), which ensure simultaneously the internal stability (in fact, exponential stability) and the desired L 1 -performance. The obtained results are only dependent on the Lipschitz-like constant matrices without regard to the nonlinear forms. A simple approach to the design of a robust controller is presented based on the conditions. We also give a numerical example to illustrate the results obtained. We use the following notation. R is the set of all real numbers. Rn is the set of all n-tuples of real numbers, and m×n R the set of all real matrices with m rows and n columns. BR p denotes the closed unit ball on the space R p . Denote by AT and A−1 the transpose and the inverse of a matrix A (if it is invertible), and by I the unit matrix of appropriate dimensions. λ M (A), λm (A) are the maximum and minimum eigenvalues of A, respectively. 2. Preliminaries Consider the following nonlinear impulsive system x˙ = Ax + Bu + H h(x) + B1 w, t 6= tk ∆x(t) = x(t + ) − x(t − ) = E k x(tk ) + Gg(x(tk )), t = tk z = C x + Du x(0) = 0.
(1)
where x(·) : R → Rn , u(·) : R → Rm , and w(·) : R → R p are the state, the input, and external disturbance vectors, respectively. z(·) : R → R p is the controlled output. A, B, H, B1 , E k , G, C, D are known real constant matrices of appropriate dimensions, k = 1, 2, . . . . limh→0+ x(tk − h) = x(tk− ), limh→0+ x(tk + h) = x(tk+ ). 0 < t1 < · · · < tk < tk+1 < . . ., and tk → ∞ as k → ∞. Assume limh→0+ x(tk − h) = x(tk− ) = x(tk ), that is, the solution x(t) of system (1) is left continuous at tk . Assume that the admissible disturbance set is W := {w : R → BR p , w is measurable}, the L ∞ norm is defined by kwk∞ =: supt kw(t)k2 . Definition 1. f : Rn → Rn is said to be a global Lipschitz-like type function with the Lipschitz-like constant matrix M ∈ Rn×n , i.e., f satisfies k f (x1 ) − f (x2 )k ≤ kM(x1 − x2 )k for any x1 , x2 ∈ Rn . In this paper, we consider only the class of nonlinear Lipschitz-like type functions, that is, we make the following assumption. Assumption 1. h, g are global Lipschitz-like type functions with the Lipschitz-like constant matrices Mh and Mg , respectively. Recall that a set Ω is said to be positively invariant for a dynamical system, if x(0) ∈ Ω , then the trajectory x(t) of the system remains in Ω for all t > 0. The origin-reachable set (R∞ (0)) of a system is the set that the state of the system can reach from the origin. It is the minimal closed positively invariant set containing the origin. To formulate our problem, let us consider the following system x˙ = F x + H h(x) + B1 w, t 6= tk ∆x = E k x(tk ) + Gg(x(tk )), t = tk (2) z = Cx x(t0 ) = x(0) = 0.
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For a given scalar ρ > 0, we call system (2) with initial state x(0) = 0 has ρ-performance if kzk∞ ≤ ρ for all w ∈ W. Define the performance set as follows: Ω (ρ) = {x : kzk∞ = kC xk∞ ≤ ρ, ∀w ∈ W}. Thus, if R∞ (0) ⊂ Ω (ρ), then the system has ρ-performance. The objective of this brief is to find for (1) a state-feedback controller u = L x with gain matrix L ∈ Rm×n , a constant matrix, such that the resulting closed-loop system x˙ = (A + B L)x + H h(x) + B1 w, t 6= tk ∆x = E k x(tk ) + Gg(x(tk )), t = tk (3) z = (C + DL)x x(t0 ) = x(0) = 0 satisfies the following conditions: (i) The system is internally stable, namely the system without external disturbance (i.e., w = 0) is asymptotically stable; and (ii) For a given scalar ρ > 0, the system has ρ-performance. A set Ω is said to be a robust attractor of (2) with respect to (w.r.t.) w ∈ W, if all the state trajectories initiating from the exterior of Ω eventually enter and remain in Ω for all w ∈ W. Obviously, a robust attractor is also a positively invariant set. So, any one robust attractor must contain the origin-reachable set R∞ (0). The following lemmas will be useful in our discussion. Lemma 1 ([14]). Let P, B be constant matrices of appropriate dimensions, then for any scalar α > 0, it follows that 1 T x P B B T P x + αw T w, α where x, w are arbitrary dimensional vectors. 2x T P Bw ≤
Lemma 2. The Schur complement formula (see [15] or [16]), namely R11 R12 R= <0 T R12 R22 if and only if one of the following conditions holds. −1 T (1) R22 < 0 and R11 − R12 R22 R12 < 0; −1 T (2) R11 < 0 and R22 − R12 R11 R12 < 0.
3. Main results For a positive definite matrix P, let the ellipsoid Ω P = {x : x T P x ≤ 1}. We first consider (2). Theorem 1. For a given scalar ρ > 0, if there exist a positive definite matrix P and scalars α > 0, i , i = 1, 2 such that the following conditions hold, P F + F T P + α P + 1 MhT Mh P B1 P H (4) B1T P −α I 0 < 0, T H P 0 −1 I T T −P + 2 Mg Mg 0 (I + E k ) P < 0, (5) 0 −2 I GT P P(E k + I ) PG −P 2 T −ρ P C ≤ 0, (6) C −I
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then (2) is internally stable and Ω P is a robust attractor of it w.r.t. w ∈ W. Moreover, Ω P ⊂ Ω (ρ) and hence the system has ρ-performance. Proof. To prove that Ω P is a robust attractor of (2) w.r.t. w ∈ W, we only need to prove that the time derivative of V (x) = x T P x along the solution of the system is negative for any x 6∈ Ω P , where P is the positive definite solution of (4)–(6). For t ∈ (tk , tk+1 ], by Lemma 1 and Assumption 1, we have V˙ |(2) (x) = x T P(F x + H h(x) + B1 w) + (F x + H h(x) + B1 w)T P x = x T (P F + F T P)x + 2x T P B1 w + 2x T P H h(x) 1 1 ≤ x T P F + F T P + P B1 B1T P + α P + P H H T P + 1 MhT Mh x − α(x T P x − w T w). α 1 By the Schur complement formula, (4) is equivalent to P F + FT P +
1 1 P B1 B1T P + α P + P H H T P + 1 MhT Mh < 0. α 1
So, we have V˙ |(2) (x(t)) < −αx T P x = −αV (x) < 0
(7)
whenever w = 0. Furthermore, since > 1 for x 6∈ Ω P , we have 1 1 V˙ |(2) (x(t)) < x T P F + F T P + P B1 B1T P + α P + P GG T P + 1 MhT Mh x < 0, α 1 xT Px
for any w ∈ W. On the other hand, by the definition of V (x(t)) and (5), together with the Schur complement formula, we have V (x(tk+ )) − V (x(tk− )) = x T (tk+ )P x(tk+ ) − x T (tk− )P x(tk− ) = [(E k + I )x(tk ) + Gg(x(tk ))]T P[(E k + I )x(tk ) + Gg(x(tk ))] − x T (tk )P x(tk ) ≤ x T (tk )(E k + I )T P(E k + I )x(tk ) + 2x T (tk )(E k + I )T P Gg(x(tk )) − g T (x(tk ))W1 g(x(tk )) + 2 g T (x(tk ))g(x(tk )) where W1 =: 2 I − G T P G. By Lemma 1, one has 2x T (tk )(E k + I )T P Gg(x(tk )) − g T (x(tk ))W1 g(x(tk )) ≤ x T (tk )(E k + I )T P GW1−1 G T P(E k + I )x(tk ). Thus, from (5) and by using the Schur complement formula and Assumption 1, we have that V (x(tk+ )) − V (x(tk− )) ≤ x T (tk )[(E k + I )T P(E k + I ) + (E k + I )T P GW1−1 G T P(E k + I ) + 2 MgT Mg − P]x(tk ) < 0.
(8)
Therefore, by (7) and (8), we can obtain V (x(t)) ≤ V (x(t0 ))e−α(t−t0 ) ≤ V (0)e−αt .
(9)
Thus, we can obtain that system (2) is asymptotical stable, and further Ω P is a robust attractor of (2) w.r.t. w ∈ W. Moreover, by the Schur complement formula, (6) is equivalent to the following inequality ρ 2 P ≥ C T C. This inequality implies ρ 2 x T P x − kC xk2 ≥ 0. From this, it is clear that if x T P x ≤ 1, then kC xk ≤ ρ. This shows that Ω P ⊂ Ω (ρ) and hence R∞ (0) ⊂ Ω P ⊂ Ω (ρ). So, (2) has ρ-performance.
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Remark 1. For a fixed α > 0, (4) in Theorem 1 is an LMI in P, 1 . In the process of verifying the feasibility of the LMIs, we can first search α > 0, then we solve the LMIs by using the LMI tool for the obtained α. Moreover, we can transform the inequality (4) into a frequency-domain condition that the H∞ norm of a transfer function is less than 1 (see similarly [9] for more details). This gives a frequency-domain explanation of (4), thereby (4) can be verified conveniently. Notice that Theorem 1 gives a condition that ensures simultaneously the dynamical performance and the desired disturbance rejection performance. Remark 2. From Theorem 1, we can see that the result is only dependent on the Lipschitz-like constant matrices without regard to the nonlinear forms. Remark 3. From the proof of Theorem 1, it can be proved that the system is exponentially stable with exponential stability degree α2 . In fact, from (9), we have that λm (P)kxk2 ≤ V (x(t)) ≤ V (0)e−αt ≤ e−αt λ M (P)kx0 k2 . Therefore, one has s λ M (P) − α t kxk ≤ e 2 kx0 k. λm (P) This shows that the system is exponentially stable with exponential stability degree
α 2
(see [17]).
Now we give the main result of this paper as follows. Theorem 2. For system (1) and a fixed performance level ρ > 0, if there exist a matrix X ∈ Rm×n , a positive definite matrix Q and scalars α > 0, α1 , 2 such that the following conditions hold, AQ + Q AT + α Q + B X + X T B T + α1 H H T B1 Q MhT (10) B1T −α I 0 < 0, 0 −α1 I Mh Q −Q 0 Q(I + E k )T Q MgT 0 −2 I GT 0 < 0, (11) (I + E k )Q G −Q 0 0
Mg Q −ρ 2 Q C Q + DX
QC T + X T D −I
−2−1 I
0 T ≤ 0,
(12)
then the closed-loop system (3) is internally asymptotical stable and Ω Q −1 is a robust attractor of (3) w.r.t. w ∈ W, where the state-feedback gain matrix L = X Q −1 .
(13)
Moreover, Ω Q −1 ⊂ Ω (ρ) and hence (3) has ρ-performance. Proof. Take V (x(t)) = x T (t)P x(t), where P = Q −1 . For u = L x = X Q −1 x, the corresponding closed-loop system (3) reads x˙ = (A + B X Q −1 )x + H h(x) + B1 w, t = 6 tk ∆x = E k x(tk ) + Gg(x(tk )), t = tk . Letting
P N = 0 0
0 I 0
0 0 P
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and multiplying N on both sides of (10), we have P A + AT P + α P +P B X P + P X T B T P + α1 P H H T P B1T P Mh
P B1 −α I 0
MhT < 0. 0 −α1 I
Since L = X Q −1 and P = Q −1 , L = X P. Substituting L = X P into the above inequality and using the Schur complement formula, we get P(A + B L) + (A + B L)T P + α P + 1 MhT Mh P B1 P H B1T P −α I 0 <0 T H P 0 −1 I where α1 = 1−1 . Similarly, by the Schur complement formulas, (11) and (12) imply (5) and (6) respectively for the closed-loop system. From this and Theorem 1, one can readily get the desired result. Remark 4. By solving the LMIs (10)–(12) in Theorem 2, we can obtain a controller stabilizing (1) and achieving the desired performance level of persistent bounded disturbance rejection in the closed-loop system. Note that we have not assumed here the controllability of (A, B) as it is not enough for the controllability of (1) (see [4]). Clearly, if the LMIs (10)–(12) are feasible, then (1) can be stabilized by the linear state-feedback controller u = L x with L specified by (13). The above results can be extended to the following system x˙ = Ax + Bu + H h(x) + B1 w, t 6= tk ∆x = E k x(tk ) + Gg(x(tk )), t = tk z = C x + Du + D w 1 x(t0 ) = x(0) = 0.
(14)
Theorem 3. For system (14) and a fixed ρ > 0, if there exist a matrix X ∈ Rm×n , a positive definite matrix Q and a scalar α > 0 such that the inequalities (10), (11) hold, then the closed-loop system of (14) with u = L x is internally asymptotical stable, and Ω Q −1 is a robust attractor of the closed-loop system w.r.t. w ∈ W, where L is given by (13). Moreover, if there exists a scalar σ > 0 such that σ Q −1 0 (C + DL)T ≥ 0, (15) 0 (ρ 2 − σ )I D1T C + DL D1 I where Q satisfies (10) and (11), then Ω Q −1 ⊂ Ω (ρ) and hence the closed-loop system has ρ-performance.
4. Numerical example Consider impulsive system (1) with the following coefficient matrices (that is, the linear part borrowed from [9]): −1 0 0 0.3 0.1 A= , B= , B1 = , 0.2 0.1 1 0 0.1 0.3 0.1 0.01 E k = −1.5I, k = 1, 2, . . . , C= , D= . 0.3 1.8 0 sin(x1 ) sin(x1 ) h(x) = , g(x) = 0.1 ∗ , sin(x2 ) sin(x2 ) 0.1 0.01 0.2 −0.03 , G= . H= 0.02 0.1 0.01 0.1
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Fig. 1. Robust attractor.
In this case, it is easy to see that h(x), g(x) are global Lipschitz-like type functions with constant matrices Mh = I and Mg = 0.1 ∗ I . Notice that the system matrix A is unstable, and the pair (A, B) is not controllable. Here, we use Theorem 2 to find a linear state feedback controller to stabilize the system and guarantee the closed-loop system to have ρ-performance with ρ = 0.9. Taking α = 0.1 and 2 = 5, we can solve the inequalities (10)–(12) to obtain X = −0.8467 −2.5589 , 3.0036 −0.4781 Q= . −0.4781 0.2949 α1 = 5.7710. Let L = X Q −1 = [−0.6567 −2.1230], by Theorem 2, the closed-loop system of (14) with u = L x, that is, −1.0000 0 0.3 0.1 0.1 0.01 x˙ = x+ w+ h(x), t 6= tk −2.0416 −12.2114 0.02 0.1 0 0.1 0.2 −0.03 ∆x = −1.5I x(tk ) + g(x(tk )), t = tk 0.01 0.1 0.2776 −0.0231 z= x 0.3000 1.8000 x(t0 ) = x(0) = 0 is internal stable and has ρ-performance. The robust attractor for the closed-loop system is shown in Fig. 1. Consider T 1 1 the external disturbance with form w = √ sin(π t − 1) √ cos(2πt + 1) and the impulsive time step is one 2 2 second. The simulation of the state response for the system involving the disturbance effects is shown in Fig. 2, and that for the system without disturbances in Fig. 3 from the original state (0.4, −0.3). 5. Conclusion We have considered the persistent bounded disturbance rejection problem for a class of nonlinear impulsive systems through invariant set analysis using the Lyapunov function method in this paper. Conditions on a robust ellipsoidal attractor for this class of nonlinear impulsive systems were presented in terms of LMIs, which ensure simultaneously the internal stability and the desired level of persistent bounded disturbances. A simple approach to the design of a state-feedback controller has been presented. Our results did not require that the pair (A, B) was controllable. Numerical simulation showed the efficiency of our proposed approach. Further results on persistent bounded disturbance rejection for nonlinear impulsive systems with uncertainties will be presented elsewhere.
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Fig. 2. The state response of the closed-loop system with external disturbances w.
Fig. 3. The state response of the closed-loop system without disturbances w.
References [1] V. Lakshmikantham, D.D. Bainov, P.S. Simenov, Theory of Impulsive Differential Systems, World Scientific Press, Singapore, 1989. [2] Z.H. Guan, G. Chen, T. Ueta, On impulsive control of a periodically forced chaotic pendulum system, IEEE Trans. Automat. Control 45 (9) (2000) 1724–1727. [3] Z.H. Guan, C.W. Chan, A.Y.T. Leung, G. Chen, Robust stabilization of singular impulsive delayed systems with nonlinear perturbation, IEEE Trans. Circuits Syst. I 48 (8) (2001) 1011–1019. [4] S. Leela, F.A. Mcrae, S. Sivasundaram, Controllability of impulsive differential equations, J. Math. Anal. Appl. 177 (1) (1993) 24–30. [5] Z.G. Li, C.Y. Wen, Y.C. Soh, Analysis and design of impulsive control systems, IEEE Trans. Automat. Control 46 (6) (2001) 894–897. [6] M. Vidyasagar, Optimal rejection of persistent bounded disturbances, IEEE Trans. Automat. Control 31 (6) (1986) 527–535. [7] M.A. Dahleh, I.J. Diaz-Bobillo, Control of Uncertain Systems: A Linear Programming Approach, Prentice Hall, Englewood Cliffs, New Jersey, 1995. [8] J. Abedor, K. Nagpal, K. Poolla, A linear matrix inequality approach to peak-to-peak gain minimization, Internat. J. Robust Nonlinear Control 6 (1996) 899–927. [9] F. Hao, T. Chu, L. Huang, L. Wang, Persistent bounded disturbance rejection for impulsive systems, IEEE Trans. Circuits Syst. I 50 (6) (2003) 785–788. [10] F. Hao, T. Chu, L. Wang, L. Huang, An LMI approach to persistent bounded disturbance rejection for uncertain impulsive systems, in: Proc. the 42nd IEEE Conference on Decision and Control, Maui, Hawaii, USA, 2003, pp. 4068–4073.
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[11] F. Hao, L. Wang, T. Chu, L. Huang, An LMI approach to persistent bounded disturbance rejection for impulsive systems with polytopic uncertainties, Dyn. Contin. Discrete Impuls. Syst. 13 (2) (2006) 269–282. [12] C.E. De Souza, L. Xie, Y. Wang, H∞ filtering for a class of uncertain nonlinear systems, Syst. Control Lett. 20 (1993) 419–426. [13] L. Xie, C.E. De Souza, Y. Wang, Robust filtering for a class of discrete-time uncertain nonlinear systems: An H∞ approach, Internat. J. Robust Nonlinear Control 6 (1996) 279–312. [14] I.R. Petersen, V.A. Ugrinovskii, A.V. Sakin, Robust Control Design Using H∞ Metholds, Springer, New York, 2000. [15] K. Zhou, J.C. Doyle, K. Glover, Robust and Optimal Control, Prentice-Hall, Upper Saddle River, New Jersey, 1996. [16] S. Boyd, L.E. Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, PA, 1994. [17] A.N. Michel, R.K. Miller, Qualitative Analysis of Large-Scale Dynamical System, Academic Press, New York, 1977.
An LMI approach to persistent bounded disturbance rejection for a class of nonlinear impulsive systemsI Fei Hao a,∗ , Long Wang b a The Seventh Research Division, Beijing University of Aeronautics and Astronautics, Beijing, 100083, PR China b Center for Systems and Control, Department of Mechanics and Engineering Science, Peking University, Beijing 100871, PR China
Received 14 August 2005; accepted 1 March 2006
Abstract The problem of persistent bounded disturbance rejection for a class of nonlinear (Lipschitz-like) impulsive systems is considered through invariant set analysis using the Lyapunov function method. Conditions on a robust attractor for this class of systems are given in terms of linear matrix inequalities (LMIs), which ensure simultaneously internal stability and desired L 1 -performance. The obtained results are only dependent on the Lipschitz-like constant matrices without regard to the nonlinear forms. Based on the results, a simple approach to the design of a robust controller is presented. Finally, a numerical example is worked out to illustrate the efficiency of the theoretical results. c 2006 Elsevier Ltd. All rights reserved.
Keywords: Nonlinear impulsive systems; Disturbance rejection; LMI; L 1 -performance
1. Introduction There exist extensively impulsive phenomena in many areas of science and technology, such as neural networks, communication, rhythm in medicine and biology, optimal control in economics and so on. Moreover, many systems in those areas contain nonlinear terms in general. Impulsive differential equations, that is, differential equations involving impulsive effects, appear as a natural description of these phenomena of several real world problems. The theory of impulsive differential systems is still developing up to now. There have been many works devoted to the qualitative analysis of impulsive systems (see, e.g., [1] and the references therein). Recently, problems concerning control of impulsive systems have also attracted increasing attention [2–5]. Particularly, the controllability and the design problem of linear impulsive control systems have been addressed in [4] and [5]. On the other hand, it is not only theoretically interesting but also practically important to consider the effects of persistent bounded disturbances on a dynamical system. The problem of persistent bounded disturbance rejection for linear systems without impulsive effects was first formulated in [6] and has been extensively studied in recent years (see, e.g., [7,8] and the references therein). Very recently, we have investigated the problem of persistent bounded disturbance rejection control of linear I This work was supported by the National Natural Science Foundation of China under grants (No. 60504018, 60304014, and 10372002), and in part by the innovation foundation from the School of Science, Beihang University. ∗ Corresponding author. Tel.: +86 10 89183613. E-mail address: [email protected] (F. Hao).
c 2006 Elsevier Ltd. All rights reserved. 1751-570X/$ - see front matter doi:10.1016/j.nahs.2006.03.006
298
F. Hao, L. Wang / Nonlinear Analysis: Hybrid Systems 1 (2007) 297–305
impulsive systems in [9]. Furthermore, this problem for linear impulsive systems with parameter uncertainties has been studied. Sufficient conditions for the solvability of the (robust) problems were derived in terms of LMIs (see [9–11]). It is worth noting that when parameter uncertainties as well as nonlinearities appeared in a model, the H∞ problem for continuous systems (see [12]) and the corresponding discrete system (see [13]) were considered by Xie and De Souza. However, there have been few studies devoted to the problem of persistent bounded disturbance rejection for nonlinear impulsive systems. This brief intends to provide some basic results on the L 1 -control of persistently disturbed nonlinear impulsive systems through invariant set analysis using the Lyapunov function method. For a class of nonlinear impulsive systems (i.e., a linear part plus a nonlinear term with Lipschitz-like type functions, see Definition 1), we will establish conditions on a robust ellipsoidal attractor for the class of nonlinear systems in terms of linear matrix inequalities (LMIs), which ensure simultaneously the internal stability (in fact, exponential stability) and the desired L 1 -performance. The obtained results are only dependent on the Lipschitz-like constant matrices without regard to the nonlinear forms. A simple approach to the design of a robust controller is presented based on the conditions. We also give a numerical example to illustrate the results obtained. We use the following notation. R is the set of all real numbers. Rn is the set of all n-tuples of real numbers, and m×n R the set of all real matrices with m rows and n columns. BR p denotes the closed unit ball on the space R p . Denote by AT and A−1 the transpose and the inverse of a matrix A (if it is invertible), and by I the unit matrix of appropriate dimensions. λ M (A), λm (A) are the maximum and minimum eigenvalues of A, respectively. 2. Preliminaries Consider the following nonlinear impulsive system x˙ = Ax + Bu + H h(x) + B1 w, t 6= tk ∆x(t) = x(t + ) − x(t − ) = E k x(tk ) + Gg(x(tk )), t = tk z = C x + Du x(0) = 0.
(1)
where x(·) : R → Rn , u(·) : R → Rm , and w(·) : R → R p are the state, the input, and external disturbance vectors, respectively. z(·) : R → R p is the controlled output. A, B, H, B1 , E k , G, C, D are known real constant matrices of appropriate dimensions, k = 1, 2, . . . . limh→0+ x(tk − h) = x(tk− ), limh→0+ x(tk + h) = x(tk+ ). 0 < t1 < · · · < tk < tk+1 < . . ., and tk → ∞ as k → ∞. Assume limh→0+ x(tk − h) = x(tk− ) = x(tk ), that is, the solution x(t) of system (1) is left continuous at tk . Assume that the admissible disturbance set is W := {w : R → BR p , w is measurable}, the L ∞ norm is defined by kwk∞ =: supt kw(t)k2 . Definition 1. f : Rn → Rn is said to be a global Lipschitz-like type function with the Lipschitz-like constant matrix M ∈ Rn×n , i.e., f satisfies k f (x1 ) − f (x2 )k ≤ kM(x1 − x2 )k for any x1 , x2 ∈ Rn . In this paper, we consider only the class of nonlinear Lipschitz-like type functions, that is, we make the following assumption. Assumption 1. h, g are global Lipschitz-like type functions with the Lipschitz-like constant matrices Mh and Mg , respectively. Recall that a set Ω is said to be positively invariant for a dynamical system, if x(0) ∈ Ω , then the trajectory x(t) of the system remains in Ω for all t > 0. The origin-reachable set (R∞ (0)) of a system is the set that the state of the system can reach from the origin. It is the minimal closed positively invariant set containing the origin. To formulate our problem, let us consider the following system x˙ = F x + H h(x) + B1 w, t 6= tk ∆x = E k x(tk ) + Gg(x(tk )), t = tk (2) z = Cx x(t0 ) = x(0) = 0.
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For a given scalar ρ > 0, we call system (2) with initial state x(0) = 0 has ρ-performance if kzk∞ ≤ ρ for all w ∈ W. Define the performance set as follows: Ω (ρ) = {x : kzk∞ = kC xk∞ ≤ ρ, ∀w ∈ W}. Thus, if R∞ (0) ⊂ Ω (ρ), then the system has ρ-performance. The objective of this brief is to find for (1) a state-feedback controller u = L x with gain matrix L ∈ Rm×n , a constant matrix, such that the resulting closed-loop system x˙ = (A + B L)x + H h(x) + B1 w, t 6= tk ∆x = E k x(tk ) + Gg(x(tk )), t = tk (3) z = (C + DL)x x(t0 ) = x(0) = 0 satisfies the following conditions: (i) The system is internally stable, namely the system without external disturbance (i.e., w = 0) is asymptotically stable; and (ii) For a given scalar ρ > 0, the system has ρ-performance. A set Ω is said to be a robust attractor of (2) with respect to (w.r.t.) w ∈ W, if all the state trajectories initiating from the exterior of Ω eventually enter and remain in Ω for all w ∈ W. Obviously, a robust attractor is also a positively invariant set. So, any one robust attractor must contain the origin-reachable set R∞ (0). The following lemmas will be useful in our discussion. Lemma 1 ([14]). Let P, B be constant matrices of appropriate dimensions, then for any scalar α > 0, it follows that 1 T x P B B T P x + αw T w, α where x, w are arbitrary dimensional vectors. 2x T P Bw ≤
Lemma 2. The Schur complement formula (see [15] or [16]), namely R11 R12 R= <0 T R12 R22 if and only if one of the following conditions holds. −1 T (1) R22 < 0 and R11 − R12 R22 R12 < 0; −1 T (2) R11 < 0 and R22 − R12 R11 R12 < 0.
3. Main results For a positive definite matrix P, let the ellipsoid Ω P = {x : x T P x ≤ 1}. We first consider (2). Theorem 1. For a given scalar ρ > 0, if there exist a positive definite matrix P and scalars α > 0, i , i = 1, 2 such that the following conditions hold, P F + F T P + α P + 1 MhT Mh P B1 P H (4) B1T P −α I 0 < 0, T H P 0 −1 I T T −P + 2 Mg Mg 0 (I + E k ) P < 0, (5) 0 −2 I GT P P(E k + I ) PG −P 2 T −ρ P C ≤ 0, (6) C −I
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then (2) is internally stable and Ω P is a robust attractor of it w.r.t. w ∈ W. Moreover, Ω P ⊂ Ω (ρ) and hence the system has ρ-performance. Proof. To prove that Ω P is a robust attractor of (2) w.r.t. w ∈ W, we only need to prove that the time derivative of V (x) = x T P x along the solution of the system is negative for any x 6∈ Ω P , where P is the positive definite solution of (4)–(6). For t ∈ (tk , tk+1 ], by Lemma 1 and Assumption 1, we have V˙ |(2) (x) = x T P(F x + H h(x) + B1 w) + (F x + H h(x) + B1 w)T P x = x T (P F + F T P)x + 2x T P B1 w + 2x T P H h(x) 1 1 ≤ x T P F + F T P + P B1 B1T P + α P + P H H T P + 1 MhT Mh x − α(x T P x − w T w). α 1 By the Schur complement formula, (4) is equivalent to P F + FT P +
1 1 P B1 B1T P + α P + P H H T P + 1 MhT Mh < 0. α 1
So, we have V˙ |(2) (x(t)) < −αx T P x = −αV (x) < 0
(7)
whenever w = 0. Furthermore, since > 1 for x 6∈ Ω P , we have 1 1 V˙ |(2) (x(t)) < x T P F + F T P + P B1 B1T P + α P + P GG T P + 1 MhT Mh x < 0, α 1 xT Px
for any w ∈ W. On the other hand, by the definition of V (x(t)) and (5), together with the Schur complement formula, we have V (x(tk+ )) − V (x(tk− )) = x T (tk+ )P x(tk+ ) − x T (tk− )P x(tk− ) = [(E k + I )x(tk ) + Gg(x(tk ))]T P[(E k + I )x(tk ) + Gg(x(tk ))] − x T (tk )P x(tk ) ≤ x T (tk )(E k + I )T P(E k + I )x(tk ) + 2x T (tk )(E k + I )T P Gg(x(tk )) − g T (x(tk ))W1 g(x(tk )) + 2 g T (x(tk ))g(x(tk )) where W1 =: 2 I − G T P G. By Lemma 1, one has 2x T (tk )(E k + I )T P Gg(x(tk )) − g T (x(tk ))W1 g(x(tk )) ≤ x T (tk )(E k + I )T P GW1−1 G T P(E k + I )x(tk ). Thus, from (5) and by using the Schur complement formula and Assumption 1, we have that V (x(tk+ )) − V (x(tk− )) ≤ x T (tk )[(E k + I )T P(E k + I ) + (E k + I )T P GW1−1 G T P(E k + I ) + 2 MgT Mg − P]x(tk ) < 0.
(8)
Therefore, by (7) and (8), we can obtain V (x(t)) ≤ V (x(t0 ))e−α(t−t0 ) ≤ V (0)e−αt .
(9)
Thus, we can obtain that system (2) is asymptotical stable, and further Ω P is a robust attractor of (2) w.r.t. w ∈ W. Moreover, by the Schur complement formula, (6) is equivalent to the following inequality ρ 2 P ≥ C T C. This inequality implies ρ 2 x T P x − kC xk2 ≥ 0. From this, it is clear that if x T P x ≤ 1, then kC xk ≤ ρ. This shows that Ω P ⊂ Ω (ρ) and hence R∞ (0) ⊂ Ω P ⊂ Ω (ρ). So, (2) has ρ-performance.
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Remark 1. For a fixed α > 0, (4) in Theorem 1 is an LMI in P, 1 . In the process of verifying the feasibility of the LMIs, we can first search α > 0, then we solve the LMIs by using the LMI tool for the obtained α. Moreover, we can transform the inequality (4) into a frequency-domain condition that the H∞ norm of a transfer function is less than 1 (see similarly [9] for more details). This gives a frequency-domain explanation of (4), thereby (4) can be verified conveniently. Notice that Theorem 1 gives a condition that ensures simultaneously the dynamical performance and the desired disturbance rejection performance. Remark 2. From Theorem 1, we can see that the result is only dependent on the Lipschitz-like constant matrices without regard to the nonlinear forms. Remark 3. From the proof of Theorem 1, it can be proved that the system is exponentially stable with exponential stability degree α2 . In fact, from (9), we have that λm (P)kxk2 ≤ V (x(t)) ≤ V (0)e−αt ≤ e−αt λ M (P)kx0 k2 . Therefore, one has s λ M (P) − α t kxk ≤ e 2 kx0 k. λm (P) This shows that the system is exponentially stable with exponential stability degree
α 2
(see [17]).
Now we give the main result of this paper as follows. Theorem 2. For system (1) and a fixed performance level ρ > 0, if there exist a matrix X ∈ Rm×n , a positive definite matrix Q and scalars α > 0, α1 , 2 such that the following conditions hold, AQ + Q AT + α Q + B X + X T B T + α1 H H T B1 Q MhT (10) B1T −α I 0 < 0, 0 −α1 I Mh Q −Q 0 Q(I + E k )T Q MgT 0 −2 I GT 0 < 0, (11) (I + E k )Q G −Q 0 0
Mg Q −ρ 2 Q C Q + DX
QC T + X T D −I
−2−1 I
0 T ≤ 0,
(12)
then the closed-loop system (3) is internally asymptotical stable and Ω Q −1 is a robust attractor of (3) w.r.t. w ∈ W, where the state-feedback gain matrix L = X Q −1 .
(13)
Moreover, Ω Q −1 ⊂ Ω (ρ) and hence (3) has ρ-performance. Proof. Take V (x(t)) = x T (t)P x(t), where P = Q −1 . For u = L x = X Q −1 x, the corresponding closed-loop system (3) reads x˙ = (A + B X Q −1 )x + H h(x) + B1 w, t = 6 tk ∆x = E k x(tk ) + Gg(x(tk )), t = tk . Letting
P N = 0 0
0 I 0
0 0 P
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and multiplying N on both sides of (10), we have P A + AT P + α P +P B X P + P X T B T P + α1 P H H T P B1T P Mh
P B1 −α I 0
MhT < 0. 0 −α1 I
Since L = X Q −1 and P = Q −1 , L = X P. Substituting L = X P into the above inequality and using the Schur complement formula, we get P(A + B L) + (A + B L)T P + α P + 1 MhT Mh P B1 P H B1T P −α I 0 <0 T H P 0 −1 I where α1 = 1−1 . Similarly, by the Schur complement formulas, (11) and (12) imply (5) and (6) respectively for the closed-loop system. From this and Theorem 1, one can readily get the desired result. Remark 4. By solving the LMIs (10)–(12) in Theorem 2, we can obtain a controller stabilizing (1) and achieving the desired performance level of persistent bounded disturbance rejection in the closed-loop system. Note that we have not assumed here the controllability of (A, B) as it is not enough for the controllability of (1) (see [4]). Clearly, if the LMIs (10)–(12) are feasible, then (1) can be stabilized by the linear state-feedback controller u = L x with L specified by (13). The above results can be extended to the following system x˙ = Ax + Bu + H h(x) + B1 w, t 6= tk ∆x = E k x(tk ) + Gg(x(tk )), t = tk z = C x + Du + D w 1 x(t0 ) = x(0) = 0.
(14)
Theorem 3. For system (14) and a fixed ρ > 0, if there exist a matrix X ∈ Rm×n , a positive definite matrix Q and a scalar α > 0 such that the inequalities (10), (11) hold, then the closed-loop system of (14) with u = L x is internally asymptotical stable, and Ω Q −1 is a robust attractor of the closed-loop system w.r.t. w ∈ W, where L is given by (13). Moreover, if there exists a scalar σ > 0 such that σ Q −1 0 (C + DL)T ≥ 0, (15) 0 (ρ 2 − σ )I D1T C + DL D1 I where Q satisfies (10) and (11), then Ω Q −1 ⊂ Ω (ρ) and hence the closed-loop system has ρ-performance.
4. Numerical example Consider impulsive system (1) with the following coefficient matrices (that is, the linear part borrowed from [9]): −1 0 0 0.3 0.1 A= , B= , B1 = , 0.2 0.1 1 0 0.1 0.3 0.1 0.01 E k = −1.5I, k = 1, 2, . . . , C= , D= . 0.3 1.8 0 sin(x1 ) sin(x1 ) h(x) = , g(x) = 0.1 ∗ , sin(x2 ) sin(x2 ) 0.1 0.01 0.2 −0.03 , G= . H= 0.02 0.1 0.01 0.1
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Fig. 1. Robust attractor.
In this case, it is easy to see that h(x), g(x) are global Lipschitz-like type functions with constant matrices Mh = I and Mg = 0.1 ∗ I . Notice that the system matrix A is unstable, and the pair (A, B) is not controllable. Here, we use Theorem 2 to find a linear state feedback controller to stabilize the system and guarantee the closed-loop system to have ρ-performance with ρ = 0.9. Taking α = 0.1 and 2 = 5, we can solve the inequalities (10)–(12) to obtain X = −0.8467 −2.5589 , 3.0036 −0.4781 Q= . −0.4781 0.2949 α1 = 5.7710. Let L = X Q −1 = [−0.6567 −2.1230], by Theorem 2, the closed-loop system of (14) with u = L x, that is, −1.0000 0 0.3 0.1 0.1 0.01 x˙ = x+ w+ h(x), t 6= tk −2.0416 −12.2114 0.02 0.1 0 0.1 0.2 −0.03 ∆x = −1.5I x(tk ) + g(x(tk )), t = tk 0.01 0.1 0.2776 −0.0231 z= x 0.3000 1.8000 x(t0 ) = x(0) = 0 is internal stable and has ρ-performance. The robust attractor for the closed-loop system is shown in Fig. 1. Consider T 1 1 the external disturbance with form w = √ sin(π t − 1) √ cos(2πt + 1) and the impulsive time step is one 2 2 second. The simulation of the state response for the system involving the disturbance effects is shown in Fig. 2, and that for the system without disturbances in Fig. 3 from the original state (0.4, −0.3). 5. Conclusion We have considered the persistent bounded disturbance rejection problem for a class of nonlinear impulsive systems through invariant set analysis using the Lyapunov function method in this paper. Conditions on a robust ellipsoidal attractor for this class of nonlinear impulsive systems were presented in terms of LMIs, which ensure simultaneously the internal stability and the desired level of persistent bounded disturbances. A simple approach to the design of a state-feedback controller has been presented. Our results did not require that the pair (A, B) was controllable. Numerical simulation showed the efficiency of our proposed approach. Further results on persistent bounded disturbance rejection for nonlinear impulsive systems with uncertainties will be presented elsewhere.
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Fig. 2. The state response of the closed-loop system with external disturbances w.
Fig. 3. The state response of the closed-loop system without disturbances w.
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