J Control Theory Appl 2011 9 (2) 183–188 DOI 10.1007/s11768-011-9142-2
Adaptive control for a class of nonlinear systems with time-varying delays in state and input Xiaohong JIAO, Jie YANG, Qiang LI Institute of Electrical Engineering, Yanshan University, Qinhuangdao Hebei 066004, China
Abstract: This paper is concerned with the adaptive stabilization problem of uncertain input delayed systems. A solution to this problem is given for a class of uncertain nonlinear systems with time-varying delays in both state and input. An adaptive asymptotically stabilizing controller, which can guarantee the stability of the closed-loop system and the convergence of the original system state, is designed by means of the Lyapunov-Krasovskii functional stability theory combined with linear matrix inequalities (LMIs) and nonlinear adaptive techniques. Some numerical examples are presented to demonstrate the effectiveness of the derived controller. Keywords: Adaptive stabilization; Input delay; State delay; Uncertain nonlinear delay systems; Lyapunov-Krasovskii functional; Linear matrix inequalities
1
Introduction
Time-delay systems have been an active research area for the last few decades due to the existence of time delays in various engineering systems such as long transmission lines in pneumatic systems, chemical processes and biological systems. There have been a great number of research results on stability analysis (see [1∼3] and the references therein), robust control (see [4∼6] and the references therein) and adaptive control (for example, [7, 8] and the references therein) synthesis related to systems with delayed state. The importance of the study on systems with delayed control is further highlighted by the development of communication networks in engineering. Recently, the problem of controlling input-delay systems has captured the interests of many researchers. However, it has been found that controlling input-delay systems can be a challenging problem, especially in the presence of nonlinearities and uncertainties. Most of the existing results are aimed at linear input-delay systems with bounded parametric uncertainties (see, for example, [9∼17] and the references therein). The research attention is mainly focused on improving the maximum allowable value of the bounds on uncertain parameters [11∼14] and reducing the conservatism of the maximum time delay [15∼17] by properly constructing a LyapunovKrasovskii functional and employing linear matrix inequalities (LMIs) with relaxation matrices and tuning parameters. As for nonlinear input-delay systems, few results are given except some attempts in [18, 19]. The problem of stabilizing a family of nonlinear feedforward systems with delayed input is investigated in [18]. Moreover, then in [19], it is shown how the backstepping approach can be adapted to the stabilization for nonlinear systems in feedback form with a delay in control. However, it can be seen from [18] that the relatively strict assumptions are imposed on the system even if the stabilization for the nominal system by backstepping technique is considered. Considering the above analysis, this paper tries to solve
the adaptive stabilization problem of uncertain nonlinear input-delay systems. As well known, the adaptive control can deal with the case with the unknown bounds on the uncertain parameters. However, so far, most adaptive controllers proposed for the time-delay systems result in the uniform ultimate boundedness of the resulting closed-loop systems (see, for example, [7, 20∼22] and the references therein). While in fact, the adaptive asymptotical stabilization is generally required for process control systems, which is to find an adaptive controller such that the resulting closed-loop system is stable in the sense of Lyapunov at the equilibrium and the original system state can converge to the equilibrium. Therefore, the problem addressed in this paper is the adaptive asymptotical stabilization for uncertain nonlinear input-delay systems. A solution to this problem is given for a class of uncertain nonlinear input-delay systems with different time-varying delays in the states. The nonlinearities of the system are assumed to be bounded by a weighted norm of the state and delayed-state vectors, and the upper bound of the weight is assumed to be unknown. An asymptotically stabilizing adaptive controller is derived by means of the construction of an appropriate Lyapunov-Krasovskii functional and the use of linear matrix inequalities techniques and the adoption of an adaptive factor. Finally, the effectiveness of the proposed controller is explained and demonstrated on some numerical examples. Notation Throughout this paper, the superscripts ‘−1’ and ‘T’ stand for the inverse and transpose of a matrix, respectively. Rn denotes an n-dimensional Euclidean space and Rn×m is the set of all n × m real matrices. · stands for either the Euclidean vector norm or its induced matrix 2-norm. P > 0 means that the matrix P is positive definite, I is an appropriately dimensioned identity matrix, diag{ ·, ·, · } denotes a block-diagonal matrix and the symmetric terms in a symmetric matrix are denoted by ∗, e.g., X Y . ∗ Z
Received 9 July 2009; revised 28 July 2010. This work was supported by the National Natural Science Foundation of China (No. 60774018). c South China University of Technology and Academy of Mathematics and Systems Science, CAS and Springer-Verlag Berlin Heidelberg 2011
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Problem formulation
A class of uncertain nonlinear systems with time-varying delays in both state and input is described by the form ⎧ x(t) ˙ = A1 x(t) + A2 x(t − h1 (t)) + B1 u(t) ⎪ ⎪ ⎨ + B2 u(t − h2 (t)) (1) ⎪ + Δf (x(t), x(t−h1 (t)), u(t), u(t−h2 (t))), ⎪ ⎩ ¯ 1, h ¯ 2 }, 0], t 0, x0 (τ ) = ψ(τ ), τ ∈ [− max{h where x(t) ∈ Rn and u(t) ∈ Rm are the state and control, respectively. ψ is a continuously differential initial function, A1 , A2 , B1 and B2 are known constant real matrices with appropriate dimensions, and h1 (t) and h2 (t) are time¯ i , h˙ i (t) varying bounded delays satisfying 0 hi (t) h ¯ ¯ d < 1, i = 1, 2. h1 , h2 > 0 are constants that represent the largest value of delay in the state and input, respectively, which may be unknown. d is a known constant. Δf ( · ) is a continuous nonlinear vector function with appropriate dimensions with Δf (0, 0, 0, 0) = 0 and satisfies the following condition Δf (x(t), x(t − h1 (t)), u(t), u(t − h2 (t))) β1 x(t)+β2 x(t−h1 (t))+β3 x(t−h2 (t)) (2) with unknown bounds βi , i = 1, 2, 3. Remark 1 It can be noted that in the assumption for the uncertainties, the matching condition is removed, which is usually required in the stabilization for uncertain input delay systems [9∼17]. Moreover, the uncertainties arising from the state, delayed state and input are involved and assumed to be nonlinear perturbations with unknown bounds of the weighted norm of the state and delayed state vectors. The assumption on the unknown bounds is more significant than that on the known bounds for the nonlinear uncertainties Δf . Meanwhile, the assumption on presenting only the weighted norm of the state and delayed state is reasonable since the control law is a function on the state for a system with state feedback. The adaptive stabilization problem addressed in this paper is as follows: for the uncertain nonlinear time-delay system (1) with (2), to find an adaptive controller ˆ ˆ˙ = β(x(t)), u(t) = α(x(t), θ(t)), θ(t) (3) such that the resulting closed-loop system coordinated by ˆ is stable in the sense of Lyapunov and lim x(t) = 0 (x, θ) t→∞ for any given initial condition. To this end, the following lemmas will serve as a basis for the derivation of the adaptive asymptotically stabilizing controller. Lemma 1 [23] Consider time-delay systems described by (4) x(t) ˙ = f (xt ), x0 (τ ) = φ(τ ), τ ∈ [−r, 0], where x(t) ∈ Rn , xt (τ ) ∈ C ([−r, 0], Rn ), r is a given positive constant that represents the largest time delay. If there exist a continuous functional V : C → R and continuous nondecreasing functions κ1 ( · ) and κ2 ( · ), κ1 (s), κ2 (s) > 0, ∀s > 0 with κ1 (0) = κ2 (0) = 0, such that (5) κ1 (x(t)) V (xt ) κ2 (xt c ), ˙ V (xt ) 0 (6) with xt c := sup x(t + τ ); then, the solution x = 0 −rτ 0
is Lyapunov stable. Furthermore, it follows that xt (φ) → M as t → ∞, where M is the largest invariant set in E
defined by E = {φ ∈ G : V˙ (φ) = 0} with G ⊆ C. Lemma 2 (Schur complement) Given constant matrices Ω1 , Ω2 , Ω3 where Ω1 = Ω1T and Ω2 = Ω2T > 0, then Ω1 + Ω3T Ω2−1 Ω3 < 0 if and only if −Ω2 Ω3 Ω1 Ω3T < 0, or < 0. Ω3 −Ω2 Ω3T Ω1
3 Main result This section presents the derivation of an adaptive stabilizing controller by properly constructing a LyapunovKrasovskii functional, adopting an adaptive factor and employing linear matrix inequalities techniques. The main contribution is stated in the following Theorem 1 and its proof. Theorem 1 For system (1) with (2), if there exist posi¯ 1 ∈ Rn×n , Q ¯ 2 ∈ Rn×n , matrix tive matrices P¯ ∈ Rn×n , Q m×n and scalars ε > 0, > 0 such that the following Y ∈R LMI holds: Σ11 Σ12 < 0, (7) ∗ Σ22 where ⎤ ⎡ B2 Y M A2 P¯ ¯1 ⎦, Σ11 = ⎣ ∗ −(1 − d)Q (8) 0 ¯2 ∗ ∗ −(1 − d)Q ⎤ ⎡¯ P 0 0 B2 In (9) Σ12 = ⎣ 0 P¯ 0 0 ⎦ , 0 0 P¯ 0 ⎡ ⎤ − I 0 0 0 ⎢ ∗ − I 0 ⎥ 0 ⎥ Σ22 = ⎢ (10) ⎣ ∗ ∗ − I ⎦ 0 ∗ ∗ ∗ −ε(1 − d)I T ¯1 +Q ¯ 2 , then an ¯ with M = P A1 +A1 P¯ +B1 Y +Y T B1T + Q adaptive asymptotically stabilizing controller designed as ⎧ u(t) = Kx(t)+μIn x(t) = Y P¯ −1 x(t)+μIn x(t), ⎪ ⎪ ⎨ 1 1 ˆ T P P x, (11) μ(t) ˙ = −xT P B1 In x− εμxT x− μθx ⎪ 2 2 ⎪ ⎩ ˆ˙ θ(t) = μ2 xT P P x can render the resulting closed-loop system to be globally ˜ = 0 and the original system Lyapunov stable at (x, μ, θ) state x → 0 as t → ∞, where In ∈ Rm×n is a matrix whose elements are all ones, K ∈ Rm×n is a state feedback gain with K = Y P¯ −1 , μ(t) ∈ R is an adaptive factor guaranteeing the closed-loop system stabilization, which satisfies the following projection: ⎧ ⎨ μ(t), if |μ(t)| 1, μ(t) = (12) 1, if 0 μ(t) < 1, ⎩ −1, if − 1 < μ(t) < 0, ˆ is an estimate of the uncertain parameter and θ˜ = θˆ−θ, θ(t) θ = β12 + β22 + β32 . Proof Construct a Lyapunov-Krasovskii functional as 2 t V (t) = xT (t)P x(t) + xT (s)Qj x(s)ds +ε
t
j=1
t−hj (t)
1 μ2 (s)xT (s)x(s)ds+μ2 (t)+ θ˜2 (t), t−h2 (t) 2 (13)
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with 0 < P = P T ∈ Rn×n , 0 < Qj ∈ Rn×n (j = 1, 2), and the time derivative of V (t) along the trajectory of system (1) and control law (11) can be calculated as follows: V˙ (t) = 2xT P [(A1 + B1 K)x + A2 xt1 + B2 Kxt2 +(B1 μIn x + B2 μt2 In xt2 ) + Δf (x, xt1 , xt2 )] T +xT Q1 x − (1 − h˙ 1 (t))xT t1 Q1 xt1 + x Q2 x −(1 − h˙ 2 (t))xT Q2 xt + εμ2 xT x t2
2
˙ −ε(1 − θ)θˆ (14) with the definition xt1 = x(t − h1 (t)), xt2 = x(t − h2 (t)), μt2 = μ(t − h2 (t)). Moreover, there is the inequality 2xT P B2 μt2 In xt2 −1 −1 T T T ε(1−d)μ2t2 xT t2 xt2+ε (1−d) x PB2 In In B2 P x. (15) Also, using condition (2), we can get 2xT P Δf (x, xt1 , xt2 ) 2xT P (β1 x + β2 xt1 + β3 xt2 ) T xT P P xθ + −1 (xT x + xT t1 xt1 + xt2 xt2 ). (16) Then, combining (14)∼(16), substituting the adaptive law (11) and noting that h˙ i (t) < d (i = 1, 2) and |μ(t)| 1, the time derivative of V is obtained as V˙ (t) ξ T (t){Ξ + −1 (Γ1T Γ1 + Γ2T Γ2 + Γ3T Γ3 ) +ε−1 (1 − d)−1 Γ4T Γ4 }ξ(t) + μ2 xT P P xθ 1 ˙ +(θˆ − θ)θˆ + 2μ(xT P B1 In x + εμxT x + μ) ˙ 2 ξ T (t){Ξ + −1 (Γ1T Γ1 + Γ2T Γ2 + Γ3T Γ3 ). +ε−1 (1 − d)−1 Γ4T Γ4 }ξ(t), (17) where ⎤ ⎤ ⎡ ⎡ P B2 K H PA2 x(t) ⎦, ξ(t) =⎣ x(t−h1 (t)) ⎦, Ξ =⎣ ∗ −(1−d)Q1 0 x(t−h2 (t)) ∗ ∗ −(1−d)Q2 Γ1 = [I 0 0], Γ2 = [0 I 0], Γ3 = [0 0 I], Γ4 = [InT B2T P 0 0] with H defined as H = (A1 +B1 K)T P +P (A1 +B1 K)+Q1 +Q2 . (18) By the Schur complements, it can be shown that ⎡ ⎤ Ξ Γ1T Γ2T Γ3T Γ4T ⎢ ∗ − I 0 0 ⎥ 0 ⎢ ⎥ ⎢ ⎥ (19) 0 ⎢ ∗ ∗ − I 0 ⎥ < 0, ⎢ ⎥ ⎣ ∗ ∗ ∗ − I ⎦ 0 ∗ ∗ ∗ ∗ −ε(1 − d)I h˙ 2 (t))μ2t2 xT ˙ θˆ − t2 xt2 +2μμ+(
which implies Ξ + −1 (Γ1T Γ1 + Γ2T Γ2 + Γ3T Γ3 ) + ε−1 (1 − d)−1 Γ4T Γ4 < 0. In order to obtain a controller gain K, premultiplying and postmultiplying both sides of (19) with diag{P −1 , P −1 , P −1 , I, I, I, I} and defining P¯ = P −1 , ¯ i = P¯ Qi P¯ (i = 1, 2) yield Y = K P¯ , Q ⎡ ⎤ M A2 P¯ B2 Y P¯ 0 0 B2 In ⎢ ∗ −(1−d)Q ⎥ ¯1 0 0 P¯ 0 0 ⎢ ⎥ ⎢ ∗ ⎥ ¯ 2 0 0 P¯ 0 ∗ −(1−d)Q ⎢ ⎥ ⎢ ⎥ ∗ ∗ − I 0 0 0 ⎢ ∗ ⎥ ⎢ ⎥ ⎢ ∗ ⎥ ∗ ∗ ∗ − I 0 0 ⎢ ⎥ ⎣ ∗ ⎦ ∗ ∗ ∗ ∗ − I 0 ∗ ∗ ∗ ∗ ∗ ∗ −ε(1−d)I < 0, (20)
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T T ¯ ¯ ¯ with M = P¯ AT 1 + A1 P + B1 Y + Y B1 + Q1 + Q2 . The inequality (20) is just the inequality (7) with (8) and (10). Therefore, it follows if (7) holds that V˙ ξ T (t){Ξ + −1 (Γ1T Γ1 + Γ2T Γ2 + Γ3T Γ3 ) +ε−1 (1 − d)−1 Γ4T Γ4 }ξ(t) < 0. (21) From (13) and (21), in view of Lemma 1, the proof can be completed. Remark 2 It should be noted that the asymptotically stabilizing adaptive controller proposed in Theorem 1 is independent of delays in both state and input, but the variation rate of delays is required to satisfy 0 d < 1, namely, the delays in the state and input are required to be constants or slow varying. Remark 3 It is well known that proper selection of Lyapunov-Krasovskii functional is crucial for deriving stability conditions, and different Lyapunov-Krasovskii functionals may result in different stability conditions with different conservatism and advantage. Thus, many research efforts in the existing literature of the robust stabilization for input delay systems have been focused on reducing the conservatism by developing new Lyapunov-Krasovskii functionals with more terms on the inner product of involved cross-terms and incorporating more relaxation matrices into the linear matrix inequalities (LMIs). Whereas in this paper, due to the adoption of the adaptive technique, there is no need to develop new Lyapunov-Krasovskii functionals for improving the maximum allowable value of the bounds of the uncertainties. However, with the help of constructing appropriate Lyapunov-Krasovskii functionals with additional useful terms as in [3], the restriction on d < 1 may be removed.
4 Numerical example Two numerical examples are used to illustrate the effectiveness of the proposed adaptive stabilizing controller. Example 1 Consider a linear system with uncertainty and input delay discussed in [12∼14]. x˙ = (A + ΔA)x(t) + B1 u(t − τ (t)), t 0, (22) x(0) = x0 , u(t) = φ(t), t ∈ [−0.2, 0], with 0 1 0 0 0 A= , ΔA = , |q| γ, B = , −1.25 −3 q 0 1 namely, corresponding to the form (1), A1 = A, A2 = 0, B1 = 0, B2 = B1 , Δf = ΔAx(t). Here, similar to references [12, 14], we also consider two cases for the time delay τ (t). Case 1 τ (t) is a constant and 0 τ (t) 0.2. The maximum allowable value of the bounds γ on uncertain parameters that guarantees the stabilizability of system (22) by a delayed feedback controller is γmax = 7.2568 in [12]. In [13], γmax = 8.4602 and in [14] the maximum allowable value is γmax = 9.1574 for the robust controller. While, in this paper, applying Theorem 1 and solving LMI (7) via MATLAB yield = 0.5, ε = 0.3 and the state feedback controller gain K = [−1.9650 − 0.3547]. The maximum allowable value of the bound γ on uncertainty can be arbitrarily large. Fig. 1 shows the response of the closed-loop system with the uncertain bound γ = 14 that exceeds the maximum required in [12, 14].
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Fig. 1 Response of the system with 0 τ (t) 0.2 and γ = 14.
Case 2 τ (t) is time-varying and 0 τ (t) 0.2, τ˙ (t) d < 1. When the varying rate d of the time delay τ (t) is larger, d = 0.9, the maximum allowable value of γ is γmax = 9.8506 given in [12]. Moreover, the maximum allowable value given by [14] is γmax = 13.4176. Similar to the Case 1, in this paper, the maximum allowable value of the bound
γ on uncertainty can be also arbitrarily larger than those in [12, 14]. Fig. 2 shows the response of the closed-loop system with controller gain K = [−1.1341 − 0.4532], the uncertain bound γ = 14 and the time varying is chosen as τ (t) = 0.9 sin t as well as the adaptive tuning parameters = 0.5, ε = 3.
Fig. 2 Response of the system with 0 τ (t) 0.2, τ˙ (t) d < 1, and γ = 14.
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From the above simulation comparison, it can be found that the method in this paper can lead to much less conservative results for this example. Example 2 Consider a nonlinear system with timevarying input and state delays described as the form (1) with ⎧ 0 0.3 −1.2 0 ⎪ ⎪ ⎪ A1 = , A2 = , ⎪ ⎪ −1.2 1.4 0 −1.2 ⎪ ⎪ ⎨ 0 0 (23) , B2 = , B1 = ⎪ 0 1 ⎪ ⎪ ⎪ ⎪ x1 (t − h1 (t)) sin x2 + x2 (t − h1 (t)) ⎪ ⎪ . ⎩ Δf = x2 (t − h1 (t)) cos x2 + x1 (t − h2 (t))
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Applying Theorem 1 and solving LMI (7) via MATLAB yield = 1, ε = 3 and the state feedback controller gain K = [−1.8043 − 0.4558]. Fig. 3 shows the response of the closed-loop system when the time-varying delays in state and input are chosen as h1 (t) = 0.2 sin t and h2 (t) = 0.9 cos t; the initial conditions are chosen as ˆ = 1.2, μ(0) = 2. x1 (0) = 1, x2 (0) = −1, θ(0) Note that the original system states x1 and x2 converge to zero, and the adaptive parameter θˆ and adaptive factor μ are bounded.
Fig. 3 Response of the system with time-varying delays in state and input.
From the simulation results in this section, we can conclude that the adaptive controller designed can stabilize a class of uncertain nonlinear systems with time-varying state and input delays and render the original system state to converge to zero. Moreover, the bounds of the uncertainties can be arbitrarily large due to introducing the adaptive estimate parameter, and a suitable gain of the controller can be easily obtained due to incorporating some tuning parameters into adaptive law.
tainties can be arbitrarily large. The numerical examples presented in the paper have also shown that this adaptive controller is feasible and efficient.
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[3] Y. He, Q. Wang, L. Xie, et al. Further improvement of free-weighting matrices technique for systems with time-varying delay[J]. IEEE Transactions on Automatic Control, 2007, 52(2): 293 – 299.
Conclusions
A state feedback adaptive stabilizing controller design method is presented for uncertain systems with timevarying state and input delays based on the LyapunovKrasovskii functional stability theory and linear matrix inequalities framework. The adaptive controller proposed can guarantee the Lyapunov stability of the resulting closedloop system and the convergence of the original system state. The adoption of the adaptive technique results in the asymptotical stabilization for a larger class of uncertain input delay systems since the bounds of the nonlinear uncer-
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[20] F. Zheng, Q. Wang, T. H. Lee. Adaptive robust control of uncertain time delay systems[J]. Automatica, 2005, 41(8): 1375 – 1383. [21] D. W. C. Ho, J. Li, Y. Niu. Adaptive neural control for a class of nonlinearly parametric time-delay systems[J]. IEEE Transactions on Neural Networks, 2005, 16(3): 625 – 635. [22] S. J. Yoo, J. B. Park, Y. H. Choi. Adaptive dynamic surface control for stabilization of parametric strict-feedback nonlinear systems with unknown time delays[J]. IEEE Transactions on Automatic Control, 2007, 52(12): 1697 – 1701. [23] J. K. Hale, S. M. Verduyn Lunel. Introduction to Functional Differential Equations[M]. New York: Springer-Verlag, 1993. Xiaohong JIAO received her B.E. and M.S. degrees in Automatic Control from Northeast Heavy Machinery Institute, China, in 1988 and 1991, respectively, and Ph.D. degree in Mechanical Engineering from Sophia University, Tokyo, Japan, 2004. She is a professor at the Institute of Electrical Engineering, Yanshan University, China. Her research interests include robust adaptive control of nonlinear systems and time-delay systems. E-mail:
[email protected]. Jie YANG received her B.S. and M.S. degrees from Yanshan University, in 2006 and 2009, respectively. She is currently working toward the Ph.D. degree with School of Electrical Engineering and Automation, Tianjin University. Her current research interests include robust adaptive control of nonlinear systems and intelligent control systems. E-mail: student
[email protected]. Qiang LI received his B.S. degree from Beihua University, in 2006, and the M.S. degree from Yanshan University in 2009. His research interests include robust control of nonlinear systems and control applications. E-mail:
[email protected]. ———– ——— ——— ——— ——————————————— —
