Small Gain Theorem with Restrictions for Uncertain Time-varying Nonlinear Systems

This paper gives three versions of the small gain theorem with restrictions for uncertain time-varying nonlinear systems. The result can be viewed as an extension of the small gain theorem with restrictions for time-invariant nonlinear systems or the small gain theorem without restrictions for time-varying nonlinear systems. The result can be applied to study the stabilization problem or the output regulation problem of uncertain nonlinear systems. Index Terms – small gain theorem, nonlinear control, nonlinear systems. The small gain theorem is an important tool to ascertain the stability of two interconnected systems assuming each of the individual systems is stable in some sense. Small gain theorem has many different versions under various stability concepts [2] to [14], [20]. In this paper, we will focus on the small gain theorem in the context of inputto-state and/or input-to-output stability [15] to [19]. The first small gain theorem for nonlinear time-varying systems in the input-to-state stability (ISS) framework was established by Jiang et al [7]. The resulting small gain condition is given in terms of two inequalities. Recently, Chen and Huang further considered the small gain theorem for uncertain time-varying nonlinear system [2]. They presented a simplified small gain condition which is in a familiar form of the contraction mapping known for time-invariant nonlinear systems [7]. In [20], Teel introduced the concept of ISS with restrictions on the initial states and inputs and established a small gain theorem with restrictions for time-invariant systems. In Appendix B of [6], relying upon the separation property for ISS with restrictions, Isidori et al established a more general small gain theorem with restrictions for time-invariant systems. Nevertheless, the proof of [6] cannot be carried over to the case of time-varying systems, because the separation property for ISS does not hold for time-varying systems [2]. This paper is to establish three versions of the small gain theorem with restrictions for uncertain time-varying nonlinear systems, thus filling the gap between the small gain theorem with restrictions for time-varying nonlinear systems and that for time invariant nonlinear systems. Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong, E-mail: [email protected] Corresponding author: Jie Huang, Chinese University of Hong Kong. The work described in this paper was partially supported by the Hong Kong Research Grant Council under CUHK 412006. E-mail: [email protected] 115 116 MINGHUI ZHU AND JIE HUANG 1. Preliminary. Throughout the paper, let L∞ be the set of all piecewise continuous bounded functions u : [t0,∞) 7→ R m with a finite supremum norm ‖u[t0,∞)‖ = supt≥t0 ‖u(t)‖. Denote the supremum norm of the truncation of u(t) in [t1, t2] by ‖u[t1,t2]‖ = supt1≤t≤t2 ‖u(t)‖. And denote ‖u‖a = lim supt→∞ ‖u‖. A continuous function γ : R≥0 7→ R≥0 is of class K if it is strictly increasing and γ(0) = 0; and a continuous function β(s, t) : R≥0 ×R≥0 7→ R≥0 is of class KL if, for each fixed t ≥ 0, the function β(s, t) belongs to class K with respect to s and, for each fixed s, the function β(s, t) is decreasing with respect to t, and β(s, t) → 0 as t → ∞. Consider the following time-varying uncertain nonlinear system ẋ = f(x, u, d, t), y = h(x, u, d, t) t ≥ t0 ≥ 0 (1) where x ∈ R is the plant state, u ∈ R the input, y ∈ R the output, t0 the initial time, the functions f : R × R × Rd × [t0,∞) 7→ R n and h : R × R × Rd × [t0,∞) 7→ R p are piecewise continuous in t and locally Lipschitz in col(x, u, d). And d(t) : [t0,∞) 7→ R nd is a family of piecewise continuous functions of t, representing the external disturbance and/or the internal uncertainty. Definition 1.1. System (1) is said to be (uniformly) robust input-to-state stable (RISS) with restrictions X ⊂ R and ∆ > 0 on the initial state x(t0) and the input u respectively if there exist class KL function β and class K function γ, independent of d(t), such that, for any initial state x(t0) ∈ X and any input function u(t) ∈ L∞ satisfying ‖u[t0,∞)‖ < ∆, the solution of (1) exists and satisfies, for all t ≥ t0, ‖x(t)‖ ≤ max{β(‖x(t0)‖, t− t0), γ(‖u[t0,t]‖)}. Definition 1.2. System (1) is said to be robust input-to-output stable (RIOS) with restrictions X and ∆ on the initial state x(t0) and the input u respectively if there exist classKL function β and classK function γ, independent of d(t), such that, for any initial state x(t0) ∈ X , any input function u(t) ∈ L m ∞ satisfying ‖u[t0,∞)‖ < ∆, the output of (1) exists and satisfies, for all t ≥ t0, ‖y(t)‖ ≤ max{β(‖x(t0)‖, t− t0), γ(‖u[t0,t]‖)}. Definition 1.3. System (1) is said to be semi-uniformly RISS with restrictions X and ∆ on the initial state x(t0) and the input u respectively if there exist class K functions γ and γ, independent of d(t), such that for any initial state x(t0) ∈ X SMALL GAIN THEOREM WITH RESTRICTIONS 117 and input u ∈ L∞ satisfying ‖u[t0,∞)‖ < ∆, the solution of (1) exists and satisfies, for all t ≥ t0, ‖x(t)‖ ≤ max{γ(‖x(t0)‖), γ u(‖u[t0,∞)‖)} ‖x‖a ≤ γ (‖u‖a). (2) Remark 1.1. In [17], it was shown that, for the class of time-invariant systems, ISS is equivalent to semi-uniformly ISS. Such equivalence is called separation property. This equivalent relation can also be extended to ISS with restrictions and semi-uniformly ISS with restrictions (Appendix B of [6]). Unfortunately, the separation property does not hold for the time-varying nonlinear systems [2]. Definition 1.4. System (1) is said to satisfy robust asymptotic gain (RAG) property with restrictions X and ∆ on the initial state x(t0) and the input u respectively if there exists a class K function γ, independent of d(t), such that for any initial state x(t0) ∈ X and input u ∈ L m ∞ satisfying ‖u‖a ≤ ∆, the solution of (1) exists and satisfies, for all t ≥ t0, ‖x‖a ≤ γ (‖u‖a). (3) Definition 1.5. The output function of (1) is said to satisfy robust asymptotic L∞ stability (RALS) with restrictions X and ∆ on the initial state x(t0) and the input u respectively if there exist class K functions γ and γ, independent of d(t), such that for any initial state x(t0) ∈ X and input u ∈ L m ∞ satisfying ‖u[t0,∞)‖ < ∆, the output of (1) exists and satisfies, for all t ≥ t0, ‖y(t)‖ ≤ max{γ(‖x(t0)‖), γ u(‖u[t0,∞)‖)} ‖y‖a ≤ γ (‖u‖a). (4) Definition 1.6. System(1) is said to satisfy output robust asymptotic gain (o-RAG) property with restrictions X and ∆ on the initial state x(t0) and the input u respectively if there exists class K function γ, independent of d(t), such that for any initial state x(t0) ∈ X and input u ∈ L m ∞ satisfying ‖u‖a ≤ ∆, the output of (1) exists and satisfies, for all t ≥ t0, ‖y‖a ≤ γ (‖u‖a). (5) 118 MINGHUI ZHU AND JIE HUANG 2. Small Gain Theorem with Restrictions for Uncertain Nonlinear Time-varying Systems. 2.1. The Case of Time Invariant Nonlinear Systems. Consider the feedback interconnection as depicted in Figure 1, ẋ1 = f1(x1, v1, u1), y1 = h1(x1, v1, u1) (6) ẋ2 = f2(x2, v2, u2), y2 = h2(x2, v2, u2) (7) subject to the following interconnection: v1 = y2, v2 = y1 (8) ẋ2 = f2(x2, v2, u2) y2 = h2(x2, v2, u2) ẋ1 = f1(x1, v1, u1) y1 = h1(x1, v1, u1) u1 v1