Adaptive Switched Linear Systems with Average Dwell Time to Compensate External Disturbances

Given a linear switched system composed of both Hurtwiz stable and unstable subsystems, and with additive unknown constant disturbance, a switching law with average dwell time between Hurwitz stable and unstable subsystems is proposed, together with an adaptive algorithm to compensate the unknown additive disturbance. Using Lyapunov theory, exponential stability of a desired degree of the plant state is assured and bounded-ness of the adaptive dynamic too.


Introduction
There exist many practical problems where the available choice is to switch among a finite set of subsystems [1], [2], [3]. Also, in some practical problems [4], the only way to incorporate a logic-based decision system is by switching among a family of subsystems. Switched systems are of variable structure or multi-modal class [5]. According to [6], a switched system can be viewed as a hybrid dynamical system which is composed of a family of continuous-time subsystems along with a switching law among them. In comparison to sliding mode systems (a type of variable structure one), which is an approach recognized as an efficient tool to design robust systems, switching systems present chattering behavior. This behavior, in some situations, would not be considered acceptable [7]. Although there exist solutions to reduce this phenomenon [8] (including the well known super-twisting algorithm [9]), dwell switching design can be viewed as an alternative tool of switching laws without chattering. Moreover, dwell average switching systems do not exhibit Zenoness and beating effects. These effects can appear on hybrid dynamical systems [10]. Adaptive control provides adaptation to adjust a system with parametric uncertainties [11]. Motivated by the adaptive control law presented in [12] to face with additive constant disturbance, we propose a switching law with average dwell time along with an adaptive algorithm to compensate it. This scheme is applied to linear switched systems consisting of both Hurtwiz stable and unstable subsystems. Exponential stability of a desired degree is guaranteed for the states of the plant. Bounded-ness of the adaptive dynamic is assured too. Although in [6] it is proposed a dwell average switching law for linear systems in the presence of a class of nonlinear additive perturbation satisfying a kind of Lipschitz condition, the constant additive case is not considered. This class of constant additive perturbation is presented in some engineering problems [12]. The remainder of this article is organized as follows. Section 2 gives the main theory and notation on switched systems with average dwell time. Section 3 states our main result. A numerical example supporting our theory is given in Section 4. Finally, Section 5 gives the conclusions.

Switching design with average dwell time
Consider the linear switched system with additive constant disturbance given by: where x(t) ∈ R n is the state, t 0 is the initial time, d ∈ R n is the additive constant disturbance, and x 0 is the initial state. The piecewise constant function of time . See Figure 1. The matrices A i (i ∈ I N ) are assumed of appropriate dimension, and N > 1 represents the number of subsystems. It is also assumed the switched system (1) is composed of both Hurwitz stable and unstable subsystems.
The control objective consists to find a switching law with average dwell time, along with a dynamic estimationφ(t) to d, such that the plant states goes to the equilibrium point as time goes to infinity exponentially with a stability degree, while keeping bounded the esti- . Given a switching signal σ(t) and any t > τ ≥ 0, the number of switchings of σ(t) on the interval (τ, t), denoted by N σ (τ, t), satisfies: for a given constants N 0 and τ a . The constant N 0 is named after the chatter bound, and τ a is called the average dwell constant time [4], [6]. Basically, there may exist consecutive switchings separated by less than τ a , but the average time interval between consecutive switchings is not less than τ a [6] 1 . Let us introduce the notation S a [τ a , N 0 ] to denote the set of all switching signals satisfying (2). Definition 1 [4], [6]. The state of the switched system (1) is said to be globally exponentially stable with stabil- and where the matrices P ′ i s are symmetric and positive definite.

Main result
To begin with the main result, consider the next switching law: (S 1 ) Switching law [6]: Determine the switching signal σ(t) so that holds for any given initial time t 0 . Theorem 1 Under the switching law (S 1 ), there is a finite constant τ * a such that the switched system (1), x(t) is globally exponentially stable with stability degree λ over S a [τ a , N 0 ], for any average dwell time τ a ≥ τ * a , any chatter bound N 0 ≥ 0 , and withφ(t) a solution to: where β is a given positive constant and P σ(t) are the corresponding solutions to the Lyapunov equations (3) and (4), applied over the switching law (S 1 ). Moreover, ϕ(t) remains bounded (i.e., there exist a constant value v such that ∥φ(t) ∥≤ v ∀t ≥ t 0 ). Furthermore, if: where λ M (·) and λ m (·) denote the largest and smallest eigenvalues of a symmetric matrix, then . (8) Proof. Similarly to the proof of (Theorem 1, [6]) but using the Lyapunov function: A sketch is as follows. 1) Each V i in (9) is continuous and its time derivative along the solutions of the corresponding subsystems and (6), satisfies:V 2) There exist constant scalars α 2 ≥ α 1 > 0 such that: ∀x ∈ R n and ∀i ∈ I N .

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Adaptive Switched Linear Systems with Average Dwell Time to Compensate External Disturbances ∀x ∈ R n and ∀i, j ∈ I N . For a discussion on properties 2) and 3), see [6] (in fact, a value for µ is given by (7) [6]). So, for any piecewise constant switching signal σ(t), and any t > t 0 , let t 1 < t 2 < ... < t N σ(t 0 ,t) be the switching points of σ(t) over the time interval (t 0 , t), and suppose that the p-th subsystem is activated during [t i , t i+1 ) [6]. Then, for any ζ ∈ [t i , t i+1 ), and from (11), we have: According to [6], and under the Theorem 1 conditions, the above means that: where α = N0ln(µ) 2 with N 0 ≥ 0 being an arbitrary value. This implies that the plant state (x(t)) converges exponentially with a prescribed degree. Using this fact, and from (11), we conclude thatV (t) ≤ 0 meaning thatφ(t) is bounded. This conclude the sketch of our proof. ♢

Conclusions
Essentially, in this paper, we extended the results presented in [6]. This extension involves the inclusion of additive constant perturbation to the switched system instead of dealing with additive nonlinear perturbation [6]. According to the given numerical experiments, for the slow time varying disturbance case, our propose goes well too.