Abstract

The issue of output tracking for switched cascade nonlinear systems is discussed in this paper, where not all the linear parts of subsystems are stabilizable. The conditions of the solvability for the issue are given by virtue of the structural characteristics of the systems and the average dwell time method, in which the total activation time for stabilizable subsystems is longer than that for the unstabilizable subsystems. At last, a simulation example is used to demonstrate the validity and advantages of the proposed approach.

1. Introduction

With the development of society, the control theory also confronts lots of challenges. The famous control effect cannot be achieved, if the technique which simply relies on continuous-time or discrete-time system is employed. Therefore, hybrid systems have attracted much attention [1–4]. Switched system is a particular class of hybrid system, which includes a finite number of subsystems and a rule specifying the switching among these subsystems. Stability analysis and control synthesis of switched systems have been two important topics, and the average dwell time approach is an effective tool to handle the stability problem for switched systems [5–9].

On the other hand, the problem of output tracking for nonlinear systems has an important theoretical and practical significance, because it is a fundamental problem in control theory and has widespread application in engineering, such as aeronautics, robot control, and flight control [10–13]. However, the output tracking problem for nonlinear systems is harder to investigate than stability. This is because output tracking requires the output of systems to track the reference signal besides the internal stability. Furthermore, for switched systems, due to the interaction between the continuous and discrete dynamics, the problem of output tracking becomes more difficult.

As far as we know, there are only a few results about the output tracking problem of switched systems in the literature. Reference [14] studied the tracking control problem for linear switched systems based on a reference model. Reference [15] gave sufficient conditions for output tracking problem of linear switched systems to be solvable under asynchronous switching. For switched cascade nonlinear systems, References [16, 17] addressed the output tracking problem with external disturbance using a variable structure control method, Reference [18] investigated the observer-based model tracking problem, Reference [19] discussed the output tracking problem based on the method of multiple Lyapunov function, and [20] studied the output tracking problem based on the average dwell time method, in which all subsystems of switched systems are stabilizable.

Prompted by the above results, this paper studies the solvability of the output tracking control problem of switched cascade nonlinear systems. The main contribution of this paper is that the extended results of output tracking problem for switched cascade nonlinear systems are obtained by relaxing the conditions that all subsystems are stabilizable. The solvability conditions for the output tracking problem are proposed, if the total activation time of stabilizable and unstabilizable subsystems satisfies certain relations. A numerical example is demonstrated to verify the effectiveness of the main results.

2. Problem Formulation and Preliminaries

In this paper, we consider a switched cascade nonlinear system:where and , , , and are the states, the control input, the measurable output, and the external disturbance, respectively. The switching law is a right continuous piecewise constant function of time, and the switching sequence means that the th subsystem is active when . , , , and are known matrices, and are known smooth vector fields with appropriate dimensions.

In order to solve the output tracking problem, motivated by [20], we apply the controller as follows:where is the tracking error, is the reference input, and , , and are the gain matrices, which will be determined later.

Combining (1) and (3), we get the following augmented system [20]:where , , , and

In the development to follow, we introduce a definition first.

Definition 1 (see [14]). System (1) is said to satisfy weighted tracking performance, if the following conditions are satisfied:
(i) Internal stability: the system is asymptotically (or exponentially) stable.
(ii) Tracking performance: given the performance index asthe tracking performance index can meet certain upper bound, where and are positive semidefinite matrices and is a positive definite matrix.

Definition 2 (see [21]). For any switching signal and any , let denote the number of switching of over . If holds for and , then is called the average dwell time. In general, we assume
For nonswitched cascade nonlinear system (7), we give a lemma as follows:

Lemma 3 (see [20]). For a given constant , if there exist constants , , , and , positive definite matrices , , and , a matrix , and a function , such that the following conditions are satisfied for system (7) then the following inequality is satisfiedwhere is a constant, is a Lyapunov function for system (7), and and

3. Main Results

The objective of this paper is to design switching scheme such that system (1) has output tracking performance. We first consider the nonswitched nonlinear system (7).

Lemma 4. For a given constant , if there exist positive constants , , , and , matrices , , , and , and a function , such that the following conditions are satisfied for system (7) then following inequality is satisfiedwhere is a constant and is a Lyapunov function for system (7).

Proof. According to the proof of Lemma 3 and the condition of Lemma 4, we obtainwhere and . Thus, Consider the switched cascade nonlinear system (1). For the problem of output tracking, suppose that not all the linear parts of the subsystems of system (4) are stabilizable. Without loss of generality, we suppose that the first subsystems are stabilizable (the positive integer satisfies ) and the other subsystems are unstabilizable. For any switching law and any , we let (resp., ) denote the total activation time of stabilizable (resp., unstabilizable) subsystems during . Let be a specified sequence of time instants.

A sufficient condition for output tracking performance of system (1) is proposed as follows.

Theorem 5. Consider the switched cascade nonlinear system (1). For a given positive constant , if there exist positive constants , , , , and , matrices , , , and , and functions , such that the following conditions are satisfied for the system (4) (1) (2) then, under the average dwell time schemeand the conditionthe output tracking control problem is solved, where , , , , and .

Proof. Consider the following Lyapunov function for the switched system (4):When th subsystem is active, according to Lemmas 3 and 4 and the conditions of Theorem 5, there exist constants and , such that the Lyapunov function (21) satisfiesWhen , according to the above inequality, we haveFrom (16), (17), and (21), we getTherefore, on every switching instant , we get thatAccording to (20), we haveBecause , according to (19), it holds thatThus, .
Therefore, From (17), it easily holds that there exist constants and such that ; furthermore, . Thus, , which implies asymptotic stability of the closed-loop system (4) when .
When , , by virtue of (22), we get According to (20) and (24), on every switching instant , we have The above inequality and conditions (19)-(20) give thatIntegrating both sides of the above inequality from to , we obtainwhere , , , and .
Therefore,To conclude, the output tracking problem is solved.

4. Example

In this section, we will illustrate the effectiveness of the proposed results by an example.

Consider the cascade switched nonlinear system (4) with , , , , , , , , , , and , , , , and .

Let , let , and let ; solving inequalities (14)–(16) yieldsTherefore, and .

Obviously, the first subsystem is stabilizable and the second subsystem is unstabilizable.

We choose and and define . Selecting , , and , we can obtain , , and . . Let and let ; then, and