Immersion and invariance stabilization for a class of nonlinear switched systems with average dwell time

Abstract

An immersion and invariance (I&I) stabilization theorem for nonlinear switched systems with average dwell time is set up which provides a tool for analyzing the behavior of switched systems. Also, a method integrating the I&I technique and viability method is proposed to stabilize nonlinear switched systems with state constraints. Based on the proposed theorem, this paper further investigates the problem of I&I stabilization of nonlinear switched systems in feedback form and with state constraints represented by inequalities, when all target subsystems are stable as well as only some target subsystems are stable and others are not. State-feedback controllers for subsystems are obtained constructively and a class of switching signals with average dwell time is derived simultaneously. Finally, an application to a numerical example is given to illustrate the effectiveness of the proposed method.

Introduction

Nonlinear switched systems consist of several nonlinear continuous (discrete) sub-models (subsystems) and a switching rule. Under the guidance of the switching rule, the system switches between these nonlinear sub-models. The stability problem of nonlinear switched systems is an important research topic in the field of switched systems, and some valuable results have been achieved [1], [2], [3], [4], [5], [6].

However, it is worth pointing out that most existing results about the stability problem of nonlinear switched systems are based on Lyapunov function of the controlled system. As we all know, it is very difficult to find a Lyapunov function. On the other hand, to be emphasized, presented by Astolfi and Ortega in [7] and was further improved in [8], the immersion and invariance (I&I) control method plays an important role in the study of the stability of nonlinear systems. The basic idea of the I&I method is to attain the control aim by immersing the controlled system into an order-reduced target system, which has the desired performance. Different from the other nonlinear control method, a considerable benefit of this approach is that the Lyapunov function of the controlled system is not required and thus the difficult problem of finding the Lyapunov function can be avoided. Therefore, this method has attracted the attention of many researchers, and a large number of related studies have appeared in recent yeas. The upward stability of vehicle pendulum system was studied by means of I&I method in [9]. In [10], for the synchronous generator system with controllable series capacitor, an asymptotic stabilization controller was designed by using I&I method, and the transient performance of the system was improved. A robust output feedback controller design method based on I&I principle is proposed for speed and altitude tracking of an air breathing hypersonic vehicle [11]. Recently, the I&I approach has been extended to nonlinear cascaded discrete systems [12], which has greatly broadened the range of applications of this method. In [13] compression principle was adopted to improve the conditions of the attractivity of the manifold in I&I theorem and thus significantly increase the flexibility in controller design.

As a powerful research tool in the study of non-switch nonlinear systems, the I&I method is naturally expected to be applied to nonlinear switched systems. However, to the best of our knowledge, few results on the application of this method to nonlinear switched systems have been reported up to now. Recently, an immersion and invariance stabilization theorem for nonlinear switched systems under arbitrary switchings was established in [14]. But in [14], the target switched system was required to be asymptotically stable under arbitrary switchings. The requirement is relatively harsh, making it much more difficult to find a suitable target switched system. In addition, the average dwell time (ADT) method [15], as a very important design method for the switching laws, has greater design flexibility and strong maneuverability in the stability analysis and stabilization of switched systems [16], [17] . Therefore, It is more natural and reasonable that each target subsystem is allowed to have its own Lyapunov function and the target switched system is asymptotically stable under a class of switching signals with average dwell time. In view of this, the problem of I&I stabilization for nonlinear switched systems with average dwell time will be studied in this paper.

However, utilizing the I&I method to stabilize nonlinear switched systems still poses a challenging task mainly for the following difficulties:

• The trajectory boundedness is a prerequisite in the I&I stabilization theorem, which is very hard to check and thereby greatly restricts the further development and application of I&I theory. This problem still exists for nonlinear switched systems.

• Due to the interaction between subsystems and the switching signal, a switched system may exhibit a more complex behavior. Hence, the I&I method cannot be simply extended to switched systems. For instance, when none of individual subsystems can be I&I stabilized, the existing methods may no longer ensure the stability of the overall system if the designed switching law is not suitable. In this case, how to design a suitable switching law and I&I controllers for subsystems to stabilize the whole closed-loop system? In addition, the target switched dynamics, the off-the-manifold switched dynamics and the closed-loop controlled plant must switch synchronously, which makes it more difficult to design a suitable switching law.

On the other hand, the states of most actual physical systems are usually constrained due to the need to consider the limitations of the security or the device itself. If these state constraints are not considered at the controller design stage, the closed-loop stability may not be guaranteed or the performance of closed-loop systems may deteriorate dramatically. Although a lot of literature has studied the stabilization problem for both non-switched systems [18], [19], [20] and switched systems [21], [22], [23] with state constraints. However, one common feature of the control methods proposed in the above literatures is that the Lyapunov functions of the controlled system are all required either in the controller design or in the stability analysis of the closed-loop system. In addition, it is sometimes difficult or unnecessary to construct the Lyapunov function of the plant. Therefore, how to explore an alternative and efficient approach to stabilize this kind of systems is of urgent practical significance.

Viability theory that was first formally proposed by Aubin [24] is usually used to investigate the evolution of systems with state constraints. The study on the viability of the system mainly focus on two aspects. One is to determine the viability of the controlled system on a certain region [25], [26] or to construct a controller to make the closed-loop system viable on a given region [27], [28]. The other is to work out the viability kernel of the system. The viability theory is also applied to solve some other control problems such as stabilization problem [27], [29], the reachability problem [30] and differential games [31]. A systematic introduction to viability theory can be found in [24] or [32].

In order to overcome the above difficulties in stabilizing switched systems with state constraints, we also propose a new controller design method based on the integration of I&I and viability theory. Although an individual use of the two methods has been applied successfully to control various systems as we mentioned in the above literature, unfortunately, few studies on how to use the two methods together to solve the stabilization problem for switched systems have been reported so far.

Motivated by the aforementioned considerations, this paper first gives a sufficient condition for I&I stabilization of a general nonlinear switched system under arbitrary switching signals satisfying certain average dwell time. Then, combined with the viability method, a sufficient condition for I&I stabilization of a state-constrained nonlinear switched system with average dwell time is derived. Further, the obtained results are applied to nonlinear switched systems in feedback form and with state constraints expressed by inequalities, both when all target subsystems are stable as well as when only some of target subsystems are stable and others are not. Compared with the existing results of nonlinear switched systems, the results of this paper have three distinct features. First, based on the average dwell time method, the I&I method for non-switched systems is extended to switched systems, and an I&I stabilization theorem for nonlinear switched systems is established which provides a new tool for analyzing the behavior of nonlinear switched systems. Second, a new method integrating the I&I theory and viability theory is proposed to stabilize the nonlinear switched system with state constraints, which not only ensures the trajectories boundedness condition, but also guarantees the state constraints not be violated. Finally, we simultaneously construct state-feedback controllers for subsystems and give a class of switching signals with certain average dwell time.