Stability analysis for a class of nonlinear switched systems using variational principle

https://doi.org/10.1016/j.jfranklin.2016.08.001Get rights and content

Abstract

This paper presents a new theorem for stability of all possible trajectories starting at a given initial condition of a class of nonlinear switched systems with an arbitrary switching law. By utilizing the idea of variational principle, it is shown that a nonlinear switched system is asymptotically stable if and only if a critical trajectory is asymptotically convergent to the origin. In this regard, a necessary and sufficient condition is obtained. In order to investigate the stability property, the critical trajectory, characterized as a solution that maximizes a cost function, should be calculated. To this end, an affine control system, instead of the original switched system, and optimal control techniques are utilized. Afterwards, to calculate the critical trajectory, the initial condition x(0) is assumed given, and an algorithm is introduced for stability of all possible trajectories starting at a given initial condition of n-dimensional nonlinear switched systems under arbitrary switching with several operating modes. Simulation results reveal the efficiency of proposed technique.

Introduction

Generally, the systems that experience both discrete and continuous dynamic behavior are called “hybrid systems” [1]. Hybrid systems have been seriously considered by many researchers during the past decade. This importance comes from the main role of these systems in practical applications and modeling of physical systems. A special and main class of hybrid systems is switched systems, which are suitable for modeling natural and human-made systems with several operation modes through which any mode has a different dynamic behavior [2], [3]. Also, in switched systems, as a special class of hybrid systems, the behavior in any mode is described by a continuous subsystem and transition between the modes is described as a discrete-time event. The function that determines which mode is active at each point in time is called switching signal. In other words, the switched system consists of subsystems that at each time, only one of them is active [4], [5].

As an example of hybrid and switched systems is a vehicle which has a different dynamic behavior in each gear transmission and this behavior is changed by automatic gearbox. Also communication systems in [6], [7], power electronic in [8], flight control [9], chemical processes in [10] and automated highway systems [11] are other applications of hybrid systems.

Definitively, stability of switched systems is one of the main issues that should be analyzed and investigated for utilizing such systems. So far, several works have been done on stability analysis of these systems and different approaches have been developed.

The primary idea is to consider a nonlinear switched system as a nonlinear time-varying system and use the idea of time-varying Lyapunov function for stability analysis of the systems [12]. However, this idea needs a time-varying Lyapunov function satisfying the stability conditions for time-varying nonlinear systems. In the meantime, the switching signal should be known. Finding such a Lyapunov function is very difficult and almost impossible. Also, the assumption of switching signal being known is very restrictive. Therefore, this idea is not practically helpful and instrumental.

Until now, the main developed approaches for stability analysis of switched systems are “Common Lyapunov Function (CLF)”, “Multiple Lyapunov Functions (MLF)” and also “Dwell-time” stability approaches.

In the methods based on CLF a sufficient condition has been presented for uniformly globally asymptotically stability (UGAS) of an equilibrium point of switched systems. It has been shown that a switched system consists of a finite family of subsystems, globally asymptotically stability (GAS) of any subsystem and binary commutation of subsystems are sufficient conditions for globally asymptotically stability [13]. Also, it has been demonstrated that by combining these results with the inverse Lyapunov theorem, the conditions are sufficient for existence of a Common Lyapunov Function [13]. In [14], stability analysis for a special and limited class of switched nonlinear systems based on construction of a CLF is presented and sufficient conditions to guarantee the existence of a CLF are expressed in terms of linear inequalities.

The results obtained from the methods based on MLF determine a sufficient condition for stability as follows: All subsystems must be stable and a slow-switching condition must be held between subsystems. In fact, this slow-switching condition is in the form of an upper bound on the probability mass function of the number of switching which happens between the first instance and current time [15]. In fact, the main idea of this method is that if there is a separated Lyapunov function for any subsystem, then it is sufficient to apply some limitations to switching signal to guarantee the general system stability. In the method based on MLF, finding Lyapunov functions and satisfying the conditions are easier than the method based on CLF; thus, this method is more customary than the CLF method.

And, in the methods based on Dwell-time, stability condition is applied to the switching signal so that any subsystem must stay in a mode for a sufficient time. In other words, the condition is applied to the switching frequency or switching time between the two modes [16]. Also, [17] studied stability analysis of switched linear systems with nonlinear disturbances using generalized Gronwall–Bellman inequalities and presented sufficient conditions for stability analysis of special class of switched systems with constrained conditions on both linear and nonlinear sections.

Also, in [18], necessary and sufficient conditions for stability of nonlinear switched systems with predefined and known switching path are presented where the switching signal is not arbitrary. In [19], stability analysis for switched systems with an arbitrary switching signal, however only for linear systems, is investigated.

As can be observed, in CLF method, it is difficult and restrictive to find such a function or to solve linear matrix inequalities. In the meantime, only sufficient conditions for stability can be obtained. In the other two approaches, a condition is applied to the switching signal in addition to the existing conditions on subsystems.

Since, switching signal may be time-dependent, state-dependent, pre-determined or random, applying conditions to the switching signal in order to guarantee stability is not practical in many applications and systems. So, it is desirable to investigate and analyze stability of switched systems independent from switching signal; in other words, stability analysis of switched systems with an arbitrary switching signal is demanded. The only restriction is that the switching signal must have finite number of switching points in any bounded time interval. That is a standard assumption in switched systems and does not violate the generality of results.

Variational principle is based on the following simple idea: If the “most unstable solution” or “critical trajectory” of switched system is convergent, then all other solutions of system are also convergent [4]. This idea has been previously introduced in [4], [20], [21], [22]. In this approach, by finding the critical solution of system, the complexity of stability problem is reduced from the behavior analysis of a system to behavior analysis of a specific and single trajectory. In [4], this idea has been used for stability analysis of linear switched systems and also related theorems have been presented only for linear switched systems with n=2 theoretically and n=3 numerically. Additionally, the variational approach is utilized to derive the stability analysis of positive linear switched systems with two subsystems [20], [21], [22].

In this paper, using the mentioned idea, a stability criterion is introduced that obtains a necessary and sufficient condition for stability analysis of nonlinear switched systems with an arbitrary switching signal. On this view, a nonlinear switched system is asymptotically stable if and only if a critical trajectory is asymptotically convergent to the origin. In order to calculate the critical trajectory, the scope of presented stability theorem is restricted to a class of nonlinear switched systems with a given continuous state initial condition and the critical trajectory is calculated utilizing an affine control system instead of the original switched system. So, a stability theorem and an algorithm are presented for the stability of all trajectories starting at a given initial condition of a class of n-dimension nonlinear switched systems with an arbitrary switching signal. Therefore, the basic idea of variational approach is utilized to obtain a necessary and sufficient condition for stability of all trajectories starting at a given initial state of nonlinear switched systems and an algorithm is introduced for this purpose.

The contribution of this paper is in the line of presenting and proving a new stability criterion for nonlinear switched systems (Theorem 1) and also, introducing an algorithm for stability of all possible trajectories starting at a given initial condition of a class of nonlinear switched systems under arbitrary switching (Theorem 2) through which the optimal control techniques is utilized to find a worst-case switching signal that maximizes a cost function while an affine nonlinear control system is used instead of the original nonlinear switched system (Lemma 1).

The rest of this paper is organized as follows. In Section 2, stability of switched systems using the variational approach is presented. Section 3 presents calculation of the critical trajectory of switched system. The main result of this paper and details of the presented new method for stability analysis of hybrid systems are described in Section 4. In 5 Simulation results, 6 Conclusions, the simulation and comparison results and conclusion are mentioned.

Notations: The notations used in this paper are standard. The vectors are column vectors, xn×1. The superscript ‘T’ used for matrix transpose. For a scalar f, f/x=[f/x1,,f/xn]T is a column vector. The notation P0(0) means that matrix P is positive (semi) definite and P0(0) means that matrix P is negative (semi) definite. · represents the Euclidean norm.

Section snippets

Stability of nonlinear switched systems

A mathematical model for switched systems can be considered as follows:ẋ(t)=fσ(t)(x(t))where x(t)Rn, fi:RnRn are vector fields and σ:[0,){0,1,,m} is a piecewise constant function of time which is called the switching signal. Any operation mode is related to a specific subsystem asẋ(t)=fi(x(t))i{0,1,,m}and the switching signal determines which subsystem is active at any time. It is usually assumed that the switching signal has a finite number of switches on any limited time interval,

Calculation of the critical trajectory

As mentioned, a switched system with an arbitrary switching signal has an infinite number of responses for any initial condition; so, a basic idea is based on determination of the critical trajectory. If this solution is convergent to the origin, then all other solutions are convergent and thus, the nonlinear switched system would be asymptotically stable and vice versa.

In this approach, finding the critical trajectory, x(t), is necessary. In the following subsection, an affine nonlinear

Main result

As mentioned in previous sections, with regard to Theorem 2 in Section 2, the “critical” trajectory, x(t), should be determined for stability of all trajectories starting at a given initial condition of nonlinear switched systems with an arbitrary switching signal. To characterize x(t), by utilizing Lemma 1, Problem 2 should be solved. Then, according to Section 3, this problem can be restated as a “Input-affine and Constrained Infinite Time Nonlinear Optimal Control” problem, i.e.;

Find the

Simulation results

In this section, in order to show ability of the presented method in extracting the necessary and sufficient condition for stability of all possible trajectories starting at a given initial condition of switched systems, four examples are simulated. Comparing its generality and simplicity with other methods exhibits capabilities of the proposed approach.

Example 1

Consider a linear switched system with n=2 as follows [4]:ẋ(t)={f0(x),Mode=0f1(x),Mode=1where f0(x)=A0x,A0=(0121)f1(x)=A1x,A1=(01(2+k)1)

Conclusions

A new method for stability of all possible trajectories starting at a given initial condition of nonlinear switched systems under arbitrary switching is presented. While the method presented in [4] can be used only for stability analysis and acquires global stability property of two-dimension linear switched systems (analytically) and three-dimension linear switched systems (numerically), the proposed approach here can analyze analytically the stability property of all possible trajectories

Acknowledgment

The authors would like to thank members of Advanced Control Systems Laboratory of the University of Tehran for the fruitful discussion on the subject.

References (28)

  • Z. Li et al.

    Exponential stability analysis and stabilization of switched delay systems

    J. Frankl. Inst.

    (2015)
  • V. Manousiouthakis et al.

    On constrained infinite-time nonlinear optimal control

    Chem. Eng. Sci.

    (2002)
  • D. Liberzon

    Switching in Systems and Control

    (2003)
  • M. Margalioat

    Stability analysis of switched systems using variational principlesan introduction

    Automatica

    (2006)
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