Exponential Stabilizability of Switched Systems with Polytopic Uncertainties

Abstract

The exponential stabilizability of switched nonlinear systems with polytopic uncertainties is explored by employing the methods of nonsmooth analysis and the minimum quadratic Lyapunov function. The switchings among subsystems are dependent on the directional derivative along the vertex directions of subsystems. In particular, a sufficient condition for exponential stabilizability of the switched nonlinear systems is established considering the sliding modes and the directional derivatives along sliding modes. Furthermore, the matrix conditions of exponential stabilizability are derived for the case of switched linear system and the numerical example is given to show the validity of the synthesis results.

1. Introduction

In the last two decades, there has been increasing research in stability analysis and control design for switched systems and many results have been studied on stability and stabilizability problems for various types of switched systems [1–9]. Many interesting results for different kinds of problems of switched systems can be found in some books [10, 11]. The motivation of studying the switched system is out of the fact that many practical systems are inherently multimodal, and several dynamical subsystems are required to describe their behavior which may depend on various environmental factors. Sometimes there are some systems that cannot be asymptotically stabilized by a single continuous feedback control rule but can be stabilized by switching rule [3].

It is very important to investigate switched systems which contain uncertainties due to modelling errors, aging disturbance, and complex environment in realistic problems. One important type of uncertain systems are switched systems which are composed of polytopic uncertainty subsystems. As pointed out in [12], polytopic uncertainties exist in many real systems, and most of the uncertain control systems can be approximated by systems with polytopic uncertainties. The polytopic uncertain systems are less conservative than systems with norm bounded uncertainties [13]. Recently, the stability and stabilization problems for both continuous-time and discrete-time switched systems with polytopic uncertainties are investigated in [14, 15]. In particular, the paper [14] investigated the quadratic stabilizability problem via state feedback, and provided sufficient conditions to be quadratically stabilized for the switched systems which being composed of two subsystems. More recently, necessary and sufficient conditions for continuous-time case via state feedback are proved in the paper [16, 17].

Lyapunov theory is a very important approach to stability analysis or stabilization for switched systems, and the construction of Lyapunov functions is one of the fundamental problem in system theory. The most popular types of Lyapunov function are quadratic functions, piecewise-linear functions, and piecewise affine functions [9, 18–20]. Furthermore, there are some results for the systems with time-delay [21, 22]. Paper [22] investigates the stability and stabilization properties of linear switched time delay dynamic systems subject to; in general, multiple uncommensurate known internal point delays based on Lyapunovs stability analysis via appropriate Krasovsky-Lyapunovs functionals and the related stability study is performed to obtain both delay independent and delay dependent results. In [9], the authors studied the exponential stabilization problem for discrete-time switched linear systems based on a special control Lyapunov function which can make the hybrid-control policy of the related switched system be derived analytically and computed efficiently. Hu and Lin [23] proposed a composed quadratic Lyapunov function for constrained control systems. The composite quadratic Lyapunov function turned out to be very effective in dealing with some constrained control systems as well as a class of more general nonlinear systems [4, 24–26].

Motivated by the methods in [4], we consider the exponential stabilizability problems of switched nonlinear systems with polytopic uncertain subsystems using the composite quadratic functions. We mainly use the minimum quadratic function. The contributions in our work is that, the analysis of possible sliding modes in the switched systems and employing the nonquadratic Lyapunov functions to reduce the conservatism in references [14, 16, 17]. We extend the main result of [14] and establish the matrix conditions of exponential stabilizability for the switched linear systems.

The remainder of this paper is organized as following. In Section 2, we briefly review some conceptions especially the switched systems with polytopic uncertainties and the minimum quadratic functions. Then, in Section 3, the exponential stabilizability results based on the minimum functions are established, and the conditions as matrix inequalities are derived for the switched linear system. Moreover an example is given to demonstrate the effectiveness of our method. Finally, concluding remarks are given in Section 4 and some definitions or conclusions from nonsmooth analysis are listed in the appendix.

Notations. We use 𝐼[𝑘₁,𝑘₂] to represent the set of integers {𝑘₁,𝑘₁+1,𝑘₁+2,…,𝑘₂};  ∇𝑉(𝑥) denotes the gradient of 𝑉 at 𝑥 and 𝜕𝑉(𝑥) the subdifferential of 𝑉 at 𝑥; 𝑉̇(𝑥;𝜁) stands for the one-sided directional derivative of 𝑉 at 𝑥 along 𝜁; co{𝑆} denotes the convex hull of a set 𝑆.

2. Switched Systems and the Minimum Quadratic Function

Switched system is a hybrid dynamical system composed by a family of continuous-time or discrete-time subsystems with a rule orchestrating the switching between the subsystems [1, 2]. In this paper, we consider the time-invariant switched nonlinear systems 𝑥̇(𝑡)=𝑓_{𝜎(𝑥,𝑡)}(𝑥(𝑡)), (2.1) where 𝑥(𝑡)∈𝑅^𝑛 is the state vector and 𝜎(𝑥,𝑡) is the switching rule defined by 𝜎(𝑥,𝑡)∶𝑅^𝑛×𝑅⁺→𝐼[1,𝑁], 𝑅⁺ denotes nonnegative real numbers. Therefore, the switched system is composed of 𝑁 continuous time subsystems which are expressed as 𝑥̇(𝑡)=𝑓_𝑖(𝑥(𝑡)),𝑖∈𝐼[1,𝑁]. (2.2)

We assume that all subsystems are uncertain systems of polytopic type described as 𝑓_𝑖=^𝑁𝑖_𝑗=1𝜆_𝑖𝑗𝑓_𝑖𝑗,𝑖∈𝐼[1,𝑁], (2.3) where 𝑓_𝑖𝑗∶𝑅^𝑛→𝑅^𝑛, 𝑗∈𝐼[1,𝑁_𝑖] are known and in locally Lipschitzian. They are called vertex directions of the subsystems. 𝑁_𝑖, 𝑖∈𝐼[1,𝑁] are integers. 𝜆_𝑖𝑗≥0, 𝑗∈𝐼[1,𝑁_𝑖] are polytopic uncertain parameters for 𝑖∈𝐼[1,𝑁] and satisfy ∑^𝑁𝑖_𝑗=1𝜆_𝑖𝑗=1.

Define the minimum function 𝑉_min(𝑥)=min𝑉_𝑗(𝑥)∣𝑗∈𝐼[1,𝐽], (2.4) where {𝑉₁,𝑉₂,…,𝑉_𝐽} is a set of differentiable, positive definite, and radially unbounded functions, which are zero at 𝑥=0.

The function 𝑉_min(𝑥) is positive definite and homogeneous of degree two [23]. It is established from the theory of nonsmooth analysis that the function is not convex and not differentiable everywhere even if the functions 𝑉_𝑖(𝑥) are all differentiable.

For a vector field 𝑥̇=ℎ(𝑥), we have 𝑉(𝑥(𝑡+Δ𝑡))≈𝑉(𝑥+ℎ(𝑥)Δ𝑡) for Δ𝑡>0 small enough. Thus directional derivative 𝑉̇(𝑥;ℎ(𝑥)) measures the time derivative of 𝑉 at 𝑥 along the trajectory. For this reason the exponential stabilizability of the switched systems in our paper can utilize the directional derivative of the minimum quadratic function.

3. Exponential Stabilizability of the Switched Systems

There are many methods to construct switching rule for the stabilizability of switched systems [4, 14, 27]. Just as the paper [4], our switching rule is constructed by employing the minimum function and its directional derivative. We select the minimum function (2.4), which is constructed from 𝐽 general functions 𝑉_𝑗(𝑥), where each 𝑉_𝑗(𝑥) is continuously differentiable.

Now we consider the stability of a closed loop system 𝑥̇=𝑓_{𝜎(𝑥,𝑡)}(𝑥) by examining the directional derivative of 𝑉_min along its trajectories.

Proposition 3.1. Consider the closed loop systems (2.1). Construct the switching rule as follows: 𝜎(𝑥)=argmin_{[]𝑖∈𝐼1,𝑁}max_{𝑗∈𝐼1,𝑁𝑖}𝑉̇_min𝑥;𝑓_𝑖𝑗(𝑥). (3.1) Define 𝜇(𝑥)∶=min_{𝑖∈𝐼[1,𝑁]}max_{𝑗∈𝐼1,𝑁𝑖}𝑉̇_min𝑥;𝑓_𝑖𝑗(𝑥). (3.2) When 𝑥 is not in a sliding mode, 𝑉̇_min(𝑥;𝑥̇)=𝑉̇_min(𝑥;𝑓_𝜎(𝑥)(𝑥))≤𝜇(𝑥).

Proof. When 𝑥 is not in a sliding mode, then 𝑥̇ may equal to a 𝑓_𝑖(𝑥). 𝑖 is chosen by the switching rule 𝜎(𝑥). Then 𝑉̇_min(𝑥;𝑥̇)=𝑉̇_min𝑥;^𝑁𝑖_𝑗=1𝜆_𝑖𝑗𝑓_𝑖𝑗(𝑥)=^𝑁𝑖_𝑗=1𝜆_𝑖𝑗𝑉̇_min𝑥;𝑓_𝑖𝑗(𝑥)≤max_{𝑗∈𝐼1,𝑁𝑖}𝑉̇_min𝑥;𝑓_𝑖𝑗(𝑥)=min_{[]𝑖∈𝐼1,𝑁}max_{𝑗∈𝐼1,𝑁𝑖}𝑉̇_min𝑥;𝑓_𝑖𝑗(𝑥)=𝜇(𝑥). (3.3) The proof is completed.

From a practical point of view, a sliding mode is very important and it is often unavoidable in the switched systems. So we need to pay particular attention to sliding modes. When there is a sliding mode, we assume that 𝐼_sm be the set of indices of subsystems involved in the sliding mode. Then 𝑥̇ may not equal to any of 𝑓_𝑖(𝑥), instead 𝑥(𝑡) stay on the switching surface and 𝑥̇ is a convex combination of those 𝑓_𝑖(𝑥)’s, that is, 𝑥̇=_{𝑖∈𝐼sm}𝛼_𝑖𝑓_𝑖(𝑥),𝑖∈𝐼_sm, (3.4) where 𝛼_𝑖≥0, ∑_{𝑖∈𝐼sm}𝛼_𝑖=1 (see [28]).

Proposition 3.2. Consider the closed loop systems (2.1) with the switching rule (3.1). Assuming a sliding mode involving subsystems 𝑥̇=𝑓_𝑖(𝑥), 𝑖∈𝐼_sm, then for each 𝑥₀ in this sliding mode, 𝑉̇_min𝑥₀;𝑓_𝑖𝑥₀≤𝜇𝑥₀,∀𝑖∈𝐼_sm. (3.5) Moreover, along the sliding direction ∑_{𝑖∈𝐼sm}𝛼_𝑖𝑓_𝑖(𝑥₀), one has 𝑉̇_min𝑥₀;_{𝑖∈𝐼sm}𝛼_𝑖𝑓_𝑖𝑥₀≤𝜇𝑥₀, (3.6) where 𝛼_𝑖≥0, ∑_{𝑖∈𝐼sm}𝛼_𝑖=1.

Proof. If 𝑉_min is differentiable at 𝑥₀, then there is an integer 𝑘 such that 𝑉_min(𝑥₀)=𝑉_𝑘(𝑥₀)<𝑉_𝑠(𝑥₀) for all 𝑠≠𝑘, and 𝑉̇_min(𝑥₀;𝑓_𝑖(𝑥₀))=𝑉̇_𝑘(𝑥₀;𝑓_𝑖(𝑥₀)). One has 𝑉̇_𝑘𝑥₀;𝑓_𝑖𝑥₀=^𝑁𝑖_𝑖=1𝜆_𝑖𝑗𝑉̇_𝑘𝑥₀;𝑓_𝑖𝑗𝑥₀≤max_{𝑗∈1,𝑁𝑖}𝑉̇_𝑘𝑥₀;𝑓_𝑖𝑗𝑥₀=min_{[]𝑖∈1,𝑁}max_{𝑗∈1,𝑁𝑖}𝑉̇_𝑘𝑥₀;𝑓_𝑖𝑗𝑥₀=𝜇𝑥₀,∀𝑖∈𝜎𝑥₀,∀𝑠∉𝜎𝑥₀. (3.7) For the structure of the switching rule (3.1), we have max_{𝑗∈𝐼[1,𝑁𝑖]}𝑉̇_𝑘𝑥₀;𝑓_𝑖𝑗𝑥₀<max_{𝑗∈𝐼[1,𝑁𝑠]}𝑉̇_𝑘𝑥₀;𝑓_𝑠𝑗𝑥₀,∀𝑖∈𝜎𝑥₀,𝑠∉𝜎𝑥₀. (3.8)
Since 𝑉_𝑘(𝑥;𝑓_𝑖(𝑥)) is continuous in 𝑥₀, max_{𝑗∈[1,𝑁𝑖]}{𝑉_𝑘(𝑥;𝑓_𝑖𝑗(𝑥))} is continuous in 𝑥₀ too. There is a small neighborhood of 𝑥₀, say 𝑈(𝑥₀), where max_{𝑗∈𝐼[1,𝑁𝑖]}𝑉̇_𝑘𝑥;𝑓_𝑖𝑗(𝑥)<max_{𝑗∈𝐼[1,𝑁𝑠]}𝑉̇_𝑘𝑥;𝑓_𝑠𝑗(𝑥),∀𝑖∈𝜎𝑥₀,𝑠∉𝜎𝑥₀, (3.9) which implies 𝜎(𝑥)⊂𝜎(𝑥₀). We can conclude that if 𝑠∉𝜎(𝑥₀),𝑓_𝑠(𝑥) will not be chosen by the switching rule for 𝑥∈𝑈(𝑥₀), that is 𝐼_sm⊂𝜎(𝑥₀), and (3.5) comes from (3.7).
Moreover, along the sliding direction ∑_{𝑖∈𝐼sm}𝛼_𝑖𝑓_𝑖(𝑥₀) we have 𝑉̇_min𝑥₀;_{𝑖∈𝐼sm}𝛼_𝑖𝑓_𝑖𝑥₀=∇𝑉_𝑘𝑥₀^𝑇⋅_{𝑖∈𝐼sm}𝛼_𝑖^𝑁𝑖_𝑗=1𝜆_𝑖𝑗𝑓_𝑖𝑗𝑥₀=_{𝑖∈𝐼sm}𝛼_𝑖^𝑁𝑖_𝑗=1𝜆_𝑖𝑗∇𝑉_𝑘𝑥₀^𝑇𝑓_𝑖𝑗𝑥₀=_{𝑖∈𝐼sm}𝛼_𝑖^𝑁𝑖_𝑗=1𝜆_𝑖𝑗𝑉̇_𝑘𝑥₀;𝑓_𝑖𝑗𝑥₀≤_{𝑖∈𝐼sm}𝛼_𝑖𝜇𝑥₀=𝜇𝑥₀. (3.10)
Now we consider the case that 𝑉_min is not differentiable at 𝑥₀; that is, 𝐽_min(𝑥₀) has two or more integers. Suppose that 𝐽_min(𝑥₀)=𝐼[1,𝑘], then 𝑉₁(𝑥₀)=𝑉₂(𝑥₀)=⋯=𝑉_𝑘(𝑥₀)<𝑉_𝑠(𝑥₀) for all 𝑠>𝑘. When 𝑘=2, then 𝑉₁(𝑥₀)=𝑉₂(𝑥₀)<𝑉_𝑗(𝑥₀), for all 𝑗>0. Furthermore, there must exist two 𝑓_𝑖’s, say, 𝑓_𝑖1, and 𝑓_𝑖2. In this situation 𝐼_sm={𝑖₁,𝑖₂}.
Defining Φ_𝑗=𝑥∈𝑅^𝑛∣𝑉_𝑗(𝑥)<𝑉_𝑘(𝑥),∀𝑘≠𝑗, (3.11) and within the vicinity of 𝑥₀, we have two cases.(a)If 𝑥∈Φ₁, then 𝑖₁∈𝜎(𝑥) and a trajectory of 𝑥̇=𝑓_𝑖1(𝑥) would enter Φ₂ from Φ₁.(b)If 𝑥∈Φ₂, then 𝑖₂∈𝜎(𝑥) and a trajectory of 𝑥̇=𝑓_𝑖2(𝑥) would enter Φ₁ from Φ₂.
We interpreter cases (a) and (b) below. Case (a) implies that 𝑓_𝑖1(𝑥) is chosen by the switching law in Φ₁, thus one has max_{𝑗∈𝐼[1,𝑁𝑖1]}𝑉̇₁𝑥;𝑓_𝑖1𝑗(𝑥)≤max_{𝑗∈𝐼1,𝑁𝑠}𝑉̇₁𝑥;𝑓_𝑠𝑗(𝑥) (3.12) for all 𝑠∈𝐼[1,𝑁]. In particular, one has max_{𝑗∈𝐼[1,𝑁𝑖1]}𝑉̇₁𝑥;𝑓_𝑖1𝑗(𝑥)≤max_{𝑗∈𝐼[1,𝑁𝑖2]}𝑉̇₁𝑥;𝑓_𝑖2𝑗(𝑥). (3.13)
Taking 𝑥→𝑥₀, then max_{𝑗∈𝐼[1,𝑁𝑖1]}𝑉̇₁𝑥₀;𝑓_𝑖1𝑗𝑥₀≤max_{𝑗∈𝐼1,𝑁𝑠}𝑉̇₁𝑥₀;𝑓_𝑠𝑗𝑥₀,∀𝑠. (3.14) When 𝑠=𝑖₂, we have max_{𝑗∈𝐼[1,𝑁𝑖1]}𝑉̇₁𝑥₀;𝑓_𝑖1𝑗𝑥₀≤max_{𝑗∈𝐼[1,𝑁𝑖2]}𝑉̇₁𝑥₀;𝑓_𝑖2𝑗𝑥₀. (3.15) To ensure that a trajectory of 𝑥̇=𝑓_𝑖1(𝑥) would enter Φ₂ from Φ₁, we must have max_{𝑗∈𝐼[1,𝑁𝑖1]}𝑉̇₂𝑥₀;𝑓_𝑖1𝑗𝑥₀≤max_{𝑗∈𝐼[1,𝑁𝑖1]}𝑉̇₁𝑥₀;𝑓_𝑖1𝑗𝑥₀. (3.16) Similarly, from case (b), we have max_{𝑗∈𝐼[1,𝑁𝑖2]}𝑉̇₂𝑥₀;𝑓_𝑖2𝑗𝑥₀≤max_{𝑗∈𝐼1,𝑁𝑠}𝑉̇₂𝑥₀;𝑓_𝑠𝑗𝑥₀,∀𝑠,max_{𝑗∈𝐼[1,𝑁𝑖2]}𝑉̇₂𝑥₀;𝑓_𝑖2𝑗𝑥₀≤max_{𝑗∈𝐼[1,𝑁𝑖1]}𝑉̇₂𝑥₀;𝑓_𝑖1𝑗𝑥₀,max_{𝑗∈𝐼[1,𝑁𝑖2]}𝑉̇₁𝑥₀;𝑓_𝑖2𝑗𝑥₀≤max_{𝑗∈𝐼[1,𝑁𝑖2]}𝑉̇₂𝑥₀;𝑓_𝑖2𝑗𝑥₀. (3.17)(3.18)(3.19) Combining the inequalities from (3.15) to (3.19), one has max_{𝑗∈𝐼[1,𝑁𝑖1]}𝑉̇₁𝑥₀;𝑓_𝑖1𝑗𝑥₀≤max_{𝑗∈𝐼[1,𝑁𝑖2]}𝑉̇₁𝑥₀;𝑓_𝑖2𝑗𝑥₀≤max_{𝑗∈𝐼[1,𝑁𝑖2]}𝑉̇₂𝑥₀;𝑓_𝑖2𝑗𝑥₀≤max_{𝑗∈𝐼[1,𝑁𝑖1]}𝑉̇₂𝑥₀;𝑓_𝑖1𝑗𝑥₀≤max_{𝑗∈𝐼[1,𝑁𝑖1]}𝑉̇₁𝑥₀;𝑓_𝑖1𝑗𝑥₀. (3.20) Thus, the following inequalities: max_{𝑗∈𝐼[1,𝑁𝑖1]}𝑉̇₁𝑥₀;𝑓_𝑖1𝑗𝑥₀=max_{𝑗∈𝐼[1,𝑁𝑖2]}𝑉̇₁𝑥₀;𝑓_𝑖2𝑗𝑥₀=max_{𝑗∈𝐼[1,𝑁𝑖2]}𝑉̇₂𝑥₀;𝑓_𝑖2𝑗𝑥₀=max_{𝑗∈𝐼[1,𝑁𝑖1]}𝑉̇₂𝑥₀;𝑓_𝑖1𝑗𝑥₀, (3.21) are valid. They are all equal to 𝜇(𝑥₀). So we have 𝑉̇_min𝑥₀;𝑓_𝑖1𝑥₀=min𝑉̇₁𝑥₀;𝑓_𝑖1𝑥₀,𝑉̇₂𝑥₀;𝑓_𝑖1𝑥₀=min𝑉̇₁𝑥₀;^𝑁𝑖1_𝑗=1𝜆_𝑖1𝑗𝑓_𝑖1𝑗𝑥₀,𝑉̇₂𝑥₀;^𝑁𝑖1_𝑗=1𝜆_𝑖1𝑗𝑓_𝑖1𝑗𝑥₀≤minmax_{𝑗∈𝐼[1,𝑁𝑖1]}𝑉̇₁𝑥₀;𝑓_𝑖1𝑗𝑥₀,max_{𝑗∈𝐼[1,𝑁𝑖1]}𝑉̇₂𝑥₀;𝑓_𝑖1𝑗𝑥₀=𝜇𝑥₀. (3.22) Similarly, we obtain 𝑉̇_min(𝑥₀;𝑓_𝑖2(𝑥₀))≤𝜇(𝑥₀). Thus the relation (3.5) follows for 𝐼_sm={𝑖₁,𝑖₂}.
The above argument can be extended to the case 𝑘>2. Then 𝑉₁,𝑉₂,…,𝑉_𝑘 are involved in the sliding motion with corresponding indices 𝑖₁,𝑖₂,…,𝑖_𝑘, with 𝑓_𝑖1(𝑥₀) pointing from Φ₁ to Φ₂,…,𝑓_𝑖𝑘(𝑥₀) pointing from Φ_𝑘 to Φ₁. Similar procedure can be used to derive the conclusion extended from (3.21). max_{𝑗∈𝐼[1,𝑁𝑖𝑠]}𝑉̇_𝑠𝑥₀;𝑓_𝑖𝑠𝑗𝑥₀=max_{𝑗∈𝐼[1,𝑁𝑖𝑡]}𝑉̇_𝑙𝑥₀;𝑓_𝑖𝑡𝑗𝑥₀=𝜇𝑥₀,∀𝑠,𝑡,𝑙∈𝐼[1,𝑘]. (3.23) Thus we have 𝑉̇_min𝑥₀;𝑓_𝑖1𝑥₀=min𝑉̇_𝑡𝑥₀;𝑓_𝑖1𝑗𝑥₀,𝑡∈𝐼[1,𝑘]=min^𝑁𝑖1_𝑗=1𝜇_𝑖1𝑗𝑉̇_𝑡𝑥₀;𝑓_𝑖1𝑗𝑥₀,𝑡∈𝐼[1,𝑘]≤minmax_{𝑗∈𝐼[1,𝑁𝑖1]}𝑉̇_𝑡𝑥₀;𝑓_𝑖1𝑥₀[𝑥,𝑡∈𝐼1,𝑘]=𝜇₀. (3.24) Similarly, we have 𝑉̇_min(𝑥₀;𝑓_𝑖2(𝑥₀))≤𝜇(𝑥₀),…,𝑉̇_min(𝑥₀;𝑓_𝑖𝑘(𝑥₀))≤𝜇(𝑥₀). Thus the relation (3.5) is obtained for 𝐼_sm={𝑖₁,𝑖₂,…,𝑖_𝑘}.
Now we consider there are three elements in the sliding direction. Let 𝜈₁, 𝜈₂, 𝜈₃ be positive numbers, such that 𝜈₁+𝜈₂+𝜈₃=1. Define 𝜉=(𝜈₁𝑓_𝑖1+𝜈₂𝑓_𝑖2+𝜈₃𝑓_𝑖3)(𝑥₀). Since sliding mode stays in the set where 𝑉₁(𝑥)=𝑉₂(𝑥)=𝑉₃(𝑥), we have 𝑉̇₁𝑥₀;𝜉=𝑉̇₂𝑥₀;𝜉=𝑉̇₃𝑥₀;𝜉=𝑉̇_min𝑥₀,𝑉;𝜉₁𝑥₀=𝑉₂𝑥₀=𝑉₃𝑥₀=𝑉_min𝑥₀. (3.25) For Δ𝑡>0, let 𝑥₁=𝑥₀+Δ𝑡𝜈₁𝑓_𝑖1(𝑥₀), 𝑥₂=𝑥₀+Δ𝑡(𝜈₁𝑓_𝑖1(𝑥₀)+𝜈₂𝑓_𝑖2(𝑥₀)), and 𝑥₃=𝑥₀+Δ𝑡𝜉. Due to the sliding motion for Δ𝑡 sufficient small, 𝑥₁∈Φ₂, 𝑥₂∈Φ₃ and by (3.25), 𝑉_𝑗(𝑥₃)−𝑉_min(𝑥₃)=𝑜(Δ𝑡), 𝑗=1,2,3: 𝑉̇_min𝑥₀;𝜉=lim_Δ𝑡↓0𝑉_min𝑥₀+Δ𝑡𝜉−𝑉_min𝑥₀Δ𝑡=lim_Δ𝑡↓0𝑉_min𝑥₃−𝑉_min𝑥₀Δ𝑡=lim_Δ𝑡↓0𝑉_min𝑥₃−𝑉_min𝑥₂Δ𝑡+lim_Δ𝑡↓0𝑉_min𝑥₂−𝑉_min𝑥₁Δ𝑡+lim_Δ𝑡↓0𝑉_min𝑥₁−𝑉_min𝑥₀Δ𝑡≤lim_Δ𝑡↓0𝑉₃𝑥₃−𝑉₃𝑥₂Δ𝑡+lim_Δ𝑡↓0𝑉₂𝑥₂−𝑉₂𝑥₁Δ𝑡+lim_Δ𝑡↓0𝑉₁𝑥₁−𝑉₁𝑥₀.Δ𝑡 (3.26) Since each function 𝑉_𝑗, 𝑗=1,2,3 is continuously differentiable, then 𝑉̇_min𝑥₀;𝜈₁𝑓_𝑖1𝑥₀+𝜈₂𝑓_𝑖2𝑥₀+𝜈₃𝑓_𝑖3𝑥₀=𝑉̇_min𝑥₀;𝜉≤𝜈₁𝑉̇₁𝑥₀;𝑓_𝑖1𝑥₀+𝜈₂𝑉̇₂𝑥₀;𝑓_𝑖2𝑥₀+𝜈₃𝑉̇₃𝑥₀;𝑓_𝑖3𝑥₀=𝜈₁𝑉̇₁𝑥₀;^𝑁𝑖1_𝑗=1𝜈_𝑖1𝑗𝑓_𝑖1𝑗𝑥₀+𝜈₂𝑉̇₂𝑥₀;^𝑁𝑖2_𝑗=1𝜈_𝑖2𝑗𝑓_𝑖2𝑗𝑥₀+𝜈₃𝑉̇₃𝑥₀;^𝑁𝑖3_𝑗=1𝜈_𝑖3𝑗𝑓_𝑖3𝑗𝑥₀≤𝜈₁max_{𝑖∈𝐼[1,𝑁𝑖1]}𝑉̇₁𝑥₀;𝑓_𝑖1𝑗𝑥₀+𝜈₂max_{𝑖∈𝐼[1,𝑁𝑖2]}𝑉̇₂𝑥₀;𝑓_𝑖2𝑗𝑥₀+𝜈₃max_{𝑗∈𝐼[1,𝑁𝑖3]}𝑉̇₃𝑥₀;𝑓_𝑖3𝑗𝑥₀=𝜇𝑥₀. (3.27) Above step can be extended to the case where 𝐼_sm has more than three elements. Therefore, the inequality (3.6) is satisfied and the proof is completed.

To summarize Propositions 3.1 and 3.2 we have the result that if 𝑥 is not in a sliding mode, then 𝑥̇=𝑓_𝜎(𝑥)(𝑥) and 𝑉̇_min(𝑥;𝑥̇)=𝑉̇_min(𝑥;𝑓_𝜎(𝑥)(𝑥))≤𝜇(𝑥). If 𝑥 is in a sliding mode involving subsystems 𝑥̇=𝑓_𝑖(𝑥), 𝑖∈𝐼_sm, then there exist 𝛼_𝑖, 𝑖∈𝐼_sm, satisfying 0≤𝛼_𝑖≤1, and ∑_{𝑖∈𝐼sm}𝛼_𝑖=1, such that 𝑥̇=∑_{𝑖∈𝐼sm}𝛼_𝑖𝑓_𝑖(𝑥). We also have 𝑉̇_min(𝑥;𝑥̇)≤𝜇(𝑥) from (3.6). In view of these arguments, we can establish a sufficient condition of stability for the switched systems in terms of 𝜇(𝑥).

Theorem 3.3. For the switched system (2.1) under the switching rule (3.1), if there exists an 𝜂∈𝑅, such that min_{𝑖∈𝐼[1,𝑁]}max_{𝑗∈𝐼1,𝑁𝑖}𝑉̇_min𝑥;𝑓_𝑖𝑗(𝑥)=𝜇(𝑥)≤𝜂𝑉_min(𝑥) (3.28) for all 𝑥∈𝑅^𝑛⧵{0}, then 𝑉_min(𝑥(𝑡))≤𝑉_min(𝑥(0))𝑒^𝜂𝑡 (3.29) for every solution 𝑥(⋅).

Remark 3.4. This theorem indicates that when a minimum function is used, it is sufficient to use the directional derivatives along all vertex directions of subsystems to characterize stability of the switched systems with polytopic uncertainties, and the existence of sliding modes has no effect.
When we consider the the case of the switched linear systems, then the systems (2.1) turn to 𝑥̇(𝑡)=𝐴_{𝜎(𝑥,𝑡)}𝑥(𝑡),𝐴_𝑖=^𝑁𝑖_𝑗=1𝜆_𝑖𝑗𝐴_𝑖𝑗,𝑖∈𝐼[1,𝑁], (3.30)(3.31) where 𝐴_𝑖𝑗, 𝑗∈𝐼[1,𝑁_𝑖], 𝑖∈𝐼[1,𝑁] are constant matrices.
We can establish a matrix condition for exponential stabilizability of the systems.

Theorem 3.5. The switched linear system (3.30) is exponentially stabilizable via state switching if there exist real matrices 𝑃_𝑗=𝑃^𝑇_𝑗>0, 𝑗∈𝐼[1,𝐽], and real number 𝜂, 0≤𝜇_{𝑖𝑠𝑖𝑗}≤1, ∑^𝑁_𝑖=1𝜇_{𝑖𝑠𝑖𝑗}=1, 𝛽_𝑗𝑘≥0, 𝑗,𝑘∈𝐼[1,𝐽], 𝑠_𝑖∈𝐼[1,𝑁_𝑖], 𝑖∈𝐼[1,𝑁], such that ^𝑁_𝑖=1𝜇_{𝑖𝑠𝑖𝑗}𝐴_𝑖𝑠𝑖^𝑇𝑃_𝑗+𝑃_𝑗^𝑁_𝑖=1𝜇_{𝑖𝑠𝑖𝑗}𝐴_𝑖𝑠𝑖≤^𝐽_𝑘=1𝛽_𝑗𝑘𝑃_𝑗−𝑃_𝑘+𝜂𝑃_𝑗. (3.32)

Proof. Let 𝑉_min(𝑥)=min{𝑥^𝑇𝑃_𝑗𝑥∣𝑗∈𝐼[1,𝐽]}. Consider 𝑥≠0 and assume that 𝐽_min(𝑥)={1,2,…,𝐽₀} for a certain integer 𝐽₀≤𝐽. Then 𝑥^𝑇𝑃_𝑘𝑥=𝑉_min(𝑥) for 𝑘≤𝐽₀ and 𝑥^𝑇𝑃_𝑘𝑥>𝑉_min(𝑥) for 𝑘>𝐽₀. Hence 𝑥^𝑇(𝑃_𝑗−𝑃_𝑘)𝑥≤0, for all 𝑗≤𝐽₀. According to (3.32), the following inequalities: 2𝑥^𝑇𝑃_𝑗^𝑁_𝑖=1𝜇_{𝑖𝑠𝑖𝑗}𝐴_𝑖𝑠𝑖𝑥≤^𝐽_𝑘=1𝛽_𝑗𝑘𝑥^𝑇𝑃_𝑗−𝑃_𝑘𝑥+𝜂𝑥^𝑇𝑃_𝑗𝑥≤𝜂𝑥^𝑇𝑃_𝑗𝑥=𝜂𝑉_min(𝑥),∀𝑗≤𝐽₀, (3.33) hold.
Since 0≤𝜇_{𝑖𝑠𝑖𝑗}≤1, we have min2𝑥^𝑇𝑃_𝑗𝐴_𝑖𝑠𝑖𝑥∶𝑠_𝑖∈𝐼1,𝑁_𝑖,𝑖∈𝐼[1,𝑁]≤𝜂𝑉_min(𝑥),∀𝑗≤𝐽₀. (3.34) It follows that min_{𝑖∈𝐼[1,𝑁]}max_{𝑠𝑖∈𝐼1,𝑁𝑖}2𝑥^𝑇𝑃_𝑗𝐴_𝑖𝑠𝑖𝑥≤𝜂𝑉_min(𝑥),∀𝑗≤𝐽₀. (3.35) By Lemma A.1 in the appendix, min_{[]𝑖∈𝐼1,𝑁}max_{𝑠∈𝐼1,𝑁𝑖}𝑉̇_min𝑥;𝐴_𝑖𝑠𝑖𝑥=min_{[]𝑖∈𝐼1,𝑁}max_{𝑠𝑖∈𝐼1,𝑁𝑖}min𝑉̇_𝑗𝑥;𝐴_𝑖𝑠𝑖𝑥,𝑗∈𝐽_min(𝑥)=min_{[]𝑖∈𝐼1,𝑁}max_{𝑠𝑖∈𝐼1,𝑁𝑖}min2𝑥^𝑇𝑃_𝑗𝐴_𝑖𝑠𝑖𝑥,𝑗∈𝐽_min(𝑥)≤min_{[]𝑖∈𝐼1,𝑁}max_{𝑠𝑖∈𝐼1,𝑁𝑖}2𝑥^𝑇𝑃₁𝐴_𝑖𝑠𝑖𝑥≤𝜂𝑉_min(𝑥). (3.36)
Define the switching rule as 𝜎(𝑥)=argmin_{[]𝑖∈𝐼1,𝑁}max_{𝑗∈𝐼1,𝑁𝑖}𝑉̇_min𝑥;𝐴_𝑖𝑗𝑥. (3.37) Under this switching rule we have 𝑉_min(𝑥(𝑡))≤𝑉_min(𝑥(0))𝑒^𝜂𝑡 for every solution 𝑥(⋅) from Theorem 3.3. The proof is completed.

Remark 3.6. If 𝑃_𝑗=𝑃 for all 𝑗∈𝐼[1,𝐽] and 𝑁=2, then 𝑉_min can be reduced to a quadratic function and (3.32) can be reduced to the matrix inequalities 𝜇_𝑠𝑡𝐴_1𝑠+1−𝜇_𝑠𝑡𝐴_2𝑡^𝑇𝑃+𝑃𝜇_𝑠𝑡𝐴_1𝑠+1−𝜇_𝑠𝑡𝐴_2𝑡<𝜂₀𝑃. (3.38) If 𝜂₀=0, then this corresponds to the stability condition given in [14]. In this sense, Theorem 3.5 is the extension of the main result in [14].

Example 3.7. Consider the switched linear system (3.30) composed of two subsystems, where 𝐴₁₁=212−4,𝐴₁₂=,𝐴211−6₂₁=−4−102.1,𝐴₂₂=.−5−10.12 (3.39) The eigenvalues of 𝐴₁₁, 𝐴₁₂, 𝐴₂₁, 𝐴₂₂ are {2.3166,−4.3166}, {−6.1231,2.1231}, {−4,2.1}, and {−4.9857,1.9857}, respectively. They are all unstable. Therefore, neither subsystems is quadratically stable.
We turn to use 𝑉_min composed of two quadratic functions and minimize 𝜂 subject to (3.32). The minimal 𝜂 is given as −1.4752 when fix the parameters as following: 𝜇₁₁=0.5, 𝜇₁₂=0.5, 𝜇₂₁=0.4, 𝜇₂₂=0.6. That is to say a unified convergence rate is guaranteed. For verification, other parameters are provided as following: 𝑃₁=11.56080.46780.46782.1290,𝑃₂=,𝛽11.55910.47230.47232.1177₁₂=8.9,𝛽₂₁=5.8. (3.40)
Let us investigate the system state trajectory using the two special subsystems 𝐴₁=0.4𝐴₁₁+0.6𝐴₁₂=,𝐴211.4−5.2₂=0.1𝐴₂₁+0.9𝐴₂₂=.−4.9−10.092.01 (3.41) They are both unstable.
We suppose that the initial state is 𝑥₀=[3,4]^𝑇. According to the switched rule (3.37), the dynamic system can exponential stabilizability and the typical result is plotted in Figure 1, which show that the system state converge to zero very quickly.

Figure 1 

The state of the switched system in example.

4. Conclusion

In this paper, the exponential stabilizability of switched nonlinear systems with polytopic uncertainties is considered employing the methods of nonsmooth analysis and nonquadratic Lyapunov functions. The function is formed by taking the pointwise minimum of a family of quadratic functions. We establish the switching rule to stabilize the switched systems by utilizing the directional derivative along the vertex directions of subsystems. In this process, we take a lot of effort in examining the case that the control systems involving sliding modes. The matrix conditions for exponential stabilizability of the switched linear systems are also obtained. As numerical example demonstrate in this paper, our synthesis result is effective. Future efforts will be devoted to the switched systems with time delay.

Appendix

For convenience we briefly list some definitions and conclusions from nonsmooth analysis. One can refer to [29, 30] for more detail.

Suppose 𝑓 is defined from 𝑅^𝑛 to 𝑅=𝑅⋃{−∞,+∞}, and 𝑓(𝑥) is finite. The one-sided directional derivative of 𝑓 at 𝑥 in direction 𝜁 can be expressed by 𝑓̇(𝑥;𝜁)=lim_Δ𝑡↓0𝑓(𝑥+𝜁Δ𝑡)−𝑓(𝑥).Δ𝑡 (A.1)

Suppose 𝑓 is a convex function in 𝑅^𝑛, and finite at 𝑥. The set 𝜕𝑓(𝑥)=𝑥^∗∣𝑓(𝑧)≥𝑓(𝑥)+𝑥^∗𝑇(𝑧−𝑥),∀𝑧∈𝑅^𝑛 (A.2) is called the subdifferential of 𝑓 at 𝑥 and 𝑥^∗ is a subgradient of 𝑓 at 𝑥 [30]. For a convex function 𝑓 in 𝑅^𝑛, 𝑓 is differential at 𝑥₀ if and only if 𝜕𝑓(𝑥₀) has only one vector. In this case we have 𝜕𝑓(𝑥₀)=∇𝑓(𝑥₀).

If 𝑓 is locally Lipschitz near 𝑥 and 𝑆 is any set of Lebesgue measure 0 in 𝑅^𝑛. The set of points at which 𝑓 fails to be differentiable is denoted by Ω_𝑓. Then the generalized gradient of 𝑓 at 𝑥 in the sense of Clarke denoted by 𝜕_𝐶𝑓(𝑥) is defined as 𝜕_𝐶𝑓(𝑥)=colim_{𝑥𝑖→𝑥}∇𝑓𝑥_𝑖∣𝑥_𝑖∉𝑆,𝑥_𝑖∉Ω_𝑓. (A.3)

The following lemma can be obtained by combining some results from [29, 31].

Lemma A.1. Suppose 𝑥₀∈𝑅^𝑛. Then one has(1)For 𝜁∈𝑅^𝑛, the directional derivative of 𝑉_min at 𝑥₀ along 𝜁 is given by 𝑉̇_min𝑥₀;𝜁=min𝑉̇_𝑗𝑥₀;𝜁∣𝑗∈𝐽_min𝑥₀, (A.4) where the index set 𝐽_min(𝑥₀)={𝑗∈𝐼[1,𝐽]∣𝑉_𝑗(𝑥₀)=𝑉_min(𝑥₀)}.(2)If 𝑉_𝑗(𝑥)=𝑥^𝑇𝑃_𝑗𝑥,𝑗∈𝐼[1,𝐽], then 𝜕_𝐶𝑉_min(𝑥₀)=co{2𝑃_𝑗𝑥₀∣𝑗∈𝐽_min(𝑥₀)}.

Acknowledgment

This work was supported by the National Natural Science Foundation of China (under Grant: 10971187, 11071029, and 11201362).