Stability analysis of singularly perturbed control systems with actuator saturation

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Abstract

Stability of singularly perturbed systems (SPSs) with actuator saturation is investigated. A set invariance condition with guaranteed stability bound is proposed. Based on this condition, two optimization problems are formulated separately. One is used to get the best estimate of the stability bound with guaranteed basin of attraction, while the other is used to get the best estimate of the basin of attraction with guaranteed stability bound. The proposed algorithms to solve the optimization problems are independent of the singular perturbation parameter and thus well conditioned. Two examples are included to show the advantages and effectiveness of the proposed results.

Introduction

Actuator saturation is ubiquitous in practical engineering systems and can deteriorate the performance of control systems. Thus continual efforts have been devoted to address actuator saturation and fruitful results have been reported [1], [2], [3], [4]. Unfortunately, a direct application of these results to singularly perturbed systems (SPSs) may lead to ill-conditioned numerical issues because of the two-time-scale nature of the systems [5]. This paper will study stability of the following SPS with actuator saturation[ẋεż]=[A11A12A21A22][xz]+[B1B2]sat(u),where ε>0 denotes the singular perturbation parameter, xRn1 and zRn2 are the state variables, uRm is the control input. A11,A12,A21,A22,B1 and B2 are constant matrices of appropriate dimensions, and sat(·) is a componentwise saturation map RmRm defined bysat(uj)=sign(uj)min{1,|uj|},j=1,2,,m.

Stability problem of system (1) involves two indexes. One is the stability bound ε0, which is the upper bound for the singular perturbation parameter ε such that stability of the system preserves for all ε(0,ε0]. Stability bound characterizes the robustness of system stability with respect to the perturbation parameter ε. Many methods have been proposed to calculate stability bound of linear SPSs [6], [7], [8], [9]. To enlarge the stability bound, numerous controller design methods for SPSs without actuator saturation have been reported [10], [11], [12], [13], [14]. On the other hand, it is known that the global stability is difficult to achieve when system (1) is open-loop unstable [15], [16]. Thus the basin of attraction of the closed-loop system is also an important stability index. For control systems without two-time-scale behavior, many approaches to estimate or optimize the basin of attraction of the closed-loop system have been proposed by using Lyapunov stability theory and LaSalle׳s invariance principle [17], [18], [19].

Recently, analysis and design of system (1) have attracted much attention [20], [21], [22], [23], [24]. In [20], a composite stabilizing controller was designed by combining the reduced-order controllers for the slow and fast subsystems and a convex optimization problem was formulated to estimate the basin of attraction of SPSs. A full-order controller design method which is independent of system decomposition was proposed in [21]. In [22], [23], the so-called reduced-order adjoint systems were introduced under the assumption that the state feedback controller gain was given, by which a reduced-order method was proposed to estimate the basin of attraction of SPSs. The results in [20], [21], [22], [23] pay more attention to estimating the basin of attraction than computing the stability bound. They can guarantee the existence of the stability bound, but cannot determine the value of the stability bound. Thus in practice, the effectiveness of the controllers and the estimate of the basin of attraction have to be validated by trial and error. To deal with this problem, an ε-dependent controller was designed, which can achieve a desired stability bound and satisfactory estimate of the basin of attraction of SPSs [24]. However, since the controller and the estimate of the basin of attraction depends on the singular perturbation parameter ε, the results are not valid when ε is not available or undergoes an unknown shift.

The aim of this paper is to study the stability of system (1) under a given linear state feedback controller. A set invariance condition is established by using a ε-dependent Lyapunov function. The condition considers both the stability bound and basin of attraction simultaneously. It is shown that a larger stability bound will lead to a smaller basin of attraction. To cope with such a competition, two optimization problems are proposed to optimize one of the stability indexes by fixing the other. Compared with the existing results [20], [21], [22], [23], [24], the proposed methods have the following features: (1) the stability bound and basin of attraction are quantitatively addressed simultaneously; (2) the estimate of the basin of attraction is independent of the singular perturbation parameter ε; (3) the best estimate of one of the stability indexes which guaranteed the other is obtained.

The rest of this paper is organized as follows. In Section 2, the problems under consideration are formulated. The main results are proposed in Section 3. In Section 4, two examples are given to show the effectiveness and advantages of the proposed methods. Section 5 makes a conclusion of this paper.

Notations: The superscript T stands for matrix transpose. For a matrix M, the notation MT denotes the transpose of the inverse matrix of M and M(i) denotes ith row of M. For a square matrix M, denote He(M)M+MT. Let QRn×n be a positive definite matrix. An ellipsoid Ω(Q) is defined as Ω(Q){ηRn|ηTQη1}.

Section snippets

Problem formulation

System (1) can be written as the following compact formη̇=Aεη+Bεsat(u),where η=[xz]Rn,Aε=[A11A121εA211εA22],Bε=[B11εB2].

Under the state feedback controller u=Kη, the closed-loop system can be described byη̇=Aεη+Bεsat(Kη).

The aim is to evaluate the stability of the closed-loop system (4) quantitatively. Specifically, we will determine a stability bound ε0 and an ellipsoid Ω(Q) contained in the basin of attraction, such that for any ε less than or equal to ε0, the trajectories starting from Ω(Q)

Main results

A set invariance condition is derived, by which solutions to Problem 1, Problem 2 are proposed.

Examples

In this section, two examples are given to illustrate various features of the proposed methods and show their advantages.

Example 1

Consider the following system[ẋεż]=[5111][xz]+[01]sat(u).

In [24], the ε-dependent state feedback controlleru=[2.24966.2612][0.0071+0.8137ε1.0908ε1.09081.46910.3860ε]1[xz]was designed, under which the ellipsoid Ω(P(ε)) with P(ε)=[0.0071+0.8137ε1.0908ε1.0908ε1.4691ε0.3860ε2]1is an estimate of the basin of attraction for any ε(0,0.1].

It can be seen that the

Conclusion

This paper proposed a set invariance condition for SPSs with actuator saturation. By this condition, a bisectional search algorithm was formulated to get the best estimate of the stability bound with guaranteed basin of attraction. A convex optimization problem which can be solved by the existing tools was proposed to get the best estimate of the basin of attraction with guaranteed stability bound. The presented examples illustrated the advantages of the proposed methods.

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    This work was supported by the National Natural Science Foundation of China (61374043, 60904009, 61020106003), the China Scholarship Council (201406425030), the Jiangsu Provincial Natural Science Foundation of China (BK20130205), the China Postdoctoral Science Foundation funded project (2013M530278, 2014T70558), the Fundamental Research Funds for the Central Universities (2013QNA50, 2013RC10, 2013RC12) and the Natural Science Foundation of Liaoning Province (201202201).

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