Stability analysis of singularly perturbed control systems with actuator saturation☆
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 saturationwhere denotes the singular perturbation parameter, and are the state variables, is the control input. and B2 are constant matrices of appropriate dimensions, and is a componentwise saturation map defined by
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 . 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 M−T denotes the transpose of the inverse matrix of M and denotes ith row of M. For a square matrix M, denote . Let be a positive definite matrix. An ellipsoid is defined as .
Section snippets
Problem formulation
System (1) can be written as the following compact formwhere
Under the state feedback controller , the closed-loop system can be described by
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 contained in the basin of attraction, such that for any ε less than or equal to ε0, the trajectories starting from
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
In [24], the ε-dependent state feedback controllerwas designed, under which the ellipsoid with is an estimate of the basin of attraction for any .
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.
References (27)
- et al.
Fault tolerant control for singular systems with actuator saturation and nonlinear perturbation
Automatica
(2010) - et al.
Output feedback gain scheduled control of actuator saturated linear systems with applications to the spacecraft rendezvous
J. Frankl. Inst.
(2014) Characterization and computation for the bound in linear time-invariant singularly perturbed systems
Syst. Control Lett.
(1988)New stability/performance results for singularly perturbed systems
Automatica
(1996)- et al.
Stabilization bound of discrete two-time-scale systems
Syst. Control Lett.
(1992) - et al.
Semi-global exponential stabilization of linear systems subject to input saturation via linear feedbacks
Syst. Control Lett.
(1993) - et al.
An analysis and design method for linear systems subject to actuator saturation and disturbance
Automatica
(2002) - et al.
Stabilization of linear systems with distributed input delay and input saturation
Automatica
(2012) - et al.
Stabilization bound of singularly Perturbed systems subject to actuator saturation
Automatica
(2013) - et al.
Soft variable structure controller design for singular systems
J. Frankl. Inst.
(2015)
Control Systems with Actuator SaturationAnalysis and Design
Gain scheduled control of linear systems subject to actuator saturation with application to spacecraft rendezvous
IEEE Trans. Control Syst. Technol.
Singular Perturbation Methods in Control: Analysis and Design
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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).