Analysis and design of singular linear systems under actuator saturation and disturbances☆
Introduction
Singular linear systems have been drawing continual interest from control theorists for decades now (see, for example, [4], [8], [13], [15], [20] and the references therein). This is in part due to the many applications of singular systems in engineering, social and economic, and biological systems and in part due to the additional challenges that these systems present. For example, the algebraic constraints on system states that are inherent in singular systems lead to considerable complexities, such as impulses in system response.
The presence of actuator saturation imposes additional constraints on the analysis and design of singular systems. Addressing actuator saturation has been well recognized to be practically imperative yet theoretically challenging. This is reflected in the large body of literature on regular linear systems in the presence of actuator saturation (see, for example, [1], [11], [16], [29] and the references therein). This literature covers a wide range of control problems, including stabilization [17], [26], [28], [30], output regulation [5], [10], input output stability [2], [3], [18], the rejection of and disturbances [6], [7], [9], [23], [24], [25], [27], [31], [32], and stabilization in the presence of delays in the input [21], [22].
In comparison, analysis and design for singular linear systems subject to actuator saturation have been less studied. A few exceptions include [14], [19]. In particular, it is established in [14] that a singular linear system subject to actuator saturation can be semi-globally asymptotically stabilizable by linear state feedback if its reduced system under actuator saturation is semi-globally asymptotically stabilized by linear feedback. The latter was earlier established in [17], [16] to be possible under the assumptions that the open-loop system is semi-stable (i.e., with all its poles located in the closed left-half plane) and stabilizable in the usual linear system sense. More recently, we in [19] studied the stabilization problem for general, not necessarily semi-stable, singular linear systems by a saturated linear feedback law. A set of conditions were established under which the closed-loop system is regular and impulse free within an elliptical cylinder, and its cross-sectional ellipsoid is contractively invariant. With these set invariance conditions, which can be expressed in terms of linear matrix inequalities, the stabilization problem can be formulated and solved as an optimization problem with LMI constraints and the objective of maximizing the contractively invariant ellipsoid.
This paper continues from our earlier work [19] on the stabilization of singular linear systems under actuator saturation and carries out an analysis of the gain and performance for such systems. We will carry out such an analysis as follows. We will first introduce the notion of bounded state stability (BSS) with respect to the influence of or disturbances. We will then characterize the bounded state stability in terms of linear matrix inequalities (LMIs). We will measure the disturbance tolerance capability of the system by the bound on the or norms of the disturbances under which the system remains bounded state stable and the disturbance rejection capability by the restricted gain from the disturbance to the system output or norm of the system output. Based on the BSS conditions, assessment of the disturbance tolerance and rejection capabilities of the system under a given state feedback law is formulated and solved as LMI constrained optimization problems. By viewing the feedback gain as an additional variable, these optimization problems can be readily adapted for control design. Our analysis and design reduce to the existing results for regular linear systems in the degenerate case where the singular linear system reduces to a regular system [7], and to the existing results for singular systems in the absence of actuator saturation or when the disturbance is weak enough to not cause saturation [20].
More specifically, we consider the following singular linear systems subject to actuator saturation and disturbances, where is the state, is the control input, is the disturbance, is the output, and is a vector valued standard saturation function defined as Here we have slightly abused the notation by using to stand for both scalar valued and vector valued saturation functions. Also, we have assumed, without loss of generality, unity saturation level. Non-unity saturation levels can be absorbed into the matrices and . We also assume that the disturbance belongs to one of the following two classes for some positive number .
Our analysis is aimed to answer two questions. First, what is the maximal value of such that, under the given feedback law , the state remains bounded for all (or )? Such a question is related to the disturbance tolerance of the closed-loop system. When such a maximal , say , has been determined, our second question to answer is what the restricted gain (or norm of the output of the closed-loop system) is over a given (or ) with an . This second question is related to the disturbance rejection capability of the closed-loop system. We will also be interested in the design of feedback gain to increase the disturbance tolerance and rejection capabilities of the closed-loop system.
The remainder of the paper is organized as follows. Section 2 defines and characterizes the bounded state stability. Based on the characterization of bounded state stability, the disturbance tolerance and disturbance rejection capabilities are analyzed in Sections 3 Disturbance tolerance, 4 , respectively. Some numerical and simulation results are presented in Section 5 to illustrate the effectiveness of our analysis. Section 6 concludes the paper.
Section snippets
Bounded state stability
Consider a singular linear system under a saturated linear state feedback (1). Let . Without loss of generality, assume that are in the following form, We will also partition the state vector accordingly as
For the singular linear system under actuator saturation (1) with the matrices in the form of (4), we are interested in the boundedness of its state trajectories. Such a system is
Disturbance tolerance
The disturbance tolerance capability of the closed-loop system under a given feedback gain can be measured by the largest , denoted as , such that the closed-loop trajectory under any disturbance or remains bounded. We will denote the maximal value of over the choice of as , i.e., .
gain and performance
The singular system subject to actuator saturation (1) is said to have a restricted gain less than or equal to if, for and any compatible with , On the other hand, the output of the system has its norm less than or equal to if, for and any compatible with ,
The following theorem characterizes the conditions under which the system has a restricted gain less than or equal to or the
Example 1: disturbance tolerance and rejection
Let us consider system (1) with and .
Let the set of initial conditions be specified by . Then, solving the optimization problem (22) leads to the following results: with and Shown in Fig. 1 is the ellipsoid and a trajectory starting from a point on the boundary of the
Conclusions
This paper considered the problems of disturbance tolerance and rejection for singular linear systems subject to actuator saturation and disturbances. Under a given feedback, we developed methods for assessing the disturbance tolerance and rejection capability of the closed-loop system. Our analysis methods can be readily adapted for control design. Our analysis and design reduce to the existing results for regular linear systems in the degenerate case where the singular linear system
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Work supported in part by National Natural Science Foundation of China under the Overseas Young Investigator Program (Class B: Overseas Collaboration).