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Static Output Feedback Control for Discrete-Time Switched Systems via Improved Path-Following Method
This paper focuses on the problems of static output feedback control and controller design for discrete-time switched systems. Based on piecewise quadratic Lyapunov functions and a new linearization method, new sufficient conditions for system stability and controller design are obtained. Then, an improved path-following algorithm is built to solve the problems. Finally, the merits and effectiveness of the proposed method are shown by two numerical examples.
Recently, there has been an increasing interest in the study of switched systems because a wide class of nonlinear systems are naturally written as switched systems . Moreover, many other types of nonlinear systems can also be modeled as switched systems approximately . Switched systems are a particular class of hybrid systems which are bounded together by a switching rule. Such systems can be used to describe a wide range of physical and engineering systems in practice .
There are a large number of literatures about the stability analysis and design of switched systems during the last few years [4–9]. For discrete-time systems, several sufficient conditions have been presented based on different Lyapunov functions [6, 7, 10, 11], which are different in the conservative level and in the numerical difficulties. Previous work has concentrated on the output feedback controller design methods for switched systems based on piecewise quadratic Lyapunov function [12, 13]. Lyapunov-based controller synthesis is formulated as a biconvex optimization problem which is nonconvex, NP-hard, and very expensive to solve globally . Although there exist some results in solving this problem that the corresponding conditions can be determined by checking a set of linear matrix inequalities (LMIs) [12, 13], most of them are very restrictive.
Path-following method, which is an effective method for solving the biconvex optimization problem, was proposed by Hassibi et al.  and employed to solve mixed control [16, 17] and other control problems . An improved path-following method  has enhanced the convergence and the performance of the algorithm. As a step-by-step method, implying linearization approach at its key step, it gradually shows enormous potential in solving control problems.
In this paper, the problem of static output feedback (SOF) control for discrete-time switched systems is studied. Based on piecewise quadratic Lyapunov functions [12, 13] and a new linearization method , the piecewise quadratic stability conditions are linearized around some points. As a result, less conservative conditions for system stability are derived. The problems of control design can be readily treated as well. Then, based on an improved path-following method, an iterative algorithm is built. Finally, two examples are given to show the merits and effectiveness of our work.
This paper is organized as follows. Section 2 is the problem formulation and preliminaries. Section 3 gives the SOF controller design for switched systems. Section 4 extends the method to SOF control. Section 5 provides two numerical examples to show the merits and effectiveness of the results and Section 6 concludes this paper.
Notation. denotes the -dimensional Euclidean space; the superscripts and denote the matrix inverse and transpose, respectively; means that is positive definite (positive semi-definite); is the spectral norm; the star denotes the symmetric term in a matrix; .
2. Problem Formulation and Preliminaries
Consider the discrete-time switched system:where , , and are the state, the control input, and the measured output, respectively; , , denotes a partition of the output space into a number of closed polyhedral regions. For future use, define a set to represent all possible switches from one region to itself or another region; that isThe set can be determined by the reachability analysis for mixed logical dynamical systems. The system is allowed to switch arbitrarily between subsystems.
We study the problem of designing a static output feedback controller:where such that the closed-loop switched systemis stable.
Lemma 2 (see ). If there exists a fixed point such that the LMIholds for some , then the point is a feasible solution to the bilinear matrix inequality (BMI)
3. SOF Controller Design
In this section, based on a piecewise quadratic Lyapunov function and the new linearization method, we will give new sufficient conditions for solving this problem.
Theorem 3. If there exist points , , such that the following inequalities: hold for some , where , then the points are feasible solutions to inequalities (5).
Proof. By Schur complement, (5) are equivalent toWrite , , and , where and are fixed matrices. The left side of inequality (9) is expanded around asthat isThus, by Lemma 2, inequalities (8) hold.
With Theorem 3, the nonlinear SOF optimization problem can be replaced by solving the following linear optimization problem :
Remark 4. In , additional inequalities are added. This inequalities are the variation of inequalities . Due to our new linearization method, the prescribed scalar should be small . Otherwise, the conservation will increase.
4. Extension to Control
Consider the discrete-time switched system:where , , , , and are the state, the control input, the measured output, the control output, and the disturbance input, respectively; , , denotes a partition of the output space into a number of closed polyhedral regions. Let be the set of all possible switches from one region to itself or another region; that is,
In this section, new sufficient conditions for SOF control design for the switched system (14) in the framework will be present. Given a scalar , assuming , the exogenous signal is attenuated by if for each integer and for every
Lemma 5 (see ). Consider the switched system (14), if there exists a function , with satisfying the following inequality:then the closed-loop switched system (16) is exponentially stable with performance .
Obviously, inequality (18) is equivalent to the following inequalities:where .
Now, the sufficient conditions to obtain SOF control gains with performance are given in the following theorem.
Theorem 6. If there exist points , , such that the LMIs hold for some , where and , then the points are feasible solutions to inequalities (19).
Proof. By Schur complement, inequalities (19) are equivalent toWrite , , and , where and are fixed matrices. The left side of inequality (21) is expanded around as That is, the following inequalities hold: Thus, by Lemma 2, inequalities (20) hold.
With Theorem 6, the nonlinear optimization problem for solving SOF control with performance can be replaced by solving the following linear optimization problem :
Algorithm 7. Consider the following.
Step 1 (initialization step). At initial, we need to obtain initial values of and .
Firstly, let . Then, solve the optimization problem (or instead for case) with respect to and .
If , stop; else let , , , and , and set .
Step 2 (small perturbation step). Set , where is a prescribed small value. Solve LMI optimization problem (or instead for case) with respect to , , and .
Step 3 (update step). Let , , , and . For fixed , compute new by solving (or instead for case), and then compute new and by solving (or instead for case).
If , stop; else if the relative improvement in is more than a desired accuracy, set , and go to Step . Else, set , and let , , , and .
Step 4 (wide perturbation step). Set . Solve LMI optimization problem (or instead for case) with respect to , , and .
Step 5 (update step). Let , , , and . For fixed , compute new by solving (or instead for case), and then compute new and by solving (or instead for case).
If , stop; else if the relative improvement in is more than a desired accuracy, set , and let , , , and , and go to Step 2. Else if the relative improvement in is inferior to the desired accuracy and , where is a prescribed integer, set , and go to Step 4. Else, stop.
Remark 8. The wide perturbation step is a crucial step in improved path-following method. The purpose of this step is to broaden the search scope during each iteration so that the algorithm has the opportunity to escape form the local optimum. However, the enlarged search scope may cause nonconvergence. So the iteration number of wide perturbation step should not be too large. As long as the objective function has been improved significantly, the wide perturbation step will be replaced by a small perturbation step immediately.
5. Numerical Examples
In this section, two examples are given to show the effectiveness of our method. Example 1 is with respect to the SOF control problem for switched systems. Example 2 is concerning the controller design problem for switched systems.
Example 1. Consider system (1) with the following parameters: It cannot be stabilized by the method in [12, 20]. However, using our method, set ; after iterations, output feedback stabilizing controller matrices are computed to be
Example 2. Consider a system with the following parameters:
Using our method, set ; after iterations, output feedback control matrices are computed to be In this case, and the closed-loop system has the performance . The result is better than the solution solved by the method in  which gives , which is inferior compared to our result and
This paper studies the problems of static output feedback control and controller synthesis for discrete-time switched systems. Based on piecewise quadratic Lyapunov functions and a new linearization method, new sufficient conditions for system stability and controller design are obtained. Then, an improved path-following algorithm is built to solve the problems. Finally, the merits and effectiveness of the proposed method are shown by two numerical examples. Compared to the existing methods, the proposed method is less conservative.
Important future research work will be applying the results to some real-world systems. How to reduce the design conservatism is an important research topic that deserves further investigation.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported in part by the National Natural Science Foundation of China (nos. 61174033 and 61473160) and in part by the Natural Science Foundation of Shandong Province, China (ZR2011FM006).
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Copyright © 2015 Jian Chen and Chong Lin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.