www.ietdl.org Published in IET Control Theory and Applications Received on 9th June 2012 Revised on 2nd December 2012 Accepted on 16th December 2012 doi: 10.1049/iet-cta.2012.0428
ISSN 1751-8644
Brief Paper Feedback control for switched positive linear systems Junfeng Zhang1, Zhengzhi Han1, Fubo Zhu1, Jun Huang2 1
School of Electronic Information and Electrical Engineering, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China 2 School of Mechanical and Electrical Engineering, Soochow University, Suzhou, Jiangsu 215021, People’s Republic of China E-mail:
[email protected]
Abstract: This study investigates the feedback control for a class of switched positive linear systems (SPLSs). By means of the linear programming approach, output-feedback and state-feedback controllers for the underlying systems with average dwell time are designed, respectively. Under these controllers, the closed-loop systems are positive and asymptotically stable. These results obtained provide a way to solve the control synthesis problems of SPLSs by multiple linear copositive Lyapunov functions. Finally, an example is given to illustrate the validity of the present design.
1
Introduction
Over past decades, positive systems have become a hot topic in control community. There are many practical systems that can be modelled by positive systems such as ecology [1], industrial engineering [2], chemical engineering [3] economics, and so on. Switched positive linear systems (SPLSs) are a class of hybrid systems that consists of positive linear subsystems and a rule orchestrating when and how the switching occurs among them. SPLSs have extensive applications in real life [4, 5]. It is well known that stability is fundamental for an engineering system. Thus, stabilisation is of importance in the design of control systems. Hence, there are lots of researches made on the problem of control systems, as well as positive systems and SPLSs [6–11]. So far, there are three commonly used methods for the problem of positive linear systems. The first one is based on the quadratic Lyapunov function and the linear matrix inequality (LMI) [12–14]. The second is based on the linear copositive Lyapunov function (LCLF) and linear programming (LP) [15–18]. The third is called as D-stability strategy [19, 20]. Turning to SPLSs, the major research concerns on the stability analysis of non-autonomous systems. In [21], the switched LCLF method was proposed, and a necessary and sufficient condition for the existence of the LCLF had been established. Zappavigna et al. [22] defined dwell time for SPLSs. By considering the vector fields and geometric characteristics, the stabilisability of second-order SPLSs was studied in [23]. As the first attempt, the stability design of SPLSs in both continuous-time and discrete-time cases with the average dwell time (ADT) were considered in [24, 25], respectively. By illustrating the above results and reading the relevant literature on SPLSs, we find that there are a few conclusions 464 & The Institution of Engineering and Technology 2013
on the non-autonomous SPLSs based on the state-feedback or output-feedback design. The feedback control of SPLSs is more challenging than general switched systems. The target of feedback contains that the closed-loop systems are not only stable, but also positive. This directly leads to that the issue is not a simple one. Although there existed several works on the feedback design of positive systems [13, 14, 17, 26], those conclusions could not be applied directly to the feedback design of SPLSs. These motivate us carry out the present work. This paper focuses on the output-feedback and state-feedback design for SPLSs with ADT based on the LP approach which is more easier to implement than LMI. Multiple linear copositive Lyapunov functions (MLCLFs) are defined and applied to the design of SPLSs. We first design the output-feedback controllers for SPLSs. Based on ADT switching, the stabilisation of SPLSs is obtained. Then, the results are extended to the state feedback design for SPLSs. The present conditions are solvable by LP. The reminder of the paper is organised as follows: Section 2 gives the preliminaries. The output-feedback design is proposed in Section 3. In Section 4, the state-feedback design is given. Section 5 gives an simulation example. Section 6 concludes the paper. Notations: < denotes the set of real numbers,
