Reliable linear-quadratic control for symmetric composite systems

In recent years there has been great interest in the study of symmetric composite systems. Symmetric composite systems are systems composed of identical subsystems which are symmetrically interconnected. The motivation for studying this class of systems is due to its very diverse application areas, such as in electric power systems, industrial manipulators and computer networks (Lunze 1986, Sundareshan and Elbanna 1991, Hovd and Skogestad 1994). It is shown that many analysis and control problems for symmetric composite systems can be simpli® ed because of their special structure. Lunze (1986) ® rst proposed the state-space model of symmetric composite systems and studied some fundamental properties of the systems. Sundareshan and Elbanna (1991) considered the decentralized control of the systems. Liu (1992) studied the output regulation for symmetric composite systems. Lam and Yang (1996) investigated the model reduction problem of such systems. H2-, H1and ·-optimal control problems were considered by Hovd and Skogestad (1994) and Hovd et al. (1997). The centralized and decentralized control for uncertain symmetric composite systems were studied by Yang and Zhang (1995) and Bakule and Rdellar (1996) respectively. Sometimes, control systems may result in unsatisfactory performance or even instability in the event of control component failures. Recently, Veillette et al. (1992) and Veillette (1995) considered the design of reliable control systems. The resulting control systems provide guaranteed stability and satisfy an H1-norm disturbance attenuation bound or given performance bound not only when all control components are operational, but also in the case of actuator or sensor outages in the systems. The outages were restricted to occur within a pre-selected subset of available measurement or control inputs. Reliable control using redundant controllers was studied by Yang et al. (1998). For the reliable control of symmetric composite systems, Yang et al. (1996) studied the primary contingency case of a reliable H1 controller design problem; Huang et al. (1999) considered the decentralized H1 control problem and the fault tolerance of the designed systems. This paper is concerned with the reliable linear± quadratic (LQ) controller design for symmetric composite systems. Both centralized and decentralized reliable controller design problems are studied. Several cases of actuator outages are considered. By using the special structure of symmetric composite systems and the results of Veillette (1995), necessary and su cient conditions for a state feedback controller to be a reliable LQ controller are given and some simple controller design methods are presented. The rest of this paper is organized as follows. In } 2 the state-space description of the system concerned is given and the problem statement is presented. Some notation and lemmas are given in } 3. Necessary and su cient conditions for several kinds of reliable controller to exist are given and some reliable controller design methods are presented in } 4. In } 5, an example is given to illustrate the methodology. Finally, } 6 concludes the paper.