Performance analysis for a class of decentralized control systems

This paper is concerned with decentralized controller design for large-scale interconnected systems of pseudo- hierarchical structure. Given such a system, one can use the existing techniques to design a decentralized controller for the reference hierarchical model, which is obtained by eliminating certain weak interconnections of the original system. Although this indirect controller design is often fascinating as far as the computational complexity is concerned, it may not provide a satisfactory performance for the original pseudo-hierarchical system. A LQ cost function is defined in order to evaluate the discrepancy between the pseudo-hierarchical system and its reference hierarchical model under the designed decentralized controller. A discrete Lyapunov equation should then be solved to compute this performance index. However, due to the large- scale nature of the system, this equation can by no means be handled for many real-world systems. Thus, attaining an upper bound on this cost function can be much more desirable than finding its exact value. For this purpose, a novel technique is proposed, which only requires solving a simple LMI optimization problem with three variables. The problem is then reduced to a scalar optimization problem, for which an explicit solution is provided. It is also proved that as the pseudo-hierarchical system approaches its reference hierarchical model, the bounds obtained from the LMI and scalar optimization problems will both go to zero. In the particular case, when the two models are identical (i.e., the original system is exactly hierarchical), both upper bounds will be zero.