Stability of Decentralized Systems Using Non-LTI Controllers

Abstract

Up to this point in the book, we have focused our attention on the design of LTI controllers for the control of LTI decentralized systems. Now, we explore the possible benefits of using a more complicated nonlinear time-varying (NLTV) or linear time-varying controllers. We first demonstrate, using a somple example, how a linear time-varying controller. We first demonstrate, using a simple example, how a linear time-varying controller can stabilize the plant while on LTI controller cannot. We then use directed graphs to explain exactly when this occurs in the general case: we first introduce the notion of a quotient system and its corresponding fixed modes, labeled the quotient fixed modes (QFMs), and then show that there exists a NLTV decentralized controller which provides closed-loop stability if and only if the decentralized plant has no unstable QFMs. We explore several ways to compute the QFMs in an efficient manner. Last of all, we provide several systematic design methodologies for the construction of a linear periodic controller which stabilizes a decentralized plant which has no unstable QFMs but possibly an unstable DFM.