Time-Scale and Eigenstructure Assignment via State Feedback

Abstract

We present in this chapter one of the major applications of the structural decomposition techniques of linear systems in modern control system design, namely, the asymptotic time-scale and eigenstructure assignment (ATEA) design method using state feedback. The concept was originally proposed in Saberi and Sannuti [117,118] and developed fully in Chen [18] and Chen et al. [27]. It is decentralized in nature and is in fact rooted in the concept of singular perturbation methods of Kokotovic et al. [75]. It uses the structural decomposition of a given linear system characterized by a matrix quadruple (A, B, C, D) to design a state feedback gain F such that the resulting closed-loop system matrix A + BF possesses pre-specified time-scales and eigenstructures. The specified finite eigenstructure of A + BF is assigned appropriately by working with subsystems which represent the finite zero structure of the given system, whereas the specified asymptotically infinite eigenstructure of A + BF is assigned appropriately by working with the subsystems which represent the infinite zero structure of the given system. Such a design method has been utilized intensively to solve many control problems, such as H∞ control (see, e.g., Chen [22]), H2 optimal control (see, e.g., Saberi et al. [120]), loop transfer recovery (see, e.g., Chen [18], and Saberi et al. [116]), and the disturbance decoupling problem (see, e.g., Chen [22], Lin and Chen [86], and Ozcetin et al. [106,107]). It will be seen shortly that the ATEA design technique is a good way of capturing the core differences between H2 and H∞ control.