Chapter 12 - Stability of Feedback Control Systems

Abstract

This chapter focuses on the stability of feedback control systems. It utilizes the transfer function modeling and an approach similar to the one of Chapter 7, which introduces the concept of stability applied to open-loop (uncontrolled) dynamic systems. The closed-loop transfer function is employed for single-input, single-output (SISO) systems, and the closed-loop transfer function matrix is used for multiple-input, multiple-output (MIMO or multivariable) systems to determine the pole position in the complex plane and to establish stable, marginally stable, or unstable behavior of feedback systems. The state-space model is also applied to determine the stability behavior of dynamic control systems. Analytical or MATLAB methods can be used to solve the characteristic equation related to the closed-loop transfer function or the closed-loop transfer function matrix when such solutions are available. The Routh–Hurwitz test or criterion is the analytical tool of choice for designing control systems with parameters, such as gains, that are originally undetermined. The method is illustrated by several examples. The root-locus method and the Nyquist criterion are introduced here to monitor the closed-loop pole migration between the left-hand complex plane and the right-hand complex plane for feedback system with a variable gain by analyzing the poles and zeros of the open-loop transfer function. For the same purpose, the Bode plots are utilized, as well as the phase and/or gain margins.