Linear and Nonlinear Multivariable Feedback Control

5Absolute stability of nonlinear MIMO systems

5.1 INTRODUCTION

The problem of nonlinear systems stability has always attracted the attention of researchers and engineers. Rigorous analytical methods for investigating the stability of nonlinear systems are those based on the works of prominent Russian scientist A. Lyapunov (1892, 1907). In his works, Lyapunov considered in general form the stability of the solution of nonlinear differential equations, and gave precise notions of stability, asymptotical stability, etc. Lyapunov’s theorems of stability at a first approximation, actually reducing the issue of the stability analysis of a nonlinear system to the analysis of a linear (linearized) system, constitute the basis of linear control theory. The second (or direct) Lyapunov method, which allows finding sufficient conditions for motion stability, has had a huge impact on the development of nonlinear control theory. At the same time, the necessity for representing nonlinear systems in state-space, as Lyapunov’s second method presumes, has restricted to a certain extent its application in practice. In many respects, it is because the basis of the classical control theory is formed by the frequency-domain methods. This led to the interest of technicians in the search for frequency stability criteria of nonlinear systems motion, which finally resulted in a great breakthrough in that field.

The beginning of impetuous development of the frequency-oriented direction in the theory of absolute ...