Introduction

As we know, the stability is the most fundamental problem in the design of automatic control systems, since only a stable system can keep working properly under disturbances [3, 4, 6, 8–10]. In fact, automatic control theory began from Maxwell’s study on the stability of Watt centrifugal governor. When one designs a control system, one first needs to consider some type of stability for the system and then investigate other problems.

Among various stability theories, the Lyapunov stability is still the most important one [56–63, 101–109, 128, 164, 186, 187]. However, the main difficulty in analyzing Lyapunov stability is how to determine a Lyapunov function for a given system. There does not exist general rules for constructing Lyapunov functions, but are merely based on a researcher or designer’s experience and some particular techniques. The first-order approximation method and many results obtained for the first and second critical cases demand very restrictive requirements on nonlinear terms, which cause difficulties in applications. Moreover, it should be pointed out that the Lyapunov stability theory is mainly applicable for local stability, while many practical problems need to consider globally asymptotic stability or even globally exponential stability.