Lyapunov-based stability of dynamical systems

The concepts of stability and convergence (asymptotic, exponential, uniform, etc.) play a fundamental role in control theory. Given that numerical computation tools are becoming increasingly powerful, one might think that an intensive simulation campaign combined with sound intuition could provide sufficient evidence to draw conclusions about the behavior of a dynamic system. However, it is impossible to draw direct conclusions from simulations to prove fundamental properties—such as stability, reachability, etc.—of a complex system, as certain very specific phenomena may be overlooked. Instead, simulation can be used to illustrate or supplement the mathematical analysis.

Another approach might then consist of calculating the system’s trajectories—and thus gaining a complete understanding of them—and applying definitions and theorems based on these elements to draw conclusions. However, these calculations can prove very complicated because they are directly linked to the system’s structure: see, for example, the case of nonlinear or hybrid systems .

To overcome these difficulties, the idea is to consider Lyapunov theory, which is based on the notion that if a system’s energy is dissipated, then the system is stable. This method formalizes the fact that studying the behavior of a function allows one to conclude that the system is stable without having to explicitly calculate its trajectories.

In this article, we provide an overview of the stability of dynamical systems as addressed through Lyapunov theory.

In this context, we focus on deriving stability conditions for various classes of systems; that is, we assume a control law is known. Beginning with nonlinear systems (which encompass the classes of systems considered in this article), we define the main concepts of stability (stability of an equilibrium point, attractor, region of attraction). Next, we propose generic conditions for certifying stability using Lyapunov functions. Time-invariant linear systems and linear systems subject to isolated nonlinearity at the input are also addressed. In both cases, conditions formulated in terms of linear matrix inequalities (LMIs) are proposed.

We also briefly discuss how stability certificates...