Constructions of Strict Lyapunov Functions: Stability, Robustness, Delays, and State Constraints

The construction of strict Lyapunov functions is important for proving stability and robustness properties for nonlinear control systems. In some cases, stabilization problems can be solved with the help of nonstrict Lyapunov functions, which are proper and positive definite functions whose time derivatives are nonpositive along all solutions of the closed loop system. By properness and positive definiteness of a function V , we mean that V is zero at the equilibrium, positive at all other states, and satisfies V (x) → ∞ as |x| → ∞ or as x approaches the boundary of the state space. However, nonstrict Lyapunov functions by themselves are insufficient to solve asymptotic stabilization problems, since they do not ensure convergence to the equilibrium. Instead, one often uses nonstrict Lyapunov functions, combined with LaSalle invariance or a Matrosov approach [6]. However, even if one uses LaSalle invariance or standard Matrosov approaches, there is usually no guarantee of robustness, e.g., under control or model uncertainty. This helped motivate the lecturer’s joint research with Frederic Mazenc on the ‘strictification’ approach for converting nonstrict Lyapunov functions into strict ones [2]. A strict Lyapunov function is a proper positive definite function whose time derivative is negative along all solutions of the system outside the equilibrium. Strict Lyapunov functions also allow us to robustify controls, e.g., to prove robustness in the key sense of input-to-state stability (or ISS). To see how this ‘robustification’ approach can be done in the special case of time invariant nonlinear control affine systems of the form ẋ = f(x) + g(x)u(x), assume that we found a control u(x) such that the closed loop system is globally asymptotically stable to 0, and that we have a strict Lyapunov function V for the closed loop system such that −V̇ (x) is proper and positive definite, or equivalently, there is a class K∞ function α such that V̇ (x) ≤ −α(|x|) holds along all trajectories of the closed loop system [2]. Then the closed loop system ẋ = f(x) + g(x)(u](x) + δ) has the ISS property with respect to the set of all measurable essentially bounded functions δ when we use u](x) = u(x)− (∇V (x)g(x))>, i.e., we get ISS with respect to actuator errors δ [2]. However, to use u], one needs formulas for the gradient ∇V of the strict Lyapunov function, and this was another motivation for the strictification approach, but there are other motivations. For instance, having closed form strict Lyapunov functions leads to explicit formulas for comparison functions in ISS estimates, and can make backstepping possible. Analogous results can be shown for systems with delays, where strict Lyapunov functions for undelayed systems are replaced by strict Lyapunov-Krasovskii functionals, whose domains are infinite dimensional sets of functions, and this makes it possible to quantify the effects of input delays on the control performance.