Global Stabilization of a Bounded Controlled Lorenz System

Abstract

In this work, we present a method for the Global Asymptotic Stabilization (GAS) of an affine control chaotic Lorenz system, via admissible (bounded and regular) feedback controls, where the control bounds are given by a class of (convex) polytopes. The proposed control design method is based on the control Lyapunov function (CLF) theory introduced in [Artstein, 1983; Sontag, 1998]. Hence, we first recall, with parameters including those in [Lorenz, 1963], that these equations are point-dissipative, i.e. there is an explicit absorbing ball${ℬ}$ given by the level set of a certain Lyapunov function, $V_{\infty }(x)$. However, since the minimum point of $V_{\infty }(x)$ does not coincide with any rest point of Lorenz system, we apply a modified solution to the “uniting CLF problem” (to unify local (possibly optimal) controls with global ones, proposed in [Andrieu & Prieur, 2010]) in order to obtain a CLF $V(x)$ for the affine system with minimum at a desired equilibrium point. Finally, we achieve the GAS of “any” rest point of this system via bounded and regular feedback controls by using the proposed CLF method, which also contains the following controllers: (i) damping controls outside ${ℬ}$, so they collaborate with the beneficial stable free dynamics, and (ii) (possibly optimal) feedback controls inside ${ℬ}$ that stabilize the control system at “any” desired rest point of the (unforced) Lorenz system.