Regulating Constraint Obedience for Fuzzy Mechanical Systems Based on $\beta$-Measure and a General Lyapunov Function

We consider an uncertain mechanical system. The uncertainty includes the initial condition and system parameter. The uncertain parameter in the system is (possibly fast) time varying. The only known information about the uncertainty is that it lies in a fuzzy set. The mechanical system is to follow a set of constraints, which may include many engineering applications, even in the presence of uncertainty. For this purpose, we propose a <inline-formula> <tex-math notation="LaTeX">$\beta$</tex-math></inline-formula>-measure for constraint obedience, which reflects how much this constraint is obeyed. Based on a very general Lyapunov function, a control scheme is proposed to render a twofold performance: guaranteed and optimal. In the guaranteed phase, the <inline-formula><tex-math notation="LaTeX"> $\beta$</tex-math></inline-formula>-measure is assured to be uniformly bounded and uniformly ultimately bounded, regardless of the actual value of the uncertainty. In the optimal phase, a fuzzy-theoretic-based performance, by which both the “average” <inline-formula><tex-math notation="LaTeX">$\beta$</tex-math></inline-formula>-measure and control effort are considered, is minimized. As a result, the control serves the practical engineering purposes: The mechanical system is guaranteed to follow the desired task with the minimum cost. This paper is part of a unique endeavor to cast both the descriptions of the uncertainty and desired performance index into a fuzzy framework.