Optimal control of a class of pseudo Euler-Lagrange systems

This paper presents a solution of the optimal control problem for a class of pseudo Euler‐Lagrange systems and proposes a systematic approach to find a Lyapunov function for stability analysis and controller synthesis for such systems. There are three main contributions of the paper. First, a systematic procedure is proposed and proved to construct a Lyapunov function for pseudo Euler‐Lagrange system directly from the mathematical structure of the differential equations, without the need to determine any kinetic or potential energy of the system first. Second, control methodologies for pseudo Euler‐Lagrange systems are also developed. In particular, an optimal controller is found for the case of second order dynamics yielding the same structure for the closed‐loop Lyapunov function as the one derived from the systematic procedure outlined as the first contribution. Finally, the optimal control methodology is extended to systems with order higher than two for a class of triangular systems. The method proposed here works for any mathematical model in the class of pseudo Euler‐Lagrange systems and is therefore not restricted to models of physical systems. Several examples illustrate the application of the novel approach, including mass‐spring‐damper systems and Van der Pol oscillators.