Design of stabilizing control laws for mechanical systems based on Lyapunov's method

Abstract

Design methods of stabilizing control laws are discussed for general mechanical systems. Hamiltonians of the mechanical systems are assumed not to depend on time explicitly, but the mechanical systems may embody elastic bodies and nonlinear motions. The methods are based on Lyapunov's second method for stability analysis, and the equations of motion of the controlled systems are not necessary for control design. A candidate for the Lyapunov function is the composition of the Hamiltonians of the subsystems in a given system. A sufficient condition of stabilizing control laws is derived and is extended so as to allow a class of frequency-dep endent control laws with positive real filters, observers, or prefilters. The multiplier method (augmented Lagrangian method) is also introduced to attain desirable equilibrium in the closed-loop systems and results in integral control action. Slew maneuver of a flexible beam is employed as a design example to demonstrate the closed-loop characteristics of several control laws based on the present method. YAPUNOV S second method is a powerful tool for the stability analysis of dynamical systems. One of the preferable features of the method is that it is not necessary to obtain the explicit solutions of the differential equations governing the dynamical systems and, therefore, it can often be applied successfully to nonlinear systems and distributed parameter systems. The method is suitable especially for stability analysis of mechanical systems, since the Hamiltonians of mechanical systems are natural candidates of the Lyapunov function. Lyapunov's method is employed in Ref. 1 for designing control laws for rigid manipulators that are nonlinear systems. The Lyapunov function consists of the Hamiltonian of the whole system and a potential function introduced to achieve the desirable configuration of the manipulators. Lyapunov's method is utilized in Ref. 2 to analyze the stability of direct velocity feedback for active vibration suppression of flexible space structures, where the system energy is chosen as a candidate of the Lyapunov function. In Ref. 3, flexible multilink manipulators are treated, and the closedloop stability of proportional- derivative (PD) and strain feedback is analyzed applying Lyapunov's method to an ordinary differential equation obtained through modal expansion. It is shown in Ref. 4 that Lyapunov's method is directly applicable without modal expansion to coupled partial and ordinary differential equations that describe the slew maneuver of a flexible beam, and the PD control law for the attitude angle is shown to be stable by employing the Lyapunov function that consists of the Hamiltonian of the whole system and a potential function to attain a desirable attitude angle. In Refs. 5 and 6, which consider a problem similar to Ref. 4, the Lyapunov function consists of a weighted sum of Hamiltonians of the rigid hub and the flexible beam, not simply the Hamiltonian of the whole system, and a potential function to attain desirable equilibrium. The coupled partial and ordinary differential equations are treated directly without modal expansion in the analysis, and the resultant control law feeds back the attitude angle, the angular rate, the shear force, and the bending moment at the root of the beam. It is shown that the feedback law with a constant gain results in satisfactory