Integral manifolds of singularly perturbed systems with application to rigid-link flexible-joint multibody systems

In this paper, we first review results of integral manifolds of singularly perturbed non-linear differential equations. We then outline the basic elem ents of the integral manifold method in the context of control system desig n, namely, the existence of an integral manifold, its attractivity, and sta bility of the equilibrium while the dynamics are restricted to the manifold . Toward this end, we use the composite Lyapunov method and propose a new e xponential stability result which gives, as a by-product, an explicit range of the small parameter for which exponential stability is guaranteed. The results are applied to the control problem of multibody systems with rigid links and flexible joints in which the inverse of joint stiffness plays the role of the small parameter. The proposed controller is a composite contro l law that consists of a fast component, as well as a slow component that w as designed based on the integral manifold approach. We show that: the prop osed composite controller has the following properties: (i) it enables the exact characterization and computation of an integral manifold, (ii) it mak es the manifold exponentially attractive, and (iii) it forces the dynamics of the reduced flexible system on the integral manifold to coincide with th e dynamics of the corresponding rigid system (i.e. the one obtained by maki ng stiffness very large) implying that any control law that stabilizes the rigid system would stabilize the dynamics of the flexible system on the man ifold. We finally present a detailed stability analysis and give an explici t range of the joint stiffness, in terms of system parameters and controlle r gains, for which the established exponential stability is guaranteed. (C) 1999 Elsevier Science Ltd. All rights reserved.