Copyright © IFAC Nonlinear Control Systems, Stuttgart, Germany, 2004 A NOTE ON DISCRETE-TIME STABILIZATION OF HAMILTONIAN SYSTEMS Dina Shona Laila and Alessandro Astolfi 1 Electrical and Electronic Engineering Department Imperial College, Exhibition Road, London S W7 2A Z, UK. E-mail: {d .laila, a. astolf i }<Oimperial. ac . uk ELSEVIER IFAC PUBLICATIONS www.e1sevier.com/1ocate/ifac Abstract: In this paper we present some preliminary results on the stabilization problem for Hamiltonian systems using approximate discrete-time models. The issues of constructing a discrete-time model for Hamiltonian system are in general different from those for dissipative systems. We propose an algorithm for constructing an approximate discrete-time model, which guarantees the Hamiltonian conservation, and apply the algorithm to a class of port-controlled Hamiltonian systems. We illustrate the usefulness of the algorithm in designing a discrete-time controller to stabilize the angular velocity of a rigid body. Copyright © 2004 IFAC Keywords: Hamiltonian systems; Discrete-time systems; Stabilization; Nonlinear systems; Conservation. 1. INTRODUCTION One of the main issues of direct discrete- time de- sign for a nonlinear sampled-data control system is to find a good model to use. Even if the continuous- time model of the plant is known, we cannot in general compute the exact discrete-time model of the plant, since it requires computing an explicit analytic solution of a nonlinear differential equa- tion. A way to solve the problem of finding a good model is by using an approximate model of the plant. A quite general framework for stabilization of sampled-data nonlinear systems via their approxi- mate discrete-time model was presented in (Nesic and Teel, 2004; Nesic et al., 1999b). In these works, the problems are seen from the framework of dissipative systems. In this context, approximate discrete-time models can be obtained using var- ious numerical algorithms, such as Runge-Kutta and multistep methods. Consistency properties are used to measure the discrepancies between the 1 This work is supported by the EPSRC Portfolio Award, Grant No. GR/S61256/01. 967 approximate and the exact models (see (Nesic and Teel, 2004; Nesic et al., 1999b; Stuart and Humphries, 1996)). Unfortunately, most numerical methods that apply to dissipative systems do not apply to conservative systems, in the sense that numerical approximation in most cases will destroy the conservative property of the systems. As a results, we may not be able to use these methods to obtain a good model for conservative systems, especially when control design exploits the conser- vative property of the plant. Hamiltonian systems are an important class of nonlinear systems commonly used to model con- servative physical systems, e.g. electrical networks, rigid manipulators etc. There has been a lot of research to develop numerical algorithms that pre- serve important properties of Hamiltonian conser- vative systems, particularly conservation of the symplectic mapping and Hamiltonian conserva- tion, which are two of the most important prop- erties of these systems (see for example (Stuart and Humphries, 1996; Gonzalez, 1996) and refer- ences therein). However, to the best of the authors knowledge, there is only very little research done