Linearization Algorithm for a Reduced Order H∞ Control Design of an Active Suspension System

This paper proposes a method for the design of reduced-order controllers and the proposed method is applied to the active suspension system in the Laboratoire d’Automatique de Grenoble, France. All performance specifications are posed as a single constraint on the H∞ norm of a certain frequencyscaled closed loop transfer function along with the requirement that a controller should be strictly proper. The number of states in the generalized plant is twenty-nine. In order to solve the proposed discrete-time H∞ control problem, we derived necessary and sufficient conditions for the design of strictly proper H∞ discrete-time controllers. The resulting conditions turned out to be the necessary and sufficient conditions for standard (non strictly proper) H∞ control plus one additional matrix inequality. As in the standard H∞ control problem, when the controller has the same order as the plant to be controlled, these conditions become convex sets on the design variables. When the order of the controller is specified to be less than the order of the plant to be controlled, the conditions for the design of H∞ controllers are concave. In order to compute controllers with very low order, we propose a numerical algorithm by linearizing the concave matrix inequalities so as to generate a sequential semi-definite programming problems with monotonically decreasing cost. At each iteration of this algorithm, a linearized version of the original set of (nonconvex) matrix inequalities is solved using semi-definite programming. Our methodology provides a controller of order nine which satisfies all performance requirements. Reduced order controllers from order two to eight are also designed. The controller of order three appears to be the one that well compromises performance with complexity. Experiments using the proposed third order controller have been performed on the actual suspension system. The proposed third order controller performs better than the best third order controller shown in [15].