Reduced order ℋ ∞ controller design for vibration control using genetic algorithms

The design of vibration controllers for flexible structures requires special attention due to the size of structural models, generally with a high number of degrees of freedom. The implementation of full order controllers for structures with high numbers of degrees of freedom often requires a high computational processing effort and advanced hardware. To avoid this, it is desirable to use reduced order controllers. The design of reduced order H _∞ controllers characterizes a nonconvex optimization problem. In this context, this work presents a direct minimization method to design reduced order H _∞ controllers in the controllable canonical form. An optimization problem is formulated to minimize the H _∞ norm with an additional constraint to consider stability of the closed-loop system. The solution of the optimization problem is obtained using genetic algorithms, exploiting the advantage of this point of view in the solution of nonconvex problems. This formulation is verified in the active vibration control of a cantilever beam. A comparison of the proposed formulation with the formulation that uses linear matrix inequalities and the Augmented Lagrangian method is presented in this work and some numerical aspects of the problem are discussed.