Fixed Order H∞-synthesis: Computing Optimal Values by Robust Performance Analysis

The computation of optimal H∞ controllers with a prescribed order is important for their real-time implementation. This problem is well-known to be non-convex, and only algorithms that compute upper bounds on the global optimal value are known. We present a method to compute lower bounds by re-formulating the problem as a robust analysis problem, where the controller variables are the “uncertain” parameters. This allows to apply the wide spectrum of robust analysis techniques to the fixed order controller design problem. The solution to the robust analysis problem is a global lower bound on the optimal closed loop H∞ performance. We construct a family of robust analysis problems and relax them to convex optimization problems using the S-procedure. The optimal values of this family converge from below to the globally optimal fixed order H∞-norm. This allows verification of global optimality of controllers. The number of complicating variables in our robust analysis problem is small if we optimize over a few controller parameters. The technique is therefore computationally feasible for optimization over few controller variables, e.g. PID-tuning of systems with large Mc-Millan degree. The method is applied to the tuning of two controller parameters of a 4-block H∞ design of an active suspension system, with a Mc-Millan degree of the plant of 27.