Mixed H 2 =h 1 Control

In this article we provide a solution to the mixed H 2 =H 1 problem with reduced order controllers for time-varying systems in terms of the solvability of diierential linear matrix inequalities and rank conditions, including a detailed discussion of how to construct a controller. Immediate specializations lead to a solution of the full order problem and the mixed H 2 =H 1 problem for linear systems whose description depends on unknown but in real-time measurable time-varying parameters. As done in the literature for the H 1 problem, we resolve the quadratic mixed H 2 =H 1 problem by reducing it to the solution of a nite number of algebraic linear matrix inequalities. Moreover, we point out directions how to overcome the conservatism caused by assuming a particular parameter dependence or by using constant solutions of the diierential matrix inequalities. For linear time-invariant systems, we reveal how to incorporate robust asymptotic tracking or disturbance rejection as an objective in the mixed H 2 =H 1 problem. Finally, we address the specializations to the fully general pure H 1 or generalized H 2 problem, and provide quadratically convergent algorithms to compute optimal values. Our techniques do not only lead to insights into the structure of the solution sets of the corresponding linear matrix inequalities, but they also allow to explicitly describe the innuence of various system zeros on the optimal values.