Mixed H2/H∞ control for time‐varying and linear parametrically‐varying systems

This paper provides a solution of the mixed H 2 =H 1 problem with reduced order controller for time-varying systems in terms of solvability of diierential linear matrix inequalities and rank conditions, including a detailed discussion of how to construct a controller. Immediate specialization lead to a solution of the full order problem and the mixed H 2 =H 1 problem for linear systems whose description depend on an unknown but in real-time measurable time-varying parameter. As known for the H 1 problem, we completely resolve the quadratic mixed H 2 =H 1 problem (under certain hypotheses on the parameter dependence) by reducing it to the solution of nitely many algebraic linear matrix inequalities. However, we also point out directions how to overcome the conservatism introduced by using constant solutions of the diierential inequalities. Finally, we clarify that a simple specialization leads to a linear matrix inequality approach to the pure H 2 problem for general LTI systems, and we reveal how to compute the optimal value. 1 Notation R n is equipped with the Euclidean norm, and R nm with the corresponding induced norm, both denoted as k:k. L p denotes the signal space L n p 0; 1) (for an appropriate n) and is equipped with the standard norm k:k p (deened with the Euclidean spatial norm). Functions are tacitly assumed to be continuous and bounded, and smooth functions are, in addition, continuously diierentiable. Time functions are functions deened on 0; 1). For a symmetric valued function X(s) deened on a set S, X is said to be strictly positive (X 0) if there exists an > 0 such that X(s) I for all s 2 S. For the system or input output mapping _ x = Ax + Bu; y = Cx + Du with x(0) = 0 we use the notation " A B C D #. If A B C D ! is a constant matrix, the system is called LTI, if it is a time function, the system is called LTV. The time function A is exponentially stable if the system _ x = Ax has this property.

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