Robust Control Theory pp 81-103 | Cite as

# Generalized *H*_{2}/*H*_{∞} Control

## Abstract

In this paper we consider the problem of finding a controller that minimizes an upper bound on the worst case overshoot of a controlled output in response to arbitrary but bounded energy exogenous inputs subject to an inequality constraint on the *H* _{∞} norm of another closed loop transfer function. This problem, which we define as the *generalized* *H* _{2}/*H* _{∞} *control problem*, can be interpreted and motivated as a problem of optimal nominal performance subject to a robust stability constraint. We consider both state-feedback and output feedback problems. It is shown that in the state-feedback case one can come arbitrarily close to the optimal performance measure using memory-less state-feedback. Moreover, the state-feedback generalized *H* _{2}/*H* _{∞} problem can be converted into a convex optimization problem over a set of matrices defined in terms of affine matrix inequalities. For output feedback problems, we show that the generalized *H* _{2}/*H* _{∞} control problem is equivalent to a state-feedback generalized *H* _{2}/*H* _{∞} control problem of a suitably constructed auxiliary plant. An output feedback controller that solves the generalized *H* _{2}/*H* _{∞} synthesis problem may be readily constructed from the solution to the auxiliary state-feedback problem and the solution to the standard *H* _{∞} filtering algebraic Riccati equation. The structure of this controller is similar to that of the so-called central *H* _{∞} controller.

## Key words

*H*

_{2}and

*H*

_{∞}controller design state-space methods convex programming

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