Multi-objective MIMO Optimal Control Design Using The H2 and the `1 norms

The H2 norm has been widely employed in controller design as an e ective measure on the performance of a system particularly when spectral information is available on exogenous disturbances. The `1 norm provides a measure on the performance of the system to persistent magnitude bounded disturbances. The `1 norm also provides stability margins with respect to uncertainty in the system. Optimal control techniques which minimize the H2(`1) norm over all stabilizing controllers can fail to address the concerns of the `1(H2) objectives. In this paper we provide e ective techniques to synthesize controllers which address the multiple objective concerns of the H2 and the `1 measures. Two problems, the combination problem and the `1=H2 problem are studied. The combination problem setup is suitable for studying the tradeo s between the possible `1 and H2 norms of the closed-loop map. The `1=H2 problem provides a tool whereby a speci ed level of nominal performance is guaranteed while minimizing the `1 norm. In the solution method the use of zero interpolation to characterize achievable closedloop maps is avoided. Thus many di culties existing in the numerical techniques employing zero-interpolation are eliminated. In particular the knowledge of multivariable zeros and their directions is not needed. Also the di culties in the existing methods that arise due to pole-zero cancellations which can occur when retrieving the optimal controller from the optimal closed-loop map are avoided.