Reduction of H_∞ state feedback control problems for the MIMO servo systems

We deal with H_∞ state feedback control problem for the multi-input-multi-output (MIMO) servo system and discuss the advantages of the facial reduction (FR) to the resulting linear matrix inequality (LMI) problems. In fact, as far as our usual setting, the dual of the LMI problem is not strictly feasible because the generalized plant has always stable invariant zeros. Thus FR is available to such LMI problems, and we can reduce and simplify the original LMI problem to a smaller-size LMI problem. As a result, we observe that the numerical performance of the SDP solvers is improved. Also, as a by-product, we obtain the best performance index of the reduced LMI problem with a closed-form expression. This helps the H_∞ performance limitation analysis. Another contribution is to reveal that the resulting LMI problem obtained from H_∞ control problem has a finite optimal value, but no optimal solutions under an additional assumption. This is also confirmed in the numerical experiment of this paper. FR also plays an essential role in this analysis.