Existence of Similarity Transformation Converting BMIs to LMIs

Under a mild condition, we investigate the existence of a similarity transformation regarding robust H_infin stability analysis and observer-controller synthesis problems. The observer-based controller is capable of disturbance-rejection in the presence of unknown but bounded disturbance. We present results, which illustrate plainly the role of the similarity transformation in converting a BMI problem into an LMI problem in a unified fashion applicable to both continuous- and discrete-time problems with or without uncertainty. Finally the validity and applicability of the approach are illuminated by examples.