On logarithmic integrals and performance bounds for MIMO systems, Part I: Sensitivity integral inequalities

In this two-part series we extend the classical Bode integral relations to multi-input multi-output systems. Our main contributions include Bode and Poisson type integral inequalities for the sensitivity and complementary sensitivity functions, together with bounds on the sensitivity and complementary sensitivity magnitudes, which are obtained for both continuous-time and discrete-time systems. Part I presents Bode type integral inequalities. These inequalities may be viewed as some variants to the previously available results, but are also significantly different. The main difference is two fold. First, while the previous integral equalities are valid only under a restriction assumption, the results herein hold in general. Secondly, in spite of the fact that the present results are based upon techniques and assumptions similar to those used in deriving Bode type integral inequalities, the in-depth study carried out here leads to more explicable expressions particularly indicative of the adverse effects of open loop unstable poles. As a direct outcome, the new results confirm that in a multivariable system design limitation and tradeoff depends on both the locations and directions of open loop unstable poles, and in particular, in how such directions may be aligned. The latter is characterized by angles measuring the mutual orientation between the pole directions.