Achievable performance of multivariable systems with unstable zeros and poles

This paper examines the fundamental limitations imposed by unstable (right half plane; RHP) zeros and poles in multivariable feedback systems. We generalize previously known controller-independent lower bounds on the H-infinity-norm of closed-loop transfer functions WXV, where X is input or output sensitivity or complementary sensitivity. The weights W and V may be unstable and non-minimum phase and may depend on the plant G. The bounds are tight for cases with only one RHP-zero or pole. For plants with RHP-zeros we obtain bounds on the output performance for reference tracking and disturbance rejection. For plants with RHP-poles we obtain new bounds on the input performance. This quanitifies the minimum input usage needed to stabilize an unstable plant in the presence of disturbances or noise. For a one degree-of-freedom controller the combined effect of RHP-zeros and poles further deteriorate the output performance, whereas there is no such additional penalty with a two degrees-of-freedom controller where also the disturbance and/or reference signal is used by the controller.