Output Feedback Sensitivity Functions Under Signal to Noise Ratio Constraint

We analyse the control output feedback sensitivity function which results from imposing the minimal signal-to-noise ratio requirement needed for stabilisability. The signal to noise ratio constraint used here includes the effect of a band limited, additive coloured Gaussian noise communication link and a (non) minimum phase unstable continuous (discrete) linear time invariant (LTI) plant with time delay. Target performance is introduced in a form that directly implies an additional term in the required SNR compared to the case of stabilisability only. In a first approach performance is sought as output disturbance rejection giving guaranteed reference tracking at zero frequency (integral action) and compensation for perturbations over a range of frequencies. It is noted that the resulting additional cost can be made vanishingly small by making the closed loop arbitrarily slow. In a second approach we define frequency bounds for the magnitude of the output sensitivity function. These bounds will guarantee low error reference tracking and rejection of disturbances over a non trivial range of frequencies. Based on this, a lower bound for the required extra SNR that satisfies the mentioned sensitivity magnitude bounds is proposed and its tightness discussed.