Constraints on the Sensitivity and Complementary Sensitivity Functions

Even though S (s) and T (s) are functions of the complex variable s, and their frequency responses are functions of jω, the sum of the two functions is a constant. Closed-loop stability and performance issues will impose other constraints as well. In these notes, we will look at the constraints imposed by the relative degree of the loop gain L(s) and by the presence of right-half plane zeros in L(s). We can see from (2) and (3) that in order to have good tracking of the reference signal and good rejection of the disturbance signal, we need the conditions that S(s) ≈ 0, T (s) ≈ 1. Both of these conditions can be satisfied by having |L(s)| À 1. However, in order to prevent propagation of measurement noise to the error and output signals, we need T (s) ≈ 0⇒ S(s) ≈ 1. These conditions are only satisfied by |L(s)| ¿ 1. In order to achieve all of these objectives, there must be a frequency separation between the reference and disturbance signals on the one hand and the measurement noise signal on the other. In many real situations, the reference and disturbance signals contain mostly low frequencies, and the measurement noise contains a wide band of frequencies, including high frequencies. Therefore in general, |L(jω)| À 1 is required at low frequencies and |L(jω)| ¿ 1 is required at high frequencies. In addition to a large loop gain at low frequencies, steady-state error constraints might require that |L(j0)| be infinitely large, with the slope of the magnitude curve as ω → 0 depending on the specific steady-state error characteristics that are desired. For example, if zero steady-state error is required for a constant reference signal or disturbance signal, then the low-frequency slope of |L(jω)| must be at least −20 db/decade (or −1 on a log-log plot). If zero steady-state error is required for a linearly increasing reference or disturbance signal, then the low-frequency slope of |L(jω)| must be at least −40 db/decade (or −2 on a log-log plot).