64 Digital Control Systems

Figure 2.37. The input sensitivity function and its inverse

78 Digital Control Systems

Figure 2.38. Input sensitivity function for the plant model q^-1 /(1 – 0.8 q^-1) as a function of the desired closed loop performances

Equation 2.6.8 gives a sufficient condition for the accepted (additive) tolerance in terms of real plant transfer function parameters variations (or uncertainties) with respect to the nominal plant transfer function.

Moreover, we may be interested in the evaluation of the accepted relative tolerance with respect to the nominal plant transfer function magnitude. It follows from Equation 2.6.8 that

where Syb is the “noise-output sensitivity function” and Syr is the complementary sensitivity function.

The noise-output sensitivity function S_yb allows the definition of a frequency “template” to ensure that the “delay margin” constraint is fulfilled. Let consider the case of a delay margin The noise-output sensitivity function Syb allows the definition of a frequency “template” to ensure that the “delay margin” constraint is fulfilled. Let consider the case of a delay margin s ∆τ=1⋅T_S.
It follows that
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and consequently



Equation 2.6.9 becomes



or in dB units






Figure 2.39. Frequency template for the noise-output sensitivity function and ∆τ=T_S.

80 Digital Control Systems

This expression defines a frequency robustness template for the sensitivity function S_yb. This template corresponds to the frequency response of a digital integrator and is represented in Figure 2.39.

As the modulus margin introduces a frequency template on the output sensitivity function we are interested in finding what template is introduced by the delay margin on | S_yp|.

From Equations 2.5.14 and 2.5.17 it results that

S_yp (z^-1) = 1+S_yp (z^-1) 2.6.14

and by means of the triangle inequality it follows that

Taking into account the frequency bound on Syb given by Equation 2.6.12, the following condition is obtained :