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Z/j is positive. 121) implies that if log jS j is negative over some frequency range so that, equivalently, jS j is less than 1, then necessarily jS j is greater than 1 over a complementary frequency range. This we already concluded from Bode’s sensitivity integral. The Freudenberg-Looze equality strengthens the Bode integral because of the weighting function wz included in the integrand. /d! may be viewed as a weighted length of the frequency interval. The weighted length equals the extra phase added by the right-half plane zero z over the frequency interval.

34(a). 8. 143) Inspection shows that the open-loop plant zeros re-occur in the uncompensated closed-loop transfer function H0 but that instead of the roots of Y (the compensator zeros) now the roots of X (the compensator poles) appear as zeros. Hence, the precompensator design problem for this configuration is different from that for the configuration of Fig. 33. In fact, if the compensator has right-half plane poles or zeros, or both, it is impossible to achieve identical overall closed-loop transfer functions for the two configurations.

2 (Input sensitivity function for oscillatory plant). 1. Compute and plot the resulting input sensitivity function M and discuss its behavior. 6. Measurement noise To study the effect of measurement noise on the closed-loop output we again consider the configuration of Fig. 21. F r m z/ for the output z 31 1. 96) This shows that the influence of the measurement noise m on the control system output is determined by the complementary sensitivity function T . For low frequencies, where by the other design requirements T is close to 1, the measurement noise fully affects the output.

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