Question

The plant H(s)=40/(S^2+4) is now put in a unity-feedback connection with a proportionalderivative compensator

Cpd(s) = K(1 + sT), where K and T are real constants to be determined. The closed-loop is stable with a constant step-response error of +20% in steady state. Ignore implementation issues arising from the improperness of the compensator. (a)Determine K.

(b)What range of values can T take?

(c) What is the gain margin?

(d) If the gain cross-over frequency is 10 rad/s, determine the phase margin in radians.

(e) Assuming gain cross-over at 10 rad/s, sketch the Bode diagrams for the open-loop transfer function. Indicate and evaluate all relevant features: asymptotes, asymptotic slopes, any breakpoints, and the exact magnitude and phase at each breakpoint.

(f) Suppose there is a delay ∆ > 0 (s) in the plant, in series with H(s). i. Briefly explain whether or not this this would change the gain cross-over frequency. ii. What range of ∆ could be tolerated? What would happen to the closed-loop system for ∆ values outside this range?

(g) A colleague of yours decides to instead use an alternative compensator of form Ca(s) = k(1 + sτ )(s 2 + 4) (1 + sτ0) 3 in a unity-feedback configuration with H(s). He claims this effectively ‘removes’ the plant dynamics, and simplifies his controller design. He chooses positive constants k, τ and τ0 that ensure stability of the transfer function from the reference r to output y, and implements his design on the physical plant. However, the physical system does not behave as he anticipated. Explain what he observes, and why.

Answer 1