For the system of Problem 7.2-5 and Fig. P7.2-5, let K = 1 .

(a) Determine the stability of the system.

(b) Sketch the Bode diagram, and use this diagram to sketch the Nyquist diagram.

(c) If the system is stable, determine the gain and phase margins. If the system is unstable, find the value of K that gives a phase margin of 45° .

(d) From the Nyquist diagram, determine the value of K >0 for which the system is marginally stable.

(e) Find the frequency ω at which the marginally stable system will oscillate.

Problem 7.2-5

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Consider the system of Fig. P7.2-5 with T = 1 s. Let the digital controller be a variable gain K such that D(z)= K. Hence m(kT) = Ke(kT). (a) Write the closed-loop system characteristic equation. (b) Determine the range of K for which the system is stable. (c) Suppose that K is set to the lower limit of the range in part (b) such that the system is marginally stable. Find the natural-response term that illustrates the marginal stability. (d) Repeat part (c) for the upper limit of the range of K. R(s) + E(s) T = 1 s D(z) 1--T's S M(s) Gp(s) 1 s+1 C(s)