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) Use the Routh–Hurwitz criterion to determine the range of K for stability.

(c) Check the results of part (b) using the Jury test.

(d) Determine the location of all roots of the characteristic equation in both the w-plane and the z-plane for the value of K 7 0 for which the system is marginally stable.

(e) Determine the s-plane frequency at which the system will oscillate when marginally stable, for K > 0.

(f) Consider the system with all sampling removed and with G_p(s) = K > (s + 1). Find the range of K for which the analog system is stable.

(g) Comparing the ranges of K from parts (b) and (f), give the effects on stability of adding sampling to the analog system.

Transcribed Image Text:

R(s) + E(s) T = 1 s D(z) 1-g-Ts S M(s) 1 s+1 C(s)