91Ó°ÊÓ

Suppose the closed-loop poles of a feedback system satisfy \\[\frac{1}{(s+2)(s+3)}=-\frac{1}{K}.\\] Use the root-locus method to determine the values of \(K\) for which the feedback system is guaranteed to be stable.

Suppose the closed-loop poles of a feedback system satisfy $$\frac{s-1}{(s+1)(s+2)}=-\frac{1}{K}.$$ Use the root-locus method to determme the negative values of \(K\) for which the feedback system is guaranteed to be stable.

Suppose the closed-loop poles of a feedback system satisfy \\[\frac{(s+1)(s+3)}{(s+2)(s+4)}=-\frac{1}{K}.\\] Use the root-locus method to determine whether there are any values of the adjustable gain \(K\) for which the system's impulse response has an oscillatory component of the form \(e^{-a t} \cos \left(\omega_{0} t+\phi\right),\) where \(\omega_{0} \neq 0\).

Suppose the closed-loop poles of a discrete-time feedback system satisfy \\[\frac{z^{-2}}{\left(1-\frac{1}{2} z^{-1}\right)\left(1+\frac{1}{2} z^{-1}\right)}=-\frac{1}{K}.\\] Using the root-locus method, determine the positive values of \(K\) for which this system is stable.

cConsider a continuous-time feedback system whose closed-loop poles satisfy \\[G(s) H(s)=\frac{1}{(s+1)}=-\frac{1}{K}.\\] Use the Nyyuist plot and the Nyquist stability criterion to determine the range of values of \(K\) for which the closed-loop system is stable. Hin: In sketching the Nyquist plot, you may find it useful to sketch the corresponding Bode plot first. It also is helpful to determine the values of \(\omega\) for which \(G(j \omega\\} H(j \omega)\) is real.

Consider a continuous-time feedback system whose closed-loop poles satisfy \\[G(s) H(s)=\frac{1}{(s+1)(s / 10+1)}=-\frac{1}{K}.\\] Use the Nyquist plot and the Nyguist stability criterion to determine the range of yaiues of \(K\) for which the closed-loop system is stable.

Consider a feedback system, either in continuous-time or discrete-time, and suppose that the Nyquist plat for the system passes through the point \(-1 / K\). Is the feedback system stabie or unstable for this value of the gain? Explain your answer.

Consider a feedback system with \\[H(s)=\frac{s+2}{s^{2}+2 s+4}, \quad G(s)=K.\\] (a) Sketch the root locus for \(K > 0\) (b) Sketch the root locus for \(K < 6 \) (c) Find the smallest positive value of \(K\) for which the elosed-loop impulse response does not exhibit any oscillatory behavior.

(a) Consider a feedback system with \\[H(z)=\frac{z+1}{z^{2}+z+\frac{1}{4}}, \quad G(z)=\frac{K}{z-1}.\\] (i) Write the closed-loop system function explicitly as a ratio of two polynomials. (The denominator polynomial witl have coefficients that depend on \(K .)\) (ii) Show that the sum of the closed-loop poles is independent of \(K\) (b) More generaliy, consider a feedback system with system function \\[G(z) H(z)=K \frac{z^{m}+b_{m-1} z^{m-1}+\cdots+b_{0}}{z^{n}+4 n-1 z^{n-1}+\cdots+a_{0}}.\\] Show that if \(m \leq n-2,\) the sum of the closed-loop poles is independent of \(K\).