(a) Consider the discrete-time feedback system of Figure P11.59. Suppose that 1 H(~ =

Chapter 11, Problem 11.59

(choose chapter or problem)

(a) Consider the discrete-time feedback system of Figure P11.59. Suppose that

\(H(z)=\frac{1}{(z-1)\left(z+\frac{1}{2}\right)} \text {. }\)

 

Show that this system can track a unit step in the sense that if x[n] = u[n], then

\(\lim _{n \rightarrow \infty} e[n]=0\)       (P11.59-l)

(b) More generally, consider the feedback system of Figure P11.59, and assume that the closed-loop system is stable. Suppose that H(z) has a pole at z = 1. Show that the system can track a unit step. [Hint: Express the transform E(z) of e[n] in terms of H(z) and the transform of u[n]; explain why all the poles of E(z) are inside the unit circle.]

(c) The results of parts (a) and (b) are discrete-time counterparts of the results for continuous-time systems discussed in 11.57 and 11.58. In discrete time, we can also consider the design of the systems that track specified inputs peifectly after a finite number of steps. Such systems are known as deadbeat feedback systems.

Consider the discrete-time system of Figure P 11.59 with

\(H(z)=\frac{z^{-1}}{1-z^{-1}}\)

Show that the overall closed-loop system is a deadbeat feedback system with the property that it tracks a step input exactly after one step: that is, if x[n] = u[n], then e[n] = 0, n \(\geq\) 1.

(d) Show that the feedback system of Figure P11.59 with

\(H(z)=\frac{\frac{3}{4} z^{-1}+\frac{1}{4} z^{-2}}{\left(1+\frac{1}{4} z^{-1}\right)\left(1-z^{-1}\right)}\)

is a deadbeat system with the property that the output tracks a unit step perfectly after a finite number of steps. At what time step does the error e[n] first settle to zero?

(e) More generally, for the feedback system of Figure P11.59, find H(z) so that y[n] perfectly tracks a unit step for n \(\geq\) N and, in fact, so that

\(e[n]=\sum_{k=0}^{N-1} a_{k} \delta[n-k],\) (P11.59-2)

where the ai are specified constants. Hint: Use the relationship between H(z) and E(z) when the input is a unit step and e[n] is given by eq. (P11.59-2).

(f) Consider the system of Figure P11.59 with

\(H(z)=\frac{z^{-1}+z^{-2}-z^{-3}}{\left(1+z^{-1}\right)\left(1-z^{-1}\right)^{2}}\)

Show that this system tracks a ramp x[n] = (n + 1)u[n] exactly after two time steps.