Draw the poles and zeros for the closed loop transfer

Question

Problem 1:

Obtain the z transform of the curve x(t) shown in Figure below. Assume that sampling period is 1 sec. Provide the expression in closed form.

Problem 2:

Given the following analog signal

x(t) = 3cos (500 Πt + Π/3)+ 2cos (3500 Πt - Π/4)

(a) Determine the minimum Nyquist sampling rate f_s.

(b) Determine the sampled data sequence x(n) for the sampling frequency f_s = 1000 Hz

(c) What is the reconstructed analog signal from the above samples? Show the mathematical expression for the reconstructed signal. Is the reconstructed signal same as the original signal?

(d) Does aliasing or folding occur during this sampling and reconstruction process? Explain why'?

Problem 3:

Find the inverse Z transform of the following: (not by long division)

X(z) = z²/(z -2)(z² +4z +5)

Draw first 10 samples from the inverse Z transform to show that the system is unstable.

Problem 4:

Albo is a sophisticated entertainment robot. Albo looks like a Chihuahua and wags its tail, does tricks, goes for walk and grabs the ball when he is told to do so. Albo depends on a wide range of sensors including touch, color CCO camera, and range finders. Alba is a digitally controlled system as shown below and the transfer function of the Albo has been adjusted as follows:

A step input r(t) rs applied to Albo through a remote controlled device in order to grab a ball Assuming sampling interval is 0 5 sec. do the following:

(i) From G_p(s) determine G_Z(s) with zero order hold

(ii) Determine overall transfer function T(z) for the system with unity feedback

(iii) Determine the expression for output Y(z) for stop input r(t) = 1

(iv) Determine the final value of Y(k) as k approaches to ∞.

Problem 5:

a) Determine the closed loop transfer function for the following system

Assume the transfer function Gp(s) =

(s +1)/(s² + 5s + 6)

(b) Draw the poles and zeros for the closed loop transfer function and determine whether the system is stable or unstable.