1. Determine the Z-transform of the following discrete sequences





2. Consider the digital control system as shown in Figure 1. The transfer function of the plant is Design a lead compensator in the w-plane such that the phase margin is the gain margin is at least 10 dB, and the velocity error constant kv is 2. Assume that the sampling period is 0.2 sec.

<img src='./qimages/13210-2.jpg'>

3. Determine the inverse Z-transform of the following functions

3z^2



4. Obtain the state variable model of the given transfer function in Jordan canonical form



5. Consider a unity feedback system with the plant

Ax bu,

y cx,

Where

A 0 1
0 2

b 0
k

c 1 0

Find the range of values of k for which the closed-loop system is stable.

6. Investigate the controllability of the following system:


1 0 0


0 1

7. Determine the stability of the following system by solving Lyapunov matrix equation:




8. Determine the stability of the sampled data control system described by the following characteristic equation

z^3 -0.2z^2 0.25z 0.05 0

9. Write short notes on any two of the following:

Discrete Euler-Lagrange equation for optimal digital control system

Root locus plots

Stability of discrete system