1. Derive the impulse response of a first order hold from fundamentals in terms of ro and T.

State and prove Right Shift (Time delay) Theorem with function and its Z..transform,

State and prove the Final Value Theorem with function and its Z-transform,

3. Explain Routh stability criterion on rth plane used to analyse the stability of discrete system.

4. Given the transfer function

0·3679 z 0·2642/z2 0·6321

obtain the linear constant coefficient difference equation.

5. Derive the expression for steady state error in the output of a type-1 discrete system in z-domain in case of unit ramp input.

6. A discrete-time system has state equation given by
1
-10

Find out its State Transition Matrix.

7. A second order digital control system is represented by x(k B Where A is

0
B is



Obtain the feedback gain matrix G with state feedback control law to place the closed loop eigenvalues at z1 0·4 and z2 =0·6.

8. Consider the following characteristic equation of a discrete data system: z^3 3·3 z^2 4z 0·8 Apply Jury's test and comment upon the stability of the system.

9. A discrete data control system is given as
x(k

where
[0.5 0
0 0.2]
B is



The state feedback control law is


Obtain the optimal control to minimize the performance index


where is the Lyapunov function such that

10. Write short notes on any four of the following:

Shannon's Sampling Theorem

Eigenvectors

The Cayley Hamilton Theorem Method

Complete Observability

Phase Lag-Lead Controllers

Pole Placement by State Feedback