1. State and explain Lyapunov stability theorem. Investigate the system described by

x1 0 1 x1
x2 1 x2

for stability.

2. Using Routh's criteria check the stability of a system whose characteristic equation is given by
s^6 8s^4 12s^3 20s^2 16s 16 =O.

3. Explain Jury stability criterion in detail for discrete systems.

What do you mean by by Fuzzy logic Explain it with an example.

What is the significance of membership functions in fuzzy logic?

5. A system is characterized by transfer function s^3 +l1s+6·

Find the state and output equation in matrix form and also test the controllability and observability of the systems.

6. Obtain the state transition matrix in the form e^At and determine the time response for the system,

where A 0 1
0
and x1 x2 10.

7. Write down the general form of "Steady state Ricatti Equation". How are these equations important? How do we solve these equations

8. Determine the Z transform and ROC of the signal

where is a discrete time signal.

9. Explain the methods for generating Lyapunov's function for discrete systems in detail.

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

Popov's Criterion

Self Tuning Regulators

Sample and Hold Circuit