# Modern Control System 2006-07

Note : Attempt all questions.

1. Attempt any two parts of the following :

(a) Derive the state space representation of the speed control system shown in Fig. 1(a) Assume torque constant as Kj, and back emf constant as Kb.

(b) Consider the state space model of a system with

A =

 0 1 0 0 0 -1 1 , B = 0 0 -1 -10 10

C = [l 0 0] Obtain characteristic polynomial and transfer function of the system.

(c) Consider the system

 0 0 -2 0 X = 0 1 0 X x(0) = 1 1 0 3 5 0

Obtain the free response of the system.

2. Attempt any two of the following

(a) Consider the difference equation y{k + 2) + axy{k + 1) + a2y(k) =

bQr(k + 2) + b-f{k + 1) + b2r(k)

(b) Assuming that the system is initially at rest and r(k) = 0 for k < 0 obtain the transfer function G(z) = Y(z)/K(z)

(c) The state variable model of a plant is given by x = Ax + Bu y = Cx

where A =

 fo 1 1 i© i 1© 1 cn i IIcq 1h-1 1

C = [l 0]

Obtain its discrete time state model for T = 0.1 sec. Obtain the pulse transfer function of the system shown in Fig. 2(a) for a sampling period T = 1 sec.

3. Attempt any four parts of the following :

(a) Consider the system

 0 0 -1 1 X = 1 0 -2 x + 2 0 1 -2 0

(b)

u

y = [0 0 l]x

(b) Design an observer that has eigen values at s = -4 – 5 and -6. Show that the following system is uncontrollable :

u

 Xj -0.5 0 Xj + 0 x2 0 -2 x2 1

y = [o i]

X-,

Xn

(c) Is the following system observable ?

 Xj 11 h-1 o 1 Xj 10 1 X2 0 -2 X2 1h-1 1

u X-, Xa

(d) Explain the procedure for designing state observer. Use Liapunov’s method to find conditions for the stability of linear system described by the state matrix

-a P-P -a

(e) Distinguish between state and output controllability and give method to test it.

4. Attempt any four parts of the following :

(a) The system x = —x + u is to be transferred from lj. Ox(0) = 5 to x(l) = 0 such that J = — I (u) dt is minimized. Find the optimal control.

(b) Find the optimal control for the system

 0 1 0 X = -10 0 x + 10 u

which minimizes the performance index

 1 0 given x(0) = 1 , X£ — 0

(c) State the two point boundary value problem and give its solution in terms of Euler-Lagrange equation.

(d) How will you formulate an optimal control problem ? What are standard optimal control problems ?

(e)  State and explain Hamilton Jacobi equation.

(f)  Explain :

(2) Constrained optimization

5. Attempt any four parts of the following :

(a) What is fuzzy logic based control system ? Explain with example.

(b) Define the term membership function and fuzzy PI controller.

(c) What is the need for an adaptive control system ?

(d) Compare MRAC with self tuning regulator.

(e) List some of the advantages and disadvantages of sampled data control system ?

(f)  Give the various controller structures used in adaptive control system.