*EJM020*

II Semester M.E. Degree Examination, January 2013 (Control and Instrumentation) 2K8 CI 212 : OPTIMAL CONTROL SYSTEMS Time : 3 Hours

Max. Marks : 100

Instructions : 1) Answer any five full questions. 2) Missing data can be assumed suitably. 1. a) Define optimal control problem. Explain the formulation of optimal control problem with an example. b) Derive Euler-Lagrange equation to solve fixed end time and fixed end state problem. Indicate the different cases of Euler-Lagrange equation.

10 10

2. a) Find the extremal of a functional π

J(x ) =

4

∫ 0

[x (t) + x ( t) x (t) + x (t ) ] dt . 2 1

1

2

2 2

The functions x1 and x2 are independent and the boundary conditions are

( )

( )

x1(0 ) = 1, x 2 (0 ) = 3 , x 1 π = 2, x 2 π free . 2 4 4

b) Explain the solution of Extrema of functions with conditions using Direct method and Lagrange Multiplier Method.

10 10

3. a) Derive the Weierstrass-Erdmann necessary corner conditions for extremal. 10 b) Explain procedure summary of Pontragin principle for solving BOLZA problem. 10 4. a) Consider a simple first order system x(t) = − 3 x(t) + u ( t) and the cost function (CF) as

J=

∞

∫ 0

[x ( t) + u (t )] dt 2

2

where x(0) = 1 and the final state x( ∞ ) = 0. Find the open loop and closed loop optimal controllers. b) State and Derive Hamilton-Jacobi equation.

12 8

P.T.O.

*EJM020*

EJM – 020

5. a) What is principle of optimality ? What are the differences between Forward and Backward Dynamic Programming ? Explain Forward dynamic programming with an example. 12 b) Consider I-D regulator problem with system equations X (k + 1) = 2X (k) + u (k) 2 and the performance index to be minimized as J = x (k ) +

1

∑

2u2 (k )

k=0

subject to 0 .0 ≤ X(k ) ≤ 1 .0 k = 0, 1, 2 and − 1 .0 ≤ u(k ) ≤ 2 .0 k = 0, 1 Find the optimal control sequence u*(k) and X*(k) using Dynamic Programming. 6. a) Explain Liapunov Approach for Quadratic Optimal Control.

8 8

b) Explain the general discrete optimization problem and also obtain optimal control law for sub-optimal feedback regulator problem with the functions of final state fixed. 12 7. Write explanatory notes on the following : a) H2 and H∞ control and optimal estimation b) Bang-Bang control c) Tracking performance Analysis d) Kalman filter. ____________________

(5×4=20)