II Semester M.E. (Control and Instrumentation) Degree Examination, January/February 2014 (2K8 Scheme) CI 212 : OPTIMAL CONTROL SYSTEMS Time : 3 Hours
Max. Marks : 100
Instruction : Answer any five full questions. 1. a) State the optimal control problem. How the optimal control systems are classified according to the performance measures ? Give practical example for each of classification. 10 b) Derive Euler-Lagrange equation to solve fixed end time and fixed end state problem. Indicate different cases of Euler-Lagrange equation.
10
2. a) Given a second order system as x 1 (t ) = x 2 (t )
x 2 (t ) = u(t )
1 and the performance index as J = 2
tf
∫
u2 (t ) dt
to
Find the optimal control and optimal state for the following boundary conditions. x(0 ) = [ 1
1 ]1 and x(2 ) = [ 0
0 ]1
Assume that the control and state are unconstrained. b) Define functional and how the minimization of functional can be decided.
10 4
c) Solve the following using Lagrange multiplier method. 2
2
2
Minimize z = x 1 + x 2 + x 3 Subject to x1 + x2 + 3x3 = 2 5x1 + 2x2 + x3 = 5.
6
3. a) Derive the Weierstrass – Erdmann necessary corner conditions for an extremal. 10 b) Explain the recurrence relation of dynamic programming for optimal control problem.
10 P.T.O.
*JUP014*
JUP – 014
4. a) What is principle of optimality ? Write the differences between forward and backward dynamic programming. Explain any one method of dynamic programming with an example. 12 b) Obtain feedback control law and optimal cost for the system with x 1 = x 2 , x 1(0 ) = 2, x 2 = − 2 x 1 + x 2 + u, x 2 (0 ) = − 3 which minimizes the
performance index J =
1 2
∞
∫ 0
[ 2x
2 1
]
2
+ 6x 1x 2 + 5 x 2 + 0.25 u2 dt .
8
5. a) State discrete time optimal control system with free final point condition. Explain the solution procedure to solve the problem.
10
b) Find the closed loop optimal control for the first order system x = − 2x + u with the performance index J =
∞
∫
0
[x
2
]
+ u2 dt . Assume that J∗ = f x 2 .
6. a) Explain Liapunov approach for quadratic optimal control.
{
10 8
}
b) Derive Bang-off-Bang control law u∗ (t ) = − DEZ B′ λ∗ (t ) for a minimum Fuel optimal control system. 12 7. a) What is optimal control law ? Derive the Matrix Differential Riccati equations. 12 b) Write explanatory notes on any two of the following : a) Kalman filter and Guassian control b) BOLZA PROBLEM c) Hamilton-Jacobi equation d) H2 and H∞ control. _______________________
How optimal control problems are classified ? Give the practical examples for each classification. 10. b) Find the extremal for the following functional dt. 2t. x (t) J.
... time and fixed end state. problem. Indicate the different cases of Euler-Lagrange equation. 10. 2. a) Find the extremal of a functional. J(x) [ ] x (t) x (t) x (t) x (t) dt.
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