Second Semester M.E. (Control and Instrumentation) Degree Examination, June/July 2015 (2K8 Scheme) CI 212 : OPTIMAL CONTROL SYSTEM Time : 3 Hours
Max. Marks : 100
Instruction : Answer any five full questions. 1. a) Define optimal control problem. How optimal control problems are classified ? Give the practical examples for each classification. 10 b) Find the extremal for the following functional J =
2 2
∫
1
x (t) 2t 3
dt with x(1) = 1
and x(2) = 10.
10
2. a) Derive Euler-Lagrange Equation to solve fixed end time and fixed end state problem. Indicate the different cases of Euler Lagrange Equation.
10
b) Solve the following using Lagrange multiplier method. i) Minimize Z = x12 + 2 x 22 + 3 x 23 Subject to x1 + x2 + 3x3 = 2 2x2 + 4x3 = 4 5x1 + 2x2 + x3 = 5 ii) Minimize Z = 6 x12 + 5 x 22 Subject to x1+5x2=3 x1, x2 ≥ 0
10
3. a) With usual notations, derive Matrix Differential Riccatti Equation as applicable to linear quadratic control problems. 10 b) Consider a simple first order system x (t ) = −3x ( t ) + u( t) and the cost function ∞
as J =
∫ [x 0
2
]
( t ) + u2 ( t ) dt .
Where x(0) = 1 and the final state x(∞ ) = 0 . Find open-loop and closed loop optimal controllers.
10
P.T.O.
*ME055*
ME – 055
4. a) What is principle of optimality ? Explain forward dynamic programming with an example.
8
b) Consider I-D regulator problem with system equation X(k + 1) = X(k) + u(k) and performance index to be minimized as J = 2x 2 (2) +
1
∑ 2u2 (k) subject to
k =0
0.0 ≤ x(k ) ≤ 1.0; k = 0, 1, 2 and − 1.0 ≤ u(k) ≤ 1.0; k = 0,1. Find optimal control
sequence u∗ (k ) and x∗(k ) using dynamic programming.
12
5. a) Explain the general discrete optimization problem obtain optimal control law for a sub optimal feedback regulator problem. b) State and derive Hamilton-Jacobi Bellmann equation.
10 10
6. a) Explain the solution steps of Liapunov approach of steady state quadratic control.
10
b) Obtain the minimization of a functional J (x(k 0 ), k 0 ) = J =
k f −1
∑
k =k 0
[ X(k)X(k + 1) + X (k) ] 2
Subject to boundary conditions X(0) = 2 and X(10) = 5.
10
7. Write explanatory notes on : i) Piece-wise smooth extremals ii) Kalman filter iii) H2 and H∞ control iv) Bang of Bang 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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