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. ____________________
... 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.
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.
during experiments with Brain Machine Interfaces ... Data analysis: Overall neural modulations are defined as the variance of the underlying rate, and expressed ...
on the system dynamics. Here, the improved accuracy in trajectory tracking results has led to the development of various control schemes, such as proportional ...
Xin Zhanga â . Ming Zhoub. aSchool of ... Di Masi, Kabanov and Runggaldier (1994), Yao, Zhang, and Zhou (2001), Buffington and. Elliott (2002), Graziano and ...
system to obtain a robust quasi optimal adaptive control law. Such an approach is used ..... therefore reduces to the tracking of the singular value xsing given by eq. (8). For the .... [7] I. Smets, G. Bastin, and J. Van Impe. Feedback stabilisation
I, 4th Edition, read online Dynamic Programming and Optimal Control, Vol. .... been instrumental in the recent spectacular success of computer Go programs.
IRISA/INRIA Campus de Beaulieu 35042 Rennes Cedex, France npapadak ... a term related to the discrepancy between the state vari- ables evolution law and ...
recognition, data processing, filtering, clustering, blind signal separation, compression, system identification and control, pattern recognition, medical diagnosis, financial applications, data mining, visualisation and e-mail spam filtering [5], [4
Sep 27, 2006 - 2005) Recent examples of best practice include work at the New Zealand ... (Brayton and Tinsley, 1996), the Bank of Canada (Black et al, 1994) and the Bank ..... the domestic interest rate and the overseas interest rate, rw.
