![JNTUW
ORLD
Code No: D7501
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
M.Tech II - Semester Examinations, March/April 2011
OPTIMAL CONTROL THEORY
(CONTROL SYSTEMS)
Time: 3hours Max. Marks: 60
Answer any five questions
All questions carry equal marks
- - -
1. a) State and explain the Bellman’s principle of optimality.
b) Explain the concept of applying the principle of optimality in decision making process
to get optimal trajectory. [6+6]
2. a) Derive recurrence relation for time-invariant systems of dynamic programming for N-
stage process.
b) Explain the characteristics of dynamic programming solution. [6+6]
3. A first – order linear system is given )()(10 tutxx +−=
•
is to be controlled to
minimize the performance measure
dt(t)]u
2
1
(t)x
4
1
[(0.04)x
2
1 2
0.04
0
22
++= ∫J
The admissible state and control values are not constrained by any boundaries. Find
the optimal control law by using the Hamilton-Jacobi-Bellman equation. [12]
4. a) State and explain the fundamental theorem of the calculus of variations.
b) Determine an extremal for the functional
dt(t)]x)()(2)(4[)(
2
0
2
2
∫
••
++= txtxtxxJ
The boundary conditions are x(0)=1, and x(2) is free. [5+7]
5. The system is given by:
)()( 21 txtx =
•
)()()](1[)()( 2
2
112 tutxtxtxtx +−+−=
•
is to be transferred from the origin to the surface
9]2[]5)([]4)([ 22
2
2
1 =−+−+− ttxtx
with minimum fuel expenditure. The final time is free, and 0.1)( ≤tu
a) Determine the costate equations.
b) Determine the control that minimizes the Hamiltonian.
c) Determine the boundary conditions at t=tf. [4+4+4]
6. State the Two-point boundary value problems and explain how it can be solved
though variation of extremals. Also summarize the algorithm of variation of
extremals. [12]
Contd…..2
R09
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![JNTUW
ORLD
::2::
7. State and explain the state estimation problem and with suitable block diagram derive
mathematical model of the non-statistical full order estimator. Also summarize its
algorithm. [12]
8. Consider the plant vtxtx +=
•
)()( , y(t) = x(t)+w, with
)()]()([)]()([ τδττ −== twtwEvtvE and v and w independent. Suppose that at time
zero, x(0) is known to be zero. Design an optimal estimator. [12]
*****
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R09 optimal control theory
- 1. JNTUW ORLD Code No: D7501 JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD M.Tech II - Semester Examinations, March/April 2011 OPTIMAL CONTROL THEORY (CONTROL SYSTEMS) Time: 3hours Max. Marks: 60 Answer any five questions All questions carry equal marks - - - 1. a) State and explain the Bellman’s principle of optimality. b) Explain the concept of applying the principle of optimality in decision making process to get optimal trajectory. [6+6] 2. a) Derive recurrence relation for time-invariant systems of dynamic programming for N- stage process. b) Explain the characteristics of dynamic programming solution. [6+6] 3. A first – order linear system is given )()(10 tutxx +−= • is to be controlled to minimize the performance measure dt(t)]u 2 1 (t)x 4 1 [(0.04)x 2 1 2 0.04 0 22 ++= ∫J The admissible state and control values are not constrained by any boundaries. Find the optimal control law by using the Hamilton-Jacobi-Bellman equation. [12] 4. a) State and explain the fundamental theorem of the calculus of variations. b) Determine an extremal for the functional dt(t)]x)()(2)(4[)( 2 0 2 2 ∫ •• ++= txtxtxxJ The boundary conditions are x(0)=1, and x(2) is free. [5+7] 5. The system is given by: )()( 21 txtx = • )()()](1[)()( 2 2 112 tutxtxtxtx +−+−= • is to be transferred from the origin to the surface 9]2[]5)([]4)([ 22 2 2 1 =−+−+− ttxtx with minimum fuel expenditure. The final time is free, and 0.1)( ≤tu a) Determine the costate equations. b) Determine the control that minimizes the Hamiltonian. c) Determine the boundary conditions at t=tf. [4+4+4] 6. State the Two-point boundary value problems and explain how it can be solved though variation of extremals. Also summarize the algorithm of variation of extremals. [12] Contd…..2 R09 www.jntuworld.com www.jntuworld.com
- 2. JNTUW ORLD ::2:: 7. State and explain the state estimation problem and with suitable block diagram derive mathematical model of the non-statistical full order estimator. Also summarize its algorithm. [12] 8. Consider the plant vtxtx += • )()( , y(t) = x(t)+w, with )()]()([)]()([ τδττ −== twtwEvtvE and v and w independent. Suppose that at time zero, x(0) is known to be zero. Design an optimal estimator. [12] ***** www.jntuworld.com www.jntuworld.com