APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY
Second Semester M.Tech Degree Examination, May 2016
Branch: Electrical and Electronics Engineering
Streams: Control Systems, Guidance and Navigational Control

OIEE6102: OPTIMAL CONTROL THEORY

Time: 3 hrs Max. Marks: 60

Instruction: Answer any two fun questions from each part =
PART A

1. (a) How will you formulate an optimal control problem? How the perforrpance measure (4)
is selected in various physical problems?

(b) The pitch angle (t) of a manned spacecraft is to be controlled by a gas expulsion (5)
; system. The differential equation that describes the motion is
d29(t)
where 115 the moment of inertia and Mt) is the + = A(t) torque produced by gas jets. Selecting Cl(t) =
Q(t) and C2(t) = O(t) as state variables and u(t) dt = as the control gives the state equations as ~

21(६)= c2(t)
2ಬ ಇ)
The primary objective of the control system is to maintain the angular position near zero. This is to be

accomplished with small acceleration (control input). Formulate the optimal control problem.

2. (3) Derive Weierstrass- Erdmann corner conditions for piecewise smooth extremals. (6) (0) Find the necessary conditions that must
be satisfied by the curve of smallest (3) length which lies on the sphere + w3(t) + t 2 = R?for t e [to, tf] and joins the specified points
wo, to and and t!

3. (a) Find the extremal of (4)
) =(c2(t) +> (t)}de
if (0) = 1; 0(0 is free.
(b).Suppose that the system (5)
(1) = C2(t) — ct(t)
മ(0- - 211(0 — 3C2(t) + u(t) is to
be controlled to minimise the performance measure = -
[C12(t) + + 7>
Find the necessary conditions for optimal control.
PART B

4. In which context Pontryagin's Minimum Principle is used in an optimal control prob- (9) lem?
State and prove Pontryagin's Minimum principle.

5. (a) Compare Bang Bang control and Bang off Bang control (4)