Optimal Control Assignment (Series 1)
1) Let
Find
a)
,
,
?
b)
?
c) adj A ?
d) Is A singular?
e) Rank of A?
f) A-1 ?
g) exp(A)
h) The definiteness type of A?
2) Given a linear system described by the following dynamic equation
Find
a) The transfer function
?
b) The state-transition matrix
?
c) Response of the system for when the components of the input vector u(t) are all
unit-step functions?
d) Check the controllability and observability of the system ?
Good luck
Dr. Abbas Chatraei
Optimal Control Assignment (Series 2)
(Chapters 2 and 3)
1) Find the extremal of the following functional
a)
b)
2) A first order system is given by
and performance index is
where, x(0)=x0 and x(tf) is free and tf being fixed. Show that the optimal state x*(t)
is given by
3) Find the optimal control for the plant
with performance criterion
And x(0)=[1,2]T . The additional conditions are given below:
a) F11=0, F22=0 and fixed-final conditions tf=2, x(2)=[4,6]T.
b) F11=3, F22=5, and free-final conditions x(tf)=[4,6]T and tf is free.
4) A DC motor speed control system is described by a second order state equation
where, xl(t) = the speed of the motor, and x2(t) = the current in the armature
circuit and the control input u(t) = the voltage input to an amplifier supplying the
motor.
Find the open-loop optimal control to keep the speed constant at a particular
value and the system to respond the regulated value with minimum energy
consumption.
5) The linearized state equations of an inverted pendulum on a cart are as follows:
u(t)
Cart
where, xl (t) = is horizontal linear displacement of the cart, x2(t) = is linear
velocity of the cart, x3(t) = is angular position of the pendulum from vertical line,
x4(t) = is angular velocity, and u(t) = is the horizontal force applied to the cart.
Find the open-loop, optimal control to keep the pendulum in a vertical position
with minimum energy consumption.
Good Luck
Dr. A. Chatraei
Optimal Control Assignment (Series 3)
kl’
Optimal Control Assignments, Series 4
Good Luck
Dr. A. Chatraei
Optimal Control Assignment
Series 5
Problem2. Solve the following OCP using dynamic programming method
Problem3. Solve the following OCP using dynamic programming method
 0  x1 (k )  5
Constraints: 
; 2  u( k )  3
0

x
(
k
)


3
2

Good Luck
Dr. A. Chatraei