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
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:
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
Cart u(t)
Optimal Control Assignment (Series 3)
Good Luck Dr. A. Chatraei
Series 5
Problem2.
Solve the following OCP using dynamic programming method
Problem3. Solve
the following OCP using dynamic programming method1 2
0 ( ) 5
Constraints: ; 2 ( ) 3
0 ( ) 3
x k u k
x k
Good Luck Dr. A. Chatraei
