E ENG 433, Control Systems
E E 581 Optimal Control, Spring 2014Instructor: Hossein Jula
Homework #9
Assigned: Thu., 04/03/2014Due: Thu., 04/17/2014, at 11:30 a.m.
(a two-week homework assignment)
1) Lewis, Problem 3.2-3.
2) Lewis, Problem 3.4-6. Let the system
have performance index of
.a. For use the eigen-structure of the Hamiltonian matrix to
determine the steady-state Riccati solution. Hence find the
steady-state Kalman gain and closed-loop poles.b. For use the
Chang-Letov equation to find the optimal feedback gain .c. Plot the
root locus for the closed-loop plant as varies from to 0.
3) Given the system
And the cost function:
Do the following.a) Write the Hamiltonian matrix.b) Write the
Riccati equation.c) Find the optimal feedback gain (this can be
done numerically)d) Find the steady-state optimal gain using
eigenvalue method.e) What is the pole of the steady state, closed
loop system?f) How does this pole change if the state weighting is
increasing from 2 to 10?g) If , what is the optimal cost?
4) Design a system for controlling the shaft angle of a dc
motor. The desired shaft angle is zero degree. A block diagram for
the dc motor is given below,
where the is the motor shaft angle, and is the dc voltage
applied to the motor. The cost function is
a) Design a control system to minimize the cost function.
Simulate the closed-loop system with an initial shaft angle of 10
degrees and initial shaft velocity of zero. Using the steady-state
gains, simulate the closed-loop system with the same initial
conditions. Plot the plant state and the control input for those
two simulations and compare the results. Compute the cost for each
of these two controllers and compare.b) Repeat the above when the
final time in the cost function is changed to 0.2. Comment on the
results.c) Repeat the above when the final time in the cost
function is changed to 0.1. Comment on the results.
