Linear Control Design – Final Exam

Linear Control Design – Final Exam
Consider the following data associated to a CTLTI model:
% Aircraft model AIRC. From Appendix F of:
%
% Y. S. Hung and A. G. J. MacFarlane, "Multivariable Feedback: a quasi
% classical approach", Lectures Notes in Control and Information
% Sciences, vol. 40, Berlin, Springer-Verlag, 1982.
%
% apud J. M. Maciejowski, "Multivariable Feedback Design", Cambridge
% University Press, 1989.
%
% Constants:
% h0 -> datum altitude = 3000 m (from sea level)
% v0 -> cruise forward speed = 500 Km/H
%
% Inputs:
% u1 -> spoiler angle (tenths of degree)
% u2 -> forward acceleration (m/s^2)
% u3 -> elevator angle (degrees)
%
% States:
% x1 -> altitude relative to h0 (m)
% (x1 = altitude - h0)
% x2 -> forward speed relative to v0 (m/s)
% (x2 = forward speed - v0)
% x3 -> pitch angle (degrees)
% x4 -> pitch rate (degrees/s)
% x5 -> vertical speed (m/s)
%
% Outputs:
% y1 -> x1
% y2 -> x2
% y3 -> x3
%
% Linearized continuous-time plant parameters:
A = [0 0 1.1320 0 -1;
0 -0.0538 -0.1712 0 0.0705;
0 0 0 1 0;
0 0.0485 0 -0.8556 -1.013;
0 -0.2909 0 1.0532 -0.6859]
Bu = [0 0 0;
-0.120 1 0;
0 0 0;
1

4.4190 0 -1.665;
1.5750 0 -0.0732]
Cy = [eye(3) zeros(3,2)]
1. Study the controllability properties of the inputs (is the plant controllable
from individual inputs?) Study the observability properties of the outputs
(is the plant observable from individual outputs?)
2. State the following problem as an H2 norm minimization problem: using
only the spoiler angle (u1) as a control input, design a controller that
minimizes the sum of the variance of the pitch angle (y3) and the variance
of the control input (u1) when the aircraft is subject to zero mean unity
variance white-noise perturbations on the altitude, forward and vertical
speeds (x1, x2, x5).
3. State an appropriate SDP (LMI set and linear objective) that can be used
to solve the above problem. Solve the problem by computing a solution
to this SDP.
4. Solve the same problem using a Riccati equation (matlab function lqr).
Are you having trouble with this? Can you explain why? Did you have
this problem with the SDP solution? If not, how can you tell something is
“wrong” by looking at the matrices that came as a solution to the SDP?
5. The design specifications for this aircraft are:
Desired Pitch Angle Variance (x3) 1
Desired Altitude Variance (x1) 0.25
Can you attain that performance with the above design? If not, why?
6. Consider now that you can also use the elevator angle (u3) as a control
input. In order to prevent saturation of the actuators, the variance of the
control inputs should be limited to the following values:
Maximum Spoiler Angle Variance (u1) 100
Maximum Elevator Angle Variance (u3) 100
State a multi-objective H2 norm minimization control design problem that
can be used to check the existence of a controller satisfying simultaneously
all performance and control requirements. State this problem as an SDP.
For the given values, does such a controller exist? In the affirmative
case design a controller that satisfies the requirements and minimizes the
elevator angle (u3) variance.
7. You would make the life of the person who designs the wiring of this
aircraft easier if the spoiler angle control input (u1) feedback only the
2

pitch angle (x3) and the pitch rate (x4), and the elevator angle input (u3)
feedback only the altitude (x1), the forward speed (x2) and the vertical
speed (x5). How can you modify the SDP obtained in the previous item
to account for this particular control structure? Can you design such a
structured controller satisfying all performance and control specifications?
If not, relax, and relax the performance requirements a little bit so as to
obtain a feasible controller. Design a controller that satisfies the relaxed
requirements and minimizes the elevator angle (u3) variance. Compute
the actual performance of this controller. Does the actual performance
satisfies the requirements?
8. Are you happy with the above designs? (Justify with plots, simulations,
numbers, or whatever you think necessary.)
9. Are you happy with this course? Do you have any comments or suggestions?
Do not expect that the answer given to this question be able to
change your score. Not even positively :).
10. (*) If you some extra time, try the design of item 6) with a full order dynamic
output feedback controller. Assume you are measuring the altitude
(y1) the forward speed (y2) and the pitch angle (y2).
Some other details:
• This exams and all homeworks are due on March 21 (Wednesday), at
noon. No delays, please.
• This exam will be assigned a grade in the range 0−10. If you do questions
1) to 8) 100% correctly you have a 10. Item 10) grants you 3 extra points.
That is, your exam will be assigned a grade in the range 0 − 13, with the
final grade “saturating” at 10.
• Your final grade will be a mix of the results of the final and the homeworks.
• If you think you can do it by Friday, leave it on my mailbox. If you come to
my office on Monday I can hand it back to you with a “preliminary grade”.
This way, you’ll have a change to correct your mistakes till Wednesday.
The final grade will be computed on the Wednesday version.
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