Linear Control Design – Final Exam

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