MECH 6323 HW 5

MECH 6323 HW 5 
yes

Final Project in lieu of Final Exam Please start thinking about your final project. Your
project should use the techniques and tools covered in this course on a particular physical system.
This may or may not be related to your research. In the project, you can use analysis tools to assess
the robustness of a given feedback system. Alternatively, you can use optimal control techniques
(LQR, H1, H2) to synthesize a controller. For now, please specify the system that you will consider.
Give a brief 1-page summary describing the physical system that you will study. This should include
a description of the (linearized) dynamics as well as some initial simulation results (e.g., open-loop
step response if the system is stable). You may work individually or in a group of two. Just make
sure you mention the names of both group members on the summary. Deadline: Please email me
your 1-page project summaries before the beginning of class on April 6th.
1. Consider the feedback configuration in the figure below.
Prove that

I −K
−G22 I
−1
=

(I − KG22)
−1
(I − KG22)
−1 K
(I − G22K)
−1 G22 (I − G22K)
−1

| {z }
.
H(G22, K)
2. Consider the block diagram in the previous exercise. Suppose G22 and K have minimal state
space realizations 
A B2
C2 D22 
and 
AK BK
CK DK

. Let
T =












A1
z }| {
A 0
0 AK
B
z }| {
B2 0
0 BK
0 −CK
−C2 0
| {z }
−C
I −DK
−D22 I
| {z }
D












.
Thus,
T
−1 =
"
A1 + BD−1C BD−1
D−1C D−1
#
=: "
A¯ B¯
C¯ D¯
#
1
where
D¯ = D−1 =

I −DK
−D22 I
−1
=

I + (I − D22DK)
−1D22 DK(I − D22DK)
−1
(I − D22DK)
−1D22 (I − D22DK)
−1

=

I 0
0 0 
+

(I − D22DK)
−1D22 DK(I − D22DK)
−1
(I − D22DK)
−1D22 (I − D22DK)
−1

=

I 0
0 0 
+

DK
I

(I − D22DK)
−1

D22 I

.
Thus,
A¯ = A1 + BD−1C =

A B2CK
0 AK

+

B2DK
BK

(I − D22DK)
−1

C2 D22CK

.
Prove that the following are equivalent:
(a) (A, ¯ B, ¯ C, ¯ D¯) is stabilizable and detectable;
(b) (A, B2, C2, D22) and (AK, BK, CK, DK) are stabilizable and detectable.
3. For the feedback configuration in the first exercise, prove that:
1. If K is stable then the closed loop interconnection is stable if and only if G22 (I − KG22)
−1
is stable.
2. If G22 is stable then the closed loop interconnection is stable if and only if K (I − G22K)
−1
is stable.
4. Consider the standard negative feedback loop with the nominal plant dynamics P(s) = 1
s+1
and controller K(s) = 20. Assume the “true” dynamics lie within the following multiplicative
uncertainty set:
M :=
n
Pˆ = P (1 + Wu∆) : k∆k∞ < 1 and ∆ stableo
.
Assume the uncertainty weight is Wu(s) = 2s + 1
s + 10 .
(a) Provide an interpretation for the uncertainty described by the weight Wu.
(b) Is the nominal feedback system stable? What are the gain and phase margins of the nominal
loop L = PK?
(c) The robust stability condition for this type of multiplicative uncertainty is stated as: K
stabilizes all Pˆ ∈ M if and only if kWuTk∞ ≤ 1. Does K robustly stabilize all models in
M based on this condition?
(d) We can construct the uncertainty set M in MATLAB using the following commands:
>> Delta = ultidyn(0Delta0
, [1 1]);
>> Phat = P ∗ (1 + Wu ∗ Delta);
>> Lhat = Phat ∗ K;
>> That = feedback(Lhat, 1);
2