MECH 6323 HW 6

MECH 6323 HW 6 
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1. Let M and N be matrices of suitable dimension and let ∆ be a structured uncertainty. Prove
or disprove the follow:
(a) µ∆(M) = 0 =⇒ M = 0.
(b) µ∆(M1 + M2) ≤ µ∆(M1) + µ∆(M2).
(c) µ∆(αM) = |α| µ∆(M).
(d) µ∆(I) = 1.
(e) µ∆(MN) ≤ σ¯(M) µ∆(N).
(f) µ∆(MN) ≤ σ¯(N) µ∆(M).
2. Let ∆ = 
∆1 0
0 ∆2

, where ∆i are structured uncertainties. Show that
µ∆
 M11 M12
0 M22  = max { µ∆1
(M11), µ∆2
(M22) }.
3. Work through the exercise on Computing Complex µ. The attached zip file includes relevant
m-files along with the pdf for this exercise.
4. Consider a simple model for a car:
m v˙ = −b v + F
where v is the velocity, m is the mass, b is the wind drag coefficient, and F is the force generated
due to the engine. Assume that F is proportional to the engine throttle angle: F = cu, where u
is the engine throttle and c is the force constant. The vehicle model can be written as
m v˙ = −b v + c u.
Moreover, assume the throttle actuator dynamics from ucmd to u can be modeled as a
first-order lag, 1
τs + 1 . Thus, the nominal vehicle model from ucmd to v is given by P(s) =
c
ms + b
1
τs + 1 . The parameter values for the nominal model are given by m = 2150 kg, b =
20 N s/m, c = 150 N/deg, and τ = 0.1 sec.
(a) A cruise control algorithm is designed to control the vehicle velocity v to track a desired
speed vdes set by the driver. Let this tracking objective be specified as the requirement
|S(jω)| ≤ |B(jω)| for all ω with the performance bound
B(s) = s + 0.02
0.5 s + 1
.
Provide a brief interpretation for the performance objective specified by the bound B(s).
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(b) Design a simple proportional control law that achieves nominal performance. What is the
time constant of the closed-loop system? Submit a single Bode magnitude plot with S(jω)
and |B(jω)|.
(c) Next we consider the effect of model uncertainty. The vehicle mass will vary depending on
the number of passengers, etc. Assume m = 2150 ± 150 kg. The other parameters will also
have some uncertainty. Assume b is 20% uncertain and c is 10% uncertain. Finally, assume
the actuator time constant is in the range τ ∈ [0.05, 0.2]. Denote the set of all models that
arise over these parameter ranges by A. For simplicity, we will “cover” the uncertainty with
a multiplicative model. Specifically, we will choose an uncertainty weight Wu such that the
multiplicative uncertainty set M described by Wu contains all models in A, i.e., A ⊂ M.
Generate 20 samples Pˆ
i ∈ A by randomly sampling the uncertain parameter values. The
relative error between these samples and the nominal model is given by Ri
:=
|P − Pˆ
i|
|P|
. Plot
the relative error vs frequency for all 20 samples. Choose first order uncertainty weight
Wu(s) = a1s + a2
s + a3
such that |Wu(jω)| ≥ maxi Ri(ω) for all ω. This will ensure that M
contains all samples and hence we approximately have A ⊂ M. Plot the magnitude of your
weight |Wu(jω)| on the same plot as the relative errors.
(d) Does your proportional control law robustly stabilize all plants in M for the weight Wu
designed in the previous part?
(e) Construct the closed-loop sensitivity function Sˆ
i for each of the 20 samples Pˆ
i generated in
part (c). Hand in a single Bode magnitude plot with Sˆ(jω) (for i = 1, . . . , 20) and |B(jω)|.
Does your proportional control law achieve robust performance on these samples of the
plant dynamics, i.e., does K achieve the performance objective for all plants {Pˆ
i}
20
i=1 ⊂ M?
Comment: the steps covered in the last problem can be automated using MATLAB commands
ureal and ucover. While we will be using these commands later on in the course, for now, you
can complete this problem without them.
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