ME 506 – Advanced Control Systems
Rose-Hulman Institute of Technology
Homework 4
Due: 1 October 2015
1) SVD practice with non-square matrices
Consider the following 3x2 and 2x3 matrices:
4 3
A =  1 − 1
− 1 1 
1 0 2
B=

0 4 0 
a) Determine the rank of each matrix. Compute (by hand) the SVD for each matrix and
check the MATLAB answer.
b) Explain the geometric properties (sketch ellipsoids, etc.) of the linear equation y=Ax
where x and y have the appropriate dimensions, in terms of the singular values and the
singular vectors (defined by the unitary matrices U and V).
c) repeat part b) for the linear equation y=Bx.
d) Use the SVD to compute the pseudo-inverse of A and B.
2) SVD for real matrix
Consider the linear equation: y=Ax
(Z)
where the real matrix A is given by
3 − 1 0 
A = 1 3 − 2
0 4 − 1 
a) Carry out the singular value decomposition (SVD) of A using the MATLAB software of
the form
A=UΣ
ΣVT
(Y)
and state the numerical values of the matrices in Eq. (3). Provide a brief geometrical
description, including a rough 3-D sketch, of the transformation defined by Eq (Z).
Also, find the SVD of A-1 without explicitly computing it.
b) Let ui and vi, i = 1,2,3, denote the column vectors of the matrices U and V, respectively, in
Eq. (Y). Suppose that
x=v1+2v2+3v3
then find y using the singular values fo A and the vectors ui. Next suppose that
y=u1+u2+3u3
then find x using the singular values of A and the vectors vi
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Rose-Hulman Institute of Technology
3) Eigenstructure Assignment Pre-Lab
Given the following state-space model of the two mass, two actuator ECP setup:
=
0
0
0
1
0
0
80.9074
0
433.22
0
= −433.22 −5.5089
+
0
0
0
1
0
0
287.338
0
−287.338 −3.9477
0
−89.0056
Design a static state feedback controller such that the mass 1 response is decoupled from
the mass 2 response, mass 1 has eigenvalues , ∈ #−30, −32$ and mass 2 has
eigenvalues
,
= −6 ± 9&
Implement your solution in a Matlab script that displays the feedback gain matrix,
closed-loop eigenvalues and eigenvectors. Also form two SISO transfer functions with
poles
,
, and
,
respectively, and use the step command to plot theoretical step
responses for each on a common axis.
ME 406 – Control Systems – Lab 2
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