EE 580WC
EXAM I
February 2020
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Section:
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Problem Weight Score
1
25
2
25
3
25
4
25
Total
100
INSTRUCTIONS
1. You have 48 hours to complete this exam.
2. You may use the text book and class notes.
3. You must complete the exam by hand and show all calculations. No credit is given for answers obtained directly
from MATLAB or any other CAD tool.
4. Solve each part of the problem in the space following the question. If you need more space, insert an additional
page that includes the question number; for example, Problem 1.2 Continued. NO credit will be given to
solutions that do not meet this requirement.
5. The quality of your analysis and evaluation is as important as your answers. Your reasoning must be precise
and clear. complete sentences should convey what you are doing. To receive credit, you must show your
work.
1
Problem 1: (25 Points)
1. (16 points) A system with input u(t) and output y(t) is represented by the ordinary differential equation
d4 y
d3 y
d2 y
dy
du
+6 3 +4 2 +2 +y = 6
− u.
4
dt
dt
dt
dt
dt
(a) (8 points) Draw an all-integrator diagram from which the controllable canonical state-space model can be
derived.
2
(b) (8 points) State the controllable canonical state-space representation by assigning x1 (t) and x4 (t) to the
output of the last and first integrators, respectively.
3
2. (9 points) A SISO system has the state-space representation
−1 6
0
ẋ =
x+
u
0 2
1
y =
1 1 x.
Determine the transfer function representation in terms of the A, B, and C matrices of the given state-space
representation. No credit will be given if you obtain the transfer function by unbundling the matrices and
taking the Laplace transform of the individual state equations for ẋ1 and ẋ2 .
4
Problem 2: (25 Points)
1. (14 points) Let V be a linear space consisting of all 2 × 2 matrices with real elements defined over the field <.
(a) (4 points) The trace of a matrix A is denoted by T r(A). For the 2 × 2 matrix
a11 a12
A =
,
a21 a22
the trace T r(A) is the sum of the diagonal elements
T r(A)
=
a11 + a22 .
If A and B are both n × n square matrices with real elements defined over <, then
T r(A)
T r(AB)
T r(AB)
T r(A + B)
= T r(AT )
= T r(BA)
= T r(B T AT )
= T r(A) + T r(B).
Show that < A, B > = T r(AT B) is a valid inner product.
5
6
(b) (4 points) A basis for this space is the set of matrices
2
1
1
1
−2
,
,
−2 −1
−1 −1
4
−1
2
Using the inner product defined in part (a), is the basis orthogonal?
7
,
−1 −1
2
2
(c) (6 points) Using the given basis, find the representation of the identity matrix
1 0
I=
.
0 1
8
9
2. (11 points) Determine all solutions of the equation Ax = y where

0 1
A= 1 2
2 0

 
1 2 −1
0
3 4 −1  and y =  1  .
2 0 2
2
10
11
Problem 3: (25 Points)
The differential equation
ÿ + 2ẏ + ẏ3 + 6y = u
describes a nonlinear mechanical system with input force u(t) and output displacement y(t).
1. (8 points) Determine a state-space representation of the form
ẋ =
y =
f(x, u)
g(x, u)
using the state vector x(t) with x1 (t) = y(t) and x2 (t) = ẏ(t).
2. (8 points) For a constant input force u(t) = uo = 6, determine the physically meaningful static equilibrium
state and the corresponding output ye .
12
3. (9 points) At the static equilibrium state in part 2, determine the linearized state-space model
δ ẋ =
Aδx + Bδu
=
Cδx + Dδu
δy
by calculating the numeric values of the matrices A, B, C, and D.
13
Problem 4: (25 Points)
1. (13 points) A system is represented using the state-space model
ẋ =
y
=




−1
1
1
0
 0 −2
1 x +  1 u
0
0 −3
0
0 1 2 x + u.
(a) (6 points) Determine the rank of the controllability matrix.
(b) (7 points) Determine the range space of the controllability matrix.
14
2. (12 points) Consider the same system as in part 1,
ẋ =
y
=




−1
1
1
0
 0 −2
1 x +  1 u
0
0 −3
0
0 1 2 x + u.
(a) (6 points) Determine the nullity of the observability matrix.
(b) (6 points) Determine the null space of the observability matrix.
15