Stochastic Processes in Dynamic Systems
ECE 272A
Midterm Exam-Solution
Instructor: Behrouz Touri
1. For each of the following statements, determine if the statement is true or false. If
you believe the statement is wrong, provide a counterexample, otherwise, prove the
statement.
a. If a linear system x˙ = A(t)x has a solution for some x(0) = x0 ∈ Rn, the solution is
always unique.
b. A dynamics x˙ = f(x) cannot have multiple marginally stable equilibrium.
Solution:
a. False.
Consider the one-dimensional linear system defined by x˙ = A(t)x, where
A(t) =
{
2
t t > 0
0 t = 0
and let x(0) = 0. For this linear system you can verify that x(t) = 0, for all t, is a
solution. Also, note that x(t) = t2 for all t ≥ 0 is another solution.
b. False.
The dynamics x˙ = f(x) where f(x) = 0 for all x ∈ R has multiple marginally stable
equilibrium (namely all the points on the real line).
2. A model for the unicycle’s movement on the plain whose linear velocity is v and angular
velocity is ω is given by
p˙ =
(
v cos(θ)
v sin(θ)
)
, θ˙ = ω, (1)
where p(t) ∈ R2 is the the Cartesian coordinates of unicycle and θ its orientation. Let
the input to this system be u =
(
v
ω
)
∈ R2.
(a) Let
x =
x1x2
x3
=
p1 cos(θ) + (p2 1) sin(θ) p1 sin(θ) + (p2 1) cos(θ)
θ
.
Write down the state-space formulation of this system using state variable x and
input variable u, i.e., find out f(x, u) such that x˙ = f(x, u).
1
Solution: By calculating x˙ and plugging in p˙ and θ˙, we have
x˙ =
p˙1 cos(θ) p1θ˙ sin(θ) + p˙2 sin(θ) + (p2 1)θ˙ cos(θ) p˙1 sin(θ) + p1θ˙ cos(θ) + p˙2 cos(θ) (p2 1)θ˙ sin(θ)
θ˙
