Description
1: 20 points
An LTI system is described by the equations
x˙ =
a 0
1 −1
x.
Use Lyapunov’s direct method to determine the range of variable a for which the system is
asymptotically stable.
3: 30 points
For the three systems given below, determine the stability (i.e. Lyapunov, asymptotic, or BIBO).
(a)
x(k + 1) =
1 0
−0.5 0.5
x(k) +
1
−1
u(k)
y(k) =
5 5
x(k)
(b)
x˙ =
−7 −2 6
2 −3 −2
−2 −2 1
x +
1 1
1 −1
1 0
u
y =
−1 −1 2
1 1 −1
x
(c) Provide answers for both DT and CT interpretations of the following matrices.
A =
2 −5
−4 0
, B =
1
−1
, C =
1 1
, D = 0
4: 20 points
Use Lyapunov’s stability theorem to determine the stability of the following system.
x˙ 1 = x2 − x1x
2
2
x˙ 2 = −x
3
1
HINT: Consider the Lyapunov function V (x1, x2) = x
4
1 + 2x
2
2
.
5: 20 points
Use instability criterion with the function V (x1, x2) = x
2
1 − x
2
2
to prove that the origin of the
following system is unstable.
x˙ 1 = 3×1 + x
3
2
x˙ 2 = −x2 + x
2
1
6: 20 points
Consider the equation of motion for a simple pendulum
¨θ +
g
L
sin θ = 0
(a) Using the total energy of the system as a Lyapunov function, show that θ0 = 0 is stable in the
sense of Lyapunov.
(b) Using the system energy as a Lyapunov function, show that the equilibrium point θ0 = π is
unstable. For this you will need to use a change of variables x = θ − π to give an equivalent
system with x0 = 0 the relevant equilibrium point.
