24-677 (LCS) Homework 5 solution

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
.
1