ELEG 5443
Nonlinear Systems and Control
Lecture # 1
Introduction
p. 1/1
Nonlinear State Model
x 1 = f1 (t, x1 , . . . , xn , u1 , . . . , up )
x 2 = f2 (t, x1 , . . . , xn , u1 , . . . , up )
.
.
.
.
x n = fn (t, x1 , . . . , xn , u1 , . . . , up )
x i d

ELEG 5443
Nonlinear Systems and Control
Lecture # 2
Examples of Nonlinear Systems
p. 1/1
Pendulum Equation
l
mg
ml = mg sin kl
x1 = ,
x2 =
p. 2/1
x 1 = x2
x 2 =
g
l
sin x1
k
m
x2
Equilibrium Points:
0 = x2
0 =
g
l
sin x1
k
m
x2
(n, 0) for n = 0, 1,

ELEG 5443
Nonlinear Systems and Control
Lecture # 5
Limit Cycles
p. 1/?
Oscillation: A system oscillates when it has a nontrivial
periodic solution
x(t + T ) = x(t), t 0
Linear (Harmonic) Oscillator:
"
#
0
z =
z
0
z1 (t) = r0 cos(t + 0 ),
q
r0 = z12 (0

ELEG 5443
Nonlinear Systems and Control
Lecture # 9
Lyapunov Stability
p. 1/1
Quadratic Forms
V (x) = xT P x =
n X
n
X
pij xi xj ,
P = PT
i=1 j=1
min (P )kxk2 xT P x max (P )kxk2
P 0 (Positive semidefinite) if and only if i (P ) 0 i
P > 0 (Positive defin

ELEG 5443
Nonlinear Systems and Control
Lecture # 4
Qualitative Behavior Near
Equilibrium Points
&
Multiple Equilibria
p. 1/?
The qualitative behavior of a nonlinear system near an
equilibrium point can take one of the patterns we have seen
with linear s

ELEG 5443
Nonlinear Systems and Control
Lecture # 3
Second-Order Systems
p. 1/?
x 1 = f1 (x1 , x2 ) = f1 (x)
x 2 = f2 (x1 , x2 ) = f2 (x)
Let x(t) = (x1 (t), x2 (t) be a solution that starts at initial
state x0 = (x10 , x20 ). The locus in the x1 x2 plan

ELEG 5443
Nonlinear Systems and Control
Lecture # 8
Lyapunov Stability
p. 1/1
Let V (x) be a continuously differentiable function defined in
a domain D Rn ; 0 D . The derivative of V along the
trajectories of x = f (x) is
V (x) =
n
X
V
i=1
=
=
h
xi
x i =

ELEG 5443
Nonlinear Systems and Control
Lecture # 10
The Invariance Principle
p. 1/1
Example: Pendulum equation with friction
x 1 = x2
x 2 = a sin x1 bx2
V (x) = a(1 cos x1 ) +
1
2
x22
V (x) = ax 1 sin x1 + x2 x 2 = bx22
The origin is stable. V (x) is no

ELEG 5443
Nonlinear Systems and Control
Lecture # 11
Exponential Stability
&
Region of Attraction
p. 1/1
Exponential Stability:
The origin of x = f (x) is exponentially stable if and only if
the linearization of f (x) at the origin is Hurwitz
Theorem: Le

ELEG 5443
Nonlinear Systems and Control
Lecture # 7
Stability of Equilibrium Points
Basic Concepts & Linearization
p. 1/?
x = f (x)
f is locally Lipschitz over a domain D Rn
Suppose x
D is an equilibrium point; that is, f (
x) = 0
Characterize and study