Boundedness and Ultimate Boundedness
Definition 4.3
The solutions of x = f (t, x) are
uniformly bounded if there exists c > 0, independent of
t0 , and for every a (0, c), there is > 0, dependent on
a but independent of t0 , such that
kx(t0 )k a kx(t)k , t

V (x) = xT P x,
P = P T > 0,
c = cfw_V (x) c
If D = cfw_kxk < r, then c D if
c < min xT P x = min (P )r 2
kxk=r
If D = cfw_|bT x| < r, where b Rn , then
min xT P x =
|bT x|=r
r2
bT P 1 b
Therefore, c D = cfw_|bTi x| < ri , i = 1, . . . , p, if
c < min
1ip

Example 4.8
x 2 = (1 + x21 )x1 x2 + M cos t,
x 1 = x2 ,
With M = 0,
T
V (x) = x
M 0
x 2 = (1 + x21 )x1 x2 = h(x1 ) x2
1
2
1
2
1
2
1
V (x) = xT
x+ 2
3
2
1
2
1
2
1
Z
x1
(y + y 3 ) dy (Example 3.7)
0
x + 1 x41 def
= xT P x + 12 x41
2
Nonlinear Control Lec

Perturbed Systems: Nonvanishing Perturbation
Nominal System:
x = f (x),
f (0) = 0
Perturbed System:
x = f (x) + g(t, x),
g(t, 0) 6= 0
Case 1:(Lemma 4.3) The origin of x = f (x) is exponentially
stable
Case 2:(Lemma 4.4) The origin of x = f (x) is asymptot

Let 1 and 2 be class K functions such that
1 (kxk) V (x) 2 (kxk)
V (x) c 1 (kxk) c kxk 11 (c)
If Br D,
c = 1 (r) c Br D
kxk V (x) 2 ()
= 2 () B
What is the ultimate bound?
V (x) 1 (kxk) kxk 11 () = 11 (2 ()
Nonlinear Control Lecture # 5 Time Varying and

Example 3.2
x = x3
The origin is asymptotically stable
x(0)
x(t) = p
1 + 2tx2 (0)
x(t) does not satisfy |x(t)| ket |x(0)| because
|x(t)| ket |x(0)|
e2t
k2
1 + 2tx2 (0)
e2t
=
t 1 + 2tx2 (0)
Impossible because lim
Nonlinear Control Lecture # 2 Stability o

A set M is an invariant set with respect to x = f (x) if
x(0) M x(t) M, t R
Examples:
Equilibrium points
Limit Cycles
A set M is a positively invariant set with respect to x = f (x)
if
x(0) M x(t) M, t 0
Example; The set c = cfw_V (x) c with V (x) 0 in c

The distance from a point p to a set M is defined by
dist(p, M) = inf kp xk
xM
x(t) approaches a set M as t approaches infinity, if for each
> 0 there is T > 0 such that
dist(x(t), M) < , t > T
Example: every solution x(t) starting sufficiently near a st

Converse Lyapunov Theorems
Theorem 3.8 (Exponential Stability)
Let x = 0 be an exponentially stable equilibrium point for the
system x = f (x), where f is continuously differentiable on
D = cfw_kxk < r. Let k, , and r0 be positive constants with
r0 < r/k

Local input-to-state stability of x = f (x, u) is equivalent to
asymptotic stability of the origin of x = f (x, 0)
Lemma 4.7
Suppose f (x, u) is locally Lipschitz in (x, u) in some
neighborhood of (x = 0, u = 0). Then, the system
x = f (x, u) is locally i

T
T
V (x) = x P x = x
2 1
1 1
x
V (x) is negative definite in cfw_|x1 + x2 | 1
bT = [1 1],
c=
min
|x1 +x2 |=1
xT P x =
1
=1
bT P 1 b
The region of attraction is estimated by cfw_V (x) 1
Nonlinear Control Lecture # 3 Stability of Equilibrium Points

Remarks
The ultimate bound is independent of the initial state
The ultimate bound is a class K function of ; hence, the
smaller the value of , the smaller the ultimate bound.
As 0, the ultimate bound approaches zero
Nonlinear Control Lecture # 5 Time Vary

Case 2: The origin of the nominal system is asymptotically
stable
V
V
V
V (t, x) =
f (x) +
g(t, x) W3 (x) +
g(t, x)
x
x
x
Under what condition will the following inequality hold?
V
x g(t, x) < W3 (x)
Special Case: Quadratic-Type Lyapunov function
V

Example 4.12
x = x3 + u
The origin of x = x3 is globally asymptotically stable
V = 21 x2
V
= x4 + xu
= (1 )x4 x4 + xu
1/3
(1 )x4 , |x| |u|
0<1
The system is ISS with (r) = (r/)1/3
Nonlinear Control Lecture # 5 Time Varying and Perturbed Systems

Example 4.6
x = x3 + g(t, x)
V (x) = x4 is a quadratic-type Lyapunov function for x = x3
V
V
3
6
= 4|x|3
(x ) = 4x ,
x
x
(x) = |x|3 ,
c3 = 4,
Suppose |g(t, x)| |x|3 , x,
c4 = 4
with < 1
V (t, x) 4(1 )2 (x)
Hence, the origin is a globally uniformly as

Example 4.4
x = Ax + g(t, x);
Q = QT > 0;
A is Hurwitz;
P A + AT P = Q;
kg(t, x)k kxk
V (x) = xT P x
min(P )kxk2 V (x) max (P )kxk2
V
Ax = xT Qx min (Q)kxk2
x
V
T
2
x g = k2x P gk 2kP kkxkkgk 2kP kkxk
V (t, x) min (Q)kxk2 + 2max (P )kxk2
The origin is

Proof
D
0 < r , Br = cfw_kxk r
B
r
B
= min V (x) > 0
kxk=r
0<
= cfw_x Br | V (x)
kxk V (x) <
Nonlinear Control Lecture # 2 Stability of Equilibrium Points

Compute the integral
V (x) =
Z
x
T
g (y) dy =
0
Z
0
x
n
X
gi (y) dyi
i=1
over any path joining the origin to x; for example
Z x1
Z x2
g2 (x1 , y2 , 0, . . . , 0) dy2
g1 (y1 , 0, . . . , 0) dy1 +
V (x) =
0
0
Z xn
gn (x1 , x2 , . . . , xn1 , yn ) dyn
+ +
0

Lemma 4.6
If the systems = f1 (, ) and = f2 (, u) are input-to-state
stable, then the cascade connection
= f1 (, ),
= f2 (, u)
is input-to-state stable. Consequently, If = f1 (, ) is
input-to-state stable and the origin of = f2 () is globally
asymptotic

V (x) xT Qx + 2kP G(x)k kxk2
Given any positive constant k < 1, we can find r > 0 such that
2kP G(x)k < kmin (Q), kxk < r
xT Qx min(Q)kxk2 xT Qx min (Q)kxk2
V (x) (1 k)min (Q)kxk2 , kxk < r
V (x) = xT P x is a Lyapunov function for x = f (x)
Nonlinear Con

Lemma 3.1
If a solution x(t) of x = f (x) is bounded and belongs to D for
t 0, then its positive limit set L+ is a nonempty, compact,
invariant set. Moreover, x(t) approaches L+ as t
LaSalles Theorem (3.4)
Let f (x) be a locally Lipschitz function define

Theorem 3.9 (Asymptotic Stability)
Let x = 0 be an asymptotically stable equilibrium point for
x = f (x), where f is locally Lipschitz on a domain D Rn
that contains the origin. Let RA D be the region of
attraction of x = 0. Then, there is a smooth, posit

Terminology
V (0) = 0, V (x) 0 for x 6= 0 Positive semidefinite
V (0) = 0, V (x) > 0 for x 6= 0
Positive definite
V (0) = 0, V (x) 0 for x 6= 0 Negative semidefinite
V (0) = 0, V (x) < 0 for x 6= 0
Negative definite
kxk V (x)
Radially unbounded
Lyapunov

Theorem 3.7
A matrix A is Hurwitz if and only if for every Q = QT > 0
there is P = P T > 0 that satisfies the Lyapunov equation
P A + AT P = Q
Moreover, if A is Hurwitz, then P is the unique solution
Nonlinear Control Lecture # 3 Stability of Equilibrium

Example: Pendulum equation without friction
x 1 = x2 ,
x 2 = a sin x1
V (x) = a(1 cos x1 ) + 21 x22
V (0) = 0 and V (x) is positive definite over the domain
2 < x1 < 2
V (x) = ax 1 sin x1 + x2 x 2 = ax2 sin x1 ax2 sin x1 = 0
The origin is stable
Since V (

V (x) = a on L+ and L+ invariant V (x) = 0, x L+
L+ M E
x(t) approaches L+ x(t) approaches M (as t )
Nonlinear Control Lecture # 3 Stability of Equilibrium Points

Alternative Bound on :
V (x) = kxk2 + 2xT P g(x) kxk2 + 18 x32 ([2 5]x)
kxk2 +
29
x22 kxk2
8
Over c , x22 (1.8194)2 c
V (x) 1 829 (1.8194)2c kxk2
c
kxk2
= 1
0.448
If < 0.448/c, the origin will be exponentially stable and c
will be an estimate of the regi

x22 + |x2 | |u| 0 for |x2 | |u|/ and has a maximum
value of u2/(4) for |x2 | < |u|/
x21
|u|
u2
or x22 2
2
kxk2
|u| u2
+ 2
2
x41 x22 + |x2 | |u| 0
x41 x22 + |x2 | |u| 0
(r) =
r
r2
r
+ 2
2
V (1 )[x41 + x22 ], kxk (|u|)
The system is ISS
Nonlinear Contr