Chapter 3
Fundamentals of Lyapunov
Theory
3.1 Nonlinear Systems and
Equilibrium Points
• Nonlinear Systems
A nonlinear dynamic system
x  f  x, t 
f : n×1 nonlinear vector function
x: n×1 state vector.
n : numbers of states, order of the systems.
(3.1)
—A solution x(t):
(i) a curve in state space as t varies
from 0 to ∞,
(ii) a state trajectory or a system
trajectory.
Closed-Loop Dynamics
If the plant dynamics is
x  f  x, u, t 
and some control law has been selected
u  g  x, t 
then the closed-loop dynamics is
x  f  x, g  x, t  , t 
Linear systems: A special class of
nonlinear systems
—Its dynamics:
x  A t  x
A(t): n×n matrix
• Autonomous
and Non-autonomous
systems
Linear systems are classified as either timevarying or time-invariant, depending on
whether the system matrix A varies with time or
not.
Autonomous and Nonautonomous
Definition 3.1: The nonlinear system
(3.1) is said to be autonomous if f does not
depend explicitly on time, i.e., if the
system’s state equation can be written as
x  f  x
Otherwise, the system is called nonautonomous.
---Linear time-invariant (LTI) systems:
autonomous.
—Linear time varying (LTV) systems:
non-autonomous.
Equilibrium Points
1. Definition 3.2: A state x* is an
equilibrium state (or equilibrium point)
of the system if once x(t) is equal to x*,
it remains equal to x* for all future time.
2. Example 3.1: The Pendulum
Consider the pendulum
MR   b  MgR sin   0
2
(3.5)
R: the pendulum’s length
M: its mass
b: the friction coefficient at the hinge
g: the gravity constant.
Letting
x1 = , x2 =  , the corresponding statespace equations is
x1  x2
b
g
x2  
x  sin x1
2 2
MR
R
—The equilibrium points:
(0 [π], 0)
x2 = 0, sin x1 = 0
the pendulum resting exactly at the
vertical up and down positions.
•Perturbation Dynamics
—Let x* (t) be the nominal motion trajectory,
corresponding to initial condition x*(0) = xo.
— Perturb x(0) = xo + δxo
x*  f  x* 
x  0  x0
x  f ( x)
x  0  x0   x0
Figure
3.2
—The motion error due to initial condition
perturbation:
e(t) = x(t)－x*(t)
and
e = f(x* + e,t)－f(x*, t) = g(e, t)
with
e(0) = δxo.
represent the perturbation dynamics
—g(0, t) = 0: the new dynamic system,
with e as state and g in place of f, has
an equilibrium point at the origin of the
state space.
- Studying the deviation of x(t) from x*(t) for
the original system  study the stability of
the perturbation dynamics with respect to the
equilibrium point 0.
- Perturbation dynamics is non-autonomous,
due to the presence of the nominal trajectory
x*(t).
•Stability and Instability
Definition 3.3：The equilibrium state x = 0 is
said to be stable (or Lyapunov stable) if , for
any R > 0, there exists r > 0, such that if
||x(0)|| < r, then ||x(t)|| < R for all t 0 .
Otherwise, the equilibrium point is unstable.
Figure 3.3：Concepts of stability
Phase Portrait of a Mass-spring System
Mass-spring system
x  x  0 , x(0)  x0
x(0)  0
Solution:
x(t )  x0 cos t
x(t )   x0 sin t
Equation of the trajectories:
x  x  x0
2
2
2
Figure 2.1: A mass-spring system and its phase
portrait
Example 2.5: A satellite control system
Figure 2.5:Satellite control using on-off thrusters
If u = -U, then
  2U  c1
2
Example 3.3: Instability of the Van
der Pol Oscillator
The Van der Pol oscillator
x1  x2
x2   x1  1  x12  x2
Origin: an unstable equilibrium point
Figure 3.4：Unstable origin of the Van der
Pol Oscillator
—Observation: system trajectories starting
from any non-zero initial states all
asymptotically approach a limit cycle.
—Illustration: if we choose R to be small
enough  the circle of radius R fall completely
within the closed-curve of the limit cycle, then
system trajectories starting near the origin will
eventually get out of this circle
Definition 3.4：An equilibrium point 0 is
asymptotically stable if it is stable, and if, in
addition, there exists some r > 0 such that
||x(0)|| < r implies that x(t) 0 as t  
—Asymptotic stability means that the
equilibrium is stable, and that in addition,
started close to 0 actually converge to 0 as time
t goes to ∞.
—marginally stable: An equilibrium point
which is Lyapunov stable but not
asymptotically stable.
Figure 3.3：Concepts of stability
—Domain of attraction of the equilibrium point:
the largest region such that trajectories initiated
at the points in the region eventually converge to
the origin.
•Example 2.2: A nonlinear second-order
system
The system
x  0.6 x  3x  x  0
2
has two singular points: (0, 0) and (-3, 0).
- The trajectories move towards the point (0,0)
while moving away from the point (-3,0).
—State convergence stability.
—Example: a simple system studied by
Vinograd.
— The origin is unstable in the sense of
Lyapunov, despite the state
convergence.
Figure 3.5: State convergence does not imply
stability
— Calling such a system unstable is quite
reasonable, since a curve such as C may be
outside the region where the model is valid.
— Example: the subsonic and supersonic
dynamics of a high-performance aircraft are
radically different, while, with the problem
under study using subsonic dynamic
models, C could be in the supersonic
range.
In many engineering applications, there is a
need to estimate how fast the system
trajectory approaches 0.
Definition 3.5 ：An equilibrium point 0 is
exponentially stable if there exists two
strictly positive numbers  and  such that
t  0, x  t    x  0  e
 t
in some ball Br around the origin.
λ: rate of convergence
—The state vector of an exponentially stable
system converges to the origin faster than an
exponential function.
—Example:
x   1  sin x  x
2
is exponentially convergent to x = 0 with a rate
 = 1,because
t
therefore

x  t   x  0  exp   1  sin 2  x    d
0
x  t   x  0  et

—Exponential stability  asymptotic stability.
— Asymptotic stability  exponential
stability
—Eample:
2
x(0)  1
x  x ,
whose solution is x = 1/(1+t), a function
slower that any exponential function e   t (with
 > 0).
Definition 3.6: If asymptotic (or exponential)
stability holds for any initial states, the
equilibrium point is said to be asymptotically
(or exponentially) stable in the large. It is also
called globally asymptotically (or
exponentially) stable.
—Examples:
—Linear time-invariant (LTI) systems are
either asymptotically stable, or marginally
stable, or unstable
— Asymptotic stability of LTI system is
always global and exponential
— Instability of LTI system always implies
exponential blow-up.
— This explains why the refined notions of
stability are explicitly needed only for
nonlinear systems.
3.3 Linearization and
Local Stability
Consider the autonomous system x  f ( x) assuming
'
f(x)  C is continuously differentiable and f(0)=0.
Then the system dynamics can be written as
 f 
x   
 x  x 0
x + fh. o. t. (x)
where fh. o. t. stands for higher-order terms in x.
—Let A denote
 f 


 x  x 0
Then, the system
x  Ax
is called the linearization (or linear
approximation) of the original nonlinear
system at the equilibrium point 0.
—For non-autonomous nonlinear
system with a control input u
x  f ( x, u )
such that f(0,0)=0, we can write
f
x  ( ) ( x0,u0)
x
f
x  ( ) ( x0,u0 )
u
u  fh.o.t ( x,u )
—Let
f
f
A(
) ( x0 ,u 0 ) B  (
) ( x0 ,u 0 )
x
u
— x  Ax  Bu
: the linearization (or linear
approximation) of the original nonlinear
system at (x = 0, u = 0).
Control law u = u(x) (with u(0) = 0)
transforms the non-autonomous system into
an autonomous closed-loop system, having x
= 0 as an equilibrium point.
—Linearly approximating the control law as
du
u  ( ) x0x=Gx
dx
the closed-loop dynamics can be linearly
approximated as
x  f ( x, u ( x))  ( A  BG) x
Alternative Linear Approximation
—Considering the autonomous closed-loop
system
x  f ( x, u( x))  f1 ( x)
and linearizing the function f1with respect
to x at its equilibrium point x = 0.
Example 3.4：consider the system
x1  x  x1 cos x2
x2  x2  ( x1  1) x1  x1 sin x2
2
2
—Linear approximation about x = 0:
x1 1 x1  0 x2  x1
x2 1 x1 1 x2  x1  x2
—The Linearized system:
1 0
x  
x

1
1


—Example:consider the system
x  4 x5  ( x 2  1)u  0
— Linear approximation about x = 0:
x  0  (0  1)u  0
— The linearized system:
x  u
—Control law:
u  sin x  x 3  x cos 2 x
— The linearized closed-loop dynamic:
x x x  0
Theorem 3.1 (Lyapunov’s linearization
method)
•If the linearized system is strictly stable (i.e ,if
all eigenvalues of A are strictly in the left-half
complex plane), then the equilibrium point is
asymptotically stable (for the actual nonlinear
system).
•If the linearized system is unstable (i.e , if at
least one eigenvalue of A is strictly in the
right-half complex plane), then the equilibrium
point is unstable(for the nonlinear system).
•If the linearized system is marginally stable (i. e,
all eigenvalues of A are in the left-half complex
plane, but at least one of them is on the j axis),
then one cannot conclude anything from the
linear approximation (the equilibrium point may
be stable, asymptotically stable, or unstable for
the nonlinear system).
Example 3.6：Consider the first order system
x  ax  bx
5
—The origin 0 is an equilibrium point
— The linearization around 0:
x  ax
—The stability properties of the nonlinear
system:
a < 0：asymptotically stable;
a > 0：unstable;
a = 0：cannot tell from linearization.
—What shall we do when a=0?
3.4 Lyapunov’s Direct Method
Basic idea from Physical
Observation
If the total energy of a mechanical (or
electrical) system is continuously
dissipated, then the system, where linear or
nonlinear, must eventually settle down to an
equilibrium point.
–Example: Consider the nonlinear mass-damperspring system:
mx  b x x  k0 x  k1 x 3  0
:nonlinear dissipation or damping
(k0 x  k1x3 ) :nonlinear spring term
bx x
–Assume the mass is pulled away from the natural
length of the spring by a large distance, and then
released (i) linearization method does not apply
(ii) the linearized system is marginally stable only
Figure 3.6：A nonlinear mass-damperspring system
–The total mechanical energy of the system :
x
1 2
3


V (x)  mx   (k0 x  k1 x )dx
0
2
1 2 1
1 4
2
 mx  k0 x  k1 x (3.14)
2
2
4
– Comparing the definitions of stability and
mechanical energy:
‧zero energy at the equilibrium point (x  0, x
‧asymptotic stability the convergence
of mechanical energy to zero
-- instability is related to the growth of
mechanical energy
 0)
–These relations indicate
(i) the mechanical energy indirectly reflects the
magnitude of the state vector
(ii) the stability properties can be characterized
by the variation of the mechanical energy
–The rate of energy variation:
V (x)  m x x  (k0 x  k1 x 3 ) x
 x(b x x )  b x
3
shows the energy of the system is continuously
dissipated by the damper until x  0 .
Definition 3.7： A scalar continuous function
V(x) is said to be locally positive definite if
V(0) = 0 and, in a ball B R
0
x  0  V (x)  0
If V(0) and the above property holds over the
whole state space, then V(x) is said to be
globally positive definite.
– A function V(x) is negative definite if
V(x) is positive definite
– V(x) is positive semi-definite if V(x)  0
and V(x)=0 for some x  0
– V(x) is negative semi-definite if (–V(x)) is
positive semi-definite.
Figure 3.7：Typical shape of a positive
definite function V ( x1 , x2 )
Figure 3.8：Interpreting positive definite
functions using contour curves
–V(x) represents an implicit function of time t.
– Assuming that V(x) is differentiable:
dV (x) V
V
V

x
f(x)
dt
x
x
Definition 3.8 If, in a ball B R , the function
V(x) is positive definite and has continuous
partial derivatives, and if its time derivative
along any state trajectory of system x  f ( x)
is negative semi-definite, i. e.,
0
V (x)  0
then V(x) is said to be a Laypunov function
for the system x  f ( x) .
Figure 3.9：Illustrating Definition 3.8 for n= 2.
Theorem 3.2 (Lyapunov Theorem for Local Stability)
If, in a ball B R , there exists a scalar
function V(x) with continuous first partial
derivatives such that
V(x) is positive definite (locally in B R ).
V ( x) is negative semi-definite (locally in B R ).
then the equilibrium point 0 is stable. If,
actually, the derivative V ( x) is locally
negative definite in B R0, then the stability is
asymptotic.
0
0
0
Figure 3.11: Illustrating the proof of Theorem
3.2 for n =2
Example 3.7：Local stability
A simple pendulum with viscous damping
    sin   0
Consider a locally positive definite
V (x)  (1-cos ) 
2
2
:total energy of the pendulum
V (x)   sin     － 2  0
–The origin is a stable equilibrium point
–Physical insight for V (x)  0 :the damping
term absorbs energy.
– V is the power dissipated in the pendulum.
–However, with this Lyapunov function, one
cannot draw conclusions on the asymptotic
stability of the system, because V (x) is only
negative semi-definite.
Figure 3.6：A nonlinear mass-damperspring system
–The total mechanical energy of the system :
x
1 2
3


V (x)  mx   (k0 x  k1 x )dx
0
2
1 2 1
1 4
2
 mx  k0 x  k1 x (3.14)
2
2
4
–These relations indicate
(i) the mechanical energy indirectly reflects the
magnitude of the state vector
(ii) the stability properties can be characterized
by the variation of the mechanical energy
–The rate of energy variation:
V (x)  m x x  (k0 x  k1 x 3 ) x
 x(b x x )  b x
3
shows the energy of the system is continuously
dissipated by the damper until x  0 .
Example 3.8：Asymptotic stability
Consider the nonlinear system
x1  x1 ( x1  x2 －2 )－4 x1 x2
2
2
2
x2  4 x1 x2  x2 ( x1  x2 －2 )
2
2
2
Define the positive definite function
V ( x1 , x2 )  x1  x2
2
2
– V  2( x12  x22 )( x12  x22  2)
–Locally negative definite in


B2   x1 , x2  x  x2  2
2
1
2
 the origin is asymptotically stable.
Theorem 3.3 (Lyapunov Theorem for Global
Stability ) Assume that there exists a scalar
function V of the state x, with continuous first
order derivatives such that
• V(x) is positive definite
• V (x) is negative definite
• V (x)   as x   (V(x) must be radially
unbounded)
then the equilibrium at the origin is globally
asymptotically stable.
Reason for Radial Unboundedness
–To assure the contour curves V (x)  V
correspond to closed curves.
(why?) If the curves are not closed, it is
possible for the state trajectories to drift away
from the equilibrium point, even though the
state keeps going through contours
corresponding to smaller and smaller V' s .
Figure 3.12：Motivation of the radial
unboundedness condition
3.4.3 Invariant Set Theorems
–Motivation: it often happens that V , is only
negative semi-definite.
–Fortunately, it is still possible to draw
conclusions on asymptotic stability, with the
help of the powerful invariant set theorems,
attributed to La Salle.
–Additional feature: To describe convergence
to a limit cycle
Definition 3.9： A set G is an invariant set for a
dynamic system if every system trajectory which
starts from a point in G remains in G for all future
time.
–Example: any equilibrium point.
–The domain of attraction of an equilibrium point
–The whole state-space
–any trajectories of an autonomous system
–limit cycles
Theorem 3.4(Local Invariant Set Theorem)
Consider an autonomous system x  f ( x), with f
continuous, and let V(x) be a scalar function with
continuous first partial derivatives. Assume that
• for some l > 0, the region  l defined by
V (x)  l is bounded
• V (x)  0 for all x in l
Let R be the set of all points within  l where
V (x)  0 , and M be the largest invariant set in R.
Then, every solution x(t) originating in  l tends to
M as t  .
Remarks
–The word “largest” means M is the union of all
invariant sets (e. g., equilibrium points or limit
cycles) within R.
–If the set R is itself invariant (i. e., if once
V  0, then V  0 for all future time), then M = R.
–V, although often still referred to as a Lyapunov
function, is not required to be positive definite.
Figure 3.14：Convergence to the largest
invariant set M
Figure 3.6：A nonlinear mass-damperspring system
Example 3.7：Local stability
A simple pendulum with viscous damping
    sin   0
Consider a locally positive definite
V (x)  (1-cos ) 
2
2
:total energy of the pendulum
V (x)   sin     － 2  0
–The origin is a stable equilibrium point
Example 3.12：Limit Cycle
Consider the system
x1  x2 
7
x1
x
4
1
 2 x2  10
2


x2   x1  3 x2 x1  2 x2  10
3
5
4
2

Notice first that the set defined by x  2x 10  0
is an invariant set, because
4
1
d
( x14  2 x22  10)
dt
 (4 x110  12 x26 )( x14  2 x22  10)  0
2
2
on the set. The motion on this invariant set is
described (equivalently) by either of the
equations
x1  x2
x2   x13
Therefore, we see that the invariant set
actually represents a limit cycle, along which
the state vector moves clockwise.
Figure 3.15：Convergence to a limit cycle
Global Invariant Set theorems
The above invariant set theorem can be
simply extended to a global result, by
requiring the radial unboundedness of the
scalar function V rather than the
existence of a bounded  l
Theorem 3.5 (Global Invariant Set Theorem)
Consider the autonomous system
, with f
x  function
f ( x)
continuous, and let V(x) be a scalar
with continuous first partial derivatives.
Assume that
•
•
V (x)   as x  
 (x)  0
V
Let R be the set of all points where
and M be the largest invariant set inVR.
(x)Then
 0,
all solutions globally asymptotically converge
to M as
t 
Example 3.14：A class of second-order
nonlinear systems
Consider a second-order system
x  b( x )  c ( x )  0
where b and c are continuous functions
satisfying the sign conditions
 
x b  x   0 for x  0
 
xc  x   0 for x  0
–Example systems: mass-damper-spring
system, nonlinear R-L-C electrical circuit
–If the functions b and c are
linear  b( x)   x, c( x)   x  , the sign
conditions are the necessary and sufficient
conditions for the system’s stability (1  0,0  0).
1
0
–Assume b(0) = 0 and c(0) = 0
Figure 3.16：A nonlinear R-L-C circuit
–
1 2 x
V  x   c( y )dy
0
2
the sum of the kinetic and potential energy of
the mass-damper-spring system
– V  x x  c ( x ) x   x b( x )  x c ( x )  c ( x ) x
  x b( x )  0
power dissipated in the system
– xb( x)  0 only if x  0
– x  0  x  c ( x )
which is nonzero as long as x  0. Thus M
contains only one point: origin.
– The local invariant set theorem  the origin
is a locally asymptotically stable point.
Figure 3.17：The functions b( x ) and c( x )
Furthermore, if the integral

x
o
c( y)dy is unbounded as x  
, then V is a radially unbounded function
and the equilibrium point at the origin is
globally asymptotically stable, according to
the global invariant set theorem.
3.5 System Analysis Based on
Lyapunov’s Direct Method
There is no general way of finding
Lyapunov functions for nonlinear systems.
This is the fundamental drawback of the
direct method. Thus, a case by case
approach is usually taken.
•Definition 3.10
T
A square matrix M is symmetric if M  M
(in other words, if i, j, Mij  M ji ). A
square matrix M is skew-symmetric if
T
M  - M (i. e., if i, j, Mij  M ji ).
–Any square n n matrix M
MM
M-M
M

2
2
T
T
–M is an n n skew symmetric matrix 
x Mx  0, x
T
(3.16)
–In linear systems, we often use xT Mx as
Lyapunov function.
–We can always assume, without loss of
generality, that M is symmetric.
Definition 3.11：A square n n matrix M is
positive definite (p. d.) if
x  0  x Mx  0
T
–A matrix M is positive definite  x Mx is
a positive definite function.
T
•Lyapunov Functions for Linear
Time-Invariant Systems
Consider a linear system x  Ax, let the
Lyapunov function candidate be
V  xT Px
P: a symmetric positive definite matrix.
If
V  x Px  x Px  -x Qx
T
T
T
A
P  PA  -Q
then
is the Lyapunov equation.
T
(3.18)
(3.19)
Lyapunov Equation
–Symmetric matrix Q defined by AT P  PA  -Q
is p. d.  the origin is globally asymptotically
stable.
–This “natural” approach may lead to
inconclusive result, i. e., Q may be not positive
definite even for stable systems.
Example 3.17：consider a second-order linear
system whose A matrix is
4 
0
A


8

12


If we take P = I, then
4 
0
T
 Q  PA  A P  


4

24


Q is not positive definite  don’t know whether
the system is stable or not.
Thinking in Opposite Direction
–To derive a positive definite matrix P from a
given positive definite matrix Q, i. e.,
• choose a positive definite matrix Q
• solve for P from the Lyapunov equation
• check whether P is p. d
• If P is p. d., global asymptotical stability is
guaranteed.
Theorem 3.6：A necessary and sufficient
condition for a LTI system x  Ax to be strictly
stable is that, for any symmetric p. d. matrix Q,
the unique matrix P, solution of the Lyapunov
equation (3.19), be symmetric positive definite.
Example 3.18：consider again the secondorder system of Example 3.17. Let us take Q =
I and denote P by
 p11 p12 
P

p
p
 21
22 
where, due to the symmetry of P, p21  p12
Then the Lyapunov equations is
 p11
p
 21
p12   0
4  0  8   p11




p22   8  12 4  12  p21
p12   1 0 


p22   0  1
whose solution is
p11  5 / 16, p12  p 22  1 / 16
The corresponding matrix
1 5 1
P 
16 1 1
is positive definite  the linear system is
globally asymptotically stable.
3.5.3 The Variable Gradient Method
–A formal approach to constructing Lyapunov
functions.
–Assuming the gradient V of an unknown V
–Integrating the assumed gradient
V x   0 Vd x
x
where
V  [V / x1,........,V / xn ] .
T
curl conditions
Vi V j

x j
xi
where
(i, j  1, 2,....., n)
V
Vi 
(i,j=1,2,......n)
xi
for n=2,
V1 V2

x2
x1
Procedure for Seeking V
•Assume
n
Vi   aij x j ,　i  1,...., n
j 1
where
aij
•Solve for
are coefficients to be determined.
aij
•Restrict aij so that V is negative semi-definite
(at least locally)
•Compute V from V
•Check whether V is positive definite
–It is usually convenient to obtain V by
integrating along a path which is parallel to
each axis, i. e.,
V  x    V1  x1,0,.....,0  dx1   V2  x1 , x2 ,0,.....,0  dx2  ...
x1
x2
0
0
  Vn  x1 , x2 ,....., xn  dxn
xn
0
THEOREM 9.9
Test for Path Independence
Let P and Q have continuous first partial derivatives in
an open simple connected region. Then  C P dx  Q dy
is independent of the path C if and only if
P Q

y  x
for all (x, y) in the region.
Example 3.20：To find a Lyapunov function for
x1  2 x1
x2  2 x2  2 x x
2
1 2
and discuss the stability of the e.p. (0,0)
Assume
V  a x  a x
1
11 1
12 2
V2  a21x1  a22 x2
The curl equation is
V1 V2

or
x2
x1
a11
a12
a21 a22
x1  a12  x2
 a21  x1

x2
x2
x2
x1
x1
If the coefficients are chosen to be
a11  a22  1, a12  a21  0
which leads to
V1  x1
V2  x2
then
V  V x  2 x12  2 x22 1  x1x2 
locally negative definite in the region
(1  x1 x2 )  0
Then
V x   0 x1dx1  0
x1
x2
x12  x22
x2 dx2 
2
positive definite  the asymptotic stability
–The Lyapunov function obtained by the
variable Gradient Method is not unique.
(3.22)
Taking
a11  1, a12  x22
a21  3 x22 , a22  3
we obtain the positive definite function
x12 3 2
V   x2  x1 x23
2 2
whose derivative is
V  2 x12  6 x22  2 x22 ( x1 x2  3x12 x22 )
Which is locally negative definite.
3.5.4 Physically Motivated Lyapunov
Functions
1 T
T
V  [q Hq  q K p q]
2
•A Simple Convergence Lemma
Lemma：If a real function W(t) satisfies the
inequality
W t   W t   0
(3.26)
where  is a positive real number. Then
 t
W (t )  W (0)e
Proof：Let us define a function Z(t) by
Z (t )  W  W
(3.27)
t
W (t )  W (0)e   e
0
then
W (t )  W (0)et
 t
 ( t  )
Z ( )d
The above lemma implies that, if W is a nonnegative function, the satisfaction of (3.26)
guarantees the exponential convergence of W
to zero.
4.5 Lyapunov-Like Analysis
Using Barbalat's Lemma
4.5.2 Barbalat's Lemma
Lemma 4.2 (Barbalat)
If the differentiable function f(t) has a finite
limit as t  , and if f is uniformly
continuous, then f (t )  0 as t  .
-- A function g is said to be uniformly
continuous on [0, ) if
R  0, ( R)  0, t  0, t  t1    | g (t )  g (t1 ) | R
-- Alternative approach: examine the
function's derivative.
A differentiable function is uniformly
continuous if its derivative is bounded.
Using Barbalat's Lemma for
Stability Analysis
Lemma 4.3 ("Lyapunov-Like Lemma")
If a scalar function V(x, t) satisfies the
following conditions
• V (x, t ) is lower bounded
• V (x, t ) is negative semi-definite
• V (x, t ) is uniformly continuous in time
then V (x, t )  0 as t  .
Example 4.13：The closed-loop error
dynamics of an adaptive control system for a
first-order plant with one unknown parameter :
e  e  w(t )
  ew(t )
e , : two states of the closed-loop dynamics,
representing tracking error and
parameter error.
w(t): a bounded continuous function.
Consider the lower bounded
function
2
2
V e 
Its derivative is
V  2e(e  w)  2(ew(t ))  2e 2  0
Then V(t)  V(0), which implies V(t) has a
finite limit as t goes to infinity, and e and  are
bounded.
But the invariant set theorem cannot be used to
conclude the convergence of e, because the
dynamics is non-autonomous.
The derivative of V :
V  4e(e  w)
This shows that V is bounded, since w is
bounded by hypothesis, and e and  were
shown above to be bounded. Hence, V is
uniformly continuous.
Application of Barbalat's lemma: e  0 as t  
Note that, although e converges to zero, the
system is not asymptotically stable, because  is
only guaranteed to be bounded.