Applied Nonlinear Control - Chapter 3: Fundamentals of Lyapunov Theory - Nguyễn Tân Tiến

in. Let A(x) denote the Jacobian matrix of the system, 
i.e., 
x
fA ∂
∂=)(x 
If the matrix TAAF += is negative definite in a 
neighborhood Ω , then the equilibrium point at the origin is 
asymptotically stable. A Lyapunov function for this system is 
)()()( xfxfx TV = 
If Ω is the entire state space and, in addition, ∞→)(xV as 
∞→x , then the equilibrium point is globally 
asymptotically stable. 
Example 3 .19 ______________________________________ 
Consider the nonlinear system 
211 26 xxx +−=& 
3
2212 262 xxxx −−=& 
We have 



−−
−
∂
∂= 2
2662
26
xx
fA 


−−
−=+= 2
212124
412
x
TAAF 
The matrix F is easily shown to be negative definite. Therefore, 
the origin is asymptotically stable. According to the theorem, a 
Lyapunov function candidate is 
23
221
2
21 )262()26()( xxxxxV −−++−=x 
Since ∞→)(xV as ∞→x , the equilibrium state at the 
origin is globally asymptotically stable. 
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The applicability of the above theorem is limited in practice, 
because the Jcobians of many systems do not satisfy the 
negative definiteness requirement. In addition, for systems of 
higher order, it is difficult to check the negative definiteness of 
the matrix F for all x. 
Theorem 3.7 (Generalized Krasovkii Theorem) Consider 
the autonomous system defined by (3.2), with the equilibrium 
point of interest being the origin, and let A(x) denote the 
Jacobian matrix of the system. Then a sufficient condition for 
the origin to be asymptotically stable is that there exist two 
symmetric positive definite matrices P and Q, such that 
0≠∀x , the matrix 
QPAPAxF ++= T)( 
is negative semi-definite in some neighborhood Ω of the 
origin. The function )()()( xfxfx TV = is then a Lyapunov 
function for this system. If the region Ω is the whole state 
space, and if in addition, ∞→)(xV as ∞→x , then the 
system is globally asymptotically stable. 
3.5.3 The Variable Gradient Method 
The variable gradient method is a formal approach to 
constructing Lyapunov functions. 
To start with, let us note that a scalar function )(xV is related 
to its gradient V∇ by the integral relation 
∫∇= x xx 0)( dVV 
where TnxVxVV }/,,/{ 1 ∂∂∂∂=∇ K . In order to recover a 
unique scalar function V from the gradient V∇ , the gradient 
function has to satisfy the so-called curl conditions 
),,2,1,( nji
x
V
x
V
i
j
j
i K=∂
∂∇=∂
∂∇ 
Note that the ith component iV∇ is simply the directional 
derivative ixV ∂∂ / . For instance, in the case 2=n , the above 
simply means that 
1
2
2
1
x
V
x
V
∂
∂∇=∂
∂∇ 
The principle of the variable gradient method is to assume a 
specific form for the gradient V∇ , instead of assuming a 
specific form for a Lyapunov function V itself. A simple way 
is to assume that the gradient function is of the form 
∑
=
=∇
n
j
jiji xaV
1
 (3.21) 
where the ija ’s are coefficients to be determined. This leads 
to the following procedure for seeking a Lyapunov functionV 
• assume that V∇ is given by (3.21) (or another form) 
• solve for the coefficients ija so as to sastify the curl 
equations 
• assume restrict the coefficients in (3.21) so that V& is 
negative semi-definite (at least locally) 
• computeV from V∇ by integration 
• check whetherV is positive definite 
Since satisfaction of the curl conditions implies that the above 
integration result is independent of the integration path, it is 
usually convenient to obtain V by integrating along a path 
which is parallel to each axis in turn, i.e., 
++∇+∇= ∫∫ KKK 21 0 2120 111 )0,,0,()0,,0,()( xx dxxVdxxVV x 
∫ ∇nx nn dxxV0 1 )0,,0,( K
Applied Nonlinear Control Nguyen Tan Tien - 2002.3 
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Chapter 3 Fundamentals of Lyapunov Theory 
17 
Example 3 .20 ______________________________________ 
Let us use the variable gradient method top find a Lyapunov 
function for the nonlinear system 
11 2xx −=& 
2
2122 22 xxxx +−=& 
We assume that the gradient of the undetermined Lyapunov 
function has the following form 
2121111 xaxaV +=∇ 
2221212 xaxaV +=∇ 
The curl equation is 
1
2
2
1
x
V
x
V
∂
∂∇=∂
∂∇ ⇒ 
1
21
121
2
12
212 x
axa
x
axa ∂
∂+=∂
∂+ 
If the coefficients are chosen to be 0,1 21122211 ==== aaaa 
which leads to 11 xV =∇ , 22 xV =∇ then V& can be computed 
as 
2
)(
2
2
2
1
0
22
0
11
21 xxdxxdxxV
xx +=+= ∫∫x (3.22) 
This is indeed p.d., and therefore, the asymptotic stability is 
guaranteed. 
If the coefficients are chosen to be ,,1 221211 xaa == 
3,3 22
2
221 == axa , we obtain the p.d. function 
3
21
2
2
2
1
2
3
2
)( xxxxV ++=x (3.23) 
whose derivative is )3(262 22
2
121
2
2
2
2
2
1 xxxxxxxV −−−−=& . 
We can verify that V& is a locally negative definite function 
(noting that the quadratic terms are dominant near the origin), 
and therefore, (3.23) represents another Lyapunov function for 
the system. 
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3.5.4 Physically motivated Lyapunov functions 
3.5.5 Performance analysis 
Lyapunov analysis can be used to determine the convergence 
rates of linear and nonlinear systems. 
A simple convergence lemma 
Lemma: If a real function )(tW satisfies the inequality 
0)()( ≤+ tWtW α& (3.26) 
where α is a real number. Then teWtW α−≤ )0()( 
The above Lemma implies that, if W is a non-negative 
function, the satisfaction of (3.26) guarantees the exponential 
convergence of W to zero. 
Estimating convergence rates for linear system 
Let denote the largest eigenvalue of the matrix P by )(max Pλ , 
the smallest eigenvalue of the matrix Q by )(min Qλ , and their 
ratio )(/)( minmax QP λλ by γ . The p.d. of P and Q implies 
that these scalars are all strictly positive. Since matrix theory 
shows that IPP )(maxλ≤ and QIQ ≤)(minλ , we have 
VxIPx
P
QxQx γλλ
λ ≥≥ ])([
)(
)(
max
max
min TT 
This and (3.18) implies that VV γ−≤& .This, according to 
lemma, means that .)0( tT e γ−≤ VxQx This together with the 
fact 2min )()( t
T xPxPx λ≥ , implies that the state x 
converges to the origin with a rate of at least 2/γ . 
The convergence rate estimate is largest for IQ = . Indeed, let 
0P be the solution of the Lyapunov equation corresponding to 
IQ = is 
IAPPA −=+ 00T 
and let P the solution corresponding to some other choice of 
Q 
1QPAPA −=+T 
Without loss of generality, we can assume that 1)( 1min =Qλ 
since rescaling 1Q will rescale P by the same factor, and 
therefore will not affect the value of the corresponding γ . 
Subtract the above two equations yields 
)()()( 100 I-QAP-PP-PA −=+T 
Now since )(1)( max1min IQ λλ == , the matrix )( 1 I-Q is 
positive semi-definite, and hence the above equation implies 
that )( 0P-P is positive semi-definite. Therefore 
)()( 0maxmax PP λλ ≥ 
Since )(1)( min1min IQ λλ == , the convergence rate estimate 
)(/)( maxmin PQ λλγ = 
corresponding to IQ = the larger than (or equal to) that 
corresponding to 1QQ = . 
Estimating convergence rates for nonlinear systems 
The estimation convergence rate for nonlinear systems also 
involves manipulating the expression of V& so as to obtain an 
explicit estimate of V . The difference lies in that, for 
nonlinear systems, V and V& are not necessarily quadratic 
function of the states. 
Example 3 .22 ______________________________________ 
Consider again the system in Example 3.8 
2
21
2
2
2
111 4)2( xxxxxx −−+=& 
)2(4 22
2
122
2
12 −++= xxxxxx& 
Choose the Lyapunov function candidate 2x=V , its 
derivative is )1(2 −= VVV& . That is dt
VV
dV 2
)1(
−=− . The 
solution of this equation is easily found to be 
Applied Nonlinear Control Nguyen Tan Tien - 2002.3 
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Chapter 3 Fundamentals of Lyapunov Theory 
18 
dt
dt
e
eV 2
2
1
)( −
−
+= α
αx , where 
)0(1
)0(
V
V
−=α . 
If 1)0()0( 2 <=Vx , i.e., if the trajectory starts inside the 
unit circle, then 0>α , and tetV 2)( −<α . This implies that 
the norm )(tx of the state vector converges to zero 
exponentially, with a rate of 1. 
However, if the trajectory starts outside the unit circle, i.e., if 
1)0( >V , then 0<α , so that )(tV and therefore x tend to 
infinity in a finite time (the system is said to exhibit finite 
escape time, or “explosion”). 
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3.6 Control Design Based on Lyapunov’s Direct Method 
There are basically two ways of using Lyapunov’s direct 
method for control design, and both have a trial and error 
flavor: 
• Hypothesize one form of control law and then finding a 
Lyapunov function to justify the choice 
• Hypothesize a Lyapunov function candidate and then 
finding a control law to make this candidate a real 
Lyapunov function 
Example 3 .23 Regulator design_______________________ 
Consider the problem of stabilizing the system uxxx =+− 23&&& . 
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