Applied Nonlinear Control - Chapter 4: Advanced Stability Theory - Nguyễn Tân Tiến

ore from the second 
equation that xCy && = is bounded. Thus the system output y is 
uniformly continuous. 
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Using Barbalat’s lemma for stability analysis 
To apply Barbalat’s lemma to the analysis of dynamic systems, 
one typically uses the following immediate corollary, which 
looks very much like an invariant set theorem in Lyapunov 
analysis: 
Lemma 4.3 (Lyapunov-Like Lemma) If a scalar function 
),( tV x satisfies the following conditions 
• ),( tV x is lower bounded 
• ),( tV x& is negative semi-definite 
• ),( tV x& is uniformly continuous in time 
then 0),( →tV x& as ∞→t . 
Indeed, V the approaches a finite limiting value ∞V , such that 
)0),0((xVV ≤∞ (this does not require uniform continuity). The 
above lemma then follows from Barbalat’s lemma. 
Example 4.13_______________________________________ 
Consider the closed-loop error dynamics of an adaptive 
control system for a first-order plant with unknown parameter 
)(twee θ+−=& 
)(twe−=θ& 
where e andθ are the two states of the closed-loop dynamics, 
representing tracking error and parameter error, and )(tw is a 
bounded continuous function. 
Consider the lower bounded function 
22 θ+= eV 
 Its derivative is 
02)]([2)]([2 2 ≤−=−++−= etwetweeV θθ& 
This implies that )0()( VtV ≤ , and therefore, that e andθ are 
bounded. But the invariant set cannot be used to conclude the 
convergence of e , because the dynamics is non-autonomous. 
To use Barbalat’s lemma, let us check the uniform continuity 
of V& . The derivative of V& is )(4 weeV θ+−−=&& . This shows 
that V&& is bounded, since w is bounded by hypothesis, and e 
and θ were shown above to be bounded. Hence, V& is 
uniformly continuous. Application of Babarlat’s lemma then 
indicates that 0→e as ∞→t . 
Note that, although e converges to zero, the system is not 
asymptotically stable, because θ is only guaranteed to be 
bounded. 
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⊗ Note that: Such above analysis based on Barbalat’s lemma 
shall be called a Lyapunov-like analysis. There are two 
important differences with Lyapunov analysis: 
- The function V can simply be a lower bounded function 
of x and t instead of a positive definite function. 
- The derivative V& must be shown to be uniformly 
continuous, in addition to being negative or zero. This is 
typically done by proving that V&& is bounded. 
4.6 Positive Linear Systems 
In the analysis and design of nonlinear systems, it is often 
possible and useful to decompose the system into a linear 
subsystem and a nonlinear subsystem. If the transfer function 
of the linear subsystem is so-called positive real, then it has 
important properties which may lead to the generation of a 
Lyapunov function for the whole system. In this section, we 
study linear systems with positive real transfer function and 
their properties. 
4.6.1 PR and SPR transfer function 
Consider rational transfer function of nth-order SISO linear 
systems, represented in the form 
0
1
1
0
1
1)(
apap
bpbpbph n
n
n
m
m
m
m
+++
+++= −−
−−
K
K 
The coefficients of the numerator and denominator 
polynomials are assumed to be real numbers and mn ≥ . The 
difference mn − between the order of the denominator and 
that of the numerator is called the relative degree of the system. 
Definition 4.10 A transfer function h(p) is positive real if 
0)](Re[ ≥ph for all 0]Re[ ≥p (4.33) 
It is strictly positive real if )( ε−ph is positive real for 
some 0>ε 
Condition (4.33) is called the positive real condition, means 
that )( ph always has a positive (or zero) real part when p has 
positive (or zero) real part. Geometrically, it means that the 
rational function )( ph maps every point in the closed RHP (i.e., 
including the imaginary axis) into the closed RHP of )( ph . 
Applied Nonlinear Control Nguyen Tan Tien - 2002.4 
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Chapter 4 Advanced Stability Theory 
24 
Example 4.14 A strictly positive real function_____________ 
Consider the rational function λ+= pph
1)( , which is the 
transfer function of a first-order system, with 0>λ . 
Corresponding to the complex variable ωσ jp += , 
22)()(
1)( ωλσ
ωλσ
λωσ ++
−+=++=
j
j
ph 
Obviously, 0)](Re[ ≥ph if 0≥σ . Thus, )( ph is a positive real 
function. In fact, one can easily see that )( ph is strictly 
positive real, for example by choosing 2/λε = in Definition 
4.9. 
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Theorem 4.10 A transfer function )( ph is strictly positive real 
(SPR) if and only if 
i. )( ph is a strictly stable transfer function 
ii. the real part of )( ph is strictly positive along the ωj axis, 
i.e., 
 0)](Re[0 >≥∀ ωω jh (4.34) 
The above theorem implies necessary conditions for asserting 
whether a given transfer function )( ph is SPR: 
• )( ph is strictly stable 
• The Nyquist plot of )( ωjh lies entirely in the RHP. 
Equivalently, the phase shift of the system in response to 
sinusoidal inputs is always less than 900 
• )( ph has relative degree of 0 or 1 
• )( ph is strictly minimum-phase (i.e., all its zeros are in 
the LHP) 
Example 4.15 SPR and non-SPR functions______________ 
Consider the following systems 
bapp
pph ++
−= 21
1)( 
1
1)( 22 +−
−=
pp
pph 
bapp
ph ++= 23
1)( 
1
1)( 24 ++
+=
pp
pph 
The transfer function 21,hh and 3h are not SPR, because 1h is 
non-minimum phase, 2h is unstable, and 3h has relative degree 
larger than 1. 
Is the (strictly stable, minimum-phase, and of relative degree 
1) function 4h actually SPR ? We have 
222
2
24 )1(
)1)(1(
1
1)( ωω
ωωω
ωω
ωω +−
+−−+=++−
+= jj
j
jjh 
(where the second equality is obtained by multiplying 
numerator and denominator by the complex conjugate of the 
denominator) and thus 
222222
22
4
)1(
1
)1(
1)](Re[ ωωωω
ωωω +−=+−
++−=jh 
which shows that 4h is SPR (since it is also strictly stable). Of 
course, condition (4.34) can also be checked directly on a 
computer. 
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⊗ The basic difference between PR and SPR transfer 
functions is that PR transfer functions may tolerate poles on 
the ωj axis, while SPR functions cannot. 
Example 4.16_______________________________________ 
Consider the transfer function of an integrator .1)(
p
ph = Its 
value corresponding to ωσ jp += is 22)( ωσ
ωσ
+
−= jph . We 
can easily see from Definition 4.9 that )( ph is PR but not SPR. 
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Theorem 4.11 A transfer function )( ph is positive real if, and 
only if, 
• )( ph is a stable transfer function 
• The poles of )( ph on the ωj axis are simple (i.e., distinct) 
and the associated residues are real and non-negative 
• 0)](Re[ ≥ωjh for any 0≥ω such that ωj is not a pole of 
)( ph 
The Kalman-Yakubovich lemma 
If a transfer function of a system is SPR, there is an important 
mathematical property associated with its state-space 
representation, which is summarized in the celebrated 
Kalman-Yakubovich (KY) lemma. 
Lemma 4.4 (Kalman-Yakubovich) Consider a controllable 
linear time-invariant system 
ubxAx +=& 
xcTy = 
The transfer function 
bAIc 1][)( −−= pph T (4.35) 
is strictly positive real if, and only if, there exist positive 
matrices P and Q such that 
-QPAPA =+T (4.36a) 
cbP = (4.36b) 
In the KY lemma, the involved system is required to be 
asymptotically controllable. A modified version of the KY 
lemma, relaxing the controllability condition, can be stated as 
follows 
Lemma 4.5 (Meyer-Kalman-Yakubovich) Given a scalar 
0≥γ , vector b and c , any asymptotically stable matrix A , 
and a symmetric positive definite matrix L , if the transfer 
function 
bAIp 1][
2
)( −−+= pcH Tγ 
Applied Nonlinear Control Nguyen Tan Tien - 2002.4 
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Chapter 4 Advanced Stability Theory 
25 
is SPR, then there exist a scalar 0>ε , a vector q , and a 
symmetric positive definite matrix P such that 
Lq-qAPPA ε−=+ TT 
qcbP γ+= 
This lemma is different from Lemma 4.4 in two aspects. 
• the involved system now has the output equation 
uy T
2
γ+= xc 
• the system is only required to be stabilizable (but not 
necessary controllable) 
4.6.3 Positive real transfer matrices 
The concept of positive real transfer function can be 
generalized to rational positive real matrices. Such generation 
is useful for the analysis and design of MIMO systems. 
Definition 4.11 An mm× matrix )( pH is call PR if 
• )( pH has elements which are analytic for 0)Re( >p 
• *)()( pp THH + is positive semi-definite for 0)Re( >p 
where the asterisk * denote the complex conjugate transpose. 
)( pH is SPR if )( ε−pH is PR for some 0>ε . 
4.7 The Passivity Formalism 
4.8 Absolute Stability 
4.9 Establishing Boundedness of Signal 
4.10 Existence and Unicity of Solutions