Systems & Control Letters 56 (2007) 167 – 172
www.elsevier.com/locate/sysconle
Dichotomy property and Lagrange stability for uncertain pendulum-like
feedback systems夡
Jinzhi Wang ∗ , Zhisheng Duan, Lin Huang
State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Engineering Science, Peking University, Beijing 100871, China
Received 7 April 2004; received in revised form 19 September 2006; accepted 5 October 2006
Available online 13 November 2006
Abstract
In this note, property of dichotomy and Lagrange stability of the uncertain pendulum-like systems with additional, multiplicative and H∞
uncertainty are considered, respectively. By using some results of H∞ theory the property of dichotomy and Lagrange stability of uncertain
pendulum-like systems with multiple equilibria are transformed into L∞ or H∞ norm condition of some transfer functions. The convergence
problem of all bounded solutions for some uncertain pendulum-like systems can be converted into the internal stability of another system under
unstructured perturbations.
© 2006 Elsevier B.V. All rights reserved.
Keywords: Uncertainty; Pendulum-like systems; Multiple equilibria; Dichotomy; Lagrange stability
1. Introduction
Note that a lot of practical systems are nonlinear systems
with multiple equilibria and periodic nonlinearity which are
called pendulum-like systems, such as phase-locked loops,
mathematical pendulums and synchronous machines [5,6,4].
The pendulum-like systems can be divided into two kinds,
one is certain systems and the other is uncertain systems. A
large amount of systems arisen in physics or engineering are
uncertain systems where the uncertainty may be caused by
parameter changes or by neglected dynamics, or by a host of
other unspecified effects.
A number of necessary conditions for various forms of stability including stability and attractivity with the meaning of
Lyapunov as well as ultimate boundedness for nonlinear system ẋ = f (x) were given in [7] and Brockett’s necessary condition for stabilizability of nonlinear systems ẋ = f (x, u) was
shown to be true for continuous functions f and u. In [9] the
夡 This work is supported by The National Science Foundation of China
under Grant 60334030, 10472001 and The Foundation of Engineering Research Institute of Peking University under Grant 204035.
∗ Corresponding author. Tel.: +86 1062765037; fax: +86 1062762266.
E-mail address: [email protected] (J. Wang).
0167-6911/$ - see front matter © 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.sysconle.2006.10.001
corresponding results were shown under the cases of smooth
feedback and discontinuous feedback. For the pendulum-like
systems with multiple equilibria there appear new types of stability problems which are different from the system with a single stationary point. The issues we have to consider are some
global properties of solutions such as Lagrange stability (i.e.,
all the solutions of the systems are bounded) and dichotomy
property (means all bounded solutions are convergent) as well
as gradient-like property (means all solutions are convergent)
and so on. The frequency-domain methods established in the
fifties and sixties of last century were firstly applied and developed to the frames of absolute stability theory for investigation
of global stability of the stationary point [1,3,8]. In the seventies and eighties of last century the frequency-domain methods
were applied successfully to investigate the global properties of
solutions for the pendulum-like systems with multiple equilibria [5,6,4]. A series of frequency-domain inequality conditions
guaranteeing the systems with various global properties were
obtained [5,6,4]. However, all these results are focussed on
the certain pendulum-like systems and there is almost no result
on global properties of the pendulum-like systems with uncertainties.
In this note we consider dichotomy property and Lagrange stability of the uncertain pendulum-like systems with
168
J. Wang et al. / Systems & Control Letters 56 (2007) 167 – 172
additional, multiplicative and H∞ uncertainty, respectively. By
using some results of H∞ theory the property of dichotomy
and Lagrange stability of uncertain pendulum-like systems with
the above uncertain structures are transformed into L∞ or H∞
norm condition of some transfer functions and thus it becomes
simple and easy to test the property of dichotomy and Lagrange
stability of the uncertain pendulum-like systems directly. The
obtained results can be viewed as generalization of results in
[6,4] where the systems have no uncertainty. At the same time,
by using small gain theorem the equivalence between the given
condition described by H∞ norm for dichotomy of an uncertain
pendulum-like system and internal stability of another system
under unstructured perturbations is established. Therefore the
convergence problem of all bounded solutions for the uncertain pendulum-like system can be converted into the internal
stability of some system under unstructured perturbations. The
results given in this paper can possibly be used to investigate
the problem of synchronisation for the pendulum-like systems
which will be discussed in another paper.
In this paper we use the following notations: for n × m
matrix A with real elements is denoted by A ∈ R n×m . AT and
A∗ denote the transpose and complex conjugate transpose of
A, respectively. (A)
¯
denotes the largest singular values of the
matrix A. RL∞ is the set of all proper and real rational transfer
matrices with no poles on the imaginary axis. RH ∞ is the set
of all stable transfer matrices.
Define the Redheffer star product
F (P , Q11 )
S(P , Q)
Q21 (I −P22 Q11 )−1 P21
P12 (I −Q11 P22 )−1 Q12
,
Fu (Q, P22)
where F (P , Q11 ) = P11 + P12 Q11 (I − P22 Q11 )−1 P21 and
Fu (Q, P22 ) = Q22 + Q21 P22 (I − Q11 P22 )−1 Q12 are the lower
linear fractional transformation and the upper linear fractional
transformation, respectively.
The following property holds [2,12]:
F (P , F (Q, K)) = F (S(P , Q), K).
ẋ = Ax + b(),
Consider nonlinear system of differential equations:
ẋ = f (t, x) (f : R+ × R → R ).
n
n
(1)
We will use the following definitions throughout the paper.
Definition 1 (Leonov et al. [6]). A solution x(t) of (1) is said
to be convergent if x(t) → c as t → +∞ where c is an
equilibrium point of (1).
Definition 2 (Leonov et al. [6]). Eq. (1) is said to be dichotomous if its every bounded solution is convergent. Eq. (1) is said
to be gradient-like if its every solution is convergent. Eq. (1) is
said to be Lagrange stable if its every solution is bounded.
Definition 3 (Khalil [3], Sun et al. [10]). A transfer matrix
G(s) is said to be strictly positive real (SPR) if G(s) is analytic in Re s 0 and satisfies G(j) + GT (−j) > 0 for ∈
[0, ∞). A transfer matrix G(s) is said to be extended strictly
positive real (ESPR) if it is strictly positive real and G(j∞) +
GT (−j∞) > 0.
Definition 4 (Green and Limebeer [2], Zhou and Doyle
[12]). Let G(s) ∈ RL∞ and the L∞ norm of a matrix rational
transfer function G is defined as
Definition 5 (Green and Limebeer [2], Zhou and Doyle
[12]). Let
P11 P12
Q11 Q12
P=
, Q=
.
P21 P22
Q21 Q22
(2)
Consider a class of autonomous pendulum-like systems with
multiple equilibria:
2. Preliminaries
G∞ = sup {G(j)}.
¯
Fig. 1. The pendulum-like feedback system.
= cx,
(3)
where A ∈ R n×n is a constant real matrix with det A = 0, b ∈
R n , cT ∈ R n are real vectors. () is continuous for all ∈ R
and satisfies
( + ) = (),
() = 0,
(4)
−1
where is a period of (). Let P (s) = c(A − sI ) b be the
transfer function of the linear part of system (3) from −()
to which is non-degenerate, that is, (A, b) is controllable and
(A, c) is observable. The pendulum-like feedback system (3)
can be described by Fig. 1.
Lemma 1 (Leonov et al. [6]). Suppose that the transfer function P (s) of the linear part of (3) is non-degenerate and has
n − 1 poles with negative real parts besides the zero pole of
multiplicity one. Suppose also that
Re[iP (i)] = 0 for all ∈ R
lim 2 Re[iP (i)] = 0.
→∞
and
(5)
Then system (3) is dichotomous.
Lemma 2 (Leonov et al. [6]). Suppose that the conditions of
Lemma 1 are fulfilled and
() d = 0,
(6)
0
where is a period of (). Then system (3) is gradient-like.
J. Wang et al. / Systems & Control Letters 56 (2007) 167 – 172
169
For system (3), since det A = 0, P (s) can be expressed as
P (s) = (1/s)G(s). Therefore the condition (5) of Lemma 1 is
satisfied if
Re G(i) = 0
for all ∈ R ∪ ∞
(7)
which is equivalent to the condition that there exists a scalar
∈ R, = 0 such that
G(i) + GT (−i) > 0,
∈ R ∪ ∞.
(8)
This means the transfer function G(s) is extended strictly
positive real and thus we can obtain the following result from
Lemma 1.
Lemma 3. Suppose that the transfer function P (s) of the linear
part of (3) is non-degenerate and has n − 1 poles with negative
real parts besides the zero pole of multiplicity one. If there
exists a scalar ∈ R, = 0 such that G(s) is ESPR, then
system (3) is dichotomous.
For Lagrange stability, we consider the pendulum-like feedback system in the form
ẋ = Ax + b,
= cx,
= (t, ),
(9)
where A ∈ R n×n is a constant real matrix and detA = 0, b ∈
R n×1 and cT ∈ R n×1 are real vectors and : R+ × R → R
is continuous and locally Lipchitz continuous in the second
argument and there exists a > 0, such that
(t, + ) = (t, ),
t ∈ R+ , ∈ R.
(10)
Furthermore, we assume that belongs to the sector M[1 , 2 ],
i.e.
1 (t, )
2 ,
(11)
where 1 is either a certain negative number or −∞ and 2 ia
either a certain positive number or +∞. In the case 1 = −∞
−1
(resp. 2 = +∞) we assume that −1
1 = 0 (resp. 2 = 0).
Let P (s) = c(A − sI )−1 b be non-degenerate, i.e., (A, b) is
controllable and (A, c) is observable.
Lemma 4 (Leonov et al. [6]). Suppose that there exists a number > 0 such that the following conditions for systems (9)–(11)
are fulfilled:
(i) the matrix A + I has n − 1 eigenvalues with negative real
parts.
(ii)
Fig. 2. The uncertain feedback system.
Since (12) holds if and only if (G0 (j))∗ G0 (j) 2 which
means G0 (s) ∈ RL∞ and G0 (s)∞ , where
G0 (s) = P (s − ) + ,
=
−1
−1
1 + 2
.
2
=
−1
−1
2 − 1
,
2
(13)
Therefore the result of Lemma 4 can be rewritten as:
Lemma 5. Suppose that the transfer function P (s) of the linear
part of (9) is non-degenerate, and there exists a scalar > 0
such that P (s) has a zero pole of multiplicity one and P (s − )
has n − 1 poles with negative real parts. If G0 (s)∞ ,
where G0 (s) and are defined by (13), then the pendulum-like
feedback system (9)–(11) is Lagrange stable.
3. Dichotomy of uncertain feedback systems
In this section, our aim is to find some conditions such that
some uncertain feedback systems defined by Fig. 2 are dichotomous. Here the uncertainty P (s) can be divided into three
cases, that is, additive uncertainty, multiplicative uncertainty
and H∞ uncertainty.
3.1. Additive uncertainty
Consider additive uncertainty:
P (s) = P (s) + W10 (s)0 (s)W20 (s),
(14)
where W10 (s) and W20 (s) are transfer functions that characterize the spatial and frequency structure of the uncertainty. P (s)
is the transfer function of nominal model. The uncertainty may
be caused by parameter changes or by neglected dynamics, or
by a host of other unspecified effects.
Assume that:
(A1) P (s) is non-degenerate.
(A2) P (s) has a zero pole.
From (A1) and (A2), P (s) can be expressed as
−1
−1
−1
2
−1
1 2 + (1 + 2 ) Re P (i − ) + |P (i − )| 0
∀ ∈ R.
(12)
P (s) =
Then the pendulum-like feedback system (9)–(11) is Lagrange stable.
where (s) is uncertainty and W1 (s) and W2 (s) are weighted
functions.
1
s
[G(s) + W1 (s)(s)W2 (s)],
(15)
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J. Wang et al. / Systems & Control Letters 56 (2007) 167 – 172
if the condition that there exists a scalar = 0 such that G(s)
is ESPR is substituted by S(s)∞ < 1.
Theorem 1. Assume that:
(i) (A1) and (A2) hold.
(ii) G(s), W1 (s), W2 (s), (s) are stable transfer functions
and ∞ < 1.
(iii) W1 (j) = 0 or W2 (j) = 0 for ∈ R ∪ ∞.
(iv) There exists a scalar ∈ R, = 0, G∞ < 1 such that
3.2. Multiplicative uncertainty
1 + G(∞) = 0,
1 + W2 (∞)(1 + G(∞))−1 W1 (∞)(∞) = 0.
(v) TG (s)∞ 1, where
(G − 1)(G + 1)−1
TG =
W2 (G + 1)−1
−1
Consider multiplicative uncertainty:
2(G + 1) W1
.
−W2 (G + 1)−1 W1
(16)
Then the uncertain feedback system with additive uncertainty
defined by Fig. 2 is dichotomous.
Proof. Let
P (s) = (1 + W1 (s)(s)W2 (s))P (s).
(17)
where TG is defined by (16). Condition (iv) guarantees that
F (TG , ) is well-posed [12] and TG ∈ RH ∞ . By conditions (ii), (iii), (v) and Lemma 15.1 in [12] we can imply that
S ∞ < 1. Let
T = (1 − S )−1 (1 + S ).
(18)
Then
2(T∗ (j) + T (j))
.
(T∗ (j) + 1)(T (j) + 1)
(A3) P (s) has a zero pole.
(A4) P (s) is non-degenerate.
By (A3), P (s) can be expressed as P (s) = (1/s)G(s) and
thus P (s) can be rewritten as
for ∈ R ∪ ∞. On another aspect, by
(2), we have
1
T = 1 + 2(1 − S )−1 S = F
1
1 2
= F
, F (TG , )
1 1
G
2W1
= F
,
1
2 W2 0
= G + W1 W2 ,
1
s
(1 + W1 W2 )G(s).
(21)
Along the line of Theorem 1 we can show that:
Theorem 2. Assume that:
(i) (A3) and (A4) hold.
(ii) G(s), W1 (s), W2 (s), (s) are stable transfer functions
and (s)∞ < 1.
(iii) W1 (j) = 0 or W2 (j)G(j) = 0 for ∈ R ∪ ∞.
(iv) There exists a scalar = 0, G∞ < 1 such that
1 + G(∞) = 0, 1 + W2 (∞)G(∞)(1 + G(∞))−1
× W1 (∞)(∞) = 0.
Thus by S ∞ < 1, we have
T∗ (j) + T (j) > 0
(20)
We make the following assumptions:
P (s) =
S = F (TG , ),
1 − S∗ (j)S (j) =
Remark 2. If () satisfies (6) in Lemma 2 and the conditions
of Theorem 1 are fulfilled, then the uncertain feedback system
with additive uncertainty defined by Fig. 2 is gradient-like.
(19)
(17), (18) and formula
2
, S
1
which implies T is a stable transfer function by condition (ii)
and thus combining with (19) T is ESPR. Consequently, the
uncertain feedback system with additive uncertainty defined by
Fig. 2 is dichotomous by Lemma 3. Remark 1. When there is no uncertainty in the systems, i.e.,
= 0, we have S = (G − 1)(G + 1)−1 by (17). From the
proof of Theorem 1 we derive that G(s) is ESPR if and only
if S(s) = S = (G − 1)(G + 1)−1 is strictly bounded real,
i.e., S(s)∞ < 1. Consequently, the result of Lemma 3 is true
(v) TG (s)∞ 1, where
(G − 1)(G + 1)−1
TG =
W2 G(G + 1)−1
2(G + 1)−1 W1
.
−W2 G(G + 1)−1 W1
(22)
Then the uncertain feedback system with multiplicative uncertainty defined by Fig. 2 is dichotomous.
Remark 3. If () satisfies (6) in Lemma 2 and the conditions
of Theorem 2 are fulfilled, then the uncertain feedback system
with multiplicative uncertainty defined by Fig. 2 is gradientlike.
3.3. H∞ uncertainty
Consider the uncertain feedback system defined by Fig. 3,
where the transfer function P of linear part for the system is
described as
P = F (P , 0 ) = P11 + P12 0 (1 − P22 0 )−1 P21 ,
(23)
P11 P12
where P =
is a nominal model with size 2 × 2.
P21 P22
0 is an uncertainty and () is continuous and satisfies (4).
J. Wang et al. / Systems & Control Letters 56 (2007) 167 – 172
171
Fig. 4. The interconnected system.
Fig. 3. The closed-loop feedback system with H∞ uncertainty.
Assume that:
(A5) P (s) is non-degenerate.
(A6) P (s) has n − 1 poles with negative real parts besides the
zero pole of multiplicity one.
In this case, P can be rewritten as
P =
1
s
(G11 + G12 (1 − G22 )−1 G21 ).
(24)
Remark 6. In Theorems 1–3, since TG ∈ RH ∞ , uncertainty
∈ RH ∞ and ∞ < 1 it is obvious that condition (v) of
Theorems 1–3, i.e., TG ∞ < 1, is equivalent to the condition
that the interconnected system shown in Fig. 4 is internally stable by small gain theorem [12, Theorem 8.1]. It is also equivalent to the condition that (I − TG )−1 ∈ RH ∞ . Therefore
in terms of small gain theorem the convergence problem of all
bounded solutions for uncertain pendulum-like systems can be
converted into the internal stability of another system under
unstructured perturbations.
Theorem 3. Suppose that:
4. Lagrange stability of an uncertain feedback system
(i) (A5), (A6) hold. G11 G12
(ii) G=
∈ RH ∞ , G22 ∞ < 1 and ∈ RH ∞ ,
G21 G22
∞ < 1.
(iii) G12 (j) = 0 or G21 (j) = 0 for ∈ R ∪ ∞.
(iv) There exists a scalar = 0, G11 ∞ < 1 such that
In this section the conditions of Lagrange stability for the
uncertain feedback systems with additive uncertainty and multiplicative uncertainty defined by Fig. 2 are given. Note that the
nonlinear function () can be replaced by (t, ).
Consider the uncertain feedback system defined by
Fig. 2, where the uncertainty is described by additive uncertainty, i.e.,
1 + G11 (∞) = 0, 1 + (G21 (∞)(1 + G11 (∞))−1
× G12 (∞) − G22 (∞))(∞) = 0.
(v) TG (s)∞ 1, where
(G11 − 1)(G11 + 1)−1
TG =
G21 (G11 + 1)−1
P (s) = P (s) + W1 (s)(s)W2 (s).
(25)
2(G11 + 1)−1 G12
.
−G21 (G11 + 1)−1 G12 + G22
Then the uncertain feedback system with H∞ uncertainty
defined by Fig. 3 is dichotomous.
Proof. Let
G = G11 + G12 (1 − G22 )−1 G21 .
Assumption (ii) guarantees that G ∈ RH ∞ from [2]. Let
S = (G − 1)(G + 1)−1 .
Then S can be expressed as S = F (TG , ) from Lemma
5.1 of [11]. Similar to the proof of Theorem 1 it is not difficult
to verify that G is ESPR and thus the result follows. Remark 4. If () satisfies (6) in Lemma 2 and the conditions
of Theorem 3 are fulfilled, then the uncertain feedback system
with H∞ uncertainty defined by Fig. 3 is gradient-like.
Remark 5. When there is no uncertainty in the systems, that is,
= 0, the results of Theorems 1–3 are identical and degenerate
into the results of Remark 1 which is equivalent to Lemma 3.
Theorem 4. Suppose that:
(i) P (s) is non-degenerate.
(ii) P (s) has a zero pole of multiplicity one and there exists
a scalar > 0 such that P (s − ) has n − 1 poles with
negative real parts.
(iii) W1 (j − ) = 0 or W2 (j − ) = 0 for ∈ R ∪ ∞.
(iv) (s − ) ∈ RL∞ and (s − )∞ < 1.
(v) TG (s − )∞ 1, where
1
P (s − ) + 1
W1 (s − )
TG (s − ) = (26)
W2 (s − )
0
and and are defined by (13).
Then the uncertain feedback system with additive uncertainty
defined by Fig. 2 is Lagrange stable.
Proof. (1/
)P (s − ) + /
can be written as
1
P (s − ) + = F (TG (s − ), (s − )),
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J. Wang et al. / Systems & Control Letters 56 (2007) 167 – 172
where TG (s − ) is defined by (26). In terms of Lemma 5 and
Lemma 15.1 of [12] the result follows. of Theorem 2 hold. The uncertain feedback system defined by
Fig. 2 is dichotomous.
For the uncertain feedback system defined by Fig.2 where
the uncertainty is described by multiplicative uncertainty, i.e.,
In special, Let (s)=/(1+s), where and are uncertain
scalar parameters with || < 1, = 0, 0. In this case, the
uncertain feedback system defined by Fig. 2 is dichotomous.
P (s) = (1 + W1 (s)(s)W2 (s))P (s)
(27)
6. Conclusions
we have:
Theorem 5. Suppose that:
(i) P (s) is non-degenerate.
(ii) P (s) has a zero pole of multiplicity one and there exists
a scalar > 0 such that P (s − ) has n − 1 poles with
negative real parts.
(iii) W1 (j − ) = 0 or W2 (j − )P (j − ) = 0 for ∈
R ∪ ∞.
(iv) (s − ) ∈ RL∞ and (s − )∞ < 1.
(v) TG (s − )∞ 1, where
1
1
P (s − ) + W1 (s − )
TG (s − ) =
(28)
W2 (s − )P (s − ) 0
and and are defined by (13).
Then the uncertain feedback system with multiplicative uncertainty defined by Fig. 2 is Lagrange stable.
Remark 7. The result of Lagrange stability for an uncertain
feedback system with H∞ uncertainty defined by Fig. 3 can
also be given in the same as Theorems 4 and 5.
Remark 8. When there is no uncertainty in the systems, that
is, = 0, the condition TG (s − )∞ 1 in Theorems 4 and
5 becomes (1/
)P (s − ) + /
∞ 1 and thus the results of
Theorem 4 or 5 are identical to that of Lemma 5.
5. Examples
Example 1. Consider an uncertain feedback system defined by
Fig. 2, where
1
P (s) = (1 + (s)).
s
(29)
Assume that (s) ∈ RH ∞ , (s)∞ < 1, (∞) = 0 and
(1 + (s))−1 has no zero pole. The system (29) can be viewed
as an uncertain system with multiplicative uncertainty, where
G(s)=1, W1 (s)=W2 (s)=1. It is easy to verify that there exists
a scalar =0.5 such that TG (S)∞ =1 and thus the conditions
In this paper some conditions ensuring the property of dichotomy and Lagrange stability of the uncertain pendulum-like
systems with additional, multiplicative and H∞ uncertainty are
given, respectively. The problems of testing the property of
dichotomy and Lagrange stability of uncertain pendulum-like
systems with the above uncertain structures are transformed
into the problem of testing L∞ or H∞ norm of some transfer
functions. It is obvious that the latter can be dealt with easily.
At the same time the equivalence between the given norm condition for the dichotomy of an uncertain pendulum-like system
and the internal stability of another system under unstructured
perturbations is established.
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