- Research Article
- Open access
- Published:
About Robust Stability of Dynamic Systems with Time Delays through Fixed Point Theory
Fixed Point Theory and Applications volume 2008, Article number: 480187 (2008)
Abstract
This paper investigates the global asymptotic stability independent of the sizes of the delays of linear time-varying systems with internal point delays which possess a limiting equation via fixed point theory. The error equation between the solutions of the limiting equation and that of the current one is considered as a perturbation equation in the fixed- point and stability analyses. The existence of a unique fixed point which is later proved to be an asymptotically stable equilibrium point is investigated. The stability conditions are basically concerned with the matrix measure of the delay-free matrix of dynamics to be negative and to have a modulus larger than the contribution of the error dynamics with respect to the limiting one. Alternative conditions are obtained concerned with the matrix dynamics for zero delay to be negative and to have a modulus larger than an appropriate contributions of the error dynamics of the current dynamics with respect to the limiting one. Since global stability is guaranteed under some deviation of the current solution related to the limiting one, which is considered as nominal, the stability is robust against such errors for certain tolerance margins.
1. Introduction
Time-delay dynamic systems are an interesting field of research in dynamic systems and functional differential equations. Their intrinsic related theoretical interest is due to the fact that the necessary formalism lies in that of functional differential equations, being infinite dimensional. Another reason for their interest relies on the wide range of their applicability in modelling a number of physical systems like, for instance, transportation systems, queuing systems, teleoperated systems, war/peace models, biological systems, finite impulse response filtering, and so on [1–4]. Important particular interest has been devoted to stability, stabilization, and model-matching of control systems where the object to be controlled possesses delayed dynamics and the controller is synthesized incorporating delayed dynamics or its structure may be delay-free (see, e.g., [1, 4–14]). The properties are formulated as either being independent of or dependent on the sizes of the delays. An intrinsic problem which generated analysis complexity is the presence of infinitely many characteristic zeros because of the functional nature of the dynamics. This fact generates difficulties in the closed-loop pole-placement problem compared to the delay-free case [14], as well as in the stabilization problem [2, 4–6, 8–11, 13, 15–20], including the case of singular time-delay systems where the solution is sometimes nonunique and impulsive because of the dynamics associated to a nilpotent matrix [15]. The properties of the associated evolution operators have been investigated in [2, 6, 11]. This paper is devoted to obtain results relying on a comparison and an asymptotic comparison of the solutions between a nominal (unperturbed) functional differential equation involving wide classes of delays and a perturbed version (describing the current dynamics) with some smallness in the limit assumptions on the perturbed functional differential equation. The nominal equation is defined as the limiting equation of the perturbed one since the parameters of the last one converge asymptotically to those of its limiting counterpart. The problem of interest arises since very often the perturbations related to a nominal model in dynamic systems occur during the transients while they are asymptotically vanishing in the steady state or, in the most general worst case, they grow at a smaller rate than the solution of the nominal differential equation. In this context, the nominal differential equation may be viewed as the limiting equation of the perturbed one. The comparison between the solutions of the limiting differential equation and those of the perturbed one based on Perron-type results has been studied classically for ordinary differential equations and more recently for the case of functional equations [10, 21, 22]. Particular functional equations of interest are those involving both point and distributed delays potentially including the last ones Volterra-type terms [2, 5–7, 23]. On the other hand, fixed point theory [2, 21, 24] is a very powerful mathematical tool to be used in many applications where stability is required. At a theoretical level, fixed point theory is being of an increasing interest along the last years. For instance, the concept of weak contractiveness is discussed in [25] where the contraction constant is allowed to be unity but a negative vanishing term associated with some continuous nondecreasing function is also allowed. Weak contractiveness still ensures the existence of a unique fixed point. The existence of a unique fixed point has also been proved for asymptotic contractions [26]. Also, the existence of a nonempty fixed point set in a self-map of where
is a complete metric space allows guaranteeing the
-stability of iteration procedures [27]. In this paper, linear time-varying functional differential equations with point constant delays are investigated. Based on the contraction mapping principle, it is first proved the existence of a unique fixed point. The related proofs are based on the convergence of the parameters of the current equation to their counterparts of the limiting equation. The existence of such a fixed point requires that a relevant matrix of the limiting equation (either that of the delay-free dynamics or that of the zero-delay dynamics) be a stability matrix. Furthermore, an inequality concerning the parameters of the absolute value of such a matrix with a measure of all the remaining dynamics (formulated in terms of norms) has to be fulfilled. Once the existence of a unique fixed point has been proved, simple extra conditions ensure that such a point is a globally stable zero equilibrium point of the state-trajectory solution. This leads immediately to prove the global asymptotic stability independent of the sizes of the delays of the dynamic system. The analysis is then extended to the case of closed-loop systems obtained via state or output linear feedback from the original uncontrolled dynamic system. A method to synthesize both the time-invariant parts and the incremental ones of the controller matrices is given so that the existence of a fixed point of the closed-loop system is guaranteed. The obtained results are of robust stability type since the global asymptotic stability is guaranteed under a certain deviation from the current solution with respect to the limiting one, which is considered the nominal dynamics.
1.1. Notation
and
are the sets of complex, real, and integer numbers, respectively.
and
are the sets of positive real and integer numbers, respectively;
is the set of complex numbers with positive real part.
where
is the complex unity,
and
and
are the sets of negative real and integer numbers, respectively;
is the set of complex numbers with negative real part.
where
is the complex unity,
and
"
" is the logic disjunction, and "
" is the logic conjunction.
is the integer part of the rational quotient
denotes the spectrum of the real or complex square matrix
(i.e., its set of distinct eigenvalues).
denotes any vector or induced matrix norm. Also,
and
are the
-norms of the vector
or (induced) real or complex matrix
and
denote the
measure of the square matrix
[4]. The matrix measure
is defined as the existing limit
which has the property
for any square
-matrix
of spectrum
An important property for the investigation of this paper is that
if
is a stability matrix, that is, if
denotes the supremum norm on
or its induced supremum metric, for functions or vector and matrix functions without specification of any pointwise particular vector or matrix norm for each
If pointwise vector or matrix norms are specified, the corresponding particular supremum norms are defined by using an extra subscript. Thus,
and
are, respectively, the supremum norms on for vector and matrix functions of domains in
respectively, in
defined from their
pointwise respective norms for each
is the n th identity matrix.
is the condition number of the matrix
with respect to the
-norm.
2. Linear Systems with Point Constant Delays and the Contraction Mapping Theorem
Consider the following time-varying linear system subject to constant point delays:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F480187/MediaObjects/13663_2008_Article_1088_Equ1_HTML.gif)
where are the
(in general incommensurate delays)
subject to the system piecewise continuous bounded matrix functions of dynamics
which are decomposable as a (nonunique) sum of a constant matrix plus a matrix function of time
Equation (2.1) is assumed subject to any piecewise continuous real vector function of initial conditions
with
that is,
Thus, it has a unique solution
satisfying
and the differential system (2.1),
for any bounded piecewise continuous set
  
what follows from Picard-Lindeloff's theorem, [4, 11, 15]. Such a unique solution is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F480187/MediaObjects/13663_2008_Article_1088_Equ2_HTML.gif)
According to Lyapunov's stability theory, global stability means that the state-trajectory solution is uniformly bounded for any bounded function of initial conditions. Global asymptotic stability implies also that there is a unique asymptotically stable equilibrium point which is then a global attractor. See, for instance, [2, 4–11, 13, 16, 21, 24, 28]. Generic relations of stability with fixed point theory have been reported in [2, 21, 24, 27, 29, 30]. It turns out that a system whose state-trajectory solutions are all bounded and converge to a unique point is globally asymptotically stable to its equilibrium in Lyapunov's sense, provided that such equilibrium is unique. The following simple result is well known. Assume the system (2.1) with then the resulting linear time-invariant delay-free system (2.1) is globally asymptotically stable if
is a stability matrix so that if
Nonasymptotic stability is guaranteed if
The subsequent result is concerned with global stability independent of the sizes of the delays and it is obtained from the contraction mapping theorem for the case when (2.1) has a limiting equation with a unique asymptotically stable equilibrium point. It is assumed that the matrices defining the delayed dynamics have sufficiently small norms and that the norm of the error matrix of the delay-free dynamics with respect to its limiting value is also sufficiently small.
Theorem 2.1.
The following properties hold.
-
(i)
Assume that
is a stability matrix of
-matrix measure
and that
for some real constants
and
and any real constant
such that the
-semigroup of the infinitesimal generator
satisfies
Assume also that
and that
is nonsingular. Then, the system (3.1) is globally asymptotically stable independent of the sizes of the delays.
-
(ii)
If all the eigenvalues of
are distinct, then global asymptotic stability independent of the sizes of the delays delay holds if
since
with the remaining conditions being identical.
Proof. (i) The pointwise difference between the two solutions and
of (2.1) subject to respective initial conditions
and
is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F480187/MediaObjects/13663_2008_Article_1088_Equ3_HTML.gif)
Define the complete metric space with the supremum metric
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F480187/MediaObjects/13663_2008_Article_1088_Equ4_HTML.gif)
where is the set of bounded continuous n-vector functions on
Now, define
as the subsequent bounded continuous function:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F480187/MediaObjects/13663_2008_Article_1088_Equ5_HTML.gif)
Since is an infinitesimal generator of the
-semigroup of the infinitesimal generator
there exist real constants
(which is norm dependent) and
satisfying
since
is a stability matrix, such that for any matrix norm
Then, one gets from (2.4)-(2.5) that the supremum metric, induced by the supremum norm, is then the supremum norm
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F480187/MediaObjects/13663_2008_Article_1088_Equ6_HTML.gif)
for any vector of matrix norms on Now,
is a contraction if
and then there is a unique point
such that
from the contraction mapping theorem [21, 24].
is also a contraction if
holds. The above conditions may be also tested with any supremum norm associated with the supremum metric. For instance, the
supremum real vector function norm
for any
and its induced real matrix function norm
for
provided that such norms exist, where
denotes the maximum (real) eigenvalue of the square symmetric matrix (
). Note that
is a contraction if
holds for any
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F480187/MediaObjects/13663_2008_Article_1088_Equ7_HTML.gif)
for some where
is the spectrum of
of cardinal
and any given vector norm and corresponding induced matrix norm. The limiting equation of (3.1) is
Since
is nonsingular,
Thus, the unique fixed point
in
of the limiting equation, and then that of (3.1) whose uniqueness follows from the contraction mapping theorem, is
As a result, the unique fixed point
is a global attractor so that (2.1) is globally asymptotically stable. Property (i) has been proven.
(ii) If all the eigenvalues of are distinct, that is,
then Property (i) holds for all
and
A stronger result than Theorem 2.1 with the replacement in the relevant first inequality is now given. In other words,
may be taken as unity and
may be zeroed.
Corollary 2.2.
Assume that is a stability matrix of
-matrix measure
and that
Assume also that
and that
is nonsingular. Then, the system (2.1) is globally asymptotically stable independent of the sizes of the delays.
Proof.
Rearrange for any
Then,
where
depends on
and
Redefine the bounded continuous function
replacing
of Theorem 2.1, with the set in (2.4) being redefined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F480187/MediaObjects/13663_2008_Article_1088_Equ8_HTML.gif)
so that the fixed point is looked for any potential perturbation in and not in
First, note that
is still continuous everywhere in its definition domain and also uniformly bounded since
being a stability matrix and
being bounded imply
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F480187/MediaObjects/13663_2008_Article_1088_Equ9_HTML.gif)
for some finite
since
is finite. Thus, (3.11) may be replaced with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F480187/MediaObjects/13663_2008_Article_1088_Equ10_HTML.gif)
so that is a contraction if
since
may be chosen either fulfilling
(the stability abscissa of
is associated with a multiple eigenvalue), but arbitrarily close to
or
(the dominant eigenvalue of
is single). As a result,
has a unique fixed point in
Since any positive and arbitrarily close to zero real constant
may be used,
is a contraction if
In addition, since (3.1) converges to a limiting equation and since
the unique fixed point is zero which is a global asymptotic attractor independent of the sizes of the delays. Therefore, no state-trajectory solution may converge to a distinct point or to be oscillatory since the attractor is global and asymptotic, and no state-trajectory solution may be unbounded (since
is bounded). Therefore, the constraint
leads to a contraction and then to a fixed point for the mapping
and the result has been proven.
Similar results to Theorem 2.1 and Corollary 2.2 may be obtained by comparing the dynamic time-delay system (3.1) with the obtained one for zero delays. The system (2.1) is equivalently written as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F480187/MediaObjects/13663_2008_Article_1088_Equ11_HTML.gif)
By stating the analogy with (2.2), the state-trajectory solution of (2.11), being equivalent to (2.2), is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F480187/MediaObjects/13663_2008_Article_1088_Equ12_HTML.gif)
where the delay-free system is given by of limiting counterpart
with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F480187/MediaObjects/13663_2008_Article_1088_Equ13_HTML.gif)
Use again the complete metric space with the supremum metric
and
defined in (2.4) and replace the continuous mapping (2.5), using (2.12), by
defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F480187/MediaObjects/13663_2008_Article_1088_Equ14_HTML.gif)
The constraint (2.6) changes to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F480187/MediaObjects/13663_2008_Article_1088_Equ15_HTML.gif)
where (norm-dependent) and
(provided that
is a stability matrix) are such that, for instance, for the supremum on
of the
vector (and induced matrix) norm,
so that (2.15) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F480187/MediaObjects/13663_2008_Article_1088_Equ16_HTML.gif)
Thus, Theorem 2.1 and Corollary 2.2 are modified as follows.
Theorem 2.3.
The following properties hold.
-
(i)
Assume that
is a stability matrix of
-matrix measure
and that
for some real constants
and
and any real constant
such that the
-semigroup of the infinitesimal generator
satisfies
Assume also that
Then, the system (2.1) is globally asymptotically stable independent of the sizes of the delays.
-
(ii)
If all the eigenvalues of
are distinct, then global asymptotic stability independent of the sizes of the delays delay holds if
since
with the remaining conditions being identical.
Corollary 2.4.
Assume that is a stability matrix of
-matrix measure
and that
Assume also that
Then, the system (2.1) is globally asymptotically stable independent of the sizes of the delays.
Note that the requirement that is nonsingular is imposed in Theorem 2.3 and Corollary 2.4, since
is directly nonsingular as it is a stability matrix. Note also that
being a stability matrix is also a direct consequence of Theorem 2.1 and Corollary 2.2, which give a result of asymptotic stability independent of the delays thus being valid for zero delays. However, such condition of nonsingularity of
(and even the strongest one of
being a stability matrix) is neither required to apply of the contraction mapping principle [21, 24], nor a direct consequence of it in Theorem 2.1 and Corollary 2.2. As a result, it cannot be invoked prior to stability but only being a consequence after stability has been proven.
Remark 2.5.
Note that concerning the system matrices of the delay-free limiting systems and with
and of zero
delayed dynamics and zero delays, respectively, one has the respective
-matrix measures
Provided they are stable, those limiting systems possess the respective Lyapunov's functions
and
with respective time-derivatives:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F480187/MediaObjects/13663_2008_Article_1088_Equ17_HTML.gif)
so that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F480187/MediaObjects/13663_2008_Article_1088_Equ18_HTML.gif)
As a result, a stronger result than Theorem 2.1(i) holds by replacing and also a stronger result than Theorem 2.1(ii) holds by replacing
In the same way, a stronger result than Theorem 2.3(i) holds by replacing
and a stronger result than Theorem 2.3(ii) holds by replacing
Then, Corollaries 2.2 and 2.4 follow directly as stronger results than Theorem 2.1, respectively, Theorem 2.3 via a very short modified proof by using simple Lyapunov's theory. In other words, global asymptotic stability of the current system holds under asymptotic stability of the respective auxiliary limiting delay-free systems by taking
and
in the relevant inequalities of norms of Theorems 2.1 and 2.3 just as proven in Corollaries 2.2 and 2.4.
3. Feedback Linear Systems with Point Constant Delays and the Contraction Mapping Theorem
The fixed point theory and associated stability results of Section 2 are used and extended directly to state-feedback controlled systems as follows. Instead of the dynamic system (2.1), consider the controlled dynamic system:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F480187/MediaObjects/13663_2008_Article_1088_Equ19_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F480187/MediaObjects/13663_2008_Article_1088_Equ20_HTML.gif)
where and
are piecewise continuous bounded matrix functions,
  
and the control
is generated according to the state-feedback linear control law:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F480187/MediaObjects/13663_2008_Article_1088_Equ21_HTML.gif)
where   
are piecewise continuous bounded matrix functions and
  
The substitution of (3.3) into (3.1) leads to a closed-loop system identical to (2.1) through the identities:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F480187/MediaObjects/13663_2008_Article_1088_Equ22_HTML.gif)
Important properties of dynamic systems are those of controllability, observability, stabilizability, and detectability. For the linear time-invariant dynamic systems
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F480187/MediaObjects/13663_2008_Article_1088_Equ23_HTML.gif)
of state of
state variables and control
and output
of respective dimensions
and
those properties are easily tested through the appropriate Popov-Belevitch-Hutus rank tests [31]. Thus,
-
(1)
the dynamic system is controllable (or, simply, the pair
is controllable) if and only if
An equivalent test is that
is controllable if and only if
The meaning of this property is that for any bounded
there exists a piecewise continuous control
such that
for some finite
An equivalent property is the existence of a controller of gain
such that a linear state-feedback control defined by
makes the matrix
feedback obtained system
to possess a prescribed spectrum
-
(2)
The dynamic system is stabilizable (or, simply, the pair
is stabilizable) if and only if
Its meaning is that there exists
such that the matrix of dynamics
of the closed-loop feedback system is a stability matrix; that is,
and any state-trajectory solution with bounded initial conditions is uniformly bounded and converges asymptotically to the zero equilibrium, as a result. By comparing the controllability and stabilizability tests, it turns out that controllability implies stabilizability but the converse is not true in general.
-
(3)
The dynamic system is observable (or, simply, the pair
is observable) if and only if the pair
is controllable. If
and
are admitted to be complex matrices, then transposes are replaced with conjugate transposes. Observability is related to the ability of calculating the past-state vector from output measurements (usually
Similarly, the dynamic system is detectable (or, simply, the pair
is detectable) if and only if the pair
is stabilizable.
The above concepts are extendable with more involved tests to time-varying and nonlinear dynamic systems. Related results have also been investigated related to fixed point theory (see, e.g., [32, 33]). Recent stability results based on fixed point theory are provided in [34, 35]. The following result follows directly from the controllability property of linear systems. It will be then used for obtaining small left-hand side terms in the norms inequalities of Theorems 2.1 and 2.3 via feedback under assumptions of controllability of relevant matrix pairs.
Lemma 3.1.
The following properties hold.
-
(i)
Assume that the pair
is controllable for any given
Then, for any prescribed set of nonnecessarily distinct complex numbers
there exists a controller matrix
such that the spectrum of the obtained
via (3.4) is
As a result,
is also predefined according to the prescribed set
-
(ii)
If the pair
is controllable, then there exists a controller matrix
such that the
for any given prescribed set of complex numbers
As a result,
is also predefined according to
If the pair is controllable then there exists a controller matrix
such that the
for any given prescribed set of complex numbers
As a result,
becomes also predefined accordingly.
Corollary 2.2, through Lemma 3.1(i), leads directly to the subsequent result.
Theorem 3.2.
Assume that
-
(1)
and that
is nonsingular,
-
(2)
are controllable pairs, and
-
(3)
for some
Then, there exist (in general, nonunique) constant controller gains such that the closed-loop system (3.1)–(3.3) is globally asymptotically stable independent of the sizes of the delays.
Proof.
Theorem 2.1 holds if   
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F480187/MediaObjects/13663_2008_Article_1088_Equ24_HTML.gif)
Note the following.
-
(1)
Since
is controllable, there exists
such that
is a stability matrix and with prescribed spectrum
then with prescribed matrix measure
-
(2)
Since
is controllable for
there exists
such that
has any prescribed spectrum
Then, fix
Since the eigenvalues of
are distinct, it always exists a nonsingular real
-matrix
such that
where
As a result, one gets from (3.6) that for some real
-matrices
provided that
under the incremental controller gains choice
The proof follows from Theorem 2.1(i), since
is independent of
and thus on
so that
is independent of
so that it can be fixed fulfilling
by appropriately selecting
for any given
Theorem 3.2 is useful to guarantee closed-loop stabilization of the system (2.1) under controllability conditions of the time-invariant dynamics by first stabilizing the delay-free dynamics of the limiting equation via linear feedback with sufficiently large stability abscissa. The result is achievable irrespective of the norms of the incremental matrices of dynamics of (2.1) with respect to its limiting equation. Theorem 3.2 is now extended by replacing the time-invariant controllers by time-varying ones.
Corollary 3.3.
Suppose that the assumptions (1)–(3) of Theorem 3.2 hold. Then, there exist nonzero (nonunique) controller gain matrix functions such that the closed-loop system (3.1)–(3.3) is globally asymptotically stable independent of the sizes of the delays.
Furthermore, if then the controller gains
defined with any members of sets of constant controller gains
chosen according to Theorem 3.2 and incremental controller gains
guarantee the global asymptotic stability independent of the sizes of the delays of the closed-loop system (3.1)-(3.2).
Proof.
It follows from Theorem 3.2, provided that the incremental controller gains satisfy
after replacing
and
Such nonzero controller gains always exist since
from Theorem 3.2. To simplify the subsequent notation define the matrix function
by
Now, taking into account (3.6), define the nonnegative real functional
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F480187/MediaObjects/13663_2008_Article_1088_Equ25_HTML.gif)
Note by construction, (3.6) and Theorem 3.2, that Thus, it follows from Theorem 3.2 that there is an open ball
of
centered at cero fulfilling
so that the closed-loop system (3.1)–(3.3) is globally asymptotically stable independent of the delays for sets of nonzero incremental controller gains since the same property is fulfilled for sets of constant controller gains.
If then the incremental controller gains
fulfill
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F480187/MediaObjects/13663_2008_Article_1088_Equ26_HTML.gif)
from least squares minimization. As a result, the matrix function on incremental controllers
and, furthermore,
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F480187/MediaObjects/13663_2008_Article_1088_Equ27_HTML.gif)
guarantee the global asymptotic stability independent of the sizes of the delays of the closed-loop system (3.1)–(3.3).
Note that if and
then
and
guarantee the closed-loop stability from Corollary 3.3. If
and
then from Kronecker-Capelli's theorem, see, for instance, [11, 15], there exist infinitely many solutions of the incremental controller gains which make
so that the closed-loop stability is guaranteed under
with
On the other hand, Corollary 3.3 allows obtaining a subsequent direct result under a weaker Condition 2 of Theorem 3.2. In particular, only the controllability of
and not that of the remaining pairs
is requested for selecting an appropriate negative value of
and the static controller gains are chosen for least-squares minimization of the associated term in
Corollary 3.4.
Assume that
-
(1)
and that
is nonsingular,
-
(2)
is a controllable pair,
-
(3)
for some
-
(4)
Then, the closed-loop system (3.1)–(3.3) is globally asymptotically stable independent of the sizes of the delays provided that the controller gains are synthesized as follows:
-
(5)
-
(6)
-
(7)
is synthesized so that such that
satisfies the constraints
(3.10)
In the same way as Theorem 3.2 is obtained from Corollary 2.2 (a refinement of Theorem 2.1), Theorem 2.3 leads to the subsequent result which is obtained based on a comparison of the delayed dynamics with the delay-free limiting dynamics.
Theorem 3.5.
Assume that
-
(1)
and that
is a stability matrix,
-
(2)
is a controllable pair,
-
(3)
for some
Then, there exist sets of nonunique controller gain matrix functions defined by
such that the closed-loop system (3.1)–(3.3) is globally asymptotically stable independent of the sizes of the delays and
has an arbitrary spectrum of distinct eigenvalues of modulus not larger than a prescribed upper bound
Proof.
The substitution of (3.3) into (3.2) yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F480187/MediaObjects/13663_2008_Article_1088_Equ29_HTML.gif)
being controllable
(a nonsingular real
-matrix) such that
for any given
Define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F480187/MediaObjects/13663_2008_Article_1088_Equ30_HTML.gif)
Then, stability of the closed-loop system (3.1)–(3.3) holds if is a stability matrix and
-
(a)
the set of static controller gain matrices
satisfies that
consists of
distinct complex numbers of modulus not larger than any prescribed
and the nonsingular matrix
defines the similarity transformation
-
(b)
The set of incremental controller gain matrix functions
is chosen so that
with
being defined in (3.12).
Corollaries to Theorem 3.5 might be obtained directly based on the ideals of Corollaries 3.3-3.4 for Theorem 3.2.
4. Further Extensions
The following definitions and associate properties are well known in control theory of linear and time-invariant dynamic systems.
-
(1)
A pair of complex matrices
is said to be stabilizable (or asymptotically controllable) if
such that
-
(2)
Stabilizability of
is a weaker property than the controllability of such a pair what means that
such that
for each prescribed set of
numbers (possibly repeated)
An equivalent characterization of the controllability of
is
-
(3)
If an open-loop system is stabilizable but not controllable, all its uncontrollable open-loop modes are invariant and stable under any state-feedback law.
This section gives extensions of some results of Section 3 for the case when some controllability conditions are lost but stabilizability still holds and some further extensions for the case of output feedback controllers. The following result is weaker but more general than Theorem 3.2.
Corollary 4.1.
Assume that
-
(1)
and that
is nonsingular,
-
(2)
is a stabilizable, but not controllable, pair and
are all controllable pairs,
-
(3)
for some
Then, there exist (in general, nonunique) constant controller gains such that the closed-loop system (3.1)–(3.3) is globally asymptotically stable independent of the sizes of the delays provided that
where
is defined in (3.6) and
is the stability abscissa of the uncontrollable dominant eigenvalue of
The stability property also holds if
with
being defined in the proof of Theorem 3.2.
Proof.
It is similar to that of Theorem 3.2 by noting that where
is the in general open and not simply connected domain of stabilizable static controller gains
of the pair
Theorem 3.2 may also be extended straightforwardly by replacing controllable by
stabilizable,
and
Corollaries 3.3-3.4 are also directly extendable based on Corollary 4.1. On the other hand, Theorem 3.5 extends directly to the subsequent result which is weaker in the sense that the matrix measure cannot be prefixed since controllability of
is replaced by its stabilizability.
Corollary 4.2.
Assume that
-
(1)
and that
is a stability matrix,
-
(2)
is a stabilizable pair,
-
(3)
for some
Then, there exist sets of nonunique controller gain matrix functions defined by
such that the closed-loop system (3.1)–(3.3) is globally asymptotically stable independent of the sizes of the delays provided that
with
defined in (3.12) and
is the stability abscissa of the uncontrollable (stable and invariant under state feedback) dominant eigenvalue of
Now, assume that the control law (3.3) is replaced with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F480187/MediaObjects/13663_2008_Article_1088_Equ31_HTML.gif)
for some set of output matrices with
The interpretation of (4.1) is that the controller has not access to all the state components of the system but only to some linear combinations of them, namely, the output vector defined by
This situation is very realistic under the constraint
that is, the numbers of input and output components are less than the number of state components. The following further definitions and related properties features are well known from basic control theory [28].
-
(4)
Observability is a dual property to controllability in the sense that the pair
  
  
is said to be observable if the pair
is controllable and conversely.
-
(5)
The triple
is said to be controllable and observable if
is observable and
is controllable. If the
is controllable and observable, then there exists
such that
has
values arbitrarily close to any prescribed subset of
of cardinal
with possibly repeated members provided that
and
are full rank. The remaining
members of
cannot be allocated arbitrarily close to prefixed values.
Detectability is a dual property to stabilizability in the sense that is detectable if
is stabilizable.
The above properties lead to the fact that the control output feedback law (3.1) is unable to reallocate all the eigenvalues of respectively, those of
to exact prescribed positions if the triples
respectively,
are controllable and observable even if
and
and
are both full rank. However, under this constraint, all the eigenvalues of the matrix
respectively, of the matrix
can be allocated arbitrarily close to any prefixed set of
complex numbers, by some choice of the static controller gain
Also, if
and
are full rank and
respectively,
are controllable and observable triples, then
of the eigenvalues of
respectively, of
may be allocated arbitrarily close to prescribed complex sets by some
Corollary 4.1 is reformulated as follows for the case of linear output feedback (4.1) by taking into account the above properties of linear time-invariant output feedback.
Corollary 4.3.
Assume that
-
(1)
and that
is nonsingular,
-
(2)
is a stabilizable and detectable triple, and
are all controllable triples,
-
(3)
for some
-
(4)
(4.2)
Then, there exist (in general, nonunique) constant controller gains
such that the closed-loop system (3.1)–(3.3) is globally asymptotically stable independent of the sizes of the delays provided that
If the above Condition (2), replaced with
is a controllable and observable triple instead of stabilizable and detectable and, furthermore,
and
then constant controller gains
can be found so that the closed-loop stability is guaranteed if
for a prefixed
Corollary 4.2 is reformulated as follows for the case of linear output feedback (4.1).
Corollary 4.4.
Assume that
-
(1)
and that
is a stability matrix,
-
(2)
is a stabilizable pair,
-
(3)
for some
-
(4)
(4.3)
Then, there exist sets of nonunique controller gain matrix functions
defined by
such that the closed-loop system (3.1)–(3.3) is globally asymptotically stable independent of the sizes of the delays provided that
The stability property of the closed-loop system also holds if
If the above Condition (2), replaced with
is a controllable and observable triple instead of stabilizable and detectable and, furthermore,
and
then constant controller gains
can be found so that the closed-loop stability is guaranteed if
for a prefixed matrix measure.
References
De la Sen M, Luo N: Discretization and FIR filtering of continuous linear systems with internal and external point delays. International Journal of Control 1994, 60(6):1223-1246. 10.1080/00207179408921518
Burton TA: Stability and Periodic Solutions of Ordinary and Functional-Differential Equations, Mathematics in Science and Engineering. Volume 178. Academic Press, Orlando, Fla, USA; 1985:x+337.
Hale JK, Verduyn Lunel SM: Introduction to Functional-Differential Equations, Applied Mathematical Sciences. Volume 99. Springer, New York, NY, USA; 1993:x+447.
Niculescu S-I: Delay Effects on Stability. A Robust Control Approach, Lecture Notes in Control and Information Sciences. Volume 269. Springer, London, UK; 2001:xvi+383.
De la Sen M, Luo N: On the uniform exponential stability of a wide class of linear time-delay systems. Journal of Mathematical Analysis and Applications 2004, 289(2):456-476. 10.1016/j.jmaa.2003.08.048
De la Sen M: Stability of impulsive time-varying systems and compactness of the operators mapping the input space into the state and output spaces. Journal of Mathematical Analysis and Applications 2006, 321(2):621-650. 10.1016/j.jmaa.2005.08.038
Datko RF: Time-delayed perturbations and robust stability. In Differential Equations, Dynamical Systems, and Control Science, Lecture Notes in Pure and Applied Mathematics. Volume 152. Marcel Dekker, New York, NY, USA; 1994:457-468.
Alastruey CF, De la Sen M, González de MendÃvil JR: The stabilizability of integro-differential systems with two distributed delays. Mathematical and Computer Modelling 1995, 21(8):85-94. 10.1016/0895-7177(95)00041-Y
Ahmed NU: Optimal control of infinite-dimensional systems governed by integrodifferential equations. In Differential Equations, Dynamical Systems, and Control Science, Lecture Notes in Pure and Applied Mathematics. Volume 152. Marcel Dekker, New York, NY, USA; 1994:383-402.
Kulenović MRS, Ladas G, Meimaridou A: Stability of solutions of linear delay differential equations. Proceedings of the American Mathematical Society 1987, 100(3):433-441.
De la Sen M: On impulsive time-varying systems with unbounded time-varying point delays: stability and compactness of the relevant operators mapping the input space into the state and output spaces. The Rocky Mountain Journal of Mathematics 2007, 37(1):79-129. 10.1216/rmjm/1181069321
Jiang Y, Jurang Y: Positive solutions and asymptotic behavior of delay differential equations with nonlinear impulses. Journal of Mathematical Analysis and Applications 1997, 207(2):388-396. 10.1006/jmaa.1997.5276
Sengadir T: Asymptotic stability of nonlinear functional-differential equations. Nonlinear Analysis: Theory, Methods & Applications 1997, 28(12):1997-2003. 10.1016/S0362-546X(96)00029-6
De la Sen M: An algebraic method for pole placement in multivariable systems with internal and external point delays by using single rate or multirate sampling. Dynamics and Control 2000, 10(1):5-31. 10.1023/A:1008351026971
De la Sen M: On positivity of singular regular linear time-delay time-invariant systems subject to multiple internal and external incommensurate point delays. Applied Mathematics and Computation 2007, 190(1):382-401. 10.1016/j.amc.2007.01.053
Chen F, Shi C: Global attractivity in an almost periodic multi-species nonlinear ecological model. Applied Mathematics and Computation 2006, 180(1):376-392. 10.1016/j.amc.2005.12.024
Meng X, Xu W, Chen L: Profitless delays for a nonautonomous Lotka-Volterra predator-prey almost periodic system with dispersion. Applied Mathematics and Computation 2007, 188(1):365-378. 10.1016/j.amc.2006.09.133
White L, White F, Luo Y, Xu T: Estimation of parameters in carbon sequestration models from net ecosystem exchange data. Applied Mathematics and Computation 2006, 181(2):864-879. 10.1016/j.amc.2006.02.014
Sarkar RR, Mukhopadhyay B, Bhattacharyya R, Banerjee S: Time lags can control algal bloom in two harmful phytoplankton-zooplankton system. Applied Mathematics and Computation 2007, 186(1):445-459. 10.1016/j.amc.2006.07.113
Bakule L, Rossell JM: Overlapping controllers for uncertain delay continuous-time systems. Kybernetika 2008, 44(1):17-34.
Burton TA: Stability by Fixed Point Theory for Functional Differential Equations. Dover, Mineola, NY, USA; 2006:xiv+348.
Pituk M: A Perron type theorem for functional differential equations. Journal of Mathematical Analysis and Applications 2006, 316(1):24-41. 10.1016/j.jmaa.2005.04.027
Da Prato G, Iannelli M: Linear integro-differential equations in Banach spaces. Rendiconti del Seminario Matematico della Università di Padova 1980, 62: 207-219.
Burton TA: Stability and Periodic Solutions of Ordinary and Functional Differential Equations. Dover, Mineola, NY, USA; 2005.
Dutta PN, Choudhury BS: A generalisation of contraction principle in metric spaces. Fixed Point Theory and Applications 2008, 2008(1):-8.
Arav M, Santos FEC, Reich S, Zaslavski AJ: A note on asymptotic contractions. Fixed Point Theory and Applications 2007, 2007:-6.
Qing Y, Rhoades BE:
-stability of Picard iteration in metric spaces. Fixed Point Theory and Applications 2008, 2008:-4.
Mackenroth U: Robust Control Systems. Theory and Case Studies. Springer, Berlin, Germany; 2004:xvi+519.
Jung S-M, Rassias JM: A fixed point approach to the stability of a functional equation of the spiral of Theodorus. Fixed Point Theory and Applications 2008, 2008:-7.
Wu M, Huang N-J, Zhao C-W: Fixed points and stability in neutral stochastic differential equations with variable delays. Fixed Point Theory and Applications 2008, 2008:-11.
Ionescu V, Oară C, Weiss M: Generalized Riccati Theory and Robust Control. A Popov Function Approach. John Wiley & Sons, Chichester, UK; 1999:xxii+380.
Chalishajar DN: Controllability of nonlinear integro-differential third order dispersion system. Journal of Mathematical Analysis and Applications 2008, 348(1):480-486. 10.1016/j.jmaa.2008.07.047
Sakthivel K, Balachandran K, Sritharan SS: Exact controllability of nonlinear diffusion equations arising in reactor dynamics. Nonlinear Analysis: Real World Applications 2008, 9(5):2029-2054. 10.1016/j.nonrwa.2007.06.013
Cho YJ, Kang SM, Qin X: Convergence theorems of fixed points for a finite family of nonexpansive mappings in Banach spaces. Fixed Point Theory and Applications 2008, 2008:-7.
Cădariu L, Radu V: Fixed point methods for the generalized stability of functional equations in a single variable. Fixed Point Theory and Applications 2008, 2008:-15.
Acknowledgments
The author is very grateful to the Spanish Ministry of Education for its partial support of this work through Project DPI2006-00714. He is also grateful to the Basque Government for its support through GIC07143-IT-269-07, SAIOTEK SPED06UN10, and SPE07UN04. The author is grateful to the reviewers who helped very much in clarifying the manuscript content in the revised version.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
De la Sen, M. About Robust Stability of Dynamic Systems with Time Delays through Fixed Point Theory. Fixed Point Theory Appl 2008, 480187 (2008). https://doi.org/10.1155/2008/480187
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1155/2008/480187