1. Introduction
Stability and convergence properties are very important topics when dealing with both continuous- and discrete-time controlled dynamic systems. In this context, one of the most important design tools is the closed-loop stabilization of control systems via the appropriate incorporation of stabilizing controllers; see, for instance, [
1,
2,
3,
4] and references therein. In particular, in [
1], and in some references therein, the robust stable adaptive control of tandem of master-slave robotic manipulators using a multi-estimation scheme is discussed. There are several questions of interest in the analysis, such as the fact that the dynamics may be time-varying and imperfectly known, and the fact that a parallel multi-estimation with eventual switching through time is incorporated into the adaptive controller to improve the transient behavior. The speed estimation and stable control of an induction motor based on the use of artificial neural networks is analyzed in [
2]. Strategies of decentralized control, including several applications and stabilization tools, are given in [
3,
4]. In particular, decentralized control is useful when the various subsystems which are integrated in a whole integrated system are located in separate areas, or when the amount of information needed presents difficulties with regards to obtaining completely optimal suitable performance. Thus, the individual controllers associated with the various subsystems get local information about the corresponding subsystems, and eventually some extra partial information about the remaining ones to achieve stabilization, provided that the neglected coupling dynamics are weak enough. Stabilizing decentralized control designs are described in [
3] for networked composite systems. Some technical aspects and the results of non-negative matrices of usefulness to describe the properties and behavior of positive dynamic systems, the robustness of matrices against numerical parameterization perturbations of their entries, and the properties of linear dynamic systems are discussed in [
5,
6,
7,
8].
This paper focuses on the study of sequences of Hermitian matrices of increasing order which are built via block partition aggregation at each iteration, in such a way that both the current iteration and the next one are Hermitian matrices. The basic mathematical tool is the use of the interlacing Cauchy’s theorem of the matrix eigenvalues of the matrices of the sequence, which orders the sequences of the eigenvalues as the iteration progresses [
8]. Our main objective is to adapt the interlacing theorem in order to use it to derive stability or convergence conditions of the sequence of matrices, and to use the results for the stability of a large-scale discrete aggregation-type dynamic system [
9,
10,
11,
12,
13,
14]. The paper is organized as follows.
Section 2 is devoted to investigating the properties of boundedness and convergence of the sequences of the determinants and the sequences of eigenvalues as the iteration progresses by aggregation of the updated information while maintaining a Hermitian structure. In the particular case when the matrices describing the problem are real, the updated information has a symmetrical structure. The results are used, in particular, to give stability or anti-stability (in the sense that all the matrix eigenvalues of the matrices of the iterative sequence are unstable) conditions to the matrices used in the standard factorization of Hermitian positive definite matrices.
Section 3 extends some of the above results to the convergence of sequences of partitioned Hermitian matrices constructed by aggregation of the updated information. Note that the concept of the convergence of matrices is a discrete counterpart of the matrix stability property in the continuous-time domain, since matrices are stability matrices if all their eigenvalues are in the open complex left-half plane. The basic idea that complex square matrices are convergent if their eigenvalues are within the open unit circle centered at zero is taken into account. An example is discussed concerning a SIR epidemic model with contagions between populations of adjacent clusters in
Section 4.
Section 5 is devoted to developing an application for the stability of an aggregation discrete-time dynamic controlled system whose order increases by successive incorporations of new subsystems as the iteration index progresses, and whose structure keeps a symmetry. Finally, some conclusions are presented at the end the paper. The relevant mathematical proofs are given in the
appendix in order to facilitate a direct reading of the manuscript. The system is assumed to be parameterized by real parameters and controlled by linear output-feedback control laws; it is also assumed that the former whole aggregation system and each new aggregated subsystem at each iteration might eventually be coupled.
Notation and Mathematical Symbols
If
is a square Hermitian matrix, then
denotes that it is positive definite and
denotes it is positive semidefinite. Also,
denotes, that it is negative definite.
is an identity matrix specified by if it denotes the -th identity matrix, denotes that the square matrix is positive definite (positive semidefinite), denotes that the square matrix is negative definite (respectively, negative semidefinite), , , , denote, respectively, that , , and , and denote, respectively, the minimum and maximum eigenvalue of a square real symmetric matrix , is the spectral radius of any square complex matrix , is the set of eigenvalues of the Hermitian matrix . If such a set is ordered with respect to the partial order relation then the ordered spectrum is denoted by . The superscripts * and stand, respectively, for complex conjugates or transposes of any vector or matrix, is the Kronecker product of the matrices , if then its vectorization is a vector whose components are all the rows of written in column in its order and respecting the order of its respective entries, is the Moore-Penrose pseudoinverse of the matrix .
2. Technical Results on Partitioned Hermitian Matrices, Cauchy’s Interlacing Theorem and Stability
The subsequent result relies on the conditions for the non-singularity of a partitioned Hermitian matrix of order
which is built by aggregation from a principal Hemitian sub-matrix of order
. Mathematical proof is given in
Appendix A.
Lemma 1. Consider the partitioned matrixfor any, whereis Hermitian,and . Then,
- (i)
is non-singular if and only if, equivalently, if and only if .
- (ii)
Assume thatand. Then,if and only if. Ifandthenif. Ifandthenif .
The subsequent result relies on some conditions which guarantee the boundedness of the determinant and eigenvalues of a recursive sequence of Hermitian matrices which were obtained and supported by Lemma 1 and Cauchy’s interlacing theorem.
Lemma 2. Consider the recursive sequence of Hermitian matricesfor a given initialfor some given arbitrary, where ;, defined by ;and assume that there is a real sequencesuch that, equivalently ;, where ;; ; withand ;. Then, the following properties hold:
- (i)
, for any givenif, andsatisfy the constraint; with, which becomes; ifand; .
- (ii)
Assume thatand thatwith, , for some given. Then, the following relations hold: If, furthermore,andfor some then
- (iii)
Assume that the constraints of Property (ii) hold with; and, furthermore,and, which is guaranteed if. Then,and ;, is bounded and the sequenceis bounded, ifis finite, and then .
Remark 1. Concerning Lemma 2 (i), we can focus on the following particular cases of interest A:
- (a)
andfails for alland some. Then,so thatandso that, a contradiction. Thus, one hasfor anysuch thatand also .
- (b)
andfails for alland some. Note from the definition of the recursive sequence that Sinceand thenandsince, otherwise, ifthen, a contradiction to. Note that ifthenunder the given constraints so that if ; - (c)
and somethen .
Now, one gets from Lemma 2 [(ii), (iii)] the subsequent dual result concerning the recursion obtained from the inverse of . The use of this result will make it possible to give sufficiency-type conditions regarding the non-singularity of the recursive calculation for any positive integer , and also as tends to infinity.
Lemma 3. For some given arbitraryand all, define:and assume that: - (1)
there is a real sequencesuch that ;, where - (2)
, and, which is guaranteed if .
Then,and ;, is bounded, the sequenceis bounded, ifis finite, and then .
One gets by combining Lemma 2 and Lemma 3 the two subsequent direct results:
Lemma 4. Assume thatfor some given arbitraryand assume also that the conditions of Lemma 2 (iii) and Lemma 3 hold. Then, ; .
Lemma 5. Assume that, for some finite ,is a stability matrix and construct a sequenceaccording to the recursive rule:with initial condition. Assume also thatand the sequence of its inverses satisfy the constraints of Lemma 2 [(ii),(iii)] and Lemma 3. Then, is a sequence of stability matrices.
The above result can be directly extended for the case when is antistable, that is, when all its eigenvalues have positive real parts and . Then, by using similar arguments, as in the proof of Lemma 5 based on the continuity of the matrix eigenvalues with respect to its entries and supported by Lemmas 2,3, according to Cauchy’s interlacing theorem, one concludes that consists of antistable members.
Lemma 6. Lemma 5 holds “mutatis-mutandis” if is antistable.
3. Some Extended Results Related to Sequences of Convergent Matrices
In order to be able to adapt the above results to discrete dynamic systems, the well-known result that that the stability domain of a convergent matrix (i.e., a “stable” discrete matrix) is the open unit circle of the complex plane centered at zero has to be taken into account. Note that, in particular,
is convergent if
so that
as
. It turns out that convergent matrices describe the stability property in the discrete sense. In other words, the solution of the discrete difference vector equation
, where
, converges to
for any given
if and only if
is convergent. The relevant results of
Section 2 can be extended to this situation as follows, provided that
is also Hermitian. Consider the following cases:
Case a: so that . Then, it is convergent if and only if . The proof is direct since if then for any . Thus, by taking any of eigenvector , one determines that if and directly fulfills the constraint. This proves the sufficiency part. The “only if part” follows, since if fails, there is of eigenvector such that then and is not convergent.
Case b: so that . Then, it is convergent if and only if . The proof is direct, since if , then for any . Thus, by taking any of eigenvector , one determines that . The remainder of the proof follows Case a closely.
Case c: so that so that . Then, it is convergent if and only if according to Case a by replacing . Note that Case c is included Case a and Case b.
Now, for Case a, replace , defined in Lemma 2, by and it has to be guaranteed that if is Hermitian, then is also Hermitan, and for some then ; .
For Case b, replace and it has to be guaranteed that if is Hermitian, then is also Hermitan, and for some then ; .
For
Case c, note that
so that
then, replace
and it has to be guaranteed that if
is Hermitian and
for some
then
;
. Since
is Hermitian, it is of the form
for some full rank
-matrix
. Then,
. If
then and for some full rank -matrix . Then, Cases a and b can be dealt with using Case c by replacing .
By taking advantage from the fact that a complex square matrix
is convergent (i.e., stable in the discrete sense) if and only if the Hermitian matrix
is convergent, we now build a sequence
of Hermitian matrices as follows, in order to discuss the convergence of its members, provided that
is convergent for some given
or, with no loss in generality, provided that
is convergent. Then,
with
.Then,
which holds if
or,
;
, provided that
;
, that is
is strictly decreasing, so
, and
;
.
Now, assume that the iterations to build
do not add a new row and column to obtain
from
via the contribution of the members of an updating sequence
;
but a set of the, in general. Then, one may get that:
so that
which holds by complete induction if
is convergent and
or,
;
, provided that
, that is
is strictly decreasing, so
, and
;
. This implies that
is convergent.
In the particular case that for some , ; , such a is a forgetting factor of the iteration.
We now consider the matrix factorization
;
. By construction,
is Hemitian (then square), even if
is not square;
. In the case when
is not square, and since its order strictly increases as
increases, it is possible to consider the convergence of the sequence
(without invoking the values of its eigenvalues) as the following property
is asymptotically convergent if
for any given
.
is convergent if
. The following related results are direct of simple proofs given in
Appendix A:
Lemma 7. Ifis convergent then it is asymptotically convergent. The inverse is, in general, not true.
Lemma 8. Assume that; is a complex square matrix of any arbitrary order. Then:
- (i)
If ;thenand are convergent sequences.
- (ii)
If ;thenand are convergent sequences.
- (iii)
For any, if and only if .is convergent if and only if is convergent.
5. Dynamic Linear Discrete Aggregation Model with Output Delay and Linear Feedback Control
In this section, the convergence results of
Section 3 are applied to a dynamic discrete system which is built by the aggregation of discrete dynamic subsystems subject to linear output feedback control. Since we are dealing with a physical system, it turns out that the formalism of
Section 2 can be developed by invoking conditions related to real symmetric systems, rather than to complex Hermitian ones, when necessary. It would suffice to describe the state by expressing the matrix of dynamics in the real canonical form and to transform the control and output matrices by the appropriate similarity matrix. The necessary mathematical proof is given in
Appendix A.
Consider the aggregation linear discrete dynamic system subject to
point delays under linear output-feedback:
, with initial conditions
, where
is a sequence of positive integer numbers,
is the “a priori” vector state at the
n-th iteration,
is the aggregated “a priori” new substate at the
-th iteration (that is basically, the new information needed to update the state vector and its dimension) and
is the “a priori” whole state at the
-th iteration. Also,
,
and
are, respectively, the “a priori” input and measurable output vectors at the
-th iteration and
is a sequence of delays influencing the global dynamics. The sequences of matrices of dynamics
and
, control
, output-state coupling
for
are of members
,
, and
and
for
and the output matrix
. The sequences of matrices
, with
for
, are the output-feedback control gains which generate the control law sequence
.
The dynamics of the new dynamics at the
-th iteration aggregated to the former global aggregation system of state
obtained at the
-th iteration, are assumed to be described by:
, where
is the “a posteriori” state of the aggregated subsystem at the
-th iteration whose “a priori” value is
,
,
,
,
,
,
for
;
,
,
for
, and
and
,
for
;
.
Note that the aggregated subsystem (22)–(24) is coupled to the former global state
describing the total system’s dynamics prior to the aggregation action. It can be seen that the coupling terms do not necessary demonstrate infinite memory requirements as
tends to infinity, since the matrices
,
and
can contain nonzero columns associated with the most recent state/output data related to the previous aggregation system; see, for instance, [
12]. Note also that, due to the coupling between the a priori whole state at the
-th iterations with the a priori new aggregated substate, it can happen that the a posteriori vector after the new aggregated substate has a higher dimension than its a priori version. The various dynamics, control and output matrices have the appropriate orders.
After incorporating the control law, we can write this whole system of extended states
;
in a compact way:
so that
and
and
imply that
,
.
In order to construct a state vector which includes delayed dynamics, we now define the modified extended state
defined by
;
. Thus, one determines from (25) that:
where
, with
. Now, consider the symmetric matrices:
where the relations between the a priori dynamics of the new iteration after the aggregation of a new substate to the whole dynamics with the a posteriori dynamics of the former iteration are given by:
. which are built in order to complete a square a priori matrix of dynamics of the
- the aggregated system which was obtained after the aggregation of the
-th subsystem.
The stability of the aggregation dynamic system (19) to (24) under discrete delays is now discussed via the modified extended system (26), subject to (27), which can be obtained via Lemmas 7,8 from the convergence of the symmetric matrix (28), subject to (29),(30). The following result holds:
Theorem 1. The following properties hold:
- (i)
andare convergent, and also asymptotically convergent, if and only if ; .
- (ii)
andare asymptotically convergent if and only iffor any given .
- (iii)
If(and then) is convergent, then the state of the modified extended system, (26), converges asymptotically to zero, i.e.and alsoasfor any given initial conditionand anyso that the aggregation system is globally asymptotically stable.
- (iv)
If(and then) is asymptotically convergent thenasfor any givenand any given initial conditionand alsoasfor any given initial conditionand any givenso that the incremental aggregation system is globally asymptotically stable.
- (v)
Assume thatis convergent and that ;for some strictly decreasing real sequence. Then, ;andand are convergent sequences.
It is of interest to now discuss how the stability properties of the aggregation system of Theorem 1 can be guaranteed or addressed by the synthesis of the basic controller (21) on the current aggregated system, and how its updated rule (24) can be applied to the new aggregated subsystem to generate the aggregated system for the next iteration step. This discussion invokes conditions to guarantee that the equation of dimensionally compatible real matrices
is solvable in
for a given quadruple
with
and
being square,
being convergent (basically stable in the discrete context) and defining the closed-loop system dynamics after linear output-feedback control
via the linear stabilizing controller of gain
;
,
and
are the open-loop dynamics (i.e., the one being got for
) and
and
are the control and output matrices. Equation (31) is written in equivalent vector form for the unknown
as follows:
It turns out that (31) is solvable in
if and only if (32) is solvable in
, that is, if
according to the Rouché-Froebenius theorem for solvability of linear systems of algebraic equations. Note that if
satisfies the constraint
, for some square matrix
of the same order as
, then
(so that
is convergent) if
. In particular, if
with
then
is convergent if
. A preliminary technical result concerning the solvability if the concerned algebraic system (31), or equivalently (32), is (either indeterminate or determinate) compatible to be then used follows:
Lemma 9. Assume that , and . Then, the following properties hold:
- (i)
linear output-feedback controller exists which stabilizes the closed-loop matrix of dynamicsfor somewith, [5,7], which satisfies the rank constraint: If (33) holds, then the set of stabilizing linear-output feedback controllers of gains which solve (32), equivalently (31), which is a compatible algebraic linear system, for , are given bywithbeing any arbitrary real vector of the same dimension as. Assume that(a necessary condition being). Then (32) foris a compatible determinate, and the unique solution to (33) is Ifwiththen (33), (34) and (35) become, in particular,and - (ii)
Assume that and Then, (32), equivalently (31), is an algebraically incompatible system of equations, and
i.e., Equation (34) for , is the best least-squares approximated solution to (32) in the sense that the corresponding controller gain minimizes the norm error . If (39) holds for any of the form then there is no solution to (31) in ; only best approximation solutions exist.
Particular cases of interest which are well-known from basic Control Theory (see e.g., [
13]) are:
- (1)
,
is non-singular and
is stabilizable, i.e., any unstable or critically unstable mode of the open-loop dynamics can be closed-loop stabilized under linear state feed-back control. Thus,
;
with
(discrete form of Popov-Belevitch-Hautus stabilizability test [
6,
13,
14]). Then (31) becomes
, which is solvable in
, and there is always an output-feedback stabilizing linear controller generating a stabilizing controller of gain
, generating a control
, such that the closed-loop dynamics is defined by a convergent matrix
.
- (2)
In Case 1, . Then, the control law is a linear state-feedback control, and a state-feedback stabilizing linear controller generating a control exists, leading to closed-loop dynamics defined by the convergent matrix .
Lemma 9 is useful to guarantee the relevant results of Theorem 1 in terms of the controller gains choices under certain algebraic solvability conditions. This feature is addressed in the subsequent result:
Theorem 2. Assume that:
(1)so thatis solvable infor some convergent matrixof appropriate order; ,so thatis solvable infor some matrixof appropriate order; ,so thatis solvable infor some matrixof appropriate order; ,
(2) and that subsequent rank conditions hold:so that the following matrix equations are solvable in the delayed controller gains ,and:
Then, the matrix equationsare solvable in the controller gains, and; ;leading to the solutionswith ,, , ,and; ; being arbitrary matrices of appropriate orders for the corresponding equation (above) in each case whose equivalent vector expressions are denoted by . It should be pointed out that it can be of interest to apply the results on interlacing Cauchy’s theorem and some of its extensions (see e.g., [
20,
21,
22]) to the stability of aggregation models based on dynamic systems formulated via differential, difference or hybrid differential/difference equations.