Dimensionalinvariance principles in coupled dynamical systems
Abstract
In this paper we study coupled dynamical systems and establish several invariance principles relating to the dimensions of the subspace spanned by solutions of each individual system. We consider two types of coupled systems, one with scalar couplings and the other with matrix couplings. Via the rankpreserving flow theory, we prove that scalarcoupled dynamical systems possess the dimensionalinvariance principles, in that the dimension of the subspace spanned by the individual systems’ solutions remains invariant. For coupled dynamical systems with matrix coefficients/couplings, necessary and sufficient conditions are given to characterize dimensionalinvariance principles. The established invariance principles provide additional characterizations and insights to analyze the transient behaviors and solution evolution for a large family of coupled systems, such as multiagent consensus dynamics, distributed coordination systems, formation control systems, among others.
1 Introduction
In this paper we consider the following coupled dynamical systems consisting of individual systems
(1) 
where is the state for system , is a coupling weight scalar or matrix between systems and (when , then it is a coefficient scalar/matrix for system ). The coupling/coefficient weights could be constant, timevarying, or statedependent. The system (1) serves as a very general model to describe many different types of coupled/networked systems, such as formation control systems [1, 2, 3], network computation systems [4, 5], multiagent consensus dynamics [6], [7].
Coupled dynamical systems are often operated in a networked manner, where each individual systems interact with other systems to perform a global or common task. Networked control systems in the general model (1) have been attracting increasing attention in the recent decade and can be found in a variety of applications. Depending on the actual control task, the coupling terms can be designed to reflect the information flow between spatially distributed systems, communication requirements or constraints, or cooperative interactions incorporating local tasks to achieve a global task [7].
In this paper, we aim to identify several invariance principles for the solutions of each system arising from different couplings and interactions between individual systems. We discuss two types of coupled dynamical systems that can be represented by (1), one with scalar couplings, and the other with matrix couplings, respectively. These principles relate to the invariance of the dimensions of the subspaces spanned by the solutions of each individual system, which are thus termed as dimensionalinvariance principles. The core concept is based on the rankpreserving flow theory, which is used to characterize the rank of the solution to a matrix differential system. Based on the rankpreserving flow theory and its extensions, we prove that coupled dynamical systems with scalar couplings (i.e., in (1) being scalars), the dimensions of the spanned subspace of individual systems’ solutions remain invariant. For coupled dynamical systems with matrix couplings, necessary and sufficient conditions are given to guarantee such dimensionalinvariance principles.
The invariance principles developed in this paper are fundamental yet universal properties for coupled dynamical systems. We note that in most papers on coupled/networked control systems, the focus has been on the stability and convergence analysis, while transient behaviors are largely ignored. The results revealed from the dimensionalinvariance principles provide us with additional insights on the transient behaviors and evolutions of all individual solutions, and could assist the convergence and stability analysis of the overall coupled dynamical systems. An example is the distancebased formation control system described by gradient flows from potential functions of interest, which show that an initially collinear formation remains collinear for all time under such flows (see e.g. [2]).
The invariance principles also provide feasible coupling conditions to guarantee that the solutions of individual systems are constrained in some smaller dimensional spaces, which could find particular applications in several practical scenarios. For example, collinear solutions of a coupled dynamical system are of particular interests. In [1], a line formation, in which individual systems’ states are confined in a 1D subspace, is studied with insights to more general formations on other dimensions. As another example, for a coupled dynamical system that describes the coordination of multiple mobile antennas, collinear solutions have practical significance to align the direction of all antennas in a single line [8]. Motivated by these practical applications, the theory of collinear dynamical systems was studied in [9]. The dimensionalinvariance principles established in this paper will provide a unified and general theory and framework to facilitate these applications.
The remaining parts of this paper are structured as follows. In Section 2, we prove that the solutions of scalarcoupled dynamical systems have the dimensionalinvariance (and furthermore, subspacepreserving) principles. In Section 3, matrixcoupled dynamical systems are discussed, for which necessary and sufficient conditions are given to guarantee the dimensionalinvariance principle. Applications of the invariance principles in formation control systems are shown in Section 4. Section 5 presents the conclusions of this paper. In the appendix sections, we present preliminary background on rankpreserving flows, some extensions and proofs, and a brief review of several popular networked dynamical systems that fit in the general model (1).
1.1 Assumptions and solution issues of (1)
To address the solution issue of the coupled dynamical system (1), we impose the following mild assumption.
Assumption 1.
The coefficient/coupling terms are continuous scalar/matrix functions.
The above mild assumption guarantees the existence and uniqueness of the solutions for system (1) [10, Chapter 1.2]. Note that we do not impose additional assumptions on ’s. They can be static, timevarying, statedependent or other general continuous functions. Note also that the system (1) is not necessarily a timeinvariant or timevarying linear system, sice the coupling term may also depend on the state which may result in a coupled nonlinear system.
2 Coupled dynamical systems with scalarweighted couplings
Consider the following coupled dynamical systems with scalar couplings
(2) 
where is a scalar (static or timevarying) coupling weight between agents and . Note that we do not require , i.e., the coupling weight could be asymmetric.
2.1 Main results
In this section we show that the coupled dynamical system (2) has the following dimensionalinvariance principle.
Theorem 1.
The coupled dynamical system (2) has the dimensionalinvariance principle in the sense that
{empheq}[box=]align
rank(X(t)) = r, (r ≤d), & ∀t ≥0,
& if rank(X(0)) = r.
where .
Proof.
Define the composite vector . In order to obtain a compact form of the system , we define the matrix . Therefore, a compact form of (2) can be written as
(3) 
where denotes the Kronecker product. The vector differential equation (3) on the real vector space can be stated equivalently as the following differential flow on the matrix space (without involving the Kronecker product term)
(4) 
Since the solution of (2) is well defined, the existence and uniqueness of the solution to (4) is also well guaranteed. Then according to Lemma 2 (in Appendix I), the rankpreserving property of the matrix flow (4) follows by observing and , which implies the dimensionalinvariance property of the solutions to (2) in the sense of (1). ∎
We now show a stronger result, that if initial conditions are chosen from some (affine) subspace, then the solutions of the coupled system (2) will always be in that (affine) subspace.
Corollary 1.
In addition to the rankinvariance principle proved above, the solutions of the coupled dynamical systems (2) are subspacepreserving in the sense that
{empheq}[box=]align
& span([x_1(t), x_2(t), …, x_n(t)])
& = span([x_1(0), x_2(0), …, x_n(0)]) ∀t ≥0.
2.2 Interpretations and implications
The system (2) is a very general form of coupled dynamical systems which encompass many control systems that have been actively studied in the literature. Examples include the distributed system for networked function computation [4], [5], multiagent consensus systems initialled in [6] (undirected graphs), and developed in e.g. [7] (directed graphs) and [11] (timevarying couplings), and distributed formation control systems [1, 2, 3]. The results established in this section indicate that, the solutions for individual systems coupled in the form (2) will span a subspace of the same dimension as that spanned by initial conditions, and solutions will be constrained in that subspace over time.
The dimensionalinvariance principle for a particular distancebased formation control system [2] has been proved in our previous paper [12]. We note that such a principle also holds for a large family of formation control systems, including those covered in [3]. In a later section we will show, by this example, how this invariance principle could assist our understanding on the evolutions of agents’ positions in a multiagent formation system.
For some typical coupled dynamical systems with scalar couplings reported in the literature that can be described by the general form (2), see Table 1 in the Appendix. As a consequence of Theorem 1 and Corollary 1, all systems reviewed in Table 1 satisfy the dimensionalinvariance and subspacepreserving property.
3 Coupled dynamical systems with matrixweighted couplings
In this section we consider the following dynamical systems with matrix couplings
(5) 
where is the state coefficient or coupling matrix: i.e., is the coefficient matrix for system , and is the coupling matrix from systems to system .
3.1 Main results
The main result in this section is the following theorem:
Theorem 2.
The proof can be found in the Appendix III. We note that in [9], an alternative but equivalent condition for coupled dynamical systems’ solutions to be collinear (or to be coplanar with dimension ) was obtained, via a somewhat more complicated proof.
The following corollary further characterizes the solution property for the case of coupled dynamical systems.
Corollary 2.
Suppose the coupled system (3) consists of individual systems, and initial conditions for all the coupled systems are chosen to satisfy (i.e. the real symmetric matrix space). Then the coupled dynamical systems (3) have both the dimensionalinvariance principle and signaturepreserving property
(8)  
(9) 
for some matrix .
Proof.
By invoking Lemma 3 (in Appendix II), the above condition can be proved by modifying in the proof of Theorem 2 as . ∎
3.2 Interpretations and implications
The coupled dynamical systems (3) with matrix weights are also very general that can describe many different types of distributed/networked control systems. Examples include the matrixweighted consensus dynamics [13], bearingbased formation control systems [14], or networked linear systems for synchronization [15].
To guarantee the invariance of the dimensions of the subspaces spanned by individual systems’ solutions, each individual system should have the same matrix structure in their state coefficients, with the difference being a scalar multiplier of an identity matrix. Furthermore, the couplings should also be a scalar multiplier of an identity matrix. Since the condition is necessary and sufficient, for other types of couplings between individual systems that are not in the forms of (6) and (7), the dimensionalinvariance principles cannot be guaranteed.
We also note a difference of the invariance principles between the scalarcoupling case and the matrixcoupling case. As proved in Corollary 1, the solutions of coupled system (3) not only span a subspace of the same rank to that of their initial conditions, but also evolve in that particular subspace spanned by initial conditions. However, this subspacepreserving property is not guaranteed for the solutions of the coupled system (3). Theorem 2 only shows the invariance of the dimension of the spanned subspace, while the solutions may also evolve in a different subspace with the same dimension. To be precise, we introduce the concept of Grassmannian subspace to illustrate the difference. The Grassmannian, denoted as , is a space which parameterizes all linear subspaces of a given dimension in a vector space (in this paper, we restrict our attention of to the dimensional Euclidean space ) [16, Page 21]. For example, for , the Grassmannian is the space of lines through the origin in the dimensional space, and it is the same as the projective space of dimensions. For the solutions of coupled dynamical system (3), they will remain collinear if they start collinearly, but the line that passes through the solutions of all individual systems may not be identical over time. In other words, the solutions will evolve in if they start at a subspace of dimension .
4 Applications in convergence analysis for formation control systems with generalized controllers
Consider a particular formation control system in the following form
(10) 
where is the position of agent that lives in , denotes agent ’s neighboring set, and is the desired distance that agents and agent aim to achieve. In the literature, the above control system (10) is usually called distancebased formation control system [17], since the target formation shape is described by a set of interagent distances.
The collinearitypreserving property for the solutions of the formation control systems (10) was observed in several previous papers (e.g., see [2, 1, 18]). In [12], we have generalized this collinearitypreserving property, and proved a general dimensionalinvariance principle for the formation control system (10). Inspired by the results in Theorem 1, one can also consider the following formation control systems with generalized controllers
(11) 
where is a continuous function of the distance error , which is defined as . The local exponential stability of the general formation control system (11) has been discussed in [3].
Corollary 3.
For 2D formations, if all the agents start with collinear positions, they will always be in that collinear subspace under the general control law described by (11). Similarly, for 3D formations, if all the agents start with coplanar (resp. collinear) positions, then they will always be in that coplanar (resp. collinear) subspaces under the control law (11).
Conversely, one can also obtain the following dimensionalinvariance principle for formation systems (11) with noncollinear/noncoplanar initial positions.
Corollary 4.
For 2D/3D formations, if all the agents start with noncollinear/noncoplanar positions, they will always be noncollinear/noncoplanar under the general control law described by (11).
The global analysis of stability and convergence for the formation control system (10) has been discussed in several papers (e.g. [18], [3], [19]), which turns out to be a very challenging problem. The dimensionalinvariance (and subspacepreserving) principles as shown in the above two corollaries will hopefully present additional insights for the convergence and stability analysis of general formation control systems (11). In addition, we can conclude that for any formation control system, if it can be written in the form of (2), then agents cannot escape collinear/coplanar positions if they start with collinear/coplanar positions. If one needs to design formation controllers to avoid such invariance property and to enable agents to escape collinear/coplanar positions even if they start collinearly/coplanarly, then one needs to modify the formation controllers such that they cannot be described by (2). For typical examples of formation control systems without the collinear/coplanar invariance property, see [20, 21].
5 Conclusions
In this paper we establish several invariance principles for coupled dynamical systems (with scalar couplings and with matrix couplings), in relation to the dimensions of the subspaces spanned by their individual solutions. For coupled dynamical systems with scalar couplings, we prove that their individual solutions satisfy the dimensionalinvariance principle (and furthermore, the subspacepreserving principle). For coupled dynamical systems with matrix coefficients/couplings, necessary and sufficient conditions are given to guarantee the dimensionalinvariance principle. The interpretations and implications for the obtained invariance principles are also discussed, with an application to the convergence analysis of formation control systems.
Appendix I: Background on rankpreserving matrix flow
In this section we will briefly review some background on the rankpreserving flow theory [22, Chapter 5].
For integers , let
(12) 
denote the set of real matrices of fixed rank . The following results will be useful in later analysis.
Lemma 1.
is a smooth and connected manifold of dimension , if . The tangent space of at an element is
(13) 
The proof can be found in [22, Page 133]. A matrix differential equation evolving on the matrix space is said to be rankpreserving if the rank of every solution is constant as a function of , that is, for all . The following lemma characterizes such rankpreserving flows (cf. Lemma 1.22 in Chapter 5 of [22]).
Lemma 2.
Let be an interval and let , with be a continuous timevarying family of matrices. Then
(14) 
is rankpreserving. Conversely, every rankpreserving differential equation on is of the form (14) for matrices and .
The proof of Lemma 2 is based on the fact that (14) defines a time varying vector field on the subset of the tangent space of described by (13). The full proof can be found in [22, Page 139].
Remark 1.
Note that the above lemma on rankpreserving flows also implies that the limit value (if it exists) has rank less than or equal to .
Appendix II: Extensions on rankpreserving matrix flow
This section presents some extensions on the rankpreserving flow theory. The following lemma further characterizes rankreserving flows on a symmetric matrix space. Let denote the real symmetric matrix space. For integers , let
(15) 
denote the set of real symmetric matrices of fixed rank .
Lemma 3.
Let be an interval and let with be a continuous timevarying family of matrices. Then
(16) 
is a rankpreserving (and hence signaturepreserving) flow on . Conversely, every rankpreserving (and hence signaturepreserving) differential equation on is of the form (16).
Proof.
The rankpreserving property of is obvious from Lemma 2. The tangent space of at an element is
(17) 
Therefore (16) defines a time varying vector field on each subset of the tangent space of . Thus for any initial condition , the solution of (16) satisfies , for . The signaturepreserving property is therefore a direct consequence of the rankpreserving property and the fact that . Conversely, suppose is rankpreserving and (and therefore is signaturepreserving). Then it defines a vector field on , with as in (17). Letting completes the proof. ∎
In the following we present a more refined principle, termed subspacepreserving principle, for matrix differential systems.
Lemma 4.
Let be an interval and let with be a continuous timevarying family of matrices. Then
(18) 
is subspacepreserving in the sense that . Conversely, every subspacepreserving differential equation on is of the form (14) for some matrices .
Proof.
We rewrite (18) as , which has a unique solution given by , where is the state transition matrix associated with the coefficient matrix (see [10, Chapter 1.3]). Therefore, the solution to the system (18) can be written as . Since the state transition matrix is nonsingular [10, Chapter 1.3], this implies that , . For the converse statement, note that implies that there exists a nonsingular matrix such that . In the context of matrix differential equation, the transpose of the matrix is the state transition matrix associated with a matrix in a matrix differential equation in the form of (18). ∎
Remark 2.
Correspondingly, one can also show that a matrix differential equation in the form , where is a continuous matrix, is rowsubspacepreserving, in the sense that . The proof is similar to that of Lemma 4 and is omitted here.
Appendix III: Proofs of Theorem 2
In this section we present the proof for Theorem 2.
Proof.
Define . We now determine conditions for the coefficient/coupling matrix such that the coupled linear dynamical system (3) possesses the required dimensionalinvariance property. From Lemma 2, this is equivalent to saying that the matrix differential system should take the following form
(19) 
for some and .
Expanding the expression of in (Appendix III: Proofs of Theorem 2), one can obtain the equivalent formula in (Appendix IV: A brief review of coupled systems that can be described by (2) and ()) (in the next page).
Note also that from (3) the matrix differential system (Appendix III: Proofs of Theorem 2) can be written as (22) (in the next page). In order to guarantee the dimensionalinvariance principle, each coefficient term in the system (22) should take the identical form as in (Appendix IV: A brief review of coupled systems that can be described by (2) and ()), which implies
(20) 
which is the necessary and sufficient condition to guarantee the dimensionalinvariance property for the coupled dynamical system (3). ∎
Appendix IV: A brief review of coupled systems that can be described by (2) and (3)
We review and summarize in Table 1 and Table 2 several popular coupled dynamical systems reported in the vast literature, which can be described by (2) and (3), respectively. As a consequence of Theorem 1 and Corollary 1, for all the coupled or networked control systems with scalar couplings reviewed in Table 1, dimensionalinvariance (and furthermore, subspacepreserving) principles are guaranteed. For coupled/networked control systems reviewed in Table 2, if the matrix condition in Theorem 2 is satisfied, then they also possess the dimensionalinvariance property. For example, for the synchronization control of identical networked linear systems with matrix coefficients/couplings [15] (i.e., for the Type I system, with and ), the matrix condition of Theorem 2 is satisfied and the solutions of such networked control systems possess the dimensionalinvariance principle. In contrast, for the Type II coupled systems for linear system synchronization, the condition (9) in Theorem 2 would be violated and thus the dimensionalinvariance property is not guaranteed.
(21) 
(22) 
Ref.  Coupled/networked control systems  System dynamics equation  Coefficient/coupling term  

[4], [5], etc.  Network distributed computation  
[6], [7], [11], etc.  Multiagent consensus  
[1], [2], [3], etc. 


[24], [25], [26], etc. 

Ref.  Coupled/networked control systems  System dynamics equation  Coefficient/coupling term  

[9]  Collinear dynamical systems  
[13], [27], etc.  Matrixweighted consensus  
[15], [28], [29], etc. 

Footnotes
 The signature of a real symmetric matrix refers to the number (counted with multiplicity) of its positive, negative and zero eigenvalues.
 One typical example of comes from the formation control problem with unrealizable shapes [23]: If the triangle inequality does not hold for the desired distances in a triangular shape control problem, then all the agents will converge to a stable collinear equilibrium for which , even if they start with noncollinear positions with . Note that for such flows the rankpreserving property still holds for any finite time but at the limit the rank reduces.
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