Second-order observability of matrix-weight-based networks☆
Introduction
Observability is an important and fundamental concept in modern control theory and can be shown in many seemingly different but mathematically equivalent forms [1] for LTI systems. One expression was given in [2] referring to that the relative outputs have synchronized solutions for the identical LTI systems. In recent years, synchronization and consensus have become popular topics in systems, closely related to other advanced problems such as flocking, formation control, sensor networks, spacecraft and collaborative surveillance [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27]. So far, most of the existing studies on synchronization and consensus for LTI systems mainly aimed at individual state components being irrelevant, that is, the information interaction between individuals is a constant (scalar-weighted networks), adapting to the state dimension of the individual through Kronecker product, which indicates that the different state dimensions between the individuals are independent of each other (see [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27] and references therein). In [15], a scale-free cooperation protocol was designed through a scalar-weighted graph, and then made the states of homogeneous multi-agent systems converge asymptotically to a common trajectory according to any given convergence rate. In [22], the authors investigated the consensus of multi-agent systems with unknown time-varying communication delays and achieved the asymptotic bounded consensus of the system by using scalar weight-values and sliding mode control protocols.
However, in real life, there can be a certain correlation between the multidimensional components of individuals, and the state connection between individuals is often not independent of each other. For example, the unity of human opinions is not only influenced by the views themselves, but also influenced by people’s actions, speech, expression and other factors [28]. Matrices can be used to represent the dependencies between these related topics. Thus, the matrix interaction way provides a mechanism to capture finer associations in social networks. Matrix-coupled networks are the natural extension of scalar networks, extending numerical weights into correlations to characterize the interdependencies between the multidimensional states of neighboring individuals. For scalar-weight-based networks, the state components are independent of each other and match multidimensional states by Kronecker product. Matrix-weight-based networks can describe the state coupling relationship between agents more accurately than scalar weights [29], and appear in some scenarios such as effective resistance in generalized electrical networks [30] and social networks [31]. Therefore, the matrix weights can describe the edge weights between agents in multi-agent networks more effectively, accurately and completely, which are of important theoretical and practical value for the modeling, analysis and coordination control of multi-agent systems and other real networks. Tuna [2] modified the scalar weight values in first-order integrator model of multi-agent systems into matrix-weighted values, established a Laplacian coupled LC oscillator model, thus extended to the synchronization problem of LTI systems, and designed a special feedback gain matrix to make the system achieve synchronization. The authors in [32] designed a consensus protocol using rotation matrix value weights and solved the multi-agent loop tracking and direction estimation problems, respectively. Trinh et al. [33] studied the single integrator model under fixed undirected topology, proposed the necessary and sufficient conditions for the global exponential convergence based on the null space of the matrix-weight Laplacian, and then discussed the matrix-weighted group clustering and graph connectivity to guarantee the system consensus. In [34], the authors studied the bipartite consensus problem of matrix-valued weighted networks in the case of structure equilibrium, but ignored that properties of the Laplacian were greatly changed under the matrix-value weights, and did not discuss the case of structure non-equilibrium. At the same time, they also studied the consensus problem of matrix-weighted switching networks, gave the sufficient and necessary conditions for the system to achieve the average consensus, and also discussed the consensus of matrix-valued weighted networks with periodic switching topology in [35]. The synchronization problem of identical oscillators based on matrix-weighted links with sampled-data was studied in [36]. The authors discussed the consensus of discrete-time multiagent networks interacting couplings by nonnegative definite matrices in [37]. More recently, the bipartite consensus, hybrid consensus and second-order consensus of matrix-weighted multiagent systems were studied in [38], [39], [40], [41], [42].
At present, the research in this direction is still in its infancy. The study of matrix-coupled networks helps us to examine the properties of general scalar networks in higher dimensions. Compared with the connectivity of multi-agent systems in scalar networks, the evolution behavior of agents in multi-agent systems with matrix-coupled networks is more complex. Therefore, a new theory different from traditional methods must be developed.
Inspired by Tuna [2], this paper considers the observability of second-order LTI systems with incommensurable output matrices through a matrix-weighted graph by using the space analysis, spectral analysis and matrix decomposition, respectively. Compared with the existing works, the main contributions and novelties of this work include fivefold: (1) A new second-order LTI system with matrix-weighted network is established; (2) The definition of the observability of second-order LTI systems is given; (3) Compared with the observability of identical first-order LTI systems in [2], the observability of second-order LTI systems is discussed, in which the observability or synchronization for positions and velocities can be satisfied simultaneously; (4) The space analysis, spectral analysis and matrix decomposition are introduced to determine the observability of second-order LTI systems with incommensurable output matrices through a matrix-weighted graph, which are very valid to compute the observable subspace using spectral analysis tool by Matlab; and (5) The orthogonal complement space of the observable subspace is obtained by means of space analysis. The main differences between [2] and this paper lie in: (1) Different from the first-order LTI system model considered in [2], this paper considers the second-order LTI system one with positions and velocities. (2) The observability (synchronization) for positions in [2] was discussed by means of the orthonormal basis, but the observability (synchronization) for positions and velocities in this paper is studied by the spectral analysis and matrix decomposition, which can use the lower dimensional matrices to obtain the spectrum and eigenmatrix of matrices for judging observability. Besides, compared with most of the existing related results [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], the incommensurable output is specially considered in our work.
Table 1 shows some useful symbols and notations in this paper.
The rest of this work is arranged as follows. In Section 2, some preliminaries and problem statement are presented. Section 3 presents main results. Examples and simulations are given in Section 4. Finally, the conclusions are summarized in Section 5.
Section snippets
Model
Consider a second-order LTI system with output matrices through a matrix-weighted graph aswhere and are the position and velocity states of the th system with and , respectively; is the th relative output with and being any positive integer. Throughout this paper, and for are needed to ensure the Hermitian property for the Laplace matrix of the eigengraph
Observability and connectivity
By means of space analysis and spectral analysis, we can get some sufficient and necessary conditions for the observability of system Eq. (3). Proposition 2 For pair , the assertions below are equivalent. (1) Pair is detectable. (2) . (3) For every eigenvalue with nonnegative real part of , ifboth hold, then
To study the observability of system Eq. (3), let , then we can have the following results. Theorem 1 System Eq. (3) is detectable if and only if
Examples and simulations
To more intuitively see the validity of the theoretical results in this work, we will give some numerical examples and simulations in the following. Example 1 This example will show the validity of Theorem 1. LetBy computing, we can getthen the eigenvalues of and their corresponding eigenmatrices are and
Conclusion
In this paper, the detectability and observability of a second-order LTI system with output matrices have been discussed, respectively. By means of the spectral analysis and matrix decomposition, the spectrum and eigenmatrix for judging observability can be obtained by the lower dimensional matrix. Some necessary and sufficient conditions for the detectability of the second-order LTI system have been given. On this basis, the equivalent conditions between the observability of such system and
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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This work was supported by the National Natural Science Foundation of China under Grant Nos. 62173355, 119610652, 61991412, the Program for Innovative Research Team in Universities of Inner Mongolia Autonomous Region Grant No. NMGIRT2317 and theNatural Science Foundation of Inner Mongolia under Grant No. 2021MS01006.