Finite-Level Quantized Consensus of Multi-agent Networked Systems

This paper is concerned with the analysis of consensus multi-agent networked system. Adopted in the analysis is the finite-level logarithmic quantizer, for the transmission of the networked-agents state. Two protocols are utilised in the analysed multi-agent networked system: the consensus protocol, which is determined from the outputs and states of a set of encoder-decoder pair that is employed in the system, and convergence rate protocol that is precisely characterized via the use of a dynamic scaling factor. With information exchange among neighbouring agents, the asymptotic consensus can be reached. The proof of protocols is based on proper selection of parameters of the logarithmic quantizer chosen for the connected network. As a demonstra-tion of the validity of the protocols, a four-agent networked system is used. It is shown that an undirected network exchange of information via a communication channel that is equipped with a set of encoder and decoder can lead to attainment of estimates of neighbour state protocol for the networked system. Furthermore, desired asymptotic convergence can be reached through appropriate choice of parameters of the logarithmic quantizer.


Introduction
The problem of consensus of Multi-Agent Systems (MASs) has attracted the attention of many researchers involved in control systems and allied scientific fields, such as unmanned vehicles, automated systems, mini-satellites communication networks, etc. [27,14]. Previous efforts into the study of consensus tivated by the average consensus problem on a network of digital links, [4] proposed sets of algorithms established on pairwise "gossip" communication.
A new design method is introduced in [2], this depends on how sensitive the quantizer changes while the system evolves, interconnected with the given system resulting in a hybrid system. The joint significance of dynamics of agents, the topology of the network and communication on discrete-time linear MASs is investigated by [29]; through the introduction of few necessary and sufficient conditions. The consensus control of an average undirected network comprising several discrete-time first-order agents that are communication constraints is considered in [15]. The problem is addressed through a design of a distributed protocol that establishes dynamic encoding and decoding of agents states. In a related effort, [31] approached the problem of distributed consensus of discrete-time agents through the introduction of agents' states broadcast via infinite logarithmic quantization. In addition, a controller based on the states and outputs of the channel is projected.
This paper studies the quantized logarithmic consensus of MASs with finite bits rate. Each agent has a real-valued state at which the exchange of important information with neighbours can be realized. For protocol specification, consideration is given to error compensation defined by the bit rate of MASs, control system gain and scaling factor. The main focus is on finite-level logarithmic quantized networked of a discrete-time system based on the enhanced procedure of the states and outputs of a set of encoders and decoders. However, for appropriately selected control parameters with a one-bit quantizer, consensus can be reached. Also examined is the connection between the number of quantization levels and the convergence rate; faster convergence requires more bits. Hence, the asymptotic rate of convergence of quantized systems is dependent on the number of bits, the consensusability of the system and the number of MASs involved in the network.
The rest of the paper is structured as follows: while section 2 presents an overview of the graph theory and quantization, section 3 highlights the problem under study. The convergence analysis of finite-level quantized consensus is contained in section 4. This paves the way to section 5, where numerical results are presented to corroborate the validity of the protocols formed using the four-agent system in the illustration. Section 6 concludes the paper.

Notation and preliminaries
Network topology of a system of N -linked agents can be modelled either as a directed or undirected graph denoted by G = (V, E). The communication network modelled through a graph G, consisting of a vertex/node set (G)V = 1, 2, ...N also, an edge set E(G) ⊂ V × V. Each vertex represents an agent, the edge (i, j) shows that the agents i and j share the information about their states, if i, j ∈ V and i, j ∈ E then i and j are adjacent, that is, agents connected by an edge referred to as neighbours, the relation, is represented by i j and assumes that i i always holds.
Various matrices are by nature connected with graphs, such as the adjacency matrix, the diagonal matrix, and the Laplacian matrix. Given a graph G, the formed matrix contains information about the structured graph.
The graph adjacency matrix G = (V, E), described by A, is the rows and columns of the integer matrix that is indexed by vertices of G, such that a ij = 1, a ii = 0 if (j, i) ∈ E or else 0. The graph Laplacian matrix G = (V, E), symbolized by L, is the rows and columns of the matrix is indexed by the vertices of G, such that L ii = j̸ =i a ij , L ij = −a ij for i ̸ = j. This matrix is closely related to adjacency matrix A of G. Given D to be diagonal matrix of a graph G = (V, E), whose rows and columns are indexed by vertices of G, with diagonal entries D ii = d i , defined as: D ij = deg(v i ), alternatively, or else 0, which is equivalent to: L = D − A The adjacency and Laplacian matrices are positive semidefinite by definition; furthermore, for directed graphs, they show symmetric structure. The eigenvalues of the Laplacian matrix L in the increasing order represented by 0 = λ 1 (L) ≤ λ 2 (L) ≤ ...λ N (L), and the spectral radius given as λ N (L) of (L). A ⊗ B denotes the Kronecker product or tensor product of A and B. Resultant matrices satisfy the following prop- [25], if a Laplacian matrix L ∈ R n×n , l ij ≤ 0, l ij ≥ 0, ∀i ̸ = j, and N j=1 l ij = 0 for each j, then L has at least one zero eigenvalues, in which, all the nonzero eigenvalues occupy the open left half plane. Besides, L would have precisely one zero eigenvalues, given the spanning tree in L is a directed graph.
Linear quantization has some advantages but is not the ideal choice for many applications, while logarithmic quantization is intended for a high dynamic range. Each channel has a set of decoder and encoder where all agent receives the signalled information from; then the estimated neighboring state is obtained by a decoding algorithm. The logarithmic quantizer is adopted such that the state of each agent is quantized. q l (x) given as [11,22,16,30]: where ν = 1−ι 1+ι determines the quantization density of q l (x). q l (x) is the set such that U = {±u i : u i = ι i u 0 , i = ±1, ±2, ...}∪{±u 0 }∪{0} 0 < ι < 1, u 0 > 0 the quantizer is easily bounded by a sector bound such as: Hence, the lower the value of ν, the higher the precision the quantized logarithmic signal.
The logarithmic quantizer is illustrated in Figure 1 which shows the sector bound. The primary objective is to use the quantizer information to reach the stability of the overall closed-loop system. The Lyapunov function of the quadratic stabilization problem V (x) = x T P x, P T = P > 0 is deployed to determine the system feedback stability. Text with citations [?] and [?].

Problem Statement
Considering a class of network of MASs comprising (N ) agents, with the linear dynamic system, the ith agents can be expressed as where x i (k) ∈ R N , denote the real-valued state of the agents, and u i (k) ∈ R P is the control input. Matrices A ∈ R N ×N is assumed unstable, and B ∈ R N ×M , is non-varying, while both are stabilisable matrices. For a controllable system A, B the feedback control law should be designed as u i = −Kx i for each subsystem, such that A − BK is stable. The possible control law for controller is given where K is the controller gain, and the Laplacian matrix L elements l ij in the same notion The controller uses all accessible information around the neighbourhood from the analysis in (2).

Consensus Control Feedback System
Using the given control law, the closed loop system gives a state feedback algorithm described by (2) and (4) The quantized logarithmic in (1) examines the number of infinite quantized level. Based on the limitation in (1) it feasible to vary the quantizer inputoutput signal to achieve asymptotic stability via introduction of logarithmic quantizer finite-level. Introducing an 2Q + 1-level with a quantization density, q l (x) is given as [4,11]: The digital channels transmit signal data, which is assumed to be reliable, i.e. noiseless, due to the requirement in (4) of the communication channel in which the actual state valued of each agent can not be accessed by the neighboring agents at each time interval. Each channel has a pair of encodersdecoders. When each agent accepts the signal data from the neighbors, then the estimated neighbor state is obtained by a decoding algorithm. The logarithmic quantizer is adopted to determine the state of each of the agents. The encoder ϕ i of ith agent sent to its neighbours is given as provided x i (k + 1) indicates the inner state and is the output of ϕ i . The q l (.) is the finite-level logarithmic quantizer, and g(k) > 0 is the scaling factor (gain). Initializing g(0) to be positive integer, the constants γ 1 , γ 2 ∈ (0, 1) define g(k + 1) whenever t ≥ 0 in which [10] The q l (.) : R → U, is considered the symmetric quantizer.

Remark 1
The scaling factor influences the encoder ϕ i and affects the magnitude of the predicted error, such that it is less than the state itself, which is expressed by fewer bits. If consensus is reached asymptotically, hence, the prediction error x i (k + 1) −x i → 0 as t → ∞.

Remark 2
Here γ 1 and γ 2 (scaling parameters) keeps the scaled input g(k)x(k) within the quantization range. The scaling parameters γ 1 and γ 2 either scale down or scale up g(k+1) while g(k) is zoomed-out, or zoomed-in. Note stability is determined by the scaling parameter.
For every communication channel (j, i) ∈ ε, the ith agent accepts information from the neighbor j using the outputs of decoder ψ ij to update the state estimates of x i (k) as follows [31] x ij (0) = 0 wherex ij (k) is the output of ψ ij (k). Note that the output of the quantizer determines the signal flow among the agents. Hence, status of an agent is a function of its quantized state and those of its neigbhours.

Cnvergence analysis of finite-level quantized consensus
The consensusability with the state feedback is subject to finite-level logarithmic quantized communication data rate of the protocol (14). This analysis can be carried out by considering the following in reaching consensus: choice of the finite-level quantizer parameters for the system; the number of bits needed for each agent to broadcast within their neighborhood. With the following assumptions [15]: A1 G is connected, A2 max 1≤i≤N ∥x i (0)∥ ≤ C x , max 1≤i≤N ∥δ i (0)∥ ≤ C δ such that C x and C δ are noted nonnegative constant.

Remark 4
The performance of MASs is influenced; when the initial state is considerable large, the input signal tends to be saturated, causing a rapid decrease in g(t). Thus, resulting in a period of overshoot, hence the immediate reduction in g(t) stops saturation while the state decays exponentially.
Remark 5 [15] The set of consensus conditions γ ∈ (ρ τ , 1) with ρ τ = max 2≤i≤N |A− BKλ i | where τ < Kλ N determines the convergence analysis of MASs, considering the decomposition matrix, eigenvalue spectrum of an undirected symmetric network L and the associated Laplacian matrix L.

Remark 6
The connection between the scaling factor (42) and consensus error of the closed-loop system determines the convergence rate of the closed-loop system. Hence, in Theorem 1 the scaling factor with the quantizer decays exponentially. Thus, with (9) and (11) average consensus can be reached asymptotically. Note for any specified τ and γ; the number of bits is approximately conservative. Besides, the required number of bits, control gain τ and scaling factor γ shows how the number K 1 (τ, γ) are related. Likewise, based on the logarithmic quantizer parameter given in Theorem 1 as γ reduces, consensus error converges to zero faster.

Simulation Result
Consider a dynamic undirected network systems of four agents such that the adjacent matrix A is 0 -1. This implies that a ij = 1, if (i, j) ∈ E, otherwise, a ij = 0 shown in Figure 2 with the system matrices in (2)

Conclusions
The consensus of an undirected network of MASs has been studied under a finite quantized information network. The undirected network exchange information through the communication channel which has a couple of encoder and decoder that estimate and generates the neighbour state protocol for the system. Besides, the desired asymptotic convergence can be reached by rightly choosing the appropriate parameters of the logarithmic quantizer. For future research, the focus will be on observer-based consensusability using finite-level logarithmic-quantized feedback control. x 2 x 3 x 4 x 5 x 6 x 7