Nonlinear Consensus Protocol Modified from Doubly Stochastic Quadratic Operators in Networks of Dynamic Agents

Abstract

This article explores nonlinear convergence to limit the effects of the consensus problem that usually occurs in multi-agent systems. Most of the existing research essentially considers the outline of linear protocols, using complex mathematical equations in various orders. In this work, however, we designed and developed an alternative nonlinear protocol based on simple and effective mathematical approaches. The designed protocol in this sense was modified from the Doubly Stochastic Quadratic Operators (DSQO) and was aimed at resolving consensus problems. Therefore, we called it Modified Doubly Stochastic Quadratic Operators (MDSQO). The protocol was derived in the context of coordinated systems to overcome the consensus issue related to multi-agent systems. In the process, we proved that by using the proposed nonlinear protocol, the consensus could be reached via a common agreement among the agents (average consensus) in a fast and easy fashion without losing any initial status. Moreover, the investigated nonlinear protocol of MDSQO realized the reaching consensus always as well as DSQO in some cases, which could not reach consensus. Finally, simulation results were given to prove the validity of the theoretical analysis.

1. Introduction

The consensus problem for distributed systems has developed increasingly growing attention in various research areas. One of the essential problems linked to multi-agent systems (MASs) is the consensus in convergence to a common value via a mathematical distributed model of discrete-time. Such problems are solved by linear and nonlinear consensus algorithms. This research is applicable to distributed computing, multi-agent systems, game theory, sensor network, flocking, population genetics, economics, management science, sociology, and robotics by providing the modified nonlinear protocol DSQOs. Previous studies have established a general scheme of the consensus problem using various models [1,2,3,4,5,6].

The central problem of multi-agent systems lies in the coordination ability of the agents when executing a specific task [7]. Consequently, how will the agents converge or agree among themselves to achieve a common goal at the same time? Another concern is how do all agents in one group converge or agree on a consensus objective at the same time. To resolve these type of problems, past studies have been based on the linear consensus model developed by DeGroot, which achieves the consensus only after long iterations. Some authors have tackled this consensus problem using nonlinear models while others have focused on improving the existing models theoretically [8]. The key concern in this respect is that the process of designing a nonlinear mathematical model is considerably complex.

This paper contributes to the body of existing knowledge in several aspects. Firstly, the handling of the nonlinear consensus problem is proposed based on actual mathematical nonlinear systems of DSQOs. This approach will modify the existing works of Linear consensus of [9,10,11,12] including the nonlinear consensus protocols of Lyapunov function of [13,14] and the nonlinear consensus protocols’ equations and calculus equations of [15,16] that are being applied in order to evaluate the consensus problem. Since the building of the existing Lyapunov function for nonlinear consensus and equation of the second order was difficult, the proposed work in [17,18,19] investigated the consensus behaviour of nonlinear MAS using stochastic quadratic operators based on the majorization technique. Linear and nonlinear consensus for multi-agent systems were addressed in [20]. Hence, in the best interest of this work, it can be stressed that the aim of this research was to build non-complicated mathematics, with nonlinear protocols, that is effective and easier to appreciate. Using the designed nonlinear protocols, we demonstrated that the consensus among a group’s agents was reached through local interaction of stochastic networks. Furthermore, the final group decision value would be achieved through a common value of an initial status. By using the nonlinear consensus protocols proposed in this research, we proved that the agents reached an agreement on the value of the group’s decision.

Secondly, we investigated nonlinear protocols, which are as powerful as the linear protocols that additionally fulfill the information requirements of physical frameworks and also the state of data transmission among agents are saved without any loss of generality. Accordingly, we contended that these nonlinear protocols were perceptible to development over present protocols. What more is that these nonlinear agreement protocols generalize the consensus of linear protocols and may be utilized to upgrade the performance of the dynamic consensus algorithm.

The objectives to achieve the agreement means were:

• To find the optimal decision of the group.
• To investigate simple mathematical nonlinear consensus protocol.

The proposed nonlinear model has an effective consensus compared to a linear protocol and it is easy to calculate this than traditional nonlinear protocols.

Thirdly, our results were general, and the modified nonlinear model of MDSQO could be applied for many applications where DSQO and QSO have applications in biology, etc. Meanwhile, the modified nonlinear model of MDSQO will break a new background where especially DSQO cannot do.

The paper is organized as follows. In Section 2 we introduce the background of the consensus problem for MAS. Section 3 shows the materials and methods of the DeGroot linear model, QSO, DSQO, in addition to the solution domain of the proposed nonlinear model (MDSQO). Section 4 proves the consensus result of MDSQO theoretically, while Section 5 discusses several simulation examples to demonstrate the effectiveness of the nonlinear MDSQOs protocols proposed. In Section 6 the evaluation of the nonlinear consensus model MDSQO with the linear consensus model of DeGroot is presented. Section 7 contains the Comparison of the Convergence of MDSQO with DSQO, while Section 8 includes a discussion and critical reflection of the research. Finally, Section 9 concludes the paper.

2. Background

Most recent research has successfully demonstrated that input constraints can be realized in real-world physical systems using the nonlinear consensus model [21]. For instance, the input that is necessary to overcoming the behaviour orientation problem of robots is bounded, and the controls of required inputs cannot be secured by linear protocols. Another disadvantage of the linear protocols lies in the fact that they cannot limit the data transmission rate as characterized by flexible communications and exact transfer data among agents [22]. From this point of view, we were motivated to investigate nonlinear protocols for consensus problems in MAS. The strength of nonlinear mathematical models lies in the fact that they converge faster than linear models when applying the feedback process [23,24,25,26,27,28]. However, the nonlinear protocol for multi-agent systems prevents any possible loss of energy during controls consensus in the case of uncertain measurements [29] since nonlinear models need to make fewer iterations to reach a consensus. The proven efficiency of the nonlinear models shields them from the weaknesses inherent to the processes applied in linear models.

However, most nonlinear protocols are based on the Lyapunov function theory, which means that it has remained hard to shape nonlinear protocols based on a realistic Lyapunov function for MAS [5,23,30,31]. Furthermore, the recent research completed in the study of nonlinear protocols was constructed on fractional second-order and second-order of calculus such as in [32,33,34,35,36,37,38,39]. It has been established that the equations of the second order constitute highly complex calculation systems, whereas for the third and higher-orders clearly-defined solutions have yet to be developed.

Control consensus on nonlinear heterogeneous MAS is also a challenging issue because the problem of regulating the power of nonlinear systems itself is more difficult. Different types of nonlinearities make it difficult to discover a universal control ruling or a requirement to achieve consensus. However, this type of research would be beneficial and encouraging as the most practical systems are nonlinear in real-life environments.

Traditional approaches to control consensus problems are frequently based on linear models founded on DeGroot’s model [40]. More recently, researchers have proposed nonlinear models since they converge faster, require a lesser number of iterations, and are capable of converging to optimal consensus.

Many researches have endeavored to contemplate nonlinearity processes for evolving the reaching consensus of MAS. Jian et al. [41] have considered the nonlinear consensus case for MAS where all agents are divided into a group set consisting of information. It was exposed that the reaching consensus could be done by this nonlinear control in the continuous and discrete-time. A real-time nonlinear control approach via a new system was investigated in [42] for MAS under mutually associated exchanging topologies. In [43] sufficient conditions were addressed according to an integrated nonlinear control under laws of duplex control. A finite-time nonlinear consensus algorithm was established for heterogeneous MAS where the consensus was guaranteed to be zero in a finite time. Two general nonlinear consensus algorithms of higher-order models have been solved for the MAS where the agreement has been achieved in a finite time also [44]. The ensured execution of the agreement control for MAS frameworks with Lipschitz nonlinear flow and coordinated interaction topologies was explored in [45], where the coordinated interaction topology contained a crossing tree. Onother researches via a nonlinear operator was proposed in [46,47,48,49] in order to converge the individual network connectivity agents at a common value. However, Zhu and Martínez [50] have studied a new method of a discrete-time dynamic that the output stability properties are inculcated for convergence. Furthermore, Meng et al. [5] achieved iterative learning control by allowing the agent to interact with its neighbours and by using stochastic matrices that resulted in an agreement formation nonlinear control of MAS. Their approach utilized optimistic optimization to control the agents behaviour and designed a black box with an unknown nonlinearities model system [51]. Ajorlou et al. [52] completed a study on nonlinear class of continuous time for agreement whereby each agent was controlled by sharing its neighbour’s information in the graph. Their consensus algorithm provided sufficient conditions for converging the agents at a common point.

However, one of the weaknesses of nonlinear models is that they are often associated with higher complexity and restrictions. The present concern was to assess and evalaute possible nonlinear models with faster convergence and optimal consensus, while offering relatively more flexible system conditions and low complexity.

3. Materials and Methods

In this work, we aimed to provide certain nonlinear protocols for convergence in MAS and further study the problem of consensus for the intended nonlinear protocols. It is desirable to provide a linear protocol amendment process under the stated environment that is invariable in time and structured synchronously presented at [1,53] first.

(a) The Problem Domain

Consider a swarm of $m$ agents $\left(x_1, x_2,\dots ,x_m\right)$ taking on values of states $x_i^{\left(0\right)}\in R$ as given. From the background, the cases of agents can be as velocities, opinions, estimates, values, positions, and so on [54]. The interaction distribution of the agent swarm systems can be described by a network distribution of a transition matrix $p_{ij}={\displaystyle \sum }_{i,j=1}^m{a}_{ij}$, where $a_{ij}$ represents the weight value of the interaction between $x_i$ and $x_j$. The problem that we are focusing on in this work is the decentralization control of limited information.

In the model of DeGroot, it is supposed that the distribution of $(x_1^{\left(0\right)},x_2^{\left(0\right)},\dots ,x_m^{\left(0\right)})$ has a linear integration distribution for interaction to reach an agreement [40]. It was supposed that $p_{ij}\ge 0$ and ${\displaystyle \sum }_{i=1}^m{p}_{ij}=1$ (where all the elements of the matrix are equal to one). Therefore, after each agent has started informing its distribution of initial state $x_i^{\left(0\right)}$, then the state of the agent is updated from $x_i^{\left(0\right)}$ to $x_i^{\left(1\right)}={\displaystyle \sum }_{i=1}^m{p}_{ij}{x}_i^{\left(0\right)}$. Let $p_{ij}$ be the transition square matrix, where $p_{ij}$ are the elements of the matrix and $i$ and $j$ represent the number of row and column respectively. The subscripts “ $i,j$” in this sense denote the numbers of agents updating their states between each other, that is, agent $i$ updates its state with agent $j$ by weight value of $a_{ij}$. It is clear that the matrix is stochastic since the sum of each row is equal to one. The process step here is that the updated states of agents are iterated. After the state of the agent is updated from $x_i^{\left(0\right)}$ to $x_i^{\left(1\right)}={\displaystyle \sum }_{i=1}^m{p}_{ij}{x}_i^{\left(0\right)}$, then the process is repeated for the next update from $x_i^{\left(1\right)}$ to $x_i^{\left(2\right)}={\displaystyle \sum }_{i=1}^m{p}_{ij}{x}_i^{\left(1\right)}$. The process continues in this way.

$$x_i^{\left(t+1\right)}={\displaystyle \sum }_{i=1}^m{p}_{ij}{x}_i^{\left(t\right)},$$

(1)

The process is repeated for $t$ iterations. By DeGroot’s model the states of $(x_1^{\left(t\right)},x_2^{\left(t\right)},\dots ,x_m^{\left(t\right)})$ reach a consensus if only all of $x_i^{\left(0\right)}$ converge to the same limit, where $t\to \infty$.

On the opposing side, we have a nonlinear consensus of DSQO [55,56]. In this model, it was supposed that the distribution be $(x_1^{\left(t\right)},x_2^{\left(t\right)},\dots ,x_m^{\left(t\right)})$ in every iteration, with ${\displaystyle \sum }_{i=1}^m{x}_i^{\left(t\right)}$ = 1 and $x_i^{\left(t\right)}\ge 0$ for all $=\overline{1,m}$ $t\in N$. It is assumed also that $S^{m-1}$ is an $\left(m-1\right)$-dimensional simplex, where

$$S^{m-1}=\{x_i^{\left(t\right)}=(x_1^{\left(t\right)},x_2^{\left(t\right)},\dots ,x_m^{\left(t\right)})\in R^m:x_i^{\left(t\right)}\ge 0, \forall i=\overline{1,m,} {\displaystyle \sum }_{i=1}^m{x}_i^{\left(t\right)}=1\},$$

(2)

Definition 1.

A matrix${\left(p_{ij,k}\right)}_{ij,k=1}^m$is called stochastic matrix if all its entries are nonnegative and the sum of all m matrices is a matrix that has all elements equal to 1, i.e.,

$${\displaystyle \sum }_{k=1}^m{P}_{ij,k}=1, p_{ij,k}\ge 0, \forall k=\overline{1,m},$$

(3)

Definition 2.

The$\underset{t\to \infty }{lim}{x}_k^{\left(t\right)}\in S^{m-1}$is a sequence of the trajectories of the stochastic matrix$P=\left(P_1,P_2,\dots ,P_m\right)$starting from initial points$x_k^{\left(0\right)}$if

$$x_i^{\left(t+1\right)}={\displaystyle \sum }_{ij=1}^m x_i^{\left(t\right)}{P}_{ij,k} x_j^{\left(t\right)},$$

(4)

where$P_{ij,k}$is astochastic matrix defined by Equation (3) and the$x_i^{\left(t\right)}$the row vectors and$x_j^{\left(t\right)}$the column vectors.

Considerably, the nonlinear model of the quadratic stochastic operator (QSO) $V:S^{m-1}\to S^{m-1}$ is as follows:

$${\left(Vx\right)}_k=\left({\displaystyle \sum }_{ij=1}^m{p}_{ij,1} x_i^{\left(t\right)} x_j^{\left(t\right)}, {\displaystyle \sum }_{ij=1}^m{p}_{ij,2} x_i^{\left(t\right)} x_j^{\left(t\right)},\dots , {\displaystyle \sum }_{ij=1}^m{p}_{ij,m} x_i^{\left(t\right)} x_j^{\left(t\right)} \right)$$

(5)

The DSQO refers to Quadratic Stochastic Operators (QSO) theory. However. the history of QSOs can be refered to the work of Bernshtein [57] who first modeled an application of QSOs in population genetics [58]. QSOs were defined based on one-dimensional simplex, simplex being a set of points [59]. However, higher-dimensional simplex is more challenging to study, and two-dimensional simplex has remained unsolved [60]. Lyubich studied QSOs on one-dimensional simplex and proved that it contained a finite set for the $\omega -limit$ from any initial point [61]. Vallander succeeded in investigating and producing the results for some cases of QSOs on two-dimensional simplex [62] that were developed on finite-dimensional simplex by Ganikhodzhaev in [59,63]. Subclasses have been investigated from the class of QSOs such as DSQOs [60,64], dissipative doubly stochastic operators [65], and complementary stochastic quadratic operators [66].

QSOs have been developed through the technique of majorization where the majorization concept of vectors has risen to become a very beneficial method for the classification of QSOs into their related sub-classes [55,67,68]. DSQOs have also been defined using the majorization concept as introduced in [63]. The QSOs are called DSQOs if $Vx\prec x$ where $“\prec ”$ commonly denotes the majorization concept [69]. The structural properties of DSQOs were studied in [67]. As the class of DSQOs is considerably large the study of limited behaviour constitutes a rather challenging task [60]. The set is ordered after comparing the coordinates’ partial sums and rearranging them in a non-increasing order [70]. In turn, the extreme DSQOs using majorization were examined in the works of [71].

Definition 3.

The stochastic operator$V:S^{m-1}\to S^{m-1} V:S^{m-1}\to S^{m-1}$is called doubly stochastic [59], if

$$Vx\prec xforallx\in S^{m-1}$$

(6)

In the class of DSQOs, element $x\in S^{m-1}$ is the rearrangement of non-increasing $x_{\downarrow }=\left(x_{\left[1\right]}, \dots , x_{\left[m\right]}\right)\in S^{m-1}$, where $x_{\left[1\right]}\ge \dots \ge x_{\left[m\right]}$.

If we have two elements $x,y \in S^{m-1}$, and if

$${\displaystyle \sum }_{i=1}^k{x}_{\left[i\right]}\le {\displaystyle \sum }_{i=1}^k{y}_{\left[i\right]}, k=1, \dots , m,$$

(7)

Then, we can say that $x$ is majorized by $y$, and write $x\prec y.$

It has defined that $x\prec y$ if a doubly stochastic matrix $P$ is $x=Py$ [68].

Hence, if $P$ is a doubly stochastic matrix, then $Px\prec x$ for any element $x\in S^{m-1}.$

The DSQO should has a doubly stochastic matrix $P$ under majorization rules [68].

It is proved that $V:S^{m-1}\to S^{m-1}$ is a DSQO if the coefficient $P_{ij,k}$ in QSO satisfies the condition $V\left(x\right)\prec x$ [61,70]. We refer to the conditions of the majorization concept as a set of $U_1$ [70].

$$U_1=\left\{A=\left(a_{ij}\right):a_{ij}=a_{ji}\ge 0, {\displaystyle \sum }_{i,j\in \alpha }{a}_{ij}\le \left|\alpha \right|,{\displaystyle \sum }_{i,j\in I}{a}_{ij}=m\right\}$$

(8)

Since the DSQO has not reached consensus in some cases of extreme points as it was analyzed in the previous research [70,71], and since it has converged for the consensus problem in multi-agent systems [72] we figured then to introduce our model of nonlinear protocol. This model has some new modified notions and notations of DSQO to achieve a consensus always in any case of the stochastic matrix (we mean here that each transition matrix for each agent could be either of the following: not stochastic, stochastic or doubly stochastic).

The consensus problem is a special part of cooperative control. It is defined as a team of agents who reach an agreement on a common value by negotiating with their neighbours.

(b) The Solution Domain

Now, we have presented our modified nonlinear protocol for a consensus problem in MAS, which we will call modified doubly stochastic quadratic operators (MDSQOs).

Definition 4.

$MAX(x_k^t)$is the maximum status and$MIN(x_k^t)$is the minimum status of$\{x_k^t=\left(x_1^t,x_2^t, \dots , x_m^t\right)\in R^m:0\le x_k^t\le 1, \forall k=\overline{1,m, }{\displaystyle \sum }_{k=1}^m{x}_k^t=1\}$. The difference between$MAX(x_k^t)$and$MIN(x_k^t)$is$d(x_k^t)$. We say the$\underset{t\to \infty }{lim}{x}_k^{\left(t\right)}\in S^{m-1}$converges to the center$\left(\frac{1}{m}, \dots , \frac{1}{m}\right)$of the simplex$S^{m-1}$if$\underset{t\to \infty }{lim}d(x_k^t)= \underset{t\to \infty }{lim}MAX(x_k^t)- \underset{t\to \infty }{lim}MIN(x_k^t)=0$, where$t:0\to \infty$.

Protocol MDSQOs: In MAS, each agent has an initial status sharing a dynamic controlled by the following MDSQOs nonlinear protocol where it is a trajectory of the stochastic matrix

$$x_i^{\left(t+1\right)}= x_i^{\left(t\right)}\left(p_{ij,k}^{\left(t+1\right)}\right) x_j^{\left(t\right)}=P_{ij,k}^{\left(t+1\right)} x_k^{\left(t\right)}$$

(9)

where $x_i^{\left(t\right)}=(x_1^{\left(t\right)},x_2^{\left(t\right)},\dots ,x_m^{\left(t\right)})$ is the probabilistic distribution of the agents’ group after $t$ iterations for $x_i^{\left(t\right)}$ row vectors and $x_j^{\left(t\right)}$ column vectors, while $p_{ij,k}=\left(p_{ij,1},p_{ij,2},\dots ,p_{ij,m}\right)$ is a stochastic matrix according to the Equation (3).

New notions and notations for MDSQOs: A matrix ${\left(p_{ij,k}\right)}_{ij,k=1}^m$ is called a stochastic matrix in MDSQOs protocol if the entries of elements $a_{ij.k}$ are positive, with the sum of the elements of each matrix equal to $m$, while each matrix is symmetric and the sum of all m matrices is a matrix that has all elements equal to 1 as follows

$$p_{ij,k}=\left\{\begin{array}{c}{a}_{ij,k}>0, a_{ij,k}=a_{ji,k}, {\displaystyle {\displaystyle \sum }_{ij=1}^m}{a}_{ij,k}=m\\ , {\displaystyle {\displaystyle \sum }_{k=1}^m}{a}_{ij,k}=1\end{array}\right\}$$

(10)

where $i$ and $j$ represent the matrix’s row and column numbers respectively, while $k$ is the matrix’s number.

However, in stochastic matrix we get some cases of these matrices that are:

(i)

The sum of all m matrices is a matrix with all entries as ones as follows:

$${\displaystyle \sum }_{i=1}^m{a}_{ij,k}\ne {\displaystyle \sum }_{j=1}^m{a}_{ij,k}\ne 1 and {\displaystyle \sum }_{k=1}^m{a}_{ij,k}=1, a_{ij,k}>0, \forall i,j,k=\overline{1,m},$$

(11)

(ii)

The sum of each row or column in each matrix is stochastic and the sum of all m matrices is a matrix with all entries as ones as follows:

$${\displaystyle \sum }_{i=1}^m{a}_{ij,k} or {\displaystyle \sum }_{j=1}^m{a}_{ij,k}=1 and {\displaystyle \sum }_{k=1}^m{a}_{ij,k}=1, a_{ij,k}>0, \forall i, j,k=\overline{1,m},$$

(12)

(iii)

Triple stochastic matrix, where the sum of each row and column in each matrix is stochastic and the sum of all m matrices is a matrix with all entries as ones as follows:

$${\displaystyle \sum }_{i=1}^m{a}_{ij,k}={\displaystyle \sum }_{j=1}^m{a}_{ij,k}=1 and {\displaystyle \sum }_{k=1}^m{a}_{ij,k}=1, a_{ij,k}>0, \forall i,j,k=\overline{1,m},$$

(13)

Hence, the initial status sharing dynamics can be written as follows:

$$x_i^{\left(t+1\right)}=\left({\displaystyle \sum }_{i,j=1}^m{p}_{ij,1} x_i^{\left(t\right)} x_j^{\left(t\right)}, {\displaystyle \sum }_{i,j=1}^m{p}_{ij,2} x_i^{\left(t\right)} x_j^{\left(t\right)},\dots , {\displaystyle \sum }_{i,j=1}^m{p}_{ij,m} x_i^{\left(t\right)} x_j^{\left(t\right)} \right)$$

(14)

We mention here that the initial status for agents should be positive as follows:

$$x_i^0=\left\{x_i^0=\left(x_1^0,x_2^0,\dots ,x_m^0\right), where x_i^0\ge 0, {\displaystyle \sum }_{i=1}^m{x}_i^0=1\right\}$$

(15)

This means that the sum of positive initial status should be equal to one.

Here, our model by MDSQO protocol was a modification that generalized the linear DeGroot and nonlinear DSQOs models and showed that the MAS reached a consensus if agents had a positive initial status. In our proposed protocol of MDSQOs, we aimed to investigate a simple nonlinear model with faster convergence than the traditional linear protocols. In turn, we established these results with a stochastic matrix.

4. Results

Theorem 1.

In the case of$n$agents$\left(x_1, x_2, \dots , x_n\right)$of the system ordered to execute the same task simultaneously, multi-agent systems have to make a decision in order to reach a consensus and complete it. In the MDSQO protocol, the multi-agent systems converge to an average consensus after updating their status to confirm that each agent has a positive symmetric interaction with all other group’s agents.

Proof. 

Let the agents $\left(x_1,x_2,\dots ,x_m\right)$ be with any initial positive status difference as/ states be with any positive initial status difference as $\left(x_1^0,x_2^0,\dots ,x_m^0\right)\in R$. Let $x$ be the row vector containing the status of all agents and $Y$ be the matrix of interactions in the current state. Then, using the MDSQO protocol, each state is updated based on the simple matrix multiplication defined by $xY{x}^t$ where $x^t$ is the transpose of $x$ (column vector),

$$x_i^{\left(t+1\right)}= \begin{array}{ccc}\left(\begin{array}{cccc}{x}_1^{\left(t\right)}& x_2^{\left(t\right)}& \dots & x_m^{\left(t\right)}\end{array}\right)& \left(\begin{array}{c}\begin{array}{cc}\begin{array}{ccc}{a}_{11,k}& a_{12,k}& \cdots \\ a_{21,k}& a_{22,k}& \dots \\ ⋮& ⋮& \ddots \end{array}& \begin{array}{c}{a}_{1m,k}\\ a_{2m,k}\\ ⋮\end{array}\end{array}\\ \begin{array}{ccc}{a}_{m1,k}& a_{m2,k}& \begin{array}{cc}\dots & a_{mm,k}\end{array}\end{array}\end{array}\right)& \left(\begin{array}{c}{x}_1^{\left(t\right)}\\ x_2^{\left(t\right)}\\ \begin{array}{c}⋮\\ x_m^{\left(t\right)}\end{array}\end{array}\right)\end{array}$$

where each agent has a positive symmetric interaction a_ij,k with other agents and the sum of these interactions equals to the number of agents ${\displaystyle \sum }_{ij=1}^m{a}_{ij,k}=m$ and $t$ is the sequence of the states.

Then

$$x_i^{\left(t+1\right)}=x_i^{*}$$

(16)

where $x_i^{*}$ is a new state, and still ${\displaystyle \sum }_{k=1}^m{x}_i^{*}=1$, without losing the sum properties for maintaining the properties related to the “sums” for the general states.

Therefore

$$MAX(x_i^0)>MAX(x_i^1)>MAX(x_i^2) >\dots$$

(17)

$$MIN(x_i^0)<MIN(x_i^1)<MIN(x_i^2) <\dots$$

(18)

In effect, it can be clearly seen that

$$MAX(x_i^0)\mathrm{is}\mathrm{decreasing}\mathrm{monotonically},$$

while

$$MIN(x_i^0)\mathrm{is}\mathrm{increasing}\mathrm{monotonically}.$$

Consequently, converging to the same point

$$MAX(x_i^0)=MIN(x_i^0)$$

(19)

In fact, this entails that

$$MAX(x_i^0) and MIN(x_i^0)\mathrm{are}\mathrm{bounded},$$

which is attributed to the fact that the sum of states of the agents equals to 1, ${\displaystyle \sum }_{i=1}^m{x}_i^{\left(t\right)}=1$, and due to the equal stochastic distribution for each agent, they are bounded by the common average value of $\left(\frac{1}{m}\right)$.

Then, the multi-agent systems converge to a common consensus with a value of ($\frac{1}{m})$.  □

Corollary 1.

Theorem 1 implies that the nonlinear model of MDSQO converges faster than the linear model of DeGroot.

Proof. 

Let a team of agents $\left(x_1^0,x_2^0,\dots ,x_m^0\right)$ start with an initial random positive status where $0\le x_i^0\le 1,x_i^0\in S^{m-1}$.  □

Then applying the nonlinear model of MDSQO using the equation

$$a_{ij,k}\ast (x_i^0)\ast (x_j^0)$$

(20)

and the linear model of DeGroot using the equation

$$a_{ij}\ast (x_j^0)$$

(21)

Hence, the initial value of $\left(x_1^0,x_2^0,\dots ,x_m^0\right)$ is $MAX$ or $MIN$ in the Equations (17) and (18). It is then clear that the initial value for the case of Equation (20) will decrease or increase faster than that of Equation (21).

Therefore, the nonlinear model of MDSQO converges faster than that of a linear model.

5. Numerical Solution

In this section several simulated examples are given to determine the efficiency of the proposed nonlinear protocols of MDSQOs. The simulation results confirm that reaching consensus for multi-agent systems can be achieved if (and only if) the communication among the agents is positive.

Let us consider 10 different initial states $\left(x_1^0, x_2^0,x_3^0\right)$ and transition matrices $\left(p_{ij,1},p_{ij,2},p_{ij,3}\right)$ to show the consensus of the MDSQO protocol for multi-agent systems. (See Appendix A).

Now, recalling Equation (9) then we have

$$\begin{array}{l}{x}_1^{\left(t+1\right)}=a_{11,1}{x}_1^{\left(t\right)}{x}_1^{\left(t\right)}+a_{12,1}{x}_1^{\left(t\right)}{x}_2^{\left(t\right)}+a_{13,1}{x}_1^{\left(t\right)}{x}_3^{\left(t\right)}+a_{21,1}{x}_2^{\left(t\right)}{x}_1^{\left(t\right)}+a_{22,1}{x}_2^{\left(t\right)}{x}_2^{\left(t\right)}+a_{23,1}{x}_2^{\left(t\right)}{x}_3^{\left(t\right)}+a_{31,1}{x}_3^{\left(t\right)}{x}_1^{\left(t\right)}+a_{32,1}{x}_3^{\left(t\right)}{x}_2^{\left(t\right)}+a_{33,1}{x}_3^{\left(t\right)}{x}_3^{\left(t\right)},\\ x_2^{\left(t+1\right)}=a_{11,2}{x}_1^{\left(t\right)}{x}_1^{\left(t\right)}+a_{22,2}{x}_1^{\left(t\right)}{x}_2^{\left(t\right)}+a_{13,2}{x}_2^{\left(t\right)}{x}_3^{\left(t\right)}+a_{21,2}{x}_2^{\left(t\right)}{x}_1^{\left(t\right)}+a_{22,2}{x}_2^{\left(t\right)}{x}_2^{\left(t\right)}+a_{23,2}{x}_2^{\left(t\right)}{x}_3^{\left(t\right)}+a_{31,2}{x}_3^{\left(t\right)}{x}_1^{\left(t\right)}+a_{32,2}{x}_3^{\left(t\right)}{x}_2^{\left(t\right)}+a_{33,2}{x}_3^{\left(t\right)}{x}_3^{\left(t\right)},\\ x_3^{\left(t+1\right)}=a_{11,3}{x}_1^{\left(t\right)}{x}_1^{\left(t\right)}+a_{12,3}{x}_1^{\left(t\right)}{x}_2^{\left(t\right)}+a_{13,3}{x}_1^{\left(t\right)}{x}_3^{\left(t\right)}+a_{21,3}{x}_2^{\left(t\right)}{x}_1^{\left(t\right)}+a_{22,3}{x}_2^{\left(t\right)}{x}_2^{\left(t\right)}+a_{23,3}{x}_2^{\left(t\right)}{x}_3^{\left(t\right)}+a_{31,3}{x}_3^{\left(t\right)}{x}_1^{\left(t\right)}+a_{32,3}{x}_3^{\left(t\right)}{x}_2^{\left(t\right)}+a_{33,3}{x}_3^{\left(t\right)}{x}_3^{\left(t\right)},\end{array}$$

We obtained the simultation results for the cases 1–10 as presented in the graphs below:

The simulations of the cases 1–10 are presented in Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10 respectively. The combined experimental examples confirmed that the MAS reached a consensus from any initial status in a shorter time period and from any initial values of the agents interaction under the MDSQO protocol. We noted that even though the cases (1, 2, 3, 5, 6, 7, 8, 9) were not under conditions of the DSQO protocol in which any sub-block of $m by m$ for each matrix should be less than $m$, ${\displaystyle \sum }_{i,j=1}{a}_{ij,k}\le \left|\alpha \right| for \alpha \subset \left\{1,2,\dots , m\right\}$, and still, the operator converged. Moreover, there were some cases (like 4 and 5) that were under the conditions of DSQO and MDSQO protocols, and this meant that the protocol of MDSQO could sometimes be DSQO and this is when the matrix satisfied the condition related to the sub-block.

The simulation of the consensus of MDSQO validated that the MAS consisting of 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, 300, 400 and 500 agents converged to an average consensus of $(\frac{1}{m})$ as depicted in Figure 11. Moreover, more agents in the systems made a fast convergence as is shown in Figure 11. The group of agents consisting of more than 10 agents converged after the processing of only one iteration, while the group of agents consisting of less than 10 agents converged by two or more iterations. This is because of the nonlinear stochastic advantages.

6. Comparison Study between the Nonlinear Model of MDSQO and DeGroot’s Linear Model

In this section, the evaluation of the nonlinear consensus model MDSQO with the linear consensus model of DeGroot is presented. The top row of figures in Figure 12 presents the consensus of the DeGroot linear model for both cases of SM and DSM (see Figure 12a,b). The lower row shows the convergence of DSQO in the case of SM and DSM, respectively (see Figure 12c,d).

It can be shown that the linear models of DeGroot reached consensus after 16 iterations using SM (Figure 12a), and 50 iterations using DSM (Figure 12)). It was obvious that there were fewer iterations using SM, yet the convergence was not to the center (unbounded value) (Figure 12b). However, the consensus of a nonlinear model of MDSQO was achieved by 12 iterations using SM with convergence to the center (average) (Figure 12c), and only three iterations by using DSM (Figure 12d).

Initial states:

$$x_1^0= 0.438744, x_2^0= 0.214114, x_3^0= 0.347142.$$

7. The Comparison of the Convergence of MDSQO with DSQO

DSQO has some cases that cannot reach consensus, while MDSQO reaches a consensus always and that why MDSQO has modified to avoid that cases of the unreachable consensus of DSQO.

The cases that DSQO cannot reach a consensus that when the equation is periodic and fixed. The periodic case is the case when the limit of the equation goes to somewhere the comeback to the same point. However, the fixed case is the case when the limit does not change for any time.

The simulation solution of the four examples in Figure 13 show the cases of convergence and nonconvergence of DSQO. Figure 13 for V1-2D shows the case of the convergent of DSQO. Figure 13 for V2-2D shows the fixed case of DSQO where DSQO never converged. Figure 13 for V3-2D shows the periodic case of DSQO which also never reached a consensus. However, Figure 13 for V4-2D shows a mixed case of convergent which was fixed and where some points had reached a consensus while others were fixed or periodic. The conclusion of this simulation was that DSQO had some cases which could not reach a consensus. The matrices of the cases in Figure 13 have been presented next.

If we go through MDSQO we can see that the adding condition between the connection among agents must be positive where the values of the matrices are a positive lead and the MDSQO converges for any time and never has cases of periodic and fixed. From another point of view, the interaction among agents must be positive. The proof of this is shown in Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12.

The operator V1-2D is DSQO on a two-dimensional simplex, which means there are three agents. In the case of DSQO, the consensus was reached. The operator of V1 was:

$x_1= x_1{x}_2+x_1{x}_3+x_2{x}_3;$

$x_2= x_1^2+ x_1{x}_3+x_2{x}_3;$

$x_3= x_2^2+ x_3^2+x_1{x}_2;$

However, the operators V2, V3, and V4 were also on 2D and had not reached a consensus that because either the DSQO here had a fixed point case as in V2 and V4 or periodic points case as in V3.

V2:

$x_1= x_1^2+x_1{x}_2+x_1{x}_3;$

$x_2= x_2^2+x_1{x}_2+x_2{x}_3;$

$x_3= x_3^2+x_1{x}_3+x_2{x}_3;$

V3:

$x_1= x_3^2+x_1{x}_3+x_2{x}_3;$

$x_2= x_1^2+x_1{x}_2+x_1{x}_3;$

$x_3= x_2^2+x_1{x}_2+x_2{x}_3;$

V4:

$x_1= x_1^2+x_1{x}_2+x_1{x}_3;$

$x_2= x_3^2+x_1{x}_2+x_2{x}_3;$

$x_3= x_2^2+x_1{x}_3+x_2{x}_3;$

From this analysis, we concluded that DSQO had some cases that were impossible to reach consensus.

8. Discussion and Critical Reflection of the Research

MDSQO was analyzed in Section 4 and Section 5 where it reached consensus in all cases. A central problem was focused on multi-agent systems (MAS). To solve this problem a mathematical model is required to do a convergence among the agents. The mathematical model could be linear or nonlinear. The nonlinear model has more efficiency to achieve a consensus among agents. To design a nonlinear model remains very complicated and one of the challenges and issues in math area. Most of the consensus model was linear because of its easy structure. In this work, a new nonlinear model called MDSQO referred to DSQO and was investigated for the consensus in MAS. The process and structure of MDSQO follow the linear model of DeGroot. As a result, it has been shown that MDSQO reaches a consensus faster than the linear model of DeGroot and achieved the consensus in the cases that DSQO could not. DSQO had cases that never reached a consensus, these are proven in Figure 13.

9. Conclusions

In this paper, we established a nonlinear protocol for the problem of consensus in MAS which generalized linear and nonlinear protocols. The interaction among agents has been proposed here to be characterized by stochastic matrices that are called MDSQO. We have demonstrated that a consensus can be reached faster than the DeGroot linear model by the respective agents in the multi-agent system. In this sense, the consensus between agents can be reached from any given initial status for the agents interaction. This has shown the MDSQO reaches a consensus always if each member agents of the group have positive interactions with the other members compared to DSQO where it has some cases impossible to reach a consensus.