Stability study of network communication systems supported by fusion topology control techniques

With the rapid development of the communication system, people are no longer satisfied with the traditional network communication technology, and a more rapid and convenient communication system has become the pursuit of people. In order to ensure the normal operation of the network communication system, the network communication system architecture is essential, as a way to solve a series of technical problems in the network communication system. Under normal circumstances, network applications by no time limit when the application system, there is no need for network quality of service control mechanisms. However, when the user imposes time limits or other restrictions on specific applications, the network communication system architecture is required to set up the corresponding control mechanisms. If the network information system delay, overload, congestion and other conditions, this time the network quality of service control mechanism should play a role in order to effectively avoid the network delay caused by data loss and other conditions, in order to better ensure the stability of the network communication system [1-2].

Topology control is a technology that reduces network energy consumption and bandwidth consumption, improves network interference immunity, and optimizes the overall network connectivity, throughput, and transmission delay and other key indicators by adjusting the topological attributes of nodes and wireless communication links in a fixed or mobile wireless network under the constrained conditions of network resources [3-4]. Existing topology control methods are mainly categorized into three types, i.e., expanding communication coverage by adjusting node transmit power, moving nodes to change the distribution of node locations, and changing the number of nodes by node dormancy or adding additional nodes [5]. When designing the network topology, factors such as coverage, connectivity stability, network lifetime, transmission efficiency, interference and competition, and network delay need to be considered [6].

Coutinho, R. W. et al. examined the role played by topology control algorithms in the communication of Underwater Wireless Sensor Networks (UWSNs), which effectively facilitated the propagation of wireless communications in underwater environments and improved the performance of network services and protocols designed for UWSNs, by performing an appropriate, autonomous and dynamic organization [7]. Fu, X. et al. proposed a local search operator MA-TOSCA based on the sink-oriented cascade model of wireless sensor networks (WSNs), which improves network robustness by inscribing the cascading process of wireless sensor networks to quickly discover more stable network topologies [8]. Cao, B. et al. investigated the high-speed wireless topology applicable to wireless data center networks, transformed the topology optimization problem into a multi-objective optimization problem, and established a heat map-based radio wave propagation model to solve the network topology architecture through a multi-objective evolutionary algorithm [9]. Ochoa, M. N. et al. pointed out that different radio configurations and network topologies can optimize the network energy consumption, and by comparing the effect of star and mesh topologies on the energy consumption of LoRa transceivers with different radio configurations, it was found that the star topology was able to ensure low energy consumption while obtaining a higher data rate or a longer transmission distance [10]. Ma, L. et al. developed a distributed filtering strategy for nonlinear time-lag systems in sensor networks, which introduces link multiplicative random noise to reflect random perturbations during information exchange between sensor nodes, introduces a switching topology to switch the communication according to a predetermined rule, and obtains the desired filtering gain by solving for the filtering error dynamics [11].

Bhattacherjee, D. et al. in their study on designing low-latency and high-capacity inter-satellite networks utilized repetitive patterns in the network topology in order to avoid costly link changes over time and to increase the global Internet service network to two times throughput and achieve almost minimal latency [12]. Cheng, T. H. et al. designed a decentralized controller based on event-triggered communication scheduling under fixed or time-varying network topology conditions, with the basic strategy of determining when the intelligences need to update their states and eliminating continuous communication between them [13]. Huang, J. et al. showed that topology control provides a promising approach for designing large-scale, energy-efficient IoT, proposing a systematic approach to IoT topology construction that effectively addresses the scalability and energy-efficiency challenges faced by IoT [14]. Wang, X. et al. investigated an adaptive and reliable coordinated control strategy for multi-intelligent body systems to achieve global tracking error convergence of the system in the presence of intermittent communication constraints and actuator failures by updating the parameters of the local controllers within a single node and utilizing a topology assignment method to deal with network switching jumps [15]. Korkali, M. et al. compared the robustness of a simple topological network model and a cascading dynamic model reflecting network coupling in a specific situation using an example of an electric power communication system, and experiments showed that network interconnections with complementary capabilities are beneficial to improve the reliability and security of the infrastructure in the presence of restricted network failure propagation modes [16].

Multi-intelligent body network communication cooperative control system needs to solve the problems of communication delay and diversified communication topology, in order to solve these problems, the mathematical model of multi-intelligent body network communication cooperative control system based on state prediction is established. First, the vehicle linear dynamics model is established by linearizing the vehicle nonlinear dynamics equations through the feedback linearization method. Then, to address the communication delay problem in the multi-intelligent body network communication cooperative control system, a state predictor is designed to compensate for the delay effects.Then, to address the problem of diverse communication topologies, the model of communication topology is developed with the help of the relevant theory of graph theory to describe communication between vehicles. Finally, all the vehicle dynamics equations and communication topologies are integrated to construct a cooperative control system for multi-intelligent body network communication, and simulation experiments and practical applications are carried out in the leader-follower case and the leaderless case to explore the effectiveness of the design.

2

Preparatory knowledge

2.1

Algebraic graph theory

Multi-intelligent body network communication system is, as the name suggests, a system composed of multiple intelligences, which can be categorized into leaderless system, leader-follower system and multi-leader system according to the number of leaders [17]. Considering the content of this paper, the leader-follower system is taken as an example here: a multi-intelligence network communication system consisting of one leader intelligence as well as N follower intelligence, the communication topology of this system can be described by $\bar{G} = (E, \bar{V})$ , where $\bar{V} = {v_{0}, v_{1}, \dots, v_{N}}$ is the set of nodes and $E \subseteq \bar{V} \times \bar{V}$ is the set of edges. Its structure is shown in Fig. 1.

In this paper, we assume that the leader does not receive information from the follower, and so the Laplace matrix corresponding to Fig. $\bar{G}$ can be defined as: (1) $L = [\begin{matrix} 0 & 0_{1 \times N} \\ L_{2} & L_{1} \end{matrix}]$

The following definition of a jointly connected topology is given in the context of switching signals, time intervals:

Definition 1: Let a topological graph G₁,G₂,…,G_M with the same vertex set V, and record their concatenation as a new topological graph G_1–M, which still has vertex set V and edge set G₁,G₂,…,G_M the concatenation of all edges, and is said G₁,G₂,…,G_M to be jointly connected if the joint graph G_1–M is connected or contains a directed spanning tree.

If the following one-to-one correspondence is satisfied between switching signals and sub-time intervals: (2) ${\begin{array}{l} σ (t) = 1, t \in [t_{p}^{0}, t_{p}^{1}) \\ σ (t) = 2, t \in [t_{p}^{1}, t_{p}^{2}) \\ ⋮ \\ σ (t) = M, t \in [t_{p}^{m_{p} - 1}, t_{p}^{m_{p}}) \end{array}$ and ${{\bar{G}}_{σ (t)} : t \in [t_{p}, t_{p + 1})}$ is jointly connected, then the topology is said to be jointly connected in the time interval [t_p,t_p+1).

Lemma 1: If G(t) is an undirected connected graph and there is at least one follower in the graph who has access to the leader's information, then L₁(t) > 0, and there exists an orthogonal matrix U(t) such that U^T(t)L₁(t)U(t) = Λ(t), where Λ(t) = diag{λ₁(t),λ₂(t),…λ_N(t)} satisfies 0 < λ₁(t) ≤ λ₂(t) ≤ … ≤ λ_N(t), and λ₁(t),λ₂(t),…λ_N(t) is an eigenvalue of L₁(t).

2.2

Matrix theory

The Kronecker product is a generalization of the matrix product AB, independent of the number of rows and columns of the matrix, and is used to represent the solution of matrix equations very succinctly, and has very important applications in the analysis of multi-intelligent body network communication systems [18].

Definition 2: Let $A = [a_{i j}] \in ℝ^{m \times n}$ , $B = [b_{i j}] \in ℝ^{p \times q}$ , then call the following matrix: (3) $A \otimes B = [\begin{matrix} a_{11} B & a_{12} B & \dots & a_{1 n} B \\ a_{21} B & a_{22} B & \dots & a_{2 n} B \\ ⋮ & ⋮ & \dots & ⋮ \\ a_{m 1} B & a_{m 2} B & \dots & a_{m n} B \end{matrix}] \in ℝ^{m p \times n q}$ is the Kronecker product of A and B, or the direct or tensor product of A and B, which can be simply notated as $A \otimes B = {[a_{i j} B]}_{m p \times n q}$ . It can be shown that A⊗B is a m×n-block partitioned matrix, which unfolds to a matrix of mp×nq.

Definition 3: If V is a linear space on C and for any vector x of V, there is a corresponding real-valued function ∥x∥ satisfying the following conditions:

1) Nonnegativity: ∥x∥ > 0 when x ≠ 0. ∥x∥ = 0 if and only if x = 0.

2) Chirality: ∥kx∥ = |k|∥x∥, k∈C.

3) Triangular inequalities: ∥x+y∥ ≤ ∥x∥+∥y∥, x∈V, y∈V.

Then ∥x∥ is said to be the paradigm of vector x on V, referred to as the vector paradigm.

The matrix paradigm is defined similarly to the vector paradigm, with the difference that the matrix paradigm also needs to satisfy compatibility:

Definition 4: If for A∈ℂ^m×n, there exists a real-valued function ∥A∥ satisfying the following condition:

1) Non-negativity: ∥A∥ > 0 when A is not a zero matrix. ∥A∥ = 0 when and only when A is a zero matrix.

2) Chirality: ∥kA∥ = |k|∥A∥, k∈ℂ.

3) Triangular inequality: ∥A+B∥ ≤ ∥A∥+∥B∥, B∈ℂ^m×n.

4) Compatibility: for two matrices A and B, if their matrix product AB makes sense, then there is ∥AB∥ ≤ ∥A∥∥B∥.

Then ∥x∥ is said to be the matrix paradigm of A.

Obviously, there are various different vector paradigms that can be defined on Cⁿ, and their numerical magnitudes are generally different. However, the following relationship exists between different vector paradigms:

Theorem 1: If ∥x∥_a and ∥x∥_b are any two paradigms defined on a finite dimensional linear space Vⁿ, then there exist two positive constants c₁c₂ independent of x such that: (4) $c_{1} {‖ x ‖}_{b} \leq {‖ x ‖}_{a} \leq c_{2} {‖ x ‖}_{b}, \forall x \in V^{n}$

Paradigms ∥x∥_a and ∥x∥_b are then said to be equivalent.

Since matrix paradigms are often mixed with vector paradigms, the construction of matrix paradigms needs to take into account the connection with vector paradigms, which can be achieved by relying on the notion of compatibility of matrix and vector paradigms:

Definition 5: Consider a matrix paradigm ∥·∥_a and C^m on C^m×n with a similar vector paradigm ∥·∥_b on Cⁿ if: (5) ${‖ A x ‖}_{a} \leq {‖ A ‖}_{a} {‖ x ‖}_{b}, \forall A \in ℂ^{m \times n}, \forall x \in ℂ^{n}$

Then matrix-paradigm ∥·∥_a and vector-paradigm ∥·∥_b are said to be compatible.

Definition 6: Let A = [a_ij]∈C^m×n, x = (x₁,x₂,…x_n)∈Cⁿ, then the number of matrix paradigms subordinate to the 1-paradigm, 2-paradigm, and ∞-paradigm of vector x are in order:

1) Column sum paradigm: ${‖ A ‖}_{1} = \max_{j} \sum_{i = 1}^{m} | a_{i j} |$ .

2) Spectral paradigm number: ${‖ A ‖}_{2} = \sqrt{λ}$ . λ is the largest eigenvalue of A^HA.

3) Row sum paradigm: ${‖ A ‖}_{\infty} = \max_{i} \sum_{j = 1}^{n} | a_{i j} |$ .

Also, introduce the Lemma used to solve linear matrix inequalities:

Lemma 2: (Schur complement) Consider a matrix: (6) $S = [\begin{array}{l} S_{11} & S_{12} \\ S_{21} & S_{22} \end{array}] \in ℝ^{n \times n}$ where there is $S_{12} = S_{21}^{T}$ and S₁₁ is r×r-dimensional, then the following three conditions are equivalent:

1) S < 0.

2) $S_{11} < 0, S_{22} - S_{12}^{T} S_{11}^{- 1} S_{12} < 0$ .

3) $S_{22} < 0, S_{11} - S_{12} S_{22}^{- 1} S_{12}^{T} < 0$ .

2.3

Stability theory

Most of the control systems have complex structures with nonlinear and time-varying factors, and the Lyapunov stability theorem is most commonly used when analyzing the stability problems of such complex systems.

Firstly, the concept of stability in the sense of Lyapunov is introduced. Let the state equation of the system be: (7) $\dot{x} = f (x, t)$ where x∈ℝⁿ is the state variable of the system and f(x,t) is a nonlinear vector function with respect to the state variable x as well as time t. A state x_e exists if it makes the system satisfy for all t: (8) $f (x_{e}, t) \equiv 0$

Then x_e is said to be the equilibrium state of the system.

Definition 7: (Lyapunov stability) Let the solution of this system under the initial condition (x₀,t₀) be: (9) $x (t) = g (t | (x_{0}, t_{0}))$

If for any ε > 0 and any initial time t₀, there exists a corresponding neighborhood δ(ε,t₀) > 0 such that when: (10) $‖ x (t_{0}) - x_{e} ‖ < δ$

when Eq. (2) has, for all t > 0, the initial condition x = x₀ and both: (11) $‖ x (t) - x_{e} ‖ < ε$

Then the system equilibrium x_e is said to be stable in the Lyapunov sense. In this case, x_e is said to be uniformly stable if δ is independent of t₀.

Definition 8: (Lyapunov asymptotic stability) A system is said to be consistently stable in the Lyapunov sense if its equilibrium state x_e is stable in the Lyapunov sense at an initial condition of (x₀,t₀) and: (12) $\lim_{t \to \infty} x (t) = x_{e}$

Then equilibrium x_e is said to be asymptotically stable in the Lyapunov sense. Similarly, x_e is said to be uniformly asymptotically stable if δ is independent of t₀.

The definitions of positively definite, negatively definite, semi-positively definite, semi-negatively definite and indefinite scalar real functions are given first below:

Definition 9: Let V(x) be a continuously differentiable scalar function under V(0) = 0 when x ≠ 0:

1) If V(x) > 0, then V(x) is a positive definite function.

2) If V(x) ≥ 0, then V(x) is a semi-positive definite function.

3) If V(x) < 0, then V(x) is a negative definite function.

4) If V(x) ≤ 0, then V(x) is a semi-negative definite function.

5) If V(x) does not belong to any of the above four cases, then it is an indefinite function.

Based on the above, the basic method of stability analysis using Lyapunov's second method is given:

Theorem 2: Consider a nonlinear system, where x_e = [0,0,…,0]^T is the equilibrium state of the system, if there exists a positive definite function V(x) with successive first order partial derivatives and:

1) $\dot{V} (x)$ is a negative definite function, then the equilibrium state x_e of the system is asymptotically stable.

2) $\dot{V} (x)$ is a semi-negative definite function, and if $\dot{V} (x)$ is not constantly equal to zero at t > t₀ for any t₀ and x(t₀) ≠ 0, then the equilibrium state x_e of the system is asymptotically stable, otherwise it is only stable and not asymptotically stable.

3) $\dot{V} (x)$ is a positive definite function, then the equilibrium state x_e of the system is unstable.

Then the equilibrium is globally asymptotically stable if the system is asymptotically stable in equilibrium x_e and satisfies ∥x∥→∞⇒V(x)→∞.

3

State prediction based cooperative control system for network communication

Multi-intelligent body network communication cooperative control system needs to solve the problems of communication delay and diversified communication topology, in order to solve these problems, this paper will take the vehicle network communication system as an example to establish the mathematical model of multi-vehicle network communication cooperative control system based on state prediction.

3.1

Vehicle linear model based on feedback linearization

3.1.1

Vehicle nonlinear dynamics modeling

1)

Traction force of the vehicle

The formula for calculating the traction force is shown below: (13) $\begin{array}{l} F_{e, i} = \frac{T_{i}}{R_{i}} η_{T, i} \\ i = 0, 1, \dots, N \end{array}$ Where, T_i denotes the driving brake/driving torque of the vehicle i, _{η_T,i} is the mechanical efficiency of the driveline, and R_i is the tire radius.

2)

Air resistance

Since this paper only explores the longitudinal dynamics model of the vehicle, the air resistance of the vehicle in the driving process mainly refers to the component force in the direction of motion, which is mainly related to the vehicle speed and wind speed, and the calculation formula can be expressed as: (14) $\begin{array}{l} F_{a, i} = C_{A, i} {(v_{i} + v_{w})}^{2} \\ i = 0, 1, \dots, N \end{array}$ Where, C_A,i is the coefficient of collective air resistance of vehicle i, v_i is the speed of vehicle i, and v_w denotes the wind speed.

3)

rolling resistance

The rolling resistance is calculated by the formula: (15) $\begin{array}{l} F_{r, i} = m_{i, v e h} g f \cos θ \\ i = 0, 1, \dots, N \end{array}$ Where, f is the coefficient of rolling friction, which is mainly related to the construction of the tire, the material of the tire, the air pressure of the tire, the coefficient of friction of the road surface and other factors, m_i,veh is the mass of the vehicle i and g is the acceleration of gravity.

4)

Ramp resistance

Ramp resistance refers to the vehicle traveling by the gravity along the road direction downward force, can be expressed in the following form: (16) $\begin{array}{l} F_{g, i} = m_{i, v e h} g \sin θ \\ i = 0, 1, \dots, N \end{array}$

According to Newton's second law, the equation of longitudinal dynamics of vehicle i during traveling is: (17) $\begin{matrix} m_{i, v e h} {\dot{v}}_{i} = F_{e, i} - F_{a, i} - F_{r, i} - F_{g, i} \\ = \frac{T_{i}}{R_{i}} η_{T, i} - C_{A, i} {(v_{i} + v_{w})}^{2} - m_{i, v e h} g f \cos θ \\ - m_{i, v e h} g \sin θ \\ i = 0, 1, \dots, N \end{matrix}$

The longitudinal dynamics equations of the vehicle can be simplified to obtain the longitudinal dynamics model of the vehicle as shown in Fig. 2, where u_i represents the control inputs of vehicle i. From the figure, it can be seen that the relationship between the variables in the model is complex and has strong nonlinear characteristics, which brings great difficulties to the modeling of the system and the design of the controller, so the model needs to be linearized.

3.1.2

Linearization of the dynamics model

In order to obtain a feedback linearized control law, the longitudinal dynamics equations for vehicle i are first written in the following form: (18) $\begin{matrix} {\dot{p}}_{i} (t) = v_{i} (t) \\ {\dot{v}}_{i} (t) = \frac{1}{m_{i, v e h}} (η_{T, i} \frac{T_{i} (t)}{R_{i}} - C_{A, i} v_{i}^{2} - m_{i, v e h} g f) \\ T_{i, d e s} (t) = τ_{i} {\dot{T}}_{i} (t) + T_{i} (t) \\ i = 0, 1, \dots, N \end{matrix}$

The nonlinear system applies state feedback so that the resulting closed-loop system becomes linear, and the output of the feedback linearized controller is given by: (19) $\begin{matrix} T_{i, d e s} (t) = \frac{1}{η_{T, i}} (C_{A, i} v_{i} (2 τ_{i} {\dot{v}}_{i} + v_{i}) + m_{i, v e h} g f + m_{i, v e h} u_{i}) R_{i} \\ i = 0, 1, \dots, N \end{matrix}$

Assuming that all the parameters in expression (19) can be obtained in advance, the linear equations for the longitudinal dynamics of the vehicle can be obtained by substituting equation (19) into expression (18): (20) $\begin{matrix} {\dot{p}}_{i} (t) = v_{i} (t) \\ {\dot{v}}_{i} (t) = a_{i} (t) \\ τ_{i} {\dot{a}}_{i} (t) + a_{i} (t) = u_{i} (t) \\ i = 0, 1, \dots, N \end{matrix}$

For ease of analysis, the dynamics of the vehicle can be expressed in the form of state space equations as: (21) ${\dot{x}}_{i} (t) = A_{i} x_{i} (t) + B_{i} u_{i} (t), i = 0, 1, \dots, N$ where, $x_{i} (t) = [\begin{array}{l} p_{i} (t) \\ v_{i} (t) \\ a_{i} (t) \end{array}]$ , $A_{i} = [\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & - \frac{1}{τ_{i}} \end{matrix}]$ , $B_{i} = [\begin{array}{l} 0 \\ 0 \\ \frac{1}{τ_{i}} \end{array}]$ , the inertial time constants are satisfied: (22) $\begin{array}{l} {\underline{τ}}_{i} \leq τ_{i} \leq {\bar{τ}}_{i} \\ \underline{τ} = \min ({\underline{τ}}_{i}) \\ \bar{τ} = \max ({\bar{τ}}_{i}) \\ i = 0, 1, \dots, N \end{array}$

3.2

State Predictor Design for Compensation of Communication Delay

3.2.1

Ideas for solving communication delay problems

Multi-intelligent body network communication cooperative control system transmits information through wireless communication, and there is communication delay in the system due to the influence of external environment and relative distance. Previous studies have found that the delay can seriously affect the performance of the system and even make the system unstable.

To address the above problem, this paper proposes a solution idea: all the nodes send the state prediction value of the next sampling moment, and when the sampling period is known, the effect brought by the communication delay will be compensated.

3.2.2

State Predictor Design

After adding the state predictor, the vehicle sends out the predicted state value of the next sampling moment, which can compensate for the effect brought by the communication delay, and this section focuses on the design method of the state predictor [19]. Since the discretization is mainly for the state equations describing the dynamic characteristics of the system, the output equations are static algebraic equations, which should remain unchanged after discretization.

(23)

\begin{matrix} x_{i} (k + 1) = A_{i} x_{i} (k) + B_{i} u_{i} (k) \\ i = 0, 1, \dots, N \end{matrix}

A_i and B_i are the state and control matrices of the state-space equations, respectively, noting that the multi-intelligent body network communication cooperative system explored in this paper is heterogeneous, i.e., A₀ ≠ A₁ ≠ ⋯ ≠ A_N, B₀ ≠ B₁ ≠ ⋯ ≠ B_N. To make the expression succinct, a parameter ϖ_i = e^−h/τi–1 is defined, and the formulas for each matrix are: (24) $\begin{matrix} A_{i} = e^{A_{i} h} = [\begin{matrix} 1 & h & τ_{i}^{2} (ϖ_{i} + τ_{i} h) \\ 0 & 1 & - τ_{i} ϖ_{i} \\ 0 & 0 & ϖ_{i} + 1 \end{matrix}] \\ B_{i} = \int_{0}^{h} (e^{A_{i} h}) B_{i} d t = [\begin{matrix} - τ_{i}^{2} ϖ_{i} - τ_{i} h + \frac{1}{2} h^{2} \\ τ_{i} ϖ_{i} h \\ - ϖ_{i} \end{matrix}] \\ i = 0, 1, \dots, N \end{matrix}$

Combined with expression (23), the state predictor expression for each node can be obtained: (25) $\begin{matrix} {\hat{x}}_{i} (k + 1) = A_{i} x_{i} (k) + B_{i} u_{i} (k) \\ i = 0, \dots, N \end{matrix}$

3.3

Description of the communication topology

The communication topology of the multi-intelligent body network communication cooperative system describes the communication situation of each node with other nodes, and the communication situation of the nodes can be clearly represented with the help of directed graph in graph theory [20]. According to the directed graph theory in graph theory, the communication situation between the following vehicles of a multi-intelligence body network communication cooperative system composed of N following vehicles and 1 guiding vehicle can be represented by the directed graph topology G = 〈V,E〉.

The communication topology represented by directed graphs G and $\tilde{G}$ can be represented by a matrix, and the communication situation between the following vehicles is represented by an adjacency matrix $Z = [z_{i j}] \in ℝ^{N \times N}$ associated with directed graph G, whose elements are defined as follows: (26) ${\begin{array}{l} z_{i j} = 1, & {γ_{j}, γ_{i}} \in E \\ z_{i j} = 0, & {γ_{j}, γ_{i}} \notin E \end{array} i, j = 1, \dots, N$ where {γ_j,γ_i}∈E denotes the existence of a directed edge between node i and node j, i.e., node i is able to receive state information from node j. Since the acquisition of the node's own information does not need to be transmitted through the wireless network, there is no self-loop in the directed graph constituted by the communication topology network, i.e., z_ii = 0. Further, a Laplace matrix L = [l_ij]∈ℝ^N×N is defined to represent a portion of the communication topology matrix constituted by the elements of: (27) $l_{i j} = {\begin{array}{l} \sum_{j = 1}^{N} z_{i j}, & i = j \\ - z_{i j}, & i \neq j \end{array}$

The communication situation between the lead vehicle and the follow vehicle is represented by the matrix P associated with the extended directed graph $\tilde{G}$ with the elemental representation shown below: (28) $P = d i a g {{\tilde{p}}_{1}, {\tilde{p}}_{2}, \dots, {\tilde{p}}_{N}}$ where if ${γ_{0}, γ_{i}} \in \tilde{E}$ , i.e., the following vehicle i is able to receive the information from the guiding vehicle, then ${\tilde{p}}_{i} = 1$ . Otherwise, ${\tilde{p}}_{i} = 0$ In summary, the communication of the entire multi-intelligent body network communication cooperative system can be expressed in the following matrix form: (29) $\begin{array}{l} G = L + P = [g_{i j}] \in ℝ^{N \times N} \\ i, j = 1, 2, \dots, N \end{array}$

3.4

Closed-loop dynamics modeling of the system

The aim of the cooperative driving control with multi-intelligent body network communication studied in this paper is to make the following vehicle follow the motion state of the guiding vehicle and keep a fixed safety distance between neighboring vehicles, i.e: (30) $\begin{array}{l} v_{i} (t) = v_{0} (t) \\ p_{i} (t) = p_{i - 1} (t) - d i = 1, 2, \dots N \end{array}$ where d denotes the desired distance between neighboring vehicles.

The selected distributed control law for the multi-intelligent body network communication cooperative system is defined as: (31) $\begin{matrix} u_{i} (k) = K {\sum_{j = 1}^{N} l_{i j} [x_{i} (k) - {\hat{x}}_{j} (k) - d_{i j}] \\ + {\tilde{p}}_{i} [x_{i} (k) - {\hat{x}}_{0} (k) - d_{i 0}]} \\ i = 1, 2, \dots N \end{matrix}$ where $K = [\begin{array}{l} k_{p} & k_{v} & k_{a} \end{array}]$ is the static feedback gain of each node controller, $d_{i j} = {[\begin{array}{l} (i - j) d & 0 & 0 \end{array}]}^{T}$ denotes the desired distance between node i and node j, and $d_{i 0} = {[\begin{array}{l} i d & 0 & 0 \end{array}]}^{T}$ denotes the desired distance between node i and the guided vehicle.

According to the control purpose of the multi-intelligent body network communication cooperative system can be obtained the state space equation of the node control error as: (32) $\begin{matrix} e_{i} (k + 1) = A_{i} e_{i} (k) + B_{i} u_{i} (k) + A_{i} x_{0} (k) - A_{0} x_{0} (k) \\ - B_{0} u_{0} (k) i = 1, \dots, N \end{matrix}$ where e_i(k) = x_i(k)–x₀(k)–d_i0, which represents the state error vector of node i. The state space equations of each node have parameters of model heterogeneity, which makes it difficult to analyze the performance of the system. To facilitate the analysis, the matrix in the expression is replaced as follows according to the robust control theory: (33) $\begin{matrix} A_{i} = A + H F_{i} H_{1} & B_{i} & = B + H F_{i} H_{2} \\ {‖ F_{i} ‖}_{\infty} \leq 1 & i & = 0, 1, \dots, N \end{matrix}$

The state space equation for a node can be written in the following form: (34) $\begin{matrix} e_{i} (k + 1) = (A + H F_{i} H_{1}) e_{i} (k) + (B + H F_{i} H_{2}) u_{i} (k) \\ + B_{d} ρ_{i} (k) i = 1, \dots, N \end{matrix}$ where $B_{d} = [\begin{array}{l} H & - B_{0} \end{array}]$ , $ρ_{i} (k) = [\begin{matrix} (F_{i} - F_{0}) H_{1} x_{0} (k) \\ u_{0} (k) \end{matrix}]$ .

In order to facilitate the establishment of the state space equations of the whole system, the expressions of the control quantities of all the following vehicles are integrated to obtain the control input vectors of the system: (35) $U (k) = (G \otimes K) \hat{E} (k)$ where $\hat{E} (k) = E (k) - Δ_{0} (k) = [\begin{matrix} x_{1} (k) - {\hat{x}}_{0} (k) - d_{10} \\ ⋮ \\ x_{N} (k) - {\hat{x}}_{0} (k) - d_{N 0} \end{matrix}]$ , Δ₀(k) = I_N ⊗ [B₀u₀(k)], $U (k) = {[\begin{array}{l} u_{1} (k) & \dots & u_{N} (k) \end{array}]}^{T}$ , the state space equation of the whole system can be obtained as: (36) $\begin{matrix} E (k + 1) = (I_{N} \otimes A) E (k) + (I_{N} \otimes H) W (k) \\ + [G \otimes (B K)] E (k) + (I_{N} \otimes B_{d}) Γ (k) \\ W (k) = F Z (k) \\ Z (k) = (I_{N} \otimes H_{1}) E (k) + [G \otimes (H_{2} K)] E (k) \end{matrix}$ where $E (k) = {[\begin{array}{l} e_{1}^{T} (k) & \dots & e_{N}^{T} (k) \end{array}]}^{T}$ , $Γ (k) = {[\begin{array}{l} ρ_{1}^{T} (k) & \dots & ρ_{N}^{T} (k) \end{array}]}^{T}$ , and F = diag{F₁,…,F_N}, and according to expression (2.22), it is clear that ∥F∥_∞ ≤ 1 holds.

4

Simulation experiments and practical applications

4.1

Stability simulation experiment

In this section, two simulations will be used to verify the efficacy and stability of the cooperative control system for network communication.The leader-follower case and the leaderless case are both instances.Each intelligent body vehicle starts the network communication cooperative control in a randomized situation and counts its position, velocity, and parameter evolution trajectory.

4.1.1

Leader-follower situation

In this simulation, a leader and four followers are considered.The communication topology contains a directed support tree, and the weights are all taken to be 1.

The positions and velocities of the leader and followers are shown in Figures 3 and 4. The state trajectory graph shows the states of all followers that can eventually track the leader. In Fig. 3, the position spacing of each intelligent body vehicle is more than 1 unit at the beginning, and the positions of multiple vehicles converge to the same position by about 6s. In Fig. 4, it can be seen that the speed of each intelligent body vehicle enters into a consistent state at about 5s, and continues steadily until the end.

The evolution of the adaptive parameters α_i and ϒ_i is shown in Figs. 5 and 6.

It can be observed that α_i and ϒ_i are converging very fast. In fact, the growth rate of α_i is at least 1, so that at the beginning α_i grows very fast. Thus it is guaranteed that the car will stabilize quickly, i.e., the stability of the network communication system is guaranteed. Considering ϒ_i again, the larger ϒ_i is, the faster the sliding mode surface = 0 will be reached. It is important to note that these parameters depend only on local information and do not have a direct influence on each other, thus these parameters will not necessarily converge to the same value.

4.1.2

No leader situation

Suppose there are 5 intelligences (labeled as 1 to 5) and all edge weights of the strongly connected communication topology are taken as 1.

The position and velocity trajectories of the 5 intelligences are shown in Fig. 7 and Fig. 8, respectively. It can be observed that consistency is finally achieved. According to Fig. 7, the position of each intelligent body reaches consistency when the time comes to about 3 seconds and remains until the end moment.

As seen in Fig. 8, the speed of each intelligent body vehicle enters a consistent state at about 5.5s, which realizes the network communication cooperative control and stably lasts until the end.

Figures 9 and 10 show the evolution trajectories of adaptive parameters φ_i and ϕ_i, respectively. In the leaderless case, both adaptive parameters φ_i and ϕ_i stabilize after about 3.5 seconds to achieve network communication cooperative control.

4.2

Practical application effects

A scientific and suitable site has been selected to apply the system for the actual control of multi-vehicle network communication for two experiments. The vehicle mass is 1700kg, transmission efficiency is 0.91. Tire radius is 0.3m, inertia time constant is 0.5s.

4.2.1

First experiment

The first trial is the leader-follower state. Figure 11 shows the first trial of a cooperative control system for network communication. Fig. 12 shows the result of the system maintaining consistent stability.

As can be seen in Fig. 11, it is obvious that the system in this paper realizes the network communication cooperative control in about 1 s, and the trajectory starts to be consistent. This is much less than the simulation experiment of 3 s. This suggests that there is a certain conservatism in the estimation during simulation. Fig. 12 illustrates that the initial connectivity of each intelligent body has been kept within the communication range, then the connectivity stability of the communication network topology graph is maintained.

4.2.2

Second test

The second time is analyzed for the practical application of the leaderless case. Figure 13 shows the first test network communication cooperative control system consistency trajectory. Figure 14 shows the results of the system keeping consistent stability.

From Fig. 13, it can be seen that still within 1 second of completing the network communication cooperative control, the trajectory will remain consistent. Figure 14 illustrates that the initially connected smart bodies will remain within the communication range and maintain the stability of the communication network topology graph.

In summary, it can be concluded that the network communication cooperative control system based on topological graph state prediction designed in this paper has the ability to instantly realize network communication cooperative control and ensure the stability of communication network topological graph.

5

Conclusion

In this paper, we construct a mathematical model of a network communication cooperative control system based on topological graph state prediction and carry out simulation experiments in two scenarios: the leader-follower case and the leaderless case, as well as practical applications. It has been found that all the followers can eventually track the leader's state. Each intelligent body enters a consistent state approximately 3 seconds later and maintains this state until the end. In the practical application of this paper, the system realizes the cooperative control of network communication in about 1 s, and the trajectory starts to be consistent, which is much smaller than the simulation experiment of 3 s. This shows that the estimation at the time of simulation has some conservatism. The intelligent bodies that support network communication are kept in the communication range, which maintains the connectivity stability of the communication network topological map. The network communication cooperative control system based on topological graph state prediction designed in this paper has the ability to instantly realize network communication cooperative control and ensure the stability of the communication network topological graph, which is in line with the design expectation.

Language:: English

Publication timeframe:: 1 times per year
Journal Subjects:: Life Sciences, Life Sciences, other, Mathematics, Applied Mathematics, General Mathematics, Physics, Physics, other

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Stability study of network communication systems supported by fusion topology control techniques

Lu Bian

Published Online: Mar 17, 2025

Received: Nov 18, 2024

Accepted: Feb 17, 2025

DOI: https://doi.org/10.2478/amns-2025-0285

KeywordsAlgebraic graph theory, Matrix theory, State predictor, Communication topology

© 2025 Lu Bian, published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Keywords
Algebraic graph theory, Matrix theory, State predictor, Communication topology