Disturbance rejection of a class of discrete-time multi-agent systems with pinning control

Abstract

For discrete-time multi-agent systems with random switching nonlinear linking and exogenous disturbances, the disturbance rejection problem is studied and a sufficient condition is presented using the Lyapunov function. In an example of five agents with pinning control, the condition is adopted to optimize disturbance rejection. A numerical simulation shows that the optimization is effective.

Keywords

Multi-agent system disturbance rejection pinning control discrete time nonlinear linking

Introduction

Multiagent (multi-robot) systems play an important role in formation control, flocking, and synchronization. For all kinds of multiagent systems, the consensus problem has received much attention^1

–6 in terms of designing neighbor-based feedback protocols (control rules) for all agents to achieve synchronization. In the last 15 years, most consensus research results have been obtained for linear multiagent systems.⁷ In recent years, however, consensus protocol designs have been developed for nonlinear multiagent systems using internal model methods. These designs include the local distributed regulator design,⁸ the semi-global consensus design,^9,10 and the global consensus design.^11,12

All the above works consider multiagent systems without disturbance. In fact, multiagent systems always suffer from exogenous disturbances. It is necessary to deal with both the consensus and the disturbance. For this reason, the disturbance rejection problem of multiagent systems has attracted interest from a number of researchers. For first-order multiagent systems with external disturbances on fixed and switching directed topologies, disturbance rejection conditions have been presented using $H_{\infty}$ control theory.¹³ For leader–follower multiagent systems consisting of first-order followers and a second-order leader subjected to measurement noise, a velocity decomposition technique has been addressed¹⁴ to design the neighbor-based tracking protocol together with distributed estimators. For high-order multiagent systems with external disturbances on switching undirected topologies, the Schur orthogonal transformation and common Lyapunov function method have been adopted¹⁵ to reject disturbances. By constructing a suitable Lyapunov–Krasovskii function and taking the reciprocally convex approach, the disturbance rejection problem of multiagent systems with a communication delay has been tackled.¹⁶ For time-varying multiagent systems subject to energy-bounded external disturbances, the disturbance rejection problem over a finite horizon has been considered and a constrained recursive Riccati difference equation approach has been proposed to codesign the time-varying controller and estimator parameters in the framework of an event-based scheme.¹⁷ The multiagent systems investigated in the cited studies^13

–17 can be seen to be linear. In the last few years, results have been obtained for the disturbance rejection of nonlinear multiagent systems as follows. A robust consensus tracking problem for an integrator-type multiagent system was addressed by Hu¹⁸ in the presence of disturbances and nonlinear unmodeled dynamics. For a class of Lipschitz multiagent systems subject to external disturbances, a two-step algorithm¹⁹ was presented to achieve global consensus with a guaranteed $H_{\infty}$ performance. Sufficient conditions were established for the disturbance rejection of nonlinear multiagent systems with intermittent information transmissions.²⁰ A sliding mode control-based algorithm²¹ was proposed to solve the finite-time consensus tracking problem for nonlinear multiagent systems with external disturbances. Finally, the effect of inherent unmodeled disturbances has been considered for nonlinear multiagent systems, and a reduced-order observer-based consensus protocol has been developed.²² The disturbance rejection problem of nonlinear multiagent systems is also studied in the present article. Unlike the nonlinear multiagent systems whose uncertainties arise in the inherent dynamics of each agent in the above references,^18–22 the model uncertainties in the present article arise in the linking topology of the multiagent system. In addition, the linking topologies are fixed in the above references,^18

–22 while the linking topology is assumed to be randomly varying in the present article.

The simplest way to make a multiagent system have the desired behavior is controlling each agent directly. However, for a multiagent system comprising many agents, the direct control of each agent is uneconomical and even impossible in practice. Because an agent is usually connected with other agents in a multiagent system, directly controlling agent A affects the behavior of each agent that has relations with agent A. By directly controlling for “key agents,” the desired behavior of a multiagent system can also be achieved under some conditions. This is the so-called pinning control.²³ Pinning control is clearly easy to implement in engineering and has thus become an important control strategy for multiagent systems. Of course, in multiagent systems with pinning control, exogenous disturbances exist and divert each agent. For continuous-time multiagent systems with fixed linear couplings which have no uncertainties, disturbance rejection was studied²⁴ in the framework of pinning control. Seeing that digital control and digital communication are used practically in multiagent systems, the discrete-time model is preferred. For discrete-time multiagent systems with pinning control, this article explores disturbance rejection with random switching nonlinear couplings that have uncertainties.

Let ℝ be the field of real numbers, ℤ the set of integers, and ℕ the set of nonnegative integers. $I$ denotes the identity matrix of appropriate dimensions. For a matrix $A$ , $A^{T}$ is the transpose of $A$ . For a square symmetric matrix $D$ , $λ_{max} (D)$ denotes the maximal eigenvalue of $D$ , and $D < 0$ indicates that $D$ is a negative definite matrix. The floor function ⌊x⌋ maps $x \in ℝ$ to the greatest integer that is less than or equal to x. $E [\cdot]$ defines the expectation. The notation # within a symmetric matrix represents a symmetric term.

Lemma 1

Let $D \in ℝ^{n \times n}$ be a square symmetric matrix. Then $\forall x \in ℝ^{n}, x^{T} D x \leq λ_{max} (D) x^{T} x$ .

The remainder of the article is organized as follows. The second section describes multiagent systems with exogenous disturbances and presents sufficient conditions of disturbance rejection. Considering an example, the fourth section presents the optimization of disturbance rejection with pinning control. The article is concluded in the fifth section.

Disturbance rejection of multiagent systems

Consider a discrete-time multiagent system consisting of n agents of one dimension. Define $N = {1, \dots, n}$ . The model of the multiagent system without control inputs is

\begin{array}{l} x_{i} (τ + 1) = x_{i} (τ) + \sum_{j \in N \ {i}} θ_{i j} (τ) q_{i j} [x_{j} (τ) - x_{i} (τ)] \\ + b_{i} w_{i} (τ) i \in N, τ \in ℕ \end{array}

where $x_{i} (τ) \in ℝ$ is the state of agent i at time $τ$ ; $w_{i} (τ) \in ℝ$ is the exogenous disturbance acting on agent i; $b_{i} \in ℝ$ is a known constant; and the nonlinear linking function $q_{i j} [\cdot]$ from agent j to agent i is unknown but is assumed to satisfy

ϕ_{i j} y^{2} \leq y q_{i j} [y] \leq ψ_{i j} y^{2} \forall y \in ℝ

q_{i j} [0] = 0

with two known positive numbers $ϕ_{i j}$ and $ψ_{i j}$ . The above condition is illustrated by Figure 1; that is, the nonlinear linking $q_{i j} [y]$ lies entirely inside the sector formed by the straight lines $ψ_{i j} y$ and $ϕ_{i j} y$ . It should be pointed out that $q_{i j} [\cdot]$ is not required to be continuous. However, at time τ, the success of $q_{i j}$ depends on a switching value $θ_{i j} (τ) \in {0, 1}$ ; $θ_{i j} (τ) = 1$ implies the linking $q_{i j}$ is successful at time $τ$ , while $θ_{i j} (τ) = 0$ implies $q_{i j}$ is unsuccessful at τ. From the point of view of engineering, $θ_{i j} (τ)$ indicates whether there is data loss in the communication from agent j to agent i at τ. What we define as data loss is the event in which one or more packets of data traveling across a communication network fail to reach their destination. To transmit information among agents, wireless communication networks are usually used in mobile multiagent systems. Data losses in a wireless communication network commonly occur owing to diverse transmission errors, such as those relating to hardware limitations, software limitations, and channel congestion. We let

Θ (τ) = diag (θ_{1} (τ), θ_{2} (τ), \dots, θ_{n} (τ)) \in ℝ^{n \times (n^{2} - n)}

with

\begin{array}{l} θ_{1} (τ) = [θ_{12} (τ) θ_{13} (τ) \dots θ_{1 n} (τ)] \in ℝ^{1 \times (n - 1)} \\ θ_{2} (τ) = [θ_{21} (τ) θ_{23} (τ) \dots θ_{2 n} (τ)] \in ℝ^{1 \times (n - 1)} \\ ⋮ \\ θ_{n} (τ) = [θ_{n 1} (τ) θ_{n 2} (τ) \dots θ_{n, n - 1} (τ)] \in ℝ^{1 \times (n - 1)} \end{array}

Figure 1.

Graph of a nonlinear linking function.

In fact, $Θ (τ)$ represents the linking topology of the multiagent system at $τ$ . Figure 2 shows a multiagent system without control and its three possible linking topologies. There are five agents in this system. The behavior of each agent is affected by a disturbance; that is, Agent i has disturbance input $w_{i} (τ)$ . As we mentioned earlier, the linking topology depends on the communication network. Because of the stochastic nature of the communication channel, for any $τ \in ℕ$ , $Θ (τ)$ is randomly one of the three matrices $Θ_{1}$ , $Θ_{2}$ , and $Θ_{3}$ with

\begin{array}{l} Θ_{1}^{T} = diag ([\begin{matrix} 0 \\ 0 \\ 0 \\ 1 \end{matrix}], [\begin{matrix} 0 \\ 0 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} 0 \\ 0 \\ 1 \\ 0 \end{matrix}], [\begin{matrix} 1 \\ 0 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} 0 \\ 0 \\ 0 \\ 0 \end{matrix}]) \\ Θ_{2}^{T} = diag ([\begin{matrix} 0 \\ 0 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} 1 \\ 0 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} 0 \\ 0 \\ 1 \\ 0 \end{matrix}], [\begin{matrix} 0 \\ 1 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} 0 \\ 0 \\ 0 \\ 0 \end{matrix}]) \\ Θ_{3}^{T} = diag ([\begin{matrix} 0 \\ 0 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} 0 \\ 1 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} 0 \\ 0 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} 0 \\ 0 \\ 0 \\ 1 \end{matrix}], [\begin{matrix} 1 \\ 0 \\ 0 \\ 0 \end{matrix}]) \end{array}

Figure 2.

(a to c) A multiagent system without control and its three possible linking topologies.

Their linking topologies are respectively illustrated in Figure 2(a), (b), and (c). According to equation (1), the dynamics of Agent 1 in topology $Θ_{1}$ can be described in Figure 3, where only across the linking function $q_{1 i}$ can the state of Agent i affect Agent 1. Because Figure 2(a) shows that only Agent 5 imposes $x_{5} (τ)$ on Agent 1, $x_{5} (τ)$ and $w_{1} (τ)$ are the inputs of Agent 1 in Figure 3.

Figure 3.

Diagram of Agent 1 without control in topology $Θ_{1}$ .

Given a fixed point $\bar{x} \in ℝ$ that is the goal of each agent in the multiagent system (1), to instruct the agents to tend toward $\bar{x}$ , control inputs

u_{i} (τ) = k_{i} (x_{i} (τ) - \bar{x}) i \in N

are added to system (1) with control gains $k_{i} \in ℝ$ . The multiagent system with control inputs is thus modeled as

\begin{array}{l} x_{i} (τ + 1) = x_{i} (τ) + \sum_{j \in N \ {i}} θ_{i j} (τ) q_{i j} [x_{j} (τ) - x_{i} (τ)] \\ + k_{i} (x_{i} (τ) - \bar{x}) + b_{i} w_{i} (τ) \end{array}

After the control is added, the multiagent system in Figure 2 becomes that in Figure 4, and the block diagram in Figure 3 becomes that in Figure 5. Clearly, in the multiagent system with control, $\bar{x}$ is included in the inputs of each agent. For system (6), we define the error $e_{i} (τ) = x_{i} (τ) - \bar{x}$ , and system (6) can then be rewritten in terms of e_i as

\begin{array}{l} e_{i} (τ + 1) = e_{i} (τ) + \sum_{j \in N \ {i}} θ_{i j} (τ) q_{i j} [e_{j} (τ) - e_{i} (τ)] \\ + k_{i} e_{i} (τ) + b_{i} w_{i} (τ) \end{array}

Figure 4.

(a to c) Multiagent system with control and its three possible linking topologies

Figure 5.

Diagram of Agent 1 with control in topology $Θ_{1}$ .

We have the error vector $e (τ) = {[\begin{matrix} e_{1} (τ) \dots e_{n} (τ) \end{matrix}]}^{T} \in ℝ^{n}$ , disturbance vector $w (τ) = {[\begin{matrix} w_{1} (τ) \dots w_{n} (τ) \end{matrix}]}^{T} \in ℝ^{n}$ , control matrix $K = diag (k_{1}, \dots, k_{n}) \in ℝ^{n \times n}$ , and input matrix $B = diag (b_{1}, \dots, b_{n}) \in ℝ^{n \times n}$ . Equation (7) can then be transformed equivalently into the simple vector form

e (τ + 1) = (I + K) e (τ) + Θ (τ) Q [C e (τ)] + B w (τ)

where

C = [\begin{matrix} - 1 & 1 & ​ & ​ & ​ \\ - 1 & ​ & 1 & ​ & ​ \\ ⋮ & ​ & ​ & ⋱ & ​ \\ - 1 & ​ & ​ & ​ & 1 \\ 1 & - 1 & ​ & ​ & ​ \\ ​ & - 1 & 1 & ​ & ​ \\ ​ & ⋮ & ​ & ⋱ & ​ \\ ​ & - 1 & ​ & ​ & 1 \\ ​ & ​ & ⋮ & ​ & ​ \\ 1 & ​ & ​ & ​ & - 1 \\ ​ & 1 & ​ & ​ & - 1 \\ ​ & ​ & ⋱ & ​ & ⋮ \\ ​ & ​ & ​ & 1 & - 1 \end{matrix}] \in ℝ^{(n^{2} - n) \times n}

$Q [\cdot]$ is a nonlinear mapping from $ℝ^{n^{2} - n}$ to $ℝ^{n^{2} - n}$ . For any $z = {[\begin{matrix} z_{1} & \dots & z_{n^{2} - n} \end{matrix}]}^{T} \in ℝ^{n^{2} - n}$

Q [z] = {[\begin{matrix} Q_{1} [z] & Q_{2} [z] & \dots & Q_{n} [z] \end{matrix}]}^{T} \in ℝ^{n^{2} - n}

with

\begin{array}{l} Q_{1} [z] = [\begin{matrix} q_{12} [z_{1}] & q_{13} [z_{2}] & \dots & q_{1 n} [z_{n - 1}] \end{matrix}] \\ Q_{2} [z] = [\begin{matrix} q_{21} [z_{n}] & q_{23} [z_{n + 1}] & \dots & q_{2 n} [z_{2 n - 2}] \end{matrix}] \\ ⋮ \\ Q_{n} [z] = [q_{n 1} [z_{n^{2} - 2 n + 2}] \dots q_{n, n - 1} [z_{n^{2} - n}]] \end{array}

Figure 4 shows that it is not necessary to exert $\bar{x}$ on each agent. Exerting $\bar{x}$ on one key agent can realize the goal with less control cost. For instance, $\bar{x}$ is imposed on only Agent 1 in Figure 6. The linking among agents provides the information of $\bar{x}$ to other agents, even if the linking is randomly time varying. Figure 6 is a case of pinning control. The pinning control strategy selects only n_c agents to which control inputs are added. In other words, the number of nonzero elements of ${k_{1}, \dots, k_{n}}$ , n_c , is as small as possible in pinning control. It is noted that a multiagent system with pinning control can still be described using equation (8). In pinning control, most diagonal elements of $K$ are zero.

Figure 6.

(a to c) Multiagent system with pinning control and its three possible linking topologies.

Before investigating stochastic consensus and disturbance rejection of equation (8), we need to make assumptions on the randomly time-varying linking topology. In this article, for equation (8), m linking topologies $Θ_{1}, Θ_{2}, \dots, Θ_{m}$ are given and $Θ (τ)$ is requested to switch within the set ${Θ_{1}, Θ_{2}, \dots, Θ_{m}}$ randomly; that is

Θ (τ) = Θ_{s_{τ}} s_{τ} \in {1, \dots, m}

It is assumed that $s_{τ}$ s are independently identically distributed ${1, \dots, m}$ -valued random variables. The probability of the mass function of $s_{τ}$ is known as

p_{h} = Prob (s_{τ} = h) h \in {1, \dots, m}

and $\sum_{h = 1}^{m} p_{h} = 1$ .

When $w = 0$ , equation (8) is referred to as a disturbance-free multiagent system. The disturbance-free multiagent system is said to achieve stochastic consensus if

lim_{τ \to \infty} E [e^{T} (τ) e (τ)] = 0

$\forall e (0) \in ℝ^{n}$ and for any $(s_{0}, s_{1}, s_{2}, \dots)$ according to (10). We let

Φ = diag (ϕ_{1}, ϕ_{2}, \dots, ϕ_{n}) \in ℝ^{(n^{2} - n) \times (n^{2} - n)}

Ψ = diag (ψ_{1}, ψ_{2}, \dots, ψ_{n}) \in ℝ^{(n^{2} - n) \times (n^{2} - n)}

with

\begin{array}{l} ϕ_{1} = diag (ϕ_{12}, ϕ_{13}, \dots, ϕ_{1 n}) \\ ψ_{1} = diag (ψ_{12}, ψ_{13}, \dots, ψ_{1 n}) \\ ϕ_{2} = diag (ϕ_{21}, ϕ_{23}, \dots, ϕ_{2 n}) \\ ψ_{2} = diag (ψ_{21}, ψ_{23}, \dots, ψ_{2 n}) \\ ⋮ \\ ϕ_{n} = diag (ϕ_{n 1}, ϕ_{n 2}, \dots, ϕ_{n, n - 1}) \\ ψ_{n} = diag (ψ_{n 1}, ψ_{n 2}, \dots, ψ_{n, n - 1}) \end{array}

Theorem 1

For multiagent system (8) stated above, suppose that there exist positive definite $M \in ℝ^{n \times n}$ , positive definite diagonal $T_{1} \in ℝ^{(n^{2} - n) \times (n^{2} - n)}$ , and positive definite diagonal $T_{2} \in ℝ^{(n^{2} - n) \times (n^{2} - n)}$ such that

[\begin{matrix} Ξ_{1} & Ξ_{2} \\ Ξ_{2}^{T} & Ξ_{3} \end{matrix}] < 0

where

\begin{array}{l} Ξ_{1} = (I + K) M (I + K) - M - 2 C^{T} T_{1} Φ C \\ Ξ_{2} = (I + K) M \sum_{h = 1}^{m} p_{h} Θ_{h} + C^{T} T_{1} + C^{T} Ψ T_{2} \\ Ξ_{3} = \sum_{h = 1}^{m} p_{h} Θ_{h}^{T} M Θ_{h} - 2 T_{2} \end{array}

The disturbance-free multiagent system then achieves stochastic consensus.

Proof. See Appendix 1

Next, for system (8) with nonzero $w (τ)$ , we consider the disturbance rejection problem. Given a positive number $γ \in ℝ$ , multiagent system (8) is said to be γ-disturbance rejective if

(i) its disturbance-free multiagent system achieves stochastic consensus and

(ii) when $e (0) = 0$ ,

E [\sum_{τ = 0}^{\infty} e^{T} (τ) e (τ) - γ \sum_{τ = 0}^{\infty} w^{T} (τ) w (τ)] \leq 0

Obviously, γ-disturbance rejection of system (8) can guarantee the stochastic consensus of system (8) without disturbances. For nonzero disturbances, under the zero initial condition, γ-disturbance rejection also guarantees

\frac{E [\sum_{τ = 0}^{\infty} e^{T} (τ) e (τ)]}{\sum_{τ = 0}^{\infty} w^{T} (τ) w (τ)} \leq γ

That is to say, the ratio between the expectation of the energy of error signals and the energy of the disturbance is less than γ. Therefore, equation (15) provides a measuring method for disturbance rejection and the performance of disturbance rejection improves with decreasing γ.

Theorem 2

[\begin{matrix} I + Ξ_{1} & Ξ_{2} & (I + K) M B \\ # & Ξ_{3} & \sum_{h = 1}^{m} p_{h} Θ_{h}^{T} M B \\ # & # & - γ I + B^{T} M B \end{matrix}] < 0

Multiagent system (8) is then γ-disturbance rejective.

Proof. See Appendix 1

Optimal disturbance rejection with pinning control: An example

This section considers how to reach a consensus on the depth of a swarm of autonomous underwater vehicles (AUVs) experiencing exogenous disturbances. There are n AUVs, each of which can be modeled as

x_{i} (τ + 1) = x_{i} (τ) + η v_{i} (τ) i \in N, τ \in ℕ,

where x_i is the depth of the ith AUV, η is a time constant, and the manipulated variable v_i is the vertical velocity of the ith AUV. Each AUV collects information related to the depth differences between it and the other AUVs and uses the information to regulate its depth. Considering the complexity of underwater measure and underwater communication, the regulation law with uncertainties and disturbances is described as

η v_{i} (τ) = \sum_{j \in N \ {i}} θ_{i j} (τ) q_{i j} [x_{j} (τ) - x_{i} (τ)] + b_{i} w_{i} (τ)

where $θ_{i j}$ , $q_{i j} [\cdot]$ , b_i , and w_i were explained in the previous section. It is easy to see that (1) can be viewed as a combination of (18) and (19). In other words, the swarm AUV system is a multiagent system studied in the previous section. In our example, the swarm AUV system is assumed to be the multiagent system shown in Figure 2 with $n = 5$ , $m = 3$ , $Θ_{1}$ , $Θ_{2}$ , and $Θ_{3}$ . In addition, let $B = 0.01 I$ ; the probabilities are given as $p_{1} = 0.3$ , $p_{2} = 0.2$ , and $p_{3} = 0.5$ . For an unknown nonlinear linking function $q_{i j} [\cdot]$ , coefficients of bounds are assigned as

\begin{array}{l} ϕ_{15} = 0.53, ψ_{15} = 0.62, ϕ_{21} = 0.40, ψ_{21} = 0.68 \\ ϕ_{23} = 0.27, ψ_{23} = 0.58, ϕ_{34} = 0.18, ψ_{34} = 0.43 \\ ϕ_{41} = 0.46, ψ_{41} = 0.56, ϕ_{42} = 0.32, ψ_{42} = 0.70 \\ ϕ_{45} = 0.50, ψ_{45} = 0.67, ϕ_{51} = 0.44, ψ_{51} = 0.62 \end{array}

The other $ϕ_{i j}$ and $ψ_{i j}$ values are equal to zero. In many practical cases, it is required that the depth of each AUV converges in the mean square sense to the desired depth $\bar{x}$ when there are no disturbances. After adding the information on $\bar{x}$ to the swarm AUV system, the model of the system becomes (6). For the multiagent system (6) with disturbances and time-varying nonlinear uncertain couplings, the γ-disturbance rejection condition in Theorem 2 is useful. Our design has two objectives. The first objective is pinning control. The information on $\bar{x}$ is added to only n_c AUVs; we should determine the value of n_c and determine which n_c AUVs are selected. Because a smaller n_c means less communication equipment, making n_c as small as possible is the most important concern. For this reason, we first consider $n_{c} = 1$ . When $n_{c} = 1$ , there are only five cases of pinning control:

\begin{matrix} Case 1 (only agent  1  is  controlled  directly) : \\ K \in K_{1} = {K = diag (k_{1},0,0,0,0) | k_{1} \in ℝ}; \\ Case 2 (only agent   2   is   controlled   directly) : \\ K \in K_{2} = {K = diag (0, k_{2},0,0,0) | k_{2} \in ℝ}; \\ Case 3 (only agent   3   is   controlled   directly) : \\ K \in K_{3} = {K = diag (0, 0, k_{3},0,0) | k_{3} \in ℝ}; \\ Case 4 (only agent   4   is   controlled   directly) : \\ K \in K_{4} = {K = diag (0, 0, 0, k_{4},0) | k_{4} \in ℝ}; \\ Case 5 (only agent   5   is   controlled   directly) : \\ K \in K_{5} = {K = diag (0, 0, 0, 0, k_{5}) | k_{5} \in ℝ} \end{matrix}

We let $K = K_{1} \cup K_{2} \cup K_{3} \cup K_{4} \cup K_{5}$ . Obviously, in this example, (8) is a pinning control system if and only if $K \in K$ . To implement optimal disturbance rejection with pinning control, for $K \in K$ , we define the objective function

μ (K) = {\begin{array}{l} min γ, & (17) is feasible for some γ \in ℝ \\ \infty, & \forall γ \in ℝ, (17) is infeasible \end{array}

From the definition of γ-disturbance rejection, we know that if $K$ cannot achieve stochastic consensus, it cannot achieve γ-disturbance rejection ((17) is infeasible) for any γ. These $K$ s that cannot achieve γ-disturbance rejection are excluded by setting their objective function value to ∞ in (20). If $μ (K) = \infty$ for any $K \in K$ , we have to abandon $n_{c} = 1$ and consider $n_{c} = 2$ . Of course, if $μ (K)$ is bounded for some $K \in K$ , we turn to the second objective, namely optimal disturbance rejection.

Because (17) is a linear matrix inequality when $K$ is known, it is convenient to compute $μ (K)$ using the mincx function in MATLAB. Figures 7 and 8 show the profiles of $μ$ in Cases 1 and 5, respectively. In Cases 2, 3, and 4, the computing results of MATLAB show that (17) is infeasible and hence $μ \equiv \infty$ . Even if the control gain is changed, for those cases failing to reach consensus, the infeasibility remains. This can be explained by Figure 2. According to all channels of (a), (b), and (c) in Figure 2, three paths cannot be constituted: a path from Agent 2 to Agent 5, a path from Agent 3 to Agent 1, and a path from Agent 4 to Agent 5. The information of $\bar{x}$ therefore cannot be received by every agent and consensus is never reached in Case 2, 3, or 4. Figure 2 also shows that from Agents 1 and 5 there exist paths to the other agents and hence both Cases 1 and 5 achieve disturbance rejection. To design the pinning controller, an optimal disturbance rejection problem can be addressed as

μ_{opt} = min_{K \in K} μ (K)

Figure 7.

Profile of μ in Case 1 (where the minimum of μ, 74.7833, is achieved when $k_{1} = - 0.9490$ ).

Figure 8.

Profile of μ in Case 5 (where the minimum of μ, 73.4520, is achieved when $k_{5} = - 0.9239$ ).

From the results of μ in Cases 1–5, it is easy to obtain $μ_{opt} = 73.4520$ and the corresponding control gain matrix $K_{opt} = diag (0, 0, 0, 0, - 0.9239)$ .

To simulate the example, we set a nonlinear linking function from Agent j to Agent i as

q_{i j} [y] = {\begin{array}{l} \frac{(ϕ_{i j} + ψ_{i j}) y + (ψ_{i j} - ϕ_{i j}) y sin y}{2}, & y \in ℝ_{1} \\ ϕ_{i j} y + \frac{2 (ψ_{i j} - ϕ_{i j}) y | arctan y |}{π}, & y \in ℝ_{2} \end{array}

where $ℝ_{1} = \underset{d \in ℤ}{\cup} [d, d + 0.5)$ and $ℝ_{2} = \underset{d \in ℤ}{\cup} [d + 0.5, d + 1)$ . Figure 9 illustrates the graph of $q_{21} [\cdot]$ , with $ϕ_{21} = 0.40$ and $ψ_{21} = 0.68$ . From (22), it is seen that $q_{21} [y] = 0.54 y + 0.14 y sin y$ when $⌊ 2 y ⌋$ is an even number, while $q_{21} [y] = 0.4 y + 0.56 y \frac{| arctan y |}{π}$ when $⌊ 2 y ⌋$ is an odd number. $q_{21} [\cdot]$ is therefore discontinuous in Figure 9. It is also seen that $0.4 y \leq 0.54 y + 0.14 y sin y \leq 0.68 y$ and $0.4 y \leq 0.4 y + 0.56 y \frac{| arctan y |}{π} \leq 0.68 y$ and hence $q_{21} [y]$ lies inside a sector in Figure 9. Figure 10 shows random numbers $s_{τ}$ generated using the probability distribution $p_{1} = 0.3, p_{2} = 0.2, p_{3} = 0.5$ . Figure 11 shows the exogenous disturbances generated using a white Gaussian noise distribution by specifying that the noise power is 0 dBW. With the initial depth $x_{1} (0) = 83.9 m, x_{2} (0) = 99.6 m, x_{3} (0) = 98.1 m, x_{4} (0) = 92.6 m, x_{5} (0) = 87.4 m$ and the desired depth $\bar{x} = 95 m$ , the depth trajectories under $K = K_{opt} = diag (0, 0, 0, 0, - 0.9239)$ and $K = diag (- 1.75, 0, 0, 0, 0)$ are displayed in Figures 12 and 13, respectively. Convergence times of the response are 40 and 100 in Figures 12 and 13, respectively. This means that the multiagent system with $K_{opt}$ is less affected by the disturbances.

Figure 9.

Graph of the linking function $q_{21} [\cdot]$ .

Figure 10.

Random numbers in the simulation.

Figure 11.

Exogenous disturbances in the simulation.

Figure 12.

Depth trajectories under $K = K_{opt} = diag (0, 0, 0, 0, - 0.9239)$ .

Figure 13.

Depth trajectories under $K = diag (- 1.75, 0, 0, 0, 0)$ .

Conclusion

We studied the γ-disturbance rejection of multiagent systems with exogenous disturbance. A sufficient condition of achieving γ-disturbance rejection for random switching coupling topologies was addressed. The condition was conveniently applied in the pinning control of $n_{c} = 1$ . A numerical example was given to verify the theoretical analysis. However, some limitations are worth noting. Although we consider the varying linking topology due to the random data loss in wireless communication, the independently identical distribution assumption is oversimple in the describing of wireless communication. Future work should therefore include follow-up work under a shadowing-pattern Markov chains²⁵ assumption which is suitable for modeling wireless communication.

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported in part by the National Key R&D Program of China (2017YFB1300400), and in part by the Science and Technology Project of Zhejiang Province (2019C01043).

ORCID iD

Jun Wu

Appendix 1

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