Handbook of Real-Time Computing pp 1-34 | Cite as

# Semi-global Consensus of Multi-agent Systems with Impulsive Approach

## Abstract

Consensus analysis is a basic issue of multi-agent systems. As an important topic of this issue, semi-global consensus problems have aroused interests since the capability of actuator is usually limited in the presence of a finite range in practice. In theory, semi-global consensus problems refer to design a one-parameter family of control protocols whose domain of attraction can tend to the entire state space. To deal with these problems, the low-gain feedback control strategy has been recently extended. The presented chapter offers a short survey of current studies on this topic, and then we develop the basic idea of low-gain feedback control strategy to apply a distributed impulsive strategy. Similarly with the low-gain feedback control, the magnitude of the proposed impulsive protocol can converge to zero as the low-gain parameter tends to zero. By utilizing the Lyapunov function and low-gain theory, a parametric discrete-time Riccati equation is developed for calculating control gain matrix. Then, based on low-and-high-gain feedback control, another distributed impulsive strategy is considered such that this control protocol can be limited in a finite range. Furthermore, two algorithms are proposed to solve the corresponding the control gain matrices. Subsequently, future research topics are discussed.

## Keywords

Multi-agent systems Semi-global consensus Low-gain feedback control Impulsive approach## 1 Introduction

In the last few decades, cooperative of autonomous systems has aroused many research interests due to several applications of real-time systems, e.g., sensor networks, distributed computation, and swarm of mobile robots (Belta and Kumar 2004; Sivrikaya and Yener 2004; Samejima and Sasaki 2015; Pérez et al. 2014; Palomares and Martínez 2014; Arieh et al. 2009; Wang et al. 2015; Prüfer 1985; Chen et al. 2014, 2016a; Tang et al. 2014; Lu et al. 2012). In these studies, control protocol is composed by some basic hypotheses. That is, all agents can approach to a final agreement as time evolves, which is the so-called consensus problem. From the dynamics performance, the consensus problem can be categorized by global consensus, semi-global consensus, and so on (Wen et al. 2015; You and Xie 2011; Zhou et al. 2012; Su et al. 2013; Li et al. 2013; Lin and Jia 2010; Chen et al. 2016b). For instance, global consensus problems have been considered for second-order nonlinear multi-agent systems by utilizing the neural-network-based adaptive control in Wen et al. (2015). In You and Xie (2011), the global consensus of discrete-time multi-agent systems has been investigated for the time-varying network topology and communication data rate. Generally, the global consensus renders that the consensus error system is controlled by a protocol whose domain of attraction is the entire state space. Different from the global consensus, the semi-global consensus means that the domain of attraction of the consensus error system can approach to the entire state space. Actually, the semi-global consensus originated from saturation issues since the capability of practical actuator is usually limited in the presence of a finite range (Lin et al. 2000). Recently, the saturation issues have been extended from the control systems to the synchronization of networks, the consensus, and output regulation of multi-agent systems (You and Xie 2011; Yang et al. 2014; Su et al. 2013; Li et al. 2016a,b; Zhou et al. 2009, 2010, 2012; Wang et al. 2012, 2014, 2016; Hou et al. 1998). For instance, the global consensus for discrete-time multi-agent systems has been examined in Yang et al. (2014), where the consensus protocol is subjected to input saturation constraints. The semi-global consensus problems have been considered for a kind of linear multi-agent systems with input saturation via low-gain feedback under the time-varying and time-variant topologies in Su et al. (2013). Therefore, a large number of studies have been developed this topic to consensus of multi-agent systems in Belta and Kumar (2004), Sivrikaya and Yener (2004), Samejima and Sasaki (2015), Pérez et al. (2014), Palomares and Martínez (2014), Arieh et al. (2009), Li et al. (2013), Wen et al. (2015), Wang et al. (2015), Lu et al. (2012), Chen et al. (2016a), You and Xie (2011), Yang et al. (2014), and Su et al. (2013).

Generally, the input saturation may induce instability or deterioration, e.g., the windup phenomena (Sussmann et al. 1994). To deal with this case, low-gain feedback control strategy has been introduced to the control systems since the traditional methods show analysis of complexity (Lin and Saberi 1993; Lin et al. 2000). The so-called low-gain feedback control strategy usually depends on the parameterized algebraic Riccati equation, which shows that the magnitude of feedback tends to zero as low-gain parameter decreases to zero. In such a setting, the control input of each agent is limited by the saturation bound interval such that the saturation phenomena can be excluded (Lin and Saberi 1993). This control strategy has been developed in Lin and Saberi (1993), Zhou et al. (2008, 2009), Saberi et al. (1995, 2000), Teel (1995), and Lin et al. (2000) and extended in Yang et al. (2014) and Su et al. (2013). For instance, in Lin et al. (2000), a low-gain feedback control for discrete-time linear systems has been improved in the presence of actuator saturation nonlinearity. In Zhou et al. (2008), a parametric Lyapunov equation approach has been developed to adapt the design of low-gain feedback. Then, in Zhou et al. (2009), the similar approach related to low-gain feedback control in Zhou et al. (2008) has been designed for discrete-time systems. In multi-agent systems, the protocols based on the low-gain feedback control have been examined in Su et al. (2013). Up to now, the low-gain feedback control has been developed for different problems of multi-agent systems, e.g., global consensus, semi-global consensus, and output regulation consensus. Based on the low-gain feedback control, there is another feedback control strategy, called high-gain feedback control, which has been proposed in Saberi et al. (2000) and Wang et al. (2012, 2016). As mentioned in Saberi et al. (2000) and Wang et al. (2012, 2016), the high-gain feedback control can achieve performance beyond stabilization in relation to robustness, disturbance rejection, or enhancing the utilization of control capacity. For instance, the high-gain feedback control has been considered in the semi-global consensus of multi-agent systems with input saturation in Su et al. (2013).

On the other hand, impulsive control strategy has been investigated in consensus problem of networked multi-agent systems due to a simple structure and some potential advantages (Liu et al. 2012, 2013; Lu et al. 2015; Zhang et al. 2014; Tang et al. 2015; Guan et al. 2012; Wang et al. 2009). The consensus problem for second-order multi-agent systems has been considered by using some kinds of impulsive control in Guan et al. (2013). In Liu et al. (2013), pinning impulsive consensus approach has been proposed for the networks of multi-agents, in which the consensus protocol can be activated by a single impulsive controller. In Liu et al. (2012), the consensus problem for multi-agent networks has been studied by using position-only measurements impulsive control. Generally, the impulsive control strategy, as a discontinuous control, is activated at a quite sparse sequence in time domain. In theory, an impulsive control protocol can promote consensus by measuring the information between neighboring impulsive instants. Under real network environments, such kind of information may not be applicable if the hardware device is limited to a finite range, i.e., actuator saturation. At this stage, it is a natural way to investigate saturation problem when designing an impulsive consensus protocol. However, unfortunately, the consideration of saturation problem into impulsive consensus protocols is still an open challenging issue since it has widely neglected in most existing works. In this chapter, a mathematical framework will be utilized to combine the impulsive control strategy and the low-gain feedback control strategy.

As mentioned above, this chapter aims to propose an impulsive consensus protocol for semi-global consensus of multi-agent systems when the consensus cannot be guaranteed by using the usual feedback protocol. Similarly (Lin et al. 2000; Teel 1995), we call an impulsive consensus protocol as *low-gain-based impulsive consensus protocol* as the control gain matrix can lead the magnitude of the protocol to approach zero. Combining the low-gain and high-gain feedback control (Saberi et al. 2000), we call an impulsive consensus protocol as *low-and-high-gain-based impulsive consensus protocol* in the case that can achieve semi-global consensus and enhance utilization of the impulsive control capacity. Based on the Lyapunov function theory and the guaranteed cost control, a parametric discrete-time Riccati equation is applied to solve the low-gain-based impulsive consensus protocol. Then, by utilizing the presented low-gain-based impulsive consensus protocol, a low-and-high-gain-based impulsive consensus protocol is considered for semi-global consensus of linear discrete-time multi-agent systems. Subsequently, two algorithms have been presented to obtain the parametric discrete-time Riccati equation such that the impulsive control gain matrices can be solved.

In what follows, a short review related on the results of the low-gain feedback control is given, and some notations and model description are provided in Sect. 2. In Sect. 3, the main results are proceeded to review the results about low-gain-based impulsive consensus protocol and low-and-high-gain-based impulsive consensus protocol, as well as two algorithms are also described. Several examples are given to show our results in Sect. 4. In Sect. 5, a conclusion and future research topics are drawn.

## 2 Preliminaries and Problem Statement

### 2.1 Notation

The standard notation is provided. In particular, \( \mathbb {R}^{n\times m}\) and \(\mathbb {R}^{n}\) indicate the set of *n* × *m* real matrix and the *n*-dimensional Euclidean space, respectively. The notation *X* ≥ *Y* (*X* > *Y* ), where *X* and *Y* are symmetric matrices, means that *X* − *Y* is positive semi-definite (positive definite). *I*_{n} denotes *n*-dimensional identity matrix. \(\mathcal {S}[a,b]\) denotes the number of a class of impulse time sequences {*k*_{m}} through the interval [*a*, *b*]. *ρ*(⋅) denotes the spectral radius of a square matrix. tr(*A*) means the trace of the matrix *A*. ∥*A*∥ is the norm of the matrix \(A\in \mathbb {R}^{n\times n}\) by the Euclidean vector norm, i.e., \(\|A\|=\sqrt {\lambda _{\max }(A^{T}A)}\), where \(\lambda _{\max }(\cdot )\) means the largest eigenvalue.

### 2.2 Graph Theory

*i*in the sense \(\mathcal {N}_{i}=\{j\in \mathcal {V}|e(i,j)\in \mathcal {E}\}\). The graph \(\mathcal {G}\) is assumed to be undirected and connected, (i.e., \(e(i,j)\in \mathcal {E}\) implies \(e(j,i)\in \mathcal {E}\)) and simple (i.e., without multiple edges and self-loops). Let \(\mathcal {L}=[\ell _{ij}]^{N}_{i,j=1}\) be the Laplacian matrix of graph \(\mathcal {G}\), which is defined as: for any pair

*i*≠

*j*,

*ℓ*

_{ij}=

*ℓ*

_{ji}= −1 if \(e(i,j)\in \mathcal {E}\), otherwise

*ℓ*

_{ij}=

*ℓ*

_{ji}= 0, i.e.,

*i*(\(i\in \mathcal {V}\)). Similarly, let \(\widehat {\mathcal {G}}=[\mathcal {V},\widehat {\mathcal {E}}]\) be a graph, in which the satisfied conditions are assumed to be the same as \(\mathcal {G}\). \(\widehat {\mathcal {N}}_{i}\) means the neighborhood of the

*i*-th agent of graph \(\widehat {\mathcal {G}}\). Let \(\mathcal {D}=[d_{ij}]^{N}_{i,j=1}\) be the associated Laplacian matrix of graph \(\widehat {\mathcal {G}}\). Moreover, the graphs \(\mathcal {G}\) and \(\widehat {\mathcal {G}}\) represent the communication topologies of feedback and impulsive protocol, respectively.

Apparently, when graph \(\mathcal {G}\) (or \(\widehat {\mathcal {G}}\)) is undirected and connected, according to the Gershgorin disk theorem (Horn and Johnson 2001), all the eigenvalues of the coupling configuration matrix \(\mathcal {L}\) satisfy \(0=\lambda _{1}(\mathcal {L})<\lambda _{2}(\mathcal {L})\leq \ldots \leq \lambda _{N}(\mathcal {L})\). And also, it is well known that \(\lambda _{2}(\mathcal {L})>0\) if and only if the graph \(\mathcal {G}\) is connected. For brevity, denote \(\lambda _{i}=\lambda _{i}(\mathcal {L})\) and \(\eta _{i}=\lambda _{i}(\mathcal {D})\).

### 2.3 Consensus Protocol via Low-Gain Feedback Approach

As mentioned above, low-gain feedback approach originated from the saturation problems of control systems, which was proposed in Lin and Saberi (1993) and developed in Zhou et al. (2008, 2009), Saberi et al. (1995, 2000), Teel (1995), and Lin et al. (2000). Generally, there are two types of approaches to consider low-gain feedback control, the parametric algebraic Riccati equation approach and the eigenstructure assignment approach. The parametric algebraic Riccati equation approach usually requires a Lyapunov function that can derive a parametric algebraic Riccati equation related on the control gain. Then, one can apply the guaranteed cost control to minimize the control gain. Such approach is proposed based on the solution of obtaining the parametric algebraic Riccati equation.

*i*-th agent, and

*sat*

_{w}(

*u*

_{i}(

*t*)) = [

*sat*

_{w}(

*u*

_{i1}(

*t*)),

*sat*

_{w}(

*u*

_{i2}(

*t*)), ⋯ , \(sat_{w}(u_{in}(t))]^{T}\in \mathbb {R}^{n}\) is a consensus protocol subjected to input saturation satisfying \(sat_{w}(u_{i}(t))=sign(u_{ij}(t))\min \{|u_{ij}(t)|,w\}\) for some constant

*w*> 0.

*x*

_{0}(

*t*) is the state vector of the leader.

### Definition 1

*γ*

^{∗}> 0 such that for any

*γ*∈ (0,

*γ*

^{∗}], the following condition holds

*i*= 1, 2, …,

*N*. Then, it is said that the multi-agent system in (1) achieves to reach semi-global consensus.

The following assumptions are required.

### Assumption 1

*For the pair*(

*A*,

*B*)

*, the following assumptions are satisfied:*

- 1.
*All eigenvalues of A are in the closed left-half s-plane,* - 2.
*The pair*(*A*,*B*)*is stabilizable.*

### Assumption 2

*The graph* \(\mathcal {G}\) *contains a spanning tree rooted at the leader.*

Recall the basic idea of low-gain feedback control. There is a low-gain parameter *γ* such that the magnitude of the considered protocol *sat*_{w}(*u*_{i}(*t*)) can tend to zero as the low-gain parameter *γ* tends to zero. In a proper setting of the low-gain parameter *γ*^{∗}∈ (0, 1], it can guarantee ∥*u*_{i}(*t*, *γ*^{∗})∥∈ [0, *w*]. In Su et al. (2013), a consensus protocol *u*_{i}(*t*) based on the low-gain feedback approach of the multi-agent system in (1) is designed in two steps:

### Algorithm 1

*The algorithm of low-gain-based consensus protocol in the continuous-time multi-agent system in*(1)

*and the leader in*(2)

*:*

- Step 1.
*Solve the parametric algebraic Riccati equation*$$\displaystyle \begin{aligned} A^{T}P_{\gamma}+P_{\gamma}A-2\lambda P_{\gamma}BB^{T}P_{\gamma}+\varepsilon I=0,{} \end{aligned} $$(3)*where*\(\lambda \leq \min \{\lambda _{1}(L_{s}+H)\}\)*is a positive constant, L*_{s}*is a Laplacian matrix, and H*=*{h*_{1}, …,*h*_{N}}.- Step 2.
*Design a consensus protocol for agent i as*$$\displaystyle \begin{aligned} u_{i}(t,\gamma)=K_{\gamma}\sum_{j\in\mathcal{N}_{i}}(x_{j}(t)-x_{i}(t))+h_{i}(x_{0}(t)-x_{i}(t)),{} \end{aligned} $$(4)*where*\(K_{\gamma }=-B^{T}P_{\gamma }\in \mathbb {N}^{p\times n}\)*is a control gain matrix and P*_{γ}*is a solution of the parametric algebraic Riccati equation in*(3)*. Thus, P*_{γ}→ 0*as γ*→ 0.

Meanwhile, there are a lot of results that examine the low-gain feedback approach on the consensus problems of continuous-time multi-agent system. For instance, the semi-global problems have been considered by some kind output regulation of nonlinear control subjected to input saturation in Li et al. (2016a). Then, various kinds of control strategies with input saturation have been considered, including event-triggered control, fuzzy adaptive control, and sliding mode control (Li et al. 2016a,b; Zhou et al. 2009, 2010, 2012; Wang et al. 2012, 2014, 2016; Hou et al. 1998). On the other hand, for discrete-time counterparts of multi-agent system, it is still an open question how to apply the low-gain feedback approach. The main reason is that the current works mainly focus on designing the *H*_{∞}-type-based Riccati inequality or another alternative approaches (Hengster et al. 2013).

### 2.4 Impulsive Consensus Protocol via Low-Gain Feedback Approach

*L*

_{∞}or

*H*

_{∞}low-gain feedback in constrained control (Teel 1995), and input saturation in multi-agent systems (You and Xie 2011; Yang et al. 2014; Su et al. 2013). However, these results cannot be easily extended into impulsive systems. Therefore, in the following, we will propose a way to design the low-gain-based impulsive consensus protocol. Particularly, we consider the linear discrete-time multi-agent system, in which the motion of agent

*i*is described by

*i*-th agent, \(A\in \mathbb {R}^{n\times n}\), and

*c*is a global coupling gain. Apparently, we can derive from the connected graph \(\mathcal {G}\) that \((\mathcal {L}\otimes I_{n})x=0\) if and only if

*ε*

_{ji}(

*k*) = 0 holds for all \(i, j\in \mathcal {V}\), where

*ε*

_{ji}(

*k*) =

*x*

_{i}(

*k*) −

*x*

_{j}(

*k*) means the consensus error of the states of agents

*i*and

*j*at time

*k*.

Recently, the consensus problems of discrete-time multi-agent systems have been widely investigated in Yang et al. (2014), You and Xie (2011), Chen et al. (2016a), and Qin et al. (2014). In these works, the matrix *A* plays an important factor to regulate consensus. For instance, in Qin et al. (2014), the consensus can be maintained when \(\rho (A)\leq (1-\chi (c\mathcal {L})^{N-1})^{1/(1-N)}\), where *ρ*(*A*) is spectral radius of matrix *A* and \( \chi (c\mathcal {L})\) means a kind of ergodic coefficient of matrix \(c\mathcal {L}\). However, the condition in Qin et al. (2014) is not applied when \(\rho (A)>(1-\chi (c\mathcal {L})^{N-1})^{1/(1-N)}\). It is to say that the consensus protocol in (5) may not guarantee consensus. For this reason, an impulsive consensus protocol will be considered if the consensus of linear discrete-time multi-agent systems in (5) cannot be guaranteed. Such impulsive consensus protocol can be regarded as a compensation control of the discrete-time multi-agent systems (5). In this case, impulsive consensus protocol to ensure semi-global consensus will be considered.

*u*

_{i}(

*k*) is defined as the state vector

*x*

_{i}(

*k*) is replaced by

*x*

_{i}(

*k*− 1) +

*Bu*

_{i}(

*k*− 1) at impulsive instant

*k*

_{m}, where

*u*

_{i}(

*k*) has the following form:

*δ*[⋅] is the Dirac discrete-time function, \(B\in \mathbb {N}^{n\times p}\) is an impulsive input matrix, \(K\in \mathbb {N}^{p\times n}\) is an impulsive control gain matrix, and the impulsive instants

*k*

_{m}satisfy

*γ*,

*μ*, and

*σ*. Moreover, it is worth mentioning that

*γ*is the so-called low-gain parameter in low-gain design strategies (Zhou et al. 2008, 2009; Lin et al. 2000). In these works, there is a close relationship between low-gain parameter

*γ*and low-gain feedback gain matrix. It is to say that the multi-agent systems realize semi-global consensus if time goes on, and then, the spectral radius of impulsive protocol tends to zero as the low-gain parameter

*γ*goes to zero. At this stage, the concept of low-gain parameter

*γ*is introduced into distributed impulsive control. To calculate how much energy should be injected by the impulsive protocol to ensure semi-global consensus, an optimal impulsive control problem can be considered for solving the minimum cost function:

*Q*≥ 0,

*R*> 0, 0 <

*γ*< 1,

*μ*> 0, and \(\sigma =\max \{k_{m}-k_{m-1}\}\geq 1\).

The following definition and assumption is required for deriving main results:

### Definition 2

*global consensus*of agents if the following condition holds

*i*= 1, 2, …,

*N*.

### Definition 3

*semi-global consensus*of agents if the following condition holds

*i*= 1, 2, …,

*N*. Particularly, when \(\mathcal {X}=\mathbb {R}^{n}\), the linear discrete-time multi-agent systems in (8) is said to achieve

*global*consensus.

### Assumption 3

*The pair* (*I*_{n}, *B*) *is stabilizable.*

## 3 Impulsive Consensus Protocol Design

In this section, the semi-global consensus of the linear discrete-time multi-agent system in (8) will be studied by utilizing the impulsive consensus protocol based on the low-gain approach. By modifying the methods in Saberi et al. (2000) and Zhou et al. (2008), the low-gain and low-and-high-gain impulsive consensus protocols are presented by parametric discrete-time Riccati equations.

### 3.1 Low-Gain-Based Impulsive Consensus Protocol

*K*in the low-gain-based impulsive consensus protocol. Different from the low-gain feedback control (You and Xie 2011; Zhou et al. 2008, 2009; Saberi et al. 2000), several fundamental questions need to be answered when designing the low-gain-based impulsive consensus protocol:

- 1.
How do we design a proper cost function to apply the control cost of distributed impulsive protocol?

- 2.
How do we derive the impulsive control gain matrix

*K*based on these derived criteria? - 3.
How do we find a proper control gain matrix

*K*to minimize the cost function?

In what follows, the low-gain-based impulsive protocol for ensuring the semi-global consensus of the linear discrete-time multi-agent system in (8) is considered.

### Theorem 1

*Consider the linear discrete-time multi-agent system in*(8)

*with the cost function*(9)

*. If there exists a positive parameter μ and a positive definite matrix*\(P\in \mathbb {R}^{n\times n}\)

*, such that the following criteria hold*

*where*2 ≤

*i*≤

*N. Then the linear discrete-time multi-agent system in*(4)

*can achieve consensus under the distributed impulsive protocol in*(6)

*. Furthermore, the cost function*(9)

*satisfies*

*where* \(\overline {\varepsilon }_{ji}(k)=\overline {x}_{j}(k)-\overline {x}_{i}(k)\).

### Proof

*k*∈ [

*k*

_{0},

*k*

_{1}− 1], we can get

*Y*

^{T}

*Y*=

*I*

_{N}and

*Λ*= diag{0,

*λ*

_{2}, …,

*λ*

_{N}}. Since

*λ*

_{1}= 0, one can construct \(y_{1}=1/\sqrt {N}(1,1,\ldots ,1)^{T}\in \mathbb {R}^{N}\). In this case, utilizing the unitary transform \(y(k)=(Y^{T}\otimes I)\overline {x}(k)=[y_{1}^{T}(k),y_{2}^{T}(k),\ldots ,y_{N}^{T}(k)]^{T}\in \mathbb {R}^{n\times N}\), it follows that

*k*=

*k*

_{1}, one gets

*k*∈ [

*k*

_{0},

*k*

_{M}], one has

Apparently, by means of a quadratic guaranteed cost control, Theorem 1 renders a condition for achieving the consensus of the linear discrete-time multi-agent system in (8). Under this environment, we will show how to obtain the impulsive control gain matrix *K* in the following proposition:

### Proposition 1

*Suppose that Assumption*3

*and all the conditions in Theorem*1

*are satisfied. If there exist positive parameters ω and μ, such that*

*then the cost function*(9)

*is minimized with*

*where P*

_{ γ }

*is the unique positive solution to the parametric discrete-time Riccati equation*

*and δ*∈ (0, 1)

*, R*

_{γ}=

*B*

^{T}

*P*

_{γ}

*B*+

*R*∕

*λ*

_{2}

*and μ satisfies*

*Particularly, when Q *= 0 *and* \(I_{n}-BR_{\gamma }^{-1}B^{T}P_{\gamma }\) *is Schur, the consensus of discrete-time multi-agent system in* (8) *can also be achieved.*

### Proof

*i*≤

*N*,

*i*≤

*N*.

*Ω*

_{i}≤ 0. At this stage, one gets

*P*satisfies \(\overline {\varOmega }_{i}\leq 0\).

*P*is the solution of the following discrete-time Riccati equation:

*P*

_{γ}=

*P*∕(1 −

*γ*) is the unique positive definite solution to the parametric discrete-time Riccati equation (20), when the positive parameter

*μ*satisfies

*i*≤

*N*.

*Q*= 0, the discrete-time multi-agent system in (9) can achieved consensus from (21), and

Note that a parameter *δ* is introduced in Proposition 1. In this case, one can regulate such parameter to solve the parametric discrete-time Riccati equation in (22). From You and Xie (2011), it is not hard to check that when (*I*_{n}, *B*) is controllable and *δ* ∈ (0, 1), the discrete-time Riccati equation in (20) has a unique positive definite solution. A special selection is *ω* = 2∕(*η*_{2} + *η*_{N}) and \(1-\delta ^{2}=4/[(\eta _{2}/\eta _{N})^{\frac {1}{2}}+(\eta _{2}/\eta _{N})^{-\frac {1}{2}}]\). For more results of *ω* and *δ* can be seen in You and Xie (2011) and Hengster et al. (2013). On the other hand, Proposition 1 shows a way to design the impulsive control gain matrix *K* in the low-gain-based impulsive consensus protocol. That is, the parameter *γ* ∈ (0, 1) in (3) is a low-gain parameter, which means that the convergence rate of the linear discrete-time multi-agent system with the distributed impulsive protocol in (8) is faster than \((\sqrt {1-\gamma })^{k}\).

*Q*= 0. In this case, the parametric discrete-time Riccati equation (22) becomes

Then, a further result of the solution *P*_{γ} in (20) is given to show that the value of *γ* can approach to zero.

### Theorem 2

### Proof

*γ*, we have

*P*

_{γ}is monotonically increasing with respect to

*γ*. In this case, the limit of

*P*

_{γ}exists as

*γ*→ 0. Denoting \(\lim _{\gamma \rightarrow 0^{+}}P_{\gamma }=P_{0}\), it can be derive that

*R*

_{0}=

*B*

^{T}

*P*

_{0}

*B*+

*R*∕

*λ*

_{2}.

Similarly with Saberi et al. (1995), if (*I*_{n}, *B*) is controllable, the discrete-time Riccati equation (29) has the unique semi-positive definite solution *P*_{0} = 0. Thus, the proof is completed. □

In the following, we will illustrate that the linear discrete-time multi-agent system in (5) can achieve semi-global consensus under the distributed impulsive protocol in (6).

### Theorem 3

*Suppose that all the conditions in Theorem* 2 *are satisfied. Then the linear discrete-time multi-agent system in* (4) *can achieve semi-global consensus as long as* \(\lim _{\gamma \rightarrow 0^{+}}P_{\gamma }=0.\)

### Proof

*ϖ*

_{L}such that

*Δ*

_{L}is a bound since \(\lim _{\gamma \rightarrow 0^{+}}P_{\gamma }=0\). In this case, for any \(\gamma \in (0,\gamma _{L}^{*}]\), the dynamics of (5) remains linear within

*L*(

*V*,

*ϖ*). Similar to Theorem 1, one knows the derivative of

*V*along the trajectories of the agents within the set

*L*(

*V*,

*ϖ*) as

*k*∈ [

*k*

_{m−1},

*k*

_{m}− 1], and

*x*∈

*L*(

*V*,

*ϖ*

_{L}) −{0},

*L*(

*V*,

*ϖ*

_{L}) can converge to the origin \(\varepsilon ^{ij}_{k}=0\) asymptotically as time evolves to infinity, i.e.,

To answer the questions in the beginning of this subsection, the relation of the results derived in the above needs to illustrate. In view of Theorem 1 and Proposition 1, it is easy to see that the impulsive control gain matrix requires the solution of the parametric discrete-time Riccati equation in (27). From Theorem 2, the positive definite matrix *P*_{γ} can approach to zero when low-gain parameter *γ* tends to zero. It is to say that the impulsive control gain matrix *K* can approach to zero matrix, which leads the magnitude of the impulsive protocol to tend to zero. Then, the linear discrete-time multi-agent system in (5) can achieve semi-global consensus when applying Theorem 3.

Different from the usual low-gain feedback design approaches (Lin et al. 2000; Teel 1995; Zhou et al. 2008), note that the parametric discrete-time Riccati equation (22) depends on many parameters, e.g., the low-gain parameter *γ*, *δ*, *μ*, and *σ*. When solving *P*_{γ} from (22), it inevitably leads some difficulties to obtain impulsive control gain matrix *K*. Thus, the following algorithm is given to show how to design the low-gain-based impulsive consensus protocol:

### Algorithm 2

*The algorithm of low-gain-based impulsive consensus protocol in linear discrete-time multi-agent systems in*(5)

- Step1.
*Solve the parametric discrete-time Riccati equation in*(28)*, i.e.,*$$\displaystyle \begin{aligned} (1-\gamma)(1+\mu)^{-\sigma+1}P_{\gamma}=&P_{\gamma}-(1-\delta^{2})P_{\gamma}BR_{\gamma}^{-1}B^{T}P_{\gamma}, \end{aligned} $$*to an optimization problem**min γ,**subject to*(28) ,

*where μ*∈ [*μ*_{m},*μ*_{M}]*and δ*∈ (0, 1)*is the initial parameters of*(28) . - Step2.
*Get P*_{γ}*when minimizing γ, and check*(6)*can hold or not; otherwise, return to Step 1.* - Step3.
*Design a consensus protocol for agent i as*$$\displaystyle \begin{aligned} u_{i}(k)=K_{\gamma}\sum^{\infty}_{m=1}\sum_{j\in\widehat{\mathcal{N}}_{i}}(x_{j}(k)-x_{i}(k))\delta[k-k_{m}+1], \end{aligned} $$*where*\(K_{\gamma }=-\omega R_{\gamma }^{-1}B^{T}P_{\gamma }\)*is a control gain matrix and P*_{γ}*is a solution of the parametric discrete-time Riccati equation**in*(28)*. Thus, P*_{γ}→ 0*as γ*→ 0.

### 3.2 Low-and-High-Gain-Based Impulsive Consensus Protocol

In this subsection, the low-and-high-gain control techniques are introduced into the design of the distributed impulsive protocol. The low-and-high-gain-based impulsive consensus protocol can be regarded as distributed impulsive control subjected to state saturation, as well as the low-gain-based impulsive consensus protocol proposed above. Different from the low-gain-based impulsive consensus protocol, the low-and-high-gain-based distributed impulsive protocol can enhance the utilization of the control capacity of linear discrete-time multi-agent systems in (8).

*R*=

*λ*

_{2}

*I*

_{n}. At this stage, the parametric discrete-time Riccati equation in (28) becomes

*β*∈ [0,

*β*

^{∗}] is a high-gain parameter and

*β*

^{∗}is a parameter to be designed.

In many works (You and Xie 2011; Saberi et al. 2000), the high-gain parameter *β* plays an important factor that can be utilized by handling external control inputs, robustness, or disturbance rejection. Obviously, if the low-gain parameter *γ* lies in a proper range, the semi-global consensus performance of the linear discrete-time multi-agent system in (5) can only depend on the high-gain parameter *β*. It is to say that the impulsive consensus protocol hinges on a complicated adaptation with offer of the low-gain parameter *γ* and high-gain parameter *β*.

- 1.
*β*∈ [0,*β*^{∗}], where*β*^{∗}is a parameter to be designed; - 2.
\(\|(1+\beta )(\mathcal {D}\otimes \omega \widetilde {R}_{\gamma }^{-1}B^{T}P_{\gamma })x_{k_{m}-1}\|\leq \varDelta _{LH} \) holds for any \(x_{k_{m}-1}\in \mathbb {R}^{Nn\times Nn}\), where

*Δ*_{LH}is a given bound; - 3.
\(\overline {\mathcal {X}}_{LH}=\{x_{k}\in \mathbb {R}^{Nn\times Nn}:x_{k}^{T}(L\otimes P_{\gamma })x_{k}\leq \varpi _{LH} \}\) is a bounded set for all

*k*∈ [*k*_{0}, +*∞*) and*ϖ*_{LH}> 0.

*P*

_{γ}is the unique positive definite solution to the parametric discrete-time Riccati equation (32).

Then, the design of low-and-high-gain-based impulsive consensus protocol will be divided into three steps. First, the low-gain parameter *γ*_{0} is needed to be designed, then we will give a way to design the low-and-high-gain parameter *β*_{0}, and last, combining the low-gain parameter *γ*_{0} and the low-and-high-gain parameter *β*_{0}, the low-and-high-gain-based impulsive control gain matrix *K* is obtained.

#### 3.2.1 Design of Low-Gain Parameter *γ*_{0}

In the following proposition, the bound of the low-gain parameter *γ* can be derived, and further the procedure of designing low-gain parameter *γ*_{0} based on this bound is given.

### Proposition 2

*Consider the linear discrete-time multi-agent system in*(5)

*with low-gain-based impulsive consensus protocol, where the impulsive control gain matrix is given by*

*Suppose that Assumption*2

*is satisfied. If there exists a positive parameter μ, such that the following criterion holds*

*where P*

_{γ}

*is the unique positive definite solution to the parametric discrete-time Riccati equation*(32)

*. Then the semi-global consensus of linear discrete-time multi-agent system in*(5)

*can be achieved. Furthermore, the low-gain parameter γ can be set as*\(\gamma =\gamma ^{*}_{LH}\)

*, such that*

*holds for all k *∈ [*k*_{0}, +*∞*).

### Proof

*k*∈ [

*k*

_{m−1},

*k*

_{m}− 1], one has

*k*=

*k*

_{m}, one can get

*k*∈ [

*k*

_{m−1},

*k*

_{m}− 1],

*x*

_{k}then amounts to look for the largest

*γ*such that

*x*

_{k}lies within the set \(\overline {\mathcal {X}}_{LH}\). Therefore, the low-gain parameter is configured as

*β*= 0. When

*Δ*

_{LH}is not very small, for brevity, one can also choose \(\gamma ^{*}_{LH}\) as 1.

#### 3.2.2 Design of High-Gain Parameter *β*_{0}

Now, the bound of the high-gain parameter *β* is estimated, and furthermore the design of high-gain parameter *β*_{0} will be provided. The results are presented as follows.

### Proposition 3

*Suppose that Assumption*2

*is satisfied. Let P*

_{γ}

*be the unique positive definite solution to the parametric discrete-time Riccati equation*(32)

*. Then, the following inequality holds*

*where* \(\widetilde {\gamma }=[(1-\gamma )(1+\mu )^{-\sigma +1}-\delta ^{2}]/(1-\delta ^{2})\in (0,1)\).

### Proof

*B*

^{T}

*P*

_{γ}

*B*)

^{−1}by

*ρ*

_{i}(

*i*= 1, …,

*p*). Using the Cayley-Hamilton Theorem (Horn and Johnson 2001), we have

### Lemma 1

*Suppose that Assumption*1

*is satisfied. Let P*

_{γ}

*be the unique positive definite solution to the parametric discrete-time Riccati equation*(32)

*. Then*\(I_{n}-(1+\beta )\omega \eta _{i} \widetilde {R}_{\gamma }^{-1}B^{T}P_{\gamma }\)

*is Schur when β*≥ 0

*satisfies*

*where* 2 ≤* i *≤* N. Moreover, β *∈ (0, *β*^{∗}) *can be estimated by* \(\beta ^{*}=(\widetilde {\gamma }^{-1}-1)^{-n}-1\)*, where* \(\widetilde {\gamma }\) *is defined in Proposition 3.*

### Proof

*γ*)(1 +

*μ*)

^{−σ+1}− 1]

*P*

_{γ}< 0 since

*γ*∈ (0, 1) and

*σ*> 1. To prove \(\widetilde {\mathcal {I}}_{i}^{T}P\widetilde {\mathcal {I}}_{i}-P< 0\), we need to show

*ξ*≤ 0. It can be easily verified that

*ξ*≤ 0 as

*β*satisfies

*β*. It is to say that one can calculate

*β*by solving (45) or estimating

*β*

^{∗}. Apparently, there is a close relationship between the selection of the high-gain parameter

*β*and low-gain parameter

*γ*. To deal with this case, when \(x_{k_{m}-1}\in \mathcal {X}\), a high-gain parameter can be configured by

*γ*replaced by

*γ*

_{0}of Eq. (41) and \(\widetilde {R}_{\gamma _{0}}=B^{T}P_{\gamma _{0}}B+I_{p}\). Obviously, it does not affect the consensus of the linear discrete-time multi-agent system in (5) when the high-gain parameter

*β*

_{0}is configured appropriately.

#### 3.2.3 Design of Low-and-High-Gain-Based Impulsive Control Gain Matrix *K*

The low-and-high-gain-based distributed impulsive protocol will be proposed to ensure the semi-global consensus of the linear discrete-time multi-agent system in (5). The main difficulty of presenting such impulsive protocol based on the low-and-high-gain feedback control is what kinds of the low-gain and high-gain parameters are to be designed. This problem will be answered in the end of this subsection.

*K*in (34) can realize the semi-global consensus of the linear discrete-time multi-agent system in (5) such that \(\|u_{k_{m}-1}\|\in [0,\varDelta _{LH}]\). For this object, considering

*γ*

_{0}in (41) and

*β*

_{0}in (47) to an impulsive control gain matrix

*K*in (34), an impulsive control gain matrix by an appropriate modification of (31) can be obtained:

*γ*replaced by

*γ*

_{0}of (41) and \(\widetilde {R}_{\gamma _{0}}=B^{T}P_{\gamma _{0}}B+I_{p}\).

Then, the parameter *ϖ*_{LH} in (40) and \(\|(1+\beta _{0})\mathcal {D}\otimes \omega \widetilde {R}_{\gamma _{0}}^{-1}B^{T}P_{\gamma _{0}})x(k_{m}-1)\|\in [0,\varDelta _{LH}]\) will be given.

### Theorem 4

*Consider the linear discrete-time multi-agent system in*(5)

*with low-high-gain-based impulsive consensus protocol, where the impulsive control gain matrix is given by*(48)

*and*∥

*u*(

*k*

_{m}− 1)∥∈ [0,

*Δ*

_{LH}]

*holds. Suppose that Proposition*2

*and*3

*are satisfied. Then the semi-global consensus of linear discrete-time multi-agent system in*(5)

*can be achieved. Furthermore, ϖ*

_{LH}

*can be given by*

### Proof

*ϖ*

_{LH}be such that

*γ*

_{0}is defined in (40).

*Proposition*3, for

*k*=

*k*

_{m}, it follows that

*ξ*is defined in (44).

*ξ*≤ 0, one has

*k*∈ [

*k*

_{0}, +

*∞*),

*ϖ*

_{LH}can be given by (48). The proof is completed.

On the one hand, the low-gain parameter *γ*_{0} in (41) depends on the bound *Δ*_{LH} in property (ii) and the upper bound of impulsive interval *σ*. Obviously, it is a different choice since the usual selections of low-gain parameter in low-gain feedback control (Saberi et al. 2000; Lin et al. 2000) cannot be directly applied to impulsive control. On the other hand, the high-gain parameter *β*_{0} in (47) requires the solution of the parametric discrete-time Riccati equation (30) with *γ* replaced by *γ*_{0}, although the form of impulsive control gain matrix *K* in (48) is similar to low-and-high-gain feedback control in Saberi et al. (2000) and Lin et al. (2000). Therefore, the parametric discrete-time Riccati equation in (30) plays an important role in designing the low-and-high-gain-based impulsive consensus protocol.

To answer the questions in the beginning of above, the relation of the results needs to be shown. Based on *Proposition* 3, we know that the high-gain parameter *β* refers to the low-gain parameter *γ* and the upper bound of impulsive interval *σ*. Furthermore, Theorem 4 renders how to obtain low-high-gain-based impulsive control gain matrix *K* in (47) such that ∥*u*(*k*_{m} − 1)∥ can be limited in [0, *Δ*_{LH}].

Then, if solving *P*_{γ} from (32), a checking process is required to judge whether (21) can hold or not. Therefore, to obtain the impulsive control gain matrix *K*, one can transform the above results to the optimization problems stated in Algorithm 3.

### Algorithm 3

*The algorithm of low-and-high-gain-based impulsive consensus protocol in linear discrete-time multi-agent systems in*(5)

- Step 1.
*From the parametric discrete-time Riccati equation in*(32)*, i.e.,*$$\displaystyle \begin{aligned} (1-\gamma)(1+\mu)^{-\sigma+1}P_{\gamma}=&P_{\gamma}-(1-\delta^{2})P_{\gamma}B\widetilde{R}_{\gamma}^{-1}B^{T}P_{\gamma}. \end{aligned} $$*Solve γ*_{0}*by*(41)*, and then, solve β*_{0}*by*(47)*, where μ*∈ [*μ*_{m},*μ*_{M}]*, σ*_{0}∈ [*σ*_{m},*σ*_{M}]*, and δ*_{0}∈ (0, 1)*is the initial parameters of*(32) .- Step 2.
*Get*\(P_{\gamma _{0}}\)*, and then check if*(6)*can hold or not; otherwise, return to Step 1.*- Step 3.
*Design a consensus protocol for agent i as*$$\displaystyle \begin{aligned} u_{i}(k)=K\sum^{\infty}_{m=1}\sum_{j\in\widehat{\mathcal{N}}_{i}}(x_{j}(k)-x_{i}(k))\delta[k-k_{m}+1], \end{aligned} $$*where*\(K=-(1+\beta _{0})\omega \widetilde {R}_{\gamma _{0}}^{-1}B^{T}P_{\gamma _{0}}\)*is a control gain**matrix and*\(P_{\gamma _{0}}\)*is a solution of the parametric discrete-time Riccati**equation in*(32)*. Thus,*\(\|u_{k_{m}-1}\|\in [0,\varDelta _{LH}]\).

## 4 Numerical Examples

In this section, two examples are provided to show the main results.

### Example 1

*N*= 50,

*m*= 5,

*p*= 0.1 in Strogatz (2001), and the other parameters are given as

*ωη*

_{i}|≤

*δ*when we choose

*ω*= 0.036 and

*δ*= 0.8. For

*γ*= 0.3,

*γ*= 0.03, and

*γ*= 0.003, by utilizing Algorithm 2 with

*μ*= 0.03 and

*σ*= 3, the impulsive control gain matrices can be computed, respectively, as follows:

*σ*and

*ρ*(

*K*) when

*μ*= 0.03. It can be seen that

*ρ*(

*K*) increases as

*σ*increases. Figure 1 shows the evolution of states in the linear discrete-time multi-agent system. Figure 2 shows the evolution of states in the linear discrete-time multi-agent system under low-gain-based impulsive consensus protocol with

*γ*= 0.03 and

*k*

_{m}−

*k*

_{m−1}= 4. Figure 3 shows the

*u*

_{i}(

*k*

_{m}) is capable of achieving semi-global consensus with

*γ*= 0.3,

*γ*= 0.03 and

*γ*= 0.003. It can be seen that this example matches our results very well.

The relationship between *σ* and *ρ*(*K*) when *μ* = 0.03

| 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|

| 0.0044 | 0.0059 | 0.0072 | 0.0085 | 0.0097 | 0.0108 |

### Example 2

*A*in the linear discrete-time multi-agent system (1) is strictly unstable, and then the matrix \(\mathcal {L}\) and the Laplacian matrix \(\mathcal {D}\) of the distributed impulsive protocol topology are chosen by the global coupling matrix. The other parameters are given as

*A*is strictly unstable since

*ρ*(

*A*) = 1.2685 > 1. It is easy to check |1 −

*ωη*

_{i}|≤

*δ*when we choose

*ω*= 0.01 and

*δ*= 0.9. By utilizing Algorithm 2 with

*μ*= 0.61,

*σ*= 4,

*Δ*

_{LH}= 10,

*k*

_{0}= 0, and

*k*

_{M}= 60, we can obtain

*γ*

_{0}= 0.001 and

*β*

_{0}= 0.4406. The impulsive control gain matrix can be computed as follows:

*γ*

_{0}= 0.001,

*β*

_{0}= 0.4406, and

*k*

_{m}−

*k*

_{m−1}= 4. Figure 6 shows that the \(u^{i}_{k_{m}}\) is capable of achieving semi-global consensus with

*γ*

_{0}= 0.001,

*β*

_{0}= 0.4406, and

*k*

_{m}−

*k*

_{m−1}= 4 and furthermore matches our results very well.

## 5 Conclusion

This chapter offers the semi-global consensus problems of a class of linear discrete-time multi-agent systems. Firstly, a short survey of the recent studies and developments in the semi-global consensus problems has been reviewed. Subsequently, two novel distributed impulsive consensus protocols are proposed, which are induced by the low-gain feedback control and low-and-high-gain feedback control strategies. Based on the guaranteed cost control and the Lyapunov function theory, a parametric discrete-time Riccati equation has been considered for designing the impulsive control gain matrices. The derived results also reveal that such distributed impulsive consensus protocols not only refer to the impulsive interval but also depend on the low-gain and low-and-high-gain parameters. Moreover, two kinds of algorithms are presented for obtaining impulsive control gain matrices. Finally, simulations are provided to illustrate the theoretical results. Future works on this topics are expected to include and address these important results to continuous-time systems and other applications, which are still a challenging problem.

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