# Scaled consensus of switched multi-agent systems

Scaled consensus of switched multi-agent systems Abstract In this paper, the scaled consensus problem for switched multi-agent systems composed of continuous-time and discrete-time subsystems is investigated. By using the nearest neighbour-interaction rules, three types of scaled consensus protocols are presented to achieve asymptotic, finite-time and fixed-time convergence rates, respectively. Based upon algebraic graph theory and Lyapunov theory, scaled consensus is shown to be reached for strongly connected networks. We explicitly express the final consensus states as some initial-condition-dependent values. Three scaled formation generation problems with different convergence rates are introduced and solved as generalizations. Simulation examples are provided to validate the effectiveness and availability of our theoretical results. 1. Introduction In the past decades, distributed control of networked multi-agent systems has received a great deal of attention in the system and control community due to its broad applications in areas such as formation control, cooperative coordination of unmanned aerial vehicles, robotic teams and sensor networks (Jadbabaie et al., 2003; Olfati-Saber et al., 2007). Consensus plays a fundamental role in distributed coordination, which aims to design appropriate control laws such that all the agents reach an agreement on some consistent quantity of interest. As each agent can only interact with those within its local areas due to limited communication ability, the designed protocols take advantages of nearest-neighbour rules rendering the multi-agent systems in a distributed network framework governed by the graph Laplacians (Mesbahi & Egerstedt, 2010). Such networked multi-agent systems cannot be tackled by standard stability theories (e.g. Rugh (1996)) in general as compared to the traditional monolithic ones. Up to now, a variety of consensus protocols, such as average consensus, leader-follower consensus, finite-time consensus, stochastic consensus and asynchronous consensus, have been reported to achieve consensus in continuous-time and discrete-time multi-agent systems; see the surveys Olfati-Saber et al. (2007); Ren et al. (2007); Cao et al. (2013) and references therein. In many practical applications, the states of agents may converge to prescribed ratios (so-called ‘scales’) instead of a common value. In simultaneous coordination control of satellites running on orbits and their simulating robots on ground, for example, huge difference between the scales of vehicles’ position and speed in space and on ground gives rise to the scaled coordination control required in practise (Guglieri et al., 2014; Sun et al., 2014). Other practical examples include compartmental mass-action systems, closed queuing networks and water distribution systems (Roy, 2015). As a generalization to standard consensus, Roy (2015) recently introduces a novel notion of scaled consensus, wherein agents’ states reach asymptotically assigned ratios in terms of different scales. The scaled consensus offers a general framework, which can be specialized to achieve standard consensus, cluster consensus (Shang, 2016a), wherein different subnetworks are allowed to reach different common values, and bipartite consensus (Altafini, 2013) or signed consensus (Li et al., 2017) by choosing appropriate scales. Scaled consensus has been studied for fixed strongly connected topology in Roy (2015) and switching topologies in Meng & Jia (2016b), where the agents are modelled by continuous-time single integrators. Several scaled consensus protocols are proposed to solve the finite-time coordination control for continuous-time multivehicle systems in Meng & Jia (2016a) and for discrete-time ones in Shang (2017a). Robust scaled consensus problems for continuous-time and discrete-time multi-agent systems with time delay are dealt with, respectively, in Shang (2017b) and Hou et al. (2016). For consensus protocol design, convergence rate is an important performance indicator. Usually, linear systems are only able to achieve an asymptotic convergence rate, meaning that no agreement can be made in finite time (Mesbahi & Egerstedt, 2010). In system theory, it is often required to develop controllers to drive agents to a given position as soon as possible. Moreover, finite-time controllers enjoy faster convergence rate, better disturbance rejection as well as more robustness to uncertainties (Bhat & Bernstein, 2000; Wang & Xiao, 2010). However, the settling time for finite-time consensus protocols heavily depends on the initial conditions, which weaken their advantages over asymptotic consensus protocols when the initial condition varies or is unavailable in advance. To overcome this drawback, a class of fixed-time consensus protocols has been presented (see e.g. Polyakov (2012); Parsegov et al. (2013); Defoort et al. (2015); Fu & Wang (2016)) for continuous-time multi-agent systems, where the settling time is bounded and independent of the initial states of agents. For example, based on the fixed-time stability theory, fixed-time average consensus problem is addressed in Parsegov et al. (2013) for multi-agent systems with integrator dynamics over fixed undirected networks. Leader–follower fixed-time consensus for continuous-time systems with nonlinear inherent dynamics is investigated in Defoort et al. (2015). In Shang (2016b), fixed-time average consensus problems for discrete-time multi-agent systems have also been discussed. It is noteworthy that all the aforementioned works are concerned with multi-agent systems composed of either only continuous-time systems or only discrete-time systems. In practise, a large-scale system can be split into multiple subsystems and a switching rule manages the switching between them leading to a switched multi-agent system (Zhai et al., 2006). A multi-agent system, for instance, controlled either by a digitally implemented regulator or a physically implemented one with a synchronous switching law between them gives rise to a switched multi-agent system. Consensus problems for switched multi-agent systems are firstly considered in Zheng & Wang (2016) over different topologies. Lin & Zheng (2017) further investigates the finite-time convergence for switched multi-agent systems by using matrix theory and Lyapunov theory. Containment control of switched multi-agent systems is addressed in Zhu et al. (2015). Nevertheless, to our knowledge, there has been no systematic study on scaled consensus problems for switched systems consisting of both continuous-time and discrete-time subsystems. The aim of this paper is to develop a systematic framework for convergence problems of scaled consensus of switched multi-agent systems. The contribution of this paper is threefold. First, we propose three scaled consensus protocols with asymptotic, finite-time and fixed-time convergence rates, respectively, for switched multi-agent systems consisting of both continuous-time and discrete-time subsystems. Second, the settling time for finite-time and fixed-time scaled consensus is explicitly obtained by utilizing algebraic graph theory and Lyapunov functions. Moreover, the final consensus states are explicitly expressed as certain initial-condition-dependent values. Third, asymptotic, finite-time and fixed-time scaled formation control problems for switched multi-agent systems are introduced and solved as the generalizations. It is noteworthy that general convergence of switched linear systems has been considered in Shang (2017c), however the focus therein is on the switch of system matrix rather than continuous and discrete subsystems. The rest of the paper is organized as follows. Section 2 provides some preliminaries and formulates the scaled consensus problems of switched multi-agent systems. Section 3 presents scaled consensus protocols and develops convergence as well as formation generation results. We provide some numerical examples in Section 4 and then conclude the paper in Section 5. 2. Preliminaries and problem statement We begin with some standard notations that will be used throughout the paper. Let $$\mathbb{Z}$$ represent the set of integers. Let $$\mathbb{R}$$ and $$\mathbb{R}^{N}$$ represent the set of real numbers and the N-dimensional real vector space, respectively. Let MT be the transpose of a matrix M. For a symmetric matrix $$M\in \mathbb{R}^{N\times N}$$, its eigenvalues are ordered as $$\lambda _{1}(M)\leqslant \lambda _{2}(M)\leqslant \cdots \leqslant \lambda _{N}(M)$$. For $$x\in \mathbb{R}$$ and $$\mu \geqslant 0$$, we define ⌊x⌉μ = sgn(x)|x|μ, where sgn(⋅) is the signum function. IN and 1N denote, respectively, the N × N identity matrix and the N-dimensional vector with elements being all ones. For p > 0, the p-norm ∥⋅∥p is defined as $$\|x\|_{p}=\big (\sum _{i=1}^{N}|x_{i}|^{p}\big )^{1/p}$$ for a vector $$x=(x_{1},\cdots ,x_{N})^{\mathrm{T}}\in \mathbb{R}^{N}$$. The following lemma connecting different norms is useful in dealing with the finite-time consensus problems, a proof of which can be found in Hardy et al. (1952). Lemma 1 (Hardy et al., 1952) Let $$x\in \mathbb{R}^{N}$$ and q > p > 0. Then $$\|x\|_{q}\leqslant\|x\|_{p}\leqslant N^{\frac{1}{p}-\frac{1}{q}}\|x\|_{q}.$$ 2.1. Algebraic graph theory The communication topology of a multi-agent system can often be characterized by a weighted directed graph (Mesbahi & Egerstedt, 2010) $$\mathscr{G}=(\mathscr{V},\mathscr{E},\mathscr{A})$$ with a node set $$\mathscr{V}=\{1,2,\cdots ,N\}$$ representing N agents, an edge set $$\mathscr{E}\subseteq \mathscr{V}\times \mathscr{V}$$ describing the information exchange amongst them, and a non-negative adjacency matrix $$\mathscr{A}=(a_{ij})\in \mathbb{R}^{N\times N}$$. aij > 0 if and only if $$(j,i)\in \mathscr{E}$$, representing the information flow from agent j to agent i. The degree of i is defined as $$d_{i}=\sum _{j\in \mathscr{V}\backslash \{i\}}a_{ij}$$ and the maximum degree $$d_{\max }=\max _{i\in \mathscr{V}}d_{i}$$. Given a sequence i1, i2 ⋯ , ik of distinct nodes, a path in $$\mathscr{G}$$ from i1 to ik consists of a sequence of edges $$\left (i_{j},i_{j+1}\right )\in \mathscr{E}$$ for j = 1, ⋯ , k − 1. A directed graph $$\mathscr{G}$$ is said to be strongly connected if between any pair of distinct nodes i, j, there exists a path from i to j. $$\mathscr{G}$$ is said to satisfy the detailed balance condition if there exist some scalars ωi > 0 ($$i\in \mathscr{V}$$) such that ωiaij = ωjaji for all $$i,j\in \mathscr{V}$$. This condition has proven to be instrumental in studying coupled dynamics (see e.g. Chu et al. (2003); Haken (1978)). Given N non-zero scalars α1, ⋯ , αN, a modified detailed balance condition is given as follows. Assumption 1 There exist some scalars ωi > 0 ($$i\in \mathscr{V}$$) such that ωi|αi|aij = ωj|αj|aji for all $$i,j\in \mathscr{V}$$. Remark 1 Note that for a general weighted directed network with aij > 0 but aji = 0 Assumption 1 is not applicable. However, for a bidirectional network, where aij > 0 if and only if aji > 0 (Mesbahi & Egerstedt, 2010), Assumption 1 would be appropriate. This condition will be used in finite-time/fixed-time convergence, c.f. Theorems 2 and 3 below. The graph Laplacian matrix of $$\mathscr{A}$$, $$\mathscr{L}(\mathscr{A})=(l_{ij})\in \mathbb{R}^{N\times N}$$, is defined by $$l_{ii}=-\sum _{j\in \mathscr{V}\backslash \{i\}}l_{ij}=d_{i}$$ and lij = −aij for i≠j. Clearly, $$\mathscr{L}(\mathscr{A})1_{N}=0$$. The graph $$\mathscr{G}$$ is called undirected if aij = aji for all $$i,j\in \mathscr{V}$$. An undirected graph is connected if there exists a path between any two distinct nodes of the graph. For a connected undirected graph $$\mathscr{G}$$, it is well known that $$\mathscr{L}(\mathscr{A})$$ is positive semidefinite and $$x^{\mathrm{T}}\mathscr{L}(\mathscr{A})x=\frac 12\sum _{i,j\in \mathscr{V}}a_{ij}(x_{j}-x_{i})^{2}$$ for any $$x=(x_{1},\cdots ,x_{N})^{\mathrm{T}}\in \mathbb{R}^{N}$$ (Horn & Johnson, 2012). The following lemma is a straightforward generalization of Lin & Zheng (2017, Lemma 4). Lemma 2 Suppose that Assumption 1 holds. Let $$\alpha _{\max }=\max _{1\leqslant i\leqslant N}|\alpha _{i}|$$. If the weighted directed graph $$\mathscr{G}=(\mathscr{V},\mathscr{E},\mathscr{A})$$ is strongly connected and $$0<h<(d_{\max }\alpha _{\max })^{-1}$$, then the matrix $$(I_{N}-h|\alpha |\mathscr{L}(\mathscr{A}))^{\mathrm{T}}\varOmega$$$$(I_{N}-h|\alpha |\mathscr{L}(\mathscr{A}))-\varOmega$$ is negative semidefinite and the zero eigenvalue is algebraically simple. Here, $$|\alpha |:=\textrm{diag}(|\alpha _{1}|,\cdots ,|\alpha _{N}|)\in \mathbb{R}^{N\times N}$$ and Ω := diag(ω1, ⋯ , ωN) is the diagonal matrix with the (i, i) element being ωi. Proof. First, we observe that $$I_{N}-h|\alpha |\mathscr{L}(\mathscr{A})=(I_{N}-h|\alpha |\mathscr{D})+h|\alpha |\mathscr{A}$$ is a row stochastic matrix, where $$\mathscr{D}=\textrm{diag}(d_{1},\cdots ,d_{N})$$ is the degree matrix. Clearly, $$(I_{N}-h|\alpha |\mathscr{L}(\mathscr{A}))^{\mathrm{T}}\varOmega (I_{N}-h|\alpha |\mathscr{L}(\mathscr{A}))$$ is non-negative. Thanks to Assumption 1, we have $$\varOmega |\alpha |\mathscr{L}(\mathscr{A})=\mathscr{L}(\mathscr{A})^{\mathrm{T}}\varOmega |\alpha |=\mathscr{L}(\mathscr{A})^{\mathrm{T}}|\alpha |\varOmega$$, and hence $$(I_{N}-h|\alpha |\mathscr{L}(\mathscr{A}))^{\mathrm{T}}\varOmega (I_{N}-h|\alpha |\mathscr{L}(\mathscr{A}))1_{N}=\varOmega (I_{N}-h|\alpha |\mathscr{L}(\mathscr{A}))1_{N}=\varOmega 1_{N}$$. This implies that 1N is a right eigenvector corresponding to eigenvalue zero for matrix $$(I_{N}-h|\alpha |\mathscr{L}(\mathscr{A}))^{\mathrm{T}}\varOmega (I_{N}-h|\alpha |\mathscr{L}(\mathscr{A}))-\varOmega$$. An application of the Gershgorin circle theorem (Horn & Johnson, 2012) shows that all eigenvalues of $$(I_{N}-h|\alpha |\mathscr{L}(\mathscr{A}))^{\mathrm{T}}\varOmega (I_{N}-h|\alpha |\mathscr{L}(\mathscr{A}))-\varOmega$$ are non-positive. Next, we show that the zero eigenvalue is algebraically simple. Since $$\mathscr{G}$$ is strongly connected and $$0<h<(d_{\max }\alpha _{\max })^{-1}\leqslant 2\lambda _{N}^{-1}(\mathscr{L}(\mathscr{A}))$$, we obtain $$\textrm{rank}(|\alpha |\mathscr{L}(\mathscr{A}))=n-1$$ and hence $$-2h|\alpha |\mathscr{L}(\mathscr{A})+h^{2}(|\alpha |\mathscr{L}(\mathscr{A}))^{2}$$ has rank n − 1. Therefore, employing the Sylvester inequality, we arrive at \begin{align} n-1\geqslant&\ \textrm{rank}\left((I_{N}-h|\alpha|\mathscr{L}(\mathscr{A}))^{ T}\varOmega(I_{N}-h|\alpha|\mathscr{L}(\mathscr{A}))-\varOmega\right)\nonumber\\ =&\ \textrm{rank}\left(\varOmega(I_{N}-h|\alpha|\mathscr{L}(\mathscr{A}))^{2}-\varOmega\right)\nonumber\\ =&\ \textrm{rank}\left(\varOmega\left(-2h|\alpha|\mathscr{L}(\mathscr{A})+h^{2}(|\alpha|\mathscr{L}(\mathscr{A}))^{2}\right)\right)\nonumber\\ \geqslant&\ \textrm{rank}(\varOmega)+\textrm{rank}\left(-2h|\alpha|\mathscr{L}(\mathscr{A})+h^{2}(|\alpha|\mathscr{L}(\mathscr{A}))^{2}\right)-n=n-1.\nonumber \end{align} This implies that zero eigenvalue can only be algebraically simple, which completes the proof. 2.2. Problem formulation Consider a group of N agents, labelled from 1 to N, composing a fixed directed network $$\mathscr{G}=(\mathscr{V},\mathscr{E},\mathscr{A})$$. The agents are controlled by a switching law between continuous-time dynamics and discrete-time dynamics. The information state of the agent i at time t is represented by $$x_{i}(t)\in \mathbb{R}$$. In the vector form, we denote x(t) = (x1(t), ⋯ , xN(t))T and let x(0) = (x1(0), ⋯ , xN(0))T be the initial value. Given any scalar scale αi≠0 for the agent i, we are devoted to addressing the scaled consensus problems with three different convergence rates in the following three cases: (i) Asymptotic convergence (Roy, 2015): $$\lim _{t\rightarrow \infty }(\alpha _{i}x_{i}(t)-\alpha _{j}x_{j}(t))=0$$ for all $$i,j\in \mathscr{V}$$ and all initial conditions x(0). (ii) Finite-time convergence (Meng & Jia, 2016a): $$\lim _{t\rightarrow T(x(0))}(\alpha _{i}x_{i}(t)-\alpha _{j}x_{j}(t))=0$$ and αixi(t) = αjxj(t) for $$t\geqslant T(x(0))$$ for all $$i,j\in \mathscr{V}$$ and all initial conditions x(0), where T(x(0)) is the settling time which depends on the initial state x(0). (iii) Fixed-time convergence (Meng & Jia, 2016a): $$\lim _{t\rightarrow T}(\alpha _{i}x_{i}(t)-\alpha _{j}x_{j}(t))=0$$ and αixi(t) = αjxj(t) for $$t\geqslant T$$ for all $$i,j\in \mathscr{V}$$ and all initial conditions x(0), where T is the settling time which is independent of the initial state x(0). The dynamical model of each agent is described by a continuous-time subsystem \begin{align} \dot{x}_{i}(t)=u_{i}(t),\quad i\in\mathscr{V} \end{align} (2.1) and a discrete-time subsystem \begin{align} x_{i}(t+1)=x_{i}(t)+hu_{i}(t),\quad i\in\mathscr{V} \end{align} (2.2) where $$u_{i}(t)\in \mathbb{R}$$ is the control input to be designed, h > 0 is the control gain and the synchronous switch is assumed to apply for all agents. At time instant t, the choice of subsystem is decided by the switching rule. The time domain for the subsystems (2.1) and (2.2) is $$t\in \mathbb{R}$$ and $$t\in \mathbb{Z}$$, respectively. When (2.2) is activated, the states of the agents are updated with time step length 1. Since the state of the discrete-time subsystem can be viewed as a piecewise constant vector between sampling points, we may consider the value of the system states in continuous-time domain. The switched multi-agent system (2.1)–(2.2) is said to achieve the asymptotic (finite-time or fixed-time, respectively) scaled consensus to (α1, ⋯ , αN) if the corresponding convergence condition in (i) ((ii) or (iii), respectively) holds. For finite-time and fixed-time convergence, we will assume a sequence of time instants $$0\leqslant t_{1}<\bar{t}_{1}<t_{2}<\bar{t}_{2}<\cdots <t_{k}<\bar{t}_{k}<\cdots$$ satisfying the following assumption. Assumption 2. When $$t\in (t_{k},\bar{t}_{k}]$$, the continuous-time subsystem (2.1) is activated; when $$t\in (\bar{t}_{k-1},t_{k}]$$, the discrete-time subsystem (2.2) is activated. Furthermore, $$\bar{t}_{k}-t_{k}\geqslant \tau$$ for some constant τ > 0. Remark 2. We assume that the synchronous switch is applied for each agent. The system described by (2.1) and (2.2) can be viewed as a system consisting of continuous-time subsystem and discrete-time subsystem at different time interval. The synchronized switch is often caused by an upper layer switching rule independent of the distributed communication of agents in the system (Zhai et al., 2006; Zheng & Wang, 2016). A typical example is to consider a switched multi-agent system whose subsystems are all continuous-time. If a computer is used to activate all the subsystems in a discrete-time manner, then the switched system is composed of both continuous-time and discrete-time subsystems. Another example of this kind is a continuous-time plant controlled either by a physically implemented regulator or by a digitally implemented one together with a switching rule between them. Also note that the switch between continuous-time control and sampled-data control under Assumption 2 guarantees the switched multi-agent system to be composed of continuous-time and discrete-time subsystems (or only the continuous-time subsystems). For finite-time consensus problems, such switching control method has been considered in Lin & Zheng (2017). 3. Scaled consensus protocols for switched multi-agent systems In this section the main results about the scaled consensus of the switched multi-agent system (1)–(2) are investigated. We adopt the nearest-neighbour rules and construct distributed algorithms for each agent $$i\in \mathscr{V}$$ as follows. (i) An asymptotic convergence protocol is given by \begin{align} u_{i}(t)=\textrm{sgn}(\alpha_{i})\sum_{j=1}^{N}a_{ij}(\alpha_{j}x_{j}(t)-\alpha_{i}x_{i}(t)), \end{align} (3.1) for both subsystems (2.1) and (2.2). At time instant t, the choice of subsystem is determined by the switching rule under consideration. (ii) A finite-time convergence protocol is given by \begin{align} u_{i}(t)=\begin{cases}\textrm{sgn}(\alpha_{i})\sum_{j=1}^{N}a_{ij}\lfloor\alpha_{j}x_{j}(t)-\alpha_{i}x_{i}(t)\rceil^{\mu_{ij}},& t\in(t_{k},\bar{t}_{k}];\\ \textrm{sgn}(\alpha_{i})\sum_{j=1}^{N}a_{ij}(\alpha_{j}x_{j}(t)-\alpha_{i}x_{i}(t)),& t\in(\bar{t}_{k-1},t_{k}], \end{cases} \end{align} (3.2) where μij = μji ∈ (0, 1). (iii) A fixed-time convergence protocol is given by \begin{align} u_{i}(t)=\begin{cases}\textrm{sgn}(\alpha_{i})\sum_{j=1}^{N}a_{ij}(\alpha_{j}x_{j}(t)-\alpha_{i}x_{i}(t))^{\frac{m}{n}}&\\ \quad+\textrm{sgn}(\alpha_{i})\sum_{j=1}^{N}a_{ij}(\alpha_{j}x_{j}(t)-\alpha_{i}x_{i}(t))^{\frac{p}{q}},& t\in(t_{k},\bar{t}_{k}];\\ \textrm{sgn}(\alpha_{i})\sum_{j=1}^{N}a_{ij}(\alpha_{j}x_{j}(t)-\alpha_{i}x_{i}(t)),& t\in(\bar{t}_{k-1},t_{k}], \end{cases} \end{align} (3.3) where m, n, p, q are positive odd integers satisfying m > n and p < q. Compared to the finite-time protocol (3.2), one more term is needed in (3.3) to ensure the fixed-time scaled consensus. For the particular case when αi = 1 for all $$i\in \mathscr{V}$$, (3.2) and (3.3) become the consensus protocols proposed in Lin & Zheng (2017). Moreover, it is easy to verify that the scaled consensus protocols (3.1)–(3.3) for continuous-time subsystem (2.1) are continuous. 3.1. Analysis of asymptotic scaled consensus We first consider the asymptotic scaled consensus of switched multi-agent system (2.1)–(2.2) employing protocol (3.1). Define $$\alpha :=\textrm{diag}(\alpha _{1},\cdots ,\alpha _{N})\in \mathbb{R}^{N\times N}$$ and $$|\alpha |:=\textrm{diag}(|\alpha _{1}|,\cdots ,|\alpha _{N}|)\in \mathbb{R}^{N\times N}$$. With these notations, the switched multi-agent system with protocol (3.1) can be recast as $$\alpha\dot{x}(t)=-|\alpha|\mathscr{L}(\mathscr{A})\alpha x(t)\qquad\textrm{and}\qquad \alpha x(t+1)=(I_{N}-h|\alpha|\mathscr{L}(\mathscr{A}))\alpha x(t).$$ The following lemma summarizes and reformulates some results presented in Roy (2015); Hou et al. (2016), and Shang (2017b). Lemma 3 Suppose that the communication network $$\mathscr{G}$$ is strongly connected. Then the continuous-time multi-agent system (2.1) with (3.1) achieves the asymptotic scaled consensus to (α1, ⋯ , αN). If h|αi|di < 1 for all $$i\in \mathscr{V}$$, then the discrete-time multi-agent system (2.2) with (3.1) achieves the asymptotic scaled consensus to (α1, ⋯ , αN). In particular, for both continuous-time and discrete-time systems, $$\lim _{t\rightarrow \infty }x(t)=(1/\alpha _{1},\cdots ,1/\alpha _{N})^{\mathrm{T}}{w^{\mathrm{T}}}x(0)$$, where $$w^{\mathrm{T}}|\alpha |\mathscr{L}(\mathscr{A})=0$$ and wT1N = 1. Theorem 1 Suppose that the communication network $$\mathscr{G}$$ is strongly connected. If h|αi|di < 1 for all $$i\in \mathscr{V}$$, then the switched multi-agent system (2.1)–(2.2) with protocol (3.1) achieves the asymptotic scaled consensus to (α1, ⋯ , αN) under arbitrary switching. Proof. First, noting that $$e^{-|\alpha |\mathscr{L}(\mathscr{A})}(I_{N}-h|\alpha |\mathscr{L}(\mathscr{A}))=(I_{N}-h|\alpha |\mathscr{L}(\mathscr{A}))e^{-|\alpha |\mathscr{L}(\mathscr{A})}$$, we obtain \begin{align} \alpha x(t)=e^{-|\alpha|\mathscr{L}(\mathscr{A})t_{c}}(I_{N}-h|\alpha|\mathscr{L}(\mathscr{A}))^{t_{d}}\alpha x(0), \end{align} (3.4) where $$t_{c}\in \mathbb{R}$$ and $$t_{d}\in \mathbb{Z}$$ represent the total duration time on continuous-time and discrete-time subsystems, respectively, and t = tc + td. It should be noted that if the agents switch between continuous- and discrete-time systems asynchronously, (3.4) may not be established. It follows from Lemma 3 that $$\lim _{t_{c}\rightarrow \infty }e^{-|\alpha |\mathscr{L}(\mathscr{A})t_{c}}=1_{N}w^{\mathrm{T}}$$ and $$\lim _{t_{d}\rightarrow \infty }(I_{N}-h|\alpha |\mathscr{L}(\mathscr{A}))^{t_{d}}=1_{N}w^{\mathrm{T}}$$, where $$w^{\mathrm{T}}|\alpha |\mathscr{L}(\mathscr{A})=0$$ and wT1N = 1. Next, we consider two cases when $$t\rightarrow \infty$$. If $$t_{c}\rightarrow \infty$$, then \begin{align} \lim_{t\rightarrow\infty}\alpha x(t)=&\lim_{t_{c}\rightarrow\infty}e^{-|\alpha|\mathscr{L}(\mathscr{A})t_{c}}(I_{N}-h|\alpha|\mathscr{L}(\mathscr{A}))^{t_{d}}\alpha x(0)\nonumber\\ =&1_{N}w^{\mathrm{T}}(I_{N}-h|\alpha|\mathscr{L}(\mathscr{A}))^{t_{d}}\alpha x(0)=1_{N}w^{\mathrm{T}}\alpha x(0).\nonumber \end{align} On the other hand, if $$t_{d}\rightarrow \infty$$, then we have \begin{align} \lim_{t\rightarrow\infty}\alpha x(t)=&\lim_{t_{d}\rightarrow\infty}e^{-|\alpha|\mathscr{L}(\mathscr{A})t_{c}}(I_{N}-h|\alpha|\mathscr{L}(\mathscr{A}))^{t_{d}}\alpha x(0)\nonumber\\ =&e^{-|\alpha|\mathscr{L}(\mathscr{A})t_{c}}1_{N}w^{\mathrm{T}}\alpha x(0)=1_{N}w^{\mathrm{T}}\alpha x(0),\nonumber \end{align} since $$e^{-|\alpha |\mathscr{L}(\mathscr{A})t_{c}}=\sum _{k=0}^{\infty }\frac{(-|\alpha |\mathscr{L}(\mathscr{A})t_{c})^{k}}{k!}$$. Therefore, the switched multi-agent system (2.1)–(2.2) with protocol (3.1) achieves the asymptotic scaled consensus to (α1, ⋯ , αN) under arbitrary switching. Remark 3 It is noteworthy that the switched multi-agent system (2.1)–(2.2) offers a unified framework for both the continuous-time scaled consensus problems (Roy, 2015; Shang, 2017b) and the discrete-time scaled consensus problems (Hou et al., 2016). When the scales are taken as αi = 1 for all $$i\in \mathscr{V}$$, we reproduce the consensus protocol for switched multi-agent systems proposed in Zheng & Wang (2016). In the following we extend the result to solve formation generation problem for switched multi-agent systems. In this problem, the aim is to design distributed protocols to guarantee that each pair of neighbouring agents reach a desired relative position with respect to each other (Cao et al., 2013; Xiao et al., 2009; Campos et al., 2016). A certain pattern is thus formed by the agents as a whole. Definition 1 Let $$f=(\ f_{1},\cdots , f_{N})\in \mathbb{R}^{N}$$. The agents in $$\mathscr{G}$$ are said to achieve the scaled formation f with respect to (α1, ⋯ , αN) if $$\lim _{t\rightarrow \infty }(\alpha _{i}x_{i}(t)-\alpha _{j}x_{j}(t))=f_{i}-f_{j}$$ for all $$i,j\in \mathscr{V}$$ and all initial conditions x(0). The above definition corresponds to the asymptotic convergence rate; the finite-time and fixed-time versions can be defined in a similar manner as in Section 2.2. It is easy to see that the agents reach the asymptotic scaled formation f if there exists a vector $$v_{c}\in \mathbb{R}^{N}$$ such that αixi(t) tends to fi + vc as time goes to infinity. To this end, we design the formation generation protocol as follows: \begin{align} u_{i}(t)=\textrm{sgn}(\alpha_{i})\sum_{j=1}^{N}a_{ij}(\alpha_{j}x_{j}(t)-f_{j}-(\alpha_{i}x_{i}(t)-f_{i})), \end{align} (3.5) for both subsystems (2.1) and (2.2). At time instant t, the choice of subsystem is determined by the switching rule under consideration. Corollary 1 Suppose that the communication network $$\mathscr{G}$$ is strongly connected. If h|αi|di < 1 for all $$i\in \mathscr{V}$$, then the switched multi-agent system (2.1)–(2.2) with protocol (3.5) achieves the asymptotic scaled formation f with respect to (α1, ⋯ , αN) under arbitrary switching. Proof. Let $$\bar{x}_{i}=x_{i}-f_{i}/\alpha _{i}$$ for $$i\in \mathscr{V}$$. Then the closed-loop system becomes $$\dot{\bar{x}}_{i}=\textrm{sgn}(\alpha _{i})\sum _{j=1}^{N}a_{ij}$$$$\left (\alpha _{j}\bar{x}_{j}(t)-\alpha _{i}\bar{x}_{i}(t)\right )$$ (continuous-time subsystem) and $$\bar{x}_{i}(t+1)\!=\!\bar{x}_{i}(t)+h\textrm{sgn}(\alpha _{i})\sum _{j=1}^{N}a_{ij}\left (\alpha _{j}\bar{x}_{j}(t)\!-\!\alpha _{i}\bar{x}_{i}(t)\right )$$ (discrete-time subsystem) for $$i\in \mathscr{V}$$. It follows from Theorem 1 that $$\bar{x}_{i}, i\in \mathscr{V}$$ will achieve the asymptotic scaled consensus to (α1, ⋯ , αN) under arbitrary switching. This in turn means that there exists a vector $$v_{c}\in \mathbb{R}^{N}$$ such that $$\lim _{t\rightarrow \infty }\alpha _{i}x_{i}(t)=f_{i}+v_{c}$$. The proof is complete. 3.2. Analysis of finite-time scaled consensus In this section we study the finite-time scaled consensus of switched multi-agent system (2.1)–(2.2) employing protocol (3.2). Let $$\omega =(\omega _{1},\cdots ,\omega _{N})^{\mathrm{T}}\in \mathbb{R}^{N}$$ and $$\omega _{\max }=\max _{1\leqslant i\leqslant N}\omega _{i}$$. Write $$\mu _{\max }=\max _{i,j\in \mathscr{V}}\mu _{ij}\in (0,1)$$, $$x_{\min }(t)=\min _{i\in \mathscr{V}}\alpha _{i}x_{i}(t)$$ and $$x_{\max }(t)=\max _{i\in \mathscr{V}}\alpha _{i}x_{i}(t)$$. Recall that $$\alpha _{\max }=\max _{1\leqslant i\leqslant N}|\alpha _{i}|$$ and we now have a sequence of time instants $$0\leqslant t_{1}<\bar{t}_{1}<t_{2}<\bar{t}_{2}<\cdots <t_{k}<\bar{t}_{k}<\cdots$$. The main result in this section reads as follows. Theorem 2 Suppose that Assumptions 1 and 2 hold, and the communication network $$\mathscr{G}$$ is strongly connected. If $$0<h<(d_{\max }\alpha _{\max })^{-1}$$, then the switched multi-agent system (2.1)–(2.2) with protocol (3.2) achieves the finite-time scaled consensus to (α1, ⋯ , αN) with settling time $$t_{k^{\ast }}$$ and $$k^{\ast}=\left\lceil\frac{4V^{\frac{1-\mu_{\max}}{2}}(\varepsilon(0))}{(1-\mu_{\max})(\rho_{1}\rho_{2})^{\frac{1+\mu_{\max}}{2}}\tau}\right\rceil+1,$$ where $$V(\varepsilon (0))=\frac 12\sum _{i=1}^{N}\omega _{i}\left (\alpha _{i}x_{i}(0)-\frac{\sum _{j=1}^{N}\omega _{j}\alpha _{j}x_{j}(0)}{\sum _{j=1}^{N}\omega _{j}}\right )^{2}$$, $$\rho _{1}=\frac{\min _{i,j\in \mathscr{V},a_{ij}\not =0} (\omega _{i}|\alpha _{i}|a_{ij})^{\frac{2}{\mu _{\max }+1}}}{\sum _{i,j\in \mathscr{V}}(\omega _{i}|\alpha _{i}|a_{ij})^{\frac{2}{\mu _{\max }+1}}}$$ ⋅ $$(x_{\max }(0)-x_{\min }(0))^{\frac{2(\mu _{ij}-\mu _{\max })}{\mu _{\max }+1}}>0$$, $$\rho _{2}=\frac{4K_{0}}{\omega _{\max }}$$, $$K_{0}=\min _{\xi \in U}\xi ^{\mathrm{T}}\mathscr{L}(\mathscr{B}_{0})\xi>0$$, $$\mathscr{B}_{0}=\left [(\omega _{i}|\alpha _{i}|a_{ij})^{\frac{2}{\mu _{\max }+1}}\right ]\in \mathbb{R}^{N\times N}$$ and $$U=\{\xi \in \mathbb{R}^{N}:\omega ^{\mathrm{T}}\xi =0,\|\xi \|_{2}=1\}$$. Proof. Define $$\gamma (t)=\frac{\sum _{i=1}^{N}\omega _{i}\alpha _{i}x_{i}(t)}{\sum _{i=1}^{N}\omega _{i}}$$. Note that $$\omega =(\omega _{1},\cdots ,\omega _{N})^{\mathrm{T}}\in \mathbb{R}^{N}$$ is a positive vector, ωi|αi|aij = ωj|αj|aji by Assumption 1, and μij = μji for all $$i,j\in \mathscr{V}$$. Therefore, we have $$\dot{\gamma}(t)=\frac{\sum_{i=1}^{N}\omega_{i}\alpha_{i}\dot{x}_{i}(t)}{\sum_{i=1}^{N}\omega_{i}}=\frac{1}{\sum_{i=1}^{N}\omega_{i}}\sum_{i=1}^{N}\omega_{i}|\alpha_{i}| \sum_{j=1}^{N}a_{ij}\lfloor\alpha_{j}x_{j}(t)-\alpha_{i}x_{i}(t)\rceil^{\mu_{ij}}=0,$$ and \begin{align} \gamma(t+1)=&\ \frac{1}{\sum_{i=1}^{N}\omega_{i}}\sum_{i=1}^{N}\omega_{i}\alpha_{i} \left(x(t)+h\textrm{sgn}(\alpha_{i})\sum_{j=1}^{N}a_{ij}\left(\alpha_{j}x_{j}(t)-\alpha_{i}x_{i}(t)\right)\right),\nonumber\\ =&\ \gamma(t)+\frac{h}{\sum_{i=1}^{N}\omega_{i}}\sum_{i=1}^{N}\omega_{i}|\alpha_{i}|\sum_{j=1}^{N}a_{ij}(\alpha_{j}x_{j}(t)-\alpha_{i}x_{i}(t))=\gamma(t).\nonumber \end{align} Let εi(t) = αixi(t) − γ(t) for $$i\in \mathscr{V}$$, and ε(t) = (ε1(t), ⋯ , εN(t))T. Recall that α = diag(α1, ⋯ , αN) and hence, ε(t) = αx(t) − γ(t)1N. By definition, we have ωTε(t) = 0 and $$\varepsilon (t+1)=(I_{N}-h|\alpha |\mathscr{L}(\mathscr{A}))\alpha$$$$\cdot x(t)-\gamma (t)1_{N}=(I_{N}-h|\alpha |\mathscr{L}(\mathscr{A}))\varepsilon (t)$$. Define the common Lyapunov function $$V(\varepsilon (t))=\frac 12\varepsilon ^{\mathrm{T}}(t)\Omega \varepsilon (t)$$$$=\frac 12\sum _{i=1}^{N}\omega _{i}{{\varepsilon _{i}^{2}}}(t)$$ for continuous-time subsystem (2.1) and discrete-time subsystem (2.2). For $$t\in (t_{k},\bar{t}_{k}]$$$$(k\geqslant 1)$$, the continuous-time subsystem (2.1) is activated. Since γ(t) is time-invariant, we obtain \begin{align} \dot{V}(\varepsilon(t))=&\sum_{i=1}^{N}\omega_{i}\varepsilon_{i}(t)\alpha_{i}\dot{x}_{i}(t)=\sum_{i=1}^{N}\omega_{i}\varepsilon_{i}(t) |\alpha_{i}|\sum_{j=1}^{N}a_{ij}\lfloor\varepsilon_{j}(t)-\varepsilon_{i}(t)\rceil^{\mu_{ij}}\nonumber\\ =&\frac12\sum_{i,j=1}^{N}\left(\omega_{i}\varepsilon_{i}(t) |\alpha_{i}|a_{ij}\lfloor\varepsilon_{j}(t)-\varepsilon_{i}(t)\rceil^{\mu_{ij}}+ \omega_{j}\varepsilon_{j}(t) |\alpha_{j}|a_{ji}\lfloor\varepsilon_{i}(t)-\varepsilon_{j}(t)\rceil^{\mu_{ji}} \right)\nonumber\\ =&-\frac12\sum_{i,j=1}^{N}\omega_{i}|\alpha_{i}|a_{ij}|\varepsilon_{j}(t)-\varepsilon_{i}(t)|^{\mu_{ij}+1}\nonumber\\ =&-\frac12\sum_{i,j=1}^{N}\left((\omega_{i}|\alpha_{i}|a_{ij})^{\frac{2}{\mu_{\max}+1}}|\varepsilon_{j}(t)-\varepsilon_{i}(t)|^{\frac{2(\mu_{ij}+1)}{\mu_{\max}+1}}\right)^{\frac{\mu_{\max}+1}{2}}\nonumber\\ \leqslant&-\frac12\left(\frac{\sum_{i,j=1}^{N}(\omega_{i}|\alpha_{i}|a_{ij})^{\frac{2}{\mu_{\max}+1}}|\varepsilon_{j}(t)-\varepsilon_{i}(t)|^{\frac{2(\mu_{ij}+1)}{\mu_{\max}+1}}}{2\varepsilon^{\mathrm{T}}(t)\mathscr{L}(\mathscr{B}_{0})\varepsilon(t)}\cdot V(\varepsilon(t))\frac{2\varepsilon^{\mathrm{T}}(t)\mathscr{L}(\mathscr{B}_{0})\varepsilon(t)}{V(\varepsilon(t))}\right)^{\frac{\mu_{\max}+1}{2}}, \end{align} (3.6) thanks to Assumption 1 and Lemma 1. We next estimate each term of (3.6). For $$t\in (\bar{t}_{k-1},t_{k}]$$, it is clear that $$I_{N}-h|\alpha |\mathscr{L}(\mathscr{A})$$ is a stochastic matrix and we write its (i, j) element as $$\bar{l}_{ij}$$. Thus, $$x_{\max }(t+1)=\max _{i\in \mathscr{V}}$$$$\sum _{j=1}^{N}\bar{l}_{ij}\alpha _{j}x_{j}(t)$$$$\leqslant$$$$\max _{i\in \mathscr{V}}\sum _{j=1}^{N}\bar{l}_{ij}$$$$\cdot x_{\max }(t)$$$$=x_{\max }(t)$$ and similarly $$x_{\min }(t+1)\geq x_{\min }(t)$$. For $$t\in (t_{k},\bar{t}_{k}]$$, it follows from (3.2) that $$\dot{x}_{\max }(t)\leqslant 0$$ and $$\dot{x}_{\min }(t)\geqslant 0$$. Hence, $$x_{\max }(t)-x_{\min }(t)$$ is decreasing for all t and $$|\varepsilon _{j}(t)-\varepsilon _{i}(t)|\leqslant |x_{\max }(t)-x_{\min }(t)|\leqslant |x_{\max }(0)-x_{\min }(0)|$$. Noting that \begin{align} &\sum_{i,j=1}^{N}\left(\omega_{i}|\alpha_{i}|a_{ij}\right)^{\frac{2}{\mu_{\max}+1}}|\varepsilon_{j}(t)-\varepsilon_{i}(t)|^{\frac{2(\mu_{ij}+1)}{\mu_{\max}+1}}\nonumber\\ &\quad\geqslant \left(\min_{i,j\in\mathscr{V},a_{ij}\not=0}\left(\omega_{i}|\alpha_{i}|a_{ij}\right)^{\frac{2}{\mu_{\max}+1}}\right)\cdot\sum_{i,j\in\mathscr{V},a_{ij}\not=0}|\varepsilon_{j}(t)-\varepsilon_{i}(t)|^{\frac{2(\mu_{ij}+1)}{\mu_{\max}+1}}\nonumber\\ &\quad\geqslant \left(\min_{i,j\in\mathscr{V},a_{ij}\not=0}\left(\omega_{i}|\alpha_{i}|a_{ij}\right)^{\frac{2}{\mu_{\max}+1}}\right)\cdot\left(\max_{i,j\in\mathscr{V},a_{ij}\not=0}|\varepsilon_{j}(t)-\varepsilon_{i}(t)|\right)^{\frac{2(\mu_{ij}+1)}{\mu_{\max}+1}}\nonumber \end{align} and \begin{align} 2\varepsilon^{\mathrm{T}}(t)\mathscr{L}(\mathscr{B}_{0})\varepsilon(t)=&\sum_{i,j\in\mathscr{V}}(\omega_{i}|\alpha_{i}|a_{ij})^{\frac{2}{\mu_{\max}+1}}(\varepsilon_{j}(t)-\varepsilon_{i}(t))^{2}\nonumber\\ \leqslant& \left(\sum_{i,j\in\mathscr{V}}(\omega_{i}|\alpha_{i}|a_{ij})^{\frac{2}{\mu_{\max}+1}}\right)\cdot \left(\max_{i,j\in\mathscr{V},a_{ij}\not=0}|\varepsilon_{j}(t)-\varepsilon_{i}(t)|\right)^{2},\nonumber \end{align} we deduce that \begin{align} &\frac{\sum_{i,j=1}^{N}\left(\omega_{i}|\alpha_{i}|a_{ij}\right)^{\frac{2}{\mu_{\max}+1}}|\varepsilon_{j}(t)-\varepsilon_{i}(t)|^{\frac{2(\mu_{ij}+1)}{\mu_{\max}+1}}}{2\varepsilon^{\mathrm{T}}(t)\mathscr{L}(\mathscr{B}_{0})\varepsilon(t)}\nonumber\\ &\quad\geqslant \frac{\min_{i,j\in\mathscr{V},a_{ij}\not=0}\left(\omega_{i}|\alpha_{i}|a_{ij}\right)^{\frac{2}{\mu_{\max}+1}}}{\sum_{i,j\in\mathscr{V}}\left(\omega_{i}|\alpha_{i}|a_{ij}\right)^{\frac{2}{\mu_{\max}+1}}}\cdot\left(\max_{i,j\in\mathscr{V},a_{ij}\not=0}|\varepsilon_{j}(t)-\varepsilon_{i}(t)|\right)^{\frac{2\left(\mu_{ij}+1\right)}{\mu_{\max}+1}-2}\geqslant\rho_{1}\nonumber. \end{align} On the other hand, since $$\mathscr{G}$$ is strongly connected, $$K_{0}=\min _{\xi \in U}\xi ^{\mathrm{T}}\mathscr{L}(\mathscr{B}_{0})\xi>0$$. It follows from the definition of U and ωTε(t) = 0 that $$\frac{2\varepsilon^{\mathrm{T}}(t)\mathscr{L}(\mathscr{B}_{0})\varepsilon(t)}{V(\varepsilon(t))}\geqslant\frac{4}{\omega_{\max}}\frac{\varepsilon^{\mathrm{T}}(t)}{\|\varepsilon\|_{2}}\mathscr{L}(\mathscr{B}_{0})\frac{\varepsilon(t)}{\|\varepsilon\|_{2}}\geqslant\rho_{2}.$$ Putting these estimations into (3.6) yields $$\dot{V}(\varepsilon (t))\leqslant -\frac 12(\rho _{1}\rho _{2})^{\frac{\mu _{\max }+1}{2}}V^{\frac{\mu _{\max }+1}{2}}(\varepsilon (t))\leqslant 0$$ for all $$t\in (t_{k},\bar{t}_{k}]$$. A direct application of the comparison theorem (see e.g. Hartman (2002, Theorem 3.1)) gives \begin{align} V^{\frac{1-\mu_{\max}}{2}}(\varepsilon(t))\leqslant-\frac{1-\mu_{\max}}{4}(\rho_{1}\rho_{2})^{\frac{\mu_{\max}+1}{2}}(t-t_{k})+V^{\frac{1-\mu_{\max}}{2}}(\varepsilon(t_{k})) \end{align} (3.7) for all $$t\in [t_{k},\bar{t}_{k}]$$. For $$t\in (\bar{t}_{k-1},t_{k}]$$$$(k\geqslant 2)$$, the discrete-time subsystem (2.2) is activated. By Lemma 2 and the Rayleigh–Ritz theorem (Horn & Johnson, 2012), we obtain \begin{align} V(\varepsilon(t+1))-V(\varepsilon(t))=&\frac12\left(\varepsilon^{\mathrm{T}}(t+1)\varOmega\varepsilon(t+1)-\varepsilon^{\mathrm{T}}(t)\varOmega\varepsilon(t)\right)\nonumber\\ =&\frac12\varepsilon^{\mathrm{T}}(t)\left((I_{N}-h|\alpha|\mathscr{L}(\mathscr{A}))^{\mathrm{T}}\varOmega(I_{N}-h|\alpha|\mathscr{L}(\mathscr{A}))-\varOmega\right)\varepsilon(t)\leqslant0. \end{align} (3.8) It follows from (3.6) and (3.8) that $$V(\varepsilon (0))\geqslant V(\varepsilon (t_{1}))\geqslant V(\varepsilon (\bar{t}_{1}))\geqslant \cdots \geqslant V(\varepsilon (t_{k}))\geqslant V(\varepsilon (\bar{t}_{k}))\geqslant \cdots$$. Hence, using (3.7) we derive \begin{align} V^{\frac{1-\mu_{\max}}{2}}(\varepsilon(t_{k}))\leqslant& V^{\frac{1-\mu_{\max}}{2}}(\varepsilon(\bar{t}_{k-1}))\nonumber\\ \leqslant& -\frac{1-\mu_{\max}}{4}(\rho_{1}\rho_{2})^{\frac{\mu_{\max}+1}{2}}(\bar{t}_{k-1}-t_{k-1})+V^{\frac{1-\mu_{\max}}{2}}(\varepsilon(t_{k-1}))\leqslant\cdots\nonumber\\ \leqslant&-\frac{1-\mu_{\max}}{4}(\rho_{1}\rho_{2})^{\frac{\mu_{\max}+1}{2}}\left(\bar{t}_{k-1}-t_{k-1}+\cdots+\bar{t}_{1}-t_{1}\right)+V^{\frac{1-\mu_{\max}}{2}}(\varepsilon(0)). \end{align} (3.9) Set $$T_{0}=\sum _{i=1}^{k^{\ast }-1}(\bar{t}_{i}-t_{i})$$. We have $$T_{0}\geqslant (k^{\ast }-1)\tau \geqslant \frac{4V^{\frac{1-\mu _{\max }}{2}}(\varepsilon (0))}{(1-\mu _{\max })(\rho _{1}\rho _{2})^{\frac{1+\mu _{\max }}{2}}}$$ by Assumption 2. Taking k = k* in (3.9), we see that $$V(\varepsilon (t_{k^{\ast }}))=0$$. Therefore, for any $$t\geqslant t_{k^{\ast }}$$, V (ε(t)) = 0 and accordingly ε(t) = 0. The proof is complete. Remark 4 Since γ(t) is time-invariant, the final states of the agents are proportional to the weighted average of their initial values, namely, the final states are explicitly expressed. Moreover, it is easy to see from the above proof that the condition $$\bar{t}_{k}-t_{k}\geqslant \tau$$ in Assumption 2 is proposed for the ease of estimation of the settling time. It is not essentially required for the finite-time scaled consensus behaviour. Similar to Corollary 1, we can extend the controller (3.2) to solve the finite-time scaled formation generation problem for the switched multi-agent system (2.1)–(2.2). Take \begin{align} u_{i}(t)=\begin{cases}\textrm{sgn}(\alpha_{i})\sum_{j=1}^{N}a_{ij}\lfloor\alpha_{j}x_{j}(t)-f_{j}-(\alpha_{i}x_{i}(t)-f_{i})\rceil^{\mu_{ij}},& t\in(t_{k},\bar{t}_{k}];\\ \textrm{sgn}(\alpha_{i})\sum_{j=1}^{N}a_{ij}(\alpha_{j}x_{j}(t)-f_{j}-(\alpha_{i}x_{i}(t)-f_{i})),& t\in(\bar{t}_{k-1},t_{k}], \end{cases} \end{align} (3.10) where μij = μji ∈ (0, 1). Corollary 2 Suppose that Assumptions 1 and 2 hold, and the communication network $$\mathscr{G}$$ is strongly connected. If $$0<h<(d_{\max }\alpha _{\max })^{-1}$$, then the switched multi-agent system (2.1)–(2.2) with protocol (3.10) achieves the finite-time scaled formation f with respect to (α1, ⋯ , αN) with settling time $$t_{k^{\ast }}$$, where $$t_{k^{\ast }}$$ is given as in Theorem 2. 3.3. Analysis of fixed-time scaled consensus To achieve fixed-time scaled consensus, we will apply the protocol (3.3) for the switched multi-agent system to remove the dependency of $$t_{k^{\ast }}$$ in Theorem 2 regarding the initial conditions. As in the finite-time case, we follow a sequence of time instants $$0\leqslant t_{1}<\bar{t}_{1}<t_{2}<\bar{t}_{2}<\cdots <t_{k}<\bar{t}_{k}<\cdots$$. The main result in this section reads as follows. Theorem 3 Suppose that Assumptions 1 and 2 hold, and the communication network $$\mathscr{G}$$ is strongly connected. If $$0<h<(d_{\max }\alpha _{\max })^{-1}$$, then the switched multi-agent system (2.1)–(2.2) with protocol (3.3) achieves the fixed-time scaled consensus to (α1, ⋯ , αN) with settling time $$t_{\hat{k}}$$ and $$\hat{k}=\left\lceil\frac{1}{\tau\hat{\rho}_{1}(c_{1}-1)}+\frac{1}{\tau\hat{\rho}_{2}(1-c_{2})}\right\rceil+1,$$ where $$\hat{\rho }_{1}=2^{(2c_{1}-1)}N^{2(1-c_{1})}\left (\frac{K_{1}}{\omega _{\max }}\right )^{c_{1}}$$, $$\hat{\rho }_{2}=2^{(2c_{2}-1)}\left (\frac{K_{2}}{\omega _{\max }}\right )^{c_{2}}$$, $$c_{1}=\frac{m+n}{2n}>1$$, $$c_{2}=\frac{p+q}{2q}<1$$, $$K_{1}=\min _{\xi \in U}$$$$\xi ^{\mathrm{T}}\mathscr{L}(\mathscr{B}_{1})\xi>0$$, $$K_{2}=\min _{\xi \in U}\xi ^{\mathrm{T}}\mathscr{L}(\mathscr{B}_{2})\xi>0$$, $$\mathscr{B}_{1}=\left [(\omega _{i}|\alpha _{i}|a_{ij})^{\frac{1}{c_{1}}}\right ]\in \mathbb{R}^{N\times N}$$, $$\mathscr{B}_{2}=$$$$\left [(\omega _{i}|\alpha _{i}|a_{ij})^{\frac{1}{c_{2}}}\right ]\in \mathbb{R}^{N\times N}$$ and $$U=\{\xi \in \mathbb{R}^{N}:\omega ^{\mathrm{T}}\xi =0,\|\xi \|_{2}=1\}$$. Proof. Since $$\omega =(\omega _{1},\cdots ,\omega _{N})^{\mathrm{T}}\in \mathbb{R}^{N}$$ is a positive vector, ωi|αi|aij = ωj|αj|aji by Assumption 1, and μij = μji for all $$i,j\in \mathscr{V}$$, we can define $$\gamma (t)=\frac{\sum _{i=1}^{N}\omega _{i}\alpha _{i}x_{i}(t)}{\sum _{i=1}^{N}\omega _{i}}$$ and show the time-invariance of γ(t) in the similar manner as in Theorem 2. Let εi(t) = αixi(t) − γ(t) for $$i\in \mathscr{V}$$, and ε(t) = (ε1(t), ⋯ , εN(t))T. Therefore, ε(t) = αx(t) − γ(t)1N. As in Theorem 2 , we have ωTε(t) = 0 and $$\varepsilon (t+1)=(I_{N}-h|\alpha |\mathscr{L}(\mathscr{A}))\varepsilon (t)$$. Define the common Lyapunov function $$V(\varepsilon (t))=\frac 12\varepsilon ^{\mathrm{T}}(t)\Omega \varepsilon (t)=\frac 12\sum _{i=1}^{N}\omega _{i}{{\varepsilon _{i}^{2}}}(t)$$ for continuous-time subsystem (2.1) and discrete-time subsystem (2.2). For $$t\in (t_{k},\bar{t}_{k}]$$$$(k\geqslant 1)$$, the continuous-time subsystem (2.1) is activated. Since γ(t) is time-invariant, we obtain \begin{align} \dot{V}(\varepsilon(t))=&\sum_{i=1}^{N}\omega_{i}\varepsilon_{i}(t)\alpha_{i}\dot{x}_{i}(t)\nonumber\\ =&\sum_{i=1}^{N}\omega_{i}\varepsilon_{i}(t) |\alpha_{i}|\sum_{j=1}^{N}a_{ij}(\varepsilon_{j}(t)-\varepsilon_{i}(t))^{\frac{m}{n}}+ \sum_{i=1}^{N}\omega_{i}\varepsilon_{i}(t) |\alpha_{i}|\sum_{j=1}^{N}a_{ij}(\varepsilon_{j}(t)-\varepsilon_{i}(t))^{\frac{p}{q}}\nonumber\\ =&-\frac12\sum_{i,j=1}^{N}\omega_{i}|\alpha_{i}|a_{ij}(\varepsilon_{j}(t)-\varepsilon_{i}(t))^{\frac{m+n}{n}} -\frac12\sum_{i,j=1}^{N}\omega_{i}|\alpha_{i}|a_{ij}(\varepsilon_{j}(t)-\varepsilon_{i}(t))^{\frac{p+q}{q}}\nonumber\\ =&-\frac12\sum_{i,j=1}^{N}\left((\omega_{i}|\alpha_{i}|a_{ij})^{\frac{2n}{m+n}}\left|\varepsilon_{j}(t)-\varepsilon_{i}(t)\right|{}^{2}\right)^{\frac{m+n}{2n}}-\frac12\sum_{i,j=1}^{N}\left((\omega_{i}|\alpha_{i}|a_{ij})^{\frac{2n}{m+n}}\left|\varepsilon_{j}(t)-\varepsilon_{i}(t)\right|{}^{2}\right)^{\frac{p+q}{2q}}\nonumber\\ \leqslant&-\frac12 N^{\frac{n-m}{n}}\left(\sum_{i,j=1}^{N}(\omega_{i}|\alpha_{i}|a_{ij})^{\frac{2n}{m+n}}\left|\varepsilon_{j}(t)-\varepsilon_{i}(t)\right|{}^{2}\right)^{\frac{m+n}{2n}}\nonumber\\ &\quad-\frac12\left(\sum_{i,j=1}^{N}(\omega_{i}|\alpha_{i}|a_{ij})^{\frac{2q}{p+q}}\left|\varepsilon_{j}(t)-\varepsilon_{i}(t)\right|{}^{2}\right)^{\frac{p+q}{2q}}, \end{align} (3.11) where we have employed Assumption 1 and Lemma 1. Similar to Theorem 2, we can bound (3.11) by noting that $$\frac{\sum_{i,j=1}^{N}(\omega_{i}|\alpha_{i}|a_{ij})^{\frac{2n}{m+n}}|\varepsilon_{j}(t)-\varepsilon_{i}(t)|^{2}}{V(\varepsilon(t))}=\frac{2\varepsilon^{\mathrm{T}}(t)\mathscr{L}(\mathscr{B}_{1})\varepsilon(t)}{V(\varepsilon(t))} \geqslant\frac{4}{\omega_{\max}}\frac{\varepsilon^{\mathrm{T}}(t)}{\|\varepsilon\|_{2}}\mathscr{L}(\mathscr{B}_{1})\frac{\varepsilon(t)}{\|\varepsilon\|_{2}}\geqslant\frac{4K_{1}}{\omega_{\max}}.$$ and analogously, $$\frac{\sum_{i,j=1}^{N}(\omega_{i}|\alpha_{i}|a_{ij})^{\frac{2q}{p+q}}|\varepsilon_{j}(t)-\varepsilon_{i}(t)|^{2}}{V(\varepsilon(t))}=\frac{2\varepsilon^{\mathrm{T}}(t)\mathscr{L}(\mathscr{B}_{2})\varepsilon(t)}{V(\varepsilon(t))} \geqslant\frac{4K_{2}}{\omega_{\max}}$$ Consequently, in light of (3.11), we obtain $$\dot{V}(\varepsilon (t))\leqslant -\hat{\rho }_{1}V^{\frac{m+n}{2n}}(\varepsilon (t))-\hat{\rho }_{2}V^{\frac{p+q}{2q}}(\varepsilon (t))$$ for all $$t\in (t_{k},\bar{t}_{k}]$$. Hence, we have $$\dot{V}(\varepsilon (t))\leqslant -\hat{\rho }_{1}V^{c_{1}}(\varepsilon (t))$$ and $$\dot{V}(\varepsilon (t))\leqslant -\hat{\rho }_{2}V^{c_{2}}(\varepsilon (t))$$. Using the comparison theorem (see e.g. Hartman (2002, Theorem 3.1)) we deduce \begin{align} V^{1-c_{1}}(\varepsilon(t))\geqslant-\hat{\rho}_{1}(1-c_{1})(t-t_{k})+V^{1-c_{1}}(\varepsilon(t_{k})) \end{align} (3.12) and \begin{align} V^{1-c_{2}}(\varepsilon(t))\leqslant-\hat{\rho}_{2}(1-c_{2})(t-t_{k})+V^{1-c_{2}}(\varepsilon(t_{k})) \end{align} (3.13) for all $$t\in [t_{k},\bar{t}_{k}]$$. For $$t\in (\bar{t}_{k-1},t_{k}]$$$$(k\geqslant 2)$$, the discrete-time subsystem (2.2) is activated. We obtain as in Theorem 2 that \begin{align} V(\varepsilon(t+1))-V(\varepsilon(t))\leqslant0. \end{align} (3.14) It follows from (3.11) and (3.14) that $$V(\varepsilon (0))\geqslant V(\varepsilon (t_{1}))\geqslant V(\varepsilon (\bar{t}_{1}))\geqslant \cdots \geqslant V(\varepsilon (t_{k}))\geqslant V(\varepsilon (\bar{t}_{k}))\geqslant \cdots$$. In the sequel, we consider the following two cases: (a) $$V(\varepsilon (0))\leqslant 1$$ and (b) $$V(\varepsilon (0))\geqslant 1$$. For (a), by using (3.13) we obtain \begin{align} V^{1-c_{2}}\left(\varepsilon\left(t_{\hat{k}}\right)\right)\leqslant& V^{1-c_{2}}\left(\varepsilon\left(\bar{t}_{\hat{k}-1}\right)\right)\nonumber\\ \leqslant& -\hat{\rho}_{2}(1-c_{2})\left(\bar{t}_{\hat{k}-1}-t_{\hat{k}-1}\right)+V^{1-c_{2}}\left(\varepsilon\left(t_{\hat{k}-1}\right)\right)\nonumber\\ \leqslant&-\hat{\rho}_{2}(1-c_{2})\left(\bar{t}_{\hat{k}-1}-t_{\hat{k}-1}\right)+V^{1-c_{2}}\left(\varepsilon\left(\bar{t}_{\hat{k}-2}\right)\right)\nonumber\\ \leqslant&-\hat{\rho}_{2}(1-c_{2})\left(\bar{t}_{\hat{k}-1}-t_{\hat{k}-1}+\bar{t}_{\hat{k}-2}-t_{\hat{k}-2}\right)+V^{1-c_{2}}\left(\varepsilon\left(t_{\hat{k}-2}\right)\right)\nonumber\\ \leqslant&\cdots\nonumber\\ \leqslant&-\hat{\rho}_{2}(1-c_{2})\left(\bar{t}_{\hat{k}-1}-t_{\hat{k}-1}+\cdots+\bar{t}_{1}-t_{1}\right)+V^{1-c_{2}}(\varepsilon(0))\leqslant0, \end{align} (3.15) where the last inequality follows from $$T_{0}\geqslant (\hat{k}-1)\tau \geqslant \frac{1}{\hat{\rho }_{1}(c_{1}-1)}+\frac{1}{\hat{\rho }_{2}(1-c_{2})}$$ if we set $$T_{0}:=\sum _{i=1}^{\hat{k}-1}(\bar{t}_{i}-t_{i})$$. Therefore, $$V(\varepsilon (t_{\hat{k}}))=0$$. For any $$t\geqslant t_{\hat{k}}$$, V (ε(t)) = 0 and then ε(t) = 0. The proof in the case of (a) is complete. For (b), by assumption, there must exist $$k^{\ast }\in \mathbb{Z}$$ and $$t_{k^{\ast }}^{\dagger }\in (t_{k^{\ast }},\bar{t}_{k^{\ast }}]$$ such that $$T_{1}:=t_{k^{\ast }}^{\dagger }-t_{k^{\ast }}+\cdots +\bar{t}_{1}-t_{1}\geqslant \frac{1}{\hat{\rho }_{1}(c_{1}-1)}$$ and $$T_{2}:=\bar{t}_{\hat{k}-1}-t_{\hat{k}-1}+\cdots +\bar{t}_{k^{\ast }}-t_{k^{\ast }}^{\dagger }\geqslant \frac{1}{\hat{\rho }_{2}(1-c_{2})}$$. In view of (3.12), we have \begin{align} V^{1-c_{1}}(\varepsilon(t_{1}))\geqslant&\, V^{1-c_{1}}(\varepsilon(0)),\nonumber\\ V^{1-c_{1}}(\varepsilon(\bar{t}_{1}))\geqslant& -\hat{\rho}_{1}(1-c_{1})(\bar{t}_{1}-t_{1})+V^{1-c_{1}}(\varepsilon(t_{1})),\nonumber\\ V^{1-c_{1}}(\varepsilon(t_{2}))\geqslant&\, V^{1-c_{1}}(\varepsilon(\bar{t}_{1})),\nonumber\\ V^{1-c_{1}}(\varepsilon(\bar{t}_{2}))\geqslant& -\hat{\rho}_{1}(1-c_{1})(\bar{t}_{2}-t_{2})+V^{1-c_{1}}(\varepsilon(t_{2})),\nonumber\\ &\cdots\nonumber\\ V^{1-c_{1}}(\varepsilon(t_{k^{\ast}}^{\dagger})) \geqslant&-\hat{\rho}_{1}(1-c_{1})(t_{k^{\ast}}^{\dagger}-t_{k^{\ast}})+V^{1-c_{1}}(\varepsilon(t_{k^{\ast}})).\nonumber \end{align} Adding the above inequalities through, we have $$V^{1-c_{1}}\left(\varepsilon\left(t_{k^{\ast}}^{\dagger}\right)\right) \geqslant-\hat{\rho}_{1}(1-c_{1})\left(t_{k^{\ast}}^{\dagger}-t_{k^{\ast}}+\cdots+\bar{t}_{1}-t_{1}\right)+V^{1-c_{1}}(\varepsilon(0))\geqslant1,$$ since $$T_{1}\geqslant \frac{1}{\hat{\rho }_{1}(c_{1}-1)}$$. As 1 − c1 < 0, we have $$V\left (\varepsilon \left (t_{k^{\ast }}^{\dagger }\right )\right )\leqslant 1$$. Now, we can argue in a similar way as (3.15) to get $$V^{1-c_{2}}(\varepsilon(t_{\hat{k}}))\leqslant -\hat{\rho}_{2}(1-c_{2})\left(\bar{t}_{\hat{k}-1}-t_{\hat{k}-1}+\cdots+\bar{t}_{k^{\ast}}-t_{k^{\ast}}^{\dagger}\right)+V^{1-c_{2}}\left(\varepsilon\left(t_{k^{\ast}}^{\dagger}\right)\right)\leqslant0,$$ where the last inequality follows from $$T_{2}\geqslant \frac{1}{\hat{\rho }_{2}(1-c_{2})}$$. Therefore, $$V\left (\varepsilon \left (t_{\hat{k}}\right )\right )=0$$. For any $$t\geqslant t_{\hat{k}}$$, V (ε(t)) = 0 and then ε(t) = 0. The proof in the case of (b) is complete, which concludes the proof of Theorem 3. Similar comments in Remark 4 can be applied here. Finally, we extend the fixed-time controller (3.3) to solve the fixed-time scaled formation generation problem for the switched multi-agent system (2.1)–(2.2). To this end, we take \begin{align} u_{i}(t)=\begin{cases}\textrm{sgn}(\alpha_{i})\sum_{j=1}^{N}a_{ij}\left(\alpha_{j}x_{j}(t)-f_{j}-(\alpha_{i}x_{i}(t)-f_{i})\right)^{\frac{m}{n}}&\\ \quad+\textrm{sgn}(\alpha_{i})\sum_{j=1}^{N}a_{ij}\left(\alpha_{j}x_{j}(t)-f_{j}-(\alpha_{i}x_{i}(t)-f_{i})\right)^{\frac{p}{q}},& t\in(t_{k},\bar{t}_{k}];\\ \textrm{sgn}(\alpha_{i})\sum_{j=1}^{N}a_{ij}\left(\alpha_{j}x_{j}(t)-f_{j}-(\alpha_{i}x_{i}(t)-f_{i})\right),& t\in(\bar{t}_{k-1},t_{k}], \end{cases} \end{align} (3.16) where m, n, p, q are positive odd integers satisfying m > n and p < q. The following corollary can be shown similarly as Corollaries 1 and 2. Corollary 3 Suppose that Assumptions 1 and 2 hold, and the communication network $$\mathscr{G}$$ is strongly connected. If $$0<h<(d_{\max }\alpha _{\max })^{-1}$$, then the switched multi-agent system (2.1)–(2.2) with protocol (3.16) achieves the fixed-time scaled formation f with respect to (α1, ⋯ , αN) with settling time $$t_{\hat{k}}$$, where $$\hat{k}$$ is given as in Theorem 3. Fig. 1. View largeDownload slide Communication topology $$\mathscr{G}$$ for Example 1. Fig. 1. View largeDownload slide Communication topology $$\mathscr{G}$$ for Example 1. Remark 5 The results of Theorem 3 and Corollary 3 can be generalized to accommodate double channel topology with adjacency matrices being $$(a_{ij})\in \mathbb{R}^{N\times N}$$ and $$(b_{ij})\in \mathbb{R}^{N\times N}$$ for the continuous-time subsystem. For example, in Theorem 3, the control input for the continuous-time subsystem can be designed as $$u_{i}(t)=\textrm{sgn}(\alpha_{i})\sum_{j=1}^{N}a_{ij}\left(\alpha_{j}x_{j}(t)-\alpha_{i}x_{i}(t)\right)^{\frac{m}{n}} +\textrm{sgn}(\alpha_{i})\sum_{j=1}^{N}b_{ij}\left(\alpha_{j}x_{j}(t)-\alpha_{i}x_{i}(t)\right)^{\frac{p}{q}},\quad t\in(t_{k},\bar{t}_{k}]$$ for $$i\in \mathscr{V}$$. And the settling time can be given similarly with Theorem 3. It is also noteworthy that the theoretical results can be directly extended to multi-agent systems with multidimensional dynamics by use of the properties of the Kronecker product. Fig. 2. View largeDownload slide (a) Switching law of the switched multi-agent system (2.1)–(2.2) with (b) asymptotic protocol (3.1), (c) finite-time protocol (3.2) and (d) fixed-time protocol (3.3) for Example 1. Fig. 2. View largeDownload slide (a) Switching law of the switched multi-agent system (2.1)–(2.2) with (b) asymptotic protocol (3.1), (c) finite-time protocol (3.2) and (d) fixed-time protocol (3.3) for Example 1. 4. Numerical simulation In this section, some simulation examples are provided to illustrate the theoretical results. Example 1 We consider a strongly connected network $$\mathscr{G}$$ with N = 5 nodes as shown in Fig. 1. We assume ω = (1, 1, 2, 1, 1)T, α = diag(1, −2, 1, 2, −1) and a periodic switching law with τ = 5 (see Fig. 2(a)). It is easy to check that Assumptions 1 and 2 hold. Furthermore, for the finite-time protocol (3.2), we take $$\mu _{ij}=\frac 12$$ for all i, j, and for the fixed-time protocol (3.3), we take m = 7, n = 5, p = 3, q = 5. The gain is chosen as h = 0.1. These designed parameters satisfy the conditions in Theorems 1, 2 and 3. For the initial condition x(0) = (5, −1, −3, 4, 1)T, the state trajectories of all the agents are shown in Fig. 2(b) (using the asymptotic scaled consensus protocol (3.1)), Fig. 2(c) (using the finite-time scaled consensus protocol (3.2)) and Fig. 2(d) (using the fixed-time scaled consensus protocol (3.3)), respectively. We observe that in all these three situations, scaled consensus is achieved as one would expect from Theorems 1 , 2, and 3. Moreover, the theoretical estimates of the settling time is $$t_{k^{\ast }}=605$$ (with k* = 121) for the finite-time scaled consensus and $$t_{\hat{k}}=80$$ (with $$\hat{k}=16$$) for the fixed-time scaled consensus. Obviously, the actual settling time is less than the theoretical estimates in these control protocols, respectively. It is remarkable that the estimate for fixed-time scaled consensus is much sharper than that for finite-time scaled consensus. This is because the estimation for (3.11) is more refined than that for (3.6). Fig. 3. View largeDownload slide Communication topology $$\mathscr{G}$$ for Example 2. Fig. 3. View largeDownload slide Communication topology $$\mathscr{G}$$ for Example 2. Fig. 4. View largeDownload slide (a) Switching law of the switched multi-agent system (2.1)–(2.2) with (b) asymptotic protocol (3.1), (c) finite-time protocol (3.2) and (d) fixed-time protocol (3.3) for Example 2. Fig. 4. View largeDownload slide (a) Switching law of the switched multi-agent system (2.1)–(2.2) with (b) asymptotic protocol (3.1), (c) finite-time protocol (3.2) and (d) fixed-time protocol (3.3) for Example 2. Example 2 In this example, we consider a strongly connected network $$\mathscr{G}$$ with N = 4 nodes as shown in Fig. 3. Note that Assumption 1 is no longer applicable; c.f. Remark 1. We take α = diag(2, 1, 1, −1), and a periodic switching law with τ = 10 (see Fig. 4(a)). As in Example 1, for the finite-time protocol (3.2), we take $$\mu _{ij}=\frac 12$$ for all i, j, and for the fixed-time protocol (3.3), we take m = 7, n = 5, p = 3, q = 5. The gain is chosen as h = 0.1. For the initial condition x(0) = (2, −1, −4, 1)T, the state trajectories of all the agents are shown in Fig. 4(b) (using the asymptotic scaled consensus protocol (3.1)), Fig. 4(c) (using the finite-time scaled consensus protocol (3.2)) and Fig. 4(d) (using the fixed-time scaled consensus protocol (3.3)), respectively. Although Fig. 4(b) displays the asymptotic scaled consensus as one would expect, the rapid convergence behaviour shown in Fig. 4(c) and (d) is remarkable. It implies that the detailed balance condition, i.e. Assumption 1, on the network topology is not necessary in general to guarantee finite-time/fixed-time scaled convergence for switched multi-agent systems. 5. Conclusion In this paper, we have considered the scaled consensus problems of a switched multi-agent system composed of both continuous-time and discrete-time subsystems. By employing the nearest neighbour-interaction rules, we propose three consensus protocols that possess asymptotic, finite-time and fixed-time convergence rates, respectively. The final consensus states are explicitly presented, and generalizations to scaled formation generation are also studied. For future work, we will focus on relaxation of topology condition (Assumption 1) and scaled consensus of switched multi-agent systems with switching topologies. Some other challenging future directions include the analysis of switched multi-agent systems with higher-order dynamics and robustness against communication delays. Acknowledgements The author would like to thank the reviewers for their insightful comments and careful reading. Part of the work has been done during a visit of the author to the Department of Mathematical Sciences at University of Essex. The author would like to thank its hospitality. Funding National Natural Science Foundation of China (11505127), the Shanghai Pujiang Program (15PJ1408300) and the Program for Young Excellent Talents in Tongji University (2014KJ036). References Altafini , C. ( 2013 ) Consensus problems on networks with antagonistic interactions . IEEE Trans. Autom. Control , 58 , 935 -- 946 . Google Scholar CrossRef Search ADS Bhat , S. P. & Bernstein , D. S. ( 2000 ) Finite-time stability of continuous autonomous systems . SIAM J. Control Optim. , 38 , 751 -- 766 . Google Scholar CrossRef Search ADS Campos , G. , Dimarogonas , D. V. , Seuret , A. & Johansson , K. H. ( 2016 ) Distributed control of compact formations for multi-robot swarms . IMA J. Math. Control Inform. , 32 , 1 -- 31 . Cao , Y. , Yu , W. , Ren , W. & Chen , G. ( 2013 ) An overview of recent progress in the study of distributed multi-agent coordination . IEEE Trans. Ind. Inf. , 9 , 427 -- 438 . Google Scholar CrossRef Search ADS Chu , T. , Zhang , C. & Zhang , Z. ( 2003 ) Necessary and sufficient condition for absolute stability of normal neural networks . Neural Networks , 16 , 1223 -- 1227 . Google Scholar CrossRef Search ADS PubMed Defoort , M. , Polyakov , A. , Demesure , G. , Djemai , M. & Veluvolu , K. ( 2015 ) Leader-follower fixed-time consensus for multi-agent systems with unknown non-linear inherent dynamics . IET Control Theory Appl ., 9 , 2165 -- 2170 . Google Scholar CrossRef Search ADS Fu , J. & Wang , J. ( 2016 ) Fixed-time coordinated tracking for second-order multi-agent systems with bounded input uncertainties . Syst. Control Lett. , 93 , 1 -- 12 . Google Scholar CrossRef Search ADS Guglieri , G. , Maroglio , F. , Pellegrino , P. & Torre , L. ( 2014 ) Design and development of guidance navigation and control algorithms for spacecraft rendezvous and docking experimentation . Acta Astronautica , 94 , 395 -- 408 . Google Scholar CrossRef Search ADS Haken , H. ( 1978 ) Synergetics . Berlin : Springer . Google Scholar CrossRef Search ADS Hardy , G. , Littlewood , J. & Polya , G. ( 1952 ) Inequalities , 2nd ed. New York : Cambridge University Press . Hartman , P. ( 2002 ) Ordinary Differential Equations . Philadelphia : SIAM . Google Scholar CrossRef Search ADS Horn , R. A. & Johnson , C. R. ( 2012 ) Matrix Analysis . Cambridge : Cambridge University Press . Google Scholar CrossRef Search ADS Hou , B. , Sun , F. , Li , H. , Chen , Y. & Liu , G. ( 2016 ) Scaled cluster consensus of discrete-time multi-agent systems with general directed topologies . Int. J. Control , 47 , 3839 -- 3845 . Jadbabaie , A. , Lin , J. & Morse , A. S. ( 2003 ) Coordination of groups of mobile autonomous agents using nearest neighbor rules . IEEE Trans. Autom. Control , 48 , 988 -- 1001 . Google Scholar CrossRef Search ADS Li , J. , Dong , W. & Xiao , H. ( 2017 ) Signed consensus problems on networks of agents with fixed and switching topologies . Int. J. Control , 90 , 148 -- 160 . Google Scholar CrossRef Search ADS Lin , X. & Zheng , Y. ( 2017 ) Finite-time consensus of switched multiagent systems . IEEE Trans. Syst. Man Cybern. Syst. , 47 , 1535 -- 1545 . Google Scholar CrossRef Search ADS Meng , D. & Jia , Y. ( 2016a ) Robust consensus algorithms for multiscale coordination control of multivehicle systems with disturbances . IEEE Trans. Ind. Electron. , 63 , 1107 -- 1119 . Google Scholar CrossRef Search ADS Meng , D. & Jia , Y. ( 2016b ) Scaled consensus problems on switching networks . IEEE Trans. Autom. Control , 61 , 1664 -- 1669 . Google Scholar CrossRef Search ADS Mesbahi , M. & Egerstedt , M. ( 2010 ) Graph Theoretic Methods in Multiagent Networks . New Jersey : Princeton University Press . Google Scholar CrossRef Search ADS Olfati-Saber , R. , Fax , J. A. & Murray , R. M. ( 2007 ) Consensus and cooperation in networked multi-agent systems . Proc. IEEE , 95 , 215 -- 233 . Google Scholar CrossRef Search ADS Parsegov , S. , Polyakov , A. & Shcherbakov , P. ( 2013 ) Fixed-time consensus algorithm for multi-agent systems with integrator dynamics . Proc. 4th IFAC Workshop on Distributed Estimation and Control in Networked Systems , 110 -- 115 . Polyakov , A . ( 2012 ) Nonlinear feedback design for fixed-time stabilization of linear control systems . IEEE Trans. Autom. Control , 57 , 2106 -- 2110 . Google Scholar CrossRef Search ADS Ren , W. , Beard , R. W. & Atkins , E. M. ( 2007 ) Information consensus in multivehicle cooperative control . IEEE Control Syst. Mag. , 27 , 71 -- 82 . Google Scholar CrossRef Search ADS Roy , S . ( 2015 ) Scaled consensus . Automatica , 51 , 259 -- 262 . Google Scholar CrossRef Search ADS Rugh , W. J. ( 1996 ) Linear System Theory . New Jersey : Prentice Hall . Shang , Y. ( 2016a ) A combinatorial necessary and sufficient condition for cluster consensus . Neurocomputing , 216 , 611 -- 616 . Google Scholar CrossRef Search ADS Shang , Y. ( 2016b ) Finite-time weighted average consensus and generalized consensus over a subset . IEEE Access , 4 , 2615 -- 2620 . Google Scholar CrossRef Search ADS Shang , Y. ( 2017a ) Finite-time scaled consensus through parametric linear iterations . Int. J. Syst. Sci. , 48 , 2033 -- 2040 . Google Scholar CrossRef Search ADS Shang , Y. ( 2017b ) On the delayed scaled consensus problems . App. Sci. , 7 , art. no. 713 . Shang , Y. ( 2017c ) Subspace confinement for switched linear systems . Forum Math. , 29 , 693 -- 699 . Google Scholar CrossRef Search ADS Sun , S. H. , Zhao , L. & Jia , Y. M. ( 2014 ) Similitude design method for motion reconstruction of space cooperative vehicles . J. Astronaut. , 35 , 802 -- 810 . Wang , L. & Xiao , F. ( 2010 ) Finite-time consensus problems for networks of dynamic agents . IEEE Trans. Autom. Control , 55 , 950 -- 955 . Google Scholar CrossRef Search ADS Xiao , F. , Wang , L. , Chen , J. & Gao , Y. ( 2009 ) Finite-time formation control for multi-agent systems . Automatica , 45 , 2605 -- 2611 . Google Scholar CrossRef Search ADS Zhai , G. , Liu , D. , Imae , J. & Kobayashi , T. ( 2006 ) Lie algebraic stability analysis for switched systems with continuous-time and discrete-time subsystems . IEEE Trans. Circuits Syst. II Exp. Briefs , 53 , 152 -- 156 . Google Scholar CrossRef Search ADS Zheng , Y. & Wang , L. ( 2016 ) Consensus of switched multiagent systems . IEEE Trans. Circuits Syst. II Exp. Briefs , 63 , 314 -- 318 . Google Scholar CrossRef Search ADS Zhu , Y. , Zheng , Y. & Wang , L. ( 2015 ) Containment control of switched multi-agent systems . Int. J. Control , 88 , 2570 -- 2577 . Google Scholar CrossRef Search ADS © The Author(s) 2018. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png IMA Journal of Mathematical Control and Information Oxford University Press

# Scaled consensus of switched multi-agent systems

, Volume Advance Article – Feb 1, 2018
19 pages

/lp/ou_press/scaled-consensus-of-switched-multi-agent-systems-79Re5Q55f0
Publisher
Oxford University Press
ISSN
0265-0754
eISSN
1471-6887
D.O.I.
10.1093/imamci/dnx062
Publisher site
See Article on Publisher Site

### Abstract

Abstract In this paper, the scaled consensus problem for switched multi-agent systems composed of continuous-time and discrete-time subsystems is investigated. By using the nearest neighbour-interaction rules, three types of scaled consensus protocols are presented to achieve asymptotic, finite-time and fixed-time convergence rates, respectively. Based upon algebraic graph theory and Lyapunov theory, scaled consensus is shown to be reached for strongly connected networks. We explicitly express the final consensus states as some initial-condition-dependent values. Three scaled formation generation problems with different convergence rates are introduced and solved as generalizations. Simulation examples are provided to validate the effectiveness and availability of our theoretical results. 1. Introduction In the past decades, distributed control of networked multi-agent systems has received a great deal of attention in the system and control community due to its broad applications in areas such as formation control, cooperative coordination of unmanned aerial vehicles, robotic teams and sensor networks (Jadbabaie et al., 2003; Olfati-Saber et al., 2007). Consensus plays a fundamental role in distributed coordination, which aims to design appropriate control laws such that all the agents reach an agreement on some consistent quantity of interest. As each agent can only interact with those within its local areas due to limited communication ability, the designed protocols take advantages of nearest-neighbour rules rendering the multi-agent systems in a distributed network framework governed by the graph Laplacians (Mesbahi & Egerstedt, 2010). Such networked multi-agent systems cannot be tackled by standard stability theories (e.g. Rugh (1996)) in general as compared to the traditional monolithic ones. Up to now, a variety of consensus protocols, such as average consensus, leader-follower consensus, finite-time consensus, stochastic consensus and asynchronous consensus, have been reported to achieve consensus in continuous-time and discrete-time multi-agent systems; see the surveys Olfati-Saber et al. (2007); Ren et al. (2007); Cao et al. (2013) and references therein. In many practical applications, the states of agents may converge to prescribed ratios (so-called ‘scales’) instead of a common value. In simultaneous coordination control of satellites running on orbits and their simulating robots on ground, for example, huge difference between the scales of vehicles’ position and speed in space and on ground gives rise to the scaled coordination control required in practise (Guglieri et al., 2014; Sun et al., 2014). Other practical examples include compartmental mass-action systems, closed queuing networks and water distribution systems (Roy, 2015). As a generalization to standard consensus, Roy (2015) recently introduces a novel notion of scaled consensus, wherein agents’ states reach asymptotically assigned ratios in terms of different scales. The scaled consensus offers a general framework, which can be specialized to achieve standard consensus, cluster consensus (Shang, 2016a), wherein different subnetworks are allowed to reach different common values, and bipartite consensus (Altafini, 2013) or signed consensus (Li et al., 2017) by choosing appropriate scales. Scaled consensus has been studied for fixed strongly connected topology in Roy (2015) and switching topologies in Meng & Jia (2016b), where the agents are modelled by continuous-time single integrators. Several scaled consensus protocols are proposed to solve the finite-time coordination control for continuous-time multivehicle systems in Meng & Jia (2016a) and for discrete-time ones in Shang (2017a). Robust scaled consensus problems for continuous-time and discrete-time multi-agent systems with time delay are dealt with, respectively, in Shang (2017b) and Hou et al. (2016). For consensus protocol design, convergence rate is an important performance indicator. Usually, linear systems are only able to achieve an asymptotic convergence rate, meaning that no agreement can be made in finite time (Mesbahi & Egerstedt, 2010). In system theory, it is often required to develop controllers to drive agents to a given position as soon as possible. Moreover, finite-time controllers enjoy faster convergence rate, better disturbance rejection as well as more robustness to uncertainties (Bhat & Bernstein, 2000; Wang & Xiao, 2010). However, the settling time for finite-time consensus protocols heavily depends on the initial conditions, which weaken their advantages over asymptotic consensus protocols when the initial condition varies or is unavailable in advance. To overcome this drawback, a class of fixed-time consensus protocols has been presented (see e.g. Polyakov (2012); Parsegov et al. (2013); Defoort et al. (2015); Fu & Wang (2016)) for continuous-time multi-agent systems, where the settling time is bounded and independent of the initial states of agents. For example, based on the fixed-time stability theory, fixed-time average consensus problem is addressed in Parsegov et al. (2013) for multi-agent systems with integrator dynamics over fixed undirected networks. Leader–follower fixed-time consensus for continuous-time systems with nonlinear inherent dynamics is investigated in Defoort et al. (2015). In Shang (2016b), fixed-time average consensus problems for discrete-time multi-agent systems have also been discussed. It is noteworthy that all the aforementioned works are concerned with multi-agent systems composed of either only continuous-time systems or only discrete-time systems. In practise, a large-scale system can be split into multiple subsystems and a switching rule manages the switching between them leading to a switched multi-agent system (Zhai et al., 2006). A multi-agent system, for instance, controlled either by a digitally implemented regulator or a physically implemented one with a synchronous switching law between them gives rise to a switched multi-agent system. Consensus problems for switched multi-agent systems are firstly considered in Zheng & Wang (2016) over different topologies. Lin & Zheng (2017) further investigates the finite-time convergence for switched multi-agent systems by using matrix theory and Lyapunov theory. Containment control of switched multi-agent systems is addressed in Zhu et al. (2015). Nevertheless, to our knowledge, there has been no systematic study on scaled consensus problems for switched systems consisting of both continuous-time and discrete-time subsystems. The aim of this paper is to develop a systematic framework for convergence problems of scaled consensus of switched multi-agent systems. The contribution of this paper is threefold. First, we propose three scaled consensus protocols with asymptotic, finite-time and fixed-time convergence rates, respectively, for switched multi-agent systems consisting of both continuous-time and discrete-time subsystems. Second, the settling time for finite-time and fixed-time scaled consensus is explicitly obtained by utilizing algebraic graph theory and Lyapunov functions. Moreover, the final consensus states are explicitly expressed as certain initial-condition-dependent values. Third, asymptotic, finite-time and fixed-time scaled formation control problems for switched multi-agent systems are introduced and solved as the generalizations. It is noteworthy that general convergence of switched linear systems has been considered in Shang (2017c), however the focus therein is on the switch of system matrix rather than continuous and discrete subsystems. The rest of the paper is organized as follows. Section 2 provides some preliminaries and formulates the scaled consensus problems of switched multi-agent systems. Section 3 presents scaled consensus protocols and develops convergence as well as formation generation results. We provide some numerical examples in Section 4 and then conclude the paper in Section 5. 2. Preliminaries and problem statement We begin with some standard notations that will be used throughout the paper. Let $$\mathbb{Z}$$ represent the set of integers. Let $$\mathbb{R}$$ and $$\mathbb{R}^{N}$$ represent the set of real numbers and the N-dimensional real vector space, respectively. Let MT be the transpose of a matrix M. For a symmetric matrix $$M\in \mathbb{R}^{N\times N}$$, its eigenvalues are ordered as $$\lambda _{1}(M)\leqslant \lambda _{2}(M)\leqslant \cdots \leqslant \lambda _{N}(M)$$. For $$x\in \mathbb{R}$$ and $$\mu \geqslant 0$$, we define ⌊x⌉μ = sgn(x)|x|μ, where sgn(⋅) is the signum function. IN and 1N denote, respectively, the N × N identity matrix and the N-dimensional vector with elements being all ones. For p > 0, the p-norm ∥⋅∥p is defined as $$\|x\|_{p}=\big (\sum _{i=1}^{N}|x_{i}|^{p}\big )^{1/p}$$ for a vector $$x=(x_{1},\cdots ,x_{N})^{\mathrm{T}}\in \mathbb{R}^{N}$$. The following lemma connecting different norms is useful in dealing with the finite-time consensus problems, a proof of which can be found in Hardy et al. (1952). Lemma 1 (Hardy et al., 1952) Let $$x\in \mathbb{R}^{N}$$ and q > p > 0. Then $$\|x\|_{q}\leqslant\|x\|_{p}\leqslant N^{\frac{1}{p}-\frac{1}{q}}\|x\|_{q}.$$ 2.1. Algebraic graph theory The communication topology of a multi-agent system can often be characterized by a weighted directed graph (Mesbahi & Egerstedt, 2010) $$\mathscr{G}=(\mathscr{V},\mathscr{E},\mathscr{A})$$ with a node set $$\mathscr{V}=\{1,2,\cdots ,N\}$$ representing N agents, an edge set $$\mathscr{E}\subseteq \mathscr{V}\times \mathscr{V}$$ describing the information exchange amongst them, and a non-negative adjacency matrix $$\mathscr{A}=(a_{ij})\in \mathbb{R}^{N\times N}$$. aij > 0 if and only if $$(j,i)\in \mathscr{E}$$, representing the information flow from agent j to agent i. The degree of i is defined as $$d_{i}=\sum _{j\in \mathscr{V}\backslash \{i\}}a_{ij}$$ and the maximum degree $$d_{\max }=\max _{i\in \mathscr{V}}d_{i}$$. Given a sequence i1, i2 ⋯ , ik of distinct nodes, a path in $$\mathscr{G}$$ from i1 to ik consists of a sequence of edges $$\left (i_{j},i_{j+1}\right )\in \mathscr{E}$$ for j = 1, ⋯ , k − 1. A directed graph $$\mathscr{G}$$ is said to be strongly connected if between any pair of distinct nodes i, j, there exists a path from i to j. $$\mathscr{G}$$ is said to satisfy the detailed balance condition if there exist some scalars ωi > 0 ($$i\in \mathscr{V}$$) such that ωiaij = ωjaji for all $$i,j\in \mathscr{V}$$. This condition has proven to be instrumental in studying coupled dynamics (see e.g. Chu et al. (2003); Haken (1978)). Given N non-zero scalars α1, ⋯ , αN, a modified detailed balance condition is given as follows. Assumption 1 There exist some scalars ωi > 0 ($$i\in \mathscr{V}$$) such that ωi|αi|aij = ωj|αj|aji for all $$i,j\in \mathscr{V}$$. Remark 1 Note that for a general weighted directed network with aij > 0 but aji = 0 Assumption 1 is not applicable. However, for a bidirectional network, where aij > 0 if and only if aji > 0 (Mesbahi & Egerstedt, 2010), Assumption 1 would be appropriate. This condition will be used in finite-time/fixed-time convergence, c.f. Theorems 2 and 3 below. The graph Laplacian matrix of $$\mathscr{A}$$, $$\mathscr{L}(\mathscr{A})=(l_{ij})\in \mathbb{R}^{N\times N}$$, is defined by $$l_{ii}=-\sum _{j\in \mathscr{V}\backslash \{i\}}l_{ij}=d_{i}$$ and lij = −aij for i≠j. Clearly, $$\mathscr{L}(\mathscr{A})1_{N}=0$$. The graph $$\mathscr{G}$$ is called undirected if aij = aji for all $$i,j\in \mathscr{V}$$. An undirected graph is connected if there exists a path between any two distinct nodes of the graph. For a connected undirected graph $$\mathscr{G}$$, it is well known that $$\mathscr{L}(\mathscr{A})$$ is positive semidefinite and $$x^{\mathrm{T}}\mathscr{L}(\mathscr{A})x=\frac 12\sum _{i,j\in \mathscr{V}}a_{ij}(x_{j}-x_{i})^{2}$$ for any $$x=(x_{1},\cdots ,x_{N})^{\mathrm{T}}\in \mathbb{R}^{N}$$ (Horn & Johnson, 2012). The following lemma is a straightforward generalization of Lin & Zheng (2017, Lemma 4). Lemma 2 Suppose that Assumption 1 holds. Let $$\alpha _{\max }=\max _{1\leqslant i\leqslant N}|\alpha _{i}|$$. If the weighted directed graph $$\mathscr{G}=(\mathscr{V},\mathscr{E},\mathscr{A})$$ is strongly connected and $$0<h<(d_{\max }\alpha _{\max })^{-1}$$, then the matrix $$(I_{N}-h|\alpha |\mathscr{L}(\mathscr{A}))^{\mathrm{T}}\varOmega$$$$(I_{N}-h|\alpha |\mathscr{L}(\mathscr{A}))-\varOmega$$ is negative semidefinite and the zero eigenvalue is algebraically simple. Here, $$|\alpha |:=\textrm{diag}(|\alpha _{1}|,\cdots ,|\alpha _{N}|)\in \mathbb{R}^{N\times N}$$ and Ω := diag(ω1, ⋯ , ωN) is the diagonal matrix with the (i, i) element being ωi. Proof. First, we observe that $$I_{N}-h|\alpha |\mathscr{L}(\mathscr{A})=(I_{N}-h|\alpha |\mathscr{D})+h|\alpha |\mathscr{A}$$ is a row stochastic matrix, where $$\mathscr{D}=\textrm{diag}(d_{1},\cdots ,d_{N})$$ is the degree matrix. Clearly, $$(I_{N}-h|\alpha |\mathscr{L}(\mathscr{A}))^{\mathrm{T}}\varOmega (I_{N}-h|\alpha |\mathscr{L}(\mathscr{A}))$$ is non-negative. Thanks to Assumption 1, we have $$\varOmega |\alpha |\mathscr{L}(\mathscr{A})=\mathscr{L}(\mathscr{A})^{\mathrm{T}}\varOmega |\alpha |=\mathscr{L}(\mathscr{A})^{\mathrm{T}}|\alpha |\varOmega$$, and hence $$(I_{N}-h|\alpha |\mathscr{L}(\mathscr{A}))^{\mathrm{T}}\varOmega (I_{N}-h|\alpha |\mathscr{L}(\mathscr{A}))1_{N}=\varOmega (I_{N}-h|\alpha |\mathscr{L}(\mathscr{A}))1_{N}=\varOmega 1_{N}$$. This implies that 1N is a right eigenvector corresponding to eigenvalue zero for matrix $$(I_{N}-h|\alpha |\mathscr{L}(\mathscr{A}))^{\mathrm{T}}\varOmega (I_{N}-h|\alpha |\mathscr{L}(\mathscr{A}))-\varOmega$$. An application of the Gershgorin circle theorem (Horn & Johnson, 2012) shows that all eigenvalues of $$(I_{N}-h|\alpha |\mathscr{L}(\mathscr{A}))^{\mathrm{T}}\varOmega (I_{N}-h|\alpha |\mathscr{L}(\mathscr{A}))-\varOmega$$ are non-positive. Next, we show that the zero eigenvalue is algebraically simple. Since $$\mathscr{G}$$ is strongly connected and $$0<h<(d_{\max }\alpha _{\max })^{-1}\leqslant 2\lambda _{N}^{-1}(\mathscr{L}(\mathscr{A}))$$, we obtain $$\textrm{rank}(|\alpha |\mathscr{L}(\mathscr{A}))=n-1$$ and hence $$-2h|\alpha |\mathscr{L}(\mathscr{A})+h^{2}(|\alpha |\mathscr{L}(\mathscr{A}))^{2}$$ has rank n − 1. Therefore, employing the Sylvester inequality, we arrive at \begin{align} n-1\geqslant&\ \textrm{rank}\left((I_{N}-h|\alpha|\mathscr{L}(\mathscr{A}))^{ T}\varOmega(I_{N}-h|\alpha|\mathscr{L}(\mathscr{A}))-\varOmega\right)\nonumber\\ =&\ \textrm{rank}\left(\varOmega(I_{N}-h|\alpha|\mathscr{L}(\mathscr{A}))^{2}-\varOmega\right)\nonumber\\ =&\ \textrm{rank}\left(\varOmega\left(-2h|\alpha|\mathscr{L}(\mathscr{A})+h^{2}(|\alpha|\mathscr{L}(\mathscr{A}))^{2}\right)\right)\nonumber\\ \geqslant&\ \textrm{rank}(\varOmega)+\textrm{rank}\left(-2h|\alpha|\mathscr{L}(\mathscr{A})+h^{2}(|\alpha|\mathscr{L}(\mathscr{A}))^{2}\right)-n=n-1.\nonumber \end{align} This implies that zero eigenvalue can only be algebraically simple, which completes the proof. 2.2. Problem formulation Consider a group of N agents, labelled from 1 to N, composing a fixed directed network $$\mathscr{G}=(\mathscr{V},\mathscr{E},\mathscr{A})$$. The agents are controlled by a switching law between continuous-time dynamics and discrete-time dynamics. The information state of the agent i at time t is represented by $$x_{i}(t)\in \mathbb{R}$$. In the vector form, we denote x(t) = (x1(t), ⋯ , xN(t))T and let x(0) = (x1(0), ⋯ , xN(0))T be the initial value. Given any scalar scale αi≠0 for the agent i, we are devoted to addressing the scaled consensus problems with three different convergence rates in the following three cases: (i) Asymptotic convergence (Roy, 2015): $$\lim _{t\rightarrow \infty }(\alpha _{i}x_{i}(t)-\alpha _{j}x_{j}(t))=0$$ for all $$i,j\in \mathscr{V}$$ and all initial conditions x(0). (ii) Finite-time convergence (Meng & Jia, 2016a): $$\lim _{t\rightarrow T(x(0))}(\alpha _{i}x_{i}(t)-\alpha _{j}x_{j}(t))=0$$ and αixi(t) = αjxj(t) for $$t\geqslant T(x(0))$$ for all $$i,j\in \mathscr{V}$$ and all initial conditions x(0), where T(x(0)) is the settling time which depends on the initial state x(0). (iii) Fixed-time convergence (Meng & Jia, 2016a): $$\lim _{t\rightarrow T}(\alpha _{i}x_{i}(t)-\alpha _{j}x_{j}(t))=0$$ and αixi(t) = αjxj(t) for $$t\geqslant T$$ for all $$i,j\in \mathscr{V}$$ and all initial conditions x(0), where T is the settling time which is independent of the initial state x(0). The dynamical model of each agent is described by a continuous-time subsystem \begin{align} \dot{x}_{i}(t)=u_{i}(t),\quad i\in\mathscr{V} \end{align} (2.1) and a discrete-time subsystem \begin{align} x_{i}(t+1)=x_{i}(t)+hu_{i}(t),\quad i\in\mathscr{V} \end{align} (2.2) where $$u_{i}(t)\in \mathbb{R}$$ is the control input to be designed, h > 0 is the control gain and the synchronous switch is assumed to apply for all agents. At time instant t, the choice of subsystem is decided by the switching rule. The time domain for the subsystems (2.1) and (2.2) is $$t\in \mathbb{R}$$ and $$t\in \mathbb{Z}$$, respectively. When (2.2) is activated, the states of the agents are updated with time step length 1. Since the state of the discrete-time subsystem can be viewed as a piecewise constant vector between sampling points, we may consider the value of the system states in continuous-time domain. The switched multi-agent system (2.1)–(2.2) is said to achieve the asymptotic (finite-time or fixed-time, respectively) scaled consensus to (α1, ⋯ , αN) if the corresponding convergence condition in (i) ((ii) or (iii), respectively) holds. For finite-time and fixed-time convergence, we will assume a sequence of time instants $$0\leqslant t_{1}<\bar{t}_{1}<t_{2}<\bar{t}_{2}<\cdots <t_{k}<\bar{t}_{k}<\cdots$$ satisfying the following assumption. Assumption 2. When $$t\in (t_{k},\bar{t}_{k}]$$, the continuous-time subsystem (2.1) is activated; when $$t\in (\bar{t}_{k-1},t_{k}]$$, the discrete-time subsystem (2.2) is activated. Furthermore, $$\bar{t}_{k}-t_{k}\geqslant \tau$$ for some constant τ > 0. Remark 2. We assume that the synchronous switch is applied for each agent. The system described by (2.1) and (2.2) can be viewed as a system consisting of continuous-time subsystem and discrete-time subsystem at different time interval. The synchronized switch is often caused by an upper layer switching rule independent of the distributed communication of agents in the system (Zhai et al., 2006; Zheng & Wang, 2016). A typical example is to consider a switched multi-agent system whose subsystems are all continuous-time. If a computer is used to activate all the subsystems in a discrete-time manner, then the switched system is composed of both continuous-time and discrete-time subsystems. Another example of this kind is a continuous-time plant controlled either by a physically implemented regulator or by a digitally implemented one together with a switching rule between them. Also note that the switch between continuous-time control and sampled-data control under Assumption 2 guarantees the switched multi-agent system to be composed of continuous-time and discrete-time subsystems (or only the continuous-time subsystems). For finite-time consensus problems, such switching control method has been considered in Lin & Zheng (2017). 3. Scaled consensus protocols for switched multi-agent systems In this section the main results about the scaled consensus of the switched multi-agent system (1)–(2) are investigated. We adopt the nearest-neighbour rules and construct distributed algorithms for each agent $$i\in \mathscr{V}$$ as follows. (i) An asymptotic convergence protocol is given by \begin{align} u_{i}(t)=\textrm{sgn}(\alpha_{i})\sum_{j=1}^{N}a_{ij}(\alpha_{j}x_{j}(t)-\alpha_{i}x_{i}(t)), \end{align} (3.1) for both subsystems (2.1) and (2.2). At time instant t, the choice of subsystem is determined by the switching rule under consideration. (ii) A finite-time convergence protocol is given by \begin{align} u_{i}(t)=\begin{cases}\textrm{sgn}(\alpha_{i})\sum_{j=1}^{N}a_{ij}\lfloor\alpha_{j}x_{j}(t)-\alpha_{i}x_{i}(t)\rceil^{\mu_{ij}},& t\in(t_{k},\bar{t}_{k}];\\ \textrm{sgn}(\alpha_{i})\sum_{j=1}^{N}a_{ij}(\alpha_{j}x_{j}(t)-\alpha_{i}x_{i}(t)),& t\in(\bar{t}_{k-1},t_{k}], \end{cases} \end{align} (3.2) where μij = μji ∈ (0, 1). (iii) A fixed-time convergence protocol is given by \begin{align} u_{i}(t)=\begin{cases}\textrm{sgn}(\alpha_{i})\sum_{j=1}^{N}a_{ij}(\alpha_{j}x_{j}(t)-\alpha_{i}x_{i}(t))^{\frac{m}{n}}&\\ \quad+\textrm{sgn}(\alpha_{i})\sum_{j=1}^{N}a_{ij}(\alpha_{j}x_{j}(t)-\alpha_{i}x_{i}(t))^{\frac{p}{q}},& t\in(t_{k},\bar{t}_{k}];\\ \textrm{sgn}(\alpha_{i})\sum_{j=1}^{N}a_{ij}(\alpha_{j}x_{j}(t)-\alpha_{i}x_{i}(t)),& t\in(\bar{t}_{k-1},t_{k}], \end{cases} \end{align} (3.3) where m, n, p, q are positive odd integers satisfying m > n and p < q. Compared to the finite-time protocol (3.2), one more term is needed in (3.3) to ensure the fixed-time scaled consensus. For the particular case when αi = 1 for all $$i\in \mathscr{V}$$, (3.2) and (3.3) become the consensus protocols proposed in Lin & Zheng (2017). Moreover, it is easy to verify that the scaled consensus protocols (3.1)–(3.3) for continuous-time subsystem (2.1) are continuous. 3.1. Analysis of asymptotic scaled consensus We first consider the asymptotic scaled consensus of switched multi-agent system (2.1)–(2.2) employing protocol (3.1). Define $$\alpha :=\textrm{diag}(\alpha _{1},\cdots ,\alpha _{N})\in \mathbb{R}^{N\times N}$$ and $$|\alpha |:=\textrm{diag}(|\alpha _{1}|,\cdots ,|\alpha _{N}|)\in \mathbb{R}^{N\times N}$$. With these notations, the switched multi-agent system with protocol (3.1) can be recast as $$\alpha\dot{x}(t)=-|\alpha|\mathscr{L}(\mathscr{A})\alpha x(t)\qquad\textrm{and}\qquad \alpha x(t+1)=(I_{N}-h|\alpha|\mathscr{L}(\mathscr{A}))\alpha x(t).$$ The following lemma summarizes and reformulates some results presented in Roy (2015); Hou et al. (2016), and Shang (2017b). Lemma 3 Suppose that the communication network $$\mathscr{G}$$ is strongly connected. Then the continuous-time multi-agent system (2.1) with (3.1) achieves the asymptotic scaled consensus to (α1, ⋯ , αN). If h|αi|di < 1 for all $$i\in \mathscr{V}$$, then the discrete-time multi-agent system (2.2) with (3.1) achieves the asymptotic scaled consensus to (α1, ⋯ , αN). In particular, for both continuous-time and discrete-time systems, $$\lim _{t\rightarrow \infty }x(t)=(1/\alpha _{1},\cdots ,1/\alpha _{N})^{\mathrm{T}}{w^{\mathrm{T}}}x(0)$$, where $$w^{\mathrm{T}}|\alpha |\mathscr{L}(\mathscr{A})=0$$ and wT1N = 1. Theorem 1 Suppose that the communication network $$\mathscr{G}$$ is strongly connected. If h|αi|di < 1 for all $$i\in \mathscr{V}$$, then the switched multi-agent system (2.1)–(2.2) with protocol (3.1) achieves the asymptotic scaled consensus to (α1, ⋯ , αN) under arbitrary switching. Proof. First, noting that $$e^{-|\alpha |\mathscr{L}(\mathscr{A})}(I_{N}-h|\alpha |\mathscr{L}(\mathscr{A}))=(I_{N}-h|\alpha |\mathscr{L}(\mathscr{A}))e^{-|\alpha |\mathscr{L}(\mathscr{A})}$$, we obtain \begin{align} \alpha x(t)=e^{-|\alpha|\mathscr{L}(\mathscr{A})t_{c}}(I_{N}-h|\alpha|\mathscr{L}(\mathscr{A}))^{t_{d}}\alpha x(0), \end{align} (3.4) where $$t_{c}\in \mathbb{R}$$ and $$t_{d}\in \mathbb{Z}$$ represent the total duration time on continuous-time and discrete-time subsystems, respectively, and t = tc + td. It should be noted that if the agents switch between continuous- and discrete-time systems asynchronously, (3.4) may not be established. It follows from Lemma 3 that $$\lim _{t_{c}\rightarrow \infty }e^{-|\alpha |\mathscr{L}(\mathscr{A})t_{c}}=1_{N}w^{\mathrm{T}}$$ and $$\lim _{t_{d}\rightarrow \infty }(I_{N}-h|\alpha |\mathscr{L}(\mathscr{A}))^{t_{d}}=1_{N}w^{\mathrm{T}}$$, where $$w^{\mathrm{T}}|\alpha |\mathscr{L}(\mathscr{A})=0$$ and wT1N = 1. Next, we consider two cases when $$t\rightarrow \infty$$. If $$t_{c}\rightarrow \infty$$, then \begin{align} \lim_{t\rightarrow\infty}\alpha x(t)=&\lim_{t_{c}\rightarrow\infty}e^{-|\alpha|\mathscr{L}(\mathscr{A})t_{c}}(I_{N}-h|\alpha|\mathscr{L}(\mathscr{A}))^{t_{d}}\alpha x(0)\nonumber\\ =&1_{N}w^{\mathrm{T}}(I_{N}-h|\alpha|\mathscr{L}(\mathscr{A}))^{t_{d}}\alpha x(0)=1_{N}w^{\mathrm{T}}\alpha x(0).\nonumber \end{align} On the other hand, if $$t_{d}\rightarrow \infty$$, then we have \begin{align} \lim_{t\rightarrow\infty}\alpha x(t)=&\lim_{t_{d}\rightarrow\infty}e^{-|\alpha|\mathscr{L}(\mathscr{A})t_{c}}(I_{N}-h|\alpha|\mathscr{L}(\mathscr{A}))^{t_{d}}\alpha x(0)\nonumber\\ =&e^{-|\alpha|\mathscr{L}(\mathscr{A})t_{c}}1_{N}w^{\mathrm{T}}\alpha x(0)=1_{N}w^{\mathrm{T}}\alpha x(0),\nonumber \end{align} since $$e^{-|\alpha |\mathscr{L}(\mathscr{A})t_{c}}=\sum _{k=0}^{\infty }\frac{(-|\alpha |\mathscr{L}(\mathscr{A})t_{c})^{k}}{k!}$$. Therefore, the switched multi-agent system (2.1)–(2.2) with protocol (3.1) achieves the asymptotic scaled consensus to (α1, ⋯ , αN) under arbitrary switching. Remark 3 It is noteworthy that the switched multi-agent system (2.1)–(2.2) offers a unified framework for both the continuous-time scaled consensus problems (Roy, 2015; Shang, 2017b) and the discrete-time scaled consensus problems (Hou et al., 2016). When the scales are taken as αi = 1 for all $$i\in \mathscr{V}$$, we reproduce the consensus protocol for switched multi-agent systems proposed in Zheng & Wang (2016). In the following we extend the result to solve formation generation problem for switched multi-agent systems. In this problem, the aim is to design distributed protocols to guarantee that each pair of neighbouring agents reach a desired relative position with respect to each other (Cao et al., 2013; Xiao et al., 2009; Campos et al., 2016). A certain pattern is thus formed by the agents as a whole. Definition 1 Let $$f=(\ f_{1},\cdots , f_{N})\in \mathbb{R}^{N}$$. The agents in $$\mathscr{G}$$ are said to achieve the scaled formation f with respect to (α1, ⋯ , αN) if $$\lim _{t\rightarrow \infty }(\alpha _{i}x_{i}(t)-\alpha _{j}x_{j}(t))=f_{i}-f_{j}$$ for all $$i,j\in \mathscr{V}$$ and all initial conditions x(0). The above definition corresponds to the asymptotic convergence rate; the finite-time and fixed-time versions can be defined in a similar manner as in Section 2.2. It is easy to see that the agents reach the asymptotic scaled formation f if there exists a vector $$v_{c}\in \mathbb{R}^{N}$$ such that αixi(t) tends to fi + vc as time goes to infinity. To this end, we design the formation generation protocol as follows: \begin{align} u_{i}(t)=\textrm{sgn}(\alpha_{i})\sum_{j=1}^{N}a_{ij}(\alpha_{j}x_{j}(t)-f_{j}-(\alpha_{i}x_{i}(t)-f_{i})), \end{align} (3.5) for both subsystems (2.1) and (2.2). At time instant t, the choice of subsystem is determined by the switching rule under consideration. Corollary 1 Suppose that the communication network $$\mathscr{G}$$ is strongly connected. If h|αi|di < 1 for all $$i\in \mathscr{V}$$, then the switched multi-agent system (2.1)–(2.2) with protocol (3.5) achieves the asymptotic scaled formation f with respect to (α1, ⋯ , αN) under arbitrary switching. Proof. Let $$\bar{x}_{i}=x_{i}-f_{i}/\alpha _{i}$$ for $$i\in \mathscr{V}$$. Then the closed-loop system becomes $$\dot{\bar{x}}_{i}=\textrm{sgn}(\alpha _{i})\sum _{j=1}^{N}a_{ij}$$$$\left (\alpha _{j}\bar{x}_{j}(t)-\alpha _{i}\bar{x}_{i}(t)\right )$$ (continuous-time subsystem) and $$\bar{x}_{i}(t+1)\!=\!\bar{x}_{i}(t)+h\textrm{sgn}(\alpha _{i})\sum _{j=1}^{N}a_{ij}\left (\alpha _{j}\bar{x}_{j}(t)\!-\!\alpha _{i}\bar{x}_{i}(t)\right )$$ (discrete-time subsystem) for $$i\in \mathscr{V}$$. It follows from Theorem 1 that $$\bar{x}_{i}, i\in \mathscr{V}$$ will achieve the asymptotic scaled consensus to (α1, ⋯ , αN) under arbitrary switching. This in turn means that there exists a vector $$v_{c}\in \mathbb{R}^{N}$$ such that $$\lim _{t\rightarrow \infty }\alpha _{i}x_{i}(t)=f_{i}+v_{c}$$. The proof is complete. 3.2. Analysis of finite-time scaled consensus In this section we study the finite-time scaled consensus of switched multi-agent system (2.1)–(2.2) employing protocol (3.2). Let $$\omega =(\omega _{1},\cdots ,\omega _{N})^{\mathrm{T}}\in \mathbb{R}^{N}$$ and $$\omega _{\max }=\max _{1\leqslant i\leqslant N}\omega _{i}$$. Write $$\mu _{\max }=\max _{i,j\in \mathscr{V}}\mu _{ij}\in (0,1)$$, $$x_{\min }(t)=\min _{i\in \mathscr{V}}\alpha _{i}x_{i}(t)$$ and $$x_{\max }(t)=\max _{i\in \mathscr{V}}\alpha _{i}x_{i}(t)$$. Recall that $$\alpha _{\max }=\max _{1\leqslant i\leqslant N}|\alpha _{i}|$$ and we now have a sequence of time instants $$0\leqslant t_{1}<\bar{t}_{1}<t_{2}<\bar{t}_{2}<\cdots <t_{k}<\bar{t}_{k}<\cdots$$. The main result in this section reads as follows. Theorem 2 Suppose that Assumptions 1 and 2 hold, and the communication network $$\mathscr{G}$$ is strongly connected. If $$0<h<(d_{\max }\alpha _{\max })^{-1}$$, then the switched multi-agent system (2.1)–(2.2) with protocol (3.2) achieves the finite-time scaled consensus to (α1, ⋯ , αN) with settling time $$t_{k^{\ast }}$$ and $$k^{\ast}=\left\lceil\frac{4V^{\frac{1-\mu_{\max}}{2}}(\varepsilon(0))}{(1-\mu_{\max})(\rho_{1}\rho_{2})^{\frac{1+\mu_{\max}}{2}}\tau}\right\rceil+1,$$ where $$V(\varepsilon (0))=\frac 12\sum _{i=1}^{N}\omega _{i}\left (\alpha _{i}x_{i}(0)-\frac{\sum _{j=1}^{N}\omega _{j}\alpha _{j}x_{j}(0)}{\sum _{j=1}^{N}\omega _{j}}\right )^{2}$$, $$\rho _{1}=\frac{\min _{i,j\in \mathscr{V},a_{ij}\not =0} (\omega _{i}|\alpha _{i}|a_{ij})^{\frac{2}{\mu _{\max }+1}}}{\sum _{i,j\in \mathscr{V}}(\omega _{i}|\alpha _{i}|a_{ij})^{\frac{2}{\mu _{\max }+1}}}$$ ⋅ $$(x_{\max }(0)-x_{\min }(0))^{\frac{2(\mu _{ij}-\mu _{\max })}{\mu _{\max }+1}}>0$$, $$\rho _{2}=\frac{4K_{0}}{\omega _{\max }}$$, $$K_{0}=\min _{\xi \in U}\xi ^{\mathrm{T}}\mathscr{L}(\mathscr{B}_{0})\xi>0$$, $$\mathscr{B}_{0}=\left [(\omega _{i}|\alpha _{i}|a_{ij})^{\frac{2}{\mu _{\max }+1}}\right ]\in \mathbb{R}^{N\times N}$$ and $$U=\{\xi \in \mathbb{R}^{N}:\omega ^{\mathrm{T}}\xi =0,\|\xi \|_{2}=1\}$$. Proof. Define $$\gamma (t)=\frac{\sum _{i=1}^{N}\omega _{i}\alpha _{i}x_{i}(t)}{\sum _{i=1}^{N}\omega _{i}}$$. Note that $$\omega =(\omega _{1},\cdots ,\omega _{N})^{\mathrm{T}}\in \mathbb{R}^{N}$$ is a positive vector, ωi|αi|aij = ωj|αj|aji by Assumption 1, and μij = μji for all $$i,j\in \mathscr{V}$$. Therefore, we have $$\dot{\gamma}(t)=\frac{\sum_{i=1}^{N}\omega_{i}\alpha_{i}\dot{x}_{i}(t)}{\sum_{i=1}^{N}\omega_{i}}=\frac{1}{\sum_{i=1}^{N}\omega_{i}}\sum_{i=1}^{N}\omega_{i}|\alpha_{i}| \sum_{j=1}^{N}a_{ij}\lfloor\alpha_{j}x_{j}(t)-\alpha_{i}x_{i}(t)\rceil^{\mu_{ij}}=0,$$ and \begin{align} \gamma(t+1)=&\ \frac{1}{\sum_{i=1}^{N}\omega_{i}}\sum_{i=1}^{N}\omega_{i}\alpha_{i} \left(x(t)+h\textrm{sgn}(\alpha_{i})\sum_{j=1}^{N}a_{ij}\left(\alpha_{j}x_{j}(t)-\alpha_{i}x_{i}(t)\right)\right),\nonumber\\ =&\ \gamma(t)+\frac{h}{\sum_{i=1}^{N}\omega_{i}}\sum_{i=1}^{N}\omega_{i}|\alpha_{i}|\sum_{j=1}^{N}a_{ij}(\alpha_{j}x_{j}(t)-\alpha_{i}x_{i}(t))=\gamma(t).\nonumber \end{align} Let εi(t) = αixi(t) − γ(t) for $$i\in \mathscr{V}$$, and ε(t) = (ε1(t), ⋯ , εN(t))T. Recall that α = diag(α1, ⋯ , αN) and hence, ε(t) = αx(t) − γ(t)1N. By definition, we have ωTε(t) = 0 and $$\varepsilon (t+1)=(I_{N}-h|\alpha |\mathscr{L}(\mathscr{A}))\alpha$$$$\cdot x(t)-\gamma (t)1_{N}=(I_{N}-h|\alpha |\mathscr{L}(\mathscr{A}))\varepsilon (t)$$. Define the common Lyapunov function $$V(\varepsilon (t))=\frac 12\varepsilon ^{\mathrm{T}}(t)\Omega \varepsilon (t)$$$$=\frac 12\sum _{i=1}^{N}\omega _{i}{{\varepsilon _{i}^{2}}}(t)$$ for continuous-time subsystem (2.1) and discrete-time subsystem (2.2). For $$t\in (t_{k},\bar{t}_{k}]$$$$(k\geqslant 1)$$, the continuous-time subsystem (2.1) is activated. Since γ(t) is time-invariant, we obtain \begin{align} \dot{V}(\varepsilon(t))=&\sum_{i=1}^{N}\omega_{i}\varepsilon_{i}(t)\alpha_{i}\dot{x}_{i}(t)=\sum_{i=1}^{N}\omega_{i}\varepsilon_{i}(t) |\alpha_{i}|\sum_{j=1}^{N}a_{ij}\lfloor\varepsilon_{j}(t)-\varepsilon_{i}(t)\rceil^{\mu_{ij}}\nonumber\\ =&\frac12\sum_{i,j=1}^{N}\left(\omega_{i}\varepsilon_{i}(t) |\alpha_{i}|a_{ij}\lfloor\varepsilon_{j}(t)-\varepsilon_{i}(t)\rceil^{\mu_{ij}}+ \omega_{j}\varepsilon_{j}(t) |\alpha_{j}|a_{ji}\lfloor\varepsilon_{i}(t)-\varepsilon_{j}(t)\rceil^{\mu_{ji}} \right)\nonumber\\ =&-\frac12\sum_{i,j=1}^{N}\omega_{i}|\alpha_{i}|a_{ij}|\varepsilon_{j}(t)-\varepsilon_{i}(t)|^{\mu_{ij}+1}\nonumber\\ =&-\frac12\sum_{i,j=1}^{N}\left((\omega_{i}|\alpha_{i}|a_{ij})^{\frac{2}{\mu_{\max}+1}}|\varepsilon_{j}(t)-\varepsilon_{i}(t)|^{\frac{2(\mu_{ij}+1)}{\mu_{\max}+1}}\right)^{\frac{\mu_{\max}+1}{2}}\nonumber\\ \leqslant&-\frac12\left(\frac{\sum_{i,j=1}^{N}(\omega_{i}|\alpha_{i}|a_{ij})^{\frac{2}{\mu_{\max}+1}}|\varepsilon_{j}(t)-\varepsilon_{i}(t)|^{\frac{2(\mu_{ij}+1)}{\mu_{\max}+1}}}{2\varepsilon^{\mathrm{T}}(t)\mathscr{L}(\mathscr{B}_{0})\varepsilon(t)}\cdot V(\varepsilon(t))\frac{2\varepsilon^{\mathrm{T}}(t)\mathscr{L}(\mathscr{B}_{0})\varepsilon(t)}{V(\varepsilon(t))}\right)^{\frac{\mu_{\max}+1}{2}}, \end{align} (3.6) thanks to Assumption 1 and Lemma 1. We next estimate each term of (3.6). For $$t\in (\bar{t}_{k-1},t_{k}]$$, it is clear that $$I_{N}-h|\alpha |\mathscr{L}(\mathscr{A})$$ is a stochastic matrix and we write its (i, j) element as $$\bar{l}_{ij}$$. Thus, $$x_{\max }(t+1)=\max _{i\in \mathscr{V}}$$$$\sum _{j=1}^{N}\bar{l}_{ij}\alpha _{j}x_{j}(t)$$$$\leqslant$$$$\max _{i\in \mathscr{V}}\sum _{j=1}^{N}\bar{l}_{ij}$$$$\cdot x_{\max }(t)$$$$=x_{\max }(t)$$ and similarly $$x_{\min }(t+1)\geq x_{\min }(t)$$. For $$t\in (t_{k},\bar{t}_{k}]$$, it follows from (3.2) that $$\dot{x}_{\max }(t)\leqslant 0$$ and $$\dot{x}_{\min }(t)\geqslant 0$$. Hence, $$x_{\max }(t)-x_{\min }(t)$$ is decreasing for all t and $$|\varepsilon _{j}(t)-\varepsilon _{i}(t)|\leqslant |x_{\max }(t)-x_{\min }(t)|\leqslant |x_{\max }(0)-x_{\min }(0)|$$. Noting that \begin{align} &\sum_{i,j=1}^{N}\left(\omega_{i}|\alpha_{i}|a_{ij}\right)^{\frac{2}{\mu_{\max}+1}}|\varepsilon_{j}(t)-\varepsilon_{i}(t)|^{\frac{2(\mu_{ij}+1)}{\mu_{\max}+1}}\nonumber\\ &\quad\geqslant \left(\min_{i,j\in\mathscr{V},a_{ij}\not=0}\left(\omega_{i}|\alpha_{i}|a_{ij}\right)^{\frac{2}{\mu_{\max}+1}}\right)\cdot\sum_{i,j\in\mathscr{V},a_{ij}\not=0}|\varepsilon_{j}(t)-\varepsilon_{i}(t)|^{\frac{2(\mu_{ij}+1)}{\mu_{\max}+1}}\nonumber\\ &\quad\geqslant \left(\min_{i,j\in\mathscr{V},a_{ij}\not=0}\left(\omega_{i}|\alpha_{i}|a_{ij}\right)^{\frac{2}{\mu_{\max}+1}}\right)\cdot\left(\max_{i,j\in\mathscr{V},a_{ij}\not=0}|\varepsilon_{j}(t)-\varepsilon_{i}(t)|\right)^{\frac{2(\mu_{ij}+1)}{\mu_{\max}+1}}\nonumber \end{align} and \begin{align} 2\varepsilon^{\mathrm{T}}(t)\mathscr{L}(\mathscr{B}_{0})\varepsilon(t)=&\sum_{i,j\in\mathscr{V}}(\omega_{i}|\alpha_{i}|a_{ij})^{\frac{2}{\mu_{\max}+1}}(\varepsilon_{j}(t)-\varepsilon_{i}(t))^{2}\nonumber\\ \leqslant& \left(\sum_{i,j\in\mathscr{V}}(\omega_{i}|\alpha_{i}|a_{ij})^{\frac{2}{\mu_{\max}+1}}\right)\cdot \left(\max_{i,j\in\mathscr{V},a_{ij}\not=0}|\varepsilon_{j}(t)-\varepsilon_{i}(t)|\right)^{2},\nonumber \end{align} we deduce that \begin{align} &\frac{\sum_{i,j=1}^{N}\left(\omega_{i}|\alpha_{i}|a_{ij}\right)^{\frac{2}{\mu_{\max}+1}}|\varepsilon_{j}(t)-\varepsilon_{i}(t)|^{\frac{2(\mu_{ij}+1)}{\mu_{\max}+1}}}{2\varepsilon^{\mathrm{T}}(t)\mathscr{L}(\mathscr{B}_{0})\varepsilon(t)}\nonumber\\ &\quad\geqslant \frac{\min_{i,j\in\mathscr{V},a_{ij}\not=0}\left(\omega_{i}|\alpha_{i}|a_{ij}\right)^{\frac{2}{\mu_{\max}+1}}}{\sum_{i,j\in\mathscr{V}}\left(\omega_{i}|\alpha_{i}|a_{ij}\right)^{\frac{2}{\mu_{\max}+1}}}\cdot\left(\max_{i,j\in\mathscr{V},a_{ij}\not=0}|\varepsilon_{j}(t)-\varepsilon_{i}(t)|\right)^{\frac{2\left(\mu_{ij}+1\right)}{\mu_{\max}+1}-2}\geqslant\rho_{1}\nonumber. \end{align} On the other hand, since $$\mathscr{G}$$ is strongly connected, $$K_{0}=\min _{\xi \in U}\xi ^{\mathrm{T}}\mathscr{L}(\mathscr{B}_{0})\xi>0$$. It follows from the definition of U and ωTε(t) = 0 that $$\frac{2\varepsilon^{\mathrm{T}}(t)\mathscr{L}(\mathscr{B}_{0})\varepsilon(t)}{V(\varepsilon(t))}\geqslant\frac{4}{\omega_{\max}}\frac{\varepsilon^{\mathrm{T}}(t)}{\|\varepsilon\|_{2}}\mathscr{L}(\mathscr{B}_{0})\frac{\varepsilon(t)}{\|\varepsilon\|_{2}}\geqslant\rho_{2}.$$ Putting these estimations into (3.6) yields $$\dot{V}(\varepsilon (t))\leqslant -\frac 12(\rho _{1}\rho _{2})^{\frac{\mu _{\max }+1}{2}}V^{\frac{\mu _{\max }+1}{2}}(\varepsilon (t))\leqslant 0$$ for all $$t\in (t_{k},\bar{t}_{k}]$$. A direct application of the comparison theorem (see e.g. Hartman (2002, Theorem 3.1)) gives \begin{align} V^{\frac{1-\mu_{\max}}{2}}(\varepsilon(t))\leqslant-\frac{1-\mu_{\max}}{4}(\rho_{1}\rho_{2})^{\frac{\mu_{\max}+1}{2}}(t-t_{k})+V^{\frac{1-\mu_{\max}}{2}}(\varepsilon(t_{k})) \end{align} (3.7) for all $$t\in [t_{k},\bar{t}_{k}]$$. For $$t\in (\bar{t}_{k-1},t_{k}]$$$$(k\geqslant 2)$$, the discrete-time subsystem (2.2) is activated. By Lemma 2 and the Rayleigh–Ritz theorem (Horn & Johnson, 2012), we obtain \begin{align} V(\varepsilon(t+1))-V(\varepsilon(t))=&\frac12\left(\varepsilon^{\mathrm{T}}(t+1)\varOmega\varepsilon(t+1)-\varepsilon^{\mathrm{T}}(t)\varOmega\varepsilon(t)\right)\nonumber\\ =&\frac12\varepsilon^{\mathrm{T}}(t)\left((I_{N}-h|\alpha|\mathscr{L}(\mathscr{A}))^{\mathrm{T}}\varOmega(I_{N}-h|\alpha|\mathscr{L}(\mathscr{A}))-\varOmega\right)\varepsilon(t)\leqslant0. \end{align} (3.8) It follows from (3.6) and (3.8) that $$V(\varepsilon (0))\geqslant V(\varepsilon (t_{1}))\geqslant V(\varepsilon (\bar{t}_{1}))\geqslant \cdots \geqslant V(\varepsilon (t_{k}))\geqslant V(\varepsilon (\bar{t}_{k}))\geqslant \cdots$$. Hence, using (3.7) we derive \begin{align} V^{\frac{1-\mu_{\max}}{2}}(\varepsilon(t_{k}))\leqslant& V^{\frac{1-\mu_{\max}}{2}}(\varepsilon(\bar{t}_{k-1}))\nonumber\\ \leqslant& -\frac{1-\mu_{\max}}{4}(\rho_{1}\rho_{2})^{\frac{\mu_{\max}+1}{2}}(\bar{t}_{k-1}-t_{k-1})+V^{\frac{1-\mu_{\max}}{2}}(\varepsilon(t_{k-1}))\leqslant\cdots\nonumber\\ \leqslant&-\frac{1-\mu_{\max}}{4}(\rho_{1}\rho_{2})^{\frac{\mu_{\max}+1}{2}}\left(\bar{t}_{k-1}-t_{k-1}+\cdots+\bar{t}_{1}-t_{1}\right)+V^{\frac{1-\mu_{\max}}{2}}(\varepsilon(0)). \end{align} (3.9) Set $$T_{0}=\sum _{i=1}^{k^{\ast }-1}(\bar{t}_{i}-t_{i})$$. We have $$T_{0}\geqslant (k^{\ast }-1)\tau \geqslant \frac{4V^{\frac{1-\mu _{\max }}{2}}(\varepsilon (0))}{(1-\mu _{\max })(\rho _{1}\rho _{2})^{\frac{1+\mu _{\max }}{2}}}$$ by Assumption 2. Taking k = k* in (3.9), we see that $$V(\varepsilon (t_{k^{\ast }}))=0$$. Therefore, for any $$t\geqslant t_{k^{\ast }}$$, V (ε(t)) = 0 and accordingly ε(t) = 0. The proof is complete. Remark 4 Since γ(t) is time-invariant, the final states of the agents are proportional to the weighted average of their initial values, namely, the final states are explicitly expressed. Moreover, it is easy to see from the above proof that the condition $$\bar{t}_{k}-t_{k}\geqslant \tau$$ in Assumption 2 is proposed for the ease of estimation of the settling time. It is not essentially required for the finite-time scaled consensus behaviour. Similar to Corollary 1, we can extend the controller (3.2) to solve the finite-time scaled formation generation problem for the switched multi-agent system (2.1)–(2.2). Take \begin{align} u_{i}(t)=\begin{cases}\textrm{sgn}(\alpha_{i})\sum_{j=1}^{N}a_{ij}\lfloor\alpha_{j}x_{j}(t)-f_{j}-(\alpha_{i}x_{i}(t)-f_{i})\rceil^{\mu_{ij}},& t\in(t_{k},\bar{t}_{k}];\\ \textrm{sgn}(\alpha_{i})\sum_{j=1}^{N}a_{ij}(\alpha_{j}x_{j}(t)-f_{j}-(\alpha_{i}x_{i}(t)-f_{i})),& t\in(\bar{t}_{k-1},t_{k}], \end{cases} \end{align} (3.10) where μij = μji ∈ (0, 1). Corollary 2 Suppose that Assumptions 1 and 2 hold, and the communication network $$\mathscr{G}$$ is strongly connected. If $$0<h<(d_{\max }\alpha _{\max })^{-1}$$, then the switched multi-agent system (2.1)–(2.2) with protocol (3.10) achieves the finite-time scaled formation f with respect to (α1, ⋯ , αN) with settling time $$t_{k^{\ast }}$$, where $$t_{k^{\ast }}$$ is given as in Theorem 2. 3.3. Analysis of fixed-time scaled consensus To achieve fixed-time scaled consensus, we will apply the protocol (3.3) for the switched multi-agent system to remove the dependency of $$t_{k^{\ast }}$$ in Theorem 2 regarding the initial conditions. As in the finite-time case, we follow a sequence of time instants $$0\leqslant t_{1}<\bar{t}_{1}<t_{2}<\bar{t}_{2}<\cdots <t_{k}<\bar{t}_{k}<\cdots$$. The main result in this section reads as follows. Theorem 3 Suppose that Assumptions 1 and 2 hold, and the communication network $$\mathscr{G}$$ is strongly connected. If $$0<h<(d_{\max }\alpha _{\max })^{-1}$$, then the switched multi-agent system (2.1)–(2.2) with protocol (3.3) achieves the fixed-time scaled consensus to (α1, ⋯ , αN) with settling time $$t_{\hat{k}}$$ and $$\hat{k}=\left\lceil\frac{1}{\tau\hat{\rho}_{1}(c_{1}-1)}+\frac{1}{\tau\hat{\rho}_{2}(1-c_{2})}\right\rceil+1,$$ where $$\hat{\rho }_{1}=2^{(2c_{1}-1)}N^{2(1-c_{1})}\left (\frac{K_{1}}{\omega _{\max }}\right )^{c_{1}}$$, $$\hat{\rho }_{2}=2^{(2c_{2}-1)}\left (\frac{K_{2}}{\omega _{\max }}\right )^{c_{2}}$$, $$c_{1}=\frac{m+n}{2n}>1$$, $$c_{2}=\frac{p+q}{2q}<1$$, $$K_{1}=\min _{\xi \in U}$$$$\xi ^{\mathrm{T}}\mathscr{L}(\mathscr{B}_{1})\xi>0$$, $$K_{2}=\min _{\xi \in U}\xi ^{\mathrm{T}}\mathscr{L}(\mathscr{B}_{2})\xi>0$$, $$\mathscr{B}_{1}=\left [(\omega _{i}|\alpha _{i}|a_{ij})^{\frac{1}{c_{1}}}\right ]\in \mathbb{R}^{N\times N}$$, $$\mathscr{B}_{2}=$$$$\left [(\omega _{i}|\alpha _{i}|a_{ij})^{\frac{1}{c_{2}}}\right ]\in \mathbb{R}^{N\times N}$$ and $$U=\{\xi \in \mathbb{R}^{N}:\omega ^{\mathrm{T}}\xi =0,\|\xi \|_{2}=1\}$$. Proof. Since $$\omega =(\omega _{1},\cdots ,\omega _{N})^{\mathrm{T}}\in \mathbb{R}^{N}$$ is a positive vector, ωi|αi|aij = ωj|αj|aji by Assumption 1, and μij = μji for all $$i,j\in \mathscr{V}$$, we can define $$\gamma (t)=\frac{\sum _{i=1}^{N}\omega _{i}\alpha _{i}x_{i}(t)}{\sum _{i=1}^{N}\omega _{i}}$$ and show the time-invariance of γ(t) in the similar manner as in Theorem 2. Let εi(t) = αixi(t) − γ(t) for $$i\in \mathscr{V}$$, and ε(t) = (ε1(t), ⋯ , εN(t))T. Therefore, ε(t) = αx(t) − γ(t)1N. As in Theorem 2 , we have ωTε(t) = 0 and $$\varepsilon (t+1)=(I_{N}-h|\alpha |\mathscr{L}(\mathscr{A}))\varepsilon (t)$$. Define the common Lyapunov function $$V(\varepsilon (t))=\frac 12\varepsilon ^{\mathrm{T}}(t)\Omega \varepsilon (t)=\frac 12\sum _{i=1}^{N}\omega _{i}{{\varepsilon _{i}^{2}}}(t)$$ for continuous-time subsystem (2.1) and discrete-time subsystem (2.2). For $$t\in (t_{k},\bar{t}_{k}]$$$$(k\geqslant 1)$$, the continuous-time subsystem (2.1) is activated. Since γ(t) is time-invariant, we obtain \begin{align} \dot{V}(\varepsilon(t))=&\sum_{i=1}^{N}\omega_{i}\varepsilon_{i}(t)\alpha_{i}\dot{x}_{i}(t)\nonumber\\ =&\sum_{i=1}^{N}\omega_{i}\varepsilon_{i}(t) |\alpha_{i}|\sum_{j=1}^{N}a_{ij}(\varepsilon_{j}(t)-\varepsilon_{i}(t))^{\frac{m}{n}}+ \sum_{i=1}^{N}\omega_{i}\varepsilon_{i}(t) |\alpha_{i}|\sum_{j=1}^{N}a_{ij}(\varepsilon_{j}(t)-\varepsilon_{i}(t))^{\frac{p}{q}}\nonumber\\ =&-\frac12\sum_{i,j=1}^{N}\omega_{i}|\alpha_{i}|a_{ij}(\varepsilon_{j}(t)-\varepsilon_{i}(t))^{\frac{m+n}{n}} -\frac12\sum_{i,j=1}^{N}\omega_{i}|\alpha_{i}|a_{ij}(\varepsilon_{j}(t)-\varepsilon_{i}(t))^{\frac{p+q}{q}}\nonumber\\ =&-\frac12\sum_{i,j=1}^{N}\left((\omega_{i}|\alpha_{i}|a_{ij})^{\frac{2n}{m+n}}\left|\varepsilon_{j}(t)-\varepsilon_{i}(t)\right|{}^{2}\right)^{\frac{m+n}{2n}}-\frac12\sum_{i,j=1}^{N}\left((\omega_{i}|\alpha_{i}|a_{ij})^{\frac{2n}{m+n}}\left|\varepsilon_{j}(t)-\varepsilon_{i}(t)\right|{}^{2}\right)^{\frac{p+q}{2q}}\nonumber\\ \leqslant&-\frac12 N^{\frac{n-m}{n}}\left(\sum_{i,j=1}^{N}(\omega_{i}|\alpha_{i}|a_{ij})^{\frac{2n}{m+n}}\left|\varepsilon_{j}(t)-\varepsilon_{i}(t)\right|{}^{2}\right)^{\frac{m+n}{2n}}\nonumber\\ &\quad-\frac12\left(\sum_{i,j=1}^{N}(\omega_{i}|\alpha_{i}|a_{ij})^{\frac{2q}{p+q}}\left|\varepsilon_{j}(t)-\varepsilon_{i}(t)\right|{}^{2}\right)^{\frac{p+q}{2q}}, \end{align} (3.11) where we have employed Assumption 1 and Lemma 1. Similar to Theorem 2, we can bound (3.11) by noting that $$\frac{\sum_{i,j=1}^{N}(\omega_{i}|\alpha_{i}|a_{ij})^{\frac{2n}{m+n}}|\varepsilon_{j}(t)-\varepsilon_{i}(t)|^{2}}{V(\varepsilon(t))}=\frac{2\varepsilon^{\mathrm{T}}(t)\mathscr{L}(\mathscr{B}_{1})\varepsilon(t)}{V(\varepsilon(t))} \geqslant\frac{4}{\omega_{\max}}\frac{\varepsilon^{\mathrm{T}}(t)}{\|\varepsilon\|_{2}}\mathscr{L}(\mathscr{B}_{1})\frac{\varepsilon(t)}{\|\varepsilon\|_{2}}\geqslant\frac{4K_{1}}{\omega_{\max}}.$$ and analogously, $$\frac{\sum_{i,j=1}^{N}(\omega_{i}|\alpha_{i}|a_{ij})^{\frac{2q}{p+q}}|\varepsilon_{j}(t)-\varepsilon_{i}(t)|^{2}}{V(\varepsilon(t))}=\frac{2\varepsilon^{\mathrm{T}}(t)\mathscr{L}(\mathscr{B}_{2})\varepsilon(t)}{V(\varepsilon(t))} \geqslant\frac{4K_{2}}{\omega_{\max}}$$ Consequently, in light of (3.11), we obtain $$\dot{V}(\varepsilon (t))\leqslant -\hat{\rho }_{1}V^{\frac{m+n}{2n}}(\varepsilon (t))-\hat{\rho }_{2}V^{\frac{p+q}{2q}}(\varepsilon (t))$$ for all $$t\in (t_{k},\bar{t}_{k}]$$. Hence, we have $$\dot{V}(\varepsilon (t))\leqslant -\hat{\rho }_{1}V^{c_{1}}(\varepsilon (t))$$ and $$\dot{V}(\varepsilon (t))\leqslant -\hat{\rho }_{2}V^{c_{2}}(\varepsilon (t))$$. Using the comparison theorem (see e.g. Hartman (2002, Theorem 3.1)) we deduce \begin{align} V^{1-c_{1}}(\varepsilon(t))\geqslant-\hat{\rho}_{1}(1-c_{1})(t-t_{k})+V^{1-c_{1}}(\varepsilon(t_{k})) \end{align} (3.12) and \begin{align} V^{1-c_{2}}(\varepsilon(t))\leqslant-\hat{\rho}_{2}(1-c_{2})(t-t_{k})+V^{1-c_{2}}(\varepsilon(t_{k})) \end{align} (3.13) for all $$t\in [t_{k},\bar{t}_{k}]$$. For $$t\in (\bar{t}_{k-1},t_{k}]$$$$(k\geqslant 2)$$, the discrete-time subsystem (2.2) is activated. We obtain as in Theorem 2 that \begin{align} V(\varepsilon(t+1))-V(\varepsilon(t))\leqslant0. \end{align} (3.14) It follows from (3.11) and (3.14) that $$V(\varepsilon (0))\geqslant V(\varepsilon (t_{1}))\geqslant V(\varepsilon (\bar{t}_{1}))\geqslant \cdots \geqslant V(\varepsilon (t_{k}))\geqslant V(\varepsilon (\bar{t}_{k}))\geqslant \cdots$$. In the sequel, we consider the following two cases: (a) $$V(\varepsilon (0))\leqslant 1$$ and (b) $$V(\varepsilon (0))\geqslant 1$$. For (a), by using (3.13) we obtain \begin{align} V^{1-c_{2}}\left(\varepsilon\left(t_{\hat{k}}\right)\right)\leqslant& V^{1-c_{2}}\left(\varepsilon\left(\bar{t}_{\hat{k}-1}\right)\right)\nonumber\\ \leqslant& -\hat{\rho}_{2}(1-c_{2})\left(\bar{t}_{\hat{k}-1}-t_{\hat{k}-1}\right)+V^{1-c_{2}}\left(\varepsilon\left(t_{\hat{k}-1}\right)\right)\nonumber\\ \leqslant&-\hat{\rho}_{2}(1-c_{2})\left(\bar{t}_{\hat{k}-1}-t_{\hat{k}-1}\right)+V^{1-c_{2}}\left(\varepsilon\left(\bar{t}_{\hat{k}-2}\right)\right)\nonumber\\ \leqslant&-\hat{\rho}_{2}(1-c_{2})\left(\bar{t}_{\hat{k}-1}-t_{\hat{k}-1}+\bar{t}_{\hat{k}-2}-t_{\hat{k}-2}\right)+V^{1-c_{2}}\left(\varepsilon\left(t_{\hat{k}-2}\right)\right)\nonumber\\ \leqslant&\cdots\nonumber\\ \leqslant&-\hat{\rho}_{2}(1-c_{2})\left(\bar{t}_{\hat{k}-1}-t_{\hat{k}-1}+\cdots+\bar{t}_{1}-t_{1}\right)+V^{1-c_{2}}(\varepsilon(0))\leqslant0, \end{align} (3.15) where the last inequality follows from $$T_{0}\geqslant (\hat{k}-1)\tau \geqslant \frac{1}{\hat{\rho }_{1}(c_{1}-1)}+\frac{1}{\hat{\rho }_{2}(1-c_{2})}$$ if we set $$T_{0}:=\sum _{i=1}^{\hat{k}-1}(\bar{t}_{i}-t_{i})$$. Therefore, $$V(\varepsilon (t_{\hat{k}}))=0$$. For any $$t\geqslant t_{\hat{k}}$$, V (ε(t)) = 0 and then ε(t) = 0. The proof in the case of (a) is complete. For (b), by assumption, there must exist $$k^{\ast }\in \mathbb{Z}$$ and $$t_{k^{\ast }}^{\dagger }\in (t_{k^{\ast }},\bar{t}_{k^{\ast }}]$$ such that $$T_{1}:=t_{k^{\ast }}^{\dagger }-t_{k^{\ast }}+\cdots +\bar{t}_{1}-t_{1}\geqslant \frac{1}{\hat{\rho }_{1}(c_{1}-1)}$$ and $$T_{2}:=\bar{t}_{\hat{k}-1}-t_{\hat{k}-1}+\cdots +\bar{t}_{k^{\ast }}-t_{k^{\ast }}^{\dagger }\geqslant \frac{1}{\hat{\rho }_{2}(1-c_{2})}$$. In view of (3.12), we have \begin{align} V^{1-c_{1}}(\varepsilon(t_{1}))\geqslant&\, V^{1-c_{1}}(\varepsilon(0)),\nonumber\\ V^{1-c_{1}}(\varepsilon(\bar{t}_{1}))\geqslant& -\hat{\rho}_{1}(1-c_{1})(\bar{t}_{1}-t_{1})+V^{1-c_{1}}(\varepsilon(t_{1})),\nonumber\\ V^{1-c_{1}}(\varepsilon(t_{2}))\geqslant&\, V^{1-c_{1}}(\varepsilon(\bar{t}_{1})),\nonumber\\ V^{1-c_{1}}(\varepsilon(\bar{t}_{2}))\geqslant& -\hat{\rho}_{1}(1-c_{1})(\bar{t}_{2}-t_{2})+V^{1-c_{1}}(\varepsilon(t_{2})),\nonumber\\ &\cdots\nonumber\\ V^{1-c_{1}}(\varepsilon(t_{k^{\ast}}^{\dagger})) \geqslant&-\hat{\rho}_{1}(1-c_{1})(t_{k^{\ast}}^{\dagger}-t_{k^{\ast}})+V^{1-c_{1}}(\varepsilon(t_{k^{\ast}})).\nonumber \end{align} Adding the above inequalities through, we have $$V^{1-c_{1}}\left(\varepsilon\left(t_{k^{\ast}}^{\dagger}\right)\right) \geqslant-\hat{\rho}_{1}(1-c_{1})\left(t_{k^{\ast}}^{\dagger}-t_{k^{\ast}}+\cdots+\bar{t}_{1}-t_{1}\right)+V^{1-c_{1}}(\varepsilon(0))\geqslant1,$$ since $$T_{1}\geqslant \frac{1}{\hat{\rho }_{1}(c_{1}-1)}$$. As 1 − c1 < 0, we have $$V\left (\varepsilon \left (t_{k^{\ast }}^{\dagger }\right )\right )\leqslant 1$$. Now, we can argue in a similar way as (3.15) to get $$V^{1-c_{2}}(\varepsilon(t_{\hat{k}}))\leqslant -\hat{\rho}_{2}(1-c_{2})\left(\bar{t}_{\hat{k}-1}-t_{\hat{k}-1}+\cdots+\bar{t}_{k^{\ast}}-t_{k^{\ast}}^{\dagger}\right)+V^{1-c_{2}}\left(\varepsilon\left(t_{k^{\ast}}^{\dagger}\right)\right)\leqslant0,$$ where the last inequality follows from $$T_{2}\geqslant \frac{1}{\hat{\rho }_{2}(1-c_{2})}$$. Therefore, $$V\left (\varepsilon \left (t_{\hat{k}}\right )\right )=0$$. For any $$t\geqslant t_{\hat{k}}$$, V (ε(t)) = 0 and then ε(t) = 0. The proof in the case of (b) is complete, which concludes the proof of Theorem 3. Similar comments in Remark 4 can be applied here. Finally, we extend the fixed-time controller (3.3) to solve the fixed-time scaled formation generation problem for the switched multi-agent system (2.1)–(2.2). To this end, we take \begin{align} u_{i}(t)=\begin{cases}\textrm{sgn}(\alpha_{i})\sum_{j=1}^{N}a_{ij}\left(\alpha_{j}x_{j}(t)-f_{j}-(\alpha_{i}x_{i}(t)-f_{i})\right)^{\frac{m}{n}}&\\ \quad+\textrm{sgn}(\alpha_{i})\sum_{j=1}^{N}a_{ij}\left(\alpha_{j}x_{j}(t)-f_{j}-(\alpha_{i}x_{i}(t)-f_{i})\right)^{\frac{p}{q}},& t\in(t_{k},\bar{t}_{k}];\\ \textrm{sgn}(\alpha_{i})\sum_{j=1}^{N}a_{ij}\left(\alpha_{j}x_{j}(t)-f_{j}-(\alpha_{i}x_{i}(t)-f_{i})\right),& t\in(\bar{t}_{k-1},t_{k}], \end{cases} \end{align} (3.16) where m, n, p, q are positive odd integers satisfying m > n and p < q. The following corollary can be shown similarly as Corollaries 1 and 2. Corollary 3 Suppose that Assumptions 1 and 2 hold, and the communication network $$\mathscr{G}$$ is strongly connected. If $$0<h<(d_{\max }\alpha _{\max })^{-1}$$, then the switched multi-agent system (2.1)–(2.2) with protocol (3.16) achieves the fixed-time scaled formation f with respect to (α1, ⋯ , αN) with settling time $$t_{\hat{k}}$$, where $$\hat{k}$$ is given as in Theorem 3. Fig. 1. View largeDownload slide Communication topology $$\mathscr{G}$$ for Example 1. Fig. 1. View largeDownload slide Communication topology $$\mathscr{G}$$ for Example 1. Remark 5 The results of Theorem 3 and Corollary 3 can be generalized to accommodate double channel topology with adjacency matrices being $$(a_{ij})\in \mathbb{R}^{N\times N}$$ and $$(b_{ij})\in \mathbb{R}^{N\times N}$$ for the continuous-time subsystem. For example, in Theorem 3, the control input for the continuous-time subsystem can be designed as $$u_{i}(t)=\textrm{sgn}(\alpha_{i})\sum_{j=1}^{N}a_{ij}\left(\alpha_{j}x_{j}(t)-\alpha_{i}x_{i}(t)\right)^{\frac{m}{n}} +\textrm{sgn}(\alpha_{i})\sum_{j=1}^{N}b_{ij}\left(\alpha_{j}x_{j}(t)-\alpha_{i}x_{i}(t)\right)^{\frac{p}{q}},\quad t\in(t_{k},\bar{t}_{k}]$$ for $$i\in \mathscr{V}$$. And the settling time can be given similarly with Theorem 3. It is also noteworthy that the theoretical results can be directly extended to multi-agent systems with multidimensional dynamics by use of the properties of the Kronecker product. Fig. 2. View largeDownload slide (a) Switching law of the switched multi-agent system (2.1)–(2.2) with (b) asymptotic protocol (3.1), (c) finite-time protocol (3.2) and (d) fixed-time protocol (3.3) for Example 1. Fig. 2. View largeDownload slide (a) Switching law of the switched multi-agent system (2.1)–(2.2) with (b) asymptotic protocol (3.1), (c) finite-time protocol (3.2) and (d) fixed-time protocol (3.3) for Example 1. 4. Numerical simulation In this section, some simulation examples are provided to illustrate the theoretical results. Example 1 We consider a strongly connected network $$\mathscr{G}$$ with N = 5 nodes as shown in Fig. 1. We assume ω = (1, 1, 2, 1, 1)T, α = diag(1, −2, 1, 2, −1) and a periodic switching law with τ = 5 (see Fig. 2(a)). It is easy to check that Assumptions 1 and 2 hold. Furthermore, for the finite-time protocol (3.2), we take $$\mu _{ij}=\frac 12$$ for all i, j, and for the fixed-time protocol (3.3), we take m = 7, n = 5, p = 3, q = 5. The gain is chosen as h = 0.1. These designed parameters satisfy the conditions in Theorems 1, 2 and 3. For the initial condition x(0) = (5, −1, −3, 4, 1)T, the state trajectories of all the agents are shown in Fig. 2(b) (using the asymptotic scaled consensus protocol (3.1)), Fig. 2(c) (using the finite-time scaled consensus protocol (3.2)) and Fig. 2(d) (using the fixed-time scaled consensus protocol (3.3)), respectively. We observe that in all these three situations, scaled consensus is achieved as one would expect from Theorems 1 , 2, and 3. Moreover, the theoretical estimates of the settling time is $$t_{k^{\ast }}=605$$ (with k* = 121) for the finite-time scaled consensus and $$t_{\hat{k}}=80$$ (with $$\hat{k}=16$$) for the fixed-time scaled consensus. Obviously, the actual settling time is less than the theoretical estimates in these control protocols, respectively. It is remarkable that the estimate for fixed-time scaled consensus is much sharper than that for finite-time scaled consensus. This is because the estimation for (3.11) is more refined than that for (3.6). Fig. 3. View largeDownload slide Communication topology $$\mathscr{G}$$ for Example 2. Fig. 3. View largeDownload slide Communication topology $$\mathscr{G}$$ for Example 2. Fig. 4. View largeDownload slide (a) Switching law of the switched multi-agent system (2.1)–(2.2) with (b) asymptotic protocol (3.1), (c) finite-time protocol (3.2) and (d) fixed-time protocol (3.3) for Example 2. Fig. 4. View largeDownload slide (a) Switching law of the switched multi-agent system (2.1)–(2.2) with (b) asymptotic protocol (3.1), (c) finite-time protocol (3.2) and (d) fixed-time protocol (3.3) for Example 2. Example 2 In this example, we consider a strongly connected network $$\mathscr{G}$$ with N = 4 nodes as shown in Fig. 3. Note that Assumption 1 is no longer applicable; c.f. Remark 1. We take α = diag(2, 1, 1, −1), and a periodic switching law with τ = 10 (see Fig. 4(a)). As in Example 1, for the finite-time protocol (3.2), we take $$\mu _{ij}=\frac 12$$ for all i, j, and for the fixed-time protocol (3.3), we take m = 7, n = 5, p = 3, q = 5. The gain is chosen as h = 0.1. For the initial condition x(0) = (2, −1, −4, 1)T, the state trajectories of all the agents are shown in Fig. 4(b) (using the asymptotic scaled consensus protocol (3.1)), Fig. 4(c) (using the finite-time scaled consensus protocol (3.2)) and Fig. 4(d) (using the fixed-time scaled consensus protocol (3.3)), respectively. Although Fig. 4(b) displays the asymptotic scaled consensus as one would expect, the rapid convergence behaviour shown in Fig. 4(c) and (d) is remarkable. It implies that the detailed balance condition, i.e. Assumption 1, on the network topology is not necessary in general to guarantee finite-time/fixed-time scaled convergence for switched multi-agent systems. 5. Conclusion In this paper, we have considered the scaled consensus problems of a switched multi-agent system composed of both continuous-time and discrete-time subsystems. By employing the nearest neighbour-interaction rules, we propose three consensus protocols that possess asymptotic, finite-time and fixed-time convergence rates, respectively. The final consensus states are explicitly presented, and generalizations to scaled formation generation are also studied. For future work, we will focus on relaxation of topology condition (Assumption 1) and scaled consensus of switched multi-agent systems with switching topologies. Some other challenging future directions include the analysis of switched multi-agent systems with higher-order dynamics and robustness against communication delays. Acknowledgements The author would like to thank the reviewers for their insightful comments and careful reading. Part of the work has been done during a visit of the author to the Department of Mathematical Sciences at University of Essex. The author would like to thank its hospitality. Funding National Natural Science Foundation of China (11505127), the Shanghai Pujiang Program (15PJ1408300) and the Program for Young Excellent Talents in Tongji University (2014KJ036). References Altafini , C. ( 2013 ) Consensus problems on networks with antagonistic interactions . IEEE Trans. Autom. Control , 58 , 935 -- 946 . Google Scholar CrossRef Search ADS Bhat , S. P. & Bernstein , D. S. ( 2000 ) Finite-time stability of continuous autonomous systems . SIAM J. Control Optim. , 38 , 751 -- 766 . Google Scholar CrossRef Search ADS Campos , G. , Dimarogonas , D. V. , Seuret , A. & Johansson , K. H. ( 2016 ) Distributed control of compact formations for multi-robot swarms . IMA J. Math. Control Inform. , 32 , 1 -- 31 . Cao , Y. , Yu , W. , Ren , W. & Chen , G. ( 2013 ) An overview of recent progress in the study of distributed multi-agent coordination . IEEE Trans. Ind. Inf. , 9 , 427 -- 438 . Google Scholar CrossRef Search ADS Chu , T. , Zhang , C. & Zhang , Z. ( 2003 ) Necessary and sufficient condition for absolute stability of normal neural networks . Neural Networks , 16 , 1223 -- 1227 . Google Scholar CrossRef Search ADS PubMed Defoort , M. , Polyakov , A. , Demesure , G. , Djemai , M. & Veluvolu , K. ( 2015 ) Leader-follower fixed-time consensus for multi-agent systems with unknown non-linear inherent dynamics . IET Control Theory Appl ., 9 , 2165 -- 2170 . Google Scholar CrossRef Search ADS Fu , J. & Wang , J. ( 2016 ) Fixed-time coordinated tracking for second-order multi-agent systems with bounded input uncertainties . Syst. Control Lett. , 93 , 1 -- 12 . Google Scholar CrossRef Search ADS Guglieri , G. , Maroglio , F. , Pellegrino , P. & Torre , L. ( 2014 ) Design and development of guidance navigation and control algorithms for spacecraft rendezvous and docking experimentation . Acta Astronautica , 94 , 395 -- 408 . Google Scholar CrossRef Search ADS Haken , H. ( 1978 ) Synergetics . Berlin : Springer . Google Scholar CrossRef Search ADS Hardy , G. , Littlewood , J. & Polya , G. ( 1952 ) Inequalities , 2nd ed. New York : Cambridge University Press . Hartman , P. ( 2002 ) Ordinary Differential Equations . Philadelphia : SIAM . Google Scholar CrossRef Search ADS Horn , R. A. & Johnson , C. R. ( 2012 ) Matrix Analysis . Cambridge : Cambridge University Press . Google Scholar CrossRef Search ADS Hou , B. , Sun , F. , Li , H. , Chen , Y. & Liu , G. ( 2016 ) Scaled cluster consensus of discrete-time multi-agent systems with general directed topologies . Int. J. Control , 47 , 3839 -- 3845 . Jadbabaie , A. , Lin , J. & Morse , A. S. ( 2003 ) Coordination of groups of mobile autonomous agents using nearest neighbor rules . IEEE Trans. Autom. Control , 48 , 988 -- 1001 . Google Scholar CrossRef Search ADS Li , J. , Dong , W. & Xiao , H. ( 2017 ) Signed consensus problems on networks of agents with fixed and switching topologies . Int. J. Control , 90 , 148 -- 160 . Google Scholar CrossRef Search ADS Lin , X. & Zheng , Y. ( 2017 ) Finite-time consensus of switched multiagent systems . IEEE Trans. Syst. Man Cybern. Syst. , 47 , 1535 -- 1545 . Google Scholar CrossRef Search ADS Meng , D. & Jia , Y. ( 2016a ) Robust consensus algorithms for multiscale coordination control of multivehicle systems with disturbances . IEEE Trans. Ind. Electron. , 63 , 1107 -- 1119 . Google Scholar CrossRef Search ADS Meng , D. & Jia , Y. ( 2016b ) Scaled consensus problems on switching networks . IEEE Trans. Autom. Control , 61 , 1664 -- 1669 . Google Scholar CrossRef Search ADS Mesbahi , M. & Egerstedt , M. ( 2010 ) Graph Theoretic Methods in Multiagent Networks . New Jersey : Princeton University Press . Google Scholar CrossRef Search ADS Olfati-Saber , R. , Fax , J. A. & Murray , R. M. ( 2007 ) Consensus and cooperation in networked multi-agent systems . Proc. IEEE , 95 , 215 -- 233 . Google Scholar CrossRef Search ADS Parsegov , S. , Polyakov , A. & Shcherbakov , P. ( 2013 ) Fixed-time consensus algorithm for multi-agent systems with integrator dynamics . Proc. 4th IFAC Workshop on Distributed Estimation and Control in Networked Systems , 110 -- 115 . Polyakov , A . ( 2012 ) Nonlinear feedback design for fixed-time stabilization of linear control systems . IEEE Trans. Autom. Control , 57 , 2106 -- 2110 . Google Scholar CrossRef Search ADS Ren , W. , Beard , R. W. & Atkins , E. M. ( 2007 ) Information consensus in multivehicle cooperative control . IEEE Control Syst. Mag. , 27 , 71 -- 82 . Google Scholar CrossRef Search ADS Roy , S . ( 2015 ) Scaled consensus . Automatica , 51 , 259 -- 262 . Google Scholar CrossRef Search ADS Rugh , W. J. ( 1996 ) Linear System Theory . New Jersey : Prentice Hall . Shang , Y. ( 2016a ) A combinatorial necessary and sufficient condition for cluster consensus . Neurocomputing , 216 , 611 -- 616 . Google Scholar CrossRef Search ADS Shang , Y. ( 2016b ) Finite-time weighted average consensus and generalized consensus over a subset . IEEE Access , 4 , 2615 -- 2620 . Google Scholar CrossRef Search ADS Shang , Y. ( 2017a ) Finite-time scaled consensus through parametric linear iterations . Int. J. Syst. Sci. , 48 , 2033 -- 2040 . Google Scholar CrossRef Search ADS Shang , Y. ( 2017b ) On the delayed scaled consensus problems . App. Sci. , 7 , art. no. 713 . Shang , Y. ( 2017c ) Subspace confinement for switched linear systems . Forum Math. , 29 , 693 -- 699 . Google Scholar CrossRef Search ADS Sun , S. H. , Zhao , L. & Jia , Y. M. ( 2014 ) Similitude design method for motion reconstruction of space cooperative vehicles . J. Astronaut. , 35 , 802 -- 810 . Wang , L. & Xiao , F. ( 2010 ) Finite-time consensus problems for networks of dynamic agents . IEEE Trans. Autom. Control , 55 , 950 -- 955 . Google Scholar CrossRef Search ADS Xiao , F. , Wang , L. , Chen , J. & Gao , Y. ( 2009 ) Finite-time formation control for multi-agent systems . Automatica , 45 , 2605 -- 2611 . Google Scholar CrossRef Search ADS Zhai , G. , Liu , D. , Imae , J. & Kobayashi , T. ( 2006 ) Lie algebraic stability analysis for switched systems with continuous-time and discrete-time subsystems . IEEE Trans. Circuits Syst. II Exp. Briefs , 53 , 152 -- 156 . Google Scholar CrossRef Search ADS Zheng , Y. & Wang , L. ( 2016 ) Consensus of switched multiagent systems . IEEE Trans. Circuits Syst. II Exp. Briefs , 63 , 314 -- 318 . Google Scholar CrossRef Search ADS Zhu , Y. , Zheng , Y. & Wang , L. ( 2015 ) Containment control of switched multi-agent systems . Int. J. Control , 88 , 2570 -- 2577 . Google Scholar CrossRef Search ADS © The Author(s) 2018. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

### Journal

IMA Journal of Mathematical Control and InformationOxford University Press

Published: Feb 1, 2018

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month ### Explore the DeepDyve Library ### Search Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly ### Organize Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place. ### Access Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals. ### Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. DeepDyve ### Freelancer DeepDyve ### Pro Price FREE$49/month
\$360/year

Save searches from
PubMed

Create lists to

Export lists, citations