Containment control of fractional-order nonlinear multi-agent systems under fixed topologies

Containment control of fractional-order nonlinear multi-agent systems under fixed topologies Abstract This article investigates the containment control problem of fractional-order nonlinear multi-agent systems under fixed topologies. The containment control problem of multi-agent systems is first transformed to the stability problem of fractional-order nonlinear systems. A distributed protocol is then proposed. By using the Lyapunov function method of fractional-order systems, it is proved that under the designed protocol, the followers can asymptotically converge to a dynamic convex hull spanned by multiple leaders. Numerical simulations are provided to illustrate the effectiveness of the proposed protocols. 1. Introduction In recent years, cooperative control of multi-agent systems has received significant attention due to its broad applications in many fields (see Caridi & Sianesi, 2000; Chen et al., 2015; Leonard et al., 2007). There are many hot topics such as formation control, consensus, rendezvous, containment control and distributed tracking, closely related to cooperative control (see Olfati-Saber & Murray, 2004; Xiao et al., 2009; Yan et al., 2011; Dong & Huang, 2013; Zhao et al., 2013; Yan & Xie, 2014; Hu et al., 2015). Distributed cooperative control means achieving control objectives of agents with local mutual effect between the individuals (see Murry, 2007; Ren & Atkins, 2007). In the consensus problem, leader-following consensus, which has been studied by many researchers, is an important topic (see Peng & Yang, 2009; Wang & Ji, 2012; Sun & Guan, 2013; Zhang & Yang, 2013; Cao et al., 2015; Du et al., 2016). In Peng & Yang (2009), the leader-following consensus problem with time-varying delays was solved. In Cao et al. (2015), the leader-following consensus of linear multi-agent systems with unknown external disturbances was investigated. A leader-following rendezvous problem of double integrator multi-agent systems was studied in Dong & Huang (2013). The leader-following finite-time consensus problem was considered in Du et al. (2015). Recently, the leader-following consensus with multiple leaders, which is called containment control problem, was studied (see Li et al., 2013, 2016; He et al., 2014; Haghshenas et al., 2015; Chu et al., 2016; Hu et al., 2016). The main objective of the containment control is to design a protocol such that all the followers can converge to the convex hull spanned by the leaders. The containment control problem for multi-agent systems with communication delay was considered in Hu et al. (2016) and Li et al. (2016). The finite-time containment control for second-order multi-agent systems under directed topology was studied in He et al. (2014). The containment control of heterogeneous linear multi-agent systems was presented in Haghshenas et al. (2015). In Chu et al. (2016), an output regulation approach was provided to solve the containment control problem of heterogeneous linear multi-agent systems. In reality, many physical systems are nonlinear in nature (see Huang et al., 2005; Cao et al., 2016a), so there are many results on nonlinear multi-agent systems such as Cao et al. (2016b), Du et al. (2014) and Wan et al. (2016). It is noted that most of the aforementioned results on cooperative control are related to integer-order dynamics. However, many real-world phenomena in biology, physics, etc. are better described by fractional-order dynamics rather than integer-order dynamics (see Bagley & Torvik, 1983; Perdikaris & Karniadakis, 2014; Taher et al., 2014). The distributed coordination of networked fractional-order systems was firstly studied in Cao et al. (2010). The consensus protocol was designed for agents with fractional-order uncertain dynamics in Yin & Hu (2013), where the fractional order is $$0 < \alpha < 2$$. Chen et al. (2016) investigated the containment control of linear uncertain fractional-order multi-agent systems. In Yin et al. (2013), the consensus protocols are designed for a set of fractional-order heterogeneous agents, which is composed of two kinds of agents differed by their dynamics. The cooperative control for fractional-order multi-agent systems with communication delay was studied in Yang et al. (2014) and Shen & Cao (2012). However, to the best of our knowledge, the containment control of fractional-order nonlinear multi-agent systems has not been investigated to date, which motivates the current study of this article. In this article, we will present a protocol for fractional-order nonlinear multi-agent systems. By adopting the Lyapunov function method of fractional-order systems, it will be proved that under some conditions, the designed protocol can make the followers converge to a dynamic convex hull spanned by multiple leaders asymptotically. The containment control problem with multi-agent systems under directed topologies is then considered. As a special case, when there is only one leader, the leader-following consensus of the multi-agent system can be achieved by the proposed protocol. The rest of this article is organized as follows. The necessary preliminaries and the problem formulation are given in section 2. In section 3, the main results of this article are presented. Some numerical simulations are given to illustrate the validity of the theoretical results in section 4. Finally, some conclusions are drawn in section 5. Notations Throughout this article, $$\mathbf{R}$$ stands for the set of all real numbers. $${\mathbf{R}^n}$$ denotes the $$n$$-dimensional real vector space and $${{\mathbf{R}}^{m \times n}}$$ is the set of $$m \times n$$ matrices. Symbol $$\otimes$$ represents the Kronecker product. The shorthand $$diag\left\{ \cdots \right\}$$ denotes the block diagonal matrix. $${\mathbf{1}}$$ and $${\mathbf{0}}$$ represent the vector with all entries being one and zero with compatible dimensions, respectively. $${I}$$ is an identity matrix with compatible dimensions. $${A^T}$$ denotes the transpose of $$A$$ while $${A^{ - 1}}$$ denotes the inverse of $$A$$, where $$A \in {{\mathbf{R}}^{n \times n}}$$. $${\lambda _{\min }}(A)$$ and $${\lambda _{\max }}(A)$$ denote the minimum eigenvalue and maximum eigenvalue, respectively. $$A > 0(A \geqslant 0)$$ if matrix $$A$$ is positive definite(positive semi-definite). Denote by $$dis\left( {x,C} \right)$$ the distance from $$x \in {{\mathbf{R}}^N}$$ to the set $$C \subseteq {{\mathbf{R}}^N}$$ in the sense of Euclidean norm, that is $$dis\left( {x,C} \right) = \mathop {\inf }\limits_{y \in C} {\left\| {x - y} \right\|}$$, where $$\left\| \cdot \right\|$$ is the Euclidean norm operator. 2. Preliminaries and problem formulation In the following, some basic preliminaries about graph theory will be given. A graph consists of a vertex set $$V\left( G \right) = \left\{ {{v_1}, \cdots ,{v_n}} \right\}$$, an edge set $$E\left( G \right) \subseteq V \times V$$ and an adjacency matrix $$A$$. An edge $$({v_i},{v_j}) \in E\left( G \right)$$ denotes that the vertex $${v_j}$$ can obtain the state of vertex $${v_i}$$. If $$({v_i},{v_j}) \in E\left( G \right)$$, $${v_i}$$ is called a neighbour of $${v_j}$$ and the set of all neighbours can be denoted by $${N_j} = \left\{ {{v_i}:({v_i},{v_j}) \in E\left( G \right)} \right\}$$. The adjacency matrix of $$G$$ is defined as $$A = {[{a_{ij}}]_{n \times n}} \in {{\mathbf{R}}^{n \times n}}$$, with $${a_{ij}} > 0$$ if and only if $$({v_j},{v_i}) \in E\left( G \right)$$ and $${a_{ij}} = 0$$ otherwise. In undirected graph,$$({v_i},{v_j}) \in E\left( G \right)$$ implies that $$({v_j},{v_i}) \in E\left( G \right)$$, but in directed graph, it does not hold. The degree matrix $$D$$ is a diagonal matrix and its $$i$$ th element is equal to $$\left| {{N_i}} \right|$$ which is the cardinality of $${N_i}$$, and we call $$\left| {{N_i}} \right|$$ the degree of $$i$$ th agent. The Laplace matrix of graph $$G$$ is defined as $$L = D - A$$. A path on $$G$$ from $${v_{{i_0}}}$$ to $${v_{{i_k}}}$$ is a sequence of distinct vertices $$\left( {{v_{{i_0}}}, \cdots ,{v_{{i_k}}}} \right)$$ where $$\left( {{v_{{i_{j - 1}}}},{v_{{i_j}}}} \right) \in E\left( G \right) for j = 1, \cdots ,k$$ (see Godsil & Royle, 2001). The communication graph among all agents can be described by $$G = \left( {V,E,A} \right)$$. Agents are abstracted as vertexes of graph and communications between agents as edges. Before giving the system model, we will review definitions of fractional calculation and some useful lemmas. There are several different definitions of fractional operators, such as Caputo and Riemann-Liouville fractional operators. In this article, Caputo operator will be adopted and the definition for fractional derivative of function $$h(t)$$ with order $$\alpha$$ is given as    t0CDtαh(t)=1Γ(α¯−α)∫t0th(α¯)(τ)(t−τ)α−α¯+1dτ, (2.1) where $$\bar \alpha$$ is the integer satisfying $$\bar \alpha - 1 < \alpha \leqslant \bar \alpha$$, $$\Gamma (x) = \int_0^\infty {{t^{x - 1}}{e^{ - t}}dt}$$ is the Gamma function, and $${t_0}$$ is the initial time. An important function frequently used in the solutions of fractional-order systems is the Mittag–Leffler function. The Mittag–Leffler function with two parameters is defined as follows (see Podlubny, 1999):   Eβ,γ(z)=∑k=0∞zkΓ(kβ+γ), (2.2) where $$\beta > 0$$, $$\gamma > 0$$. As a special case, when $$\gamma = 1$$, the Mittag–Leffler function with one parameter is defined as   Eβ(z)=∑k=0∞zkΓ(kβ+1), (2.3) where $$\beta > 0$$. Furthermore, it is noted that if $$\beta = \gamma = 1$$, $${E_{\beta ,\gamma }}\left( z \right) = {e^z}$$. We now give a property about Mittag–Leffler function with one parameter as follows. Property 1 (See Diethelm, 2010) Let $$0 < \beta \leqslant 2$$, the Mittag-Leffler function $${E_\beta }( \cdot )$$ behaves as (1) $${E_\beta }\left( {r{e^{i\phi }}} \right) \to 0$$ for $$r \to \infty $$ if $$\left| \phi \right| > \frac{{\beta \pi }}{2}$$. (2) $${E_\beta }\left( {r{e^{i\phi }}} \right)$$ remains bouned for $$r \to \infty $$ if $$\left| \phi \right| = \frac{{\beta \pi }}{2}$$. (3) $$\left| {{E_\beta }\left( {r{e^{i\phi }}} \right)} \right| \to \infty $$ for $$r \to \infty $$ if $$\left| \phi \right| < \frac{{\beta \pi }}{2}$$. The following lemmas related to Caputo operator will be used to obtain our main results. Lemma 2.1 (see Duarte-Mermoud et al., 2015) Let $$x(t) \in {{\mathbf{R}}^n}$$ be a vector of differentiable function. Then, for any time instant $$t > {t_0}$$, the following relationship holds   12 t0CDtα(xT(t)Mx(t))⩽xT(t)M t0CDtαx(t),∀α∈(0,1], (2.4) where $$M \in {{\mathbf{R}}^{n \times n}}$$ is a symmetric and positive definite matrix. Lemma 2.2 (see Liu et al., 2016) Let $$x(t) \in {{\mathbf{R}}^n}$$ be a vector of differentiable function. If a continuous function $$V:\left[ {{t_0},\infty } \right) \times {{\mathbf{R}}^n} \to {\mathbf{R}}$$ satisfies $$_{{t_0}}^CD_t^\beta V\left( {t,x\left( t \right)} \right) \leqslant - \alpha V\left( {t,x\left( t \right)} \right)$$, then   V(t,x(t))⩽V(t0,x(t0))Eβ(−α(t−t0)β), (2.5) where $$0 < \beta \leqslant 1$$, $$\alpha$$ is a positive constant. Consider a fraction-order multi-agent system composed by $$n$$ agents which include $$m$$ followers and $$\left( {n - m} \right)$$ leaders. The dynamics of agents are given as follows:    t0CDtαxi =ui+f(t,xi),i=1,⋯,m, t0CDtαvi =u0,i+f(t,vi),i=m+1,⋯,n, (2.6) where $$0 < \alpha \leqslant 1$$. $${x_i} \in {{\mathbf{R}}^N}$$ and $${u_i} \in {{\mathbf{R}}^N}$$ are the state and control input of the $$i$$ th follower, respectively. $${v_i} \in {{\mathbf{R}}^N}$$ is the state of the $$i$$ th leader and $${u_{0,i}}$$ is the preset input of the $$i$$ th leader. $$f( \cdot )\in {{\mathbf{R}}^N}$$ is an unknown nonlinear continuous vector-function with $$f(t,0) = 0$$. Remark 2.1 When $$\alpha = 1$$, (2.6) is reduced to an integer-order nonlinear multi-agent system like Mei et al. (2011). In this article, we take the unknown nonlinear uncertainties into consideration, if $$f(t,x) \equiv 0$$ and there is no leader, the model is the same as the model in Cao et al. (2010). Note that the unknown nonlinear uncertainties commonly exist in actual systems. Assumption 1 (Mei et al. (2011)) The nonlinear vector-function $$f( \cdot )$$ satisfies the following condition: Given $${\xi _1}, \cdots ,{\xi _{n - m}}$$, with $$\sum\limits_{i = 1}^{n - m} {{\xi _i} = 1}$$ and $${\xi _i} \geqslant 0,\;i = 1, \cdots ,n - m$$, there exists a positive constant $$\rho$$ such that for $$x,{y_i} \in {{\mathbf{R}}^N},\;i = 1, \cdots ,n - m$$,   ‖f(t,x)−∑i=1n−mξif(t,yi)‖⩽ρ‖x−∑i=1n−mξiyi‖. (2.7) Remark 2.2 In the leader-following consensus problem where there is only one leader, condition (2.7) becomes to the Lipschitz condition:   ‖f(t,x)−f(t,y)‖⩽ρ‖x−y‖,∀x,y∈RN. (2.8) Assumption 2 The communication topology among followers is undirected and there is no path from followers to leaders. While for each follower, there exists at least one leader that has a directed path to that follower. Under Assumption 2, the Laplace matrix of $$G$$ can be written as   L =(L1L20(n−m)×m0(n−m)×(n−m)), where $${L_1} \in {{\mathbf{R}}^{m \times m}}, {L_2} \in {{\mathbf{R}}^{m \times (n - m)}}$$. A lemma about $$L$$ will be given as follows: Lemma 2.3 (see Meng et al. (2010) Under Assumption 2, $${L_1}$$ is positive definite and symmetric. Each entry of $$- L_1^{ - 1}{L_2}$$ is nonnegative and each row sum of $$- L_1^{ - 1}{L_2}$$ is equal to one. Remark 2.3 Assumption 2 is commonly adopted in the literatures such as Zhang & Yang (2013) and Meng et al. (2010). To achieve the containment control goal, directly or indirectly connection with leader set is essential. Otherwise, sub-systems will be formed and the problem might fail to be solved. Definition 2.1 (see Rockafellar et al. (1972) A set $$C \subseteq {{\mathbf{R}}^N}$$ is a convex if $$(1 - \lambda )x + \lambda y \in C$$, for any $$x,y \in C$$ and $$\lambda \in \left[ {0,1} \right]$$. The convex hull $$Co\left( X \right)$$ of a finite set of points $$X = \left\{ {{x_1},{x_2}, \cdots ,{x_q}} \right\}$$ is defined like convex and is described as follows:   Co(X)={∑i=1qαixi|xi∈X,αi∈R,αi⩾0,∑i=1qαi=1 }. (2.9) Definition 2.2 (see Ji et al. (2008)) The containment control of fractional-order system (2.6) is achieved by the designed protocol if and only if $$\forall i \in {V_F} = \left\{ {1,2, \cdots ,m} \right\}$$,   limt→∞⁡dis(xi(t),Co(vk(t),k∈VS))=0, (2.10) where $${V_S} = \left\{ {m + 1, \cdots ,n} \right\}$$. Lemma 2.4 (see Hardy et al. (1952)) Give any scalar $$\varepsilon > 0$$, $$x \in {{\mathbf{R}}^n}$$, $$y \in {{\mathbf{R}}^n}$$, then it holds that   xTy+yTx⩽εxTx+1εyTy. 3. Main results In this section, the main results of this article will be given. In this article, we design a protocol for multi-agent systems (2.6) as follows:   ui=K(∑j=1maij(xj−xi)+∑j=m+1naij(vj−xi))+(Λi⊗IN)u0, (3.1) where $$K$$ is a matrix to be designed, $${\it\Lambda _i}$$ is the $$i$$ th row vector of $$-L_1^{ - 1}{L_2}$$, $${u_0}$$ is the column stack of $${u_{0i}}$$. Theorem 3.1 Under Assumptions 1 and 2, the containment control of multi-agent system (2.6) with the protocol (3.1) can be achieved if there exist a positive definite symmetric $$P \in {{\mathbf{R}}^{N \times N}}$$, a positive scalar $$\varepsilon$$ and a matrix $$K$$ such that $$PK + {K^T}P$$ is positive definite and   Q1=λmin(L1)λmin(PK+KTP)I−(ε(Im⊗P2)+ρ2εI)>0. (3.2) Proof. Let $${x_{_F}} = {\left( {x_1^T, \cdots ,x_m^T} \right)^T}$$ and $$v = {\left( {v_{m + 1}^T, \cdots ,v_n^T} \right)^T}$$. Define $$e = {x_F} - \left( { - \left( {L_1^{ - 1}{L_2} \otimes {I_N}} \right)v} \right)$$, then it follows from (2.6) that    t0CDtαe = t0CDtαxF+(L1−1L2⊗IN) t0CDtαv =u+Φ+(L1−1L2⊗IN)u0+(L1−1L2⊗IN)Φ0, (3.3) where $$u,\;{u_0},\;{\it}\Phi$$ and $$\;{{\it}\Phi _0}$$ are column stacks of $${u_i},\;{u_{0i}},\;f(t,{x_i})$$ and $$\;f(t,{v_i})$$, respectively. Utilizing (3.1) and (3.3) yields    t0CDtαe = −(L1⊗K)xF−(L2⊗K)v+Φ+(L1−1L2⊗IN)Φ0 = −(L1⊗K)e+Φ+(L1−1L2⊗IN)Φ0. (3.4) Consider a Lyapunov function candidate $$V = {e^T}\left( {{I_m} \otimes P} \right)e$$. From (3.4) and Lemma 2.1, we can get    t0CDtαV ⩽2eT(Im⊗P)(−(L1⊗K)e+Φ+(L1−1L2⊗IN)Φ0)  = −eT(L1⊗(PK+KTP))e+2eT(Im⊗P)(Φ+(L1−1L2⊗IN)Φ0). (3.5) Since $${L_1}$$ is positive definite and symmetric, there is an orthogonal matrix $$U$$ such that $${U^T}{L_1}U = J = diag\left\{ {{\lambda _1}, \cdots ,{\lambda _m}} \right\}$$, where $$\left\{ {{\lambda _1}, \cdots {\lambda _m}} \right\} $$ is the eigenvalue set of $${{L_1}}$$. Let $$\tilde e = \left( {{U^T} \otimes {I_N}} \right)e$$ we can get    t0CDtαV ⩽ −e~T(J⊗(PK+KTP))e~+2eT(Im⊗P)(Φ+(L1−1L2⊗IN)Φ0) = −∑i=1mλie~iT(PK+KTP)e~i+2eT(Im⊗P)(Φ+(L1−1L2⊗IN)Φ0) ⩽ −λmin(L1)λmin(PK+KTP)eTe+2eT(Im⊗P)(Φ+(L1−1L2⊗IN)Φ0) ⩽ −λmin(L1)λmin(PK+KTP)eTe+εeT(Im⊗P2)e +1ε(Φ+(L1−1L2⊗IN)Φ0)T(Φ+(L1−1L2⊗IN)Φ0), (3.6) where Lemma 2.4 was used and   e~Te~=eT(U⊗IN)(UT⊗IN)e=eTe. (3.7) We can get that   (Φ+(L1−1L2⊗IN)Φ0)T(Φ+(L1−1L2⊗IN)Φ0) =(f(t,x1)−∑j=1n−mξ1jf(t,vm+j),⋯,f(t,xm)−∑j=1n−mξmjf(t,vm+j))T(f(t,x1)−∑j=1n−mξ1jf(t,vm+j),⋯,f(t,xm)−∑j=1n−mξmjf(t,vm+j)) ⩽ρ2(xF+(L1−1L2⊗IN)v)T(xF+(L1−1L2⊗IN)v). (3.8) It follows from (3.6) and (3.8) that    t0CDtαV ⩽ −λmin(L1)λmin(PK+KTP)eTe+εeT(Im⊗P2)e +ρ2ε(xF+(L1−1L2⊗IN)v)T(xF+(L1−1L2⊗IN)v) = −eT(λmin(L1)λmin(PK+KTP)I−(Im⊗P2)−ρ2εI)e = −eTQ1e ⩽ −λmin(Q1)λmax(P)V, (3.9) where Lemma 2.4 is used. According to Lemma 2.2, we have   V⩽V(t0,e(t0))Eα(−λmin(Q1)λmax(P)(t−t0)α), (3.10) i.e.,   ‖e‖⩽V(t0,y(t0))λmin(P)(Eα(−λmin(Q1)λmax(P)(t−t0)α))12. (3.11) From (3.11) and Property 1, we have $$\mathop {\lim }\limits_{t \to \infty } e = 0$$, i.e. $${x_F} \to - \left( {L_1^{ - 1}{L_2} \otimes {I_N}} \right)v$$ as $$t \to \infty$$. The proof is completed. □ Remark 3.1 The topology of followers is undirected, then $${L_1}$$ is symmetric and with real eigenvalues. However, when the topology of followers is directed, some eigenvalues of $${L_1}$$ may be complex, and then the problem becomes more complicated. Assumption 3 The communication topology among followers is directed and there is no path from followers to leaders. While for each follower, there exists at least one leader that has a directed path to that follower. Lemma 3.1 (See Mei et al. (2011) Under Assumption 3, all eigenvalues of $${L_1}$$ have positive real parts. Each entry of $$- L_1^{ - 1}{L_2}$$ is nonnegative and each row sum of $$- L_1^{ - 1}{L_2}$$ is equal to one. Theorem 3.2 Under Assumptions 1 and 3, the containment control of multi-agent system (2.6) with the protocol (3.1) can be achieved if there exist a positive definite symmetric $$P \in {{R}^{m \times m}}$$, a positive scalar $$\varepsilon$$ and a symmetric matrix $$K$$ such that $$P{L_1} + L_1^TP$$ is positive and   Q2=(PL1+L1TP)⊗K−(ε(P2⊗IN)+ρ2εI)>0. (3.12) Proof. Since all eigenvalues of $${{L_1}}$$ have positive real parts, there is always a positive definite symmetric $$P$$ such that $$P{L_1} + L_1^TP$$ is positive. Consider a Lyapunov function candidate $$\ V= {e^T}\left( {P \otimes {I_N}} \right)e$$. From (3.4) and Lemma 2.1, we get    t0CDtαV ⩽ −eT((PL1+L1TP)⊗K)e+2eT(P⊗IN)(Φ+(L1−1L2⊗IN)Φ0). (3.13) According to (3.8) and (3.13), we can get that    t0CDtαV ⩽ −eT((PL1+L1TP)⊗K)e+εeT(P2⊗IN)e+ρ2εeTe. One has    t0CDtαV ⩽ −eT((PL1+L1TP)⊗K−ε(P2⊗IN)−ρ2εI)e ⩽ −eTQ2e ⩽ −λmin(Q2)λmax(P)V (3.14) It yields that   V⩽V(t0,e(t0))Eα(−λmin(Q2)λmax(P)(t−t0)α), then we have $$\mathop {\lim }\limits_{t \to \infty } e = 0$$, the result can be derived. □ The containment control problem is in fact a generalization of leader-following consensus problem. When $$n = m + 1$$, multi-agent system (2.6) becomes    t0CDtαxi =ui+f(t,xi),i=1,⋯,m t0CDtαv =u0+f(t,v), (3.15) it is reduced to the leader-following multi-agent system. We define that the leader is globally reachable if for each follower, there exists at least one directed path from the leader to it. Assumption 4 The communication topology among followers of multi-agent system (3.15) is undirected and the leader is globally reachable. Under Assumption 4, the Laplace matrix of $$G$$ can be written as $L = \left( {\begin{array}{*{20}{c}} {{L_1}} & {{l_2}} \\ {{{\mathbf{0}}_{1 \times n}}} & 0 \end{array}} \right)$, where $${L_1} \in {{\mathbf{R}}^{n \times n}}$$, $${l_2} \in {{\mathbf{R}}^{n \times 1}}$$. According to Lemma 2.3, $${L_1}$$ is symmetric and positive definite. Assumption 5 In multi-agent system (3.15), the leader is globally reachable. Under Assumption 5, we can get that all the eigenvalues of $${L_1}$$ is have positive real part. The protocol for multi-agent system (3.15) is designed as follows:   ui=K(∑j=1naij(xj−xi)+bi(v−xi))+u0, (3.16) where $$K$$ is a matrix to be designed. For multi-agent system (3.15) with the protocol (3.16), we have the following result. Corollary 3.1 Suppose that Assumption 4 and (2.8) hold. Then if there exists a positive definite symmetric $${P \in {{\mathbf{R}}^{N \times N}}}$$, a positive scalar $$\varepsilon$$ and a matrix $$K$$ such that   Q=λmin(L1)λmin(PK+KTP)I−(ε(Im⊗P2)+ρ2εI)>0, then the leader-following consensus for system (3.15) is achieved by protocol (3.16), i.e., $${x_i} \to v$$ as $$t \to + \infty$$, $$\forall i \in {V_F}$$. Proof. Let $${\tilde x_i} = {x_i} - v$$, we have    t0CDtαx~i=ui+f(t,xi)−u0−f(t,v). (3.17) Consider a Lyapunov function candidate $$V = {{\tilde x}^T}\left( {{I_m} \otimes P} \right)\tilde x$$, then following the proof line of Theorem 3.1, Corollary 3.1 can be derived. □ Corollary 3.2 Suppose that Assumption 5 and (2.8) hold. Then if there exists a positive definite symmetric $${P \in {{\mathbf{R}}^{m \times m}}}$$, a positive scalar $$\varepsilon$$ and a symmetric matrix $$K$$ such that   Q=P(L1⊗K)+(L1⊗K)TP−(ε(Im⊗P2)+ρ2εI)>0, then the leader-following consensus for system (3.15) is achieved by protocol (3.16), i.e., $${x_i} \to v$$ as $$t \to + \infty$$, $$\forall i \in {V_F}$$. Proof. Following the proof line of Theorem 3.2 and Corollary 3.1, Corollary 3.2 can be derived. □ Remark 3.2 It is noted that in Corollary 3.2, the restriction on nonlinear function is more relaxed than in Yu et al. (2015). 4. Simulation examples In this section, we give three examples to show the effectiveness of main results. In all three examples, if $$({v_j},{v_i}) \in E\left( G \right)$$, we choose $${a_{ij}} =1 $$. Example 4.1 The fractional-order multi-agent system consists of two leaders and five followers. Assume that the nonlinear function is $${f\left( {t,x} \right) = \frac{1}{{10}}x\sin t}$$, the fractional order is $$\alpha = 0.95$$,$${u_{01}} = {u_{02}} = 0$$, and the topology is described as Fig. 1, where the topology among followers is undirected. Fig. 1. View largeDownload slide The topology of the multi-agent system with two leaders. Fig. 1. View largeDownload slide The topology of the multi-agent system with two leaders. From Fig. 1, we have   L1=(30−1−100200−1−10100−100100−1002),L2=(−1−10000000−1). The initial of agents are set as $${v_1} = 9$$, $${v_2} = 13$$, $${x_1} = 15$$, $${x_2} = - 20$$, $${x_3} = 18$$, $${x_4} = - 25$$ and $${x_5} = - 5$$. Then $$K \in {{\mathbf{R}}^{1 \times 1}}$$ can be chose as 4. Under the designed protocol, the state trajectories of agents are shown in Fig. 2. It can be observed from Fig. 2 that the containment control objective is achieved. Fig. 2. View largeDownload slide The states trajectories of agents. Fig. 2. View largeDownload slide The states trajectories of agents. Example 4.2 The fractional-order multi-agent system consists of one leader and five followers. $${f\left( {t,x} \right) = \frac{1}{{10}}\sin x}$$, $$\alpha = 0.95$$, $${u_0} = 0$$, and the topology is described as Fig. 3. Fig. 3. View largeDownload slide The topology of the multi-agent system with one leader. Fig. 3. View largeDownload slide The topology of the multi-agent system with one leader. From Fig. 3, we have   L1=(30−1−100200−1−10100−10010−1−1002),L2=(−1−1000) The initial states of agents are set as $$v = 8$$, $${x_1} = 15$$, $${x_2} = - 20$$, $${x_3} = 18$$, $${x_4} = - 25$$ and $${x_5} = - 5$$. Then $$K \in {{\mathbf{R}}^{1 \times 1}}$$ can be designed as 3. Under the designed protocol, the state trajectories of agents are shown in Fig. 4. We can get that the consensus objective is achieved under directed topology. Fig. 4. View largeDownload slide The states trajectories of agents. Fig. 4. View largeDownload slide The states trajectories of agents. Example 4.3 The fractional-order multi-agent system consists of two leaders and five agents. $$f( \bullet )$$, $$\alpha$$ are set as Example 1, $${u_{01}} = {u_{02}} = 0$$, $$K=3.5$$. The topology of fractional-order multi-agent system is given as Fig. 5. It is noted that the topology among followers is directed. Fig. 5. View largeDownload slide The topology of the multi-agent system with two leaders. Fig. 5. View largeDownload slide The topology of the multi-agent system with two leaders. From Fig. 5, we have   L1=(2−1000−12000−10100−100100−1002),L2=(−100000−100−1). Under the designed protocol, the state trajectories of agents are shown in Fig. 6. We can get that the containment control objective is achieved under directed topology. Fig. 6. View largeDownload slide The states trajectories of agents. Fig. 6. View largeDownload slide The states trajectories of agents. 5. Conclusions In this article, we have addressed the containment control problem for fractional-order nonlinear multi-agent systems. A distributed protocol has been proposed to guarantee that the followers can asymptotically converge to a dynamic convex hull spanned by multiple leaders. When there is only one leader, the leader-following consensus of the multi-agent system can be achieved by the proposed protocol. Some numerical simulations have been provided to demonstrate the effectiveness of the proposed results. In the future, some more research topics could be considered, such as the containment control of fractional-order multi-agent systems with uncertainties or switching topologies. Funding National Natural Science Foundation of China under Grant No. 61273120. References Bagley R. L. & Torvik P. J. ( 1983) Fractional calculusła different approach to the analysis of viscoelastically damped structures. Am. Inst. Aeronaut. Astronaut. , 21, 741– 748. 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Containment control of fractional-order nonlinear multi-agent systems under fixed topologies

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Abstract

Abstract This article investigates the containment control problem of fractional-order nonlinear multi-agent systems under fixed topologies. The containment control problem of multi-agent systems is first transformed to the stability problem of fractional-order nonlinear systems. A distributed protocol is then proposed. By using the Lyapunov function method of fractional-order systems, it is proved that under the designed protocol, the followers can asymptotically converge to a dynamic convex hull spanned by multiple leaders. Numerical simulations are provided to illustrate the effectiveness of the proposed protocols. 1. Introduction In recent years, cooperative control of multi-agent systems has received significant attention due to its broad applications in many fields (see Caridi & Sianesi, 2000; Chen et al., 2015; Leonard et al., 2007). There are many hot topics such as formation control, consensus, rendezvous, containment control and distributed tracking, closely related to cooperative control (see Olfati-Saber & Murray, 2004; Xiao et al., 2009; Yan et al., 2011; Dong & Huang, 2013; Zhao et al., 2013; Yan & Xie, 2014; Hu et al., 2015). Distributed cooperative control means achieving control objectives of agents with local mutual effect between the individuals (see Murry, 2007; Ren & Atkins, 2007). In the consensus problem, leader-following consensus, which has been studied by many researchers, is an important topic (see Peng & Yang, 2009; Wang & Ji, 2012; Sun & Guan, 2013; Zhang & Yang, 2013; Cao et al., 2015; Du et al., 2016). In Peng & Yang (2009), the leader-following consensus problem with time-varying delays was solved. In Cao et al. (2015), the leader-following consensus of linear multi-agent systems with unknown external disturbances was investigated. A leader-following rendezvous problem of double integrator multi-agent systems was studied in Dong & Huang (2013). The leader-following finite-time consensus problem was considered in Du et al. (2015). Recently, the leader-following consensus with multiple leaders, which is called containment control problem, was studied (see Li et al., 2013, 2016; He et al., 2014; Haghshenas et al., 2015; Chu et al., 2016; Hu et al., 2016). The main objective of the containment control is to design a protocol such that all the followers can converge to the convex hull spanned by the leaders. The containment control problem for multi-agent systems with communication delay was considered in Hu et al. (2016) and Li et al. (2016). The finite-time containment control for second-order multi-agent systems under directed topology was studied in He et al. (2014). The containment control of heterogeneous linear multi-agent systems was presented in Haghshenas et al. (2015). In Chu et al. (2016), an output regulation approach was provided to solve the containment control problem of heterogeneous linear multi-agent systems. In reality, many physical systems are nonlinear in nature (see Huang et al., 2005; Cao et al., 2016a), so there are many results on nonlinear multi-agent systems such as Cao et al. (2016b), Du et al. (2014) and Wan et al. (2016). It is noted that most of the aforementioned results on cooperative control are related to integer-order dynamics. However, many real-world phenomena in biology, physics, etc. are better described by fractional-order dynamics rather than integer-order dynamics (see Bagley & Torvik, 1983; Perdikaris & Karniadakis, 2014; Taher et al., 2014). The distributed coordination of networked fractional-order systems was firstly studied in Cao et al. (2010). The consensus protocol was designed for agents with fractional-order uncertain dynamics in Yin & Hu (2013), where the fractional order is $$0 < \alpha < 2$$. Chen et al. (2016) investigated the containment control of linear uncertain fractional-order multi-agent systems. In Yin et al. (2013), the consensus protocols are designed for a set of fractional-order heterogeneous agents, which is composed of two kinds of agents differed by their dynamics. The cooperative control for fractional-order multi-agent systems with communication delay was studied in Yang et al. (2014) and Shen & Cao (2012). However, to the best of our knowledge, the containment control of fractional-order nonlinear multi-agent systems has not been investigated to date, which motivates the current study of this article. In this article, we will present a protocol for fractional-order nonlinear multi-agent systems. By adopting the Lyapunov function method of fractional-order systems, it will be proved that under some conditions, the designed protocol can make the followers converge to a dynamic convex hull spanned by multiple leaders asymptotically. The containment control problem with multi-agent systems under directed topologies is then considered. As a special case, when there is only one leader, the leader-following consensus of the multi-agent system can be achieved by the proposed protocol. The rest of this article is organized as follows. The necessary preliminaries and the problem formulation are given in section 2. In section 3, the main results of this article are presented. Some numerical simulations are given to illustrate the validity of the theoretical results in section 4. Finally, some conclusions are drawn in section 5. Notations Throughout this article, $$\mathbf{R}$$ stands for the set of all real numbers. $${\mathbf{R}^n}$$ denotes the $$n$$-dimensional real vector space and $${{\mathbf{R}}^{m \times n}}$$ is the set of $$m \times n$$ matrices. Symbol $$\otimes$$ represents the Kronecker product. The shorthand $$diag\left\{ \cdots \right\}$$ denotes the block diagonal matrix. $${\mathbf{1}}$$ and $${\mathbf{0}}$$ represent the vector with all entries being one and zero with compatible dimensions, respectively. $${I}$$ is an identity matrix with compatible dimensions. $${A^T}$$ denotes the transpose of $$A$$ while $${A^{ - 1}}$$ denotes the inverse of $$A$$, where $$A \in {{\mathbf{R}}^{n \times n}}$$. $${\lambda _{\min }}(A)$$ and $${\lambda _{\max }}(A)$$ denote the minimum eigenvalue and maximum eigenvalue, respectively. $$A > 0(A \geqslant 0)$$ if matrix $$A$$ is positive definite(positive semi-definite). Denote by $$dis\left( {x,C} \right)$$ the distance from $$x \in {{\mathbf{R}}^N}$$ to the set $$C \subseteq {{\mathbf{R}}^N}$$ in the sense of Euclidean norm, that is $$dis\left( {x,C} \right) = \mathop {\inf }\limits_{y \in C} {\left\| {x - y} \right\|}$$, where $$\left\| \cdot \right\|$$ is the Euclidean norm operator. 2. Preliminaries and problem formulation In the following, some basic preliminaries about graph theory will be given. A graph consists of a vertex set $$V\left( G \right) = \left\{ {{v_1}, \cdots ,{v_n}} \right\}$$, an edge set $$E\left( G \right) \subseteq V \times V$$ and an adjacency matrix $$A$$. An edge $$({v_i},{v_j}) \in E\left( G \right)$$ denotes that the vertex $${v_j}$$ can obtain the state of vertex $${v_i}$$. If $$({v_i},{v_j}) \in E\left( G \right)$$, $${v_i}$$ is called a neighbour of $${v_j}$$ and the set of all neighbours can be denoted by $${N_j} = \left\{ {{v_i}:({v_i},{v_j}) \in E\left( G \right)} \right\}$$. The adjacency matrix of $$G$$ is defined as $$A = {[{a_{ij}}]_{n \times n}} \in {{\mathbf{R}}^{n \times n}}$$, with $${a_{ij}} > 0$$ if and only if $$({v_j},{v_i}) \in E\left( G \right)$$ and $${a_{ij}} = 0$$ otherwise. In undirected graph,$$({v_i},{v_j}) \in E\left( G \right)$$ implies that $$({v_j},{v_i}) \in E\left( G \right)$$, but in directed graph, it does not hold. The degree matrix $$D$$ is a diagonal matrix and its $$i$$ th element is equal to $$\left| {{N_i}} \right|$$ which is the cardinality of $${N_i}$$, and we call $$\left| {{N_i}} \right|$$ the degree of $$i$$ th agent. The Laplace matrix of graph $$G$$ is defined as $$L = D - A$$. A path on $$G$$ from $${v_{{i_0}}}$$ to $${v_{{i_k}}}$$ is a sequence of distinct vertices $$\left( {{v_{{i_0}}}, \cdots ,{v_{{i_k}}}} \right)$$ where $$\left( {{v_{{i_{j - 1}}}},{v_{{i_j}}}} \right) \in E\left( G \right) for j = 1, \cdots ,k$$ (see Godsil & Royle, 2001). The communication graph among all agents can be described by $$G = \left( {V,E,A} \right)$$. Agents are abstracted as vertexes of graph and communications between agents as edges. Before giving the system model, we will review definitions of fractional calculation and some useful lemmas. There are several different definitions of fractional operators, such as Caputo and Riemann-Liouville fractional operators. In this article, Caputo operator will be adopted and the definition for fractional derivative of function $$h(t)$$ with order $$\alpha$$ is given as    t0CDtαh(t)=1Γ(α¯−α)∫t0th(α¯)(τ)(t−τ)α−α¯+1dτ, (2.1) where $$\bar \alpha$$ is the integer satisfying $$\bar \alpha - 1 < \alpha \leqslant \bar \alpha$$, $$\Gamma (x) = \int_0^\infty {{t^{x - 1}}{e^{ - t}}dt}$$ is the Gamma function, and $${t_0}$$ is the initial time. An important function frequently used in the solutions of fractional-order systems is the Mittag–Leffler function. The Mittag–Leffler function with two parameters is defined as follows (see Podlubny, 1999):   Eβ,γ(z)=∑k=0∞zkΓ(kβ+γ), (2.2) where $$\beta > 0$$, $$\gamma > 0$$. As a special case, when $$\gamma = 1$$, the Mittag–Leffler function with one parameter is defined as   Eβ(z)=∑k=0∞zkΓ(kβ+1), (2.3) where $$\beta > 0$$. Furthermore, it is noted that if $$\beta = \gamma = 1$$, $${E_{\beta ,\gamma }}\left( z \right) = {e^z}$$. We now give a property about Mittag–Leffler function with one parameter as follows. Property 1 (See Diethelm, 2010) Let $$0 < \beta \leqslant 2$$, the Mittag-Leffler function $${E_\beta }( \cdot )$$ behaves as (1) $${E_\beta }\left( {r{e^{i\phi }}} \right) \to 0$$ for $$r \to \infty $$ if $$\left| \phi \right| > \frac{{\beta \pi }}{2}$$. (2) $${E_\beta }\left( {r{e^{i\phi }}} \right)$$ remains bouned for $$r \to \infty $$ if $$\left| \phi \right| = \frac{{\beta \pi }}{2}$$. (3) $$\left| {{E_\beta }\left( {r{e^{i\phi }}} \right)} \right| \to \infty $$ for $$r \to \infty $$ if $$\left| \phi \right| < \frac{{\beta \pi }}{2}$$. The following lemmas related to Caputo operator will be used to obtain our main results. Lemma 2.1 (see Duarte-Mermoud et al., 2015) Let $$x(t) \in {{\mathbf{R}}^n}$$ be a vector of differentiable function. Then, for any time instant $$t > {t_0}$$, the following relationship holds   12 t0CDtα(xT(t)Mx(t))⩽xT(t)M t0CDtαx(t),∀α∈(0,1], (2.4) where $$M \in {{\mathbf{R}}^{n \times n}}$$ is a symmetric and positive definite matrix. Lemma 2.2 (see Liu et al., 2016) Let $$x(t) \in {{\mathbf{R}}^n}$$ be a vector of differentiable function. If a continuous function $$V:\left[ {{t_0},\infty } \right) \times {{\mathbf{R}}^n} \to {\mathbf{R}}$$ satisfies $$_{{t_0}}^CD_t^\beta V\left( {t,x\left( t \right)} \right) \leqslant - \alpha V\left( {t,x\left( t \right)} \right)$$, then   V(t,x(t))⩽V(t0,x(t0))Eβ(−α(t−t0)β), (2.5) where $$0 < \beta \leqslant 1$$, $$\alpha$$ is a positive constant. Consider a fraction-order multi-agent system composed by $$n$$ agents which include $$m$$ followers and $$\left( {n - m} \right)$$ leaders. The dynamics of agents are given as follows:    t0CDtαxi =ui+f(t,xi),i=1,⋯,m, t0CDtαvi =u0,i+f(t,vi),i=m+1,⋯,n, (2.6) where $$0 < \alpha \leqslant 1$$. $${x_i} \in {{\mathbf{R}}^N}$$ and $${u_i} \in {{\mathbf{R}}^N}$$ are the state and control input of the $$i$$ th follower, respectively. $${v_i} \in {{\mathbf{R}}^N}$$ is the state of the $$i$$ th leader and $${u_{0,i}}$$ is the preset input of the $$i$$ th leader. $$f( \cdot )\in {{\mathbf{R}}^N}$$ is an unknown nonlinear continuous vector-function with $$f(t,0) = 0$$. Remark 2.1 When $$\alpha = 1$$, (2.6) is reduced to an integer-order nonlinear multi-agent system like Mei et al. (2011). In this article, we take the unknown nonlinear uncertainties into consideration, if $$f(t,x) \equiv 0$$ and there is no leader, the model is the same as the model in Cao et al. (2010). Note that the unknown nonlinear uncertainties commonly exist in actual systems. Assumption 1 (Mei et al. (2011)) The nonlinear vector-function $$f( \cdot )$$ satisfies the following condition: Given $${\xi _1}, \cdots ,{\xi _{n - m}}$$, with $$\sum\limits_{i = 1}^{n - m} {{\xi _i} = 1}$$ and $${\xi _i} \geqslant 0,\;i = 1, \cdots ,n - m$$, there exists a positive constant $$\rho$$ such that for $$x,{y_i} \in {{\mathbf{R}}^N},\;i = 1, \cdots ,n - m$$,   ‖f(t,x)−∑i=1n−mξif(t,yi)‖⩽ρ‖x−∑i=1n−mξiyi‖. (2.7) Remark 2.2 In the leader-following consensus problem where there is only one leader, condition (2.7) becomes to the Lipschitz condition:   ‖f(t,x)−f(t,y)‖⩽ρ‖x−y‖,∀x,y∈RN. (2.8) Assumption 2 The communication topology among followers is undirected and there is no path from followers to leaders. While for each follower, there exists at least one leader that has a directed path to that follower. Under Assumption 2, the Laplace matrix of $$G$$ can be written as   L =(L1L20(n−m)×m0(n−m)×(n−m)), where $${L_1} \in {{\mathbf{R}}^{m \times m}}, {L_2} \in {{\mathbf{R}}^{m \times (n - m)}}$$. A lemma about $$L$$ will be given as follows: Lemma 2.3 (see Meng et al. (2010) Under Assumption 2, $${L_1}$$ is positive definite and symmetric. Each entry of $$- L_1^{ - 1}{L_2}$$ is nonnegative and each row sum of $$- L_1^{ - 1}{L_2}$$ is equal to one. Remark 2.3 Assumption 2 is commonly adopted in the literatures such as Zhang & Yang (2013) and Meng et al. (2010). To achieve the containment control goal, directly or indirectly connection with leader set is essential. Otherwise, sub-systems will be formed and the problem might fail to be solved. Definition 2.1 (see Rockafellar et al. (1972) A set $$C \subseteq {{\mathbf{R}}^N}$$ is a convex if $$(1 - \lambda )x + \lambda y \in C$$, for any $$x,y \in C$$ and $$\lambda \in \left[ {0,1} \right]$$. The convex hull $$Co\left( X \right)$$ of a finite set of points $$X = \left\{ {{x_1},{x_2}, \cdots ,{x_q}} \right\}$$ is defined like convex and is described as follows:   Co(X)={∑i=1qαixi|xi∈X,αi∈R,αi⩾0,∑i=1qαi=1 }. (2.9) Definition 2.2 (see Ji et al. (2008)) The containment control of fractional-order system (2.6) is achieved by the designed protocol if and only if $$\forall i \in {V_F} = \left\{ {1,2, \cdots ,m} \right\}$$,   limt→∞⁡dis(xi(t),Co(vk(t),k∈VS))=0, (2.10) where $${V_S} = \left\{ {m + 1, \cdots ,n} \right\}$$. Lemma 2.4 (see Hardy et al. (1952)) Give any scalar $$\varepsilon > 0$$, $$x \in {{\mathbf{R}}^n}$$, $$y \in {{\mathbf{R}}^n}$$, then it holds that   xTy+yTx⩽εxTx+1εyTy. 3. Main results In this section, the main results of this article will be given. In this article, we design a protocol for multi-agent systems (2.6) as follows:   ui=K(∑j=1maij(xj−xi)+∑j=m+1naij(vj−xi))+(Λi⊗IN)u0, (3.1) where $$K$$ is a matrix to be designed, $${\it\Lambda _i}$$ is the $$i$$ th row vector of $$-L_1^{ - 1}{L_2}$$, $${u_0}$$ is the column stack of $${u_{0i}}$$. Theorem 3.1 Under Assumptions 1 and 2, the containment control of multi-agent system (2.6) with the protocol (3.1) can be achieved if there exist a positive definite symmetric $$P \in {{\mathbf{R}}^{N \times N}}$$, a positive scalar $$\varepsilon$$ and a matrix $$K$$ such that $$PK + {K^T}P$$ is positive definite and   Q1=λmin(L1)λmin(PK+KTP)I−(ε(Im⊗P2)+ρ2εI)>0. (3.2) Proof. Let $${x_{_F}} = {\left( {x_1^T, \cdots ,x_m^T} \right)^T}$$ and $$v = {\left( {v_{m + 1}^T, \cdots ,v_n^T} \right)^T}$$. Define $$e = {x_F} - \left( { - \left( {L_1^{ - 1}{L_2} \otimes {I_N}} \right)v} \right)$$, then it follows from (2.6) that    t0CDtαe = t0CDtαxF+(L1−1L2⊗IN) t0CDtαv =u+Φ+(L1−1L2⊗IN)u0+(L1−1L2⊗IN)Φ0, (3.3) where $$u,\;{u_0},\;{\it}\Phi$$ and $$\;{{\it}\Phi _0}$$ are column stacks of $${u_i},\;{u_{0i}},\;f(t,{x_i})$$ and $$\;f(t,{v_i})$$, respectively. Utilizing (3.1) and (3.3) yields    t0CDtαe = −(L1⊗K)xF−(L2⊗K)v+Φ+(L1−1L2⊗IN)Φ0 = −(L1⊗K)e+Φ+(L1−1L2⊗IN)Φ0. (3.4) Consider a Lyapunov function candidate $$V = {e^T}\left( {{I_m} \otimes P} \right)e$$. From (3.4) and Lemma 2.1, we can get    t0CDtαV ⩽2eT(Im⊗P)(−(L1⊗K)e+Φ+(L1−1L2⊗IN)Φ0)  = −eT(L1⊗(PK+KTP))e+2eT(Im⊗P)(Φ+(L1−1L2⊗IN)Φ0). (3.5) Since $${L_1}$$ is positive definite and symmetric, there is an orthogonal matrix $$U$$ such that $${U^T}{L_1}U = J = diag\left\{ {{\lambda _1}, \cdots ,{\lambda _m}} \right\}$$, where $$\left\{ {{\lambda _1}, \cdots {\lambda _m}} \right\} $$ is the eigenvalue set of $${{L_1}}$$. Let $$\tilde e = \left( {{U^T} \otimes {I_N}} \right)e$$ we can get    t0CDtαV ⩽ −e~T(J⊗(PK+KTP))e~+2eT(Im⊗P)(Φ+(L1−1L2⊗IN)Φ0) = −∑i=1mλie~iT(PK+KTP)e~i+2eT(Im⊗P)(Φ+(L1−1L2⊗IN)Φ0) ⩽ −λmin(L1)λmin(PK+KTP)eTe+2eT(Im⊗P)(Φ+(L1−1L2⊗IN)Φ0) ⩽ −λmin(L1)λmin(PK+KTP)eTe+εeT(Im⊗P2)e +1ε(Φ+(L1−1L2⊗IN)Φ0)T(Φ+(L1−1L2⊗IN)Φ0), (3.6) where Lemma 2.4 was used and   e~Te~=eT(U⊗IN)(UT⊗IN)e=eTe. (3.7) We can get that   (Φ+(L1−1L2⊗IN)Φ0)T(Φ+(L1−1L2⊗IN)Φ0) =(f(t,x1)−∑j=1n−mξ1jf(t,vm+j),⋯,f(t,xm)−∑j=1n−mξmjf(t,vm+j))T(f(t,x1)−∑j=1n−mξ1jf(t,vm+j),⋯,f(t,xm)−∑j=1n−mξmjf(t,vm+j)) ⩽ρ2(xF+(L1−1L2⊗IN)v)T(xF+(L1−1L2⊗IN)v). (3.8) It follows from (3.6) and (3.8) that    t0CDtαV ⩽ −λmin(L1)λmin(PK+KTP)eTe+εeT(Im⊗P2)e +ρ2ε(xF+(L1−1L2⊗IN)v)T(xF+(L1−1L2⊗IN)v) = −eT(λmin(L1)λmin(PK+KTP)I−(Im⊗P2)−ρ2εI)e = −eTQ1e ⩽ −λmin(Q1)λmax(P)V, (3.9) where Lemma 2.4 is used. According to Lemma 2.2, we have   V⩽V(t0,e(t0))Eα(−λmin(Q1)λmax(P)(t−t0)α), (3.10) i.e.,   ‖e‖⩽V(t0,y(t0))λmin(P)(Eα(−λmin(Q1)λmax(P)(t−t0)α))12. (3.11) From (3.11) and Property 1, we have $$\mathop {\lim }\limits_{t \to \infty } e = 0$$, i.e. $${x_F} \to - \left( {L_1^{ - 1}{L_2} \otimes {I_N}} \right)v$$ as $$t \to \infty$$. The proof is completed. □ Remark 3.1 The topology of followers is undirected, then $${L_1}$$ is symmetric and with real eigenvalues. However, when the topology of followers is directed, some eigenvalues of $${L_1}$$ may be complex, and then the problem becomes more complicated. Assumption 3 The communication topology among followers is directed and there is no path from followers to leaders. While for each follower, there exists at least one leader that has a directed path to that follower. Lemma 3.1 (See Mei et al. (2011) Under Assumption 3, all eigenvalues of $${L_1}$$ have positive real parts. Each entry of $$- L_1^{ - 1}{L_2}$$ is nonnegative and each row sum of $$- L_1^{ - 1}{L_2}$$ is equal to one. Theorem 3.2 Under Assumptions 1 and 3, the containment control of multi-agent system (2.6) with the protocol (3.1) can be achieved if there exist a positive definite symmetric $$P \in {{R}^{m \times m}}$$, a positive scalar $$\varepsilon$$ and a symmetric matrix $$K$$ such that $$P{L_1} + L_1^TP$$ is positive and   Q2=(PL1+L1TP)⊗K−(ε(P2⊗IN)+ρ2εI)>0. (3.12) Proof. Since all eigenvalues of $${{L_1}}$$ have positive real parts, there is always a positive definite symmetric $$P$$ such that $$P{L_1} + L_1^TP$$ is positive. Consider a Lyapunov function candidate $$\ V= {e^T}\left( {P \otimes {I_N}} \right)e$$. From (3.4) and Lemma 2.1, we get    t0CDtαV ⩽ −eT((PL1+L1TP)⊗K)e+2eT(P⊗IN)(Φ+(L1−1L2⊗IN)Φ0). (3.13) According to (3.8) and (3.13), we can get that    t0CDtαV ⩽ −eT((PL1+L1TP)⊗K)e+εeT(P2⊗IN)e+ρ2εeTe. One has    t0CDtαV ⩽ −eT((PL1+L1TP)⊗K−ε(P2⊗IN)−ρ2εI)e ⩽ −eTQ2e ⩽ −λmin(Q2)λmax(P)V (3.14) It yields that   V⩽V(t0,e(t0))Eα(−λmin(Q2)λmax(P)(t−t0)α), then we have $$\mathop {\lim }\limits_{t \to \infty } e = 0$$, the result can be derived. □ The containment control problem is in fact a generalization of leader-following consensus problem. When $$n = m + 1$$, multi-agent system (2.6) becomes    t0CDtαxi =ui+f(t,xi),i=1,⋯,m t0CDtαv =u0+f(t,v), (3.15) it is reduced to the leader-following multi-agent system. We define that the leader is globally reachable if for each follower, there exists at least one directed path from the leader to it. Assumption 4 The communication topology among followers of multi-agent system (3.15) is undirected and the leader is globally reachable. Under Assumption 4, the Laplace matrix of $$G$$ can be written as $L = \left( {\begin{array}{*{20}{c}} {{L_1}} & {{l_2}} \\ {{{\mathbf{0}}_{1 \times n}}} & 0 \end{array}} \right)$, where $${L_1} \in {{\mathbf{R}}^{n \times n}}$$, $${l_2} \in {{\mathbf{R}}^{n \times 1}}$$. According to Lemma 2.3, $${L_1}$$ is symmetric and positive definite. Assumption 5 In multi-agent system (3.15), the leader is globally reachable. Under Assumption 5, we can get that all the eigenvalues of $${L_1}$$ is have positive real part. The protocol for multi-agent system (3.15) is designed as follows:   ui=K(∑j=1naij(xj−xi)+bi(v−xi))+u0, (3.16) where $$K$$ is a matrix to be designed. For multi-agent system (3.15) with the protocol (3.16), we have the following result. Corollary 3.1 Suppose that Assumption 4 and (2.8) hold. Then if there exists a positive definite symmetric $${P \in {{\mathbf{R}}^{N \times N}}}$$, a positive scalar $$\varepsilon$$ and a matrix $$K$$ such that   Q=λmin(L1)λmin(PK+KTP)I−(ε(Im⊗P2)+ρ2εI)>0, then the leader-following consensus for system (3.15) is achieved by protocol (3.16), i.e., $${x_i} \to v$$ as $$t \to + \infty$$, $$\forall i \in {V_F}$$. Proof. Let $${\tilde x_i} = {x_i} - v$$, we have    t0CDtαx~i=ui+f(t,xi)−u0−f(t,v). (3.17) Consider a Lyapunov function candidate $$V = {{\tilde x}^T}\left( {{I_m} \otimes P} \right)\tilde x$$, then following the proof line of Theorem 3.1, Corollary 3.1 can be derived. □ Corollary 3.2 Suppose that Assumption 5 and (2.8) hold. Then if there exists a positive definite symmetric $${P \in {{\mathbf{R}}^{m \times m}}}$$, a positive scalar $$\varepsilon$$ and a symmetric matrix $$K$$ such that   Q=P(L1⊗K)+(L1⊗K)TP−(ε(Im⊗P2)+ρ2εI)>0, then the leader-following consensus for system (3.15) is achieved by protocol (3.16), i.e., $${x_i} \to v$$ as $$t \to + \infty$$, $$\forall i \in {V_F}$$. Proof. Following the proof line of Theorem 3.2 and Corollary 3.1, Corollary 3.2 can be derived. □ Remark 3.2 It is noted that in Corollary 3.2, the restriction on nonlinear function is more relaxed than in Yu et al. (2015). 4. Simulation examples In this section, we give three examples to show the effectiveness of main results. In all three examples, if $$({v_j},{v_i}) \in E\left( G \right)$$, we choose $${a_{ij}} =1 $$. Example 4.1 The fractional-order multi-agent system consists of two leaders and five followers. Assume that the nonlinear function is $${f\left( {t,x} \right) = \frac{1}{{10}}x\sin t}$$, the fractional order is $$\alpha = 0.95$$,$${u_{01}} = {u_{02}} = 0$$, and the topology is described as Fig. 1, where the topology among followers is undirected. Fig. 1. View largeDownload slide The topology of the multi-agent system with two leaders. Fig. 1. View largeDownload slide The topology of the multi-agent system with two leaders. From Fig. 1, we have   L1=(30−1−100200−1−10100−100100−1002),L2=(−1−10000000−1). The initial of agents are set as $${v_1} = 9$$, $${v_2} = 13$$, $${x_1} = 15$$, $${x_2} = - 20$$, $${x_3} = 18$$, $${x_4} = - 25$$ and $${x_5} = - 5$$. Then $$K \in {{\mathbf{R}}^{1 \times 1}}$$ can be chose as 4. Under the designed protocol, the state trajectories of agents are shown in Fig. 2. It can be observed from Fig. 2 that the containment control objective is achieved. Fig. 2. View largeDownload slide The states trajectories of agents. Fig. 2. View largeDownload slide The states trajectories of agents. Example 4.2 The fractional-order multi-agent system consists of one leader and five followers. $${f\left( {t,x} \right) = \frac{1}{{10}}\sin x}$$, $$\alpha = 0.95$$, $${u_0} = 0$$, and the topology is described as Fig. 3. Fig. 3. View largeDownload slide The topology of the multi-agent system with one leader. Fig. 3. View largeDownload slide The topology of the multi-agent system with one leader. From Fig. 3, we have   L1=(30−1−100200−1−10100−10010−1−1002),L2=(−1−1000) The initial states of agents are set as $$v = 8$$, $${x_1} = 15$$, $${x_2} = - 20$$, $${x_3} = 18$$, $${x_4} = - 25$$ and $${x_5} = - 5$$. Then $$K \in {{\mathbf{R}}^{1 \times 1}}$$ can be designed as 3. Under the designed protocol, the state trajectories of agents are shown in Fig. 4. We can get that the consensus objective is achieved under directed topology. Fig. 4. View largeDownload slide The states trajectories of agents. Fig. 4. View largeDownload slide The states trajectories of agents. Example 4.3 The fractional-order multi-agent system consists of two leaders and five agents. $$f( \bullet )$$, $$\alpha$$ are set as Example 1, $${u_{01}} = {u_{02}} = 0$$, $$K=3.5$$. The topology of fractional-order multi-agent system is given as Fig. 5. It is noted that the topology among followers is directed. Fig. 5. View largeDownload slide The topology of the multi-agent system with two leaders. Fig. 5. View largeDownload slide The topology of the multi-agent system with two leaders. From Fig. 5, we have   L1=(2−1000−12000−10100−100100−1002),L2=(−100000−100−1). Under the designed protocol, the state trajectories of agents are shown in Fig. 6. We can get that the containment control objective is achieved under directed topology. Fig. 6. View largeDownload slide The states trajectories of agents. Fig. 6. View largeDownload slide The states trajectories of agents. 5. Conclusions In this article, we have addressed the containment control problem for fractional-order nonlinear multi-agent systems. A distributed protocol has been proposed to guarantee that the followers can asymptotically converge to a dynamic convex hull spanned by multiple leaders. When there is only one leader, the leader-following consensus of the multi-agent system can be achieved by the proposed protocol. Some numerical simulations have been provided to demonstrate the effectiveness of the proposed results. In the future, some more research topics could be considered, such as the containment control of fractional-order multi-agent systems with uncertainties or switching topologies. Funding National Natural Science Foundation of China under Grant No. 61273120. References Bagley R. L. & Torvik P. J. ( 1983) Fractional calculusła different approach to the analysis of viscoelastically damped structures. Am. Inst. Aeronaut. Astronaut. , 21, 741– 748. 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Journal

IMA Journal of Mathematical Control and InformationOxford University Press

Published: Mar 22, 2017

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