^{2} College of Information Science and Engineering, Northeastern University, Shenyang 110819, China;
^{3} Normal College, Shenyang University, Shenyang 110044, China
Multiagent systems (MASs), as a group of autonomous agents, can be used to undertake large and complicated tasks through coordination among agents. Switched MASs in this study mean that the dynamics of each agent can be described by a switched system. A switched system is composed of a family of continuoustime or discretetime subsystems and a rule determining the switching between the subsystems^{[1, 2]}. Many practical systems may be modeled by switched systems due to the environment or parameter changes during the process of operation of systems such as constrained robot, power systems and vehicle control systems^{[35]}, etc. This inspires the study of switched systems. The main concern in the study of switched systems is the issue of stability. Under an arbitrary switching law, a necessary and sufficient condition to achieve asymptotic stability for a switched system is the existence of a common Lyapunov function (CLF) for all subsystems^{[6, 7]}. But, a CLF does not often exist or is difficult to construct. Further, multiple Lyapunov function (MLF) approach proposed in [8] is a powerful and effective tool for designing switching. Successful applications of this approach to the stability analysis, the stabilization problem and robust
On the other hand, distributed cooperative control for MASs is a very active research area in the system and control society. Distributed control, which applies constrained information among agents to implement control objectives, achieves better desired performance similar to centralized control as well as reduces the complexity for designing controller like decentralized control^{[13]}. Several main issues on distributed cooperative control include consensus, coordinated controller tuning, flocking and formation^{[1417]}, etc.
Output regulation, which involves designing of a controller to ensure the system output tracking reference signal generated by an autonomous system meanwhile realizing interference suppression, is one of the most fundamental topics in control theory. The rich results on output regulation have appeared in [18, 19]. Many approaches on solving the output regulation problem, such as state feedback control, output feedback control and internal model principle^{[2023]}, are proposed. Furthermore, these approaches are extended to cope with the distributed output regulation problem of MASs. For nonswitched agent networks, by a dynamic full information distributed control scheme, the cooperative output regulation problem can be solved as in [24]. Further, a consensus control design approach for heterogeneous MASs is proposed to ensure that the outputs of all the agents converge to the same desired output trajectory by exploiting the internal model design strategy in [25]. Xiang et al.^{[26]} address the problem of synchronized output regulation of linear networked system where all the nodes have their outputs which track signals produced by the same exosystem and the state of exosystem is accessible only to leader nodes, while follower nodes regulate their outputs via a distributed synchronous protocol. Moreover, a host internal model (hIM) approach to distributed output regulation of reaching nonlinear leaderfollowing consensus is presented in [27]. The switching of multiagent systems involves two cases. One case is that the dynamics of each agent are described by a switched system, e.g., this paper. The other case is that the communication interconnection of agents is switching such as ^{[2830]}. In ^{[28]}, a cooperative problem of MASs with switched jointly connected interconnection topologies is addressed and a sufficient condition to make all the agents
Motivated by the above considerations, this note studies the cooperative output regulation problem of MASs in which each agent dynamics are represented by a switched heterogeneous linear system. The paper has two main contributions. First, an agentdependent multiple Lyapunov function approach to solve the output regulation problem of switched MASs is proposed. Agentdependent means the switching law for each agent is dependent on its own state and does not rely on other agents
The remainder of the paper is organized as follows. The preliminaries and problem statement are given in Section 2. In Section 3, the main results are developed. An example is given in Section 4. Section 5 draws a conclusion.
Notations. Let
The information flow among agents and exosystem is described by a directed or undirected graph and the detailed content can be found in [13]. A graph is denoted by
Lemma 1.^{[13]}
The MASs considered in our paper is composed of
$ \begin{eqnarray} &&\dot{x_{i}}=A_{i\sigma_{i}(t)}x_{i}+B_{i\sigma_{i}(t)}u_{i}+ E_{i\sigma_{i}(t)}\omega\nonumber\\ &&e_{i}=C_{i}x_{i}+F\omega, ~~i\in I_{N}=\{1, 2, \cdots, N\} \end{eqnarray} $  (1) 
where
$ \begin{eqnarray} &\dot\omega=S\omega, ~~y_{0}=F\omega \end{eqnarray} $  (2) 
where
We assume that each agent
$ u_{i}=K_{i\sigma_{i}(t)}x_{i}+G_{i\sigma_{i}(t)}\eta_{i} $  (3) 
$ \dot{\eta_{i}}=S\eta_{i}+\sum\limits_{j\in\Phi_{i}}a_{ij}(\eta_{j}\eta_{i})+ a_{i0}(\omega\eta_{i}) $  (4) 
where
Remark 1. As in [24], the dynamical system (4) is called a dynamic compensator (or a distributed observer) and can be viewed as an estimate of the exosystem state
The control objective is to design distributed controllers (3)
The cooperative output regulation problem of switched MASs. Given systems (1) and (2), if possible, find the control law (3)
1) The origin of the switched MASs (1) with controllers (3) and (4) are asymptotically stable under the designed switching law
2) The tracking error
$ \begin{eqnarray} \lim\limits_{t\rightarrow\infty} e_{i}(t)=0 \end{eqnarray} $  (5) 
for any initial condition
Throughout this paper, Assumptions 14 and Lemma 2 are needed.
Assumption 1. All the eigenvalues of
Remark 2. This assumption relaxes Assumption 1 in [24] in order to simplify the calculation. If this assumption was replaced by Assumption 1 in [24], the output regulation problem for switched multiagent systems would also be solved by the same technique under the controller in [24].
Assumption 2. For
$ \begin{eqnarray} &&X_{i}S=A_{ij_{i}}X_{i}+B_{ij_{i}}U_{ij_{i}}+E_{ij_{i}}\nonumber \\ && 0=C_{i}X_{i}+F \end{eqnarray} $  (6) 
have the solution pairs
Assumption 3.
Remark 3. Assumption 2 is a standard assumption in the literature to solve the output regulation for nonswitched linear multiagent systems^{[24]}. Assumption 4 is a standard assumption in the literature to rule out Zeno behavior for all types of switching^{[1]}.
Let
Assumption 4. The graph
Lemma 2. ^{[31]} Consider the switched linear timevarying system
$ \begin{eqnarray} &&\dot{x_{c}}=A_{c\sigma_{c}(t)}x_{c}+B_{c\sigma_{c}(t)}v \nonumber \\ &&\dot{v}=\Gamma v \nonumber\\ &&e=C_{c}x_{c}+D_{c}v \end{eqnarray} $  (7) 
where
$ \begin{eqnarray} && X_{c}\Gamma=A_{ci}X_{c}+B_{ci} \nonumber\\ &&0=C_{c}X_{c}+D_{c}, i=1, 2, \cdots, \rho \end{eqnarray} $  (8) 
then, we have
$ \begin{eqnarray} \lim\limits_{t\rightarrow \infty}e(t)=0. \end{eqnarray} $  (9) 
In this section, a sufficient condition for the solvability of output regulation problem of switched agent networks is given via the agentdependent multiple Lyapunov function approach and designing distributed controllers.
It is easy to know that the agent
Suppose that the gain matrices are
$ \begin{eqnarray} &G_{ij_{i}}=U_{ij_{i}}K_{ij_{i}}X_{i} \end{eqnarray} $  (10) 
where
Let
$ \begin{align*}& A_{j} = {\rm block \ diag} (A_{1j_{1}}, A_{2j_{2}}, \cdots, A_{Nj_{N}}) \\&B_{j} = {\rm block \ diag} (B_{1j_{1}}, B_{2j_{2}}, \cdots, B_{Nj_{N}})\\& E_{j} = {\rm block \ diag} (E_{1j_{1}}, E_{2j_{2}}, \cdots, E_{Nj_{N}}) \\& K_{j} = {\rm block \ diag} (K_{1j_{1}}, K_{2j_{2}}, \cdots, K_{Nj_{N}}) \\& G_{j} = {\rm block \ diag} (G_{1j_{1}}, G_{2j_{2}}, \cdots, G_{Nj_{N}}) \\& C = {\rm block \ diag}(C_{1}, C_{2}, \cdots, C_{N}) \\&X ={\rm block \ diag} (X_{1}, X_{2}, \cdots, X_{N}) \\& i\in I_{N}, j_{i} \in M_{i}, j\in M_{1}\times M_{2}\times \cdots \times M_{N}.\end{align*} $ 
By (6) and (10), we have
$ \begin{eqnarray} &&X(I_{N}\otimes S)=(A_{j}+B_{j}K_{j})X+(E_{j}+B_{j}G_{j}) \nonumber\\ &&0=CX+(I_{N}\otimes F). \end{eqnarray} $  (11) 
We now give the solvability condition of cooperative output regulation for switched agent networks.
Theorem 1. Let Assumptions 1
$ \begin{align} &(A_{ij_{i}}+B_{ij_{i}}K_{ij_{i}})^ \mathit{\boldsymbol{ T }} P_{ij_{i}}+P_{ij_{i}}(A_{ij_{i}}+B_{ij_{i}}K_{ij_{i}})+\nonumber\\ &\qquad\gamma_{01}I_{n_{i}}+\sum\limits_{j_{s}=1}^{m_{i}}\beta_{ij_{i}j_{s}}(P_{ij_{s}}P_{ij_{i}})<0. \end{align} $  (12) 
Then, the output regulation problem for the system (1) and exosystem (2) is solved by the controller (3) and (4) under the following agentdependent switching law
$ \begin{eqnarray} &\sigma_{i}(t)= \mathop {\arg\min }\limits_{1\leq j_{i}\leq m_{i}}\{x_{i}^{\rm T}P_{ij_{i}}x_{i}\}. \end{eqnarray} $  (13) 
Proof. Under the distributed control laws (3) and (4), the closedloop system of
$ \begin{eqnarray} &&\dot{x_{i}}=(A_{ij_{i}}+B_{ij_{i}}K_{ij_{i}})x_{i}+B_{ij_{i}}G_{ij_{i}} \eta_{i}+E_{ij_{i}}\omega\nonumber \\ &&\dot{\eta_{i}}=S\eta_{i}+\sum\limits_{j\in\Phi_{i}}a_{ij} (\eta_{j}\eta_{i})+a_{i0}(\omega\eta_{i}) \nonumber \\ &&\dot{\omega}=S\omega \nonumber\\ &&e_{i}=C_{i}x_{i}+F\omega. \end{eqnarray} $  (14) 
Let
$ \begin{eqnarray} &&\dot{x}=(A_{\sigma(t)}+B_{\sigma(t)}K_{\sigma(t)})x+B_{\sigma(t)} G_{\sigma(t)}\eta+E_{\sigma(t)}\tilde{\omega} \nonumber\\ \nonumber &&\dot{\eta}=((I_{N}\otimes S)(H\otimes I_{q}))\eta+(H\otimes I_{q}) \tilde{\omega} \\ \nonumber &&\dot{\tilde{\omega}}=(I_{N}\otimes S)\tilde{\omega} \\ &&e=Cx+(I_{N}\otimes F)\tilde{\omega}. \end{eqnarray} $  (15) 
where
First, we will prove the origin of the closedloop system
$ \begin{eqnarray} &&\dot{x}=(A_{\sigma(t)}+B_{\sigma(t)}K_{\sigma(t)})x+B_{\sigma(t)} G_{\sigma(t)}\eta \nonumber\\ &&\dot{\eta}=[(I_{N}\otimes S)(H\otimes I_{q})]\eta \end{eqnarray} $  (16) 
without the disturbance input (i.e.,
$ \begin{align} [(I_{N}\otimes S)&(H\otimes I_{q})]^{\mathit{\boldsymbol{ T }}}Q+Q[(I_{N}\otimes S)(H\otimes I_{q})]+\nonumber\\&\gamma_{02}I_{Nq}<0. \end{align} $  (17) 
Let
$ V_{i}=\sum\limits_{j_{i}=1}^{m_{i}}\iota_{ij_{i}}V_{ij_{i}} $ 
where
For the closedloop system (16), we define the following Lyapunov function
$ \begin{align} V(x, \eta) &=\sum\limits_{i=1}^{N}\sum\limits_{j_{i}=1}^{m_{i}}\iota_{ij_{i}}V_{ij_{i}}(x_{i})+kV(\eta)=\nonumber\\ & \quad\sum\limits_{i=1}^{N}\sum\limits_{j_{i}=1}^{m_{i}}\iota_{ij_{i}}x_{i}^{\rm T}P_{ij_{i}}x_{i}+k\eta^{\rm T}Q\eta \end{align} $  (18) 
where
Without loss of generality, when the
$ \begin{align} \dot{V}(x, \eta)&=\sum\limits_{i=1}\limits^{N}(\dot{x}_i^\mathit{\boldsymbol{ T }}P_{ij_{i}}x_{i}+x_{i}^{\mathit{\boldsymbol{ T }}}P_{ij_{i}}\dot{x}_i)+\nonumber\\&k(\dot{\eta}^{\mathit{\boldsymbol{ T }}}Q\eta+\eta^{\mathit{\boldsymbol{ T }}}Q\dot{\eta})=\nonumber\\ &\sum\limits_{i=1}\limits^{N}\{x_{i}^{\mathit{\boldsymbol{ T }}}[(A_{ij_{i}}+B_{ij_{i}}K_{ij_{i}})^ \mathit{\boldsymbol{ T }} P_{ij_{i}}+\nonumber\\ &P_{ij_{i}}(A_{ij_{i}}+B_{ij_{i}}K_{ij_{i}})]x_{i}+2x_{i}^{\mathit{\boldsymbol{T}}} P_{ij_{i}}B_{ij_{i}}G_{ij_{i}}\eta_{i}\}+\nonumber\\ &k\eta^{\mathit{\boldsymbol{T}}}\{[(I_{N}\otimes S)(H\otimes I_{q})]^{\mathit{\boldsymbol{ T }}}Q+\nonumber\\ &Q[(I_{N}\otimes S)(H\otimes I_{q})]\}\eta<\nonumber\\ &\sum\limits_{i=1}\limits^{N}\{\sum\limits_{j_{s}=1}\limits^{m_{i}} [\beta_{ij_{i}j_{s}}x_{i}{^{\rm T}}(P_{ij_{s}}P_{ij_{i}})x_{i}] \gamma_{01}\x_{i}\_{2}^{2}+\nonumber\\ &2x_{i}^{\mathit{\boldsymbol{T}}}P_{ij_{i}}B_{ij_{i}}G_{ij_{i}}\eta_{i}\} k\gamma_{02}\\eta\_{2}^{2}, (x, \eta)^{\rm{T}}\neq 0. \end{align} $  (19) 
We know that there exist constants
$ \x_{i}^{\mathit{\boldsymbol{T}}}P_{ij_{i}}\_{2}\leq\alpha_{i}\x_{i}\_{2}, ~~\B_{ij_{i}}G_{ij_{i}}\eta_{i}\_{2}\leq\delta_{i}\\eta_{i}\_{2}. $ 
Let
$ \begin{align} \dot{V}(x, \eta)&< \sum\limits_{i=1}\limits^{N}\{\sum\limits_{j_{s}=1}\limits^{m_{i}}[\beta_{ij_{i}j_{s}}x_{i}{^{\rm T}}(P_{ij_{s}}P_{ij_{i}})x_{i}]\nonumber\\ &\sum\limits_{i=1}\limits^{N}\gamma_{01}(\x_{i}\_{2} \dfrac{\theta}{{\gamma_{01}\parallel \eta_{i} \parallel_{2}}})^{2}\}+\nonumber\\ &\sum\limits_{i=1}\limits^{N}\dfrac{\theta^{2}}{\gamma_{01}^{2}}\\eta_{i} \^{2}_{2}k\gamma_{02}\\eta\_{2}^{2}<\nonumber\\ & \sum\limits_{i=1}\limits^{N}\left\{\sum\limits_{j_{s}=1}\limits^{m_{i}}\left[\beta_{ij_{i}j_{s}}x_{i}{^{\rm T}}(P_{ij_{s}}P_{ij_{i}})x_{i}\right]\right\}\nonumber\\ &\dfrac{k\gamma_{02}\theta^{2}{\gamma_{01}^{2}}}{\\eta\_{2}^{2}}, ~~ (x, \eta)^{\rm{T}}\neq 0. \end{align} $  (20) 
Choose
$ \begin{eqnarray} k>\frac{\theta^{2}}{\gamma_{02}\gamma_{01}^{2}}. \end{eqnarray} $  (21) 
We have
$ \begin{align} \dot{V}(x, \eta)&<\sum\limits_{i=1}\limits^{N}\sum\limits_{j_{s}=1}\limits^{m_{i}}x_{i}{ ^{\rm T}}[\beta_{ij_{i}j_{s}}(P_{ij_{s}}P_{ij_{i}})]x_{i}, \nonumber\\ &(x, \eta)^{\rm{T}}\neq 0. \end{align} $  (22) 
Under the switching law (13), we obtain that
Then, we will prove the second condition of the cooperative output regulation problem of switched MASs. Rewrite the closedloop system (15) as
$ \begin{eqnarray} &&\dot{x_{c}}=A_{cj}x_{c}+E_{cj}\tilde{\omega} \nonumber\\ &&\dot{\tilde{\omega}}=(I_{N}\otimes S)\tilde{\omega} \nonumber\\ &&e=C_{c}x_{c}+F_{c}\tilde{\omega} \end{eqnarray} $  (23) 
where
$ x_{c}= \left( \begin{array}{c} x\\ \eta\\ \end{array} \right) $ 
$ A_{cj}= \left( \begin{array}{c c} A_{j}+B_{j}K_{j}&B_{j}G_{j}\\ 0&(I_{N}\otimes S)(H\otimes I_{q})\\ \end{array} \right) $ 
$ E_{cj}= \left( \begin{array}{c} E_{j}\\ H\otimes I_{q}\\ \end{array} \right), C_{c}=\ \left( \begin{array}{c c} C&0\\ \end{array} \right) $ 
$ F_{c}=I_{N}\otimes F, j\in M_{1}\times M_{2}\times \cdots M_{N}. $ 
By (11), we have
$ \begin{align*} A_{cj}X_{c}+E_{cj}&= \left( \begin{array}{c c} A_{j}+B_{j}K_{j}&B_{j}G_{j}\\ 0&(I_{N}\otimes S)(H\otimes I_{q})\\ \end{array} \right) \\& \left( \begin{array}{c} X \\ I_{qN }\\ \end{array} \right)+ \left( \begin{array}{c} E_{j}\\ H\otimes I_{q}\\ \end{array} \right) =\\ &\left( \begin{array}{c} (A_{j}+B_{j}K_{j})X +B_{j}G_{j}+E_{j}\\ I_{N}\otimes S\\ \end{array} \right) =\\& \left( \begin{array}{c} X(I_{N}\otimes S)\\ I_{N}\otimes S\\ \end{array} \right) \left( \begin{array}{c} X\\ I_{qN}\\ \end{array} \right) (I_{N}\otimes S) \end{align*} $ 
and
Using the result of Lemma 2, we obtain
$ \begin{equation} \lim\limits_{t\rightarrow\infty}e(t)=0. \end{equation} $  (24) 
Remark 4. According to Theorem 1, even though the output regulation problem is solvable for none of the subsystems of each agent, the output regulation can still be achieved for the switched agent networks under the designed switching law. This certainly increases the possibility of the solvability of the output regulation problem for switched multiagent systems.
Remark 5. Applying the Schur complement lemma, we can convert (12) to the linear matrix inequalities (LMIs)
$ \left(\!\!\!\! \begin{array}{ccccc} X_{ij_{i}}^{\rm T}A_{ij_{i}}^{\rm T}+\\R_{ij_{i}}^{\rm T}B_{ij_{i}}^{\rm T}+\\ A_{ij_{i}}X_{ij_{i}}+\\B_{ij_{i}}R_{ij_{i}}+ \\\sum\limits_{j_{s}=1}\limits^{m_{i}}\beta_{ij_{i}j_{s}}X_{ij_{i}}~~ \!\! & \!\! \sqrt{\beta_{ij_{i}1}}X_{ij_{i}} \!\!&\!\! \sqrt{\beta_{ij_{i}2}}X_{ij_{i}} \!\!& \cdots &\!\! \sqrt{\beta_{ij_{i}m_{i}}}X_{ij_{i}}~~\\ \sqrt{\beta_{ij_{i}1}}X_{ij_{i}}&X_{i1}& 0 &\cdots &0 \\ \sqrt{\beta_{ij_{i}2}}X_{ij_{i}}&0& X_{i2} &\cdots &0 \\ \vdots&\vdots& \vdots & \cdots &\vdots \\ \sqrt{\beta_{ij_{i}m_{i}}}X_{ij_{i}}&0& 0 & \cdots& X_{im_{i}} \end{array} \!\!\!\!\!\!\right)\!<0 $ 
where
Remark 6. The conventional statedependent switching law may lead to Zeno effect, which can degrade and even damage the performance of the systems. Hysteresis switching approach can be adopted to avoid Zeno behavior, e.g., see ^{[33, 34]} and the references therein.
Remark 7. According to the proof of Theorem 1, we know that the cooperative output regulation problem of the switched MASs is solved when
In what follows, we present a numerical example to illustrate the effectiveness of our result. Consider the switched MASs described by
$ \begin{eqnarray} &&\dot{x_{i}}=A_{ij_{i}}x_{i}+B_{ij_{i}}u_{i}+E_{ij_{i}}\omega \nonumber\\ &&e_{i}=C_{i}x_{i}+F\omega, i=1, 2, ~j_{1}=1, 2, ~j_{2}=1, 2. \end{eqnarray} $  (25) 
The exosystem is given by
$ \begin{eqnarray} &\dot{\omega}=S\omega \end{eqnarray} $  (26) 
where
$ A_{11}=\left( \begin{array}{c c} 1&2\\ 3&1\\ \end{array} \right), A_{12}= \left( \begin{array}{c c} 1&2\\ 4&5\\ \end{array} \right), B_{11}=\left( \begin{array}{c} 0\\ 1\\ \end{array} \right) $ 
$ B_{12}=\left( \begin{array}{c} 1\\ 0\\ \end{array} \right), E_{11}= \left( \begin{array}{c c} 1&1\\ 2&1\\ \end{array} \right), E_{12}= \left( \begin{array}{c c} 3&4\\ 1&2\\ \end{array} \right) $ 
$ A_{21}= \left( \begin{array}{c c} 2&1\\ 0&0\\ \end{array} \right), A_{22}=\left( \begin{array}{c c} 0&1\\ 1&0\\ \end{array} \right), B_{21}=\left( \begin{array}{c} 0\\ 1\\ \end{array} \right) $ 
$ B_{22}=\left( \begin{array}{c} 1\\ 0\\ \end{array} \right), E_{21}=\left( \begin{array}{c c} 2&0\\ 1&0\\ \end{array} \right), E_{22}=\left( \begin{array}{c c} 1&0\\ 0&0\\ \end{array} \right) $ 
$ C_{1}=\left( \begin{array}{c c} 1&1\\ 0&1\\ \end{array} \right), C_{2}=\left( \begin{array}{c c} 1&0\\ 1&1\\ \end{array} \right), F=\left( \begin{array}{c c} 1&0\\ 1&1\\ \end{array} \right) $ 
$ S=\left( \begin{array}{c c} 0&1\\ 1&0\\ \end{array} \right). $ 
By Assumption 3, we give the matrix
$ \begin{array}{c} H=DA+A_{0}=\left( \begin{array}{c c} 0&0\\ 0&1\\ \end{array} \right) \left( \begin{array}{c c} 0&0\\ 1&0\\ \end{array} \right) +\\ \left( \begin{array}{c c} 1&0\\ 0&0\\ \end{array} \right)=\left( \begin{array}{c c} 1&0\\ 1&1\\ \end{array} \right). \end{array} $ 
Solving the linear matrix (6), we obtain
$ X_{1}=\left( \begin{array}{c c} 0&1\\ 1&1\\ \end{array} \right), U_{11}=\left( \begin{array}{c c} 2&6\\ \end{array} \right) $ 
$ U_{12}=\left( \begin{array}{c c} 5&2\\ \end{array} \right) $ 
$ X_{2}=\left( \begin{array}{c c} 1&0\\ 0&1\\ \end{array} \right), U_{21}=\left( \begin{array}{c c} 2&0\\ \end{array} \right) $ 
$ U_{22}=\left( \begin{array}{c c} 1&0\\ \end{array} \right). $ 
Let
$ P_{11}=\left( \begin{array}{c c} 0.013 3&0.006 3\\ 0.006 3&0.007 9\\ \end{array} \right) $ 
$ P_{12}=\left( \begin{array}{c c} 0.008 1&0.012 3\\ 0.012 3&0.037 6\\ \end{array} \right) $ 
$ P_{21}=\left( \begin{array}{c c} 0.012 9&0.002 2\\ 0.002 2&0.000 8\\ \end{array} \right) $ 
$ P_{22}=\left( \begin{array}{c c} 0.001 4&0.000 6\\ 0.000 6&0.001 6\\ \end{array} \right) $ 
$ K_{11}=\left( \begin{array}{c c} 7.217 6&1.982 4\\ \end{array} \right) $ 
$ K_{12}=\left( \begin{array}{c c} 8.935 2&19.187 8\\ \end{array} \right) $ 
$ K_{21}=\left( \begin{array}{c c} 16.631 4&3.464 9\\ \end{array} \right) $ 
$ K_{22}=\left( \begin{array}{c c} 1.875 4&0.173 0\\ \end{array} \right). $ 
Choosing
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Fig. 1. The first agent 
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Fig. 2. The first agent 
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Fig. 3. The second agent 
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Fig. 4. he second agent 
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Fig. 5. Errors of switched agents 
5 Conclusions
The cooperative output regulation problem of switched heterogeneous linear agent networks has been investigated. The switching dynamics of each agent make the cooperative output regulation problem more difficult for switched MASs. Owing to the limited information exchanges among agents, a distributed controller for each agent is established based on a distributed dynamic compensator. On the basis of the traditional multiple Lyapunov function approach, a new agentdependent multiple Lyapunov function approach is presented in order to solve the output regulation problem. Further work will be focused on the output regulation problem of MASs in which each agent dynamics are described by a switched nonlinear system.
AcknowledgementsThis work was supported by National Natural Science Foundation of China (Nos. 61304058 and 61233002), IAPI Fundamental Research Funds (No. 2013ZCX0301), and General Project of Scientific Research of the Education Department of Liaoning Province (No. L2015547).
[1] 
D. Liberzon. Switching in Systems and Control, Boston, USA: Birkhäuser, 2003.

[2] 
H. Lin, P. J. Antsaklis. Stability and stabilizability of switched linear systems:A survey of recent results. IEEE Transactions on Automatic Control, vol.54, no.2, pp.308322, 2009. DOI:10.1109/TAC.2008.2012009 
[3] 
C. Tomlin, G. J. Pappas, S. Sastry. Conflict resolution for air traffic management:A study in multiagent hybrid systems. IEEE Transactions on Automatic Control, vol.43, no.4, pp.509521, 1998. DOI:10.1109/9.664154 
[4] 
D. Jeon, M. Tomizuka. Learning hybrid force and position control of robot manipulators. IEEE Transactions on Automatic Control, vol.9, no.4, pp.423431, 1993. 
[5] 
S. M. William, R. G. Hoft. Adaptive frequency domain control of PWM switched power line conditioner. IEEE Transactions on Power Electronics, vol.6, no.4, pp.665670, 1991. DOI:10.1109/63.97766 
[6] 
D. Liberzon, A. S. Morse. Basic problems in stability and design of switched systems. IEEE Control Systems, vol.19, no.5, pp.5770, 1999. 
[7] 
X. Q. Zhang, J. Zhao. L_{2}gain Analysis and antiwindup design of discretetime switched systems with actuator saturation. International Journal of Automation and Computing, vol.9, no.4, pp.369377, 2012. DOI:10.1007/s116330120657x 
[8] 
P. Peleties, R. DeCarlo. Asymptotic stability of mswitched systems using Lyapunovlike functions. In Proceedings of the American Control Conference, IEEE, Boston, USA, USA, pp. 16791684, 1991.

[9] 
J. Lu, L. J. Brown. A multiple Lyapunov functions approach for stability of switched systems. In Proceedings of American Control Conference, IEEE, Baltimore, USA, pp. 32533256, 2010. http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=5530464

[10] 
B. Niu, J. Zhao. Robust H_{∞} control for a class of switched nonlinear cascade systems via multiple Lyapunov functions approach. Applied Mathematics and Computation, vol.218, no.11, pp.63306339, 2012. DOI:10.1016/j.amc.2011.09.059 
[11] 
J. P. Hespanha, A. S. Morse. Stability of switched systems with average dwelltime. In Proceedings of the 38th IEEE Conference on Decision and Control, IEEE, Phoenix, USA, pp. 26552660, 1999. http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=831330

[12] 
G. S. Zhai, B. Hu, K. Yasuda, A. N. Michel. Stability analysis of switched systems with stable and unstable subsystems:An average dwell time approach. International Journal of Systems Science, vol.32, no.8, pp.10551061, 2001. DOI:10.1080/00207720116692 
[13] 
W. Ren, Y. C. Cao. Distributed Coordination of MultiAgent Networks:Emergent Problems, Models, and Issues, London, UK: SpringerVerlag, 2011.

[14] 
P. P. Dai, C. L. Liu, F. Liu. Consensus problem of heterogeneous multiagent systems with time delay under fixed and switching topologies. International Journal of Automation and Computing, vol.11, no.3, pp.340346, 2014. DOI:10.1007/s1163301407981 
[15] 
M. I. Menhas, L. Wang, M. R. Fei, C. X. Ma. Coordinated controller tuning of a boiler turbine unit with new binary particle swarm optimization algorithm. International Journal of Automation and Computing, vol.8, no.2, pp.185192, 2011. DOI:10.1007/s1163301105726 
[16] 
R. OlfatiSaber. Flocking for multiagent dynamic systems:Algorithms and theory. IEEE Transactions on Automatic Control, vol.51, no.3, pp.401420, 2006. DOI:10.1109/TAC.2005.864190 
[17] 
B. Das, B. Subudhi, B. B. Pati. Cooperative formation control of autonomous underwater vehicles:An overview. International Journal of Automation and Computing, vol.13, no.3, pp.199225, 2016. DOI:10.1007/s1163301610044 
[18] 
A. Isidori. Nonlinear Control Systems Ⅱ, New York, USA: SpringerVerlag, 1999.

[19] 
J. Huang. Nonlinear Output Regulation:Theory and Applications, Philadelphia, USA: SIAM, 2004.

[20] 
B. A. Francis, W. M. Wonham. The internal model principle for linear multivariable regulators. Applied Mathematics and Optimization, vol.2, no.2, pp.170194, 1975. DOI:10.1007/BF01447855 
[21] 
J. Huang, Z. Y. Chen. A general framework for tackling the output regulation problem. IEEE Transactions on Automatic Control, vol.49, no.12, pp.22032218, 2004. DOI:10.1109/TAC.2004.839236 
[22] 
X. X. Dong, J. Zhao. Solvability of the output regulation problem for switched nonlinear systems. IET Control Theory & Applications, vol.6, no.8, pp.11301136, 2012. 
[23] 
L. J. Long, J. Zhao. Robust and decentralised output regulation of switched nonlinear systems with switched internal model. IET Control Theory & Applications, vol.8, no.8, pp.561573, 2014. 
[24] 
Y. F. Su, J. Huang. Cooperative output regulation of linear multiagent systems. IEEE Transactions on Automatic Control, vol.57, no.4, pp.10621066, 2012. DOI:10.1109/TAC.2011.2169618 
[25] 
Z. T. Ding. Consensus output regulation of a class of heterogeneous nonlinear systems. IEEE Transactions on Automatic Control, vol.58, no.10, pp.26482653, 2013. DOI:10.1109/TAC.2013.2255973 
[26] 
J. Xiang, W. Wei, Y. J. Li. Synchronized output regulation of linear networked systems. IEEE Transactions on Automatic Control, vol.54, no.6, pp.13361341, 2009. DOI:10.1109/TAC.2009.2015546 
[27] 
D. B. Xu, Y. G. Hong, X. H. Wang. Distributed output regulation of nonlinear multiagent systems via host internal model. IEEE Transactions on Automatic Control, vol.59, no.10, pp.27842789, 2014. DOI:10.1109/TAC.2014.2313171 
[28] 
Y. G. Hong, L. X. Gao, D. Z. Cheng, J. P. Hu. Lyapunovbased approach to multiagent systems with switching jointly connected interconnection. IEEE Transactions on Automatic Control, vol.52, no.5, pp.943948, 2007. DOI:10.1109/TAC.2007.895860 
[29] 
W. Y. Xu, J. D. Cao, W. W. Yu, J. Q. Lu. Leaderfollowing consensus of nonlinear multiagent systems with jointly connected topology. IET Control Theory & Applications, vol.8, no.6, pp.432440, 2014. 
[30] 
W. Y. Xu, D. W. C. Ho, L. L. Li, J. D. Cao. Eventtriggered schemes on leaderfollowing consensus of general linear multiagent systems under different topologies. IEEE Transactions on Cybernetics, to be published. http://ieeexplore.ieee.org/document/7369964/

[31] 
H. W. Jia, J. Zhao. Output regulation of switched linear multiagent systems:An agentdependent average dwell time method. International Journal of Systems Science, vol.47, no.11, pp.25102520, 2016. DOI:10.1080/00207721.2014.998747 
[32] 
A. CervantesHerrera, J. RuizLeón, C. LópezLimón, A. RamirezTrevino. A distributed control design for the output regulation and output consensus of a class of switched linear multiagent systems. In Proceedings of the 17th International Conference on Emerging Technologies & Factory Automation, IEEE, Krakow, Poland, pp. 17, 2012.

[33] 
A. S. Morse, D. Q. Mayne, G. C. Goodwin. Applications of hysteresis switching in parameter adaptive control. IEEE Transactions on Automatic Control, vol.37, no.9, pp.13431354, 1992. DOI:10.1109/9.159571 
[34] 
J. P. Hespanha, D. Liberzon, A. S. Morse. Hysteresisbased switching algorithms for supervisory control of uncertain system. Automatica, vol.39, no.2, pp.263272, 2003. DOI:10.1016/S00051098(02)002418 