ℋ_∞ leader-following consensus of multi-agent systems with channel fading under switching topologies: a semi-Markov kernel approach

2.1. Graph theory

In this paper, we employed an undirected graph $$ \mathcal{G}=\{\mathcal{V},\mathcal{E},\mathcal{A}\} $$ to depict the information interaction topology among $$ N $$ agents. $$ \mathcal{V}=\{v_{1},v_{2},...,v_{N}\} $$ stands for the node sets, in which $$ v_{i} $$ is the $$ i $$th agent. $$ \mathcal{E}\subset\mathcal{V}\times\mathcal{V} $$ represents a set of edges. The adjacency matrix associated with graph $$ \mathcal{G} $$ is denoted by $$ \mathcal{A}=[a_{ij}]\in \mathbb{R}^{N\times N} $$. If node $$ v_{i} $$ can receive information from node $$ v_{j} $$, there is an edge $$ (v_{j},v_{i}) $$ between node $$ v_{i} $$ and node $$ v_{j} $$. The elements $$ a_{ij} $$ of matrix $$ \mathcal{A} $$ is weighted coefficient of edge $$ (v_{j},v_{i}) $$, and $$ a_{ij}>0 $$, if $$ (v_{j},v_{i})\in \mathcal{E} $$, otherwise, $$ a_{ij}=0 $$. Self-loop is not considered. The set of neighbors of node $$ v_{i} $$ can be represented as $$ \mathcal{N}_{i}=\{v_{j}\in \mathcal{V}|(v_{j},v_{i})\in \mathcal{E}\} $$. The degree matrix of graph $$ \mathcal{G} $$ is denoted as $$ \mathcal{D}=diag\{d_{i}\}\in \mathbb{R}^{N\times N} $$, where $$ d_{i}=\sum_{j\in \mathcal{N}_{i}}a_{ij} $$. Then, one can obtain that the Laplacian matrix is $$ \mathcal{L}=\mathcal{D}-\mathcal{A} $$. Denote matrix $$ \mathcal{M}=diag\{m_{i}\}\in \mathbb{R}^{N\times N} $$, where $$ m_{i} $$ stands for the information exchange of node $$ v_{i} $$ and leader node. If node $$ v_{i} $$ can access the information of the leader, $$ m_{i}=1 $$, otherwise, $$ m_{i}=0 $$.

2.2. Problem formulation

Consider a linear discrete-time multi-agent system consist of one leader and $$ N $$ follower agents:

where $$ x_{i}(k)\in\mathbb{R}^{n_{x}} $$, $$ u_{i}(k)\in\mathbb{R}^{m} $$, $$ x_{0}(k)\in\mathbb{R}^{n_{x}} $$ are the state, input of the $$ i $$th follower, the state of leader, respectively. Matrices $$ A\in\mathbb{R}^{n_{x}\times n_{x}} $$ and $$ B\in\mathbb{R}^{n_{x}\times m} $$ represent known constant system matrices.

In the real world, the communication network topology among agents is more likely to be time-varying. In this paper, a switching signal $$ \gamma(k) $$ is used to characterize the topology switching among agents. $$ \{\gamma(k),\; k\in\mathbb{N}^{+}\} $$ represents a discrete-time semi-Markov chain with values in a finite set $$ \mathcal{O}=\{1,2,...,O\} $$.

To describe semi-Markov chain more formally, the following concepts are introduced. (Ⅰ) The stochastic process $$ \{U_{n},n\in\mathbb{N}^{+}\}\in \mathbb{N}^{+} $$ is denoted as the mode index of the $$ n $$th jump, in which taking values in $$ \mathcal{O} $$. (Ⅱ) The stochastic process $$ \{k_{n},n\in\mathbb{N}^{+}\}\in\mathbb{N}^{+} $$ represents the time instant of at the $$ n $$th jump. (Ⅲ) The stochastic process $$ \{S_{n},n\in\mathbb{N}^{+}\}\in\mathbb{N}^{+} $$ stands for the sojourn-time of mode $$ U_{n-1} $$ between the $$ (n-1) $$th jump and $$ n $$th jump, where $$ S_{n}=k_{n}-k_{n-1} $$.

Definition 1^[39] The stochastic process $$ \{(U_{n},k_{n}),n\in\mathbb{N}^{+}\} $$ is said to be a discrete-time homogeneous Markov renewal chain (MRC), if the following conditions holds for all $$ p,\; q\in\mathcal{O} $$, $$ \tau\in\mathbb{N}^{+},\; n\in\mathbb{N}^{+} $$:

where $$ \{U_{n},n\in\mathbb{N}^{+}\} $$ is named as the embedded Markov chain (EMC) of MRC.

Denote the matrix $$ \Pi(\tau)=[\pi_{pq}(\tau)]\in \mathbb{R}^{O\times O} $$ as the discrete-time semi-Markov kernel with

where $$ \sum_{\tau=0}^{\infty}\sum_{q\in\mathcal{O}}\pi_{pq}(\tau)=1 $$ and $$ 0<\pi_{pq}(\tau)<1 $$ with $$ \pi_{pq}(0)=0 $$. The transition probability of EMC is defined by $$ \theta_{pq}=Pr\{U_{n+1}=q|U_{n}=p\}, \forall p,q\in\mathcal{O} $$ with $$ \theta_{pp}=0 $$, and the probability density function of sojourn-time is provided by $$ \omega_{pq}(\tau)=Pr\{S_{n+1}|U_{n+1}=q,U_{n}=p\} $$, $$ \omega_{pq}(0)=0 $$.

Remark 1 References ^{[35, 37, 38]} studied the leader-following consensus and containment control problems for multi-agent systems with semi-Markov switching topologies, respectively. A continuous-time semi-Markov jump process is employed to describe the switching of the topology. Accordingly, the probability density function of the sojourn-time can only be of a fixed probability distribution type for the different modes. This limits its practical application. In this paper, a discrete semi-Markov chain is introduced to characterize topology switching among agents. The introduced probability density function of sojourn time depends on both the current mode and the next mode, so that different parameters of the same distribution or different types of probability distributions can coexist. Hence, the probability density function introduced in this paper is more applicable than that in the literature ^{[35, 37, 38]}.

Definition 2^[39] The stochastic process $$ \{\gamma(k),k\in\mathbb{N}^{+}\} $$ is said to be an semi-Markov chain associated with MRC $$ \{(U_{n},k_{n}),n\in\mathbb{N}^{+}\} $$, if $$ \gamma(k)=U_{\mathbb{N}(k)}, \forall k\in\mathbb{N}^{+} $$, $$ \mathbb{N}(k)=\max\{n\in\mathbb{N}^{+}|k_{n}\leq k\} $$.

Based on the above introduction to the semi-Markov chain, the switching topology among agents in this paper can be denoted as $$ \mathcal{G}_{\gamma(k)} $$. For convenience, let $$ \mathcal{G}_{\gamma(k)}=\mathcal{G}_{p},\; p\in\mathcal{O} $$.

Every topological graph $$ \mathcal{G}_{p} $$ is an undirected graph. Then, the Laplacian matrix of the graph $$ \mathcal{G}_{p} $$, the adjacency matrix of the leader and the follower are denoted by $$ \mathcal{L}_{p}\in\mathbb{R}^{N\times N} $$, $$ \mathcal{A}_{p}\in\mathbb{R}^{N\times N} $$, and $$ \mathcal{M}_{p}\in\mathbb{R}^{N\times N} $$.

In practice, the communication process between an agent and its neighbors is often affected by channel noise and fading. Motivated by the channel fading model in ^{[28, 29]}, in this paper we assume that each agent obtains relative state information from its neighbors through fading channels. Correspondingly, the channel fading model among agents can be expressed as

where $$ \xi_{ij}(k) $$, $$ \xi_{i0}(k) $$ represent the channel fading of the follower-to-follower and the leader-to-follower, respectively. $$ \varpi_{ij}(k) $$ and $$ \varpi_{i0}(k) $$ are disturbances in the channel. Based on the above channel fading model, we design a distributed consensus controller as follows:

where $$ K_{p}\in \mathbb{R}^{m\times n_{x}} $$ is the controller gain to be determined, $$ p\in\mathcal{O} $$.

Figure 1 shows the frame diagram of the system considered in this paper. Under the semi-Markov switching communication network topology, the channel fading between agents varies randomly with the switching of the topology. Each agent generates control inputs based on the relative information obtained from neighbors through fading and interference, thereby further controlling the entire system to achieve consensus.

Figure 1. Multi-agent systems with channel fading under semi-Markov switching topologies

Remark 2 Compared with the channel fading model established in the literature ^[27-29], the channel fading model introduced in this paper not only considers the fading influence from leader to follower and follower to follower, but also introduces the effect of channel interference on signal transmission and the influence of topology switching on the statistical characteristics of fading coefficients. This means that the models presented in this paper are more general than those in the previous literature. When $$ \varpi_{ij}(k)=\varpi_{i0}(k)=0 $$, $$ \xi_{i0}(k)\equiv1 $$ in equation (3), the model degenerates to the case in ^[27]. If the values of $$ \xi_{ij}(k) $$ and $$ \xi_{i0}(k) $$ are 0 or 1 and $$ \varpi_{ij}(k)=\varpi_{i0}(k)=0 $$, the channel fading model (3) is reduced to a packet loss model. This also indicates that the channel fading model proposed in this paper is more general.

Define the consensus error as $$ \delta_{i}(k)=x_{i}(k)-x_{0}(k) $$, $$ i=1,2,...,N $$. Combining system (1) and consensus protocol (4), the dynamics of $$ \delta_{i}(k) $$ can be given by

with $$ \delta_{i}(k)=[\delta_{1}^{T}(k),\delta_{2}^{T}(k),...,\delta_{N}^{T}(k)]^{T} $$. Assume that all channel fading are identical, ie., $$ \xi_{ij}(k)=\xi(k) $$, $$ \xi_{i0}(k)=\xi_{0}(k) $$, $$ \varpi_{ij}(k)=\varpi(k) $$, and $$ \varpi_{i0}(k)=\varpi_{0}(k) $$ for $$ k\geq 0 $$, $$ i,j=1,2,...,N $$. Then, the above equation can be rewritten in compact form as follows:

where $$ \omega(k)=[\omega_{1}^{T}(k),\omega_{2}^{T}(k),...,\omega_{N}^{T}(k)]^{T} $$, $$ \omega_{i}(k)=\sum_{j=1}^{N}a_{ij}^{p}(k)\varpi_{p}(k)+m_{i}^{p}(k)\varpi_{0p}(k) $$, $$ i=1,2,...,N $$.

Before the subsequent analysis, some definition and assumptions are introduced.

Definition 3 The systems (6) is said to achieve leader-following consensus in mean square sense, if the system (6) is $$ \sigma $$-error mean square stable.

Definition 4^[39] The dynamic system (6) is said to be $$ \sigma $$-error mean square stable, if the following conditions hold

for given the upper bound of sojourn-time $$ T_{max}^{p}\in\mathbb{N}^{+} $$ and any initial conditions $$ \delta_{i}(0) $$, $$ \gamma(0)\in\mathcal{O} $$, $$ i\in\{1,2,...,N\} $$, $$ \omega(k)=0 $$.

Assumption 1 Every possible undirected graph $$ \mathcal{G}_{\gamma(k)} $$, $$ \gamma(k)=p\in\mathcal{O} $$ is connected.

Assumption 2 The mean and variance of stochastic variables $$ \{\xi_{p}(k)\} $$ and $$ \{\xi_{0p}(k)\} $$ are $$ \mathbb{E}\{\xi_{p}(k)\}=\mu_{p} $$, $$ \mathbb{E}\{\xi_{0p}(k)\}=\mu_{0p} $$, $$ \mathbb{E}\{(\xi_{p}(k)-\mu_{p})^{2}\}=\sigma_{p}^{2} $$, and $$ \mathbb{E}\{(\xi_{0p}(k)-\mu_{0p})^{2}\}=\sigma_{0p}^{2} $$.

According to the above analysis and discussion, it can be found that the leader-following consensus of system (1) under the semi-Markov switching topology $$ \mathcal{G}_{p} $$ is equivalent to the mean square stability of system (6). Since $$ \varpi_{p}(k) $$ and $$ \varpi_{0p}(k) $$ are interference in the channel, we can treat the last term in equation (6) as a disturbance. To tackle with the disturbance, the following control output are given

with $$ z_{i}(k)=E(x_{i}(k)-x_{0}(k)) $$ and $$ z(k)=[z_{1}^{T}(k),z_{2}^{T}(k),...,z_{N}^{T}(k)]^{T} $$. $$ E\in\mathbb{R}^{n_{x}\times n_{x}} $$ is a known constant matrix. Then, the consensus problem of system (1) is transformed into a $$ \mathcal{H}_{\infty} $$ control problem of the system (6).

Consequently, the objective of this paper is to design the distributed consensus controller such that the following two conditions are satisfied:

(Ⅰ) when $$ \omega(k)=0 $$, the condition (7) holds;

(Ⅱ) the inequality $$ \mathbb{E}\bigg\{\sum\limits_{n=0}^{\infty}\sum\limits_{k=k_{n}}^{k_{n+1}-1}[\|z(k)\|^{2}-\hat{\gamma}^{2}\|\omega(k)\|^{2}]\bigg\}<0 $$ holds for zero-initial condition and any nonzero $$ \omega(k)\in L_{2}(0,+\infty) $$.

3.1. Consensus and $$ \mathcal{H}_{\infty} $$ performance analysis

In the subsection, the leader-following consensus and $$ \mathcal{H}_{\infty} $$ performance analysis of systems (6) and (8) are given. Sufficient conditions for mean square consensus are derived via stochastic Lyapunov function.

Theorem 1 Given scalar $$ T_{max}^{p}\in\mathbb{N}_{\geq 1} $$, if there exist a scalar $$ \hat{\gamma}>0 $$ and a set of symmetric matrices $$ P_{p}^{(\nu)}\in\mathbb{R}^{n_{x}\times n_{x}}\succ 0 $$, $$ P_{q}^{(\nu)}\in\mathbb{R}^{n_{x}\times n_{x}}\succ 0 $$, $$ \nu\in\mathbb{N}_{[0,T_{max}^{p}]} $$ such that the following inequalities

hold for any $$ p,q\in\mathcal{O} $$, $$ \nu\in\mathbb{N}_{[1,T_{max}^{p}]} $$, then the system (6) is leader-following consensus in mean square sense and possesses a $$ \mathcal{H}_{\infty} $$ performance index $$ \hat{\gamma} $$, where $$ \Xi=[\bar{A}_{p}^{T}\; \; \sigma_{p}\mathcal{\bar{L}}_{p}^{T}\; \; \sigma_{0p}\mathcal{\bar{M}}_{p}^{T}]^{T} $$, $$ \Psi_{23}=[(I_{N}\otimes BK_{p})^{T}\; \; 0\; \; 0]^{T} $$, $$ \bar{A}_{p}=[I_{N}\otimes A+(\mu_{p}\mathcal{L}_{p}+\mu_{0p}M_{p})\otimes BK_{p}] $$, $$ \mathcal{\bar{M}}_{p}=\mathcal{M}_{p}\otimes BK_{p} $$, $$ \mathcal{\bar{L}}_{p}=\mathcal{L}_{p}\otimes BK_{p} $$,

Proof Construct a stochastic Lyapunov function candidate as $$ V(\delta(k),\gamma(k),\nu(k))=\delta^{T}(k)(I_{N}\otimes P_{\gamma(k)}^{(\nu(k))})\delta(k) $$, where $$ \gamma(k)=p\in\mathcal{O} $$, $$ k\in[k_{n},k_{n+1}) $$. $$ \nu(k)=k-k_{n} $$ represents the running time of the current mode for topology $$ \mathcal{G}_{p} $$. Assume that $$ \gamma(k_{n})=p $$, $$ \gamma(k_{n+1})=q $$, $$ \omega(k)\equiv 0 $$. Then, one can obtain along the solution of systems (6) and (8)

where $$ \mathcal{\tilde{A}}_{p}=I_{N}\otimes A+(\xi_{p}(k)\mathcal{L}_{p}+\xi_{0p}(k)\mathcal{M}_{p})\otimes BK_{p} $$, $$ \mathcal{\tilde{A}}_{p}\in\mathbb{R}^{n_{x}N\times n_{x}N} $$. It can be found from the above equation that $$ \Delta V(\delta(k),\gamma(k),\nu(k))<0 $$ if and only if $$ \Sigma\in\mathbb{R}^{n_{x}N\times n_{x}N}<0 $$,

On the other hand, the stability of the system during mode switching needs to be considered. It can be derived along the trajectory of systems (6) and (8) that

with $$ \mathcal{P}=(I_{N}\otimes P_{q}^{(\nu(k_{n+1}))}) $$. By means of the condition (10) and Schur complement lemma, one can get that

Together with the condition (12), there exists a positive constant $$ \beta $$ such that gives

Then, it follows that

Summing both sides of the above equation from 0 to $$ l $$ yields

Let $$ l\rightarrow \infty $$, then $$ \lim\limits_{l\rightarrow \infty}\mathbb{E}\big\{\Sigma_{n=0}^{l}\delta^{T}(k_{n})\delta(k_{n})\big\}<\frac{1}{\beta}V(\delta(k_{0}),\gamma(k_{0}), $$$$\nu(k_{0})) $$. Then, we have $$ \lim\limits_{n\rightarrow \infty}\mathbb{E}\{\delta^{T}(k_{n})\delta(k_{n})\}=\lim\limits_{k\rightarrow \infty}\mathbb{E}\{\delta^{T})(k\delta(k)\}=0 $$. Furthermore, it can be shown that $$ \lim\limits_{k\rightarrow \infty}\mathbb{E}\big\{\|\delta_{i}(k)\|^{2}\big\}=0 $$. By Definition 3 and Definition 4, systems (6) and (8) with $$ \omega(k)\equiv 0 $$ is leader-following mean square consensus.

Next, the $$ \mathcal{H}_{\infty} $$ consensus performance of the underlying system is discussed. Then, for $$ \omega(k)\neq 0 $$, we have

Denote $$ \eta(k)=[\delta^{T}(k)\; \; \omega^{T}(k)]^{T} $$, then the equation (16) can be transformed into

where

Further, we can obtain

under the zero initial conditions. Thus, the $$ \mathcal{H}_{\infty} $$ performance condition (Ⅱ) holds. This proof is completed.

In Theorem 1, the leader-following consensus and $$ \mathcal{H}_{\infty} $$ performance of the system (6) with channel fading is analyzed, in which data transmission between agents takes into account not only channel fading but also channel interference. Assuming that the channel interference $$ \varpi_{ij}^{p}(k)=0 $$, $$ \varpi_{i0}^{p}(k)=0 $$, that is $$ \omega(k)=0 $$. Then, the system (6) can be reduced to

In this case, the following corollary gives the leader-following mean square consensus analysis of the system (18) under no-channel interference fading model.

Corollary 1 Given scalar $$ T_{max}^{p}\in\mathbb{N}_{\geq 1} $$, if there exist a sets of symmetric matrices $$ P_{p}^{(\nu)}\in\mathbb{R}^{n_{x}\times n_{x}}\succ 0 $$, $$ P_{q}^{(\nu)}\in\mathbb{R}^{n_{x}\times n_{x}}\succ 0 $$, $$ \nu\in\mathbb{N}_{[0,T_{max}^{p}]} $$ such that the following inequalities

hold for $$ p,q\in\mathcal{O} $$, $$ \nu\in\mathbb{N}_{[1,T_{max}^{p}]} $$, $$ \tau\in\mathbb{N}_{[1,T_{max}^{p}]} $$, then the system (18) is leader-following mean square consensus, where $$ \Xi $$, $$ \mathcal{P}_{p}^{(\nu)} $$, and $$ \mathcal{\tilde{P}}_{q}^{(0)} $$ have been defined in Theorem 1.

The proof of Corollary 1 is similar to that of Theorem 1, so it is omitted.

3.2. Consensus controller gain design

Although Theorem 1 addresses a family of leader-following consensus conditions, these conditions cannot be directly utilized to solve controller gains $$ K_{p} $$. Aiming at solving the leader-following consensus control problem of systems (6) and (8) under switching topologies, sufficient conditions on the existence of the desired controller gains are presented in the following theorem.

Theorem 2 Given a scalar $$ T_{max}^{p}\in\mathbb{N}_{\geq 1} $$, if there exist a scalar $$ \hat{\gamma}>0 $$ and sets of symmetric matrices $$ P_{p}^{(\nu)}\in\mathbb{R}^{n_{x}\times n_{x}}\succ 0 $$, $$ P_{q}^{(\nu)}\in\mathbb{R}^{n_{x}\times n_{x}}\succ 0 $$, and matrices $$ Z_{p}\in\mathbb{R}^{n_{x}\times n_{x}} $$, $$ Y_{p}\in\mathbb{R}^{n_{x}\times n_{x}} $$, $$ p,q\in\mathcal{O} $$, $$ \nu\in\mathbb{N}_{[0,T_{max}^{p}]} $$ such that the following inequalities

hold for any $$ p,q\in\mathcal{O} $$, $$ \nu\in\mathbb{N}_{[1,T_{max}^{p}]} $$, where

$$ \bar{\Xi}=[\tilde{A}_{p}^{T}\; \; \sigma_{p}\mathcal{\tilde{L}}_{p}^{T}\; \; \sigma_{0p}\mathcal{\tilde{M}}_{p}^{T}]^{T} $$, $$ \bar{\Psi}_{23}=[(I_{N}\otimes Y_{P})\; \; 0\; \; 0]^{T} $$, $$ \tilde{A}_{p}=I_{N}\otimes A+(\mu_{p}\mathcal{L}_{p}+\mu_{0p}\mathcal{M}_{p})\otimes Y_{p} $$, $$ \mathcal{\tilde{L}}_{p}=\mathcal{L}_{p}\otimes Y_{p} $$, $$ \mathcal{\tilde{M}}_{p}=\mathcal{M}_{p}\otimes Y_{p} $$, $$ \mathcal{Z}_{p}=diag\{I_{N}\otimes Z_{p},I_{N}\otimes Z_{p},I_{N}\otimes Z_{p}\} $$$$ \in\mathbb{R}^{3n_{x}N\times 3n_{x}N} $$, $$ \Upsilon_{q}^{(0)}=diag\{\Upsilon,\Upsilon,\Upsilon\} $$, $$ \Upsilon=\Sigma_{\nu=1}^{T_{max}^{p}}\Sigma_{q\in\mathcal{O}}\frac{\pi_{pq}(\nu)}{\Omega_{p}}(I_{N}\otimes P_{q}^{(0)}-I_{N}\otimes Z_{p}- $$$$ I_{N}\otimes Z_{p}^{T}) $$,

then the system (6) is leader-following consensus in mean square sense and possess a $$ \mathcal{H}_{\infty} $$ performance index $$ \hat{\gamma} $$. Moreover, the controller gains are given by $$ K_{p}=(B^{T}B)^{-1}B^{T}(Z_{p}^{T})^{-1}Y_{p} $$.

Proof Performing congruence transformations $$ diag\{I_{n_{x}N},I_{n_{x}N},\mathcal{Z}_{p}^{T},I_{n_{x}N}\} $$ to (9), $$ diag\{I_{nN},\mathcal{Z}_{p}^{T}\} $$ to (10), one can obtain

According to the literature ^[40], we can obtain that for the positive definite matrix $$ U\in\mathbb{R}^{n\times n} $$ and the real matrix $$ X\in\mathbb{R}^{n\times n} $$, there must be $$ (U-X)^{T}U^{-1}(U-X)\succeq 0 $$. Performing cholesky decomposition on the matrix $$ U $$, there must be a lower triangular matrix $$ L\in\mathbb{R}^{n\times n} $$ such that $$ U=LL^{T} $$. Further, we can get $$ (U-X)^{T}U^{-1}(U-X)=Q^{T}Q\succeq 0 $$, where $$ Q=L^{T}-L^{-1}X $$. When the matrix $$ Q $$ is full rank, $$ (U-X)^{T}U^{-1}(U-X)\succ 0 $$ holds. Based on this, the inequalities

can ensure

for any $$ p,q\in\mathcal{O} $$, $$ \nu\in\mathbb{N}_{[1,T_{max}^{p}]} $$. Then, it can be shown that the inequalities (21) and (22) can ensure that the conditions (19) and (20) hold. Furthermore, it implies that the inequalities (9) and (10) are true. By Theorem 1, we can get that systems (6) and (8) is leader-following consensus in mean square sense with a $$ \mathcal{H}_{\infty} $$ performance index $$ \hat{\gamma} $$ under the controller (4) and the controller gains $$ K_{p}=(B^{T}B)^{-1}B^{T}(Z_{p}^{T})^{-1}Y_{p} $$. This proof is completed.

Supposing that the channel inferences $$ \varpi_{ij}^{p}(k) $$ and $$ \varpi_{0i}^{p}(k) $$ in (4) are ignored, that is, $$ \varpi_{ij}^{p}(k)=0 $$, $$ \varpi_{i0}^{p}(k)=0 $$. Then, we have $$ \omega(k)=0 $$ in (6). Consequently, the leader-following consensus control of system (18) under semi-Markov switching topology is realized in the following corollary.

Corollary 2 Given a scalar $$ T_{max}^{p}\in\mathbb{N}_{\geq 1} $$, if there exist a sets of symmetric matrices $$ P_{p}^{(\nu)}\in\mathbb{R}^{n_{x}\times n_{x}}\succ 0 $$, $$ P_{q}^{(\nu)}\in\mathbb{R}^{n_{x}\times n_{x}}\succ 0 $$, and matrices $$ Z_{p}\in\mathbb{R}^{n_{x}\times n_{x}} $$, $$ Y_{p}\in\mathbb{R}^{n_{x}\times n_{x}} $$, $$ p,q\in\mathcal{O} $$, $$ \nu\in\mathbb{N}_{[0,T_{max}^{p}]} $$ such that the inequalities hold:

for any $$ p,q\in\mathcal{O} $$, $$ \nu\in\mathbb{N}_{[1,T_{max}^{p}]} $$, then the system (18) is leader-following consensus in mean square sense under the controller gains $$ K_{p}=(B^{T}B)^{-1}B^{T}(Z_{p}^{T})^{-1}Y_{p} $$, where $$ \mathcal{P}_{p}^{(\nu)} $$, $$ \bar{\Xi} $$, $$ \mathcal{Z}_{p} $$, and $$ \Upsilon_{q}^{(0)} $$ are defined in Theorem 1 and 2. By employing the same approach as Theorem 2, the Corollary 2 can be proved directly, omitting the proof process here.

Remark 3 Theorem 2 realizes the consistent controller design of systems (6) and (8) under the channel fading model (3). A set of sufficient conditions to ensure the consensus of systems and the existence of controller gains is established based on the linear matrix inequality form. To solve the controller gain matrix $$ K_{pm} $$, some unknown variables are introduced into the inequality conditions of Theorem 2. The computational complexity of solving the inequality conditions in Theorem 2 can be analyzed according to the total number of unknown variables. It can be obtained by calculation that the total number of unknown variables in Theorem 2 is $$ var_{t}=\Sigma_{p=1}^{O}[(T_{max}^{p}+1)n_{x}^{2}]+ $$$$ \Sigma_{p=1}^{O}[Nn_{x}^{2}+2(Nn_{x})^{2}] $$. It can be found that with the increase of the number $$ O $$ of topological modes, the upper bound $$ T_{max}^{p} $$ of the sojourn time of mode $$ p $$, and the number of agents $$ N $$, the computational complexity of solving Theorem 2 also increases accordingly. Assuming that the upper bound $$ T_{max}^{p},p\in\mathcal{O} $$ of the sojourn time is the same in each topological mode, the total number of unknown variables is $$ var_{t}=O(T_{max}^{p}+1)n_{x}^{2}+O[Nn_{x}^{2}+2(Nn_{x})^{2}] $$. Similarly, the computational complexity of solving the inequality conditions in Theorem 3 can also be analyzed.

3.3. Extension results

In this subsection, it is assumed that the information of the semi-Markov kernel $$ \Pi(\tau) $$ is not completely accessible. By using the similar method in ^[40], the index set $$ \mathcal{O} $$ of the semi-Markov chain $$ \gamma(k) $$ can be partitioned into the following form:

$$ \mathcal{O}_{ap}=\{q\in\mathcal{O}| $$if $$ \pi_{pq}(\tau) $$ is accessible $$ \} $$,

$$ \mathcal{O}_{bp}=\{q\in\mathcal{O}| $$if $$ \pi_{pq}(\tau) $$ is inaccessible $$ \} $$,

$$ \mathcal{O}_{cp}=\{q\in\mathcal{O}| $$if $$ \omega_{pq}(\tau) $$ is accessible, $$ \theta_{pq} $$ is inaccessible $$ \} $$,

$$ \mathcal{O}_{dp}=\{q\in\mathcal{O}| $$if $$ \omega_{pq}(\tau) $$ is inaccessible, $$ \theta_{pq} $$ is accessible $$ \} $$,

$$ \mathcal{O}_{ep}=\{q\in\mathcal{O}| $$if $$ \omega_{pq}(\tau) $$ is inaccessible, $$ \theta_{pq} $$ is inaccessible $$ \} $$,

where $$ \mathcal{O}=\mathcal{O}_{ap}\cup\mathcal{O}_{bp} $$, $$ \mathcal{O}_{bp}=\mathcal{O}_{cp}\cup\mathcal{O}_{dp}\cup\mathcal{O}_{ep} $$, $$ \mathcal{O}_{ap}\cap\mathcal{O}_{bp}=\emptyset $$, $$ \mathcal{O}_{cp}\cap\mathcal{O}_{dp}=\emptyset $$.

In this paper, only the case of $$ \mathcal{O}=\mathcal{O}_{ap}\cup\mathcal{O}_{ep} $$ for incompletely accessible semi-Markov kernel is considered. In other words, the transition probability $$ \theta_{pq} $$ of EMC and the probability density function $$ \omega_{pq}(\tau) $$ of sojourn-time are partially accessible. Before presenting the results of this subsection, we make the following assumptions, which are crucial for subsequent derivations.

Assumption 3 Given a positive scalar $$ \rho $$, the selection of the upper bounds $$ T_{max}^{p} $$ for sojourn time can be guaranteed by the following prerequisite:

Then, the following Theorem proposes the leader-following mean-square consensus conditions for systems (6) and (8) under incompletely accessible semi-Markov kernel of switching topologies.

Theorem 3 Given a scalar $$ T_{max}^{p}\in\mathbb{N}_{\geq 1} $$, if there exist a scalar $$ \hat{\gamma}>0 $$ and sets of symmetric matrices $$ \tilde{P}_{p}^{(\nu)}\in\mathbb{R}^{n_{x}\times n_{x}}\succ 0 $$, $$ \tilde{P}_{q}^{(\nu)}\in\mathbb{R}^{n_{x}\times n_{x}}\succ 0 $$, and matrices $$ \tilde{Z}_{p}\in\mathbb{R}^{n_{x}\times n_{x}} $$, $$ \tilde{Y}_{p}\in\mathbb{R}^{n_{x}\times n_{x}} $$, $$ p,q\in\mathcal{O} $$, $$ \nu\in\mathbb{N}_{[0,T_{max}^{p}]} $$ such that the following inequalities

hold for any $$ p\in\mathcal{O}_{ap} $$, $$ q\in\mathcal{O}_{ep} $$, $$ \nu\in\mathbb{N}_{[1,T_{max}^{p}]} $$, where $$ \check{\Xi}=[\check{A}_{p}^{T}\; \; \sigma_{p}\mathcal{\check{L}}_{p}^{T}\; \; \sigma_{0p}\mathcal{\check{M}}_{p}^{T}]^{T} $$, $$ \tilde{\Psi}_{23}=[(I_{N}\otimes \tilde{Y}_{P})\; \; 0\; \; 0]^{T} $$, $$ \check{A}_{p}=I_{N}\otimes A+(\mu_{p}\mathcal{L}_{p}+\mu_{0p}\mathcal{M}_{p})\otimes \tilde{Y}_{p} $$, $$ \mathcal{\check{L}}_{p}=\mathcal{L}_{p}\otimes \tilde{Y}_{p} $$, $$ \mathcal{\check{M}}_{p}=\mathcal{M}_{p}\otimes \tilde{Y}_{p} $$, $$ \mathcal{\tilde{Z}}_{p}=diag\{I_{N}\otimes \tilde{Z}_{p},I_{N}\otimes \tilde{Z}_{p},I_{N}\otimes \tilde{Z}_{p}\} $$$$ \in\mathbb{R}^{3n_{x}N\times 3n_{x}N} $$, $$ \tilde{\Xi}=[\check{A}_{p}^{T}\; \; \sigma_{p}\mathcal{\check{L}}_{p}^{T}\; \; \sigma_{0p}\mathcal{\check{M}}_{p}^{T}\; \; \check{A}_{p}^{T}\; \; \sigma_{p}\mathcal{\check{L}}_{p}^{T}\; \; \sigma_{0p}\mathcal{\check{M}}_{p}^{T}]^{T} $$, $$ \tilde{\Upsilon}_{q}^{(0)}=diag\{\tilde{\Upsilon}_{1},\tilde{\Upsilon}_{1},\tilde{\Upsilon}_{1},\tilde{\Upsilon}_{2},\tilde{\Upsilon}_{2},\tilde{\Upsilon}_{2}\} $$,

then systems (6) and (8) is leader-following mean square consensus under incompletely accessible semi-Markov kernel of switching topologies and possess a $$ \mathcal{H}_{\infty} $$ performance index $$ \hat{\gamma} $$. Moreover, the controller gains are given by $$ K_{p}=(B^{T}B)^{-1}B^{T}(\tilde{Z}_{p}^{T})^{-1}\tilde{Y}_{p} $$.

Proof Since the stability proof of the system at non-switching time of topologies is independent of the semi-Markov kernel, the corresponding proof is easy to obtain by Theorem 1. For incompletely accessible semi-Markov kernel, only the stability of the system at the switching time is given in this theorem.

Similar to inequality (13), we have

for $$ \gamma(k_{n+1})=q $$, $$ \nu(k_{n+1})=0 $$, $$ \gamma(k_{n+1}-1)=p $$, $$ \nu(k_{n+1}-1)=\nu-1=S_{n+1}-1 $$. For $$ \Pi(\nu),\; p\in\mathcal{O}_{ap},\; q\in\mathcal{O}_{ep} $$, the equation (27) can be written as

with$$ \mathcal{\bar{P}}=I_{N}\otimes P_{q}^{(0)}\in\mathbb{R}^{n_{x}N\times n_{x}N} $$. Noting the fact $$ \Sigma_{\nu=1}^{T_{max}^{p}}\Sigma_{q\in\mathcal{O}_{ep}}\pi_{pq}(\nu)=\Omega_{p}-\bar{\eta}_{p}, p\in\mathcal{O} $$, it can be found that $$ \Sigma_{\nu=1}^{T_{max}^{p}}\Sigma_{q\in\mathcal{O}_{ep}}\frac{\pi_{pq}(\nu)}{\Omega_{p}-\bar{\eta}_{p}}=1 $$, $$ 0\leq\frac{\pi_{pq}(\nu)}{\Omega_{p}-\bar{\eta}_{p}}\leq1 $$, $$ 0\leq\bar{\omega}_{p}<\Omega_{p}\leq1 $$. Further, it can be derived that

Then,

By Schur complement lemma, the above inequality can be further transformed into

where $$ \tilde{\Xi}=[\bar{A}^{T}_{p}\; \sigma_{p}\mathcal{\bar{L}}_{p}^{T}\; \sigma_{0p}\mathcal{\bar{M}}_{p}^{T}\; \bar{A}^{T}_{p}\; \sigma_{p}\mathcal{\bar{L}}_{p}^{T}\; \sigma_{0p}\mathcal{\bar{M}}_{p}^{T}]^{T} $$, $$ \hat{\mathcal{P}}_{q}^{(0)}=diag\{-\hat{\mathcal{P}}_{1},-\hat{\mathcal{P}}_{1},-\hat{\mathcal{P}}_{1},-\hat{\mathcal{P}}_{2},-\hat{\mathcal{P}}_{2},-\hat{\mathcal{P}}_{2}\} $$, $$ \hat{\mathcal{P}}_{1}=\Sigma_{\nu=1}^{T_{max}^{p}}\Sigma_{q\in\mathcal{O}_{ap}}\frac{\pi_{pq}(\nu)}{\bar{\omega}_{p}}(I_{N}\otimes P_{q}^{-(0)}) $$, $$ \hat{\mathcal{P}}_{2}=(1-\bar{\eta}_{p})(I_{N}\otimes P_{q}^{-(0)}) $$.

Applying congruence transformation $$ diag\{I_{nN},\hat{\mathcal{Z}}^{T}_{p}\} $$ to (30), we can get

with $$ \hat{\mathcal{Z}}_{p}=diag\{\tilde{\mathcal{Z}}_{p},\tilde{\mathcal{Z}}_{p}\} $$. According to condition (23), one can proof that inequality (31) can guarantee that condition (26) can hold. The rest of the proof can be directly derived in a similar way to Theorem 1 and Theorem 2. This proof is completed.

Remark 4 The controller design and consensus conditions proposed in Theorem 2 and Theorem 3 are based on the same channel fading. However, in practice, the fading variables and interference of communication channels between agents are more likely to be different, due to different complex external environments or different geographic locations of the agents. This restricts the issues considered in this paper to a certain extent. It is worth noting that although the problem of non-identical channel fading has been studied in ^{[28, 29]}, the above literature only considers leaderless multi-agent systems. They ignore the fading effects from leader to follower agents, and the edge Laplacian method introduced cannot be used to tackle the models considered in this paper. Therefore, it is interesting and meaningful to investigate the non-identical channel fading problem within the framework of the fading model proposed in this paper. No better method has been proposed to solve the problem of non-identical channel fading under model (3). This also encourages us to continue to study this issue in future work.

Remark 5 In Theorems 2 and 3, the fully known and incompletely available cases of the semi-Markov kernel for switching topologies are handled respectively, and the corresponding consensus conditions are also derived. Given parameters $$ T_{max}^{p} $$ and $$ \rho $$, the minimum $$ \mathcal{H}_{\infty} $$ performance index $$ \hat{\gamma} $$ of the system can be calculated according to the solution of the following optimization problems:

and

for fully known semi-Markov kernel and incompletely available semi-Markov kernel.

Similar to Corollary 2, the following corollary gives the mean square consensus controller design for system (18) with incompletely accessible semi-Markov kernel.

Corollary 3 Given a scalar $$ T_{max}^{p}\in\mathbb{N}_{\geq 1} $$, if there exist a sets of symmetric matrices $$ \tilde{P}_{p}^{(\nu)}\in\mathbb{R}^{n_{x}\times n_{x}}\succ 0 $$, $$ \tilde{P}_{q}^{(\nu)}\in\mathbb{R}^{n_{x}\times n_{x}}\succ 0 $$, and matrices $$ \tilde{Z}_{p}\in\mathbb{R}^{n_{x}\times n_{x}} $$, $$ \tilde{Y}_{p}\in\mathbb{R}^{n_{x}\times n_{x}} $$, $$ p,q\in\mathcal{O} $$, $$ \nu\in\mathbb{N}_{[0,T_{max}^{p}]} $$ such that the inequalities

for any $$ p\in\mathcal{O}_{ap} $$, $$ q\in\mathcal{O}_{ep} $$, $$ \nu\in\mathbb{N}_{[1,T_{max}^{p}]} $$ holds, then the system (18) with incompletely accessible semi-Markov kernel is leader-following mean square consensus under the controller gains $$ K_{p}=(B^{T}B)^{-1}B^{T}(\tilde{Z}_{p}^{T})^{-1}\tilde{Y}_{p} $$, where $$ \mathcal{P}_{p}^{(\nu)} $$, $$ \check{\Xi} $$, $$ \mathcal{\tilde{Z}}_{p} $$, $$ \tilde{\Xi} $$, and $$ \tilde{\Upsilon}_{q}^{(0)} $$ are defined in Theorem 1 and 3.

The proof of Corollary 3 can be obtained in a similar way to Theorem 3, which is omitted here.

Remark 6 Theorem 2 and Theorem 3 respectively realize the distributed consensus control of multi-agent systems under the condition that the semi-Markov kernel of switched topology is fully available and incompletely unavailable. In fact, event-triggered control and sampled data control are also excellent methods for dealing with problems related to multi-agent systems ^{[6, 11, 16]}. The advantages and disadvantages of these methods cannot be directly compared. Similarly, event-triggered control and sampled-data control methods can also be applied to the problems considered in this paper. Naturally, event-triggered control and sampled-data control can also be studied in a distributed framework.

Authors' contributions

Made substantial contributions to the research, idea generation, algorithm design, simulation, wrote and edited the original draft: Yang H

Performed critical review, commentary and revision, as well as provided administrative, technical, and material support: Zhang H, Wang Z, Zhou X

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Financial support and sponsorship

This work is supported by National Natural Science Foundation of China(61922063), Shanghai Hong Kong Macao Taiwan Science and Technology Cooperation Project (21550760900), Shanghai Shuguang Project (18SG18), Shanghai Natural Science Foundation(19ZR1461400), Shanghai Municipal Science and Technology Major Project (2021SHZDZX0100) and Fundamental Research Funds for the Central Universities. (Corresponding author: Zhang H).

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All authors declared that there are no conflicts of interest.

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© The Author(s) 2022.