2. 大连海事大学信息科学技术学院 大连 116026;
3. 韦恩州立大学电子与计算机工程系 美国底特律 48201;
4. 安徽师范大学电子与计算机工程系 芜湖 241000
2. School of Information Science and Technology, Dalian Maritime University, Dalian 116026, China;
3. Department of Electrical and Computer Engineering, Wayne State University, Detroit 48201, USA;
4. College of Physics and Electronic Information, Anhui Normal University, Wuhu 241000, China
Mobile sensor networks (MSNs) have attracted increasing research attention and found various applications in the past decade [1], [2]. For a practical MSN, energy limitation is always one of the most important and challenging issues. The reason lies in the batteries powering the MSNs, which are limited in capacity and inconvenient for replacement. Besides, a typical MSN consists of a mass of energydemanding nodes capable of sensing, computing, communicating and moving. As such, energy efficient algorithms are of utmost importance for MSN applications, hence, are the focus of many research works. To name a few, by modeling power consumption as the distance traveled by the sensors, [3][5] investigated the powerconstrained deployment and coverage control issues of MSNs. In [6], an energy aware control protocol was proposed to achieve rendezvous without depleting the energy of the nodes. Based on the energy forecast, [7] designed a clusteringtree topology control algorithm to save energy and maximize the network lifetime for heterogeneous wireless sensor networks with considering packet loss rate and the link quality.
In most cases, the cost saving problem is complicated, since it is often realized at the cost of a certain degree of performance loss. Therefore, the intention of most papers comes down to finding a tradeoff of energy usage and system performance, which refers to quality of performance (QoP) for control and estimation problems, and to quality of service (QoS) for communication problems. For instance, [8] investigated the compromise of communication cost and the convergence rate (a QoP indicator), and [9] studied the tradeoff between energy expenditure and communication delay (a QoS parameter).
Here we aim at the issue of consensus seeking of MSNs subject to limited cost supply. Of course, it is not the first time that this problem is being addressed. There have been some results reported in the literature. For instance, [10] considered the energyconstrained consensus estimation issue by activating a subset of sensors at each instant. In [11], an optimal consensus controller design method was given to minimize the power cost of sensors deployed in smart home systems. Reference [12] proposed an LQR consensus algorithm for multivehicle control. In [13], a suboptimal consensus control algorithm for MSNs was presented by modeling the energy expenditure for mobility and communication independently. Most of these results assume static communication topologies in the MSNs. In practice, however, the network topology may vary with link failures, packet dropouts or environmental disturbances. As these factors are random by nature, the switching of topologies is essentially a random event modeled as a stochastic process, in particular, a Markov chain [14]. In fact, a number of papers have studied MSNs with Markov switching topologies under different frameworks with various concerns, see, e.g., in [15][21]. However, it should be noted that several important factors have not been considered in the existing results. Firstly, most of these works concern ergodic Markov chains, while a more general Markov chain is more interesting, since when a system goes through an unsteady environment to a settled one, it will reasonably experience switching from variable topology to a certain fixed one. Secondly, almost all the existing results are applied to situations with completely known transition probabilities. In reality, however, all or part of the transition probabilities are unavailable [22][25], and only the estimated values of transition rates are accessible. Moreover, estimation errors, referred to as probability uncertainties of switching topology, may also result in instability or at least degrade performance of MSNs. Thus, it is of great importance to consider partially unavailable transition rates, which do not need any knowledge of the unavailable elements. Thirdly, existing works do not consider cost limitations.
In this paper, we study the guaranteed cost consensus seeking problem for MSNs with switching communication topologies governed by a Markov chain whose initial and transition probabilities are partially unknown. A team of mobile sensors cooperate with each other and form a network whose topologies are switching within a topology set according to a Markov chain. Each directed graph in the graph set is assumed to have a spanning tree. A novel function of team cost which penalizes the state and control input errors between communicating sensors is defined in a linear quadratic form, in which cost consumption for information receiving and sending is involved. The elements of the transition probability matrix are divided into the known and unknown parts. Then, using model transformation and graph theory, the issue of guaranteed cost consensus is transformed to the stability problem of a reduced order Markov jump systems with partially unknown initial and transition probabilities. Using stochastic Lyapunov functional method, a sufficient mean square stability condition is obtained for the reduced order jumping system, which guarantees global exponential consensus of the original MSN in the mean square sense. Then, a computational algorithm is given, by which the suboptimal controller gains and a subminimal upper cost bound of the MSN can be calculated, concurrently. The validity of the controller design method is illustrated by example results.
The remaining sections are organized as follows. Section 2 provides some preliminaries of graph theory, Markov switching topology and the problem formulation. The main results on sufficient consensus condition are presented in Section 3. Then, a controller design algorithm of sensor networks is derived. In Section 4, numerical examples are given to validate the performance of the proposed method. Section 5 draws a conclusion.
Notations:
Consider a mobile sensor network consisting of
At each instant
$ \begin{align*} {\rm Pro}\{\left. {\theta (k+1)=v} \right\theta (k)=l\}=\pi _{lv} \end{align*} $ 
where
The transition probabilities of the Markov chain are assumed to be partly available, e.g., some elements in transition probability matrix
$ \begin{align*} \pi =[\pi _{lv}]_{4\times 4} =\left[{{\begin{array}{*{20}c} {\pi _{11} } \hfill & {\pi _{12} } \hfill & ? \hfill & ? \hfill \\ ? \hfill & ? \hfill & {\pi _{23} } \hfill & ? \hfill \\ {\pi _{31} } \hfill & ? \hfill & {\pi _{33} } \hfill & ? \hfill \\ {\pi _{41} } \hfill & {\pi _{42} } \hfill & {\pi _{43} } \hfill & {\pi _{44} } \hfill \\ \end{array} }} \right] \end{align*} $ 
where ? represents the unavailable element. Define
The dynamics of each sensor in the MSN is presented as:
$ \begin{align}\label{system} x_i (k+1)=Ax_i (k)+Bu_i (k), \quad i=1, {\ldots}, N \end{align} $  (1) 
where
As in many other references [26], it is assumed that a number of mobile base stations are deployed to detect the topology information of the MSN and broadcast the information of the communication topology to the sensors in the systems, which implies it is a practical way that each sensor can know the present topology. Therefore, we use the modedependent consensus control protocol for sensor as follows.
$ \begin{align}\label{eq1} u_i \mbox{(}k\mbox{)}=K(\theta (k))\sum\limits_{j\in N_j } {a_{ij} (\theta (k))(x_i(k)x_j (k))}, \quad \theta (k)\in S \end{align} $  (2) 
where
The cost consumption for information receiving, sending and control of the MSN (1), respectively, can be defined as below
$\begin{align} \label{eq2} & C_r =\sum\limits_{k=0}^{\infty} {\sum\limits_{i=1}^N {{E}\left( {z_i^T (k)Q_1 z_i (k)} \right)} } \end{align} $  (3) 
$ \begin{align} \label{eq3} & C_s =\sum\limits_{k=0}^{\infty}{\sum\limits_{i=1}^N {{E}\left( {y_i^T (k)Q_2 y_i (k)} \right)} } \end{align} $  (4) 
$ \begin{align} \label{eq4} & C_c =\sum\limits_{k=0}^{\infty} {\sum\limits_{i=1}^N {{ E}\left( {u_i^T (k)Ru_i (k)} \right)} } \end{align} $  (5) 
where
Remark 1: In many wireless sensor network applications, data transmitting and receiving are two different procedures with different power requirement. This paper gives independent considerations of the energy costs for information receiving and sending. To cope with them more accurately, we use the
The objective of this paper is to design a set of controllers (2) for MSN (1) to achieve consensus in the sense defined below.
Definition 1: MSN (1) is said to achieve meansquare consensus (MSconsensus) with consensus protocol (2) under Markov switching topologies
$ \begin{align*} \lim\limits_{k\to \infty } {E}\left[{\left\ {x_i (k)x_j (k)} \right\^2} \right]=0 \end{align*} $ 
holds true for any
$ \begin{align} \label{eq5} C= & \sum\limits_{k=0}^\infty \sum\limits_{i=1}^N {\rm e}\left( z_i^T (k)Q_1 z_i (k)+\right. \left.y_i^T (k)Q_2 y_i (k) \right.\nonumber\\ & \left.+u_i^T (k)Ru_i (k) \right) . \end{align} $  (6) 
Specifically, we aim at designing controllers (2) such that MSNs achieve consensus seeking for
In this section, firstly, the sufficient MSconsensus conditions for MSN (1) with partially unknown switching probabilities will be derived. Then, using this condition, a controller design algorithm will be given to simultaneously calculate controller gains and subminimal cost upper bound.
Let
$ \begin{equation} \label{eq6} X(k+1)=(I_N \otimes A+L_\theta \otimes BK_\theta )X(k) \end{equation} $  (7) 
where
By the definition of
$ \begin{align} \label{eq7} C= & \ \sum\limits_{k=0}^\infty {E}\left( X^T(k)\left( \left( L_\theta ^T L_\theta \right)\otimes Q_1 +\left( (L_\theta^o)^TL_\theta^o \right)\otimes Q_2 \right.\right.\nonumber\\ & \left.\left.\ +\left( L_\theta^T L_\theta \right)\otimes K_\theta^T RK_\theta \right)X(k)\right) . \end{align} $  (8) 
Now, introduce the following state transformation [21]:
$ \begin{equation} \label{eq8} \tilde {X}(k)=(T\otimes I_n)^TX(k). \end{equation} $  (9) 
And partition
$ \begin{align} \label{eq9} \tilde {X}_1 (k+1)= & ~A\tilde {X}_1 (k)+\Theta _\theta \otimes BK_\theta \tilde {X}_2 (k) \end{align} $  (10) 
$ \begin{align} \label{eq10} \tilde {X}_2 (k+1)= & ~(I_{N1} \otimes A+{\Phi} _\theta \otimes BK_\theta)\tilde {X}_2 (k) \end{align} $  (11) 
where
$ \begin{align} \label{eq11} C= & \ \sum\limits_{k=0}^\infty {E}\left( \tilde {X}_2^T(k)\left( ( {{\Phi}_\theta^T {\Phi}_\theta } )\otimes Q_1 \right.\right.\nonumber\\ & +( {({\Phi}_\theta^o) ^T{\Phi}_\theta^o } )\otimes Q_2 \notag\\ & \left.\left. + ( {{\Phi}_\theta^T {\Phi}_\theta } )\otimes K_\theta^T RK_\theta \right)\tilde {X}_2 (k)\right) \end{align} $  (12) 
where
With the above discussion, the consensus seeking issue of system (7) is transformed to the stability problem of the reduced order system (11) with performance index (12).
We are now in the position to establish a sufficient mean square stability condition for system (11) with a guaranteed energy cost
Theorem 1: Under the consensus protocol (2), MSN (1) can achieve guaranteed cost MSconsensus with Markov switching topologies set
$ \begin{align} \label{eq12} \Psi _l^T & \sum\limits_{v\in S_\kappa ^l } {\pi _{lv} P_v } \Psi _l \pi _\kappa ^l \Big(P_l Q_l Q_l^o \nonumber\\ &  (I_{N1} \otimes K_l)^TR_l (I_{N1} \otimes K_l) \Big)<0, \quad v\in S_\kappa ^l \end{align} $  (13) 
$ \begin{align} \Psi _l^T & P_v \Psi _l P_l +Q_l +Q_l^o \nonumber\\ & +(I_{N1} \otimes K_l)^TR_l (I_{N1} \otimes K_l)<0\label{eq13}, \quad\ \ v\in S_{u\kappa }^l \end{align} $  (14) 
hold true, where
$ \begin{equation} \label{eq14} C\le \tilde {C}=\tilde {X}_2^T (0)\left( {\sum\limits_{v\in \Omega _\kappa } {\pi _{0v} } P_v +(1\pi _\kappa ^0)\sum\limits_{v\in \Omega _{u\kappa } } {P_v } } \right)\tilde {X}_2 (0). \end{equation} $  (15) 
Proof: Firstly, the stability of system (11) will be proved. The following stochastic Lyapunov functional is defined as
$ \begin{equation} \label{eq15} V(k, \theta (k))=\tilde {X}_2^T (k)P_{\theta (k)} \tilde {X}_2 (k), \quad \theta (k)\in S=\{1, \ldots, q\}. \end{equation} $  (16) 
Let
$ \begin{align} \Delta V(k)= & ~{E} (V(k+1, \left {I_k } \right.)V(k) \nonumber\\= & ~{E} (\tilde {X}_2^T (k+1)P_v \tilde {X}_2 (k+1)\left {I_k } \right.)\tilde {X}_2^T (k)P_l \tilde {X}_2 (k) \nonumber\\= & ~\tilde {X}_2^T (k)\left( {\Psi _l^T \sum\limits_{v\in s} {\pi _{lv} P_v } \Psi _l P_l } \right)\tilde {X}_2 (k).\label{eq16} \end{align} $  (17) 
Since
$ \begin{align} \label{eq17} \Delta V(k)= & ~\tilde {X}_2^T (k)\left( \Psi _l^T \left( {\sum\limits_{v\in S_\kappa ^l } {\pi _{lv} P_v } +\sum\limits_{v\in S_{u\kappa }^l } {\pi _{lv} } P_v }\right)\Psi _l \right.\nonumber \\ & \left.\left( {\sum\limits_{v\in S_\kappa ^l } {\pi _{lv} } +\sum\limits_{v\in S_{u\kappa }^l } {\pi _{lv} } } \right)P_l \right)\tilde {X}_2 (k). \end{align} $  (18) 
If (13) and (14) hold, we can have
$ \begin{align} \Psi _l^T & \sum\limits_{v\in S_\kappa ^l } {\pi _{lv} P_v } \Psi _l \pi _\kappa ^l \Big( P_l Q_l Q_l^o \nonumber\\ &  (I_{N1} \otimes K_l)^TR_l (I_{N1} \otimes K_l) \Big) \nonumber\\ & +\sum\limits_{v\in S_{u\kappa }^l } \pi _{lv} \left( \Psi _l^T P_v \Psi _l P_l +Q_l +Q_l^o \right. \nonumber\\ & \left.+ (I_{N1} \otimes K_l)^TR_l (I_{N1} \otimes K_l) \right) <0.\label{eq18} \end{align} $  (19) 
Employing (18) and (19), the difference satisfies
$ \begin{equation} \label{eq19} \Delta V(k)\le \tilde {X}_2^T (k)F_l \tilde {X}_2 (k) \end{equation} $  (20) 
where
$ \begin{align} {E} (V(k+1\left {I_k } \right.) & \le V(k)\tilde {X}_2^T (k)F_l \tilde {X}_2 (k)\nonumber\\ & \le \left( {1\frac{\lambda _{\min } (F_l)}{\lambda _{\max } (P_l)}} \right)V(k)=\zeta V(k).\label{eq20} \end{align} $  (21) 
If inequalities (13) and (14) hold, it is clear that
$ \begin{equation} \label{eq21} E(V(k)\left {I_{k1} } \right.)=\zeta V(k1). \end{equation} $  (22) 
According to smooth characteristics of conditional mean [27], by (22), we can have
$ \begin{align} \label{eq22} {E} (V(k)\left {I_{k2} } \right.)= & ~{E} ({\rm e}(V(k)\left {I_{k1} } \right.)\left {I_{k2} } \right.)\nonumber\\\le & ~\zeta {E} (V(k1)\left {I_{k2} } \right.)\nonumber\\\le & ~\zeta ^2V(k2). \end{align} $  (23) 
By recursion like (21), (22) and (23), the exponentially mean square stable condition can be derived as
$ \begin{align} \label{eq23} {E} (V(k))= & ~{E} (\tilde {X}_2^T (k)P_{\theta (k)} \tilde {X}_2 (k))\nonumber\\\le & ~\zeta ^kV(0)=\zeta ^k\tilde {X}_2^T (0)P_{\theta (0)} \tilde {X}_2 (k) \end{align} $  (24) 
from which it is straightforward that
$ \begin{equation} \label{eq24} {E} (\tilde {X}_2^T (k)P_{\theta (k)} \tilde {X}_2 (k))\le \beta \zeta ^k\tilde {X}_2^T (0)\tilde {X}_2 (0) \end{equation} $  (25) 
where
Next, the proof of the guaranteed cost will be given. Summing both sides of inequality (17) from
$ \begin{align} \label{eq25} C= & ~\sum\limits_{k=0}^\infty {{E} (\tilde {X}_2 ^T(k)F_u \tilde {X}_2 (k)})\nonumber\\\le & ~{E} \left( {V(0, \theta (0))} \right)\nonumber\\= & ~{E} (\tilde {X}_2^T (0)P_{\theta (0)} \tilde {X}_2 (0))=\tilde {J} \quad \forall l\in S. \end{align} $  (26) 
Since
$ \begin{equation} \label{eq26} \tilde {C}=\tilde {X}_2^T (0)\left( {\sum\limits_{v\in \Omega _\kappa } {\pi _{0v} } P_v +(1\pi _\kappa ^0)\sum\limits_{v\in \Omega _{u\kappa } } {P_v } } \right)\tilde {X}_2 (0). \end{equation} $  (27) 
Remark 2: If
Before closing this section, a design algorithm for the controllers (2) will be proposed. To derive the algorithm, the following sufficient conditions on the existence of the controller gains are needed.
Theorem 2: For MSN (1) under Markovian switching topologies
$ \begin{align} \label{eq27} & \begin{bmatrix} {\pi _\kappa ^l \left( {P_l Q_l Q_{l^o} } \right)} & {\sqrt {\pi _\kappa ^l } \left( {I_{N1} \otimes K_l } \right)^T} & {\Psi _l^T } \\ {\sqrt {\pi _\kappa ^l } \left( {I_{N1} \otimes K_l } \right)} & {R_l^{1} } & 0 \\ {\Psi _l } & 0 & {M_l } \end{bmatrix}<0 \nonumber\\ & ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ v\in S_\kappa ^l \end{align} $  (28) 
$ \begin{align}& \label{eq28} \left[{{\begin{array}{*{20}c} {P_l +Q_l +Q_{l^o} } & {\left( {I_{N1} \otimes K_l } \right)^T} & {\Psi _l^T } \\ {\left( {I_{N1} \otimes K_l } \right)} & {R_l^{1} } & 0 \\ {\Psi _l } & 0 & {H_v } \\ \end{array} }} \right]<0 \nonumber\\ & ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ v\in S_{u\kappa }^l \end{align} $  (29) 
hold for all
Proof: By using Theorem 1 and Schur complement lemma, Theorem 2 can be easily proved, so we omit it.
As for the cost budget of the MSNs, a feasible one has been derived in Theorem 1. Since the capacity of batteries for the MSNs is restricted, a minimal cost bound is more appropriate. Namely, we intend to seek the infimum of
$ \begin{align} \label{eq29} C^* & =\inf \limits_\Delta \tilde{C}\notag\\ & = \inf \limits_\Delta \tilde {X}_2^T (0)\left( \sum\limits_{v\in \Omega _\kappa } {\pi _{0v} } P_v +(1\pi _\kappa ^0)\sum\limits_{v\in \Omega _{u\kappa } } {P_v } \right)\tilde {X}_2 (0). \end{align} $  (30) 
The minimization problem can be presented as
$ \begin{align} & {\min}~\delta \end{align} $  (31) 
$ \begin{align} & {\rm s. t.}~~\delta \ge \tilde {X}_2^T (0)\left( {\sum\limits_{v\in \Omega _\kappa } {\pi _{0v} } P_v +(1\pi _\kappa ^0)\sum\limits_{v\in \Omega _{u\kappa } } {P_v } } \right)\tilde {X}_2 (0). \end{align} $  (32) 
By Schur complement, equivalently, it can be written as
$\begin{equation} \label{eq30} \left[{{\begin{array}{*{20}c} \delta & {\tilde {X}_2^T (0)} & {\tilde {X}_2^T (0)} \\ {\tilde {X}_2 (0)} & {M_{\Omega _\kappa }^0 } & 0 \\ {\tilde {X}_2 (0)} & 0 & {M_{\Omega _{u\kappa } }^0 } \\ \end{array} }} \right]\ge 0 \end{equation} $  (33) 
where
$ M_{\Omega _\kappa }^0 =\left( {\sum\limits_{v\in \Omega _\kappa } {\pi _{0v} P_v } } \right)^{1} $ 
and
$ M_{\Omega _{u\kappa } }^0 =\left( {(1\pi _\kappa ^0)\sum\limits_{v\in \Omega _{u\kappa } } {P_v } } \right)^{1}. $ 
The conditions in (33), (28) and (29) contain some constraints of matrix inversion, which are equivalent to the rank constrained LMIs
$ \begin{align} & \left\{\begin{aligned}\label{eq34} & \varXi_v=\begin{bmatrix} \sum\limits_{v\in \Omega_\kappa } \pi_{0v} P_v & \ast \\ I & M_{\Omega_\kappa }^0 \end{bmatrix}\ge 0 \\ & {\rm Rank} (\varXi_v)\le (N1)n, \quad v\in \Omega _k \end{aligned}\right. \end{align} $  (34) 
$ \begin{align} & \left\{\begin{aligned}\label{eq35} & \varXi_v=\begin{bmatrix} {(1\pi _{\kappa}^0)\sum\limits_{v\in \Omega _{u\kappa} } { P_v } } & \ast \\ I & {M_{\Omega _{u\kappa} }^0 } \end{bmatrix}\ge 0 \\ & {\rm Rank} (\varXi_v)\le (N1)n, \quad v\in \Omega _{u\kappa } \end{aligned}\right. \end{align} $  (35) 
$ \begin{align} & \left\{\begin{aligned}\label{eq36} & \varXi_{vl}=\begin{bmatrix} {\sum\limits_{v\in S_\kappa ^l } {\pi _{lv} P_v } } & \ast \\ I & {M_l } \end{bmatrix}\ge 0\\ & {\rm Rank} (\varXi_{vl})\le (N1)n, v\in S_\kappa ^l, \quad l\in S \end{aligned}\right. \end{align} $  (36) 
$ \begin{align} & \left\{ \begin{aligned}\label{eq37} & \varXi_{v}=\begin{bmatrix} {P_v } & \ast \\ I & {H_v } \end{bmatrix}\ge 0 \\ & {\rm Rank} (\varXi_{v})\le (N1)n, v\in S_{u\kappa }^l, \quad l\in S. \end{aligned}\right. \end{align} $  (37) 
Therefore, the above minimization problem can be reduced to
$ \begin{align}\label{eq38} & {\min}~\delta \nonumber\\ & {\rm s.t.}~(33), (28), (29), (34), (35), (36)~{\rm and}~(37). \end{align} $  (38) 
To concurrently determine the controller gain matrix (2) and the subminimal cost upper bound (6), Algorithm 1 is proposed as follows.
Algorithm 1. 
Step 1: Set the initial state 
Step 2: Find a feasible solution 
Step 3: Let g=(e+f)/2, 
Step 4: If 
Remark 3: Since the guaranteed consensus issue is transformed to the stability of Markov jumping system (11) with performance index (12), the LyapunovKrasovskii functional method is used to derive the conditions on the existence of controller gains in terms of LMIs. It indicates that the results obtained in this paper depend to a great extent on the solvability of the LMIs, which may cause more complex computation as the number of sensors and their dimensions are increasing. Thus, for a large size network of sensors with high dimensions, the results obtained in this paper may not be appropriate. As for, how to amicably reduce such computational complexity, is still a tough problem, which is worth to be further studied in our future research.
4 Numerical ExamplesSome numerical examples are presented to illustrate the validity of the presented method in this section. Consider a collection of six identical sensors, whose dynamics described as
$ \begin{equation*} x_i (k+1)=x_i (k)+u_i (k) \end{equation*} $ 
where
A group of four directed graphs is shown in Fig. 1, which describes all the possible interactions among sensors.
In the simulation, it is assumed that the topology of the MSN is switching according to a Markov chain with the following transition probability matrix
$ \begin{equation} \label{eq32} \pi =\left[{{\begin{array}{*{20}c} {0.3} & {0.2} & ? & ? \\ {0.4} & ? & ? & {0.3} \\ ? & {0.5} & {0.1} & ? \\ {0.6} & ? & ? & {0.2} \\ \end{array} }} \right] \end{equation} $  (39) 
and the initial distribution of the Markov chain is
Using the proposed design Algorithm 1, we calculate the following controller gains
As a comparison, the simulation result of the method in [30] is also presented to show the superiority of Algorithm 1 proposed in this paper. The same systems in (40), the communication topology set in Fig. 1 and the same transition probability matrix of Markov chain in (39) are considered in this comparison. Using the controller design algorithm in [30], we have the controller gains
We have investigated the consensus issue of MSNs with Markov switching communication topologies under a subminimal guaranteed cost, in which both initial and transition probabilities of the Markov chain are assumed to be partially unavailable. The stability criterions for Markov jumping systems have been derived by state transformation, which ensure MSNs reach globally exponential mean square consensus. According to these conditions, the suboptimal controller gains and subminimal cost bound have been derived synchronously by our proposed controller design method. Numerical examples have been given to validate the performance of the proposed method. In our further work, the costoptimal consensus issue for MSNs with Markov switching topologies will be studied.
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