IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 63, NO. 4, APRIL 2018 1059 Stochastic Consentability of Linear Systems With Time Delays and Multiplicative Noises Xiaofeng Zong, Tao Li , Senior Member, IEEE, George Yin , Fellow, IEEE, Le Yi Wang , Fellow, IEEE, and Ji-Feng Zhang , Fellow, IEEE
Abstract—This paper develops stochastic consentability Index Terms—Consentability, consensus, multiagent of linear multiagent systems with time delays and multi- system, multiplicative noise, time delay. plicative noises. First, the stochastic stability for stochastic differential delay equations driven by multiplicative noises is examined, and the existence of the positive definite so- I. INTRODUCTION lution for a class of generalized algebraic Riccati equations (GAREs) is established. Then, sufficient conditions are de- ULTIAGENT systems have attracted much attention in duced for the mean square and almost sure consentability recent years. The research progress in such systems has and stabilization based on the developed stochastic stabil- M ity and GAREs. Consensus protocols are designed for linear provided valuable insights and benefits in understanding, de- multiagent systems with undirected and leader-following signing, and implementing distributed controllers for such sys- topologies. It is revealed that multiagent consentability de- tems; see [1]Ð[3]. To date, the consensus problem of perfect pends on certain characterizing system parameters, in- models, which assume that each agent can obtain its neighbor in- cluding linear system dynamics, communication graph, formation timely and precisely, has reached a reasonable degree channel uncertainties, and time delay of the deterministic term. It is shown that a second-order integrator multiagent of maturity; see [4] for example. However, networked systems in system is unconditionally mean square and almost surely practical applications often operate in uncertain communication consentable for any given noise intensities and time delay, environments and are inevitably subjected to communication and that the mean square and almost sure consensus can latency and measurement noises [5]Ð[7]. Hence, time delays be achieved by carefully choosing the control gain accord- and measurement noises should be taken into consideration for ing to certain explicit conditions. examining multiagent consensus problems. Manuscript received February 20, 2017; accepted July 14, 2017. Date In the literature, Olfati-Saber and Murray [7] gave optimal of publication July 27, 2017; date of current version March 27, 2018. delay bounds for consensus under undirected graphs. Bliman This work was supported in part by the National Natural Science Foun- and Ferrari-Trecate [8] studied average consensus problems for dation of China under Grant 61522310, in part by Shanghai Rising-Star Program under Grant 15QA1402000, in part by the National Key Basic undirected networks with constant, time varying and nonuni- Research Program of China (973 Program) under Grant 2014CB845301, form time delays, and presented some sufficient conditions for in part by the Fundamental Research Funds for the Central Universities, average consensus. Different types of time delays (communi- China University of Geosciences (Wuhan) under Grant CUG170610, and in part by the U.S. Air Force Office of Scientific Research under FA9550- cation delay, identical self-delay and different self-delay) were 15-1-0131. This paper was presented in part at 12th IEEE Conference investigated in [9] for output consensus. These works have fo- on Control and Automation, Kathmandu, Nepal, June 1Ð3, 2016. Rec- cused on first-order multiagent systems. For second-order mul- ommended by Associate Editor E. Zhou. (Corresponding author: George Yin.) tiagent systems, Yu et al. [10] gave necessary and sufficient X. Zong is with the School of Automation, China University of Geo- consensus conditions related to time delays. Zhou and Lin [11] sciences, Wuhan 430074 China, and with Hubei Key Laboratory of Ad- showed that consensus problems of linear multiagent systems vanced Control and Intelligent Automation for Complex Systems, Wuhan 430074, China (e-mail: [email protected]). with input and communication delays could also be solved by T. Li was with the School of Mechatronic Engineering and Automation, truncated predictor feedback protocols. Shanghai University, Shanghai 200072 China. He is now with the Shang- For multiplicative noises (or state-dependent noise), Ni and hai Key Laboratory of Pure Mathematics and Mathematical Practice, Department of Mathematics, East China Normal University, Shanghai Li [12] studied consensus problems of continuous-time systems 200241, China (e-mail: [email protected]). in which the noise intensity functions are the absolute values of G. Yin is with the Department of Mathematics, Wayne State University, relative states. Li et al. [13] revealed that multiplicative noises Detroit, MI 48202 USA (e-mail: [email protected]). L. Y. Wang is with the Department of Electrical and Computer En- may enhance the almost sure consensus, but may have damaging gineering, Wayne State University, Detroit, MI 48202 USA (e-mail: effect on the mean square consensus, indicating a distinct fea- [email protected]). ture from additive noises which are always destabilizing factors. J.-F.Zhang is with the Key Laboratory of Systems and Control, Institute of Systems Science, Academy of Mathematics and Systems Science, These results have been established for first-order multiagent Chinese Academy of Sciences, Beijing 100190 China, and also with systems. For linear discrete-time systems, Li and Chen [14] gave School of Mathematical Sciences, University of Chinese Academy of the mean square consensus analysis by solving a modified alge- Sciences, Beijing 100149, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available braic Riccati equation. When the time delay and multiplicative online at http://ieeexplore.ieee.org. noise coexist, stochastic consensus conditions were examined Digital Object Identifier 10.1109/TAC.2017.2732823 in [15] for second-order discrete-time models. However, little is
0018-9286 © 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications standards/publications/rights/index.html for more information. 1060 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 63, NO. 4, APRIL 2018 known about the consensus for general continuous-time linear designing control protocols under time delays and measurement multiagent systems with time delays and multiplicative noises. noises. The contribution of this paper is detailed as follows. All the aforementioned works have focused on the issue 1) Two new techniques (Theorems II.1 and II.2) are devel- of convergence to consensus, which is concerned with con- oped to study the multiagent consentability and consensus ditions on the given control protocols under which the agents under measurement noises and time delays. We first ex- can achieve certain agreements asymptotically. However, they ploit the degenerate Lyapunov functional [25] to establish do not address the issue of consentability. Consentability is the mean square and almost sure exponential stability cri- concerned with conditions on system parameters under which a teria of stochastic differential delay equations (SDDEs). consensus protocol exists, and is of great importance, both theo- We show that the stability is independent of the time delay retically and practically, for cooperative control protocol devel- with stochastic influence. Then, the existence of positive opment such as flocking behavior, agent rendezvous, and robot definite solutions to GARE is obtained. The GARE plays coordination. Note that Ma and Zhang [16] considered general an important role in stochastic linear-quadratic control linear continuous-time multiagent systems and obtained nec- problems [26]Ð[28]. Here, both sufficient and necessary essary and sufficient conditions on system parameters for con- conditions are presented for guaranteeing the existence of sentability. This problem was then generalized in [17] to the case a positive definite solution to the GARE. As a byproduct, with input constraints and uncertain initial conditions in terms necessary and sufficient conditions are given for the ex- of linear matrix inequalities (LMIs). For discrete-time models, istence of stochastic feedback stabilization control laws. Zhang and Tian [18] investigated second-order multiagent sys- That is, it is sufficient that the open-loop dynamics (A, B) tems with Markov-switching topologies, and proved that such is controllable and the product of the sum of real parts of networked systems are mean square consentable under linear unstable open-loop poles and the square of noise inten- consensus protocols if and only if the union of the graphs in sity is less than 1/2, while it is necessary that (A, B) is the switching topology has globally reachable nodes. For linear stabilizable and the product of the maximal real part of multiagent systems, You and Xie [19] obtained necessary and the open-loop poles and the square of noise intensity is sufficient conditions for single-input multiagent consentability less than 1/2. and revealed how the agent dynamic, network topology, and 2) The necessary and sufficient conditions for multiagent communication data rate affect consentability. Gu et al. [20] consentability and consensus are revealed. We first de- employed a properly designed dynamic filter into local con- rive the mean square and almost sure consentability con- trol protocols to relax consensusability conditions. For the issue ditions for leader-free linear multiagent systems. a) It is of structural controllability of multiagent systems, we refer the proved that if the agent dynamics (A, B) is controllable, λu N −1 2 λ ∗ readers to [21] and references therein. At present, consentability 4 0 N σ¯ < 2 and the time delay τ1 <τ1 , then the lin- for multiagent systems with multiplicative noises, even for the ear multiagent system is mean square and almost surely λu delay-free case, remains an open problem, due to the lack of consentable, where 0 denotes the sum of the real parts suitable techniques to treat intrinsic complications from general of the unstable eigenvalues of A, N is the number of linear systems. agents, λ2 is the algebraic connectivity of the graph, σ¯ ∗ This paper aims to establish the mean square and almost sure is the bound of the noise intensities, and τ1 is a bound consentability and develop the consensus protocol design of of the time delay in the deterministic term. Especially, linear multiagent systems with time delays and multiplicative some special cases are given to show that the stabiliz- noises. Our results accommodate different time delays in deter- ability of (A, B) and the restriction on noise intensity ministic (drift) and stochastic (diffusion) terms. Departing from are necessary for the mean square consentability. b) For the feedback structure of [22], [23], where each agent’s state second-order integrator multiagent systems under undi- feedback was used, our consensus protocol requires only lo- rected graphs, we show that it is mean square and almost cal relative-state measurements. In this case, each agent’s state surely consentable regardless of noises and time delays. is not locally stabilized, leading to a more difficult consensus Some necessary conditions on the mean square consensus analysis problem. It can be verified that a consensus problem is are also obtained for the delay-free case. Then, the con- equivalent to the stability problem of the corresponding closed- sentability and consensus results are extended to leader- loop system. Hence, for consensus analysis under measurement following multiagent systems which include stochastic noises, the key is to establish stochastic stability of the corre- stabilization by delayed noisy feedback as the special sponding stochastic equations driven by additive or multiplica- case. tive noises, see [12]Ð[15], [24]. However, for linear multiagent The rest of the paper is structured in five parts. Section II systems of orders higher than one, multiplicative noises are of- develops the main tools such as stochastic stability and GARE ten degenerate and the existing stochastic stability theory does for examining the consentability of linear multiagent systems. not involve the issues for such stochastic differential equations. Section III investigates consentability conditions and pursues Time delays add further difficulty in deriving consentability consensus protocols of leader-free linear multiagent systems conditions and consensus protocols of linear multiagent sys- under undirected graphs. Section IV extends our investigation tems with multiplicative noises. New ideas and techniques are to the leader-following case and stochastic stabilization with needed to resolve these difficulties. In this paper, stochastic delayed noisy feedback. Section V presents simulation results stability and a generalized algebraic Riccati equation (GARE) to verify the theoretical analysis. Section VI concludes the paper are established for obtaining multiagent consentability and with some remarks that outline possible directions for future ZONG et al.: STOCHASTIC CONSENTABILITY OF LINEAR SYSTEMS WITH TIME DELAYS 1061 research in this field. For clarity of presentation, the proofs of Theorem II.1: Suppose Assumption II.1 and condition (2) most technical results are relegated to the appendix. hold. If there exists a matrix P>0 such that 1 Notation: The symbol N denotes the N-dimensional col- ¯T ¯ ¯T ¯ T A P + P A +(A P A + A1 PA1 )τ1 + DP < 0 (3) umn vector with all ones; ηN,i denotes the N-dimensional col- umn vector with the ith element being 1 and others being zero; where A¯ = A0 + A1 , then there exist C0 ,γ0 > 0 such that J = 1 1 1T I N N N N N ; N denotes the -dimensional identity matrix. 1 γ T 2 −γ0 t 0 For a given matrix or vector A, its transpose is denoted by A , its Ey(t) ≤ C0 e , lim sup log y(t) < − , a.s. (4) →∞ t 2 Euclidean norm is denoted by A, and its maximum real part of t the eigenvalues is denoted by max(Re(λ(A))). For a real sym- That is, the trivial solution to SDDE (1) is mean square and metric matrix A, λmax(A) and λmin(A) denote its maximum and almost surely exponentially stable. − minimum eigenvalues, respectively. For two matrices A and B, In applications, the diffusion fi (y(s τ2 ))dwi (s) may A ⊗ B denotes their Kronecker product. For symmetric matri- have various forms (see Sections III and IV). Note that d t T − − ces X and Y , the notation X ≥ Y (respectively, X>Y) means M,P0 M (t)= i=1 0 fi (y(s τ2 ))P0 fi (y(s τ2 ))ds. that the matrix X − Y is positive semidefinite (respectively, So we can use the following condition to replace (2): positive definite). For a, b ∈ R, a ∨ b represents max{a, b} and T d M,P0 M (t) ≤ y (t − τ2 )DP y(t − τ2 )dt (5) a ∧ b denotes min{a, b}.Let(Ω, F, P) be a complete proba- 0 bility space with a filtration {Ft }t≥0 satisfying the usual condi- which is a direct consequence of (2) that does not involve the tions. For a given random variable or vector X, the mathematical concrete form of diffusion. In fact, the proof of Theorem II.1 expectation of X is denoted by EX. For continuous martin- in Appendix is based on (5). Theorem II.1 also leads to the gales M(t) and N(t), their quadratic variation is denoted by following corollaries. M,N (t) (see [29]). We also denote M (t):= M,M (t). Corollary II.1: Suppose Assumption II.1 and condition (2) For the fixed τ>0,weuseC([−τ,0], Rn ) to denote the space hold. If there exists a positive definite matrix P such that n ¯T ¯ of all continuous R -valued functions ϕ defined on [−τ,0] with TP := A P + P A + DP < 0, then for any τ1 ∈ [0,τ1 (P )), the norm ϕ C =supt∈[−τ,0] ϕ(t) . the trivial solution to SDDE (1) is mean square and almost λmin(−TP ) surely exponentially stable, where τ1 (P ):= ¯T ¯ T . A P A+A 1 PA1 II. STOCHASTIC STABILITY AND GARE Corollary II.2: Suppose that A¯ is symmetric, τ1 =0, and When the control of multiagent systems with the delayed fi(y)=iy, i =1,...,d. Then, the trivial solution to SDDE and noisy measurements is studied, the consensus problem of (1) is mean square exponentially stable if and only if 2A¯ + d 2 the closed-loop systems is transformed into the stability prob- i=1 i In < 0. lem of SDDEs. Hence, the stochastic stability theorem (see Remark II.1: Concerning Theorem II.1, first it can yield Theorem II.1) should be first established. After the stability the- an explicit delay bound (see Corollary II.1). Second, it pro- orem, we need to fix how to design the feedback control based vides a delay dominated stability criterion and improves the on the delayed and noisy measurements. The solution will resort LMI stability theorems in [30], which takes the SDDE dy(t)= − − to a GARE (see Theorem II.2). This section is to establish the y(t τ1 )dt + y(t)dw(t) for example, and gave delay bound ∗ two tools. τ1 < 0.1339. Our Theorem II.1 yields a better delay bound ∗ τ1 < 0.5. Third, it shows that the stability of stochastic delay A. Stochastic Stability of SDDEs systems does not necessarily depend on the time-delay in diffu- sion. In fact, the necessary condition of the mean square stability This section aims to establish the mean square and almost is independent of such delay in Corollary II.2. sure exponential stability of the following SDDE: Remark II.2: Corollary II.1 shows that the mean square sta- ble linear time-invariant SDEs can tolerate a certain time delay, − dy(t)=[A0 y(t)+A1 y(t τ1 )]dt + dM(t) (1) which depends on the original system parameters. Especially, for the scalar SDDE: dy(t)=(a0 y(t)+a1 y(t − τ1 ))dt + σy(t − × d t ∈ Rn n − ∗ 2(a +a )+σ 2 where A0 ,A1 , M(t)= i=1 0 fi(y(s τ2 ))dwi − 0 1 τ2 )dw(t), we have a time delay bound τ1 = (a +a )2 +a2 (s), τ ,τ ≥ 0, f : Rn → Rn , d>0, {w (t)}d are indepen- 0 1 1 1 2 i i i=1 for mean square stability if the delay-free SDE dy(t)=(a + dent Brownian motions. The functions {f (x)}d satisfy the 0 i i=1 a )y(t)dt + σy(t)dw(t) is mean square stable, which is equiv- following assumption. 1 alent to 2(a + a )+σ2 < 0. Assumption II.1: f (0) = 0, i =1,...,d, and there exist 0 1 i Remark II.3: Note that the sufficient conditions for the mean positive constants { }d such that for any y ,y ∈ Rn , i i=1 1 2 square and almost sure exponential stability in Theorem II.1 f (y ) − f (y )≤ y − y , i =1,...,d. i 1 i 2 i 1 2 do not involve time delay τ in the diffusion term. Corollary We also give the initial data y(t)=ϕ(t) for t ∈ [−τ,0], τ = 2 II.2 also shows that the mean square stability is independent τ ∨ τ , and ϕ ∈ C([−τ,0], Rn ). Here, we assume that for each 1 2 of time delay τ when τ =0, which is consistent with the P > 0, there exists a D ≥ 0 such that 2 1 0 P0 scalar case in [31]. Theorem II.1 and Corollary II.2 also give ¯ d that if A is symmetric and fi (y)=i y, i =1,...,d, then the T T ≤ trivial solution to SDDE (1) is mean square and almost surely fi (y)P0 fi (y) y DP0 y. (2) λ (−(2A¯+ d 2 I )) ≤ min i =1 i n i=1 exponentially stable if τ1 ¯T ¯ T . Compared A A+A 1 A 1 1062 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 63, NO. 4, APRIL 2018 with [32, Th. 2.1], not only does it improve the stability criterion, dependent noise: but also avoids solving a complex quadratic equation. dx(t)=[Ax(t)+Bu(t)]dt + σB0 u(t)dw(t) (9) B. GARE where u(t)=Kx(t) is the input control. The authors proved In this section, we develop the existence and uniqueness of that the stochastic system (9) can be stabilized for any given the solution P>0 to the following GARE: noise intensity σ>0 if and only if (A, B) is stabilizable and the columns of input coefficient matrix B (Rank(B )
In the following sections, we apply Theorems II.1 and II.2 for t ∈ [−τ,0], and ϕ ∈ C([−τ,0]; RNn) is deterministic. Let to establish multiagent consentability, consensus, and stochastic f·(·) denotes the family of noise intensity functions {flji(·), stabilization. i =1, 2, ..., N, j ∈ Ni , l =1, 2, ..., d}. The collection of all admissible distributed protocols with different control gain III. LEADER-FREE LINEAR MULTIAGENT SYSTEMS matrices is denoted by ⎧ A. Linear Multiagent Systems ⎨ U · | ≥ We consider a system consisting of N (N ≥ 2) agents where (τ1 ,τ2 ,f·( )) = ⎩u(t) ui (t)=K zji(t),t 0, the agents are indexed by 1, 2,...,N, respectively. The dy- j=N i namics of the N agents are described by the continuous-time systems: K ∈ Rm ×n ,i=1,...,N. . (15)
x˙ i(t)=Axi(t)+Bui(t),t∈ R+ (12) Under the measurement noise, the consensus definitions are ∈ Rn ∈ Rn×n ∈ Rn×m where i =1, 2, ..., N, xi(t) , A , B , diversified due to the different asymptotic behaviors in the prob- ∈ Rm and ui(t) is the input control of the ith agent, ability sense, where the mean square and the almost sure consen- T T T respectively. Denote x(t)=[x1 (t), ..., xN (t)] and u(t)= sus are two important topics. Here, we give the definitions on the T T T [u1 (t), ..., uN (t)] . Here, the input control u(t) is to be de- mean square and the almost sure consentability and consensus. signed. The information flow structures among different agents Definition III.1: We say that the linear systems (12) are mean are modeled as a connected undirected graph G = {V, E, A}, square (or almost surely) consentable w.r.t. U(τ1 ,τ2 ,f·(·)),if where V = {1, 2, ..., N} is the set of nodes with i represent- there exists a protocol u ∈U(τ ,τ ,f·(·)) solving the mean E A 1 2 ing the ith agent, denotes the set of directed edges and = square (or almost sure) consensus, that is, it makes the agents ∈RN ×N G [aij ] is the adjacency matrix of with element aij =1 have the property that for any initial data ϕ ∈ C([−τ,0]; RNn) or 0 indicating whether or not there is an information flow from 2 and all distinct i, j ∈V, limt→∞ Exi(t) − xj (t) =0 (or agent j to agent i directly. Also, N denotes the set of the node i lim →∞ x (t) − x (t) =0, a.s.). ∈ N t i j i’s neighbors, that is, for j Ni , aij =1, and degi = j=1 aij Remark III.1: The delayed measurement (f (·) ≡ 0)orthe G lji is called the degree of i. The Laplacian matrix of is defined as noisy measurement without delay (τ = τ =0) was investi- L D−A D 1 2 = , where = diag(deg1 , ..., degN ). It is obvious that gated in [7] and [13], respectively. In fact, in the complicated L is a symmetric matrix and admits a zero eigenvalue, denoted environment, the information communication is often subject to λ λ ≤ ≤ λ by 1 ; other eigenvalues 0 < 2 ... N are positive due to time-delays and measurement noises simultaneously. The gen- G the connectivity of . eral measurement (14) is to describe this phenomenon and has We consider the distributed protocol in the following form: been examined in [15] for the first-order multiagent consensus. It is worth noting that the current results are still new even for ui(t)=K zji(t) (13) the delay-free case (τ = τ =0). j∈N i 1 2 Remark III.2: Multiagent consensus and consentability ∈ Rm ×n where K is the feedback gain matrix to be designed based on precise (noise and delay free) relative state measure-
zji(t)=xj (t − τ1 ) − xi(t − τ1 ) ments were well studied in [7] and [18], [40]. Consensus based on relative state measurements require less information than that d based on both absolute and relative state measurements, and for + f (x (t − τ ) − x (t − τ ))ξ (t),j∈ N (14) lji j 2 i 2 lji i many realistic cases, each agent may not have the ability to l=1 get the absolute state information due to the limited perception is the state measurement of the agent i from its neighbor agent j, capacity and the lack of global coordinates. Multiagent con- T ∈ Rd τ1 ,τ2 are the time delays, ξji(t)=(ξ1ji(t),...,ξdj i (t)) sentability has not been well studied in the delayed and/or noisy · Rn → Rn is the measurement noise, flji( ): is the noise environment. Here, the admissible protocol (15) can be viewed − − − intensity function. In this paper, xj ( t τ1 ) xi(t τ1 ) is as a natural extension from the precise relative state measure- d − − called the deterministic term and l=1 flji(xj (t τ2 ) ments to the delayed noisy version. The skills developed in this xi(t − τ2 ))ξlji(t) is called the stochastic term. We assume that paper can also be used for the case with absolute state feedback. the noises are independent Gaussian white noises. To be exact, Applying Theorems II.1 and II.2 produces the follow- they satisfy the following assumption. ing theorem, which gives the consentability conditions and Assumption III.1: The noise process ξji(t)=(ξ1ji(t),..., the consensus protocol design for linear systems (12) under T ∈ Rd t ≥ λ ≥ ξdj i (t)) satisfies 0 ξji(s)ds = wji(t), t 0,i=1, 2, max(Re( (A))) 0. ...,N,j ∈ Ni , where {wji(t),i=1, 2,...,N,j ∈ Ni } are in- Theorem III.1: Suppose that Assumption III.1 holds, flji(x) dependent d-dimensional Brownian motions defined on the = σljix, σlji ≥ 0, and max(Re(λ(A))) ≥ 0. Then, linear sys- complete probability space (Ω, F, P). tems (12) are mean square and almost surely consentable w.r.t. Note that the update of state x(t) depends on the past U(τ1 ,τ2 ,f·(·)) if states x(s), s ∈ [t − τ,t) with τ = τ1 ∨ τ2 . We need to de- 1) (A, B) is controllable; − λu N −1 2 λ fine the initial function on [ τ,0]. We assume that x(t)=ϕ(t) 2) 4 0 N σ¯ < 2 ; 1064 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 63, NO. 4, APRIL 2018
∈ ∗ T 3) τ1 [0,τ1 ), (0,σjix2 ) , σji > 0. Then, linear systems (12) are mean square − ∗ λ −4 N 1 σ¯ 2 (λu + ) N −1 2 1 ∧ 2 N 0 consentable w.r.t. U(τ1 ,τ2 ,f·(·)) only if λN > 4β σ , where τ1 = 2A2 P 6(λu + )λ , P>0 is the N 0 N N ∈ λu λu ∈ where σ = min σji. solution to the GARE (11) with α ( 0 , 0 + ), i,j=1 λ − N −1 2 λu 2 4 N σ¯ 0 ) 2 d N 2 (0, − ), σ¯ = max σ . In particular, 4 N 1 σ¯ 2 l=1 i,j=1 lji N B. Second-Order Integrator Multiagent Systems T −1 T the protocol (13) with K = k(Im + B PB) B P , k ∈ (k, k¯), solves the mean square and almost sure consensus, In view of Theorems III.1 and III.4, we can see that if λ − λ2 − ¯ λ λ2 − max(Re(λ(A))) > 0, the mean square consentability of linear where k =[ 2 2 2αρ]/ρ, k =[ 2 + 2 2αρ]/ρ, ρ =(2N −1 σ¯2 +3λ τ )λ . systems (12) might be destroyed by the large noise intensities. If N N 1 2 λ Remark III.3: If A =0, then condition 1) in Theorem III.1 max(Re( (A))) = 0, then the consentability of linear systems (12) holds for any given noise intensities if τ1 =0. However, we implies that B is invertible. In this case, for any given τ1 , τ2 , and need to solve the GARE (11) in order for finding the consensus σlji, i, j =1,...,N,l =1,...,d, the linear multiagent sys- protocol. Moreover, we do not know whether the restriction on tems (12) are mean square consentable w.r.t. U(τ1 ,τ2 ,f·(·)). −1 1 the time delay can be removed if max(Re(λ(A))) = 0. These In fact, K = −kB with k ∈ (0, − ) can be used to λ τ + N 1 σ¯ 2 N 1 N motivate us to give further investigation. achieve the stochastic consensus. λ In this section, we consider the second-order integrator sys- If A is stable (max(Re( (A))) < 0), then deterministic con- T 2 tems, that is, xi(t)=[yi (t),vi (t)] ∈ R with the dynamics sensus of x˙ i = Axi follows because each xi(t) tends to zero. Moreover, we can prove the following theorem, which shows y˙i(t)=vi(t), v˙i(t)=ui (t),i=1, 2, ..., N (16) that small control gain does not affect the consensus of the which can be considered as the special case of the linear system original system x˙ i = Axi . (12) with A =(01), and B =(0, 1)T . Here, y (t) and v (t) can Theorem III.2: Suppose that Assumption III.1 holds, flji(x) 00 i i be considered as the position and the velocity of the ith agent, = σljix, σlji ≥ 0, and max(Re(λ(A))) < 0. Then, the linear systems (12) are mean square and almost surely consentable respectively. It is easy to see that (A, B) is controllable and λ(A)=0. We will show that second-order integrator systems w.r.t. U(τ1 ,τ2 ,f·(·)) for any given τ1 ,τ2 ,σlji > 0. Especially, the choice K = kBT P can guarantee the mean square and al- (16) must be mean square and almost surely consentable w.r.t. U · most sure consensus of the linear system (12), where k ∈ R (τ1 ,τ2 ,f·( )) for any given system parameters. Here, we focus | |λ T 2 N −1 T | | on the choice of the protocol for solving stochastic consensus. satisfies k N PBB P (2 +σ ¯ N BB P k ) < 1, P = ∞ A T t At For this case, we consider the different uncertainties in each 0 e e dt. When τ vanishes, we can obtain the following corollary, position and velocity communication. Then, the protocol (13) 1 can be rewritten as follows: which is a direct consequence of Theorem III.1. Corollary III.1: Suppose that Assumption III.1 holds, ui(t)=k1 z1ji(t)+k2 z2ji(t) (17) f (x)=σ x, σ ≥ 0, τ =0 and max(Re(λ(A))) ≥ 0. lji lji lji 1 j∈N i j∈N i Then, linear systems (12) are mean square and almost surely ∈ R1×2 − − − consentable w.r.t. U(τ1 ,τ2 ,f·(·)) if conditions 1) and 2) in where K =[k1 ,k2 ] ,z1ji(t)=yj (t τ1 ) yi(t τ1 ) − − − − − Theorem III.1 hold. Moreover, the protocol (13) with K = + f1ji(yj (t τ2 ) yi(t τ2 ))ξ1ji(t),z2ji(t)=vj (t τ1 ) T −1 T ¯ − − − − k(Im + B PB) B P , k ∈ (k, k), solves the mean square vi(t τ1 )+f2ji(vj (t τ2 ) vi(t τ2 ))ξ2ji(t) are the and almost sure consensus, where P> 0 is the solution to position and the velocity measurements of the agent i from λ2 T ∈ λu 2 λ − λ2 − ¯ its neighbor j ∈ Ni , ξji(t)=[ξ1ji,ξ2ji] is the 2-D Gaussian GARE (11) with α ( 0 , ), k =[ 2 2 2αρ]/ρ, k = 2ρ white noise, that is, ξ and ξ are the scalar independent [λ + λ2 − 2αρ]/ρ, ρ =2N −1 σ¯2 λ . 1ji 2ji 2 2 N 2 Gaussian white noise, and f (·) and f (·) satisfy the Theorem III.1 and Corollary III.1 focus on the suf- 1ji 2ji following assumption. ficient conditions on stochastic consentability. We now Assumption III.2: For each (j, i), f (0) = f (0) = 0 examine the necessary conditions for the mean square 1ji 2ji and there exist nonnegative constants σ¯ and σ¯ consentability. 1ji 2ji such that f (y ) − f (y )≤σ¯ y − y , f (y ) − Theorem III.3: Suppose that Assumption III.1 holds and 1ji 1 1ji 2 1ji 1 2 2ji 1 f (y )≤σ¯ y − y for all y ,y ∈ R. τ =0. Then, linear systems (12) are mean square consentable 2ji 2 2ji 1 2 1 2 1 By using Theorem II.1 and choosing appropriate P in (3), we w.r.t. U(τ ,τ ,f·(·)) only if (A, B) is stabilizable. 1 2 can obtain the following consensus conditions. The following example gives the necessity of certain restric- Theorem III.5: Suppose that Assumptions III.1 and III.2 tion on the noise intensities for the mean square consentability hold, and d =2. Then, the protocol (17) solves the mean square of the two-dimensional dynamics. and almost sure consensus of the multiagent system (16) if Theorem III.4: Consider the multiagent systems (12) with k1 > 0,k2 > 0, and − 01 2 N 1 n =2,τ1 = τ2 =0,A= k 2λ τ +¯σ (2 + k λ τ )+k λ τ 0 β 1 N 1 1 N 1 N 1 1 2 1
N − 1 T λ − 2 − 2 λ2 and B =(0, 1) . Suppose that Assumption III.1 holds < 2 2 k2 1 k2 σ¯2 4k2 N τ1 (18) T N with d =2, β>0 and f1ji(x)=(σjix1 , 0) ,f2ji(x)= ZONG et al.: STOCHASTIC CONSENTABILITY OF LINEAR SYSTEMS WITH TIME DELAYS 1065
N N where σ¯1 = maxi,j=1 σ¯1ji and σ¯2 = maxi,j=1 σ¯2ji solve the stochastic leader-following consentability and con- Remark III.4: Theorem III.5 shows that for any time delays sensus problems, where we only need the subgraphs formed by { }N τ1 ,τ2 and σ1ji,σ2ji i,j=1, we can find the appropriate control the followers to be undirected. This is the following section’s gains k1 ,k2 to guarantee the mean square and almost sure con- intention. { }N sensus. In fact, for any time delays τ1 ,τ2 and σ1ji,σ2ji i,j=1, λ2 we first choose k2 such that k2 < λ2 λ 2 N −1 .Let IV. LEADER-FOLLOWING MULTIAGENT SYSTEMS 2 N τ1 + 2 σ¯ 2 N λ − 2 N −1 − k2 be fixed. Then, we have p1 := 2 2 k2 (1 k2 σ¯2 N ) A. Leader-Following Linear Multiagent Systems 2 λ2 4k2 N τ1 > 0. Based on the given k2 , we choose k1 such that k p (k )
0 − ≥ − ≥ such that σ¯1ji y1 y2 f1ji(y1 ) f1ji(y2 ) σ1ji y1 For the ith follower, the dynamics is described as (12) − − ≥ − ≥ − y2 , σ¯2ji y1 y2 f2ji(y1 ) f2ji(y2 ) σ2ji y1 with ui(t) defined by (13). Note that this is different from ∈ R y2 for all y1 ,y2 , Section III-A since for each agent i, its neighbors Ni may con- By Theorem III.5 and stochastic stability theorem of stochas- tain the leader 0. Considering the information flow from the tic ordinary differential equations, we can prove the following leader to the followers, we denote the topology by G = {V , A} necessary conditions and sufficient conditions. with V = {0, 1, 2,...,N} and Theorem III.6: Suppose that Assumptions III.1 and III.3 hold, d =2, and τ1 = τ2 =0. Then, the protocol (17) solves 00N ×N (N +1)×(N +1) the mean square consensus if A = ∈ R , a0 A − − 2 N 1 λ − 2 2 N 1λ k1 > 0,k1 σ¯1
Theorem IV.2: Suppose that Assumption III.1 holds, G forms a star topology, flji(x)=σljix, σlji ≥ 0, and max(Re(λ(A))) ≥ 0. Then, linear systems (12) and (22) are mean square and almost surely consentable w.r.t. U(τ1 ,τ2 ,f·(·)) if condition 1) holds and λu 2 2 ) 0 σ¯ < 1/2, ∗ ∗ 1−2¯σ 2 (λu + ) ∈ 1 ∧ 0 3 ) τ1 [0,τ1 ) with τ1 = 2A2 P 6(λu + ) , 0 ∈ where P>0 is the solution to the GARE (11) with α 1−2¯σ 2 λu λu λu ∈ 0 2 d N 2 ( 0 , 0 + ), (0, 2¯σ 2 ), σ¯ = l=1 maxi=1 σl0i. Espe- T −1 T cially, the protocol (13) with K = k(Im + B PB) B P , ¯ k ∈ (k, k), solves the√ mean square and almost√ sure con- sensus, where k =[1− 1 − 2αρ]/ρ, k¯ =[1+ 1 − 2αρ]/ρ, 2 ρ =(¯σ +3τ1 ). If we consider the special case: x0 (t) ≡ 0 and N =1 (one follower). Then, the stochastic consentability problem in Theorem IV.2 is actually the stochastic stabilization problem of (12), which is concluded as the following corollary and improves Fig. 1. Relative state errors of the four agents: k =0.01, k =0.065. the case with single and linear diffusion in [46]. 1 2 Corollary IV.1: Suppose that Assumption III.1 holds, x0 (t) ≡ 0, N =1, fl01(x)=σl01x, σl01 ≥ 0, and V. S IMULATIONS max(Re(λ(A))) ≥ 0. Then, linear systems (12) is mean square and almost surely stabilizable if conditions 1), 2”), We consider the almost sure and mean square consensus for 3”) hold with P>0 being the solution to the GARE (11) a second-order integrator multiagent system (16) composed of 1−2¯σ 2 λu d ∈ λu λu ∈ 0 2 2 four scalar agents. The control input ui(t) has the form of (17). with α ( 0 , 0 + ), (0, 2¯σ 2 ), σ¯ = l=1 σl01. Especially, the choice K defined in Theorem IV.2 stabilizes the We will choose the appropriate control gains k1 and k2 to guar- linear system (12). antee the mean square and almost sure consensus. Consider G = {V, E, A}, where V = {1, 2, 3, 4}, E = {(1, 2), (2, 3), (3, 4), (4, 3), (3, 2), (2, 1)} and A =[a ] × B. Leader-Following Second-Order Integrator ij 4 4 with a = a = a = a = a = a =1 and other be- Multiagent Systems 12 21 23 32 34 43 ing zero. It can be obtained the eigenvalues of the Laplacian T ∈ Consider the ith follower’s state xi(t)=[yi (t),vi (t)] matrix L: λ1 =0, λ2 =0.5858, λ3 =2, λ4 =3.4142. The ini- 2 T R with the dynamics (16), and the leader’s state x0 (t)= tial states are given by y(s)=[2, −4, −2, −5] and v(s)= T ∈ R2 T [y0 (t),v0 (t)] with the dynamic [7, 2, −4, −8] , s ∈ [−(τ1 ∨ τ2 ), 0], where τ1 =0.2 and τ2 = 0.5. Assume that the noise intensity functions f1ji(x)= y˙0 (t)=v0 (t)=v0 (23) σ¯1jix =0.5x and f2ji(x)=¯σ2jix =0.3x, i, j =1, 2, 3, 4. We now use Remark III.4 to choose the control where v0 is a constant velocity known to all followers. The gains k1 ,k2 such that the four agents achieve the mean protocol is designed as (17), which includes the leader’s infor- ∗ mation. Similarly to the linear case above, we have the following square and almost sure consensus. We compute that k2 = λ2 λ2 λ 2 N −1 =0.1246. Therefore, we choose k2 =0.065 < theorems and corollary. 2 N τ1 + 2 σ¯ 2 N ∗ λ − 2 N −1 − 2 λ2 Theorem IV.3: Suppose that Assumptions III.1, III.2, and k2 . Then, we have p1 := 2 2 k2 (1 k2 σ¯2 N ) 4k2 N τ1 = IV.1 hold, and d =2. Then, the protocol (17) solves the mean 0.0365. Based on the given k2 , we choose k1 =0.01, square and almost sure consensus of (16) and (23) if k1 > and then 0.0323 = k1 p2 (k1 )
since it is difficult work to find the necessary and sufficient condition for the stability of SDDEs. It can be observed that the consensus analysis with measurement noises and time de- lays under the general directed graphs is more difficult than the case under undirected graphs, especially for the linear (includ- ing second-order) multiagent systems. Moreover, the consensus analysis of multiagent systems with nonlinear dynamics, switch- ing topology, time-varying delays, and measurement noises has not be given. Effort can also be directed to extend our analysis to the heterogeneous multiagent systems with time delays and measurement noises.
APPENDIX A PROOF OF THEOREMS IN SECTION II-A Lemma A.1: ([41]) For any positive definite matrix P and positive constant τ, the following inequality holds: t T t t Fig. 2. States of the four agents: k1 =0.01, k2 =0.065. x(s)ds P x(s)ds ≤ τ xT (s)Px(s)ds t−τ t−τ t−τ provided that the integrals above are well defined. t Proof of Theorem II.1: Let z(t)=y(t)+A1 y(s)ds. t−τ1 We choose the degenerate Lyapunov functional (see [25]) { ∈ V (yt )=V1 (t)+V2 (t), where yt = y( t + θ ): θ T 0 t T [−τ1 , 0]}, V1 (t)=z (t)Pz(t), V2 (t)= y (θ) −τ1 t+s T A1 PA1 y(θ)dθds. Note that the elementary inequality: 2xT Oy ≤ xT Ox + yT Oy, for any positive definite matrix O ∈ Rn×n and x, y ∈ Rn .UsingIto’sˆ formula and Lemma A.1, we have T T dV1 (t)=y (t)[A¯ P + P A¯]y(t)dt + d M,PM (t) t T T +2y (t)A¯ PA1 y(s)dsdt + dM1 (t) t−τ1 t ≤ T T T y (t)S1 y(t)dt + y (s)A1 PA1 y(s)dsdt t−τ1 Fig. 3. Mean square errors of the relative states: k =0.01, 1 k2 =0.065. + d M,PM (t)+dM1 (t) (24) ¯T ¯ ¯T ¯ d taking the mean square average, we obtain Fig. 3, which shows where S1 = A P + P A + τ1 A P A, M,PM (t)= i=1 t T − − t T that the four agents achieve the mean square consensus. 0 fi (y(s τ2 ))Pfi (y(s τ2 ))ds, M1 (t)=2 0 z (s)PdM (s). Then, we get from (24) and condition (2) that dV (yt ) ≤ T − ¯T VI. CONCLUDING REMARKS y (t)S2 y(t)dt + F (t τ2 )dt + dM1 (t), where S2 = A P + ¯ ¯T ¯ T T P A +(A P A + A1 PA1 )τ1 , F (t)=y (t)DP y(t). Applying In this paper, we developed consentability for linear multi- γt the Itoˆ formula to e V (yt ), for any γ>0,wehave agent systems with multiplicative noises and time delays. By γt γt γt establishing stochastic stability theorem of SDDEs and the pos- d[e V (yt )] = γe V (yt )dt + e dV (yt ) itive definite solution of GARE, we obtained the stochastic ≤ γeγtV (y )dt + eγtyT (t)S y(t)dt consentability conditions and consensus protocols for the lin- t 2 γt γt ear multiagent systems. Our results reveal that the stochastic + e F (t − τ2 )dt + e dM1 (t). consentability depends on the essential properties of the origi- Integrating both parts of the above inequality and taking the nal systems, including (A, B), the graph, the noise intensities, expectations yield and the time delay in the deterministic term. In particular, the second-order integrator system must be mean square and almost t γtE ≤ E E γs T surely consentable regardless of the values of time delays and e V (yt ) V (y0 )+ e y (s)S2 y(s)ds 0 noise intensities. t t There are still many interesting topics deserving further in- γs γs + γe EV (ys )ds + E e F (s − τ2 )ds. (25) vestigation. We have not obtained the optimal time delay bound 0 0 1068 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 63, NO. 4, APRIL 2018
t γs γτ2 0 γτ2 t d Note that e F (s − τ2 )ds ≤ e F (s)ds + e w(t)= j dwj (t)/ is still a scalar Brownian motion. 0 −τ2 0 j=1 eγsF (s)ds. Then, we get Then, we can rewrite the equation above as t γt γs T dxi(t)=λixi(t)dt + xi(t − τ2 )dw(t),i=1,...,n. (29) e EV (yt ) ≤ C1 (γ)+ e Ey (s)S3 y(s)ds 0 Hence, the trivial solution to SDDE (1) is mean square stable t γsE if and only if (29) is mean square stable, which is equivalent to + γe V (ys )ds (26) λ 2 0 that 2 i + < 0 (see [31]). γτ2 0 where C1 (γ)=V (y0 )+e −τ F (s)ds, S3 (γ)=S2 + γτ 2 APPENDIX B e 2 DP . By the definition of the functional V (yt ) and the el- PROOF OF THEOREMS IN SECTION II-B ementary inequality (x + y)T Q(x + y) ≤ 2xT Qx +2yT Qy, n s 2 − T x, y ∈ R , Q>0,wehaveV (ys ) ≤ C2 − y(u) du + Proof of Lemma II.1: Let A(α):=A αIn .IfA P + s τ1 − 2 2 PA− 2αP B(R + BT PB) 1 BT P + Q =0 2P y(s) , where C2 =3τ1 A1 P . Substituting this has posi- T into (26) yields tive definite solution P>0, then A(α) P + PA(α)= −2α(P −1 + BR−1 BT )−1 − Q<0, where 2α[P − PB(R + t − − − − γt γs T BT PB) 1 BT P ]=2α(P 1 + BR 1 BT ) 1 is used. Hence, e EV (yt ) ≤ C1 (γ)+ e Ey (s)S3 (γ)y(s)ds 0 A(α) is Hurwitz, which implies α>max(Re(λ(A))). t s Proof of Lemma II.2: We first prove that the corresponding γs E 2 + C2 γe y(u) duds discrete-time system is mean square stabilizable with a expo- 0 s−τ1 nential convergence rate under the given conditions, then we t get the sufficiency by applying the fact [43] that the mean +2P γeγsEy(s)2 ds. (27) 0 square exponential stability of the discrete-time models yields Note that the mean square exponential stability of the continuous-time t s models. Consider the following discrete-time linear system: eγs Ey(u)2 duds xi+1 = xi + AxiΔ+BKxiΔ+σBKxiΔwi , (30) 0 s−τ1 0 u+τ1 which can be considered as the EulerÐMaruyama approximation ≤ Ey(u)2 eγsdsdu to (7), where Δ > 0 is time-stepsize and Δwk is the Brownian −τ1 u increment. Let AΔ = I + AΔ, BΔ = BΔ. Note that (A, B) is t u+τ1 controllable, then there is a Δ∗ > 0 such that for any Δ < Δ∗ , + Ey(u)2 eγsdsdu 1 1 0 u (AΔ ,B) is controllable. It can be proved that t 1 ≤ τ 2 eγτ1 ϕ2 + τ eγτ1 eγuEy(u)2 du. − λu λu 1 C 1 lim 1 u 2 /(2Δ) = Re( i (A)) = 0 0 Δ→0 Πi|λ¯ (AΔ )| i i γtE ≤ t γsE T Hence, we get e V (yt ) C3 (γ)+ 0 e y (s)S3 (γ) where {λ¯u (A)} denotes A’s eigenvalue(s) larger than one in t i i y(s)ds + C (γ)γ eγsEy(s)2 ds, where C (γ)=C (γ)+ 2 λu ∗ 4 0 3 1 absolute value. Therefore, if 2σ 0 < 1, then there exist a Δ < C (γ)γτ2 eγτ1 ϕ2 and C (γ)=C τ eγτ1 +2P .Let ∗ ∈ ∗ 1 − 1 4 1 C 4 2 1 Δ1 such that for any Δ (0, Δ ), 2 Δ > [1 |¯λu |2 ]. σ Πi i (A Δ ) S4 (γ)=S3 (γ)+γC4 (γ)In . Then, it is obtained that ∈ ∗ Hence, for any Δ (0, Δ ), the following ARE has a unique t γt γs T positive definite solution P (see [36]): e EV (yt ) ≤ C3 (γ)+ e Ey (s)S4 (γ)y(s)ds. (28) 0 T − 1 T T −1 T AΔ PAΔ 2 ΔAΔ PB(R + B PB) B PAΔ +QΔ=P Note that S4 (0) < 0 under (3). Therefore, if (3) holds, σ ∗ (31) then there must exists a γ > 0 such that for any −2 ∗ ¯T ¯ ¯T ¯ T for R>0 and Q>0. That is, the choice K = σ (R + γ<γ, S4 (γ)=A P + P A +(A P A + A1 PA1 )τ1 + T −1 T γτ2 B PB) B PAΔ guarantees the mean square stability of e DP + γC4 (γ)In < 0, which together with (28) implies ∞ γsE T − (30). Note the mean square asymptotical stability and the mean 0 e y (s)( S4 (γ))y(s)ds < C3 (γ). This also produces − square exponential stability are equivalent for the linear time- Ey(t)2 ≤ C e γ0 t . Under Assumption II.1, the mean square 0 invariant SDE. By the mean square exponential stability of the exponential stability implies the almost sure exponential numerical and exact solutions (see [43]), we can see that the stability (see [42]). That is, (4) holds. linear system (7) is mean square stabilizable. d 2 ¯ Proof of Corollary II.2: Let = j=1 j . Note that A Now we show that necessity of conditions that (A, B) is is symmetric, then there exists a unitary matrix Φ such stabilizable and 2 max(Re(λ(A)))σ2 < 1 for the mean square T −1 T ¯ that Φ =Φ and Φ AΦ=diag(λ1 , λ3 , ··· , λn )=:Λwith stabilization. If (A, B) is unstabilizable, then for any K, λ ¯ λ ≤···≤λ λ ¯ min(A)= 1 n = max(A). In view of the trans- A − BK is not Hurwitz. Let AK := A − BK. By the matrix ¯ 2 2 formation, TA := 2A + In < 0 if and only if 2λi + < 0, theorem, there exists a complex invertible matrix Q such that −1 −1 i =1,...,d. Letting x(t)=Φ y(t), then we get dxi(t)= QAK Q = J, Here, J is the Jordan normal form of AK , i.e., λ d − l ixi(t)dt + j=1 j xi(t τ2 )dwj (t),i =1,...,n. Note that J = diag(Jμ1 ,n1 ,Jμ2 ,n2 ,...,Jμl ,nl ), k=1 nk = n, where ZONG et al.: STOCHASTIC CONSENTABILITY OF LINEAR SYSTEMS WITH TIME DELAYS 1069
∗ ∗ μ1 ,μ2 ,...,μl are all the eigenvalues of AK and Jμk ,nk is Therefore, the exists a P >P0 > 0 such that P = limi→∞ Pi. the corresponding Jordan block of size nk with eigenvalue Taking the limit in (33) yields the existence of the positive t μk .LetY (t)=Qx(t) and M(t)=σQ 0 BKx(s)dw(s). definite solution. This together with [44, Th. 4.2] gives that the Then, we have from (7) that dY (t)=JY(t)dt + dM(t). positive definite solution to GARE (6) is unique. Note that AK is not Hurwitz, then there must exist an eigenvalue, denoted by μk , with the nonnegative real part APPENDIX C (Re(μk ) ≥ 0). Considering the kth Jordan block and its T PROOF OF THEOREMS IN SECTION III corresponding component ζk (t)=[ζk,1 (t),...,ζk,nk (t)] and M(k, t)=[M (t),...,M (t)]T , where ζ (t)=Y (t) For the linear multiagent systems, we denote δ(t)= k,1 k,nk k,j kj k−1 − ⊗ T T T and Mk,j(t)=Mk (t) with kj = ni + j,wehavedζk,n [(IN JN ) In ]x(t).Letδ(t)=[δ1 (t), ..., δN (t)] , where j i=1 k n δi(t) ∈ R , i =1, ..., N. Define the unitary matrix TL = (t)=μk ζk,nk (t)dt + dMk,nk (t), which together with the vari- √1N μk t [ ,φ2 , ..., φN ], where φi is the unit eigenvector of L asso- ation of constants formula implies ζk,nk (t)=e ζk,nk (0) + N t μ (t−s) 2 T T e k dM (s). Hence, we get Eζ (t) = ciated with the eigenvalue λi, that is, φ L = λiφ , φi = 0 k,nk k,nk i i t − Re(μk )t 2 E μk (t s) 2 ≥ Re(μk )t 1, i =2, ..., N. Denote φ =[φ2 , ..., φN ].Letδ(t)=(TL ⊗ e ζk,nk (0) + 0 e dMk,nk (s) e 2 I )δ (t) and δ (t)=[δ T (t), ..., δ T (t)]T , then it can be veri- ζk,nk (0) . This is in contradiction with the mean square n 1 N ≡ T T T stability. Hence, (A, B) must be stabilizable. fied that δ1 (t) 0. Denote δ(t)=[δ2 (t), ..., δN (t)] and Λ= By [38, Th. 1], if the linear system (7) is mean square diag(λ2 , λ3 , ··· , λN ). stabilizable, then there exists a positive definite solution to Proof of Theorem III.1: It suffices to show that the protocol −2 T −1 T (6) with 2α = σ . This together with Lemma II.1 implies (13) with K = k(Im + B PB) B P solves the mean square 2 max(Re(λ(A)))σ2 < 1. and almost sure consensus. The consensus problems will be Proof of Theorem II.2: The “only if” part has been proved first transformed into the stability problems of a SDDE, then in Lemma II.1. Hence, we need only prove the “if” part in the Theorems II.1 and II.2 will be applied to solve the stability two assertions, respectively. problems, which also solve the consensus problems. By Lemma II.2, we know that the system (7) is mean square Note that conditions 2) and 3) and the definition of λ2 λu stabilizable. This together with Corollary 4 in [26] can yield the imply that 2 > 2( 0 + )ρ, which guarantees that k −1/2 ¯ ∈ λu λu first “if” part by letting σ =(2α) . and k are well defined for α ( 0 , 0 + ). With the If B is invertible and α>max(Re(λ(A))), we do not protocol (13), the closed-loop system takes the form know whether the linear system is mean square stabiliz- ⊗ − L⊗ − dx(t)=(IN A )x(t) dt ( BK )x(t τ1 )dt + dM1 (t), able. Therefore, [26, Corollary 4] cannot be used to prove d N t ⊗ − where M1 (t)= l=1 i,j=1 aijσlji 0 [Si,j BK]δ(s τ2 ) the second “if” part. Here, we will develop a matrix ap- dwlji(s),Si,j =[skl]N ×N is an N × N matrix with Γ := proximate sequence to get the desired assertion. Let P sii = −aij ,sij = aij and all other elements being zero, i, j = AT P + PA− 2αP B(R + BT PB)−1 BT P + Q . It is easy to 1, 2,...,N. By the definition of δ(t),wehavedδ(t)=(IN ⊗ T ≥ see that R + B PB > 0 for P 0 since R>0. Note that A)δ(t)dt − (L⊗BK)δ(t − τ )dt + dM (t), where M − 1 2 2 A(α)=A αIn is Hurwitz. Then, there exists a P0 > 0 such d N t − ⊗ − T (t)= l=1 i,j=1 aij σlji 0 [(IN JN )Si,j BK] δ(s that A(α) P0 + P0 A(α)+Q =0. We have τ2 )dwlji(s). This together with the definition of δ(t) implies T ΓP0 = A(α) P0 + P0 A(α)+M(P0 )+Q>0 (32) − − where M(P )=2α[P − PB(R + BT PB) 1 BT P ]=2α dδ(t)=A0 δ(t)dt + A1 δ(t τ1 )dt + dM3 (t) (34) −1 −1 T −1 (P + BR B ) .LetP0 be fixed, then there exists a P1 > T ⊗ − ⊗ d 0 satisfying A(α) P1 + P1 A(α)+Q + M(P0 )=0. We can where A0 = IN −1 A, A1 = Λ BK, M3 (t)= l=1 T − − − N t T − ⊗ − see that A(α) (P1 P0 )+(P1 P0 )A(α)= M(P0 ) < 0, i,j=1 aij σlji 0 [(φ (IN JN )Si,jφ) BK] δ(s τ2 ) which implies P1 >P0 . Then, M(P1 ) >M(P0 ) and ΓP1 > dw (s). This together with the definition of δ(t) yields that ··· 1ji 0. Assume that we have obtained Pk >Pk−1 > >P0 , i = the consensus problems equal to the stability of (34). 0, 1,...,k, we now define Pk+1 as follows: Hence, we need to prove the stability of (34) un- T der conditions 1)-3). Let P¯ = IN −1 ⊗ P . Note that A(α) Pk+1 + Pk+1A(α)+Q + M(Pk )=0. (33) d N 2 t T T M3 , P¯M3 (t)= aij σ δ (s − τ2 )[((φ T − l=1 i,j=1 lji 0 Note that M(Pk ) >M(Pk−1 ), then we have A(α) (Pk+1 − T T − ⊗ T T − − − (IN JN )Si,jφ) (φ (IN JN )Si,jφ)) K B PBK ] Pk )+(Pk+1 Pk )A(α)= [M(Pk ) M(Pk−1 )] < 0, which − T − T N δ(s τ2 )ds, φ (IN JN )Si,jφ = φ Si,jφ and that i,j=1 implies Pk+1 >Pk . Repeating the same procedure iteratively, − { }∞ a (φT S φ)T (φS φ)= 2(N 1) Λ. Then, we have we can find an increasing sequence Pi i=0. It can be seen that ij i,j i,j N −1 T −1 M(Pk ) ≤ 2α(BR B ) and then T ∞ 2 d M3 , P¯M3 (t) ≤ σ¯ δ (t − τ2 )Dδ(t − τ2 )dt (35) A(α)T t A(α)t Pk+1 = e [Q + M(Pk )]e dt 0 N −1 ⊗ T T ∞ where D =2 N Λ (K B PBK). We now show that con- T − − T −1 T ≤ eA(α) t [Q +2α(BR 1 BT ) 1 ]eA(α)t dt < ∞. ditions 2) and 3) under K = k(Im + B PB) B P im- ¯T ¯ ¯ ¯ ¯T ¯ ¯ T ¯ 2 0 ply A P + P A +(A P A + A1 PA1 )τ1 +¯σ D<0, where 1070 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 63, NO. 4, APRIL 2018
t A¯ = A + A . Note that this can be guaranteed by γτi γsE 2 0 1 e 0 e δ(s) ds. Then from (39), we obtain T T t WI P + PWi + Wi PWiτ1 γt γs 2 e EV (t) ≤ C0 (γ)+h(γ) e Eδ(s) ds (40) N − 1 0 + λ2 τ +2λ σ¯2 KT (BT PB)K<0 (36) i 1 i γτ1 2 γτ2 N where C0 (γ)=V (0) + (|k|bp τ1 e + σ(b, p)k τ2 e ) 2 γτ sup ∈ − Eδ(s) , h(γ)=|k|bp − 1+|k|bp e 1 + W = A − λ BK, i =2,...,N. By the elementary inequal- s [ τ,0] i i 2 γτ2 λ | | T ≤ T T ∈ Rn σ(b, p)k e + max(P )γ. Note that h(0) = 2 k bp + ity (x + y) Q(x + y) 2x Qx +2y Qy, x, y , Q> 2 − ≥| | T ≤ T λ2 T T σ(b, p)k 1 < 0, and h(γ1 ) k bp > 0, where γ1 satis- 0, then Wi PWi 2A PA+2 i K B PBK. Substituting γ1 τ1 ∗ T −1 T fies kbp e =1. These produce that there exists γ > 0 this above into (36) and letting K = k(I + B PB) B P , ∗ m such that h(γ )=0. Therefore, we obtain from (40) that we can observe that (36) is assured by γ ∗t 2 ∗ e Eδ(t) ≤ C0 (γ ). Note that mean square exponential T T stability implies almost sure exponential stability under a linear Γi := A P + PA+2τ1 A PA growth condition (see [42]). Then, the desired assertions follow. T −1 T − ζi PB(Im + B PB) B P<0,i=2,...,N (37) Proof of Theorem III.3: For the general flji,we λ − N −1 2 λ λ 2 where ζi =2k i (2 N σ¯ +3 N τ1 ) ik . Note that P>0 can obtain form the definition of δ(t) that dδ(t)= is the solution to GARE (11). By condition 3), we can easily get ⊗ − ⊗ [( IN −1 A) (Λ BK)]δ(t)dt + dM3 (t), where M3 (t)= from (37) that d N t ⊗ − − − l=1 i,j=1 aij 0 [Q(i) BKflji(xj (s τ2 ) xi(s − T −1 T T T T Γi < (2α ζi )PB(Im + B PB) B P, (38) τ2 ))dwlji(s)]. Note that δ(t)=[δ2 (t), ..., δN (t)] , dδi(t)=(A − λiBK)δi(t)dt + dM4,i(t),i=2,...,N, Note that for k ∈ (k, k¯), 2α − ζ < 0. Then, we get from (38) i where M (t)=(η ⊗ I )M (t). In the following, we fix that Γ < 0. Hence, by Theorem II.1, there exist C > 0 and γ> 4,i N,i n 3 i 0 2 2 −γt 1 γ certain i, and show that limt→∞ Eδi(t) > 0 for δi(0) =0 0, Eδ(t) ≤ C0 e , lim sup →∞ log δ(t)≤− , a.s., t t 2 without the stabilizability condition of (A, B).If(A, B) is which together with the definition of δ(t) implies the mean not stabilizable, then for any κ =0 and K, A − κBK is not square and almost sure consensus. Hurwitz. Let κ = λ and A := A − λ BK. Then, the desired Proof of Theorem III.2: Note that max(Re(λ(A))) < 0. i K i T assertion can be proved by the similar skills used in the proof Then, there exists P>0 such that A P + PA = −In . In fact, ∞ A T t At T of Lemma II.2. P = 0 e e dt. Letting K = kB P in (34) and applying T For the 2-D dynamics (including the second-order in- Ito’sˆ formula to V (t)=δ (t)(I − ⊗ P )δ(t) yield N 1 tegrator systems in Section III-B), we denote xi(t)= [y (t),v (t)]T ∈ R2 T T 2 i i . Then, we use another variable transfor- dV (t)=(2kδ (t)(Λ ⊗ PBB P )δ(t − τ1 ) −δ(t) )dt T mation. Let y(t)=[y1 (t), ..., yN (t)] and v(t)=[v1 (t), ..., T − ⊗ vN (t)] . Denote δg (t)=(IN JN )g(t), g = y, v, and δ(t)= + d M3 , (IN −1 P )M3 (t)+dm(t) T T T T [δy (t),δv (t)] .Letδg (t)=[δg1 (t), ..., δgN (t)] , where ≤ | | − 2 | | − 2 ( k bp 1) δ(t) dt + k bp δ(t τ1 ) dt δgi(t) ∈ R, i =1, 2, ..., N.Letδq (t)=TLδq (t) and δq (t)= T 2 2 [δq1 (t), ..., δqN (t)] , TL is defined before Proof of Theorem + σ(b, p)k δ(t − τ2 ) dt + dm(t) ≡ III.1, then it can be verified that δq1 (t) 0, q = y,v. Denote t T ⊗ λ T T T T where m(t)=2 0 δ (s)(IN −1 P )dM3 (s), bp = N δq (t)=[δq2 (t), ..., δqN (t)] and δ(t)=[δy (t), δv (t)] . T 2 N −1 T PBB P , σ(b, p)=¯σ bp BB P . Hence, for any Proof of Theorem III.4: It is sufficient to show that with- N − γt γt 2 λ N 1 2 γ>0,wehaved(e V (t)) ≤ (|k|bp − 1)e δ(t) dt out the condition N > 4β N σ , the mean square con- γt γt 2 γt ∈ R2 +e dm(t)+|k|bp e δ(t − τ1 ) dt + γe V (t)dt + σ(b, p) sensus cannot be achieved for any K =[k1 ,k2 ] .We 2 γt 2 ∈ R2 k e δ(t − τ2 ) dt. Taking expectations, we obtain first show that for any given K =[k1 ,k2 ] , the mean − square consensus implies k1 > 0,k2 Λ βIN −1 > 0. Note that t dv(t)=−k Ly(t)dt − (k L−βI )v(t)dt + dM¯ (t), where eγtEV (t) ≤ V (0) + (|k|b − 1) eγsEδ(s)2 ds 1 2 N p M¯ (t)=k N a η t σ (y (s) − y (s)) dw (s)+ 0 1 i,j=1 ij N,i 0 ji j i 1ji t t k N a η σ (v (s) − v (s))dw (s). By the def- γs 2 2 i,j=1 ij N,i 0 ji j i 2ji + |k|bp e Eδ(s − τ1 ) ds 0 initions of δy and δv and τ1 = τ2 =0, it suffices to examine the t mean square behavior of the following second-order SDE: 2 γsE − 2 ⎧ + σ(b, p)k e δ(s τ2 ) ds ⎪ 0 ⎨ dδy (t)=δv (t)dt − ¯ t ⎪ dδv (t)= k1 Λδy (t)dt + dM1 (t) (41) + γ eγsEV (s)ds. (39) ⎩ − − − ¯ 0 (k2 Λ βIN 1 )δv (t)dt + dM2 (t) t ≤ λ 2 where M¯ (t)=k N a Q(i)σ (δ (s) − δ (s)) Note that V (s) max(P ) δ(s) and for i =1, 2, 1 1 i,j=1 ij 0 ji yj yi t N t eγsEδ(s − τ )2 ds ≤ τ eγτi sup Eδ(s)2 + dw (s), M¯ (t)=k a Q(i) σ (δ (s) − δ (s)) 0 i i s∈[−τi ,0] 1ji 2 2 i,j=1 ij 0 ji vj vi ZONG et al.: STOCHASTIC CONSENTABILITY OF LINEAR SYSTEMS WITH TIME DELAYS 1071 dw (s) and Q(i)=φT (I − J )η . Note that the mean Similar to (41), we have 2ji N N N,i ⎧ square consensus is equivalent to the mean square stability ⎨⎪ dδy (t)=δv (t)dt of (41). However, the fact is that Brownian motions can dδ (t)=−k Λδ (t − τ )dt + dM¯ (t) (46) not contribute positively to the mean square stability of the ⎩⎪ v 1 y 1 1 closed-loop system, that is, the unstable system can not be mean −k2 Λδv (t − τ1 )dt + dM¯ 2 (t) square stabilized by Brownian motions (see [45]). Hence, in where M¯ (t)=k N a t Q(i)f (δ (s − τ ) − order for the mean square stability of (41), the deterministic part 1 1 i,j=1 ij 0 1ji yj 2 − ¯ N t must be stable. Hence, in order for the mean square stability of δyi(s τ2 ))dw1ji(s) and M2 (t)=k2 i,j=1 aij 0 Q(i) f (δ (s − τ ) − δ (s − τ ))dw (s).LetL = L + L ¯ 0 IN −1 2ji vj 2 vi 2 2ji 0 1 the SDE (41), the matrix L =(− − ) must be with k1 Λ βIN −1 k2 Λ Hurwitz, which implies k1 > 0 and L22 := k2 Λ − βIN −1 > 0. 0 IN −1 00 Then, based on k > 0 and L := k Λ − βI − > 0,we L0 = ,L1 = . 1 22 2 N 1 00 −k Λ −k Λ will show that the mean square consensus implies there is a 1 2 − λ 2 2 N −1 λ i such that hi (k2 ):=β k2 i + k2 σ N i < 0.Infact,we Then, we have the following transformed SDDE: can choose the energy function as follows: dδ(t)=L0 δ(t)dt + L1 δ(t − τ1 )dt + dM¯ 3 (t)+dM¯ 4 (t) T 2 V1 (t)=k1 δy (t)Λδy (t)+ δv (t) . (42) (47) N t Applying the Itoˆ formula to V1 (t),wehave where M¯ (t)=k a B (s − τ )dw (s), 3 1 i,j=1 ij 0 1ij 2 1ji T ¯ N t − − ¯ ¯ M4 (t)=k2 i,j=1 aij B2ij(s τ2 )dw2ji(s),B1ij (t)= dV1 (t)= 2δv (t)L22δv (t)dt + d M1 (t)+d M2 (t) 0 T [0,b1ij (t)] , b1ij (t)=Q(i)f1ji(δyj(t) − δyi(t))), T ¯ ¯ T − +2δv (t)d[M1 (t)+M2 (t)] (43) B2ij(t)=[0,b2ij (t)] , b2ij (t)=Q(i)f2ji(δvj(t) δvi(t)). Hence, in the following, we need to prove the mean square and where M¯ (t)=k2 N a t b (s)2 ds and M¯ (t) 1 1 i,j=1 ij 0 1ij 2 almost sure stability of SDDE (47). 2 N t 2 − = k2 aij b2ij (s) ds, b1ij (t)=Q(i)σji(δyj(t) To apply Theorem II.1, we choose i,j=1 0 δyi(t))), b2ij (t)=Q(i)σji(δvj(t) − δvi(t)). Noting that (IN − T μΛ θIN −1 JN )(IN − JN )=IN − JN , then we have Q (i)Q(i)= P = (48) T − N −1 θIN −1 IN −1 ηN,i(IN JN )ηN,i = N . By the properties of undirected graphs, we have the sum-of-squares (SOS) property ([7]): with μ, θ > 0 to be designed. In fact, we need T L 1 N − 2 2 λ δq (t) δq (t)= 2 i,j=1 aij δqj(t) δqi(t) . Then, we get θ <μ 2 to guarantee the positive definiteness of P . Let Θ1 (t)= M¯ 3 ,PM¯ 3 (t), Θ2 (t)= M¯ 4 ,PM¯ 4 (t), and N − 1 N ¯ ¯ ¯ d M¯ (t) ≥ k2 σ2 a δ (t) − δ (t)2 dt M34(t)=M3 (t)+M4 (t). By the properties of the sum-of- 1 1 N ij yj yi i,j=1 squares (SOS) property ([7]), the similar estimations as (44) and (45), and Assumption III.2, we get − 2 2 N 1 T =2k1 σ δy (t)Λδy (t)dt (44) N N 2 − 2 dΘ1 (t)=k1 aij b1ij (t τ2 ) dt and similarly, i,j=1 N − 1 T ¯ ≥ 2 2 N − 1 T d M2 (t) 2k2 σ δv (t)Λδv (t)dt. (45) ≤ 2k2 σ¯2 δ (t − τ )Λδ (t − τ )dt N 1 1 N y 2 y 2 Hence, taking expectations on the both sides of (43) and noting 2 2 N −1 T and similarly, dΘ2 (t) ≤ 2k σ¯ δ (t − τ2 )Λδv (t − τ2 )dt. that M¯ 1 (t) ≥ 0, we obtain 2 2 N v ¯ ¯ ≤ T − − t Hence, d M34,PM34 (t) δ (t τ2 )Uδ(t τ2 )dt, where T EV (t) ≥ V (0) + 2E δ (s)h(k , Λ)δ (s)ds − 1 1 v 2 v k2 σ¯2 N 1 Λ0 0 U =2 1 1 N . − 2 2 N −1 − 2 2 N 1 0 k2 σ¯2 Λ where h(k2 , Λ) := βIN −1 k2 Λ+k2 σ N Λ. Hence, if N ≥ ≥ hi(k2 ) 0 for all i =1,...,N, then h(k2 , Λ) 0, and we By Theorem II.1, the mean square and almost sure stabil- E ≥ must have lim inft→0 V1 (t) V1 (0) > 0 for δ(0) =0.This ity of (47) can be assured by S¯ := LT P + PL+(LT PL+ is in conflict with the definition of the mean square consensus. T L1 PL1 )τ1 + U<0. It is easy to see Finally, we show that hi(k2 ) < 0 for certain i and k2 im- N −1 N −1 plies λ > 4β σ2 . Otherwise, λ ≤ 4β σ2 , then we −2k1 θΛ(μ − k1 − θk2 )Λ N N N N LT P + PL = − λ 2 2 N −1 λ ≥ − − − must have hi(k2 ):=β k2 i + k2 σ N i 0 for all i = (μ k1 θk2 )Λ 2(θIN −1 k2 Λ) 2,...,N and k ∈ R, which is a contradiction. 2 T T and L PL+ L PL1 = Proof of Theorem III.5: We first transform the consensus 1 problem into the stability problem of a SDDE having the form 2 2 2 2k Λ 2k1 k2 Λ − k1 θΛ of (1). Then, we use Theorem II.1 to get the stability under the 1 . 2k k Λ2 − k θΛ(μ − k θ)Λ − k θΛ+2k2 Λ2 given conditions by choosing a appropriate matrix P>0. 1 2 1 2 2 2 1072 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 63, NO. 4, APRIL 2018
Let μ = k1 + k2 θ + k1 θτ1 . Note that Hence, integrating and taking expectations on the both sides of (50) yield 2 2 2 02k1 k2 Λ 2k Λ 0 ≤ 1 t . N −1 T 2k k Λ2 0 02k2 Λ2 E ≥ − − 2 E 1 2 2 V1 (t) V1 (0) 2k2 1 k2 σ2 δv (s)Λδv (s)ds N 0 Therefore, we have ¯ ≥ ¯ ≥ − since M1 (t) 0 and M2 (t) 0. Therefore, if 1 k2 2 N −1 σ ≤ 0,wemusthavelim inft→∞ EV1 (t) ≥ V1 (0) > 0 for s11(θ)0 2 N S¯ ≤ , δ(0) =0 . This together with the definitions of δ(t) and the 0 s (θ) 22 mean square consensus gives the necessity of the condition − 2 N −1 − 2 2 2 N −1 2 1 k2 σ2 N > 0. where s11(θ)= 2k1 θΛ+4k1 Λ τ1 +2k1 N σ¯1 Λ, s22(θ)= 2 N −1 2 2 Finally, we prove the necessity of k σ
∗ ∈ − − ¯ 2 2 N −1 − It is easy to see that θ2 <θ . That is, for any θ (θ1 ,θ2 ), where h1 (μ, θ)=μ k1 θk2 , H2 (θ)=k1 σ1 N Λ k1 θΛ ¯ ¯ − 2 2 N −1 λ − we have P>0 and S<0. Hence, from Theorem II.1 and the and H3 (θ)=θIN −1 k2 Λ+k2 σ2 N Λ. Note that k2 N definition of δ(t), the mean square and almost sure consensus 2 2 N −1 λ ¯ λ − k2 σ2 N N > 0, which is proved above. Let θ1 := k2 N follow. 2 2 N −1 λ ¯ 2 N −1 ∧ ∗ ∗ k2 σ2 N N (> 0) and θ2 := k1 σ1 N θ , where θ is de- ∗ ¯ Proof of Theorem III.6: The sufficiency follows directly fined in (49) with τ1 =0. It is easy to see that θ > θ1 . from Theorem III.5. Note that condition (20) contains three 2 N −1 λ − 2 2 N −1 If k1 σ1 N
[9] U. Munz,¬ A. Papachristodoulou, and F. Allgower,¬ “Delay robustness in [33] T. Katayama, “On the matrix Riccati equation for linear systems with non-identical multi-agent systems,” IEEE Trans. Autom. Control, vol. 57, random gain,” IEEE Trans. Autom. Control, vol. AC-21, no. 5, pp. 770Ð no. 6, pp. 1597Ð1603, Jun. 2012. 771, Oct. 1976. [10] W.Yu, G. Chen, and M. Cao, “Some necessary and sufficient conditions for [34] R. Ku and M. Athans, “Further results on the uncertainty threshold prin- second-order consensus in multi-agent dynamical systems,” Automatica, ciple,” IEEE Trans. Autom. Control, vol. AC-22, no. 5, pp. 866Ð868, vol. 46, no. 6, pp. 1089Ð1095, 2010. Oct 1977. [11] B. Zhou and Z. Lin, “Consensus of high-order multi-agent systems with [35] B. Sinopoli, L. Schenato, M. Franceschetti, K. Poolla, M. I. Jor- large input and communication delays,” Automatica, vol. 50, no. 2, dan, and S. S. Sastry, “Kalman filtering with intermittent observa- pp. 452Ð464, 2014. tions,” IEEE Trans. Autom. Control, vol. 49, no. 9, pp. 1453Ð1464, [12] Y. H. Ni and X. Li, “Consensus seeking in multi-agent systems with Sep. 2004. multiplicative measurement noises,” Syst. Control Lett., vol. 62, no. 5, [36] L. Schenato, B. Sinopoli, M. Franceschetti, K. Poolla, and S. S. Sastry, pp. 430Ð437, 2013. “Foundations of control and estimation over lossy networks,” Proc. IEEE, [13] T. Li, F. Wu, and J. F. Zhang, “Multi-agent consensus with relative-state- vol. 95, no. 1, pp. 163Ð187, Jan. 2007. dependent measurement noises,” IEEE Trans. Autom. Control, vol. 59, [37] U. Haussmann, “Optimal stationary control with state control dependent no. 9, pp. 2463Ð2468, 2014. noise,” SIAM J. Control, vol. 9, no. 2, pp. 184Ð198, 1971. [14] Z. Li and J. Chen, “Robust consensus of multi-agent systems with stochas- [38] J. L. Willems and J. C. Willems, “Feedback stabilizability for stochas- tic uncertain channels,” in Proc. 2016 Amer. Control Conf., Boston, MA, tic systems with state and control dependent noise,” Automatic, vol. 12, USA, Jul. 6Ð9, 2016, pp. 3722Ð3727. pp. 277Ð283, 1976. [15] X. Zong, T. Li, and J. F. Zhang, “Consensus control of discrete-time multi- [39] V. Dragan and A. Halanay, Stabilization of Linear Systems.NewYork, agent systems with time-delays and multiplicative measurement noises (in NY, USA: Springer, 1999. chinese),” Scientia Sinica Mathematica, vol. 46, no. 10, pp. 1617Ð1636, [40] C. Ma, T. Li, and J. Zhang, “Consensus control for leader-following multi- 2016. agent systems with measurement noises,” J. Syst. Sci. Complexity, vol. 23, [16] C. Q. Ma and J. F. Zhang, “Necessary and sufficient conditions for con- no. 1, pp. 35Ð49, 2010. sensusability of linear multi-agent systems,” IEEE Trans. Autom. Control, [41] K. Gu, “An integral inequality in the stability problem of time-delay vol. 55, no. 5, pp. 1263Ð1268, May 2010. systems,” in Proc. 39th IEEE Conf. Decision Control, 2000, pp. 2805Ð [17] S. Manfredi, “On global and local consensusability of multi-agent systems 2810. with input constraint and uncertain initial conditions,” in Proc. Amer. [42] X. Mao, Stochastic Differential Equations and Their Applications (Hor- Control Conf., 2013, pp. 6102Ð6107. wood Publishing Series in Mathematics & Applications). Chichester, [18] Y. Zhang and Y. P. Tian, “Consentability and protocol design of multi- U.K.: Horwood Publishing Limited, 1997. agent systems with stochastic switching topology,” Automatica, vol. 45, [43] X. Mao, “Almost sure exponential stability in the numerical simulation no. 5, pp. 1195Ð1201, 2009. of stochastic differential equations,” SIAM J. Numer. Anal., vol. 53, no. 1, [19] K. You and L. Xie, “Network topology and communication data rate pp. 370Ð389, 2015. for consensusability of discrete-time multi-agent systems,” IEEE Trans. [44] M. A. Rami, X. Chen, J. B. Moore, and X. Y. Zhou, “Solvability and Autom. Control, vol. 56, no. 10, pp. 2262Ð2275, Oct. 2011. asymptotic behavior of generalized Riccati equations arising in indefinite [20] G. Gu, L. Marinovici, and F. L. Lewis, “Consensusability of discrete-time stochastic LQ controls,” IEEE Trans. Autom. Control, vol. 46, no. 3, dynamic multiagent systems,” IEEE Trans. Autom. Control, vol. 57, no. 8, pp. 428Ð440, Mar. 2001. pp. 2085Ð2089, Aug. 2012. [45] X. Zong, F. Wu, G. Yin, and Z. Jin, “Almost sure and p th-moment stability [21] Y.Q. Guan and L. Wang, “Structural controllability of multi-agent systems and stabilization of regime-switching jump diffusion systems,” SIAM J. with absolute protocol under fixed and switching topologies,” Sci. China Control Optim., vol. 52, no. 4, pp. 2595Ð2622, 2014. Inform. Sci., vol. 60, no. 9, 2017, Article no. 092203. [46] X. Zong, T. Li, and J. F. Zhang, “Stochastic stabilization of linear systems [22] L. Cheng, Z. G. Hou, M. Tan, and X. Wang, “Necessary and sufficient con- under delayed and noisy feedback control,” in Proc. 36th Chin. Control ditions for consensus of double-integrator multi-agent systems with mea- Conf., Dalian, China, Jul. 26Ð28, 2017, pp. 1882Ð1887. surement noises,” IEEE Trans. Autom. Control, vol. 56, no. 8, pp. 1958Ð 1963, Aug. 2011. [23] L. Cheng, Z. G. Hou, and M. Tan, “A mean square consensus proto- col for linear multi-agent systems with communication noises and fixed topologies,” IEEE Trans. Autom. Control, vol. 59, no. 1, pp. 261Ð267, Jan. 2014. [24] M. Huang, S. Dey, G. N. Nair, and J. H. Manton, “Stochastic consensus over noisy networks with Markovian and arbitrary switches,” Automatica, vol. 46, no. 10, pp. 1571Ð1583, 2010. [25] X. Z. He, “Degenerate Lyapunov functionals of a well-known prey- predator model with discrete delays,” Proc. Roy. Soc. Edinburgh, A Math., vol. 129, no. 4, pp. 755Ð771, 1999. [26] M. A. Rami and X. Y.Zhou, “Linear matrix inequalities, Riccati equations, and indefinite stochastic linear quadratic controls,” IEEE Trans. Autom. Control, vol. 45, no. 6, pp. 1131Ð1143, Jun. 2000. [27] W. Zhang, H. Zhang, and B. S. Chen, “Generalized Lyapunov equation ap- proach to state-dependent stochastic stabilization/detectability criterion,” IEEE Trans. Autom. Control, vol. 53, no. 7, pp. 1630Ð1642, Aug. 2008. Xiaofeng Zong received the B.S. degree in [28] H. Iiduka and I. Yamada, “Computational method for solving a stochastic mathematics from Anqing Normal University, linear-quadratic control problem given an unsolvable stochastic algebraic Anhui, China, in 2009, and the Ph.D. degree Riccati equation,” SIAM J. Control Optim., vol. 50, no. 4, pp. 2173Ð2192, from the School of Mathematics and Statistics, 2012. Huazhong University of Science & Technology, [29] D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Hubei, China, in 2014. vol. 293. New York, NY, USA: Springer-Verlag, 3ed., 1999. He was a Postdoctoral Researcher in the [30] L. Huang and X. Mao, “Robust delayed-state-feedback stabilization of Academy of Mathematics and Systems Science, uncertain stochastic systems,” Automatica, vol. 45, no. 5, pp. 1332Ð1339, Chinese Academy of Sciences, Beijing, China, 2009. from July 2014 to July 2016, and a Visiting As- [31] J. Appleby, X. Mao, and M. Riedle, “Geometric Brownian motion with sistant Professor in the Department of Mathe- delay: Mean square characterisation,” Proc. Amer. Math. Soc., vol. 137, matics, Wayne State University, Detroit, MI, USA, from September 2015 no. 1, pp. 339Ð348, 2009. to October 2016. He has been with the School of Automation, China Uni- [32] X. Mao, “Robustness of exponential stability of stochastic differential versity of Geosciences, Wuhan, Hubei, China, since October 2016. His delay equations,” IEEE Trans. Autom. Control, vol. 41, no. 3, pp. 442Ð current research interests include stochastic approximations, stochastic 447, Mar. 1996. systems, delay systems, and multiagent systems. 1074 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 63, NO. 4, APRIL 2018
Tao Li (M’09–SM’14) received the B.E. degree Le Yi Wang (S’85–M’89–SM’01–F’12) received in automation from Nankai University, Tianjin, the Ph.D. degree in electrical engineering from China, in 2004, and the Ph.D. degree in sys- McGill University, Montreal, QC, Canada, in tems theory from the Academy of Mathematics 1990. and Systems Science, Chinese Academy of Sci- Since 1990, he has been with Wayne State ences, Beijing, China, in 2009. University, Detroit, MI, USA, where he is cur- Since January 2017, he has been a Professor rently a Professor with the Department of Elec- with the Department of Mathematics, East China trical and Computer Engineering. His research Normal University, Shanghai, China. Before join- interests include complexity and information, ing in ECNU, he held faculty positions in AMSS, system identification, robust control, H-infinity CAS, and Shanghai University, and visiting po- optimization, time-varying systems, adaptive sitions in Nanyang Technological University, Singapore, and Australian systems, hybrid and nonlinear systems, information processing and National University. His current research interests include stochastic sys- learning, as well as medical, automotive, communications, power sys- tems, cyberphysical multiagent systems and game theory. tems, and computer applications of control methodologies. Dr. Li received the 28th “Zhang Siying” (CCDC) Outstanding Youth Dr. Wang was a Keynote Speaker at several international conferences. Paper Award in 2016, the Best Paper Award of the 7th Asian Control He serves on the IFAC Technical Committee on Modeling, Identification, Conference with coauthors in 2009, and honorable mentioned as one and Signal Processing. He was an Associate Editor of the IEEE TRANS- of five finalists for Young Author Prize of the 17th IFAC Congress. He ACTIONS ON AUTOMATIC CONTROL and several other journals. received the 2009 Singapore Millennium Foundation Research Fellow- ship and the 2010 Australian Endeavor Research Fellowship. He was entitled Dongfang Distinguished Professor by Shanghai Municipality in 2012 and received the Outstanding Young Scholar Fund from National Natural Science Foundation of China in 2015. He now serves as an Associate Editor of International Journal of System Control and Infor- mation Processing and Journal of Systems Science and Mathematical Ji-Feng Zhang (M’92–SM’97–F’14) received Sciences. He is a member of IFAC Technical Committee on Networked the B.S. degree in mathematics from Shandong Systems and a member of Technical Committee on Control and Decision University, Jinan, China, in 1985 and the Ph.D. of Cyber-Physical Systems, Chinese Association of Automation. degree from the Institute of Systems Science (ISS), Chinese Academy of Sciences (CAS), Beijing, China, in 1991. Since 1985, he has been with the ISS, CAS, and currently the Director of ISS. He is the Editor-in-Chief of Journal of Systems Science and Mathematical Sciences and All About Sys- George Yin (S’87–M’87–SM’96–F’02) received tems and Control, the Deputy Editor-in-Chief of the B.S. degree in mathematics from the Uni- Science China Information Sciences and Systems Engineering—Theory versity of Delaware, Newark, DE, USA, in 1983, and Practice. He was an Associate Editor of several other journals, in- and the M.S. degree in electrical engineering cluding IEEE TRANSACTIONS ON AUTOMATIC CONTROL, SIAM Journal on and the Ph.D. degree in applied mathematics Control and Optimization, etc. His current research interests include from Brown University, Providence, RI, USA, in system modeling, adaptive control, stochastic systems, and multiagent 1987. systems. He joined the Department of Mathematics, Dr. Zhang is also an IFAC Fellow and an Academician of the Interna- Wayne State University in 1987, and became a tional Academy for Systems and Cybernetic Sciences. He received twice Professor in 1996 and University Distinguished the Second Prize of the State Natural Science Award of China in 2010 Professor in 2017. His research interests include and 2015, respectively, the Distinguished Young Scholar Fund from Na- stochastic processes, stochastic systems theory and applications. tional Natural Science Foundation of China in 1997, the First Prize of the Dr. Yin was the Chair of the SIAM Activity Group on Control and Young Scientist Award of CAS in 1995, the Outstanding Advisor Award Systems Theory, and served on the Board of Directors of the American of CAS in 2007, 2008, and 2009, respectively. He has served as the Automatic Control Council. He assumes the responsibility of Editor-in- Vice-Chair of the IFAC Technical Board; a member of the Board of Gov- Chief of SIAM Journal on Control and Optimization in 2018, is a Senior ernors, IEEE Control Systems Society; Convenor of Systems Science Editor of IEEE CONTROL SYSTEMS LETTERS, and is an Associate Editor Discipline, Academic Degree Committee of the State Council of China; of Applied Mathematics and Optimization and many other journals. He the Vice-President of the Systems Engineering Society of China, and was an Associate Editor of Automatica in 1995Ð2011 and IEEE TRANS- the Chinese Association of Automation. He was the General Co-Chair ACTIONS ON AUTOMATIC CONTROLin 1994Ð1998. He is a Fellow of IFAC and the IPC Chair of many control conferences. andaFellowofSIAM.