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]–[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]–[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 1–3, 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]–[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