Copyright [2009] IEEE. This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of Brunel University's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to pubs-permissions@ieee.org. By choosing to view this document, you agree to all provisions of the copyright laws protecting it.This paper is concerned with the filtering problem for a class of discrete-time stochastic nonlinear networked control systems with network-induced incomplete measurements. The incomplete measurements include both the multiple random communication delays and random packet losses, which are modeled by a unified stochastic expression in terms of a set of indicator functions that is dependent on certain stochastic variable. The nonlinear functions are assumed to satisfy the sector nonlinearities. The purpose of the addressed filtering problem is to design a linear filter such that the filtering-error dynamics is exponentially mean-square stable. By using the linear-matrix-inequality (LMI) method and delay-dependent technique, sufficient conditions are derived which are dependent on the occurrence probability of both the random communication delays and missing measurement. The filter gain is then characterized by the solution to a set of LMIs. A simulation example is exploited to demonstrate the effectiveness of the proposed design procedures

He, X

Shu, H

Wang, Z

Wei, G

IEEE Transactions on Circuits & Systems II Express Briefs

English

Brunel University Research Archive

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 56, NO. 1, JANUARY 2009 71Filtering for Networked Stochastic Time-DelaySystems With Sector NonlinearityGuoliang Wei, Zidong Wang, Xiao He, and Huisheng ShuAbstract—This paper is concerned with the filtering problemfor a class of discrete-time stochastic nonlinear networked controlsystems with network-induced incomplete measurements. Theincomplete measurements include both the multiple random com-munication delays and random packet losses, which are modeledby a unified stochastic expression in terms of a set of indicatorfunctions that is dependent on certain stochastic variable. Thenonlinear functions are assumed to satisfy the sector nonlineari-ties. The purpose of the addressed filtering problem is to design alinear filter such that the filtering-error dynamics is exponentiallymean-square stable. By using the linear-matrix-inequality (LMI)method and delay-dependent technique, sufficient conditions arederived which are dependent on the occurrence probability of boththe random communication delays and missing measurement. Thefilter gain is then characterized by the solution to a set of LMIs. Asimulation example is exploited to demonstrate the effectivenessof the proposed design procedures.Index Terms—Exponential mean-square stability, networkedcontrol system (NCS), random communication delay, randompacket loss, sector nonlinearity, stochastic disturbance.I. INTRODUCTIONW ITH THE increasing application of networks in thecomplex dynamical processes such as advanced air-craft, spacecraft, automotive, and manufacturing processes,the networked control systems (NCSs) are currently receivingconsiderable attention in the literature as illustrated by recentarticles (see, e.g., [8], [12] and references therein). Commu-nication delays (or networked-induced delays) and missingmeasurements (or packet losses) have become unavoidablethat constitute the main causes for degrading the achievableperformance of the networked systems. Therefore, in the pastdecade, the filtering and control problems with communicationdelays and/or missing measurements have been extensivelyconsidered by many researchers (see, e.g., [14], [15]).Unfortunately, in most of existing references, the communi-cation delay and the data-missing problems have been studiedManuscript received May 09, 2008; revised August 26, 2008. Current ver-sion published January 16, 2009. This work was supported in part by the En-gineering and Physical Sciences Research Council (EPSRC) of the U.K. underGrant GR/S27658/01, an International Joint Project sponsored by the Royal So-ciety of the United Kingdom, by the Shanghai Natural Science Foundation ofChina under Grant 07ZR14002, by the National Natural Science Foundation ofChina under Grant 60874113, and by the Alexander von Humboldt Foundationof Germany. This paper was recommended by Associate Editor W. X. Zheng.G. Wei and H. Shu are with the School of Information Science and Tech-nology, Donghua University, Shanghai 200051, China.Z. Wang is with the Department of Information Systems and Computing,Brunel University, Uxbridge UB8 3PH, U.K., and also with the School of Infor-mation Science and Technology, Donghua University, Shanghai 200051, China(e-mail: Zidong.Wang@brunel.ac.uk).X. He is with the Department of Automation, Tsinghua University, Beijing100084, China.Digital Object Identifier 10.1109/TCSII.2008.2008522separately by using different models, despite the fact that bothproblems are caused by the limited bandwidth of communica-tion networks and may be present simultaneously. Comparedto large amount of results for linear deterministic networkedsystems, the nonlinear Itô-type stochastic NCSs have receivedmuch less attention mainly due to the mathematical complexity[11].On the one hand, Itô-type stochastic systems (also referred toas stochastic systems with multiplicative noise) have come toplay an important role in practical applications such as chem-istry, biology, ecology, control, and information systems [7].On the other hand, the sectorlike nonlinearities usually occurin practical systems that experience harsh environments such asuncontrollable elements (e.g., variations in flow rates, temper-ature, etc.) and aggressive conditions (e.g., corrosion, erosion,fouling, etc.), which give rise to possible performance degrada-tion (see, e.g., [1], [2]). It is well known that the filtering problemis one of the fundamental problems in control and signal pro-cessing that has attracted renewing research attention followingthe celebrated Kalman filtering scheme [3]–[7], [10], [12], [14],[15]. However, to the best of the authors’ knowledge, the fil-tering problem for Itô-type stochastic networked time-delay sys-tems with simultaneous random occurrences of communicationdelays and missing measurements has not been fully investi-gated yet, which still remains as a challenging research issue.In this paper, the filtering problem is addressed for a classof discrete-time stochastic nonlinear networked systems withmultiple random communication delays and random packetlosses. The communication-delay and packet-loss problems,which are frequently encountered in communication networkswith limited digital capacity, are modeled by a stochasticmechanism that combines a certain set of indicator functionsdependent on the same stochastic variable. We first derivesufficient conditions dependent on the occurrence probabilitiesof both the random communication delay and missing mea-surement, which ensure the desired filtering performance forthe networked system. Then, we characterize the filter gainin terms of the solution to a set of linear matrix inequalities(LMIs). A simulation example is exploited to demonstrate theeffectiveness of the proposed design procedures.II. PROBLEM FORMULATIONConsider the following discrete-time stochastic nonlinearNCS with multiple delays in the state:(1)(2)1549-7747/$25.00 © 2009 IEEEAuthorized licensed use limited to: Brunel University. Downloaded on March 24, 2009 at 12:49 from IEEE Xplore.  Restrictions apply.72 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 56, NO. 1, JANUARY 2009where is the state and areknown constant time delays that are assumed to satisfyfor simplicity. is a 1-D Gaussian whitenoise sequence satisfying and .is the sector nonlinearity, and is the initial state of thesystem.The measurement with random communication delays andrandom data missing is described as(3)where is the measured output vector. Forthe indicator functions and , we as-sume that and, whereare known positive scalars and . is a mutuallyindependent stochastic variable sequence used to describe, attime , the occurred delays and data missing of the measuredoutput information. represents the sector nonlinearity. ,, , , , and are constant matrices withappropriate dimensions.Remark 1: The indicator functions in (3) dependent on astochastic variable sequence are utilized to describe the twokinds of incomplete measurements (random communicationdelays and random data missing) simultaneously. That is, if, then no data missing occurs in the system; andif , the data-missing occurrence probability is, where and denote the occur-rence probability of the data missing and the communicationdelay in the networked system, respectively. These two kindsof incomplete measurements are induced by the limited band-width of communication networks, and therefore, they shouldbe considered at the same time as done in (3). This model canbe regarded as an extension of those adopted in [14] and [15]and has been used to deal with a robust filtering problem for aclass of linear deterministic discrete-time NCSs in [3].Assumption 1: The nonlinear functions (the th el-ement of ) and (the th element of ) in sto-chastic system (1)–(3) satisfy the following sector conditionsfor , , , :(4)which are equivalent to the following matrix inequalities:(5)where ,, ,and .Remark 2: The sector nonlinearities resulting from the com-plex environments exist widely in practical systems and oftenlead to the poor performance of the controlled system. Manyresearchers have investigated the analysis and synthesis prob-lems for various systems with sectorlike nonlinearities (see [1],[2]).In this paper, we are interested in designing a filter of thefollowing structure:(6)where is the state estimate and and are filterparameters to be determined.By augmenting the state variablesand combining and , we obtain the filtering-error dy-namics as follows:(7)where andIt follows from (1), (3), (4), and (7) that(8)where(9)with and.Observe the system (7) and let denote the state tra-jectory from the initial data on .Obviously, is the trivial solution of system (7), cor-responding to the initial data .Definition 1: For the system (7) and every initial conditions, the trivial solution is said to be exponentially mean-squarestable if there exist scalars and suchthat(10)The purpose of this paper is to design a filter of the form (6)for the system (1)–(3) such that, for all admissible random com-munication time delays, random data missing, and exogenousAuthorized licensed use limited to: Brunel University. Downloaded on March 24, 2009 at 12:49 from IEEE Xplore.  Restrictions apply.IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 56, NO. 1, JANUARY 2009 73stochastic disturbances, the filtering-error system (7) is expo-nentially mean-square stable.III. MAIN RESULTSTheorem 1: Consider the filtering-error system (7) with givenfilter parameters. If there exist positive-definite matrices ,, , , and such thatthe following matrix inequalities:(11)(12)hold, where(13)then the system (7) is exponentially mean-square stable.Proof: Recalling (7), we can write(14)where(15)By substituting (14) into (8), we can obtain(16)Define the following Lyapunov functional candidate for thesystem (16):(17)where .Calculating the difference of the Lyapunov functional candi-date for the system (17) according to (7) gives(18)First, we obtain(19)Authorized licensed use limited to: Brunel University. Downloaded on March 24, 2009 at 12:49 from IEEE Xplore.  Restrictions apply.74 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 56, NO. 1, JANUARY 2009From Moon’s inequality [9], we have(20)with , ,, satisfying (12).It follows from (5) and the notation of that(21)with defined in (13).Again, we can obtain from the expression (3) that, forand.(22)In addition, from (17), it follows that(23)(24)From (15) and (19)–(24), it can be seen thatwhere(25)with(26)(27)and , , , , , , , and are definedin (13).By Schur complement, we can obtain from (11) and (12) that, and therefore, there always exists a scalar suchthat , and subsequently(28)Finally, we can confirm from [15, Lemma 1 ] that the filtering-error systems (7) is exponentially stable.The following theorem shows that the desired filter param-eters can be derived by solving several LMIs. The proof isomitted to keep the paper concise.Theorem 2: Consider the system (7). If there exist matrices, , , , , , , , andsuch that the following LMIs:(29)(30)Authorized licensed use limited to: Brunel University. Downloaded on March 24, 2009 at 12:49 from IEEE Xplore.  Restrictions apply.IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 56, NO. 1, JANUARY 2009 75hold, wherethen the system (7) is exponentially mean-square stable. Theparameters of the filter are given as follows:(31)IV. ILLUSTRATIVE EXAMPLEConsider a discrete-time NCS (1)–(3) with initial stateand other system data given as follows:Using MATLAB LMI-control Toolbox to solve (29) and (30),we can calculate the filter parameters as follows:REFERENCES[1] Y. Cao, Z. Lin, and B. M. Chen, “An output feedback   controllerdesign for linear systems subject to sensor nonlinearities,” IEEE Trans.Circuits Syst. I, Fundam. Theory Appl., vol. 50, no. 7, pp. 914–921, Jul.2003.[2] Q. L. 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Absolute stability of time-delay systems with sector-bounded nonlinearity,”

An output feedback  controller design for linear systems subject to sensor nonlinearities,”

constraints,”

Control for stability and positivity: Equivalent conditions and computation,”

E. Gershon, U. Shaked, and

filtering for networked systems with multiple state delays,”

H. Gao and C. Wang, “A

Moon, P. Park, W. H. Kwon, and

P. Liu, Y. Xia, D. Rees, and W. S. Hu, “Design and stability criteria of networked predictive

Robust   filtering for uncertain systems with multiple time-varying state delays,”

Filtering for networked stochastic time-delay systems with sector nonlinearity

Filtering for networked stochastic time-delay systems with sector nonlinearity

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