Discrete Time Systems

Discrete Time Systerdited by Mario A, Jordan and Jorge L Bustamantelished by InTecha Trine 9 51000 Rijeka CroatiaCopyright e 201All chapters are Open Ake Attribution 3

0 license, which permits to copy,work is properly cited Afteryork has been published by InTech, auto make other pereferencing or pe of thest explicitly identify the original sourceStatements and opinions expressed in the chapters are thesetributessarily those of the editors or publisher No responsibility is accepterfor theation contained in the pub lished artidesassumesury to persons or property arising outof any materials, instructions, methods or ideasthe bookmanager lvatCover Designer Martina Sirotimage Copyright Emelyano, 2010 Used under license fromFirst published March 20dditional hard copies can be obtained from orders@ intechweb orgTime Systems, Edited by Mario A Jordan and Jorge L Bus8-953-307-200-5

Part 1Discretee Filterin

Real-time recursive state estimation forNonlinear Discrete Dynamic Systems withGaussian or non-Gaussian noiseKerim Demirbd Electronics EngineeringBulgari, 06531 ankaraMany systems in the real world are more accurately described by nonlinear models Sincethe original work of Kalman(Kalman, 1960; Kalman Busy, 1961), which introduces theKalman filter for linear models, extensive research has been going on state estimationnonlinear models; but there do not yet exist any optimum estimation approaches forall nonlinear models, except for certainodels; on the other handdifferent suboptimum nonlinear estimation approaches have been proposed in the literatu(Daum, 2005) These suboptimum approaches produsome sorts ofapproximations for nonlinear models

The performances and implementation complexitiesthese suboptimum approaches surely depend upon the types of approximwhichre used for nonlinear models, Model awhich affects the performances of suboptimum estimation approaches, The performance ofmum estimation approach is better than the other estimation approaches forfic models considered, that is, the performance of a suboptimum estimation approach ise nonlinear estimation approacheseries expansion(Sage &e Melsa, 1971)and the unscented Kalman filter (UKF)approximateted and deterministically chosen points (Particle filters approximates a posterior densities by a large set of weighteselectedts(called particles) in the state space(Arulampalam et al, 2002; Doucet et al2001; Ristic et al, 2004), In the nonlinear estimation982: 1984; Demirbas Leondes, 1985; 1986; Demirbas, 1988;990;2007;2010):thedisturbance noise and initial state are first approximated by ainitial state whose distribution functions the best approximate the distribution functions of thedisturbance noise and initial state, states are quantized, and then multiple hypothesis testingused for state estimation: whereas Grid-based atspace; if the state space is not finite in extent, then the state space necessitates some truncationof the state space; and grid-based estimation approachesle the availability of the state

Discrete Time Systemsdisturbane(k)lx(k-1)), which may notbe calculated for state modgeneral than grid-based approaches since 1)the state spaceneed not to be truncated, 2 )the state transition density is not needed, 3)state models can beany nonlinear functions of the disturbance noiseThis chapter presents an online recursive nonlinear state filtering and prediction scheme fortems This scheme is recently proposed in(Demirbas, 2010)andreferred to as the df throughout this chapter, The DF is very suitable for state estimation ofer either missing observations or constraints imposed on stateestimates There exist many nonlinear dynamic systems for which the DF outperforms thetended Kalman filter(EKF), saimportance resampling(SIR) particle filter(which issometimes called the bootstrap filter), and auxiliaryling (asIr)hich applte the disturbance noise and initial state, and then presents approximection 6 vields simulation results of twoexamples for which the DF outperfthe EKF, SIR, and ASIR particle filters

Section 7concludes the chat2 Problem statementThis section defines state estimation problem for nonlinear discrete dynamic systems Thedynamic systems are described bModk+1)=f(k,x(k),c(k)Observation Modelwhere k stands for the discrete time index; f: Rxrxr'-r is the state transition function;state vector at time k:8: RxR xRv(k)eR/ is the observatiector at time k(k)∈Rked to be independent with known distribution fuMoreover it is assumed that there existrecursively yields a predicted value e(k k-1)of the state x(k) given the observation seAi(kk)of the state x(k)given the observation sequence from time one to time k, that is, Zlished by first approximating the disdiscrete randoquantizing the state, that is, representing the state modevarying state machine, and an online suboptimum implementation of multiple hypo3 Approxrst discusses an approximate discrete random vector which approximates arandom vector andresents approximate models of nonlinear dynamic systems

Real-time Recursive State Estimation for nonlineaDiscreteamic Systems with Gaus- Gaussian noise31Adiscrete random vectorm vectorsthe disturbance noise and initial state throuthe chapter; moreover, a set of equawhich must be satisfied by an approximate discrete random variabLapp

roximate discrete random variables of a Gautabulatedbe an m-dimensional random vector An approximate discrete random vector with nble values of w, denoted by wd, is defined as an m-dimensional discrete random vectorvith n possible values whose distribution function the best approximates the distributionfunction of to over the distribution functions of all mmf-dimensional discrete random vectoth n possible values, that ishere d is the set of all m-dimensional discrete random vectors with n possible values, Fy(distribution fudom vector y, Fwo(a) is the distribution functionr is the m-dimensional Euclidean space

Andiscrete random vecneral, numerically, offline-calculated, stored and then usedfor estimation The possible values of wd are denoted by wdl, wand win and theobed=wdi lus random variable Then, wa isroximate discrete random variable with n possible values whose distribution functionximates the distribution function Fn(a) of w over the distribution functions ofdiscrete random variables with n possible(Fa)}which the distribution error function(the objective function)/(Fv(a)) is defined bywhere d is theall discrete randombles wissible values, Fv(a) is thedistribution function of the discrete randomable y, Fw(a) iscontinuous random variable zedR is the real line Let the distribution functioFy(a) of a discrete random variable y be given by

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ontentsPart 1 Discrete- Time Filterinwithsian or non- Gaussian noiAli Zemouche and Mohamed Boutayeb ay 19Discrete-Time Systems wapter 3 Distributed Fusion Prediction for MixedContinuous-Discrete Linear Systems 39Chapter 4 New Smoothers for Discrete-timeapter 5 on the Error Covariance Distributioor Kalman Filters with Packet DropoutsChapter 6Filtering for DiscretePart 2 Discrete-Titochastic Optimal tracking with Previewof a Discrete tiodel folm for Nonlinear Descriptor

pter 9 Output Feedback Control of DiscretetiSystems: Scaling LMI AppChapter 10 Discrete Time Mixed LQR/H, Control Problems 159-Time Systems with Delays 179Chapter 12 Quadratic D Stabilizable Satisfactory Fault-tolerantontrol with Constraints of consistene Attitude Control Systems 195d Zhang denguePart 3 Discrete-Time Adaptive Control 205Discrete-Time Adaptive Predictive Controlof Discrete-Time Multi-Agent Systems 229Chapter 15 A General Approach to Discrete-TimeAdaptive Control Systems with Perturbednned underwater vehicles28hapter 16 Stability Criterion and Stabilizationear Discrete-time SystemTime varyinhapter 17 Uncertain Discrete-Time Systems with Delayedation with perfoecification viCastro, Andre f caldeihapter 18 Stability Analysis of Grey Discrete TimeTime-Delay Systems: A Sufficient Condition 327

Chaptertability and ci Gain Analysis of Switched Lineae Descriptor Systems 33Chapter 20 Robust Stabilization for a clasiscrete-time Switched Linear Systems 355 Miscel3Its Implementation 363Chapter 22 AdaStep-size Order Statistic LMS-basedMultitone Systems 383anonChapter 23 Discrete-Time DynaSegmentation System 405teractive multiple mault Diagnosis for PEM Fuel Cell Systems 425Chapter 25 Discrete Time Systems with Event-Based DynamicsRecent Developments in Analysnd Synthesis Methods 447dgar Delgado-Eckert, Johann Reger and Klaus Schmidt

PrefaceDiscrete-Time systeresendmportant andd research field

Thedigital-based computational means in thehe field with a tremendous impact in areas like Control, Signal Processingcations, System Modelling and related AThis fact has enabcontributions and developments which are either genuinely original asme systems or are mirrors from their counterparts of previously existingThis book attempts to give a scope of the present state-of-the-art in the area of Discrete-Time Systems from selected international research groups which were specialin the fieldm framework and with a formal mathcontextIn order to facilitate the scope and global comprehension of the book, the chapters weregrouped conveniently in sections according to their affinity in 5 significant areaThe first groupFiltering that encloses above all designs of Stateators, PreThe secondis dedicated to the design of Fixed Control Systems( Chapters 7 to2,Happears designs for Tracking Control, Fault-Tolerant Control, Robust Cortrol, and designs using LMI-and mixed LQR/Hoo techniquesThe third group includes Adaptive Control Systems( Chapter 13 to 15)oriented to thespecialities of Predictive, Decentralized and Perturbed Control SystemThe fourth group collects works that address Stability Problems( Chapter 16 to 20nstance Uncertain Systems with Multiple and Time-Varying Delaysnd Switched Linear SystemsFinally, the fifth group concerns miscellaneous applications(Chapter 21 toTitone Modulation and Equalisation, Image Processing, Fatnosis, Event-Based Dynamics and Analysis of Deterministic/Stochastic and

We think that the contribution in the book whot have the intention to beall-embracing, enlargesf the Discrete-with signification in thene field we think also thatthe topics described herealso to lookme main tendencies in thenext years in the research areaMario A

Jordan and jorge L bDep, of Electrical Eng and ComputersNational University of the SouthArgentina