Hybrid SystemsState estimation for hybrid systems: applicationsto aircraft trackingI. Hwang, H. Balakrishnan and C. TomlinAbstract: The problem of estimating the discrete and continuous state of a stochastic linear hybridsystem, given only the continuous system output data, is studied. Well established techniques forhybrid estimation, known as the multiple model adaptive estimation algorithm, and the interactingmultiple model algorithm, are first reviewed. Conditions that must be satisfied to guarantee the con-vergence of these hybrid estimation algorithms are then presented. These conditions also provide ameans to predict , as a function of the system parameters, which transitions in a hybrid system arerelatively easy to detect. A new variant of hybrid estimation algorithms, called the residual-meaninteracting multiple model (RMIMM) algorithm, is then proposed and analysed. The performanceof RMIMM is demonstrated through multi-modal aircraft trajectory tracking examples.1 IntroductionHybrid systems are dynamical systems that combinecontinuous dynamics modelled by differential (or differ-ence) equations and discrete dynamics modelled by finiteautomata. Since hybrid systems can suitably model thecomplex behaviour of varied embedded control systems,such as robotic, transportation, and process controlsystems, there has been considerable interest in theestimation and control of hybrid systems among researchersin both academic and industrial communities[1–4]. Theobjective of hybrid estimation is to compute both thediscrete and continuous state estimates of a hybrid systemat any given time. Hybrid estimators usually consist of thecombination of a bank of continuous state estimators,each one designed for a different discrete state, or mode,and a mode-selecting algorithm. How the correct modeis selected depends on the type of output data available.The hybrid estimators analysed in this paper address aparticularly challenging problem: that of mode andcontinuous state estimation given only the observation ofthe continuous state. In this case the hybrid estimators usethe differences in statistical properties (such as mean andcovariance) of the outputs of the different continuous stateestimators to choose the correct mode.In air traffic surveillance, the accurate tracking of aircraftis important because all traffic advisories are based on thecurrent state estimates of the aircraft. In this domain, theobservations correspond to radar measurements of the pos-itions of the aircraft (observation of the continuous state).The flight mode of the aircraft, indicating, for example,constant velocity straight flight, or coord inate turn, is notobserved. Yet knowledge of the flight mode would bevaluable to a surveillance system, as it is one of the stron-gest indicators of the aircraft’s future trajectory. Forexample, we consider an aircraft tracking problem inwhich a possible flight trajectory of the aircraft isshown inFig. 1. The trajectory is composed of lines inwhich the aircraft flies straight and level at constant speedand circular arcs in which the aircraft maneuvres withdifferent yet constant yaw rates. The more accurate andthe faster the flight-mode detection, the more accurate theaircraft trajectory prediction, leading to safe and efficientair traffic control.A traditional continuous state estimator, such as aKalman filter designed using a single linear aircraft modelwith constant system parameters, process and measurementnoise, does not perform well when the aircraft changes itsmode unexpectedly. This difficulty arises because themodel on which the filter is based does not accurately rep-resent the behaviour of the aircraft over all of its flightregime. The flight mode changes in an aircraft depend onthe pilot’s input. In the aircraft tracking problems we con-sider in this paper, this input is usually unknown to the sur-veillance system. The lack of knowledge of the pilot’sactions makes the flight-mode changes of an aircraft non-deterministic because they cannot be determined a priori.We model this input as a random process. If a singlelinear continuous model is used for aircraft tracking theprocess noise covariance in the model has to be large toaccount for model inaccuracy. This large process noisecovariance leads to poor state estimates. A hybrid modelwith multiple modes that represent the flight regimes ofan aircraft could represent the dynamics of the aircraftmore accurately than one continuous model, and thuswould give more accurate state estimates. We model thedynamics of an aircraft as a discrete-time stochastic linearhybrid system whose modes correspond to the flightmodes of the aircraft. The flight-mode logic of the aircraftis represented by the discrete-state dynam ics and governedby a finite Markov chain. In this framework, aircrafttracking is a hybrid estimation problem that requires thecomputation of both the continuous state and discretemode estimates. Therefore, we consider a class of hybrid# The Institution of Engineering and Tech nology 2005IEE Proceedings online no. 20050053doi:10.1049/ip-cta:20050053Paper first received 22nd February 2005I. Hwang is with School of Aeronautics and Astronautics, Purdue University,IN 47907H. Balakrishnan and C. Tomlin are with Dept. of Aeronautics and Astronautics,Stanford University, CA 94305E-mail: [email protected] Proc.-Control Theory Appl., Vol. 153, No. 5, September 2006556systems, called discrete time stochastic linear hybridsystems which have continuous dynamics modelled bylinear difference equations and discrete state dynamicsmodelled by a finite Markov chain.Hybrid estimation algorithms have been developed fordiscrete time stochastic linear hybrid systems in which themode transitions are governed by finite-state Markovchains. The multiple model adaptive estimation (MMAE)algorithm[5] is an algorithm in which, during hypothesistesting, the residuals of the different Kalman filters foreach mode are used to form functions that reflect the likeli-hood that estimates of each of the different modes is thecorrect one. Th ese functions, called likelihood functions,serve as adaptive weights, and the state estimate is theweighted sum of the state estimates computed by individualKalman filters. In MMAE, the individual Kalman filtersmatched to the different modes run independently. In theinteracting multiple model (IMM) algorithm[6] which isa refinement of MMAE, a set of mode-matched Kalmanfilters interact with each other by using