1State Estimation in Stochastic Hybrid Systemswith Sparse ObservationsEugenio Cinquemani and Mario MicheliAbstractIn this paper we study the problem of state estimation for a class of sampled-measurement stochastichybrid systems, where the continuous state x satisfies a linear stochastic differential equation, and noisymeasurements y are taken at assigned discrete-time instants. The parameters of both the state andmeasurement equation depend on the discrete state q of a continuous-time finite Markov chain. Evenin the fault detection setting we consider – at most one transition for q is admissible – the switch mayoccur between two observations, whence it turns out that the optimal estimates cannot be expressed inparametric form and time integrations are unavoidable, so that the known estimation techniques cannotbe applied. We derive and implement an algorithm for the estimation of the states x, q and of thediscrete-state switching time that is convenient for both recursive update and the eventual numericalquadrature. Numerical simulations are illustrated.Index TermsJump Markov Linear Systems, Bayesian Estimation, Stochastic Hybrid Systems, Kalman Filtering,Fault Detection.The work of Eugenio Cinquemani was supported in part by the European Community through the RECSYS project of the VFramework Program.Eugenio Cinquemani is with the Department of Information Engineering, University of Padova, via Gradenigo 6/B, 35131Padova, Italy (e-mail: [email protected]).Mario Micheli is with the Division of Applied Mathematics, Box F, Brown University, Providence, RI 02912, USA (e-mail:[email protected]).March 28, 2006 DRAFT2I. INTRODUCTIONOver the last three decades, a substantial effort has been dedicated to the study of the so-calledJump Markov Linear Systems (JMLS). These are linear Gaussian systems switching among afinite number of linear modes indexed by a discrete state q. In general, measurements y of thecontinuous state x also depend on q. Switching follows the laws of a Markov chain independentof the initial state x(0) and of the system inputs. Typical linear system estimation problemscarry over to JMLS, such as continuous state filtering and prediction. In addition, being thetrajectory of q the outcome of an unobserved stochastic process, estimation of q from the availablemeasurements is also a concern. However, for any fixed trajectory of q, the system is linear time-variant. It follows that the statistics of x and y are mixtures of Gaussian distributions, and theiroptimal (Bayesian) estimates are found by averaging of conditional Kalman filters; this alsoleads to an optimal estimation of state q .Most of the literature deals with discrete-time JMLS, for which the complexity of the optimalBayesian estimates of x and q is exponential in time. Thus, a big effort has been devoted toderive effective suboptimal algorithms of reasonable (bounded) complexity, usually by elicitinga fixed number of “most likely” discrete state trajectories and obtaining approximate estimatesby suitable averaging. In the Generalized Pseudo Bayes approach of Ackerson and Fu [1],approximation is achieved by fitting a Gaussian distribution to the actual distribution of thestate. Tugnait [18] uses a Detection-Estimation strategy: at each step, the most probable modeq(t + 1) is detected out of the possible transitions of q(t) according to a new measurement ofthe system’s state. Blom & Bar-Shalom [2] keep track of a fixed number of trajectories q(t) by“merging” those that prove “undistinguishable”, and pruning the unlikely ones. Sequential MonteCarlo methods are considered in Doucet et al. [8], [9], among others, which explore at random thespace of all possible discrete trajectories based on a convenient generating distribution. Furthertecniques are illustrated in the works by Costa [6], [7], Chen & Liu [3], Elliott et al. [10],Germani et al. [12], and Logothethis & Krishnamurthy [16]. Less attention has been dedicatedto the continuous-time counterpart. The interested reader may consult the work by Hibey &Charalambous [13], Hu et al. [14], Miller & Runggaldier [17], and Zhang [19].March 28, 2006 DRAFT3Both the discrete-time and the continuous-time models are unsatisfactory whenever continuous-time information (estimates) about the system state needs to be drawn from measurementssampled at a rate comparable to that of the system dynamics. In the present paper we introducea model where the continuous state x evolves in continuous time according to a linear stochasticdifferential equation, whereas noisy measurements are acquired at given discrete time instants.The parameters of both the state equation and the measurement equation depend on a discretestate q which evolves in time as a continuous-time Markov chain. Since the discrete-stateswitch occurs almost surely between two successive measurements, ordinary JMLS estimatingtechniques such as those listed above cannot be applied. In fact, since the optimal state estimateswill rely on a continuous mixture of Gaussian densities the estimation problem cannot be solvedin a parametric manner and numerical integrations over time intervals are not avoidable. In thiswork we restrict our attention to a typical fault detection setting where all but one discrete stateare absorbing. We formulate a recursive estimation scheme that updates at each step a finitenumber of parameters and allows to isolate out of the recursion the integral approximation,which is performed at the very end of the computation. The use of such scheme is suitablefor the analysis of processes that are subject to sudden changes but observations are relativelysparse. In medical applications, for example, measurements such as blood samples cannot betaken too frequently, however the evolution of a disease or the effect of a therapy need to beclosely monitored.II. PROBLEM FORMULATIONLet T = {tk}k∈N0(where N0= {0, 1, 2, . . .}) be an arbitrary deterministic sequence, withtk< tk+1and tk→ ∞. Consider a finite state space Q = {0, 1, 2, . . . , N − 1} and let q denoteits generic element. Assume that we are given matrix functions: F : Q → Rn×n, G : Q → Rn×m,H : Q → Rp×n, and K : Q → Rp×r, which assign to each value q ∈ Q a four-tuple of matrices(Fq, Gq, Hq, Kq).March 28, 2006 DRAFT4Consider the following dynamical model:˙x(t) = Fq(t)x(t) + Gq(t)u(t)yk= Hq(tk)x(tk) + Kq(tk)vk, t ∈ R, tk∈ T , (1)where x : R → Rn, y : N0→ Rp, are stochastic processes. The