Estimation with lossy measurements: jump estimators for jump systems

Abstract

The problem of discrete time state estimation with lossy measurements is considered. This problem arises, for example, when measurement data is communicated over wireless channels subject to random interference. We describe the loss probabilities with Markov chains and model the joint plant / measurement loss process as a Markovian Jump Linear System. The time-varying Kalman estimator (TVKE) is known to solve a standard optimal estimation problem for Jump Linear Systems. Though the TVKE is optimal, a simpler estimator design, which we term a Jump Linear Estimator (JLE), is introduced to cope with losses. A JLE has predictor/corrector form, but at each time instant selects a corrector gain from a finite set of precalculated gains. The motivation for the JLE is twofold. First, the real-time computational cost of the JLE is less than the TVKE. Second, the JLE provides an upper bound on TVKE performance. In this paper, a special class of JLE, termed Finite Loss History Estimators (FLHE), which uses a canonical gain selection logic is considered. A notion of optimality for the FLHE is defined and an optimal synthesis method is given. In a simulation study for a double integrator system, performances are compared to both TVKE and theoretical predictions.

Stability of Kalman filtering with Markovian packet losses

“…In the more recent research on network models, the work [17] and [3] considered state estimation with lossy measurements resulting from time-varying channel conditions. In particular, the authors in [17] developed a suboptimal jump linear estimator for complexity reduction in computing the corrector gain using finite loss history where the loss process is modelled by a two state Markov chain.…”

“…In particular, the authors in [17] developed a suboptimal jump linear estimator for complexity reduction in computing the corrector gain using finite loss history where the loss process is modelled by a two state Markov chain. The work [3] introduced a more general multiple state Markov chain to model the loss and non-loss channel states, and the asymptotic mean square estimation error for suboptimal linear estimators is analyzed and optimized by a linear matrix inequality (LMI) approach.…”

Jump state estimation with multiple sensors with packet dropping and delaying channels

“…Smith and Seiler (2003); Sahebsara et al (2007); Peñarrocha et al (2012)). A constant gain approach leads to the lower storage requirement but also to the lower performance.…”

“…The jump linear estimator approach (Fletcher et al (2006)) improves the estimation with a set of precalculated gains that are used at each sampling time depending on the actual available measurements, requiring both storage and the implementation of a selection algorithm. If the set of gains is also a function of the history of measurement availabilities (called finite loss history estimator in Smith and Seiler (2003)) a better performance is achieved at the cost of more implementation complexity in the selection of the appropriate gain. An intermediate approach in terms of storage and selector complexity consists of a dependency on the actual available measurements and on the number of consecutive dropouts since last available measurement (Peñarrocha et al (2012); Peñarrocha et al (2014)).…”

Optimal State Estimation for Discrete-Time Markov Jump Systems with Missing Observations

“…Afterwards, by using linear matrix inequalities (LMIs) techniques, the 2 -∞ performance, ∞ performance, finite-time 2 -∞ performance, and finitetime ∞ performance have been well studied for solving filtering and control problems occurring in stochastic systems with uncertain elements [14][15][16][17][18][19][20][21]. In [22][23][24][25], the stability analysis of random Riccati equation arising from Kalman filtering with intermittent observations was investigated elaborately.…”