(553b) Hybrid System Framework for State Estimation in Systems with Wireless Devices

Wireless devices are being considered as alternatives to wired devices, in new installations and expansion projects in existing industrial control installations. When compared with wired devices, wireless devices have a number of advantages, such as mobility, flexibility in installation, configuration and maintenance. However, because of interference, power and bandwidth constraints, undesirable effects such as latency and packet loss, are introduced when these devices are used (Seiler and Sengupta 2001, Sinopoli et al. 2004, Liu and Goldsmith 2004). Hence, in addition to the requirement of good network design, reliability in the presence of stochastic effects, such as random delays and missing data, is a desirable attribute for control applications in systems which use wireless devices.

State estimation is an important component in many model-based, multivariable control techniques and has a direct impact on closed-loop performance. Optimal state estimation techniques, e.g., the Kalman filter (Kalman 1960, Sorenson 1985), have been used in a number of applications in advanced process control. In its original form, the Kalman Filter was developed for state estimation from regularly sampled data. However, in many situations, for instance in plants which use wireless devices, measurements from all sensors are not available simultaneously. Hence, there is a need for developing reliable state estimation techniques which can handle missing data and random delays. In the missing data context, optimal state estimation techniques were developed using time-varying Kalman filters (Jones 1980, Shumway and Stoffer 1982, Isaksson 1993) and the statistical convergence properties of these estimators were analyzed (Sinopoli et al. 2004, Liu and Goldsmith 2004). This problem was also studied in the hybrid system framework by modeling these systems as Jump Markov Linear Systems (Costa 2002, Nilsson 1998). However, the existing approaches for analyzing these systems lack generality and make restrictive assumptions which increase the inaccuracy of the estimates.

To overcome these issues, we propose a stochastic hybrid system framework for state estimation in systems which have wireless devices. We represent the system using a hybrid model obtained by superimposing a discrete random process, which represents the network, on the plant, which is represented using a discrete-time state-space model. We assume that the presence of the network leads to delayed or missing observations. We obtain optimal state estimates using a time-varying Kalman filter whose properties depend on the network parameters. The generality of our framework allows us to analyze the stochastic network effects on the performance of the estimator.

References

Costa, O. (2002). Stationary filter for linear minimum mean square error estimator of discrete-time markovian jump systems. IEEE Trans. Auto. Cont. 48, 1351?1356.

Isaksson, A.J. (1993). Identification of ARX-models subject to missing data. IEEE Trans. Auto. Cont. 38, 813?819.

Jones, R. H. (1980). Maximum likelihood fitting of ARMA models to time series with missing observations. Technometrics 22, 389?395.

Kalman, R.E. (1960). A new approach to linear filtering and prediction problems. Transactions of the ASME, Journal of Basic Engineering, Series D 82D, 35?45. 18

Liu, X. and A.J. Goldsmith (2004). Kalman filtering with partial observation losses. In: Proceedings of the 2004 IEEE International Conference on Decision and Control. Bahamas.

Recursive estimators of signals from measurements with stochastic delays using covariance information. Applied Mathematics and Computation. Nilsson, J. (1998). Real-time control systems with delays. Ph.D. Thesis, Dept of Automatic Control, Lund Inst. Technol, Lund, Sweden.

Seiler, P. and R. Sengupta (2001). Analysis of communication losses in vehicle control problems. In: Proceedings of the 2001 American Control Conference. Virginia, USA.

Shumway, R.H. and D.S. Stoffer (1982). An approach to time series smoothing and forecasting using the EM algorithm. Journal of Time Series Analysis 3(4), 253?264.

Sinopoli, B., L. Schenato, M. Franceschetti, K. Poolla, M.I. Jordan and S.S. Sastry (2004). Kalman filtering with intermittent observations. IEEE Trans. Auto. Cont. 49(9), 1453?1464.

Sorenson, H.W., Ed. (1985). Kalman Filtering: Theory and Application. IEEE PRESS. A volume in the IEEE PRESS Selected Reprints Series.