Optimal H2 And H∞ Mode-independent Filters For Generalised Bernoulli Jump Systems

This paper provides the optimal solution of the filtering design problem for a special class of discrete-time Markov jump linear systems whose transition probability matrix has identical rows. In the two-mode case, this is equivalent to saying that the random variable has a Bernoulli distribution. For that class of dynamic systems we design, with the help of new necessary and sufficient linear matrix inequality conditions, and optimal mode-independent filters with the same order of the plant. As a first proposal available in the literature, for partial information characterised by cluster availability of the mode, we also show it is possible to design optimal full-order linear filters. If some plant matrices do not vary within the same cluster, we show that the optimal filter exhibits the internal model structure. We complete the results with illustrative examples. A realistic practical application considering sensors connected to a network using a communication protocol such as the Token Ring is included in order to put in evidence the usefulness of the theoretical results.463405417Blair, W.P., Sworder, D.D., Feedback control of a class of linear discrete systems with jump parameters and quadratic cost criteria (1975) International Journal of Control, 21, pp. 833-841Costa, O.L.V., Val Do, J.B.R., Full information control for discrete-Time infinite markov jump parameter systems (1996) Journal of Mathematical Analysis and Applications, 202, pp. 578-603Costa, O.L.V., Val Do, J.B.R., Geromel, J.C., A convex programming approach to H 2 Control of discrete-Time markovian linear systems (1997) International Journal of Control, 66, pp. 557-579Costa, O.L.V., Fragoso, M., Marques, R., (2005) Discrete-Time Markov Jump Linear Systems, Probability and Its Applications, , London: Springer-VerlagCosta, O.L.V., Guerra, S., Robust linear filtering for discrete-Time hybrid markov linear systems (2002) International Journal of Control, 75, pp. 712-727Costa, O.L.V., Guerra, S., Stationary filter for linear minimum mean square error estimator of discrete-Time markovian jump systems (2002) IEEE Transactions on Automatic Control, 47, pp. 1351-1356Costa, O.L.V., Tuesta, E.F., H 2-Control and the separation principle for discrete-Time markovian jump linear systems (2004) Mathematics of Control, Signals and Systems, 16, pp. 320-350Souza De, C.E., A Mode-Independent H ∞ Filter design for discrete-Time Markovian jump linear systems (2003) Proceedings of the 42nd IEEE Conference on Decision and Control, 3, pp. 2811-2816Souza De, C.E., Fragoso, M.D., H ∞ Filtering for discrete-Time linear systems with markovian jumping parameters (2003) International Journal of Robust and Nonlinear Control, 13, pp. 1299-1316Fioravanti, A.R., Gonçalves, A.P.C., Geromel, J.C., H 2 Filtering of discrete-Time markov jump linear systems through linear matrix inequalities (2008) International Journal of Control, 81, pp. 1221-1231Fioravanti, A.R., Gonçalves, A.P.C., Geromel, J.C., Filter inputs with MArkovian lossy links: Zero or hold? (2011) Proceedings of the 2011 9th International Conference on Control and Automation, pp. 656-661. , Santiago, ChileGeromel, J.C., Optimal linear filtering under parameter uncertainty (1999) IEEE Transactions on Signal Processing, 47, pp. 168-175Geromel, J.C., Bernussou, J., Garcia, G., Oliveira De, M.C., H 2 and H ∞ Robust filtering for discrete-Time linear systems (2000) SIAM Journal on Control and Optimization, 38, pp. 1353-1368Geromel, J.C., Gonçalves, A.P.C., Fioravanti, A.R., Dynamic output feedback control of discrete-Time markov jump linear systems through linear matrix inequalities (2009) SIAM Journal on Control and Optimization, 48, pp. 573-593Gonçalves, A.P.C., Fioravanti, A.R., Geromel, J.C., Filtering of discrete-Time markov jump linear systems through linear matrix inequalities (2009) IEEE Transactions on Automatic Control, 54, pp. 1347-1351Gonçalves, A.P.C., Fioravanti, A.R., Geromel, J.C., Markov jump linear systems and filtering through network transmitted measurements (2010) Signal Processing, 90, pp. 2842-2850Gonçalves, A.P.C., Fioravanti, A.R., Geromel, J.C., Filtering of discrete-Time markov jump linear systems with uncertain transition probabilities (2011) International Journal of Robust and Nonlinear Control, 21, pp. 613-624Hespanha, J., Naghshtabrizi, P., Xu, Y., A survey of recent results in networked control systems (2007) Proceedings of the IEEE, 95, pp. 138-162Ji, Y., Chizeck, H.J., Feng, X., Loparo, K.A., Stability and control of discrete-Time jump linear systems (1991) Control Theory and Advanced Technology, 7, pp. 247-270Kalman, R.E., A new approach to linear filtering and prediction problems (1960) Transactions of the ASME-Journal of Basic Engineering and Automatic Control, 82, pp. 35-45Seiler, P., Sengupta, R., A bounded real lemma for jump systems (2003) IEEE Transactions on Automatic Control, 48, pp. 1651-1654Seiler, P., Sengupta, R., An H ∞ Approach to networked control (2005) IEEE Transactions on Automatic Control, 50, pp. 356-364Sinopoli, B., Schenato, L., Franceschetti, M., Poolla, K., Jordan, M.I., Sastry, S.S., Kalman filtering with intermittent observations (2004) IEEE Transactions on Automatic Control, 49, pp. 1453-1464Smith, S.C., Seiler, P., Estimation with lossy measurements: Jump estimators for jump systems (2003) IEEE Transactions on Automatic Control, 48, pp. 2163-2171Zhang, L.X., Boukas, E.K., Mode-Dependent H ∞ Filtering for discrete-Time Markovian jump linear systems with partly unknown transition probabilities (2009) Automatica, 45, pp. 1462-146