Skip to main content
Top
Published in: Automatic Control and Computer Sciences 1/2020

01-01-2020

Robust Filtering for Discrete Systems with Unknown Inputs and Jump Parameters

Authors: K. S. Kim, V. I. Smagin

Published in: Automatic Control and Computer Sciences | Issue 1/2020

Login to get access

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

The paper deals with robust filtering algorithms for discrete systems with unknown inputs (disturbances) and Markovian jump parameter. The proposed filtering algorithm is based on the separation principle, minimization of a quadratic criterion and the use of Kalman filters with unknown input and smoothing procedures. Solving a non-stationary problem is represented solving a two-point boundary value problem in kind of difference matrix equations. In the stationary case problem is represented matrix algebraic equations. Robustness ensures the stability of the filter dynamics when errors occur in identifying the jump parameter. An example is provided to illustrate the proposed approach, which showed that the use of smoothing procedures for estimating an unknown input improves the accuracy of estimates.
Literature
1.
go back to reference Blair, W.P. and Sworder, D.D., Feedback control of a class of linear discrete systems with jump parameters and quadratic cost criteria, Int. J. Control, 1975, vol. 21, pp. 833–844.MathSciNetCrossRef Blair, W.P. and Sworder, D.D., Feedback control of a class of linear discrete systems with jump parameters and quadratic cost criteria, Int. J. Control, 1975, vol. 21, pp. 833–844.MathSciNetCrossRef
2.
go back to reference Cajueiro, D.O., Stochastic optimal control of jumping Markov parameter processes with applications to finance, Ph.D. Thesis, Instituto Tecnologico de Aeronautica-ITA, 2002. Cajueiro, D.O., Stochastic optimal control of jumping Markov parameter processes with applications to finance, Ph.D. Thesis, Instituto Tecnologico de Aeronautica-ITA, 2002.
3.
go back to reference Svensson, L.E.O. and Williams, N., Optimal monetary policy under uncertainty: A Markov jump linear-quadratic approach, Fed. Reserve St. Louis Rev., 2008, vol. 90, pp. 275–293. Svensson, L.E.O. and Williams, N., Optimal monetary policy under uncertainty: A Markov jump linear-quadratic approach, Fed. Reserve St. Louis Rev., 2008, vol. 90, pp. 275–293.
4.
go back to reference Li, L., Ugrinovskii, V.A., and Orsi, R., Decentralized robust control of uncertain Markov jump parameter systems via output feedback, Automatica, 2007, vol. 43, pp. 1932–1944.MathSciNetCrossRef Li, L., Ugrinovskii, V.A., and Orsi, R., Decentralized robust control of uncertain Markov jump parameter systems via output feedback, Automatica, 2007, vol. 43, pp. 1932–1944.MathSciNetCrossRef
5.
go back to reference Ugrinovskii, V.A. and Pota, H.R., Decentralized control of power systems via robust control of uncertain Markov jump parameter systems, Int. J. Control, 2005, vol. 78, pp. 662–677.MathSciNetCrossRef Ugrinovskii, V.A. and Pota, H.R., Decentralized control of power systems via robust control of uncertain Markov jump parameter systems, Int. J. Control, 2005, vol. 78, pp. 662–677.MathSciNetCrossRef
6.
go back to reference Gray, W.S., González, O.R., and Doğan, M., Stability analysis of digital linear flight controllers subject to electromagnetic disturbances, IEEE Aerosp. Electron. Syst. Mag., 2000, vol. 36, pp. 1204–1218.CrossRef Gray, W.S., González, O.R., and Doğan, M., Stability analysis of digital linear flight controllers subject to electromagnetic disturbances, IEEE Aerosp. Electron. Syst. Mag., 2000, vol. 36, pp. 1204–1218.CrossRef
7.
go back to reference Costa, O.L.V., Fragoso, M.D., and Todorov, M.G., Continuous-Time Markov Jump Linear Systems, Springer, 2013.CrossRef Costa, O.L.V., Fragoso, M.D., and Todorov, M.G., Continuous-Time Markov Jump Linear Systems, Springer, 2013.CrossRef
8.
go back to reference Wonham, W.M., Random differential equation in control theory, in Probabilistic Methods in Applied Mathematics, Bharucha-Reid, A.T., Ed., New York: Academic Press, 1971, pp. 131–213. Wonham, W.M., Random differential equation in control theory, in Probabilistic Methods in Applied Mathematics, Bharucha-Reid, A.T., Ed., New York: Academic Press, 1971, pp. 131–213.
9.
go back to reference Shi, P., Boukas, E.K., and Agarwal, R.K., Kalman filtering for continuous-time uncertain systems with Markovian jumping parameters, IEEE Trans. Autom. Control, 1999, vol. 44, no. 8, pp. 1592–1597.MathSciNetCrossRef Shi, P., Boukas, E.K., and Agarwal, R.K., Kalman filtering for continuous-time uncertain systems with Markovian jumping parameters, IEEE Trans. Autom. Control, 1999, vol. 44, no. 8, pp. 1592–1597.MathSciNetCrossRef
10.
go back to reference Lomakina, S.S. and Smagin, V.I., Robust filtering in continuous systems with random jump parameters, Tomsk State Univ. J., 2003, vol. 280, pp. 201–203. Lomakina, S.S. and Smagin, V.I., Robust filtering in continuous systems with random jump parameters, Tomsk State Univ. J., 2003, vol. 280, pp. 201–203.
11.
go back to reference Lomakina, S.S. and Smagin, V.I., Robust filtering for continuous systems with random jump parameters and degenerate noises in observations, Avtometriya, 2005, vol. 2, pp. 36–43. Lomakina, S.S. and Smagin, V.I., Robust filtering for continuous systems with random jump parameters and degenerate noises in observations, Avtometriya, 2005, vol. 2, pp. 36–43.
12.
go back to reference Liu, W., State estimation for discrete-time Markov jump linear systems with time-correlated measurement noise, Automatica, 2017, vol. 76, pp. 266–276.MathSciNetCrossRef Liu, W., State estimation for discrete-time Markov jump linear systems with time-correlated measurement noise, Automatica, 2017, vol. 76, pp. 266–276.MathSciNetCrossRef
13.
go back to reference Costa, E.F. and De Saporta, B., Linear minimum mean square filters for Markov jump linear systems, IEEE Trans. Autom. Control, 2017, vol. 62, no. 7, pp. 3567–3572.MathSciNetCrossRef Costa, E.F. and De Saporta, B., Linear minimum mean square filters for Markov jump linear systems, IEEE Trans. Autom. Control, 2017, vol. 62, no. 7, pp. 3567–3572.MathSciNetCrossRef
14.
go back to reference Gomes, M.J.F. and Costa, E.F., On the stability of the recursive Kalman filter with Markov jump parameters, Proceeding 2010 American Control Conference Marriott Waterfront, Baltimore, 2010, pp. 4159–4163. Gomes, M.J.F. and Costa, E.F., On the stability of the recursive Kalman filter with Markov jump parameters, Proceeding 2010 American Control Conference Marriott Waterfront, Baltimore, 2010, pp. 4159–4163.
15.
go back to reference Li, F., Shi, P. and Wu, L., Control and Filtering for Semi-Markovian Jump Systems, New York: Springer, 2016.MATH Li, F., Shi, P. and Wu, L., Control and Filtering for Semi-Markovian Jump Systems, New York: Springer, 2016.MATH
16.
go back to reference Zhao, D., Liu, Y., Liu, M., Yu, J., and Shi, Y., Network-based robust filtering for Markovian jump systems with incomplete transition probabilities, Signal Process., 2018, vol. 150, pp. 90–101.CrossRef Zhao, D., Liu, Y., Liu, M., Yu, J., and Shi, Y., Network-based robust filtering for Markovian jump systems with incomplete transition probabilities, Signal Process., 2018, vol. 150, pp. 90–101.CrossRef
17.
go back to reference Terra, M.H., Ishihara, J.Y., Jesus, G., and Cerri, J.P., Robust estimation for discrete-time Markovian jump linear systems, IEEE Trans. Autom. Control, 2013, vol. 58, no. 8, pp. 2065–2071.MathSciNetCrossRef Terra, M.H., Ishihara, J.Y., Jesus, G., and Cerri, J.P., Robust estimation for discrete-time Markovian jump linear systems, IEEE Trans. Autom. Control, 2013, vol. 58, no. 8, pp. 2065–2071.MathSciNetCrossRef
18.
go back to reference Shi, P., Boukas, E.K., and Agarwal, R.K., Robust Kalman filtering for continuous-time Markovian jump uncertain systems, Proceedings of the American Control Conference, San Diego, 1999, pp. 4413–4417. Shi, P., Boukas, E.K., and Agarwal, R.K., Robust Kalman filtering for continuous-time Markovian jump uncertain systems, Proceedings of the American Control Conference, San Diego, 1999, pp. 4413–4417.
19.
go back to reference Carvalho, L.D.P., De Oliveira, A.M., and Valle Costa, O.L.D., Robust fault detection H∞ filter for Markovian jump linear systems with partial information on the jump parameter, IFAC-PapersOnLine, 2018, vol. 51, no. 25, pp. 202–207.CrossRef Carvalho, L.D.P., De Oliveira, A.M., and Valle Costa, O.L.D., Robust fault detection H∞ filter for Markovian jump linear systems with partial information on the jump parameter, IFAC-PapersOnLine, 2018, vol. 51, no. 25, pp. 202–207.CrossRef
20.
go back to reference Janczak, D. and Grishin, Yu., State estimation of linear dynamic system with unknown input and uncertain observation using dynamic programming, Control Cybern., 2006, vol. 4, pp. 851–862.MathSciNetMATH Janczak, D. and Grishin, Yu., State estimation of linear dynamic system with unknown input and uncertain observation using dynamic programming, Control Cybern., 2006, vol. 4, pp. 851–862.MathSciNetMATH
21.
go back to reference Gillijns, S. and Moor, B., Unbiased minimum-variance input and state estimation for linear discrete-time systems, Automatica, 2007, vol. 43, pp. 111–116.MathSciNetCrossRef Gillijns, S. and Moor, B., Unbiased minimum-variance input and state estimation for linear discrete-time systems, Automatica, 2007, vol. 43, pp. 111–116.MathSciNetCrossRef
22.
go back to reference Hsien, C.S., On the optimality of two-stage Kalman filter for systems with unknown input, Asian J. Control, 2010, vol. 12, no. 4, pp. 510–523.MathSciNet Hsien, C.S., On the optimality of two-stage Kalman filter for systems with unknown input, Asian J. Control, 2010, vol. 12, no. 4, pp. 510–523.MathSciNet
23.
go back to reference Koshkin, G. and Smagin, V., Filtering and prediction for discrete systems with unknown input using nonparametric algorithms, Proceeding 10th International Conference on Digital Technologies, Zilina, 2014, pp. 120–124. Koshkin, G. and Smagin, V., Filtering and prediction for discrete systems with unknown input using nonparametric algorithms, Proceeding 10th International Conference on Digital Technologies, Zilina, 2014, pp. 120–124.
24.
go back to reference Smagin, V.I., State estimation for nonstationary discrete systems with unknown input using compensations, Russ. Phys. J., 2015, vol. 58, no. 7, pp. 1010–1017.CrossRef Smagin, V.I., State estimation for nonstationary discrete systems with unknown input using compensations, Russ. Phys. J., 2015, vol. 58, no. 7, pp. 1010–1017.CrossRef
25.
go back to reference Smagin, V.I. and Koshkin, G.M., Kalman filtering and forecasting algorithms with use of nonparametric functional estimators, Springer Proc. Math. Stat., 2016, vol. 175, pp. 75–84.MathSciNetMATH Smagin, V.I. and Koshkin, G.M., Kalman filtering and forecasting algorithms with use of nonparametric functional estimators, Springer Proc. Math. Stat., 2016, vol. 175, pp. 75–84.MathSciNetMATH
27.
go back to reference Lancaster, P. and Tismenetsky, M., The Theory of Matrices, San Diego: Academic Press, 1985, 2nd ed.MATH Lancaster, P. and Tismenetsky, M., The Theory of Matrices, San Diego: Academic Press, 1985, 2nd ed.MATH
Metadata
Title
Robust Filtering for Discrete Systems with Unknown Inputs and Jump Parameters
Authors
K. S. Kim
V. I. Smagin
Publication date
01-01-2020
Publisher
Pleiades Publishing
Published in
Automatic Control and Computer Sciences / Issue 1/2020
Print ISSN: 0146-4116
Electronic ISSN: 1558-108X
DOI
https://doi.org/10.3103/S014641162001006X

Other articles of this Issue 1/2020

Automatic Control and Computer Sciences 1/2020 Go to the issue