Control of continuous-time Markov jump linear systems with partial information. (English) Zbl 1471.93256
Piunovskiy, Alexey (ed.) et al., Modern trends in controlled stochastic processes: theory and applications, V.III. Selected papers based on the presentations at the traditional Liverpool workshop on controlled stochastic processes, Liverpool, UK, July 2021. Cham: Springer. Emerg. Complex. Comput. 41, 87-107 (2021).
Summary: In this paper we study the \(H_2\) state-feedback control of continuous-time Markov jump linear systems considering that the mode of operation cannot be directly measured. Instead we assume that there is a detector that provides the only information concerning the main jump process, so that the jump processes are modelled by a continuous-time exponential hidden Markov model. We present a new convex design condition for calculating a state-feedback controller depending only on the detector which guarantees stability in the mean-square sense of the closed-loop system, as well as a suitable bound on its \(H_2\) norm. We present an illustrative example in the context of systems subject to faults and compare our results with the current literature.
For the entire collection see [Zbl 1470.93009].
For the entire collection see [Zbl 1470.93009].
MSC:
| 93E03 | Stochastic systems in control theory (general) |
| 93B52 | Feedback control |
| 93C05 | Linear systems in control theory |
| 93C30 | Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems) |
| 60J25 | Continuous-time Markov processes on general state spaces |
References:
| [1] | Dragan, V., Morozan, T., Stoica, A.-M.: Mathematical Methods in Robust Control of Linear Stochastic Systems (Mathematical Concepts and Methods in Science and Engineering). Springer, New York (2010) · Zbl 1183.93001 |
| [2] | Boukas, E.K.: Stochastic Switching Systems: Analysis and Design. Birkhäuser, Basel (2006) · Zbl 1151.93028 |
| [3] | Boyd, S., El Ghaoui, L., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory. SIAM Studies in Applied and Numerical Mathematics. SIAM, Philadelphia (1994) · Zbl 0816.93004 |
| [4] | Cardeliquio, C.B., Fioravanti, A.R., Gonçalves A.P.C.: \( \cal{H}_2\) and \(\cal{H}_{\infty }\) state-feedback control of continuous-time MJLS with uncertain transition rates. In: 2014 ECC, pp. 2237-2241 (2014) |
| [5] | Cheng, J., Ahn, C.K., Karimi, H.R., Cao, J., Qi, W.: An event-based asynchronous approach to Markov jump systems with hidden mode detections and missing measurements. IEEE Trans. Syst. Man Cybern. Syst. 49(9), 1749-1758 (2019) · doi:10.1109/TSMC.2018.2866906 |
| [6] | Costa, O.L.V., Fragoso, M.D., Todorov, M.G.: Continuous-Time Markov Jump Linear Systems. Springer, Berlin-Heidelberg-New York (2013) · Zbl 1277.60003 |
| [7] | Costa, O.L.V., Fragoso, M.D., Todorov, M.G.: A detector-based approach for the \(H_2\) control of Markov jump linear systems with partial information. IEEE Trans. Automat. Contr. 60(5), 1219-1234 (2015) · Zbl 1360.93761 · doi:10.1109/TAC.2014.2366253 |
| [8] | Costa, O.L.V., Tuesta, E.F.: \({H}_2\)-control and the separation principle for discrete-time Markovian jump linear systems. Math. Control Signals Syst. 16(4), 320-350 (2004) · Zbl 1066.93065 · doi:10.1007/s00498-004-0142-3 |
| [9] | Daafouz, J., Bernussou, J.: Parameter dependent Lyapunov functions for discrete time systems with time varying parametric uncertainties. Syst. Control Lett. 43(5), 355-359 (2001) · Zbl 0978.93070 · doi:10.1016/S0167-6911(01)00118-9 |
| [10] | de Oliveira, M.C., Bernussou, J., Geromel, J.C.: A new discrete-time robust stability condition. Syst. Control Lett. 37(4), 261-265 (1999) · Zbl 0948.93058 · doi:10.1016/S0167-6911(99)00035-3 |
| [11] | de Oliveira, A.M., Costa, O.L.V.: \({H}_2\)-filtering for discrete-time hidden Markov jump systems. Int. J. Control 90(3), 599-615 (2017) · Zbl 1359.93499 · doi:10.1080/00207179.2016.1186844 |
| [12] | de Oliveira, A.M., Costa, O.L.V.: Mixed \(\cal{H}_2/\cal{H}_{\infty }\) control of hidden Markov jump systems. Int. J. Robust Nonlinear Control 28(4), 1261-1280 (2018) · Zbl 1390.93722 · doi:10.1002/rnc.3952 |
| [13] | de Oliveira, A.M., Costa, O.L.V.: An iterative approach for the discrete-time dynamic control of Markov jump linear systems with partial information. Int. J. Robust Nonlinear Control 30(2), 495-511 (2020) · Zbl 1440.93242 · doi:10.1002/rnc.4771 |
| [14] | de Oliveira, A.M., Costa, O.L.V., Fragoso, M.D., Stadtmann, F.: Dynamic output feedback control for continuous-time Markov jump linear systems with hidden Markov models. Int. J. Control (2021, in print) |
| [15] | Ducard, G.J.J.: Fault-Tolerant Flight Control and Guidance Systems. Springer, London-New York (2009) · Zbl 1162.93001 |
| [16] | Dufour, F., Elliott, R.J.: Adaptive control of linear systems with Markov perturbations. IEEE Trans. Automat. Contr. 43(3), 351-372 (1998) · Zbl 0919.93085 · doi:10.1109/9.661591 |
| [17] | Elliott, R.J., Aggoun, L., Moore, J.B.: Hidden Markov Models. Springer, New York (1995) · Zbl 0819.60045 |
| [18] | Li, F., Xu, S., Zhang, B.: Resilient asynchronous \(H_{\infty }\) control for discrete-time Markov jump singularly perturbed systems based on hidden Markov model. IEEE Trans. Syst. Man Cybern. Syst. 50(8), 2860-2869 (2020) |
| [19] | Mahmoud, M., Jiang, J., Zhang, Y.: Active Fault Tolerant Control Systems - Stochastic Analysis and Synthesis. Springer, Germany (2003) · Zbl 1036.93002 |
| [20] | Mariton, M.: Detection delays, false alarm rates and the reconfiguration of control systems. Int. J. Control 49, 981-992 (1989) · Zbl 0675.93069 · doi:10.1080/00207178908559680 |
| [21] | Mariton, M.: Jump Linear Systems in Automatic Control. CRC Press, New York (1990) |
| [22] | Morais, C.F., Braga, M.F., Oliveira, R.C.L.F., Peres, P.L.D.: \(H_2\) and \(H_{\infty }\) control design for polytopic continuous-time Markov jump linear systems with uncertain transition rates. Int. J. Robust Nonlinear Control 26(3), 599-612 (2016) · Zbl 1332.93110 · doi:10.1002/rnc.3329 |
| [23] | Rodrigues, C.C.G., Todorov, M.G., Fragoso, D.M.: \({H}_\infty\) control of continuous-time Markov jump linear systems with detector-based mode information. Int. J. Control 90(10), 2178-2196 (2017) · Zbl 1380.93235 · doi:10.1080/00207179.2016.1238511 |
| [24] | Shen, Y., Wu, Z., Shi, P., Su, H., Huang, T.: Asynchronous filtering for Markov jump neural networks with quantized outputs. IEEE Trans. Syst. Man Cybern. Syst. 49(2), 433-443 (2019) · doi:10.1109/TSMC.2017.2789180 |
| [25] | Shi, P., Li, F.: A survey on Markovian jump systems: modeling and design. Int. J. Control Autom. Syst. 13(1), 1-16 (2015) · doi:10.1007/s12555-014-0576-4 |
| [26] | Srichander, R., Walker, B.K.: Stochastic stability analysis for continuous-time fault tolerant control systems. Int. J. Control 57(2), 433-452 (1993) · Zbl 0808.93067 · doi:10.1080/00207179308934397 |
| [27] | Stadtmann, F., Costa, O.L.V.: \(H_2\)-control of continuous-time hidden Markov jump linear systems. IEEE Trans. Automat. Contr. 62(8), 4031-4037 (2017) · Zbl 1373.93371 · doi:10.1109/TAC.2016.2616303 |
| [28] | Stadtmann, F., Costa, O.L.V.: Exponential hidden Markov models for \({H}_\infty\) control of jumping systems. IEEE Control Syst. Lett. 2(4), 845-850 (2018) · doi:10.1109/LCSYS.2018.2847741 |
| [29] | Todorov, M.G., Fragoso, M.D., Costa, O.L.V.: Detector-based \({H}_{\infty }\) results for discrete-time Markov jump linear systems with partial observations. Automatica 91, 159-172 (2018) · Zbl 1387.93148 · doi:10.1016/j.automatica.2018.01.034 |
| [30] | Val, J.B.R., Geromel, J.C., Gonçalves, A.P.C.: The \({H}_2\)-control for jump linear systems: cluster observations of the Markov state · Zbl 0991.93125 · doi:10.1016/S0005-1098(01)00210-2 |
| [31] | Liu, X., Ma, G., Pagilla, P.R., Ge, S.S.: Dynamic output feedback asynchronous control of networked Markovian jump systems. IEEE Trans. Syst. Man Cybern. Syst. 50(7), 2705-2715 (2020) · doi:10.1109/TSMC.2018.2827166 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
