Equivalent description and stability analysis for discrete-time systems with uniformly distributed uncertainty. (English) Zbl 1485.93325
Summary: We are concerned with discrete-time systems with uncertainty, which is assumed to be uniformly distributed over the unit interval \([0,1]\). To describe this uncertainty, the space \(\{0, 1\}^{\mathbf{N}}\), which is composed of all the infinite 0-1 sequences and endowed with the Haar measure, actually constitutes the underlying probability space. Therefore, it is shown that for stability analysis, such an uncertain system can be transformed into a switching system with switching sequences having maximal entropy. A stability criterion then is derived from this transformation. An illustrative example is included to show the effectiveness of the theoretical results.
MSC:
| 93C55 | Discrete-time control/observation systems |
| 93E15 | Stochastic stability in control theory |
| 93C30 | Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems) |
| 93C05 | Linear systems in control theory |
Keywords:
discrete-time systems; uniformly distributed uncertainty; switching systems; Markov chain; stability analysisReferences:
| [1] | Agrachev, A. A.; Liberzon, D., Lie-algebraic stability conditions criteria for switched systems, SIAM Journal on Control and Optimization, 40, 1, 253-269 (2001) · Zbl 0995.93064 |
| [2] | Benaïm, M.; Le Borgne, S.; Malrieu, F.; Zitt, P.-A., On the stability of planar randomly switched systems, Annals of Applied Probability, 24, 1, 292-311 (2014) · Zbl 1288.93090 |
| [3] | Berger, M. A.; Wang, Y., Bounded semigroups of matrices, Linear Algebra & Its Applications, 166, 21-27 (1992) · Zbl 0818.15006 |
| [4] | Choe, G. H., Computational ergodic theory (2005), Springer: Springer Berlin, GER · Zbl 1064.37004 |
| [5] | Cong, S., A result on almost sure stability of linear continuous-time Markovian switching systems, IEEE Transactions on Automatic Control, 63, 7, 2226-2233 (2018) · Zbl 1423.93394 |
| [6] | Costa, O. L.V.; Fragoso, M. D., Stability results for discrete-time linear systems with Markovian jumping parameters, Journal of Mathematical Analysis & Applications, 179, 1, 154-178 (1993) · Zbl 0790.93108 |
| [7] | Dai, X.; Huang, Y.; Xiao, M., Almost sure stability of discrete-time switched linear systems: a topological point of view, SIAM Journal on Control and Optimization, 47, 4, 2137-2156 (2008) · Zbl 1165.93030 |
| [8] | Daubechies, I.; Lagarias, J. C., Set of matrices all infinite products of which converge, Linear Algebra & Its Applications, 161, 227-263 (1992) · Zbl 0746.15015 |
| [9] | Fang, Y.; Loparo, K. A., On relationship between the sample path and moment Lyapunov exponents for jump linear systems, IEEE Transactions on Automatic Control, 47, 9, 1556-1560 (2002) · Zbl 1364.93562 |
| [10] | Fang, Y.; Loparo, K. A.; Feng, X., Stability of discrete time jump linear systems, Journal of Mathematical Systems, Estimation, and Control, 5, 3, 275-321 (1995) · Zbl 0855.93088 |
| [11] | Gurvits, L., Stability of discrete linear inclusion, Linear Algebra & Its Applications, 231, 47-85 (1995) · Zbl 0845.68067 |
| [12] | Leizarowitz, A., Eigenvalue representation for the Lyapunov exponents of certain Markov processes, (Arnold, L.; Crauel, H.; Eckmann, J.-P., Lyapunov exponents (LNM 1486) (1991), Springer: Springer Berlin, GER), 51-63 · Zbl 0743.60059 |
| [13] | Li, Z. G.; Soh, Y. C.; Wen, C. Y., Sufficient conditions for almost sure stability of jump linear systems, IEEE Transactions on Automatic Control, 45, 7, 1325-1329 (2000) · Zbl 0988.93059 |
| [14] | Lutz, C. C.; Stilwell, D. J., Stability and disturbance attenuation for Markov jump linear systems with time-varying transition probabilities, IEEE Transactions on Automatic Control, 61, 5, 1413-1418 (2016) · Zbl 1359.93516 |
| [15] | Margaliot, M., Stability analysis of switched systems using variational principles: an introduction, Automatica, 42, 12, 2059-2077 (2006) · Zbl 1104.93018 |
| [16] | Monovich, T.; Margaliot, M., A second-order maximum principle for discrete-time bilinear control systems with applications to discrete-time linear switched systems, Automatica, 47, 7, 1489-1495 (2011) · Zbl 1220.49013 |
| [17] | Ning, Z.; Zhang, L.; Colaneri, P., Semi-Markov jump linear systems with incomplete sojourn and transition information: analysis and synthesis, IEEE Transactions on Automatic Control, 65, 1, 159-174 (2020) · Zbl 1483.93689 |
| [18] | Ogura, M.; Cetinkaya, A.; Hayakawa, T.; Preciado, V. M., State-feedback control of markov jump linear systems with hidden-markov mode observation, Automatica, 89, 65-72 (2018) · Zbl 1387.93172 |
| [19] | Pyatnitskiy, E. S.; Rapoport, L. B., Criteria of asymptotic stability of differential inclusions and periodic motions of time-varying nonlinear control systems, IEEE Transactions on Circuits & Systems I, 43, 3, 219-229 (1996) |
| [20] | Shorten, R.; Wirth, F.; Mason, O.; Wulff, K.; King, C., Stability criteria for switched and hybrid systems, SIAM Review, 49, 4, 545-592 (2007) · Zbl 1127.93005 |
| [21] | Stroock, D. W., An introduction to markov processes (GTM 230) (2005), Springer: Springer Berlin, GER · Zbl 1068.60003 |
| [22] | Taylor, S. J., Introduction to measure and integration (1973), Cambridge University Press: Cambridge University Press Cambridge, UK · Zbl 0277.28001 |
| [23] | Vlassis, N.; Jungers, R., Polytopic uncertainty for linear systems: new and old complexity results, Systems & Control Letters, 67, 9-13 (2014) · Zbl 1288.93056 |
| [24] | Walters, P., An introduction to ergodic theory (GTM 79) (1982), Springer: Springer New York, USA · Zbl 0475.28009 |
| [25] | Xu, J.; Xiao, M., A characterization of the generalized spectral radius with Kronecker powers, Automatica, 47, 7, 1530-1533 (2011) · Zbl 1227.15013 |
| [26] | Zacchia Lun, Y.; D’Innocenzo, A.; Di Benedetto, M. D., Robust stability of polytopic time-inhomogeneous Markov jump linear systems, Automatica, 105, 286-297 (2019) · Zbl 1429.93281 |
| [27] | Zhang, L.; Cai, B.; Shi, Y., Stabilization of hidden semi-Markov jump systems: emission probability approach, Automatica, 101, 87-95 (2019) · Zbl 1415.93287 |
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