Abstract
Stability problem is essential and important in control theory and dynamic system analysis. In this chapter, the fundamental problem of stability for SMJSs with general TRMs is considered, in which the TRMs may be exactly known, uncertain, partially unknown and designed. The conditions guaranteeing a given SMJS stochastically admissible are expressed in terms of LMIs or LMIs with equation constraints which can be efficiently solved by using the standard numerical algorithms. Specifically, when TRM is given precisely, necessary and sufficient conditions in different forms are developed. Then, the robust stability of Markovian jump singularly perturbed systems with uncertain switchings and nonlinear perturbations for any perturbation parameter \(\varepsilon \in (0,\bar{\varepsilon }]\) is solved by an LMI approach. Moreover, instead of just containing \(\varepsilon \), a set of conditions guaranteeing the existence and uniqueness of a solution, as well as stochastic admissibility, is established by choosing an \(\varepsilon \)-dependent Lyapunov function and only depends on stability bound. It is worth mentioning that the stability results proposed in this chapter will play important roles in dealing with other problems discussed in this book.
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References
Drǎgan V, Morozan T (2000) Stability and robust stabilization to linear stochastic systems described by differential equations with Markovian jumping and multiplicative white noise. Stochast Anal Appl 20:33–92
Feng X, Loparo KA, Ji Y, Chizeck HJ (1992) Stochastic stability properties of jump linear systems. IEEE Trans Autom Control 37:38–53
Huang LR, Mao XR (2010) On almost srue stability of hybrid stochastic systems with mode-dependent interval delays. IEEE Trans Autom Control 55:1946–1952
Mao XR (1999) Stability of stochastic differential equations with Markovian switching. Stochastic Processes their Appl 79:45–67
Mao XR (2002) Exponential stability of stochastic delay interval systems with Markovian switching. IEEE Trans Autom Control 47:1604–1612
Boukas EK, Shi P, Benjelloun K (1999) On robust stabilization of uncertain linear systems with jump parameters. Int J Control 72:842–850
Mahmound MS, Shi P (2003) Robust stability, stabilization and \(H_\infty \) control of time-delay systems with Markovian jump parameters. Int J Robust Nonlinear Control 13:755–784
Xiong JL, Lam J (2006) On robust stabilization of Markovian jump systems with uncertain switching probabilities. Automatica 41:897–903
Xiong JL, Lam J (2006) Fixed-order robust \(H_\infty \) filter design for Markovian jump systems with uncertain switching probabilities. IEEE Trans Signal Process 54:1421–1430
Wang GL, Zhang QL (2013) Robust \(H_\infty \) control of Markovian jump systems with uncertain switching probabilities. Asian Journal Control 14:1407–1410
Zhang LX, Boukas EK (2009) Mode-dependent \(H_\infty \) filtering for discrete-time Markovian jump linear systems with partly unknown transition probability. Automatica 45:1462–1467
Zhang LX, Boukas EK (2009) Stability and stabilization of Markovian jump linear systems with partly unknown transition probability. Automatica 45:463–468
Zhang Y, He Y, Wu M, Zhang J (2011) Stabilization for Markovian jump systems with partial information on transition probability based on free-connection weighting matrices. Automatica 47:79–84
Zhang LX, Lam J (2010) Necessary and sufficient conditions for analysis and synthesis of Markov jump linear systems with incomplete transition descriptions. IEEE Trans Autom Control 55:1695–1701
Feng JE, Lam J, Shu Z (2010) Stabilization of Markovian systems via probability rate synthesis and output feedback. IEEE Trans Autom Control 55:773–777
Boukas EK (2008) Control of Singular Systems with Random Abrupt Changes. Springer, Berlin
Xu SY, Lam J (2006) Control and filtering of singular systems. Springer, Berlin
Xia YQ, Boukas EK, Shi P, Zhang JH (2009) Stability and stabilization of continuous-time singular hybrid systems. Automatica 45:1504–1509
Ghaoui LEI, Oustry F, AitRami M (1997) A cone complementarity linearization algorithm for static output-feedback and related problems. IEEE Trans Autom Control 42:1171–1176
Leibfritz F (2001) An LMI-based algorithm for designing suboptimal static \(H_2/H_\infty \) output feedback controllers. SIAM J Control Optim 39:1171–1735
Drǎgan V, Shi P (1999) Control of singularly perturbed systems with Markovian jump parameters: an \(H_\infty \) approach. Automatica 35:985–988
Liu HP, Boukas EK and Sun EK(2006) \(H_\infty \) stabilization of Markovian jump singularly perturbed delayed systems. In: Proceedings of 2006 American control conference, Minneapolis, Minnesota pp 14–16
Liu HP, Sun FC, Sun ZQ (2004) \(H_\infty \) control for Markovian jump linear singularly perturbed systems. IEE Proc Control Theory Appl 151:637–644
Wu LG, Ho DWC (2010) Sliding mode control of singular stochastic hybrid systems. Automatica 46:779–783
Wang GL, Zhang QL, Yang CY (2012) Dissipative control for singular Markovian jump systems with time delay. Optimal Control Appl Methods 33:415–432
Zhou L, Lu GP (2011) Robust stability of singularly perturbed descriptor systems with nonlinear perturbation. IEEE Trans Autom Control 56:858–863
Lu GP, Ho DWC (2006) Generalized quadratic stability for continuous-time singular systems with nonlinear perturbation. IEEE Trans Autom Control 51:813–823
Gao YB, Lu GP, Wang ZM (2010) Passivity analysis of uncertain singularly perturbed systems. IEEE Trans Circuits Syst ii Express Briefs 57:486–490
Shao ZH (2004) Robust stability of two-time-scale systesm with nonlinear uncertainties. IEEE Trans Automaic Control 49:258–261
Nguang SK, Assawinchaichote W, Shi P (2007) Robust \(H_\infty \) control design for fuzzy singularly perturbed systems with Markovian jumps: an LMI approach. IET Control Theory Appl 1:893–908
Wang GL (2013) Robust stabilization of singular Markovian jump systems with uncertain switching. Int J Control Autom Syst 11:188–193
Wang GL, Zhang QL, Yang CY (2014) Robust stability of singularly perturbed descriptor systems with uncertain Markovian switchings and nonlinear perturbations. Optimal Control Appl Methods 35:89–109
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Wang, G., Zhang, Q., Yan, X. (2015). Stability. In: Analysis and Design of Singular Markovian Jump Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-08723-8_2
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DOI: https://doi.org/10.1007/978-3-319-08723-8_2
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