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March  2019, 9(1): 39-58. doi: 10.3934/mcrf.2019002

Decay rates for stabilization of linear continuous-time systems with random switching

1. 

Institut für Mathematik, Universität Augsburg, 86159 Augsburg, Germany

2. 

Laboratoire de Mathématiques d'Orsay, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France

* Corresponding author: Guilherme Mazanti

Received  May 2017 Revised  March 2018 Published  August 2018

Fund Project: The first author is supported by DFG grant Co 124/19-1. The second author is partially supported by the iCODE Institute, research project of the IDEX Paris-Saclay, and by the Hadamard Mathematics LabEx (LMH) through the grant number ANR-11-LABX-0056-LMH in the "Programme des Investissements d'Avenir". This work was prepared while the second author was with CMAP & Inria, team GECO, École Polytechnique, CNRS, Université Paris-Saclay, 91128 Palaiseau Cedex, France.

For a class of linear switched systems in continuous time a controllability condition implies that state feedbacks allow to achieve almost sure stabilization with arbitrary exponential decay rates. This is based on the Multiplicative Ergodic Theorem applied to an associated system in discrete time. This result is related to the stabilizability problem for linear persistently excited systems.

Citation: Fritz Colonius, Guilherme Mazanti. Decay rates for stabilization of linear continuous-time systems with random switching. Mathematical Control and Related Fields, 2019, 9 (1) : 39-58. doi: 10.3934/mcrf.2019002
References:
[1]

L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[2]

Y. Bakhtin and T. Hurth, Invariant densities for dynamical systems with random switching, Nonlinearity, 25 (2012), 2937-2952.  doi: 10.1088/0951-7715/25/10/2937.

[3]

M. BenaïmS. Le BorgneF. Malrieu and P.-A. Zitt, Qualitative properties of certain piecewise deterministic Markov processes, Ann. Inst. Henri Poincaré Probab. Stat., 51 (2015), 1040-1075.  doi: 10.1214/14-AIHP619.

[4]

P. BolzernP. Colaneri and G. De Nicolao, On almost sure stability of continuous-time Markov jump linear systems, Automatica J. IFAC, 42 (2006), 983-988.  doi: 10.1016/j.automatica.2006.02.007.

[5]

A. ChailletY. ChitourA. Loría and M. Sigalotti, Uniform stabilization for linear systems with persistency of excitation: the neutrally stable and the double integrator cases, Math. Control Signals Systems, 20 (2008), 135-156.  doi: 10.1007/s00498-008-0024-1.

[6]

D. Chatterjee and D. Liberzon, On stability of randomly switched nonlinear systems, IEEE Trans. Automat. Control, 52 (2007), 2390-2394.  doi: 10.1109/TAC.2007.904253.

[7]

D. ChengL. GuoY. Lin and Y. Wang, A note on overshoot estimation in pole placements, J. Control Theory Appl., 2 (2004), 161-164.  doi: 10.1007/s11768-004-0062-2.

[8]

D. ChengL. GuoY. Lin and Y. Wang, Erratum to: "A note on overshoot estimation in pole placements" [J. Control Theory App., 2 (2004), 161-164], J. Control Theory Appl., 3 (2005), 258.  doi: 10.1007/s11768-004-0062-2.

[9]

Y. ChitourF. Colonius and M. Sigalotti, Growth rates for persistently excited linear systems, Math. Control Signals Systems, 26 (2014), 589-616.  doi: 10.1007/s00498-014-0131-0.

[10]

Y. Chitour, G. Mazanti and M. Sigalotti, Stabilization of persistently excited linear systems, in Hybrid Systems with Constraints (eds. J. Daafouz, S. Tarbouriech and M. Sigalotti), Wiley-ISTE, London, UK, 2013, chapter 4. doi: 10.1002/9781118639856.ch4.

[11]

Y. Chitour and M. Sigalotti, On the stabilization of persistently excited linear systems, SIAM J. Control Optim., 48 (2010), 4032-4055.  doi: 10.1137/080737812.

[12]

B. Cloez and M. Hairer, Exponential ergodicity for Markov processes with random switching, Bernoulli, 21 (2015), 505-536.  doi: 10.3150/13-BEJ577.

[13]

O. L. V. Costa, M. D. Fragoso and M. G. Todorov, Continuous-Time Markov Jump Linear Systems, Probability and its Applications (New York), Springer, Heidelberg, 2013. doi: 10.1007/978-3-642-34100-7.

[14]

M. H. A. Davis, Piecewise-deterministic Markov processes: A general class of nondiffusion stochastic models, J. Roy. Statist. Soc. Ser. B, 46 (1984), 353-388. 

[15]

M. H. A. Davis, Markov Models and Optimization, vol. 49 of Monographs on Statistics and Applied Probability, Chapman & Hall, London, 1993. doi: 10.1007/978-1-4899-4483-2.

[16]

A. DiwadkarS. Dasgupta and U. Vaidya, Control of systems in Lure form over erasure channels, Internat. J. Robust Nonlinear Control, 25 (2015), 2787-2802.  doi: 10.1002/rnc.3231.

[17]

A. Diwadkar and U. Vaidya, Stabilization of linear time varying systems over uncertain channels, Internat. J. Robust Nonlinear Control, 24 (2014), 1205-1220.  doi: 10.1002/rnc.2935.

[18]

Y. Fang and K. A. Loparo, Stabilization of continuous-time jump linear systems, IEEE Trans. Automat. Control, 47 (2002), 1590-1603.  doi: 10.1109/TAC.2002.803528.

[19]

X. FengK. A. LoparoY. Ji and H. J. Chizeck, Stochastic stability properties of jump linear systems, IEEE Trans. Automat. Control, 37 (1992), 38-53.  doi: 10.1109/9.109637.

[20]

J. J. Green, Uniform Convergence to the Spectral Radius and Some Related Properties in Banach Algebras, PhD thesis, University of Sheffield, 1996.

[21]

X. GuyonS. Iovleff and J.-F. Yao, Linear diffusion with stationary switching regime, ESAIM Probab. Stat., 8 (2004), 25-35.  doi: 10.1051/ps:2003017.

[22]

M. Hairer, Ergodic Properties of Markov Processes, Lecture notes from the University of Warwick, 2006.

[23]

P. R. Halmos, Measure Theory, vol. 18 of Graduate Texts in Mathematics, Springer-Verlag, 1974. doi: 10.1007/978-1-4684-9440-2.

[24]

R. A. Horn and C. R. Johnson, Matrix Analysis, 2nd edition, Cambridge University Press, Cambridge, 2013. doi: 10.1017/cbo9781139020411.

[25]

C. LiM. Z. Q. ChenJ. Lam and X. Mao, On exponential almost sure stability of random jump systems, IEEE Trans. Automat. Control, 57 (2012), 3064-3077.  doi: 10.1109/TAC.2012.2200369.

[26]

D. Liberzon, Switching in Systems and Control, 1st edition, Birkhäuser Boston, 2003. doi: 10.1007/978-1-4612-0017-8.

[27]

H. Lin and P. J. Antsaklis, Stability and stabilizability of switched linear systems: A survey of recent results, IEEE Trans. Automat. Control, 54 (2009), 308-322.  doi: 10.1109/TAC.2008.2012009.

[28]

M. Margaliot, Stability analysis of switched systems using variational principles: an introduction, Automatica, 42 (2006), 2059-2077.  doi: 10.1016/j.automatica.2006.06.020.

[29]

S. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability, 2nd edition, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511626630.

[30]

J. R. Norris, Markov Chains, vol. 2 of Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge, 1998, Reprint of 1997 original. doi: 10.1017/cbo9780511810633.

[31]

W. Rudin, Real and Complex Analysis, 3rd edition, McGraw-Hill Book Co., New York, New York, 1987.

[32]

R. ShortenF. WirthO. MasonK. Wulff and C. King, Stability criteria for switched and hybrid systems, SIAM Rev., 49 (2007), 545-592.  doi: 10.1137/05063516X.

[33]

S. Srikant and M. R. Akella, Arbitrarily fast exponentially stabilizing controller for multi-input, persistently exciting singular control gain systems, Automatica J. IFAC, 54 (2015), 279-283.  doi: 10.1016/j.automatica.2015.02.008.

[34]

Z. Sun and S. S. Ge, Switched Linear Systems: Control and Design, Communications and Control Engineering, Springer-Verlag, London, 2005. doi: 10.1007/1-84628-131-8.

show all references

References:
[1]

L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[2]

Y. Bakhtin and T. Hurth, Invariant densities for dynamical systems with random switching, Nonlinearity, 25 (2012), 2937-2952.  doi: 10.1088/0951-7715/25/10/2937.

[3]

M. BenaïmS. Le BorgneF. Malrieu and P.-A. Zitt, Qualitative properties of certain piecewise deterministic Markov processes, Ann. Inst. Henri Poincaré Probab. Stat., 51 (2015), 1040-1075.  doi: 10.1214/14-AIHP619.

[4]

P. BolzernP. Colaneri and G. De Nicolao, On almost sure stability of continuous-time Markov jump linear systems, Automatica J. IFAC, 42 (2006), 983-988.  doi: 10.1016/j.automatica.2006.02.007.

[5]

A. ChailletY. ChitourA. Loría and M. Sigalotti, Uniform stabilization for linear systems with persistency of excitation: the neutrally stable and the double integrator cases, Math. Control Signals Systems, 20 (2008), 135-156.  doi: 10.1007/s00498-008-0024-1.

[6]

D. Chatterjee and D. Liberzon, On stability of randomly switched nonlinear systems, IEEE Trans. Automat. Control, 52 (2007), 2390-2394.  doi: 10.1109/TAC.2007.904253.

[7]

D. ChengL. GuoY. Lin and Y. Wang, A note on overshoot estimation in pole placements, J. Control Theory Appl., 2 (2004), 161-164.  doi: 10.1007/s11768-004-0062-2.

[8]

D. ChengL. GuoY. Lin and Y. Wang, Erratum to: "A note on overshoot estimation in pole placements" [J. Control Theory App., 2 (2004), 161-164], J. Control Theory Appl., 3 (2005), 258.  doi: 10.1007/s11768-004-0062-2.

[9]

Y. ChitourF. Colonius and M. Sigalotti, Growth rates for persistently excited linear systems, Math. Control Signals Systems, 26 (2014), 589-616.  doi: 10.1007/s00498-014-0131-0.

[10]

Y. Chitour, G. Mazanti and M. Sigalotti, Stabilization of persistently excited linear systems, in Hybrid Systems with Constraints (eds. J. Daafouz, S. Tarbouriech and M. Sigalotti), Wiley-ISTE, London, UK, 2013, chapter 4. doi: 10.1002/9781118639856.ch4.

[11]

Y. Chitour and M. Sigalotti, On the stabilization of persistently excited linear systems, SIAM J. Control Optim., 48 (2010), 4032-4055.  doi: 10.1137/080737812.

[12]

B. Cloez and M. Hairer, Exponential ergodicity for Markov processes with random switching, Bernoulli, 21 (2015), 505-536.  doi: 10.3150/13-BEJ577.

[13]

O. L. V. Costa, M. D. Fragoso and M. G. Todorov, Continuous-Time Markov Jump Linear Systems, Probability and its Applications (New York), Springer, Heidelberg, 2013. doi: 10.1007/978-3-642-34100-7.

[14]

M. H. A. Davis, Piecewise-deterministic Markov processes: A general class of nondiffusion stochastic models, J. Roy. Statist. Soc. Ser. B, 46 (1984), 353-388. 

[15]

M. H. A. Davis, Markov Models and Optimization, vol. 49 of Monographs on Statistics and Applied Probability, Chapman & Hall, London, 1993. doi: 10.1007/978-1-4899-4483-2.

[16]

A. DiwadkarS. Dasgupta and U. Vaidya, Control of systems in Lure form over erasure channels, Internat. J. Robust Nonlinear Control, 25 (2015), 2787-2802.  doi: 10.1002/rnc.3231.

[17]

A. Diwadkar and U. Vaidya, Stabilization of linear time varying systems over uncertain channels, Internat. J. Robust Nonlinear Control, 24 (2014), 1205-1220.  doi: 10.1002/rnc.2935.

[18]

Y. Fang and K. A. Loparo, Stabilization of continuous-time jump linear systems, IEEE Trans. Automat. Control, 47 (2002), 1590-1603.  doi: 10.1109/TAC.2002.803528.

[19]

X. FengK. A. LoparoY. Ji and H. J. Chizeck, Stochastic stability properties of jump linear systems, IEEE Trans. Automat. Control, 37 (1992), 38-53.  doi: 10.1109/9.109637.

[20]

J. J. Green, Uniform Convergence to the Spectral Radius and Some Related Properties in Banach Algebras, PhD thesis, University of Sheffield, 1996.

[21]

X. GuyonS. Iovleff and J.-F. Yao, Linear diffusion with stationary switching regime, ESAIM Probab. Stat., 8 (2004), 25-35.  doi: 10.1051/ps:2003017.

[22]

M. Hairer, Ergodic Properties of Markov Processes, Lecture notes from the University of Warwick, 2006.

[23]

P. R. Halmos, Measure Theory, vol. 18 of Graduate Texts in Mathematics, Springer-Verlag, 1974. doi: 10.1007/978-1-4684-9440-2.

[24]

R. A. Horn and C. R. Johnson, Matrix Analysis, 2nd edition, Cambridge University Press, Cambridge, 2013. doi: 10.1017/cbo9781139020411.

[25]

C. LiM. Z. Q. ChenJ. Lam and X. Mao, On exponential almost sure stability of random jump systems, IEEE Trans. Automat. Control, 57 (2012), 3064-3077.  doi: 10.1109/TAC.2012.2200369.

[26]

D. Liberzon, Switching in Systems and Control, 1st edition, Birkhäuser Boston, 2003. doi: 10.1007/978-1-4612-0017-8.

[27]

H. Lin and P. J. Antsaklis, Stability and stabilizability of switched linear systems: A survey of recent results, IEEE Trans. Automat. Control, 54 (2009), 308-322.  doi: 10.1109/TAC.2008.2012009.

[28]

M. Margaliot, Stability analysis of switched systems using variational principles: an introduction, Automatica, 42 (2006), 2059-2077.  doi: 10.1016/j.automatica.2006.06.020.

[29]

S. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability, 2nd edition, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511626630.

[30]

J. R. Norris, Markov Chains, vol. 2 of Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge, 1998, Reprint of 1997 original. doi: 10.1017/cbo9780511810633.

[31]

W. Rudin, Real and Complex Analysis, 3rd edition, McGraw-Hill Book Co., New York, New York, 1987.

[32]

R. ShortenF. WirthO. MasonK. Wulff and C. King, Stability criteria for switched and hybrid systems, SIAM Rev., 49 (2007), 545-592.  doi: 10.1137/05063516X.

[33]

S. Srikant and M. R. Akella, Arbitrarily fast exponentially stabilizing controller for multi-input, persistently exciting singular control gain systems, Automatica J. IFAC, 54 (2015), 279-283.  doi: 10.1016/j.automatica.2015.02.008.

[34]

Z. Sun and S. S. Ge, Switched Linear Systems: Control and Design, Communications and Control Engineering, Springer-Verlag, London, 2005. doi: 10.1007/1-84628-131-8.

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