# American Institute of Mathematical Sciences

## Minimax joint spectral radius and stabilizability of discrete-time linear switching control systems

 1 Kharkevich Institute for Information Transmission Problems, Russian Academy of Sciences, Bolshoj Karetny lane 19, Moscow 127051, Russia 2 Kotel'nikov Institute of Radio-engineering and Electronics, Russian Academy of Sciences, Mokhovaya 11-7, Moscow 125009, Russia

Dedicated to Peter Kloeden on the occasion of his 70th birthday, friendship with whom refutes the thesis that "East is East, and West is West, and never the twain shall meet"

Received  January 2018 Revised  April 2018 Published  October 2018

Fund Project: The author is supported by the Russian Science Foundation, Project number 16-11-00063

To estimate the growth rate of matrix products $A_{n}··· A_{1}$ with factors from some set of matrices $\mathscr{A}$, such numeric quantities as the joint spectral radius $ρ(\mathscr{A})$ and the lower spectral radius $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over \rho } (\mathscr{A})$ are traditionally used. The first of these quantities characterizes the maximum growth rate of the norms of the corresponding products, while the second one characterizes the minimal growth rate. In the theory of discrete-time linear switching systems, the inequality $ρ(\mathscr{A})<1$ serves as a criterion for the stability of a system, and the inequality $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over \rho } (\mathscr{A})<1$ as a criterion for stabilizability.

Given a set $\mathscr{A}$ of $N×M$ matrices and a set $\mathscr{B}$ of $M×N$ matrices. Then, for matrix products $A_{n}B_{n}··· A_{1}B_{1}$ with factors $A_{i}∈\mathscr{A}$ and $B_{i}∈\mathscr{B}$, we introduce the quantities $μ(\mathscr{A},\mathscr{B})$ and $η(\mathscr{A},\mathscr{B})$, called the lower and upper minimax joint spectral radius of the pair $\{\mathscr{A},\mathscr{B}\}$, respectively, which characterize the maximum growth rate of the matrix products $A_{n}B_{n}··· A_{1}B_{1}$ over all sets of matrices $A_{i}∈\mathscr{A}$ and the minimal growth rate over all sets of matrices $B_{i}∈\mathscr{B}$. In this sense, the minimax joint spectral radii can be considered as generalizations of both the joint and lower spectral radii. As an application of the minimax joint spectral radii, it is shown how these quantities can be used to analyze the stabilizability of discrete-time linear switching control systems in the presence of uncontrolled external disturbances of the plant.

Citation: Victor Kozyakin. Minimax joint spectral radius and stabilizability of discrete-time linear switching control systems. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2018277
##### References:
 [1] E. Asarin, J. Cervelle, A. Degorre, C. Dima, F. Horn and V. Kozyakin, Entropy games and matrix multiplication games, in 33rd Symposium on Theoretical Aspects of Computer Science, (STACS 2016) (eds. N. Ollinger and H. Vollmer), vol. 47 of LIPIcs. Leibniz Int. Proc. Inform., Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, 2016, 14pp. doi: 10.4230/LIPIcs.STACS.2016.11. [2] M. A. Berger and Y. Wang, Bounded semigroups of matrices, Linear Algebra Appl., 166 (1992), 21-27. doi: 10.1016/0024-3795(92)90267-E. [3] V. D. Blondel and Y. Nesterov, Polynomial-time computation of the joint spectral radius for some sets of nonnegative matrices, SIAM J. Matrix Anal. Appl., 31 (2009), 865-876. doi: 10.1137/080723764. [4] J. Bochi and I. D. Morris, Continuity properties of the lower spectral radius, Proc. Lond. Math. Soc. (3), 110 (2015), 477-509. doi: 10.1112/plms/pdu058. [5] T. Bousch and J. Mairesse, Asymptotic height optimization for topical IFS, Tetris heaps, and the finiteness conjecture, J. Amer. Math. Soc., 15 (2002), 77-111. doi: 10.1090/S0894-0347-01-00378-2. [6] P. Bouyer, N. Markey, M. Randour, K. G. Larsen and S. Laursen, Average-energy games, Acta Informatica, 55 (2018), 91-127. doi: 10.1007/s00236-016-0274-1. [7] A. Czornik, On the generalized spectral subradius, Linear Algebra Appl., 407 (2005), 242-248. doi: 10.1016/j.laa.2005.05.006. [8] X. Dai, Y. Huang, J. Liu and M. Xiao, The finite-step realizability of the joint spectral radius of a pair of d×d matrices one of which being rank-one, Linear Algebra Appl., 437 (2012), 1548-1561. doi: 10.1016/j.laa.2012.04.053. [9] X. Dai, A Gel'fand-type spectral-radius formula and stability of linear constrained switching systems, Linear Algebra Appl., 436 (2012), 1099-1113. doi: 10.1016/j.laa.2011.07.029. [10] X. Dai, Some criteria for spectral finiteness of a finite subset of the real matrix space $\mathbb{R}^{d× d}$, Linear Algebra Appl., 438 (2013), 2717-2727. doi: 10.1016/j.laa.2012.09.026. [11] X. Dai, Robust periodic stability implies uniform exponential stability of Markovian jump linear systems and random linear ordinary differential equations, J. Franklin Inst., 351 (2014), 2910-2937. doi: 10.1016/j.jfranklin.2014.01.010. [12] X. Dai, Y. Huang and M. Xiao, Periodically switched stability induces exponential stability of discrete-time linear switched systems in the sense of Markovian probabilities, Automatica J. IFAC, 47 (2011), 1512-1519. doi: 10.1016/j.automatica.2011.02.034. [13] X. Dai, Y. Huang and M. Xiao, Pointwise stability of descrete-time stationary matrix-valued Markovian processes, IEEE Trans. Automat. Control, 60 (2015), 1898-1903. doi: 10.1109/TAC.2014.2361594. [14] I. Daubechies and J. C. Lagarias, Sets of matrices all infinite products of which converge, Linear Algebra Appl., 161 (1992), 227-263. doi: 10.1016/0024-3795(92)90012-Y. [15] M. Fekete, Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten, Math. Z., 17 (1923), 228-249. doi: 10.1007/BF01504345. [16] L. Gurvits, Stability of discrete linear inclusion, Linear Algebra Appl., 231 (1995), 47-85. doi: 10.1016/0024-3795(95)90006-3. [17] C. Heil and G. Strang, Continuity of the joint spectral radius: application to wavelets, in Linear algebra for signal processing (Minneapolis, MN, 1992), vol. 69 of IMA Vol. Math. Appl., Springer, New York, 1995, 51-61. doi: 10.1007/978-1-4612-4228-4_4. [18] R. Jungers, The Joint Spectral Radius, vol. 385 of Lecture Notes in Control and Information Sciences, Springer-Verlag, Berlin, 2009, Theory and applications. doi: 10.1007/978-3-540-95980-9. [19] R. M. Jungers, On asymptotic properties of matrix semigroups with an invariant cone, Linear Algebra Appl., 437 (2012), 1205-1214. doi: 10.1016/j.laa.2012.04.006. [20] R. M. Jungers and P. Mason, On feedback stabilization of linear switched systems via switching signal control, SIAM J. Control Optim., 55 (2017), 1179-1198. doi: 10.1137/15M1027802. [21] V. S. Kozyakin, On the absolute stability of systems with asynchronously operating pulse elements, Avtomat. i Telemekh., 1990, 56-63, In Russian, translation in Automat. Remote Control, 51 (1990), 1349-1355 (1991). [22] V. S. Kozyakin, Constructive stability and stabilizability of positive linear discrete-time switching systems, Journal of Communications Technology and Electronics, 62 (2017), 686-693. doi: 10.1134/S1064226917060110. [23] V. Kozyakin, An explicit Lipschitz constant for the joint spectral radius, Linear Algebra Appl., 433 (2010), 12-18. doi: 10.1016/j.laa.2010.01.028. [24] V. Kozyakin, An Annotated Bibliography on Convergence of Matrix Products and the Theory of Joint/Generalized Spectral Radius, Preprint, Institute for Information Transmission Problems, Moscow, 2013. doi: 10.13140/2.1.4257.5040. [25] V. Kozyakin, Hourglass alternative and the finiteness conjecture for the spectral characteristics of sets of non-negative matrices, Linear Algebra Appl., 489 (2016), 167-185. doi: 10.1016/j.laa.2015.10.017. [26] V. Kozyakin, Minimax theorem for the spectral radius of the product of non-negative matrices, Linear and Multilinear Algebra, 65 (2017), 2356-2365. doi: 10.1080/03081087.2016.1273877. [27] V. Kozyakin, On convergence of infinite matrix products with alternating factors from two sets of matrices, Discrete Dyn. Nat. Soc., 2018 (2018), Art. ID 9216760, 5 pp. doi: 10.1155/2018/9216760. [28] 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. [29] G.-C. Rota and G. Strang, A note on the joint spectral radius, Nederl. Akad. Wetensch. Proc. Ser. A 63 = Indag. Math., 22 (1960), 379-381. doi: 10.1016/S1385-7258(60)50046-1. [30] M.-H. Shih, J.-W. Wu and C.-T. Pang, Asymptotic stability and generalized Gelfand spectral radius formula, Linear Algebra Appl., 252 (1997), 61-70. doi: 10.1016/0024-3795(95)00592-7. [31] D. P. Stanford, Stability for a multi-rate sampled-data system, SIAM J. Control Optim., 17 (1979), 390-399. doi: 10.1137/0317029. [32] D. P. Stanford and J. M. Urbano, Some convergence properties of matrix sets, SIAM J. Matrix Anal. Appl., 15 (1994), 1132-1140. doi: 10.1137/S0895479892228213. [33] Z. Sun and S. S. Ge, Switched Linear Systems: Control and Design, Communications and Control Engineering, Springer, London, 2005. doi: 10.1007/1-84628-131-8. [34] J. Theys, Joint Spectral Radius: Theory and Approximations, PhD thesis, Faculté des sciences appliquées, Département d'ingénierie mathématique, Center for Systems Engineering and Applied Mechanics, Université Catholique de Louvain, 2005. [35] J. von Neumann and O. Morgenstern, Theory of Games and Economic Behavior, 2nd edition, Princeton University Press, Princeton, N. J., 1947. [36] F. Wirth, The generalized spectral radius and extremal norms, Linear Algebra Appl., 342 (2002), 17-40. doi: 10.1016/S0024-3795(01)00446-3.

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##### References:
 [1] E. Asarin, J. Cervelle, A. Degorre, C. Dima, F. Horn and V. Kozyakin, Entropy games and matrix multiplication games, in 33rd Symposium on Theoretical Aspects of Computer Science, (STACS 2016) (eds. N. Ollinger and H. Vollmer), vol. 47 of LIPIcs. Leibniz Int. Proc. Inform., Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, 2016, 14pp. doi: 10.4230/LIPIcs.STACS.2016.11. [2] M. A. Berger and Y. Wang, Bounded semigroups of matrices, Linear Algebra Appl., 166 (1992), 21-27. doi: 10.1016/0024-3795(92)90267-E. [3] V. D. Blondel and Y. Nesterov, Polynomial-time computation of the joint spectral radius for some sets of nonnegative matrices, SIAM J. Matrix Anal. Appl., 31 (2009), 865-876. doi: 10.1137/080723764. [4] J. Bochi and I. D. Morris, Continuity properties of the lower spectral radius, Proc. Lond. Math. Soc. (3), 110 (2015), 477-509. doi: 10.1112/plms/pdu058. [5] T. Bousch and J. Mairesse, Asymptotic height optimization for topical IFS, Tetris heaps, and the finiteness conjecture, J. Amer. Math. Soc., 15 (2002), 77-111. doi: 10.1090/S0894-0347-01-00378-2. [6] P. Bouyer, N. Markey, M. Randour, K. G. Larsen and S. Laursen, Average-energy games, Acta Informatica, 55 (2018), 91-127. doi: 10.1007/s00236-016-0274-1. [7] A. Czornik, On the generalized spectral subradius, Linear Algebra Appl., 407 (2005), 242-248. doi: 10.1016/j.laa.2005.05.006. [8] X. Dai, Y. Huang, J. Liu and M. Xiao, The finite-step realizability of the joint spectral radius of a pair of d×d matrices one of which being rank-one, Linear Algebra Appl., 437 (2012), 1548-1561. doi: 10.1016/j.laa.2012.04.053. [9] X. Dai, A Gel'fand-type spectral-radius formula and stability of linear constrained switching systems, Linear Algebra Appl., 436 (2012), 1099-1113. doi: 10.1016/j.laa.2011.07.029. [10] X. Dai, Some criteria for spectral finiteness of a finite subset of the real matrix space $\mathbb{R}^{d× d}$, Linear Algebra Appl., 438 (2013), 2717-2727. doi: 10.1016/j.laa.2012.09.026. [11] X. Dai, Robust periodic stability implies uniform exponential stability of Markovian jump linear systems and random linear ordinary differential equations, J. Franklin Inst., 351 (2014), 2910-2937. doi: 10.1016/j.jfranklin.2014.01.010. [12] X. Dai, Y. Huang and M. Xiao, Periodically switched stability induces exponential stability of discrete-time linear switched systems in the sense of Markovian probabilities, Automatica J. IFAC, 47 (2011), 1512-1519. doi: 10.1016/j.automatica.2011.02.034. [13] X. Dai, Y. Huang and M. Xiao, Pointwise stability of descrete-time stationary matrix-valued Markovian processes, IEEE Trans. Automat. Control, 60 (2015), 1898-1903. doi: 10.1109/TAC.2014.2361594. [14] I. Daubechies and J. C. Lagarias, Sets of matrices all infinite products of which converge, Linear Algebra Appl., 161 (1992), 227-263. doi: 10.1016/0024-3795(92)90012-Y. [15] M. Fekete, Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten, Math. Z., 17 (1923), 228-249. doi: 10.1007/BF01504345. [16] L. Gurvits, Stability of discrete linear inclusion, Linear Algebra Appl., 231 (1995), 47-85. doi: 10.1016/0024-3795(95)90006-3. [17] C. Heil and G. Strang, Continuity of the joint spectral radius: application to wavelets, in Linear algebra for signal processing (Minneapolis, MN, 1992), vol. 69 of IMA Vol. Math. Appl., Springer, New York, 1995, 51-61. doi: 10.1007/978-1-4612-4228-4_4. [18] R. Jungers, The Joint Spectral Radius, vol. 385 of Lecture Notes in Control and Information Sciences, Springer-Verlag, Berlin, 2009, Theory and applications. doi: 10.1007/978-3-540-95980-9. [19] R. M. Jungers, On asymptotic properties of matrix semigroups with an invariant cone, Linear Algebra Appl., 437 (2012), 1205-1214. doi: 10.1016/j.laa.2012.04.006. [20] R. M. Jungers and P. Mason, On feedback stabilization of linear switched systems via switching signal control, SIAM J. Control Optim., 55 (2017), 1179-1198. doi: 10.1137/15M1027802. [21] V. S. Kozyakin, On the absolute stability of systems with asynchronously operating pulse elements, Avtomat. i Telemekh., 1990, 56-63, In Russian, translation in Automat. Remote Control, 51 (1990), 1349-1355 (1991). [22] V. S. Kozyakin, Constructive stability and stabilizability of positive linear discrete-time switching systems, Journal of Communications Technology and Electronics, 62 (2017), 686-693. doi: 10.1134/S1064226917060110. [23] V. Kozyakin, An explicit Lipschitz constant for the joint spectral radius, Linear Algebra Appl., 433 (2010), 12-18. doi: 10.1016/j.laa.2010.01.028. [24] V. Kozyakin, An Annotated Bibliography on Convergence of Matrix Products and the Theory of Joint/Generalized Spectral Radius, Preprint, Institute for Information Transmission Problems, Moscow, 2013. doi: 10.13140/2.1.4257.5040. [25] V. Kozyakin, Hourglass alternative and the finiteness conjecture for the spectral characteristics of sets of non-negative matrices, Linear Algebra Appl., 489 (2016), 167-185. doi: 10.1016/j.laa.2015.10.017. [26] V. Kozyakin, Minimax theorem for the spectral radius of the product of non-negative matrices, Linear and Multilinear Algebra, 65 (2017), 2356-2365. doi: 10.1080/03081087.2016.1273877. [27] V. Kozyakin, On convergence of infinite matrix products with alternating factors from two sets of matrices, Discrete Dyn. Nat. Soc., 2018 (2018), Art. ID 9216760, 5 pp. doi: 10.1155/2018/9216760. [28] 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. [29] G.-C. Rota and G. Strang, A note on the joint spectral radius, Nederl. Akad. Wetensch. Proc. Ser. A 63 = Indag. Math., 22 (1960), 379-381. doi: 10.1016/S1385-7258(60)50046-1. [30] M.-H. Shih, J.-W. Wu and C.-T. Pang, Asymptotic stability and generalized Gelfand spectral radius formula, Linear Algebra Appl., 252 (1997), 61-70. doi: 10.1016/0024-3795(95)00592-7. [31] D. P. Stanford, Stability for a multi-rate sampled-data system, SIAM J. Control Optim., 17 (1979), 390-399. doi: 10.1137/0317029. [32] D. P. Stanford and J. M. Urbano, Some convergence properties of matrix sets, SIAM J. Matrix Anal. Appl., 15 (1994), 1132-1140. doi: 10.1137/S0895479892228213. [33] Z. Sun and S. S. Ge, Switched Linear Systems: Control and Design, Communications and Control Engineering, Springer, London, 2005. doi: 10.1007/1-84628-131-8. [34] J. Theys, Joint Spectral Radius: Theory and Approximations, PhD thesis, Faculté des sciences appliquées, Département d'ingénierie mathématique, Center for Systems Engineering and Applied Mechanics, Université Catholique de Louvain, 2005. [35] J. von Neumann and O. Morgenstern, Theory of Games and Economic Behavior, 2nd edition, Princeton University Press, Princeton, N. J., 1947. [36] F. Wirth, The generalized spectral radius and extremal norms, Linear Algebra Appl., 342 (2002), 17-40. doi: 10.1016/S0024-3795(01)00446-3.
Discrete-time linear switching system
Control system consisting of plant $\mathit{\boldsymbol{ \boldsymbol{\mathscr{A}} }}$ and controller $\mathit{\boldsymbol{ \boldsymbol{\mathscr{B}} }}$
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