Assignability of numerical characteristics of time-varying systems

• Autorzy: Babiarz A. , Czornik A. , Klamka J.

Identyfikatory

DOI

10.24425/bpasts.2019.130890

• Języki publikacji: EN

Abstrakty

EN

The main aim of this article is to survey and discuss the existing state of art concerning the assignability by a feedback of numerical characteristics of linear continuous and discrete time-varying systems. Most of the results present necessary or sufficient conditions for different formulation of the Lyapunov spectrum assignability problem. These conditions are expressed in terms of various controllability types and optimalizability of the controlled systems and certain properties of the free system such as: regularity, diagonalizability, boundness away, integral separation and reducibility.

Słowa kluczowe

EN

assignability; controllability; Lyapunov spectrum; linear time-varying systems

• Wydawca: Polska Akademia Nauk, Wydział IV Nauk Technicznych
• Czasopismo: Bulletin of the Polish Academy of Sciences. Technical Sciences
• Rocznik: 2019
• Tom: Vol. 67, nr 6
• Strony: 1007--1022
• Opis fizyczny: Bibliogr. 63 poz.

Twórcy

autor

Babiarz A.

• Silesian University of Technology, Institute of Automatic Control 16 Akademicka St., 44-100 Gliwice, Poland

autor

Czornik A.

• Silesian University of Technology, Institute of Automatic Control 16 Akademicka St., 44-100 Gliwice, Poland

autor

Klamka J.

• Silesian University of Technology, Institute of Automatic Control 16 Akademicka St., 44-100 Gliwice, Poland

Bibliografia

• [1] E.D. Sontag, Mathematical control theory:deterministic finite dimensional systems, Springer Science&Business Media,2013.
• [2] J. Klamka, Controll ability of dynamical systems,Kluwer Academic Publishers Dordrecht, The Netherlands, 1991.
• [3] J.Klamka,“Controllability of dynamical systems. Asurvey”, Bull.Pol.Ac.: Tech. 61(2),335-342 (2013).
• [4] J. Klamka, J. Wyrwał, and R. Zawiski, “On controllability of second or derdynamical systems-asurvey”, Bull. Pol. Ac.: Tech. 65(3),279-295 (2017).
• [5] J. Klamka, Controllability and Minimum Energy Control, Studiesin Systems, Decision and Control, Springer, 2019.
• [6] T. Tarczewski, M. Skiwski, L.J. Niewiara, and L.M. Grzesiak, “High-performance PMSM servo-drive with constrained state feedback position controller”, Bull.Pol.Ac.: Tech. 66(1),49–58(2018).
• [7] B. Porterand, R. Crossley, Modal Control: Theory and Applications, Taylor&Francis, 1972.
• [8] V.M. Popov, “Hyperstability and optimality of automatic systems with several control functions”, Rev.Roumaine.Sci.Techn., Electrotechn.et Energ.9(4),629–690(1964).
• [9] R.E. Kalman, “Contributions to the Theory of Optimal Control”, Bol.Soc.Mat. Mexicana 5(2), 102–119 (1960).
• [10] W.M. Wonham, “On pole assignment in multi input control lable linear systems”, IEEE Trans. Autom. Control 12(6),660–665 (1967).
• [11] C.Mosquera, S.Scalise,and G.Taricco,“Spectral characterization of feedback linea rperiodically time varying systems”, IEEE International Conference on Acoustics, Speech, and Signal Processing 1209–1212 (2002).
• [12] S.G. Hasnijeh, M. Poursina, B.J. Leira,H. Karimpour, and W. Chai,“Stochastic dynamics of a non linear time-varying spur gear model using an adaptive time-stepping path integration method”, J.Sound Vib. 447, 170–185 (2019).
• [13] Z. Wang,G. Mei, Q. Xiong, Z. Yin, and W. Zhang, “Motorcar-track spatial coupled dynamics model of a high-speed train with traction transmission systems”, Mech. Mach. Theory 137, 386–403 (2019).
• [14] F.L. Neerhoff, P. Van Der Kloet, A. Van Staveren, and C.J.M. Verhoeven, “Time-varying small-signal circuits for non linear electronics”, Nonlinear Dynamics of Electronic Systems 81–84, World Scientific, (2000).
• [15] V. Fromion, G. Scorletti, and J.P. Barbot, “Quadratic observers for estimation and control in induction motors”, American Control Conference 2143–2147 (1999).
• [16] H. Bourlès and B. Marinescu, Linear Time-Varying Systems: Algebraic-Analytic Approach, Lecture Notes in Controland Information Sciences, Springer, 2011.
• [17] C.M. Kang, S. Lee, and C.C. Chung, “Discrete-time LPV H2 observe with non linear bounded varying parameter and its ap-plication to the vehicles tate observer”, IEEE Trans. Ind. Electron.65 (11) ,8768–8777 (2018).
• [18] J. Klamka, A. Czornik, and M. Niezabitowski, “Stability and controllability of switched systems”, Bull.Pol.Ac.: Tech. 61(3), 547–555 (2013).
• [19] A. Babiarz, A. Czornik, J. Klamka, and M. Niezabitowski, “The selected problems of control ability of discrete-time switched linear systems with constrained switching rule”, Bull.Pol.Ac.: Tech. 63(3), 657–666 (2015).
• [20] P. Brunovsky, “Controll ability and linear closed-loop control sin linear periodic systems”, J.Differ. Equations 6(2),296–313 (1969).
• [21] N.A. Izobov, Lyapunov Exponents and Stability, Cambridge: Cambridge Scientific Publishers, 2013.
• [22] L. Barreira and Ya.B. Pesin, Lyapunov Exponents and Smooth Ergodic Theory, University Lecture Series ,American Mathematical Society, 2002.
• [23] L. Barreira, Lyapunov Exponents, Springer International Publishing AG, 2017.
• [24] E.A. Barabanov and A.V. Nyukh, “Uniform exponents of linear systems of differential equations”, Differ. Equ. (Russian), 30(10),1665–1676 (1994).
• [25] L.Y. Adrianova, Introduction to linear systems of differential equations, American Mathematical Society, 1995.
• [26] P. Bohl, “Über differentialungleichungen”, J.Reine Angew. Math.144, 284–313 (1914).
• [27] D. Hinrichsen and A.J. Pritchard, Mathematical Systems TheoryI: Modelling, State Space Analysis, Stability and Robustness, Springer Science & Business Media, 2011.
• [28] J.L. Daleckii and M.G. Krein, Stability of Solutions of Differential Equations in Banach Spaces, Translations of Mathematical Monographs, American Mathematical Society,1974.
• [29] E.K. Makarovand S.N .Popova, Controllability of asymptotic in-variants of time-dependent linear systems, Belorusskaya Nauka, 2012.
• [30] M. Ikeda, H. Maeda, and S. Kodama, “Stabilization of linear systems”, SIAMJ. Control 10(4),716–729 (1972).
• [31] B.D.O. Anderson, A. Ilchmann, nd F.R.W irth, “Stabilizability of linear time-varying systems”, Syst. Contro lLett. 62(9), 747–755 (2013).
• [32] M. Wonham ,Linear Multivariable Control: A Geometric Approach, Berlin, Springer-Verlag, 1974.
• [33] E.L. Tonkov, “A criterion of uniform control ability and stabilization of a linear recurrent system”, Differ.Equ.(Russian) 15(10), 1804–1813 (1979).
• [34] A.M. Lyapunov, “The general problem of the stability of motion”, Int. J.Control 55(3), 531–534 (1992).
• [35] L. Barreiraand C. Valls, “Stability theory and Lyapunov regularity”, J.Differ. Equations 232(2), 675–701 (2007).
• [36] A. Czornik and A. Nawrat, “On the regularity of discrete linear systems”, Linear Algebra Appl. 432(11), 2745–2753 (2010).
• [37] B.F. Bylov, R.E. Vinograd, D.M. Grobman, and V.V. Nemytskii, Theory of Lyapunov Exponents, Moscow: Nauka, 1966.
• [38] N.A. Izobov, Introduction to The Theory of Lyapunov Exponents, Minsk, 2006.
• [39] S.N. Popova, “Global reducibility of linear control systems to systems of scalar type”, Differ.Equ. 40(1), 43–49 (2004).
• [40] S.N. Popova, “On the global control ability of Lyapunov exponents of linear systems”, Differ.Equ. 43(8), 1072–1078 (2007).
• [41] J. Russell, R. Obaya, S. Novo, C. Núñez, and R. Fabbri, “Nonautonomous linear Hamiltonian systems”, Nonautonomous Linear Hamiltonian Systems: Oscillation, Spectral Theory and Control 1–75 (2016).
• [42] A.A. Kozlov, “On the control of Lyapunov exponents of two-dimensional linear systems with locally integrable coefficients”, Differ. Equ. 44(10), 1375–1392 (2008).
• [43] S.N.Popova, “Global controll ability of the complete set of Lyapunov invariants of periodic systems”, Differ.Equ. 39(12), 1713–1723 (2003).
• [44] S.N. Popova, “Simultaneous local controllability of the spectrum and the Lyapunov irregularity coefficient of regular systems”, Differ. Equ.40(3), 461–465 (2004).
• [45] E.K. Makarova nd S.N. Popova,“Sufficient conditions for the local proportional controllability of Lyapunov exponents of linear systems”, Differ.Equ. 39(2),234–245 (2003).
• [46] A. Babiarz, A. Czornik, and M. Niezabitowski, “Relations be-tween Bohl exponents and general exponent of discrete linear time-varying systems”, J. Differ. Equ. Appl. 25(4), 560–572 (2019).
• [47] A .Czornik, Perturbation Theory for Lyapunov Exponents of Discrete Linear Systems, Monografie–Komitet Automatykii Robotyki Polskiej Akademii Nauk, Wydawnictwa AGH, 2012.
• [48] I.N. Banshchikova and S.N.Popova, “On the property of integral separation of discrete-time systems”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp’yuternye Nauki 27(4), 481–498 (2017).
• [49] I.V. Ga ̆ıshun, Discrete-Time Systems, Minsk: Institut Matematiki NAN Belarusi, 2001.
• [50] V.A. Lun’kov, “On complete transformability of linear controlled systems” ,Izvestiya Instituta Matematikii Informatiki Udmurtskogo Gosudarstvennogo Universiteta 2, 15–25 (1996).
• [51] V.A. Zaitsev, S.N. Popova, and E.L. Tonkov, “On the property of uniform complete controllability of a discrete-time linear control system”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika .Komp’yuternye Nauki 24(4), 53–63 (2014).
• [52] A. Halanay and V. Ionescu, Time-Varying Discrete Linear Systems: Input-Out put Operators, Riccati Equations, Disturbance Attenuation, Springer, 1994.
• [53] V.A.Lun’kov, “The control ability and stabilizability of linear discrete systems”, Differ. Equ. 16(4), 753–754 (1980).
• [54] A.Czornik and A. Swierniak, “On the discrete JLQ and JLQG problems”, Nonlinear Anal. Theory Methods Appl. 47 (1),423–434 (2001).
• [55] A. Babiarz, A. Czornik, E. Makarov, M. Niezabitowski and S. Popova, “Pole placement theorem for discrete time-varying linear systems”, SIAMJ. Control Optim. 55(2), 671–692 (2017).
• [56] A. Babiarz, I.N. Banshchikova, A. Czornik, E. Makarov, M. Niezabitowski, and S.N. Popova, “Necessary and sufficient conditions for assignability of the Lyapunov spectrum of discrete linear time-varying systems”, IEEET. Automat. Contr. 63(11), 3825–3837 (2018).
• [57] A. Babiarz, I.N. Banshchikova, A. Czornik, E. Makarov, M. Niezabitowski, and S.N. Popova, “Proportional local assignability of Lyapunov spectrum of linear discrete time-varying systems”, SIAMJ. Control Optim. 57(2), 1355–1377 (2019).
• [58] S.N. Popova, “Assignability of certain Lyapunov in variants for linear discrete-time systems”, IFAC-Papers OnLine 51 (32), 40–45 (2018).
• [59] S.N. Popova and I.N.Banshchikova, “On the property of proportional local assignability of the Lyapunov spectrum for discrete time-varying systems”,14th International Conference“ Stability and Oscillations of Nonlinear Control Systems”,1–4 (2018).
• [60] G.R. Sell, Topological Dynamics and Ordinary Differential Equations, Van Nostrand-Reinhold, 1971.
• [61] R. Johnson, R .Obaya, S. Novo, G. Núñez, and R. Fabbri, Non autonomous Linear Hamiltonian Systems: Oscillation, Spectral Theory and Control, Springer, 2016.
• [62] G.R. Sell, “The Floquet problem for almost periodic linear differential equations”, Ordinary and Partial Differential Equations 239–251 (1974).
• [63] I.N. Sergeev, “Definition and properties of characteristic frequencies of a linear equation”, J.Math. Sci. 135 (1), 2764–2793 (2006).

Uwagi

PL

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2020).