İsmail UYANIK, Hasan HAMZAÇEBİ, M. Mert ANKARALI

State-space identification of switching linear discrete time-periodic systems with known scheduling signals

In this paper, we propose a novel frequency domain state-space identification method for switching lineardiscrete time-periodic (LDTP) systems with known scheduling signals. The state-space identification problem of lineartime-invariant (LTI) systems has been widely studied both in the time and frequency domains. Indeed, there have beenseveral studies that also concentrated on state-space identification of both continuous and discrete linear time-periodic(LTP) systems. The focus in this study is the family of LDTP systems that switch among a finite set of subsystemstriggered by known periodic scheduling signals. We address the state-space identification of such systems in frequencydomain using input–output data. We also assume that full state measurements are available for the identification process.The major difference of our study is that we explicitly model the known scheduling signals responsible for switching,which greatly reduces the parametric complexity as well as potentially increases the estimation accuracy by avoidingoverfitting. In our identification framework, we gather the Fourier transformations of input–output data, known periodicscheduling signals, and state-space system dimensions and fuse them in a linear regression framework. Later, we estimatethe Fourier series coefficients of the time-periodic system and input matrices using a least-squares solution. Finally, weillustrate the effectiveness of our method using a switching LDTP system that is based on the damped Mathieu equation.

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[1] Zhai G, Hu B, Yasuda K, Michel AN. Qualitative analysis of discrete-time switched systems. In: Proceedings of the American Control Conference, Vol. 3; Anchorage, AK, USA; 2002. pp. 1880-1885.
[2] van Wingerden JW, Felici F, Verhaegen M. Subspace identification of MIMO LPV systems using a piecewise constant scheduling sequence with hard/soft switching. In: Proceedings of the European Control Conference; Kos, Greece; 2007. pp. 927-934.
[3] Lin H, Antsaklis PJ. Stability and stabilizability of switched linear systems: a survey of recent results. IEEE Transactions on Automatic Control 2009; 54 (2): 308-322.
[4] Uyanik I, Saranli U, Morgül Ö, Ankarali MM. Parametric identification of hybrid linear time-periodic systems. IFAC-PapersOnLine 2016; 49 (9): 7-12.
[5] DaCunha JJ, Davis JM. A unified Floquet theory for discrete, continuous, and hybrid periodic linear systems. Journal of Differential Equations 2011; 251 (11): 2987-3027.
[6] Holmes P, Full RJ, Koditschek D, Guckenheimer J. The dynamics of legged locomotion: models, analyses, and challenges. SIAM Review 2006; 48 (2): 207-304.
[7] Verdult V, Verhaegen M. Subspace identification of piecewise linear systems. In: Proceedings of the IEEE International Conference on Decision and Control, Vol. 4; Paradise Island, Bahamas; 2004. pp. 3838-3843.
[8] Seipel J, Holmes P. A simple model for clock-actuated legged locomotion. Regular and Chaotic Dynamics 2007; 12 (5): 502-520.
[9] Hwang S. Frequency domain system identification of helicopter rotor dynamics incorporating models with time periodic coefficients. PhD, University of Maryland, College Park, MD, USA, 1997.
[10] Mollerstedt E, Bernhardsson B. Out of control because of harmonics-an analysis of the harmonic response of an inverter locomotive. IEEE Control Systems 2000; 20 (4): 70-81.
[11] Allen MS, Sracic MW, Chauhan S, Hansen MH. Output-only modal analysis of linear time-periodic systems with application to wind turbine simulation data. Mechanical Systems and Signal Processing 2011; 25 (4): 1174-1191.
[12] Ankarali MM, Cowan NJ. System identification of rhythmic hybrid dynamical systems via discrete time harmonic transfer functions. In: Proceedings of the IEEE International Conference on Decision and Control; Los Angeles, CA, USA; 2014. pp. 1017-1022.
[13] Hiskens IA, Reddy PB. Switching-induced stable limit cycles. Nonlinear Dynamics 2007; 50 (3): 575-585.
[14] Uyanik I, Ankarali MM, Cowan NJ, Morgül Ö, Saranli U. Toward data-driven models of legged locomotion using harmonic transfer functions. In: Proceedings of the IEEE International Conference on Advanced Robotics; İstanbul, Turkey; 2015. pp. 357-362.
[15] Hamzacebi H, Morgül Ö. On the periodic gait stability of a multi-actuated spring-mass hopper model via partial feedback linearization. Nonlinear Dynamics 2017; 88: 1237–1256.
[16] Sracic MW, Allen MS. Method for identifying models of nonlinear systems using linear time periodic approximations. Mechanical Systems and Signal Processing 2011; 25 (7): 2705-2721.
[17] Uyanik I. Identification of legged locomotion via model-based and data-driven approaches. PhD, Bilkent University, Ankara, Turkey, 2017.
[18] Bittanti S, Colaneri P. Invariant representations of discrete-time periodic systems. Automatica 2000; 36 (12): 1777- 1793.
[19] Uyanik I, Saranli U, Ankarali MM, Cowan NJ, Morgul O. Frequency-domain subspace identification of linear time periodic (LTP) systems. IEEE Transactions on Automatic Control 2019; 64: 2529-2536.
[20] Verhaegen M, Yu X. A class of subspace model identification algorithms to identify periodically and arbitrarily time-varying systems. Automatica 1995; 31 (2): 201-216.
[21] Goos J, Lataire J, Pintelon R. Estimation of affine LPV state space models in the frequency domain: extension to transient behavior and non-periodic inputs. In: Proceedings of the IEEE European Control Conference; Linz, Austria; 2015. pp. 824-829.
[22] Van Wingerden JW, Verhaegen M. Subspace identification of bilinear and LPV systems for open-and closed-loop data. Automatica 2009; 45 (2): 372-381.
[23] Van Overschee P, De Moor B. Subspace Identification for Linear Systems: Theory, Implementation, Applications. Dordrecht, the Netherlands: Springer, 2012.
[24] Katayama T. Subspace Methods for System Identification. Dordrecht, the Netherlands: Springer, 2006.
[25] Di Ruscio D. Subspace System Identification: Theory and Applications. Lecture Notes. Porsgrunn, Norway: Telemark University College, 1997.
[26] Orfanidis SJ. Introduction to Signal Processing. Upper Saddle River, NJ, USA: Prentice-Hall, Inc., 1995.
[27] Uyanik I, Ankarali MM, Cowan NJ, Saranli U, Morgül Ö. Identification of a vertical hopping robot model via harmonic transfer functions. Transactions of the Institute of Measurement and Control 2016; 38 (5): 501-511.
[28] Allievi A, Soudack A. Ship stability via the Mathieu equation. International Journal of Control 1990; 51 (1): 139-167.
[29] Mathews J, Walker RL. Mathematical Methods of Physics. Vol. 501. Menlo Park, CA, USA: WA Benjamin, 1970.
[30] Humphries S. Principles of Charged Particle Acceleration. Mineola, NY, USA: Dover Publications, Inc., 2013.
[31] Ruby L. Applications of the Mathieu equation. American Journal of Physics 1996; 64 (1): 39-44.
[32] Akaike H. A new look at the statistical model identification. IEEE Transactions on Automatic Control 1974; 19 (6): 716-723.
[33] Hausdorff JM. Gait dynamics, fractals and falls: finding meaning in the stride-to-stride fluctuations of human walking. Human Movement Science 2007; 26 (4): 555-589.
[34] Tytell ED, Hsu CY, Williams TL, Cohen AH, Fauci LJ. Interactions between internal forces, body stiffness, and fluid environment in a neuromechanical model of lamprey swimming. Proceedings of the National Academy of Sciences of the USA 2010; 107 (46): 19832-19837.
[35] Dickinson MH, Farley CT, Full RJ, Koehl M, Kram R et al. How animals move: an integrative view. Science 2000; 288 (5463): 100-106.
[36] Grizzle JW, Abba G, Plestan F. Asymptotically stable walking for biped robots: analysis via systems with impulse effects. IEEE Transactions on Automatic Control 2001; 46 (1): 51-64.
[37] Louarroudi E, Pintelon R, Lataire J. Nonparametric tracking of the time-varying dynamics of weakly nonlinear periodically time-varying systems using periodic inputs. IEEE Transactions on Instrumentation and Measurement 2012; 61 (5): 1384-1394.
[38] Logan D, Kiemel T, Jeka JJ. Using a system identification approach to investigate subtask control during human locomotion. Frontiers in Computational Neuroscience 2016; 10: 146.
[39] Burden S, Ohlsson H, Sastry SS. Parameter identification near periodic orbits of hybrid dynamical systems. In: Proceedings of the IFAC Symposium on System Identification; Brussels, Belgium; 2012. pp. 1197-1202.