Structural identifiability analysis of nonlinear time delayed systems with generalized frequency response functions

Szlobodnyik, Gergely ; Szederkényi, Gábor

Kybernetika, Tome 57 (2021) no. 6, pp. 939-957

Abstract (VO)

In this paper a novel method is proposed for the structural identifiability analysis of nonlinear time delayed systems. It is assumed that all the nonlinearities are analytic functions and the time delays are constant. We consider the joint structural identifiability of models with respect to the ordinary system parameters and time delays by including delays into a unified parameter set. We employ the Volterra series representation of nonlinear dynamical systems and make use of the frequency domain representations of the Volterra kernels, i. e. the Generalized Frequency Response Functions (GFRFs), in order to test the unique computability of the parameters. The advantage of representing nonlinear systems with their GFRFs is that in the frequency domain representation the time delay parameters appear explicitly in the exponents of complex exponential functions from which they can be easily extracted. Since the GFRFs can be symmetrized to be unique, they provide us with an exhaustive summary of the underlying model structure. We use the GFRFs to derive equations for testing structural identifiability. Unique solution of the composed equations with respect to the parameters provides sufficient conditions for structural identifiability. Our method is illustrated on non-linear dynamical system models of different degrees of non-linearities and multiple time delayed terms. Since Volterra series representation can be applied for input-output models, it is also shown that after differential algebraic elimination of unobserved state variables, the proposed method can be suitable for identifiability analysis of a more general class of non-linear time delayed state space models.

Détail

DOI : 10.14736/kyb-2021-6-0939

Classification : 93B30
Keywords: structural identifiability; Volterra series; generalized frequency response

Bibliographie

[1] Anguelova, M., Wennberg, B.: State elimination and identifiability of the delay parameter for nonlinear time-delay systems. Automatica 44 (2008), 5, 1373-1378. | DOI | MR | Zbl

[2] al., S. Audoly et.: Global identifiability of nonlinear models of biological systems. IEEE. Trans. Biomed. Engrg. 48 (2001), 55-65. | DOI

[3] Bayma, R. S., Lang, Z. Q.: A new method for determining the generalised frequency response functions of nonlinear systems. IEEE Trans. Circuits Systems I 59 (2012), 12, 3005-3014. | DOI | MR

[4] Bedrosian, E., Rice, S. O.: The output properties of Volterra systems (nonlinear systems with memory) driven by harmonic and Gaussian inputs. Proc. IEEE 59 (1971), 12, 1688-1707. | DOI | MR

[5] Belkoura, L., Orlov, Y.: Identifiability analysis of linear delay-differential systems. IMA J. Math. Control Inform. 19 (2002), 73-81. | DOI | MR

[6] Bellman, R., Aström, K. J.: On structural identifiability. Math. Biosci. 7 (1970), 3-4, 329-339. | DOI | MR

[7] al., G. Bellu et.: DAISY: A new software tool to test global identifiability of biological and physiological systems. Comput. Methods Programs Biomed. 88 (2007), 52-61. | DOI

[8] Billings, S. A., Tsang, K. M.: Spectral analysis for nonlinear systems, Part I: parametric nonlinear spectral analysis. Mechanic. Systems Signal Process. 3 (1989), 4, 319-339. | DOI

[9] Bocharov, G. A., Rihan, F. A.: Numerical modelling in biosciences using delay differential equations. J. Comput. Appl. Math. 125 (2000), 1-2, 183-199. | DOI | MR

[10] Cheng, C. M., Peng, Z. K., Zhang, W. M., Meng, G.: Volterra-series-based nonlinear system modeling and its engineering applications: A state-of-the-art review. Mech. Systems Signal Process. 87 (2017), 340-364. | DOI

[11] Chis, O. T., Banga, J. R., Balsa-Canto, E.: Structural identifiability of systems biology models: a critical comparison of methods. PloS One 6 (2011), 11.

[12] Churilov, A. N., Medvedev, A., Zhusubaliyev, Z. T.: Impulsive Goodwin oscillator with large delay: Periodic oscillations, bistability, and attractors. Nonlinear Analysis: Hybrid Systems 21 (2016), 171-183. | DOI | MR

[13] Cooke, K., Driessche, P. Van den, Zou, X.: Interaction of maturation delay and nonlinear birth in population and epidemic models. J. Math. Biology 39 (1999), 4, 332-352. | DOI | MR

[14] Denis–Vidal, L., Joly–Blanchard, G., Noiret, C.: Some effective approaches to check the identifiability of uncontrolled nonlinear systems. Math. Comput. Simul. 57 (2000), 35-44. | DOI | MR

[15] Epstein, I. R., Luo, Y.: Differential delay equations in chemical kinetics. Nonlinear models. The cross-shaped phase diagram and the oregonator. J. Chem. Phys. 95 (1991), 244-254. DOI 

[16] Fliess, M.: Fonctionnelles causales non linéaires et indéterminées non commutatives. Bull. Soc. Math. France 109 (1981), 3-40. | DOI | MR

[17] George, D.: Continuous Nonlinear Systems. MIT RLE Technical Report No. 355, 1959.

[18] Glad, T.: Nonlinear state space and input-output descriptions using differential polynomials. In Descusse. Lecture Notes in Control and Information Science J. (M. Fliess, A. Isidori and D. Leborgne, eds.), Vol. 122., Springer Berlin. | MR

[19] Hermann, K., Krener, A.: Nonlinear controllability and observability. IEEE Trans. Automat. Control, 22 (1977), 5, 728-740. | DOI | MR

[20] al., B. Huang et.: Impact of time delays on oscillatory dynamics of interlinked positive and negative feedback loops. Physical Review E 94 (2016), 5, 052413. | DOI

[21] Isidori, A.: Nonlinear Control Systems. Second edition. Springer-Verlag, Berlin 1989. | MR

[22] Kuang, Y.: Delay Differential Equations With Applications in Population Dynamics. Academic Press, Boston 1993. | MR | Zbl

[23] Lapytsko, A., Schaber, J.: The role of time delay in adaptive cellular negative feedback systems. J. Theoret. Biology 398 (2016), 64-73. | DOI

[24] Li, J., Kuang, Y., Mason, C. C.: Modeling the glucose–insulin regulatory system and ultradian insulin secretory oscillations with two explicit time delays. J. Theoret. Biology 242 (2006), 3, 722-735. | DOI | MR

[25] Liz, E., Ruiz-Herrera, A.: Delayed population models with Allee effects and exploitation. Math. Biosci. Engrg. 12 (2015), 1, 83-97. | DOI | MR

[26] Ljung, L.: System Identification: Theory for the User. Second edition. Prentice-Hall, Upper Saddle River, NJ 1999.

[27] Ljung, L., Glad, T.: On global identifiability for arbitrary model parametrizations. Automatica 30 (1994), 2, 265-276. | DOI | MR

[28] Ljung, L., Glad, T.: Modeling of Dynamic Systems. PTR Prentice Hall, 1994.

[29] Lunel, V., Sjoerd, M.: Identification problems in functional differential equations. Proc. 36th IEEE Conference on Decision and Control IEEE 5 (1997), 4409-4413.

[30] MacDonald, N.: Time-lags in Biological Models. Lecture Notes in Biomathematics, Vol. 27, Springer, Berlin 1978. | DOI | MR

[31] MacDonald, N.: Biological Delay Systems: Linear Stability Theory. Cambridge University Press, Cambridge, 1989. | MR

[32] Meshkat, N., Eisenberg, M., DiStefano, J. J.: An algorithm for finding globally identifiable parameter combinations of nonlinear ode models using Gröbner bases. Math. Biosci. 222 (2009), 61-72. | DOI | MR

[33] al, Y. Orlov et.: On identifiability of linear time-delay systems. IEEE Trans. Automat. Control 47 (2002), 8, 1319-1324. | DOI | MR

[34] Orosz, G., Moehlis, J., Murray, R. M.: Controlling biological networks by time-delayed signals. Philosoph. Trans. Royal Society A: Mathematical, Physical and Engineering Sciences 368 (1911), (2010), 439-454. | DOI | MR

[35] Palm, G., Poggio, T.: The Volterra representation and the Wiener expansion: validity and pitfalls. SIAM J. Appl. Math. 33 (1977), 2, 195-216. | DOI | MR

[36] Peng, Z. K., al, et.: Feasibility study of structural damage detection using narmax modelling and nonlinear output frequency response function based analysis. Mech. Syst. Signal Process. 25 (2011), 3, 1045-1061. | DOI

[37] Pohjanpalo, H.: System identifiability based on the power series expansion of the solution. Math. Biosci. 41 (1978), 21-33. | DOI | MR

[38] Ritt, J. F.: Differential Algebra. American Mathematical Society, Providence 1950.

[39] Roussel, M. R.: The use of delay differential equations in chemical kinetics. J. Phys. Chem. 100 (1996), 20, 8323-8330. | DOI

[40] Rugh, W. J.: Linear System Theory. Prentice Hall, New Jersey 1996. | MR | Zbl

[41] Schwaiger, J., Prager, W.: Polynomials in additive functions and generalized polynomials. Demonstratio Math. 41 (2008), 3, 589-613. | MR

[42] Silva, C. J., Maurer, H., Torres, D. F. M.: Optimal control of a tuberculosis model with state and control delays. Math. Biosci. Engrg. 14 (2017), 1, 321-337. | DOI | MR

[43] Smith, H.: An Introduction to Delay Differential Equations with Applications to the Life Sciences. Springer, New York 2011. | MR

[44] Söderström, T., Stoica, P.: System Identification. Prentice-Hall, 1989.

[45] Swain, A. K., Mendes, E. M. A. M., Nguang, S. K.: Analysis of the effects of time delay in nonlinear systems using generalised frequency response functions. J. Sound Vibration 294 (2006), 1-2, 341-354. | DOI

[46] Vághy, M., Szlobodnyik, G., Szederkényi, G.: Kinetic realization of delayed polynomial dynamical models. IFAC-PapersOnLine 52 (2019), 7, 45-50. | DOI

[47] al., S. Vajda et.: Qualitative and quantitative identifiability analysis of nonlinear chemical kinetic models. Chem. Engrg. Commun. 83 (1989), 191-219. | DOI

[48] Vajda, S., Godfrey, K., Rabitz, H.: Similarity transformation approach to identifiability analysis of nonlinear compartmental models. Math. Biosci. 93 (1989), 217-248. | DOI | MR

[49] Vajda, S., Rabitz, H.: Isomorphism approach to global identifiability of nonlinear systems. IEEE Trans. Automat. Control 34 (1989), 220-223. | DOI | MR

[50] Villaverde, A. F., Barreiro, A.: Identifiability of large non-linear biochemical networks. MATCH - Commun. Math. Comput. Chemistry 76 (2016), 2, 259-296. | MR

[51] Walter, E.: Identifiability of Parametric Models. Pergamon Press, Oxford 1987.

[52] Walter, E., Lecourtier, Y.: Unidentifiable compartmental models: what to do?. Math. Biosci. 56 (1981), 1-25. | DOI | MR

[53] Walter, E., Lecourtier, Y.: Global approaches to identifiability testing for linear andnonlinear state space models. Math. Comput. Simul. 24 (1982), 472-482. | DOI | MR

[54] Walter, E., Pronzato, L.: On the identifiability and distinguishability of nonlinear parametric models. Math. Comput. Simul. 42 (1996), 125-134. | DOI

[55] Walter, E., Pronzato, L.: Identification of Parametric Models from Experimental Data. Springer Verlag, 1997. | MR

[56] Weijiu, L.: Introduction to Modeling Biological Cellular Control Systems. Springer Science and Business Media, 2012. | MR

[57] Villaverde, A. F.: Observability and Structural Identifiability of Nonlinear Biological Systems. Complexity, 2019. | DOI

[58] Villaverde, A. F., Barreiro, A., Papachristodoulou, A.: Structural identifiability of dynamic systems biology models. PLOS Comput. Biology 12 (2016), 10. | DOI

[59] Volterra, V.: Theory of Functionals and Integral Equations. Dover, New York 1959. | MR

[60] Xia, X., Moog, C. H.: Identifiability of nonlinear systems with application to HIV/AIDS models. IEEE Trans. Automat. Control 4 (2003), 330-336. | DOI | MR

[61] Yuan, Y., Li, Y.: Study on EEG time series based on duffing equation. In: International Conference on BioMedical Engineering and Informatics, Vol. 2, Sanya S2008, pp. 516-519.

[62] Zhang, H., Billings, S. A., Zhu, Q. M.: Frequency response functions for nonlinear rational models. Int. J. Control 61 (1995), 1073-1097. | DOI | MR

[63] Zhang, J., Xia, X., Moog, C. H.: Parameter identifiability of nonlinear systems with time-delay. IEEE Trans. Automat. Control 51 (2006), 2, 371-375. | DOI | MR

[64] Zheng, G., Barbot, J. P., Boutat, D.: Identification of the delay parameter for nonlinear time-delay systems with unknown inputs. Automatica 49 (2013), 6, 1755-1760. | DOI | MR