Data-Driven Synthesis of Nonlinear State Observer for Autonomous Limit Cycle Systems through Estimation of Koopman Eigenfunctions

Chemical processes are nonlinear dynamical systems. Although the input and output signals are measurable, the variables that fully describe the evolution of the system, namely the states, must be estimated by state observers for control and monitoring purposes. The Kazantzis-Kravaris-Luenberger (KKL) observer [1], a generalization of the Luenberger observer for linear systems, offers a solution by transforming states through a nonlinear injective mapping derived from partial differential equations (PDEs). However, analytically solving these PDEs is challenging. This motivates the use of data-driven (machine learning-based) approaches for observer synthesis.

To address this challenge, we propose a data-driven, Koopman operator-based approach to explicitly construct the KKL observer. The Koopman operator acts on continuous functions to advance them with the flow of the system’s dynamics [2], and we particularly focus on its actions on the span of its eigenfunctions. These Koopman eigenfunctions can be estimated directly from measured data through a least-squares regression problem solved as an eigenvalue minimization problem. Assuming that the output function can be expressed as a linear combination of Koopman eigenfunctions, the injective mapping can be estimated similarly. Thus, the KKL observer, using the method outlined in [4], is composed solely of Koopman eigenfunctions and constructed entirely from data. The method only uses convex optimization, making it computationally efficient.

The proposed approach is implemented for the case of a planar limit cycle system, where Koopman eigenfunctions exist and are defined in terms of two transformed variables describing the latent radial and angular components of the system dynamics [5]. For such a system, the process output function can be expressed as a Fourier-Taylor series of angular and radial components. Expressing time derivatives as a finite difference approximation, we formulate a least-squares regression problem to estimate the KKL injective mapping as a linear combination of Koopman eigenfunctions, corresponding to both negative real-valued and imaginary eigenvalues, from a sample of consecutive state values over time [2]. To recover the estimated states, we construct the inverse of the injective mapping using kernel ridge regression [6]. The proposed approach was demonstrated on the Brusselator system, resulting in accurate estimations of the system states from measured data alone.

References

[1] N. Kazantzis and C. Kravaris, “Nonlinear observer design using Lyapunov’s auxiliary theorem,” Syst. Contr. Lett., vol. 34, no. 5, pp. 241-247, 1998.

[2] S. L. Brunton, M. Budišić, E. Kaiser, and J. N. Kutz, “Modern Koopman theory for dynamical systems,” SIAM Rev., vol. 64, no. 2, pp. 229–340, 2022.

[3] L. Brivadis, V. Andrieu, P. Bernard, and U. Serres, “Further remarks on KKL observers,” Syst. Contr. Lett., vol. 172, p. 105429, 2023.

[4] V. Pachy, V. Andrieu, P. Bernard, L. Brivadis, and L. Praly, “On the existence of KKL observers with nonlinear contracting dynamics (long version),” (arXiv preprint) arXiv:2402.16432, 2024.

[5] I. Mezić, “Spectrum of the Koopman operator, spectral expansions in functional spaces, and state-space geometry,” J. Nonlin. Sci., vol. 30, no. 5, pp. 2091–2145, 2020.

[6] V. Vovk, “Kernel ridge regression,” in Empirical Inference: Festschrift in Honor of Vladimir N. Vapnik, B. Schölkopf, Z. Luo, and V. Vovk. Berlin, Heidelberg: Springer, pp. 105–116, 2013.