(644g) Data-Driven Synthesis of Nonlinear State Observer through Estimation of Koopman Eigenfunctions

The design of a nonlinear observer to estimate states is an important challenge in the control and operation of chemical processes which exhibit nonlinear dynamics. The Kazantzis-Kravaris-Luenberger (KKL) observer [1] provides a solution to this problem by solving partial differential equations to determine a nonlinear injective mapping of states. However, finding this transformation directly is challenging as partial differential equations are involved. This limitation prevents the implementation of KKL observers.

To address this challenge, we propose using a Koopman operator 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]. In particular, we focus on the Koopman operator’s actions on the span of its eigenfunctions. By assuming that the process output functions can be represented as a linear combination of the Koopman eigenfunctions, a solution for the nonlinear injective mapping can be found. Thus, an explicit expression of the KKL observer, using the method outlined in [4], is composed solely of Koopman eigenfunctions. Furthermore, these Koopman eigenfunctions can be estimated directly from measured system data through a constrained least squares regression problem, resulting in a data-driven observer.

The proposed approach is implemented for the case of a planar system with a limit cycle, where the Koopman eigenfunctions 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 the angular and radial components. We formulated a constrained least squares regression problem, considering both negative real-valued and imaginary eigenvalues, to estimate the Koopman eigenfunctions and the coefficients of the linear combination representing the output function from a sample of consecutive state values over time [2]. Since the series contains an infinite amount of terms in two variables, when practically using a truncated version of the representation, the error of the truncation is bounded under regularity conditions. To demonstrate the proposed approach, a data-driven estimation of the Koopman eigenfunctions of a Brusselator system is performed, which is used to construct the observer.

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] E. M. Stein and R. Shakarchi, Fourier analysis: An introduction. Princeton University Press, 2011.