Abstract
In this chapter, we overview a new approach for nonlinear estimation based on Koopman operator-theoretic framework. We exploit Koopman eigenfunctions to create a nonlinear embedding/lifting of underlying nonlinear dynamics to synthesize observer forms (which we call Koopman observer form (KOF)) which enables the use of well-known estimation techniques developed for linear/bilinear systems in context of more general nonlinear systems. Furthermore, we present an extension of this framework for nonlinear constrained state estimation (CSE) with non-convex state constraints. Exploiting the KOF-based representation, we show that under certain conditions the CSE problem can be transformed into a higher dimensional but convex problem. We present a receding horizon estimation formulation based on this transformation, which could provide computational benefit in real-time applications. We also analyze system theoretic properties of KOF in relation to the original nonlinear system, and establish relationship between the original nonlinear estimation problem and the Koopman transformed problem. Finally, we illustrate our approach on a few examples.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Abbaszadeh, M., Marquez, H.J.: A robust observer design method for continuous-time Lipschitz nonlinear systems. In: IEEE CDC (2006)
Açıkmeşe, B., Carson, J.M., Blackmore, L.: Lossless convexification of nonconvex control bound and pointing constraints of the soft landing optimal control problem. IEEE Trans. Control Syst. Technol. 21(6), 2104–2113 (2013)
Bao, X., Sahinidis, N., Tawarmalani, M.: Semidefinite relaxations for quadratically constrained quadratic programming: a review and comparisons. J. Math. Program. Ser. B 129(1), 129–157 (2011)
Besancon, G.: Nonlinear Observers and Applications. Lecture Notes in Control and Information Sciences, vol. 363. Springer, Berlin (2007)
Boyd, S., Vandenberghe, L.: Convex optimization. Cambridge University Press, Cambridge (2004)
Brenan, K.E., Campbell, S.L., Petzold, L.R.: Numerical solution of initial-value problems in differential-algebraic equations. SIAM, New York (1995)
Budisic, M., Mohr, R.M., Mezic, I.: Applied koopmanism. Chaos 22(4), 047510 (2012)
Carleman, T.: Application de la théories des équations intégrales linéaires aux systémes déquations différentielles non linéaires. Acta Math. 59, 63–87 (1932)
Diehl, M., Ferreau, H.J., Haverbeke, N.: Efficient numerical methods for nonlinear MPC and moving horizon estimation. In: Nonlinear Model Predictive Control, pp. 391–417. Springer, Berlin (2009)
Diehl, M., Bock, H.G., Schlöder, J.P., Findeisen, R., Nagy, Z., Allgöwer, F.: Real-time optimization and nonlinear model predictive control of processes governed by differential-algebraic equations. J. Process Control 12(4), 577–585 (2002)
Domahidi, A.: Forces: fast optimization for real-time control on embedded systems. http://forces.ethz.ch (2012)
Elliott, D.: Bilinear Control Systems: Matrices in Action. Springer, Berlin (2009)
Gauthier, J., Kupka, A.: Deterministic Observation Theory and Applications. Cambridge University Press, Cambridge (1997)
Grasselli, O.M., Isidori, A.: Deterministic state reconstruction and reachability of bilinear processes. In: IEEE Joint Automatic Control Conference. IEEE, New York (1977)
Hermann, R., Krener, A.J.: Nonlinear controllability and observability. IEEE Trans. Autom. Control 22(5) (1977)
Juang, J.N., Lee, C.H.: Continuous-time bilinear system identification using single experiment with multiple pulses. Nonlinear Dyn. 69(3), 1009–1021 (2012)
Kang, W., Krener, A.J., Xiao, M., Xu, L.: A Survey of Observers for Nonlinear Dynamical Systems. Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications, vol. II. Springer, Berlin (2013)
Keller, H.: Nonlinear observer design by transformation into a generalized observer canonical form. Int. J. Control 46(6), 1915–1930 (1987)
Kelman, A., Borrelli, F.: Bilinear model predictive control of a hvac system using sequential quadratic programming. IFAC Proc. 44(1), 9869–9874 (2011)
Korda, M., Mezic, I.: Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control. Automatica 93, 149–160
Kowalski, K., Steeb, W.H.: Nonlinear Dynamical Systems and Carleman Linearization. World Scientific Publishing Co. Pte. Ltd., Singapore (1991)
Krener, A.J.: Bilinear and nonlinear realizations of input output maps. SIAM J. Control Optim. 13, 827–834 (1975)
Kühl, P., Diehl, M., Kraus, T., Schlöder, J.P., Bock, H.G.: A real-time algorithm for moving horizon state and parameter estimation. Comput. Chem. Eng. 35(1), 71–83 (2011)
Liu, X., Lu, P.: Solving nonconvex optimal control problems by convex optimization. J. Guid. Control Dyn. 37(3), 750–765 (2014)
Mao, Y., Dueri, D., Szmuk, M., Açıkmeşe, B.: Successive convexification of non-convex optimal control problems with state constraints (2017). arXiv:1701.00558
Mao, Y., Szmuk, M., Açıkmeşe, B.: Successive convexification of non-convex optimal control problems and its convergence properties. In: IEEE 55th Conference on Decision and Control, pp. 3636–3641. IEEE, Las Vegas (2016)
Mauroy, A., Mezic, I.: Global stability analysis using the eigenfunctions of the Koopman operator. IEEE Trans. Autom. Control 61(11), 3356–3369 (2016)
Mezic, I.: Analysis of fluid flows via spectral properties of the Koopman operator. Annu. Rev. Fluid Mech. 45, 357–378 (2012)
Misawa, E.A., Hedrick, J.K.: Nonlinear observers: a state-of-the-art survey. J. Dyn. Syst. Meas. Control 111(3), 344–352 (1989)
Mohr, R., Mezic, I.: Construction of eigenfunctions for scalar-type operators via Laplace averages with connections to the Koopman operator (2014). arXiv:1403.6559
Morari, M., Lee, J.H.: Model predictive control: past, present and future. Comput. Chem. Eng. 23(4), 667–682 (1999)
Nijmeijer, H., Fossen, T.I.: New Directions in Nonlinear Observer Design, vol. 244. Springer, Berlin (1999)
Pertew, A.M., Marquez, H.J., Zhao, Q.: H infinity observer design for Lipschitz nonlinear systems. IEEE Trans. Autom. Control 51(7), 1211–1216 (2006)
Phan, M.Q., Shi, Y., Betti, R., Longman, R.W.: Discrete-time bilinear representation of continuous-time bilinear state-space models. Adv. Astronaut. Sci. 143, 571–589 (2012)
Phanomchoeng, G., Rajamani, R.: Observer design for Lipschitz nonlinear systems using Riccati equations. In: Proceedings of the 2010 American Control Conference, pp. 6060–6065 (2010)
Raghavan, S., Hedrick, J.K.: Observer design for a class of nonlinear systems. Int. J. Control 59(2), 515–528 (1994)
Rajamani, R.: Observers for Lipschitz nonlinear systems. IEEE Trans. Autom. Control 43(3), 397–401 (1998)
Rao, C.V., Rawlings, J.B., Lee, J.H.: Constrained linear state estimation-a moving horizon approach. Automatica 37(10), 1619–1628 (2001)
Rao, C.V., Rawlings, J.B., Mayne, D.Q.: Constrained state estimation for nonlinear discrete-time systems: stability and moving horizon approximations. IEEE Trans. Autom. Control 48(2), 246–258 (2003)
Respondek, W.: Introduction to Geometric Nonlinear Control; Linearization, Observability, Decoupling. Lectures Given at the Summer School on Mathematical Control Theory. ICTP, Trieste (2001)
Rugh, W.J.: Nonlinear System Theory: The Volterra/Wiener Approach. Johns Hopkins University Press, Baltimore (1981)
Surana, A., Williams, M.O., Morari, M., Banaszuk, A.: Koopman operator framework for constrained state estimation. In: IEEE 56th Conference on Decision and Control, Melbourne, pp. 94–101 (2017)
Surana, A.: Koopman operator framework for observer synthesis for input-output nonlinear systems with control-affine inputs. In: IEEE 55th Conference on Decision and Control (2016)
Surana, A., Banaszuk, A.: Linear observer synthesis for nonlinear systems using Koopman Operator framework. IFAC-PapersOnLine 49(18), 716–723 (2016)
Tenny, M.J., Rawlings, J.B.: Efficient moving horizon estimation and nonlinear model predictive control. In: American Control Conference, pp. 4475–4480. Anchorage (2002)
Acknowledgements
The funding provided by UTRC is greatly appreciated.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 United Technologies Research Center
About this chapter
Cite this chapter
Surana, A. (2020). Koopman Framework for Nonlinear Estimation. In: Mauroy, A., Mezić, I., Susuki, Y. (eds) The Koopman Operator in Systems and Control. Lecture Notes in Control and Information Sciences, vol 484. Springer, Cham. https://doi.org/10.1007/978-3-030-35713-9_3
Download citation
DOI: https://doi.org/10.1007/978-3-030-35713-9_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-35712-2
Online ISBN: 978-3-030-35713-9
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)