(470d) Lie-Sobolev Nonlinear System Identification and Its Effect on Nonlinear Control

The intrinsic nonlinearity of process systems has motivated the development of nonlinear control strategies such as input—output linearizing control and model predictive control (MPC). The implementation of nonlinear control is usually heavily dependent on the accuracy of nonlinear dynamic models, and since white-box first-principle models are rarely available or directly applicable, the importance of nonlinear system identification methods is self-evident. For chemical processes, system identification usually refers to the estimation of unknown parameters in a structured dynamic model, which should be performed in combination with the state observation [1, 2]. However, the design of such an observer-estimator is challenging for general nonlinear systems due to the existence of parametric as well as structural complexities in the true underlying dynamics.

In this work, we note that the Lie derivatives constitute essential control-relevant information that is of decisive importance to the performance of nonlinear control. For input—output linearizing control, Lie derivatives are used directly in the construction of the feedback control law, while for MPC, Lie derivatives indirectly affect the prediction of future trajectories in the receding horizon through the Chen-Fliess expansion [3]. Hence the identification scheme should be designed in such a way that the identified model and the true dynamics should match not only in the direct input and output measurements, i.e., in an L²-sense, but also in the higher-order Lie derivatives, i.e., in a Sobolev sense. This argument is first motivated with a small example of nonlinear regression. Then a theoretic analysis of Luenberger observer for linear systems shows that by modifying the observer with a Lie-Sobolev criterion, the rate of convergence of observation errors to zero is increased in the absence of structural model mismatch, and in the presence of structural deviations, the ultimate bound of observation errors can be reduced.

We then formulate two different forms of the Lie-Sobolev observer-estimator. One is based on the gradient descent of a criterion function accounting for the instantaneous discrepancies between the identified model and the measured input—output trajectories, and the other is a Lie derivative-modified moving horizon estimator (MHE) that accounts for the discrepancies in the past horizon. Their convergence properties are established, and the effects of these Lie-Sobolev identification schemes on the input–output linearizing control and model predictive control (MPC) respectively are studied, based on an extension of the results for non-Lie-Sobolev estimation and control [4, 5]. Advantages of Lie-Sobolev system identification of nonlinear processes are demonstrated by two numerical examples and a glycerol etherification reactor process [6] with complex dynamics.

References

[1] Soroush, M. (1998). State and parameter estimations and their applications in process control. Computers & Chemical Engineering, 23, 229—245.

[2] Kravaris, C., et al. (2013). Advances and selected recent developments in state and parameter estimation, Computers & Chemical Engineering, 51, 111—123.

[3] Isidori, A. (1995). Nonlinear Control Systems: An Introduction. Springer.

[4] Müller, M. A. (2017). Nonlinear moving horizon estimation in the presence of bounded disturbances. Automatica, 79, 306—314.

[5] Mayne, D. Q., & Falugi, P. (2019). Stabilizing conditions for model predictive control. International Journal of Robust Nonlinear Control, 29, 894—903.

[6] Liu, J., Daoutidis, P., & Yang, B. (2016). Process design and optimization for etherification of glycerol with isobutene. Chemical Engineering Science, 144, 326—335.