(576e) Data-Driven Nonlinear State Observation with Lipschitz-Bounded Neural Networks

Chemical processes typically have nonlinear dynamics based on transport and reaction kinetics. For the control and monitoring of such nonlinear processes, state-space representations are commonly used for describing their dynamics. To estimate the states in real time, a state observer, namely an auxiliary dynamic system that takes the plant inputs and outputs as its own inputs and returns an output signal as the approximations of the states, should be used [1]. While the recent advances in nonlinear control theory have largely focused on the development of data-driven and model-free nonlinear control methods, it should be noted that many such methods, such as reinforcement learning-based and Koopman operator-based ones, in fact assume that the state information is always available [2]. In view of this status quo, the development of nonlinear state observation methods should be deemed as an imperative problem for process control in the near future.

In the seminal work of Kazantzis and Kravaris [3], a general form of nonlinear state observer was proposed as an extension of Luenberger observer for linear systems. Specifically, the observer can be designed to have a linear time-invariant controllable dynamics and a nonlinear static output map. The existence of such an observer, called Kazantzis-Kravaris/Luenberger (KKL) observer, was proved by Andrieu and Praly [4] under mild condition on the dynamics. The synthesis of KKL observer ultimately depends on the solution of a nonlinear first-order partial differential equation system, which may be difficult even when the dynamics is known. To the end of data-driven control, this work considers the problem of designing KKL observers in a data-driven manner. In particular, the training of the nonlinear static output map in the KKL observer as a neural network. This approach was considered in a few recent works, e.g., [5].

In this work, it is argued that the Lipschitz constant of the neural network trained in the KKL observer has a significant impact on the generalization loss, in the sense of an average squared state observation error. The H₂ norm of the linear time-variant dynamics also affects such a generalization loss, indicating the necessity of fine-tuning the observer gain. This is proved through a routine of statistical learning theory under a typical white noise assumption on the plant outputs, as well as by a numerical case study on the Lorenz chaotic system. In fact, the conclusion that Lipschitz constant of neural networks is associated with their sensitivity to data noises, perturbations, or attacks, is well known in the machine learning community [6].

Thus, to keep the Lipschitz constant of the neural network below an a priori given bound, the direct parameterization approach of Wang and Manchester [7] is adopted to endow a well-curated “sandwich structure” to all the comprising layers of the neural network. As a result, the neural network training problem is still in the form of unconstrained optimization, which can be efficiently implemented in PyTorch. Thus, by varying the Lipschitz bound, an optimal observer design can be approached. The case study on the Lorenz system shows satisfactory observation performance on most of the locations, yet with spurious behavior during the transition between two foils of the strange attractor. Such a behavior of Lipschitz-bounded neural network-based observer is analyzed and compared to the recent convex online optimization-based state observation approach proposed by the author [8].

References

[1] Kravaris, C., Hahn, J., & Chu, Y. (2013). Comput. Chem. Eng., 51, 111-123.

[2] Tang, W., & Daoutidis, P. (2022, June). In 2022 American Control Conference (ACC) (pp. 1048-1064).

[3] Kazantzis, N., & Kravaris, C. (1998). Syst. Control Lett., 34(5), 241-247.

[4] Andrieu, V., & Praly, L. (2006). SIAM J. Control Optim., 45(2), 432-456.

[5] Niazi, M. U. B., et al. (2023). In 2023 American Control Conference (ACC) (pp. 3048-3055).

[6] Huang, Y., et al. (2021). Advances in Neural Information Processing Systems, 34, 22745-22757.

[7] Wang, R., & Manchester, I. (2023). In International Conference on Machine Learning (pp. 36093-36110).

[8] Tang, W. (2023). AIChE J., 69(12), e18224.