(295h) Data-Driven State Observation Via Online Optimization of Chen-Fliess Series

Chemical processes are significantly nonlinear when they are operated beyond a small neighborhood of a single steady state. For nonlinear systems, the optimal control, geometric control, and recently emerging data-driven control strategies typically use a state-space representation and design the control law based on state information. This necessitates the synthesis of a state observer, which estimates the value of states in real time based on the trajectories of measurable variables. A generic form of state observer for nonlinear systems is the Kazantzis-Kravaris/Luenberger (KKL) observer, proposed by the seminal work of Kazantzis and Kravaris [1]. The conditions for its existence and extension from autonomous to controlled systems have been discussed in the literature [2].

However, the direct solution of a KKL observer requires the solution of a PDE system to find a transformation of states such that the immersed dynamics is linear and driven by the outputs. This is analytically infeasible in general and numerically difficult under high dimensions. Hence, the attention of recent research has largely been on approximating the (inverse) immersion mapping as neural networks and training them with process data (e.g., [3, 4]). Yet, the neural network training can require massive data and the problem itself is nonconvex, subject to the effect of initialization and local optima. The neural network solutions also do not provide a guaranteed performance.

This work proposes a computationally efficient and performance-provable approach for the data-driven state observation of nonlinear systems. Specifically,

1. The observer, as a map from the plant outputs to the state estimates, can be expanded on every small time interval as a Chen-Fliess series. Each term of the series contains a recursive integral of the output signal, multiplied by a coefficient, which is the corresponding Lie derivative evaluated at the initial time [5, Section 3.1]. Hence, on this interval, the Chen-Fliess series’ coefficients can be obtained through linear regression.
2. Due to the local convergence property of the Chen-Fliess series, as time proceeds, the regression should be recursive. In continuous time, this work proposes to impose a gradient flow on the least-squares estimate, thus forming an online optimization algorithm [6] for Chen-Fliess series regression. If the true states are hypothetically measurable, the resulting dynamic regret, namely the integrated squares of observation error, is proved to be finite L₂-gain stable with respect to the total variation of true states.
3. In realistic cases where the true state trajectories are not available, the immersed linear dynamics of the KKL observer is first assigned so that the transformed states can be computed. The online optimization of Chen-Fliess series is then conducted on the transformed state space, and dimensionality reduction is adopted to recover the estimates of the equivalents of the states (in the sense of a diffeomorphism).

The proposed approach is demonstrated by applications to a benchmark nonlinear process. It is worth noting that the proposed approach intrinsically does not use the model information (i.e., governing equations) and is thus model-free. Therefore, it is promising for combination with model-free control strategies (e.g., reinforcement learning and dissipativity learning control) [7].

References

[1] Kazantzis, N., & Kravaris, C. (1998). Systems & Control Letters, 34, 241-247.

[2] Bernard, P., Andrieu, V., & Astolfi, D. (2022). Annual Reviews in Control, 53, 224-248.

[3] Ramos, L. D. C., et al. (2020, December). IEEE Conference on Decision and Control (pp. 5435-5442).

[4] Niazi, M. U. B., et al. (2022). arXiv:2210.01476.

[5] Isidori, A. (1995). Nonlinear control systems. Springer.

[6] Hazan, E. (2016). Foundations and Trends in Optimization, 2, 157-325.

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