(312g) Learning to Observe, Learning to Analyze, and Learning to Control

The analysis of nonlinear dynamics and nonlinear control of process systems, based on their models built on the first principles of thermodynamics, fluid flow, and transport and reaction kinetics, has been a main theme of study in the process systems engineering community for decades. The contributions of Daoutidis (and his coworkers) are spread in many topics, including input-output linearizing control, differential-algebraic equations, partial differential equation systems, multi-time-scale systems, nonlinear model predictive control, etc. [1, 2, 3, 4, 5]. In 2018, noting the trend that machine learning is percolating into the control theory community, we started to investigate systematically the idea of data-driven control, and specifically model-free control. The aim is to control nonlinear systems without needing to identify a nonlinear model first; instead, machine learning is employed as a workforce only to obtain control-relevant properties (or information) from process data.

Nonlinear dynamics are typically described in a state-space language. For the purpose of achieving optimal control performance, handling zero dynamics, or just monitoring the process, states – variables that can fully describe the nonlinear dynamics, whether they are physical quantities or artifacts – should be estimated by a state observer. In the model-based setting, the existence of a Kazantzis-Kravaris/Luenberger (KKL) observer [6] (which has a “linear time-invariant dynamics + nonlinear output mapping” structure) is well-known. In a data-driven context, our recent works have proposed neural, online learning, and kernel-based solutions to the nonlinear state observation problem [7, 8, 9]. Using machine learning theory, probabilistic bounds are established on the observation performance.

Nonlinear dynamics are characteristic of their complex phenomena such as multi-steady-state, bifurcation, and chaos. Detecting these behaviors is of interest to depicting the nonlinearities of process systems as well as recognizing the limitation of nonlinear control schemes. Focusing on local bifurcation (around a given steady state), we proposed a data-driven method to analyze the loss of stability under parameter variations in a recent work [10]. The resulting problem is to learn a homeomorphism (continuously invertible mapping) between systems under different parameters, which is formulated as a convex optimization problem. A probabilistic bound on the prediction error was established.

Nonlinear dynamics typically have dissipative properties that can be leveraged for control, as revealed in the works of Ydstie and his coworkers [11]. For model-free control, we proposed that it suffices to learn the dissipativity from data [12, 13]. Essentially, the data collected from the process, when properly re-organized into a form called dual dissipativity parameters, forms a “duality relation” with the dissipative properties to be learned, and thus by characterizing the range of such dual dissipativity parameters (which we name as dual dissipativity set), the range of dissipative properties (which we call dissipativity set) can be estimated explicitly. Subsequently, based on the learned dissipativity set, the optimal output-feedback controller can be found. Specially in this talk, I will present a unifying framework of dissipativity learning control and the learning theory underlying it.

Clearly, data-driven control is an open problem that can be solved in a multitude of ways, and it is arguable whether there exists an “optimal” approach [14, 15]. As the process control practitioners keep pursuing higher goals of digitalized plants, process autonomy, open-source implementation, and more agile and flexible workflow, it should be believed that data-driven deserves more in-depth studies.

References

[1] Daoutidis, P., Soroush, M., & Kravaris, C. (1990). AIChE J., 36(10), 1471-1484.

[2] Christofides, P. D., & Daoutidis, P. (1996). AIChE J., 42(11), 3063-3086.

[3] Kumar, A., & Daoutidis, P. (1999). Control of nonlinear differential-algebraic equation systems with applications to chemical processes. CRC.

[4] Baldea, M., & Daoutidis, P. (2012). Dynamics and nonlinear control of integrated process systems. Cambridge University Press.

[5] Tang, W., Allman, A., Pourkargar, D. B., & Daoutidis, P. (2018). Comput. Chem. Eng., 111, 43-54.

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

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

[8] Tang, W. (2023). arXiv:2310.03187. (To appear in 2024 American Control Conference.)

[9] Weeks, C., & Tang, W. (2023). arXiv:2311.14895. (To appear in 2024 ADCHEM.)

[10] Tang, W. (2023). arXiv:2312.06634. (To appear in 6^th Annual Learning for Dynamics and Control Conference.)

[11] Alonso, A. A., & Ydstie, B. E. (2001). Automatica, 37(11), 1739-1755.

[12] Tang, W., & Daoutidis, P. (2019). Comput. Chem. Eng., 130, 106576.

[13] Tang, W., & Daoutidis, P. (2021). Syst. Control Lett., 147, 104831.

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

[15] Daoutidis, P., Megan, L., & Tang, W. (2023). Comput. Chem. Eng., 178, 108365.