In the author’s previous works [6, 7], dissipativity learning control (DLC) was proposed as a data-driven control framework, where the dissipativity property of unknown nonlinear dynamics is learned from input-output trajectories. In these works, the supply rate to be estimated from data is parameterized as a quadratic form, so that the matrix defining the quadratic form is to be learned. Naturally, quadratic forms are suitable for systems that are linear or approximately linear, which restricts their applicability to processes that exhibit severe nonlinearity. In addition, the representation of system-wide behavior by the collected trajectories often rely on restrictive assumptions and the complexity of sampling input-output trajectories is prohibitive.
Therefore, in this work, we formulate dissipativity learning in a nonparametric framework in reproducing kernel Hilbert spaces (RKHS). Specifically, (1) By exciting the system, we obtain an ergodic trajectory covering the region of interest on the state space, on which the probability distributions of inputs and states are stationary and hence the dissipativity learning can be carried out. (2) The information of inputs, outputs, and states is extracted from the trajectory and expressed in terms of their canonical features corresponding to the kernel functions. Such canonical features maps the inputs, outputs, and states nonlinearly into corresponding RKHSs, thereby accounting for the nonlinearity of the dynamics. (3) Thus, the storage and supply rate functions to be learned are represented by Hilbert-Schmidt (H-S) operators defining quadratic forms on the canonical features. This generalizes the previous works [6, 7] where quadratic forms of dissipativity (QSR-dissipativity) were used and the matrices (as finite-rank operators) parameterizing these quadratic forms are learned.
The dissipativity learning problem is hence formulated as an operator convex optimization problem with constraints that enforce dissipativity and stability properties (e.g., the L2 gain). Formally, by extending inner product and norm concepts from matrices to operators, the learning of H-S operators preserves the formulation from finite-dimensional spaces. Due to the representation theorem, our formulation is reducible to a tractable, finite-dimensional problem, despite the infinite-dimensional nature. Therefore, the proposed approach enables flexible learning of dissipative properties in general nonlinear forms, without requiring explicit knowledge of the system dynamics.
For a theoretical analysis, we adopt statistical learning theory to establish bounds on generalized performance, namely a confidence lower bound on the dissipation rate and a confidence upper bound on the guaranteed L2 gain. This work exemplifies the use of operators and kernel tricks in RKHS as generic tools for learning the system behaviors. Numerical studies are performed on two chemical processes, including a binary distillation column and a Williams-Otto reactor.
References
[1] Brogliato, B., et al. (2007). Dissipative systems analysis and control: Theory and applications. Springer.
[2] Sontag, E. D. (1998). Mathematical control theory. Springer.
[3] Alonso, A. A., & Ydstie, B. E. (2001). Stabilization of distributed systems using irreversible thermodynamics. Automatica, 37(11), 1739-1755.
[4] Bao, J., & Lee, P. L. (2007). Process control: the passive systems approach. Springer.
[5] Yan, Y., Bao, J., & Huang, B. (2023). Distributed data-driven predictive control via dissipative behavior synthesis. IEEE Trans. Autom. Control, 69(5), 2899-2914.
[6] Tang, W., & Daoutidis, P. (2019). Dissipativity learning control (DLC): A framework of input–output data-driven control. Comput. Chem. Eng., 130, 106576.
[7] Tang, W., & Daoutidis, P. (2021). Dissipativity learning control (DLC): theoretical foundations of input–output data-driven model-free control. Syst. Control Lett., 147, 104831.