(132d) Dissipativity Learning Control: Statistical and Control Theoretic Foundations

Erik Ydstie, among his numerous contributions to process control, has highlighted the potential of using irreversible thermodynamics and dissipativity/passivity analysis to design output-feedback P/PI/PID control laws for nonlinear systems (see e.g., [1]). To this end, the main challenge lies in determining the salient thermodynamic functions (supply rate, storage) from a process model. However, dissipativity properties can also be “learned” from data (see e.g. [2]). Motivated by this, we have recently extended such a dissipativity-based control framework into a data-driven model-free setting, called dissipativity learning control (DLC) [3].

In DLC, the dissipativity property, represented by a parameterization of the supply rate function, is learned from data in the form of input–output trajectories under excitations. Specifically, a convex range of the dissipativity parameters (called the dissipativity set) is first estimated from the collected trajectories by means of statistical inference. Then such a dissipativity set is incorporated into the controller synthesis, formulated as a semidefinite programming problem, the solution of which gives a controller gain matrix that is optimal in the L² sense. In this talk, we aim at providing a theoretical foundation to DLC by formalizing the statistical conditions of the sampling and inference that enable performance guarantees. Specifically, we establish that:

• If free of statistical errors, the DLC yields the optimal dissipative output-feedback control law that is in a certain nearly L²-optimal sense defined on a neighborhood of the origin;
• The errors resulting from data sampling and statistical inference of the effective estimation of the dissipativity set cause an error in the dissipativity learning result in terms of an upper bound of the L²-gain from the exogenous disturbances to the inputs and outputs;
• Under small errors in dissipativity learning, the perturbation on the resulting upper bound of L²-gain is also small, so that nearly L²-optimal control performance is still achievable.

References

[1] Ydstie, B. E. (2002). Passivity based control via the second law. Computers & Chemical Engineering, 26(7-8), 1037-1048.

[2] Romer, A., Montenbruck, J. M., & Allgöwer, F. (2017). Determining dissipation inequalities from input-output samples. IFAC-PapersOnLine, 50(1), 7789—7794.

[3] Tang, W., & Daoutidis, P. (2019). Dissipativity learning control (DLC): A framework of input–output data-driven control. Computers & Chemical Engineering, 130, 106576.