Models that couple a partial differential equation with an ordinary differential equation arise naturally in process systems, where a transport mechanism distributed in space interacts with a lumped actuator, a sensor or a well mixed unit. In many of these systems the reaction and coupling coefficients are not known exactly, and the closed loop has to remain stable despite this uncertainty of the parameters.
The paper studies a class of uncertain reaction diffusion equations that are coupled in both directions to finite dimensional dynamics, through a spatial average and through an injection term that is uniform in space. The uncertainty is modelled by a random vector with known distribution that does not change in time, and it is propagated through a polynomial chaos expansion (PCE). This yields a deterministic surrogate model that preserves the structure of the interconnection.
For this surrogate, constant deterministic feedback gains are designed that act through a direct input to the ordinary differential equation and through a Neumann boundary input proportional to the spatial average of the partial differential equation. An energy method and an inequality of the Poincaré type lead to tractable conditions in the form of linear matrix inequalities. These conditions ensure exponential stability of the surrogate and, under an explicit condition on the truncation, mean square exponential stability of the original stochastic system.
In order to accommodate coupling that is nonlinear or only partially known, a ridge regression surrogate in feature space is introduced, and it is quantified how the approximation error enlarges the coupling bound in the stability conditions. An extension to output feedback based on an observer is established with a composite Lyapunov argument. Finally, a numerical range certificate for the regression surrogate diagnoses non-normal transient behaviour and, when the numerical abscissa is negative, certifies a sharper local rate of contraction.
The study was carried out at the Institute of Control Systems at Hamburg University of Technology within the Collaborative Research Centre CRC 1615 (SMART Reactors).
Srimanta Santra (2026). Polynomial chaos expansion and feature-space regression surrogates for stabilization of uncertain coupled PDE-ODE reactor processes. Journal of Process Control 164, 103769.
