Model Predictive Control for the Fokker--Planck Equation

Abstract

This work contributes to a better understanding of Model Predictive Control (MPC) in the context of the Fokker – Planck equation. The Fokker – Planck equation is a partial differential equation (PDE) that describes the evolution of a probability density function in time. One possible application is the (optimal) control of stochastic processes described by stochastic differential equations (SDEs). Here, a macroscopic perspective is taken and instead of, e.g., individual particles (described by the SDE), all particles are controlled in terms of their density (described by the Fokker – Planck equation). This results in a PDE-constrained optimal control problem. Model Predictive Control is an established and widely used technique in industry and academia to (approximately) solve optimal control problems. The idea of the receding is easy to understand, the implementation is simple, and above all: MPC works very often in practice. The challenge, however, is to specify conditions under which this can be guaranteed or to verify these conditions for concrete systems. In this thesis it is analyzed in detail under which conditions the MPC closed loop is provably (practically) asymptotically stable, i.e., under which conditions it converges to the desired target or to a neighborhood thereof. For this purpose we first introduce the Fokker – Planck framework and show the existence of optimal space- and time-dependent controls under (weak) regularity assumptions. Subsequently, we consider both the case of stabilizing MPC and economic MPC and include both space-independent and space-dependent control functions in our analysis. In the case of stabilizing MPC, we show asymptotic stability of the MPC closed loop for a class of linear stochastic processes if the prediction horizon N is long enough. Moreover, we specify the minimal stabilizing horizon for specific stochastic processes. In the course of the analysis difficulties of the used L^2 cost function come to light and the question arises whether other cost functions allow an easier analysis. In the case of the economic MPC, we thus fix a specific stochastic process but consider different cost functions instead. Here, the crucial system property for the effective use of the MPC controller is strict dissipativity. This property is verified for different cost functions, where the main challenge is to find a suitable storage function. It turns out that for the commonly used L^2 cost it is much more difficult to find such a storage function than for another cost function we propose. Details of the numerical implementation with additional simulations and further research questions conclude the work.%%%%Diese Arbeit tragt dazu bei, Modellpradiktive Regelung (MPC) im Zusammenhang mit der Fokker – Planck Gleichung besser zu verstehen. Die Fokker – Planck Gleichung ist eine partielle Differentialgleichung (PDE), die die zeitliche Entwicklung einer Wahrscheinlichkeitsdichtefunktion beschreibt. Eine mogliche Anwendung ist die (optimale)…