Title: Sensitivity analysis and goal oriented error estimation for Model Predictive Control of PDEs
Abstract: Every step of a Model Predictive Controller includes the solution of an optimal control problem on a possibly long horizon. However, only an initial part of the optimal control is considered as a feedback for the system under control.
In this talk, we will discuss how specialized goal oriented a posteriori grid refinement techniques can be used to numerically exploit the fact that only an initial part of the control enters the MPC-feedback. In particular, we present a framework for sensitivity analysis of a wide class of optimal control problems which on the one hand allows for keeping numerical effort low when using the specialized refinement technique, and on the other hand can be used to derive a quantitative turnpike property, an important stability property of optimal control problems.
We accompany all results by means of numerical examples of linear and nonlinear parabolic PDEs with distributed or boundary control.
Many systems in areas such as electrical engineering, mechanics, or thermodynamics
can be modeled as port-Hamiltonian systems. A linear port-Hamiltonian system has the
where J encompasses energy conservation and R describes the energy loss (dissipation)
of the system. The matrix Q determines the energy potential. For example, in models
of electrical circuits, Kirchho’s laws are modeled in the matrix J, whereas the effect
of, e.g., resistors is reflected in the matrix R. In this case, the control u often stands for the voltage applied to the circuit and y is the measured current. The product u>y thus can be interpreted as the power supply rate and the integral as the energy supplied to the system.
In this talk we consider the optimal control problem (OCP) of driving a given initial state to a desired region in state space with minimal energy supply. This OCP is obviously singular, which means that standard techniques such as Riccati theory cannot be applied to study the behavior of optimal solutions. However, we are still able to prove that optimal trajectories exist and reside close to a certain subspace for the majority of the time. This behavior of optimal solutions resembles the classical turnpike phenomenon.
We also prove that optimal controls are uniquely determined by the optimal state and the adjoint state.
This is joint work with Manuel Schaller (TU Ilmenau), Timm Faulwasser (TU Dortmund),
Bernhard Maschke (Univ. CB Lyon), and Karl Worthmann (TU Ilmenau).