Model predictive control for a flexible distributed parameter system

Applied optimization and control covers a broad range of mathematical methods, in particular, those that have connection with applications. Core topics include the interplay between control and optimization, and numerical techniques for their underlying linear and nonlinear systems.  Of great interest are topics, like: numerical optimization, model predictive control, robust optimization, partial differential equations and vibrational analysis. Special interest lies on the application side of any technique. The performance of any proposed technique is to be demonstrated by solving an application problem. As an application a distributed parameter system of the stacker crane (STC) is considered. A laboratory setup with a real-time computer and a rapid prototyping tool are provided to simplify any application and gives an additional level of freedom.

Due to the weight reduction of construction systems, like the case with STC rigidity loss leads to flexibility in structure. As a result, structural vibrations within the systems are induced already for low accelerations. Therefore exact positioning can only be achieved after a specific settling time, which counteracts any fast maneuver requirements. The main control objective of the STC is to avoid vibrations as well as to ensure a corresponding robustness. To solve the problems that arise, appropriate methods are developed in the following areas:

1. Nonlinear Model Predictive Control

Model Predictive Control (MPC) is a modern and powerful framework for linear and nonlinear control, where physical constraints on both states and control are explicitly considered. Accordingly, the state of the system is predicted and then optimized over a future horizon. Complex and highly dynamical systems, e.g. mechatronic applications pose a great challenge for real-time applications. Both methodological and numerical, especially real-time capable methods are still ongoing development for addressing the vibrational issues in flexible bodies, with a focus on the practical feasibility.

2. Robust Optimization

In Model Predictive Control (MPC) constraints satisfaction is essential, especially when they are physical, which may be safety-critical. However non-ideal world contains always uncertainty, which makes constraints satisfaction a challenging task. Appropriate approaches introduce a little conservatism but leads to robust controllers that work very well in practice.

3. Partial Differential Equation Constrained Optimization

 Distributed-parameter systems are characterized by the states, which are elements of an infinite-dimensional state-time space. New methods for optimal feedforward and feedback control are strived in combination with model predictive control. Important application areas range from flexible structures till heat propagation.

4. Vibrational Analysis

Typically, dealing with vibrational damping control requires a specification of the frequencies, which need to be controlled or avoided. To study the dynamical properties of a flexible body in frequency domain the powerful approach of modal analysis is used. One pre-condition is a partial differential equation model description. Dynamical structure analyses which study the behavior of a flexible structure under both dynamical load and input is still an ongoing topic.