Characterising and approximating the boundary of the admissible set

• Applicant: Professor Dr.-Ing. Stefan Streif
• Subject Area: Automation, Mechatronics, Control Systems, Intelligent Technical Systems, Robotics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 531896505

The admissible set of a constrained control system, also known as the viability kernel, is the set of initial conditions for which there exists a control such that the constraints are satisfied for all future time. This set has various applications across many scientific fields, including the sustainable management of resources, epidemics, power systems and robotics. Additionally, it plays an important role in stability and recursive feasibility studies of model predictive control (MPC), a popular and successful control methodology. Throughout this project, we aim to exploit the minimum-like principle observed in special system trajectories that make up parts of the admissible set’s boundary, which has been shown in the in the so-called theory of barriers. In the first work package, we extend the theory of barriers to the setting of differential games and study the robust admissible set. In this setting a second input, often modelling an exogenous disturbance, appears in the problem formulation. We will also establish further theoretical foundations that connect the admissible set, optimal control and robust control. The second work package will focus on constructing the admissible set for higher dimensional systems. Since the theory of barriers calculates the lower dimensional boundary of the set, we suspect that our developed approaches will compare favourably with other algorithms. Moreover, we will explore the accurate representation of the set’s boundary as a point cloud for further usage in control algorithms. In the third work package, we will introduce a type of “sticky” invariance called the extended barrier. If the state is located on this part, all possible control functions either keep the state on the extended barrier for all time, or result in a constraint violation in finite time. Entering the interior of the admissible set without violating the constraints is then no longer possible. This part of the set’s boundary, where the state can become “stuck”, should be avoided. We will characterize this previously unknown part of the admissible set’s boundary via a study of appropriate value functions In the fourth work package, we will investigate the use of the admissible set in control methods. We will propose a safety filter-like approach that switches to the barrier control law obtained via the theory of barriers in close proximity or even directly on the admissible set’s boundary. We will study the effects of this algorithm in a switching MPC formulation, where the knowledge gained regarding the extended barrier will be used to guarantee constraint satisfaction and explore the observed “blow up” of the required stabilizing optimization horizon length as the admissible set’s boundary is approached. Finally, in the firth working package, we will apply the results obtained from the previous packages to epidemics, wheeled robotics and power systems.

• DFG Programme: Research Grants
• International Connection: France
• Cooperation Partner: Professor Dr. Jean Lévine