Efficient calculation of explicit model predictive control laws using topological equivalence classes of critical points

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Model predictive control (MPC) methods are established. A more recent variant known as explicit MPC (EMPC) differs from other MPC approaches in that no numerical optimization is required at runtime of the controller. Because no iterative algorithms are required at runtime, EMPC is suitable for systems with very high sampling frequencies and strict realtime requirements. EMPC can only be applied, however, to low order linear systems with few constraints and short prediction horizons. Both, the calculation of EMPC control laws, and the fast evaluation of these laws, are currently investigated by researchers worldwide. Today the first step, the actual calculation of the explicit laws, is the bottleneck, after methods for the fast evaluation have recently made progress.It is the goal of the project to devise a new method (the vertex-method for short) for the calculation of explicit control laws for linear systems with linear input, state and output constraints. Three aspects of this goal can be detailed as follows: (1) Development of a robust method for the calculation of complex explicit control laws: The vertex-method is computational demanding. It shares this trait with existing approaches. The calculations required in the proposed method are, however, simpler and can be carried out more reliably than for the existing methods. Assuming the proposed project can be carried out, the vertex-method will be computational demanding but can be applied to more demanding MPC problems than current methods due to its simplicity and reliability. Moreover, the vertex-method provides the explicit control law in a form which is particularly suitable for the recently proposed approaches to the fast evaluation of EMPC laws. (2) Foundations of a generalizable description of explicit laws: While the first period of the project exclusively deals with linear systems, the description of vertices with normal forms and equivalence classes is of long-term interest, because it is in principle suitable to describe the structure of explicit control laws of nonlinear MPC problems. The vertex-method is based on central ideas of the singularity theory of bifurcations, which is a nonlinear theory by origin. (3) Acceleration of online MPC by combining explicit and online MPC: A part of the project investigates a new approach to exploiting the structure of an explicit control law in online MPC. The actual explicit control law is not used at the runtime of the controller, but its structure is used to simplify the online MPC problem. The central ideas is not to store information about which constraints are active for every x(t), but about which constraints are inactive. The latter information can be stored with polytopes. These polytopes are not convex in general, but the number of these polytopes grows only linearly in the size of the control problem, in contrast to the exponential growth of computational effort and storage in EMPC.