Closed-form expressions of convex combinations

In this paper, the computation of closed-form convex combinations is considered. In many control tasks, convex combinations play a crucial role, thus requiring an efficient computation. This is the case for online control of fast dynamical systems, in which the control algorithms rely on convex combinations, for example robust control of linear parameter-varying systems. On the other hand, for formal verification, it is necessary that the closed-loop behavior can be expressed in closed-form, which is not possible if the convex combination is expressed as constraints. In this paper, we provide closed-form expressions for any kind of polytope with finitely many extreme points. For special types of polytopes, such as simplices and parallelotopes, we provide especially efficient, analytical closed-form expressions. Numerical experiments show that the closed-form expressions are significantly faster in all randomly-generated cases and thus enable shorter sampling intervals of control schemes involving convex combinations.