Closed-form expressions of convex combinations

Abstract

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 parametervarying 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 closedform 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 randomlygenerated cases and thus enable shorter sampling intervals of control schemes involving convex combinations.

Control Design of Dynamic Virtual Power Plants: An Adaptive Divide-and-Conquer Approach

“…For polytopic LPV systems, however, (21) can be reduced to a finite set of constraints [13], [17]. In particular, since the statespace matrices in (18) depend affinely on Θ(t), and Θ(t) varies in a polytope of vertices Θ(1) , ..., Θ(q) , the state-space matrices are constrained on the polytope with vertices…”

“…In particular, for the parameter polytopes we are considering in our applications, i.e. simplices and parallelotopes, there exist analytical closed form-expressions of the coefficients λ (l) (Θ) [18].…”

Set-based control for disturbed piecewise affine systems with state and actuation constraints

“…If faster computation times are required, then we can take advantage of the fact that parameters for expressing a state in a polytope can be computed in a closed-form expression [73]. By interpolating the α values of the extreme points of our zonotope, we can use the same technique to obtain them through a closed-form expression.…”

Formal LPV control for transient stability of power systems

“…with i and m denoting the i-th entry and the m-th dimension, respectively. The formula of the variable ν and the proof of (13) are detailed in our previous work [15].…”

“…After defining r := t k+1 −t k and u c as the center of U, we can express the reachable set X (t k+1 ) of the dynamicsẋ = A k x + u c , enclosed by the differential inclusion (15), based on the well-known solution of linear state-space equations where e A k r is the matrix exponential and R p (r) is the set which over-approximates the particular solution of the linear state-space equation. The reachable sets at the next point in time t k+1 , and for the time interval τ k , are evaluated by…”