Linear parameter-varying attitude controller design for a reusable launch vehicle during reentry

Closed-form expressions of convex combinations

“…Since the controllers are not known explicitly, but obtained using a convex combination, a convex decomposition problem has to be solved online in every time step. This means that every time we want to find the combination λ i (θ) for a given parameter θ, we have to solve the system of inequalities in (1). These are n + 1 equalities and p inequalities for p unknowns.…”

“…If an LPV system acts in uncertain environments and is affected by external and internal disturbances, the control algorithm must take those uncertainties and disturbances into account. For example, reusable launch vehicles, upon reentry, as discussed in [1], have to deal with many disturbances due to wind and electromagnetic interferences, as well as aerodynamic and atmospheric parameter uncertainties.…”

“…This controller approach was proposed for various applications, including missile control [4], multi-propeller airships [5], and the previously mentioned reusable launch vehicles [1]. It was also proposed for several wind power applications, including wind energy conversion systems [6], wind turbines [7], and wind power generators [8].…”

“…In the literature, closed-form expressions are only mentioned for special cases, for example for two-dimensional, axis-aligned boxes in [2], [10], [11], [7], and [12] or even axis-aligned boxes of arbitrary dimension in [1] and [8]. However, none of these papers provide proofs of the formulas.…”

“…However, none of these papers provide proofs of the formulas. All of these sources, even [1] and [8], who provide a formula for n−dimensional, axis-aligned boxes, discuss their theory for general polytopes but are only able to provide formulas for special forms of boxes. Other papers, for example [13], circumvent the problem of obtaining closed-form expressions by simply assuming that the convex combinations can be obtained in real-time.…”