General Stabilizing Formulations for Trivial Parametrization

Abstract

In the preceding chapters, several stabilizing receding-horizon formulations are proposed that involve quite general control parametrization. In the classical literature on the subject however, only the trivial piece-wise constant parametrization is studied. Recall that for systems with scalar control input u ∈ ℝ, this parametrization amounts to use the following parametrization map (see Chapter 1 for the notations):

$$ u^i (p) = p_i \in \mathbb{R}. $$

More generally, for m dimensional control input u ∈ ℝ^m, the parametrization map is defined by:

$$ u^i (p) = \left( {\begin{array}{*{20}c} {p_{1i} } \\ \vdots \\ {p_{mi} } \\ \end{array} } \right), $$

where p ∈ ℝ^m·N is the matrix of parameters. This is clearly the most commonly used piece-wise constant control parametrization. On the other hand, it is the parametrization of maximal dimension.