Model Predictive Control
Model Predictive Control
Model Predictive Control
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3.5 Operations on P-collections 43<br />
the original ones, i.e. ( q i=1 Qi) = ( p i=1 Pi), (ii) q is minimal, i.e. there exists<br />
no set {Qi}i=1,...,q with a smaller number of polyhedra, (iii) the new polyhedra are<br />
mutually disjoint, i.e. Qi = Qj for all i, j ∈ {1, . . .,q}, i = j, and (iv) the new<br />
polyhedra are formed as unions of the old ones, i.e. for each Qj, j ∈ {1, . . .,q},<br />
there exists an index set I ⊆ {1, . . .,p}, such that Qj = <br />
i∈I Pi.<br />
This problem is equivalent to an optimal merging problem. Next, we remove Requirements<br />
(iii) and (iv) thus allowing for overlaps in the resulting polyhedra.<br />
Problem 3.2 (Non-Disjoint Optimal Complexity Reduction) Given an initial<br />
set of polyhedra {Pi}i=1,...,p satisfying Assumption 3.1, the non-disjoint optimal<br />
complexity reduction problem amounts to derive a new set of polyhedra {Qi}i=1,...,q<br />
with Properties (i) and (ii) as in Problem 3.1.<br />
Strictly speaking, the second problem is not a merging problem, but a more general<br />
optimal set covering problem, which is equivalent to logic minimization frequently<br />
used in digital circuit design. Nevertheless, we will sometimes use the term merging<br />
instead of complexity reduction.<br />
Next, the assumption that the original polyhedra are cells in a hyperplane<br />
arrangement and that markings are available are removed, but require additionally<br />
that each polyhedron is represented with a minimal number of facets. This problem<br />
can be considered as the general non-disjoint optimal complexity reduction problem<br />
for PWA functions.<br />
Problem 3.3 (General Non-Disjoint Optimal Complexity Reduction) Given<br />
an initial set of polyhedra {Pi}i=1,...,p where Assumption 3.1 is not required to hold,<br />
the general non-disjoint optimal complexity reduction problem amounts to derive<br />
a new set of polyhedra {Qi}i=1,...,q with Properties (i) and (ii) as in Problem 3.1,<br />
and (iii) the number of facets for each Qi being minimal.<br />
All three tasks are non-trivial, as the union of polyhedra is in general nonconvex,<br />
and because we are aiming at deriving the optimal solution, more specifically,<br />
the set of polyhedra with the minimal cardinality. Indeed, the problems<br />
are N P-hard (see [73] and references therein). As a direct consequence, fast algorithms<br />
are unlikely to exist leaving us either with rather long computation times<br />
or suboptimal solutions.<br />
Our interest in this problem will be clear later in this book (Section 10.2) when<br />
computing the PWA state-feedback control law to optimal control problems. Once<br />
the PWA state-feedback control law has been derived, the memory requirement and<br />
the on-line computation time are linear in the number of polyhedra of the feedback<br />
law when using standard brute force search. Therefore, we will be interested in the<br />
problem of finding a minimal representation of piecewise affine (PWA) systems,<br />
or more specifically, for a given PWA system, we solve the problem of deriving a<br />
PWA system, that is both equivalent to the former and minimal in the number of<br />
regions. This is done by associating with different feedback law a different color,<br />
and we collect the polyhedra with the same color. Then, for a given color, we<br />
try to merge the corresponding P-collection by solving one of the three problem<br />
described above. If the number of polyhedra with the same affine dynamic is large,