Polyhedal Tools for Control
Jones C.N.
July 2005Abstract
Polyhedral operations play a central role in constrained control.
One of the most fundamental operations is that of projection, required both
by addition and multiplication. This thesis investigates projection and its
relation to multi-parametric linear optimisation for the types of problems
that are of particular interest to the control community.
The first part of the thesis introduces an algorithm for the projection of
polytopes in halfspace form, called Equality Set Projection (ESP). ESP has
the desirable property of output sensitivity for non-degenerate polytopes.
That is, a linear number of linear programs are needed per output facet of
the projection. It is demonstrated that ESP is particularly well suited to
control problems and comparative simulations are given, which greatly
favour ESP.
Part two is an investigation into the multi-parametric linear program
(mpLP). The mpLP has received a lot of attention in the control literature
as certain model predictive control problems can be posed as mpLPs and
thereby pre-solved, eliminating the need for online optimisation. The
structure of the solution to the mpLP is studied and an approach is
presented that eliminates degeneracy. This approach causes the control
input to be continuous, preventing chattering, which is a significant
problem in control with a linear cost. Four new enumeration methods are
presented that have benefits for various control problems and comparative
simulations demonstrate that they outperform existing codes.
The third part studies the relationship between projection and
multi-parametric linear programs. It is shown that projections can be posed
as mpLPs and mpLPs as projections, demonstrating the fundamental nature of
both of these problems.
The output of a multi-parametric linear program that has been solved for
the MPC control inputs offline is a piecewise linear controller defined
over a union of polyhedra. The online work is then to determine which
region the current measured state is in and apply the appropriate linear
control law. This final part introduces a new method of searching for the
appropriate region by posing the problem as a nearest neighbour
search. This search can be done in logarithmic time and we demonstrate
speed increases from 20Hz to 20kHz for a large example system.
Pre-Prints
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BibTex Entry
- @PhdThesis{,
- author = {Jones C.N.},
- school = {University of Cambridge},
- title = {Polyhedal Tools for Control},
- year = {2005},
- month = {July}
- }
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