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Open Access Publications from the University of California

Symmetric Constrained Optimal Control: Theory, Algorithms, and Applications

  • Author(s): Danielson, Claus
  • Advisor(s): Borrelli, Francesco
  • et al.

This dissertation develops the theory of symmetry for constrained linear systems. We use symmetry to design model predictive controllers for constrained linear systems with reduced complexity.

The dissertation is divided into three parts. In the first part we review the relevant results from model predictive control and group theory. In particular we present algorithms and data structures from computational group theory used to efficiently search groups.

In the second part we develop the theory of symmetry for constrained linear systems and model predictive control problems. Symmetries of constrained linear systems are linear transformations that preserve the dynamics and constraints. A symmetry of a model predictive control problem is a symmetry of the underlying constrained system that also preserves the cost. We use a group theoretic formalism to derive properties of symmetric constrained linear systems and symmetric model predictive control problems. We prove conditions under which the model predictive controller is symmetric and present a procedure for efficiently computing the symmetries of constrained linear systems and model predictive control problems. Our method transforms the problem of finding generators for the symmetry group into a graph automorphism problem. These symmetries are used to design model predictive control algorithms with reduced complexity.

We also present two novel explicit model predictive control designs. Both reduce memory requirements by discarding symmetrically redundant pieces of the control-law. The control-law in the eliminated pieces can be reconstructed online using symmetry. We show that storing the symmetries of the problem requires less memory than storing the controller pieces.

In the third part of this dissertation we apply our symmetry theory to the battery balancing problem. We use symmetry to reduce the memory requirements for explicit model predictive controllers for seven battery-balancing hardware designs proposed in the literature. This application demonstrates that our symmetric controller designs can significantly reduce the memory requirements of explicit model predictive control. In particular for four out of seven of the designs in our numerical study, the number of pieces in the symmetric controller did not increase as the battery pack-size was increased.

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