Geometry of Model Predictive Control

A formalization of Model Predictive Control (MPC) in Hilbert space based on concepts of Quantum Mechanics is presented. A trajectory is considered as a pure state; it is a normalized vector in Hilbert space. Control system at each state generates a set of feasible trajectories; it is a subspace in the Hilbert space of trajectories and it defines a mixed state. MPC can be described as a chain of mappings: a given (optimal in the past) trajectory to a subspace of feasible trajectories and a given subspace of feasible trajectories to a new optimal trajectory. It is shown how an integral estimate of a trajectory can be obtained with respect to a given feasible subspace (an analog of the quantum probability) and how this estimate can be used for determining an optimal trajectory. By introducing projectors in Hilbert space (a projector formalizes a (standard) measurement) these abstract constructions are transformed into the real world of numbers and actions. It is shown how an abstract trajectory can be represented by its components in different bases. Then a data matrix formed by a set of feasible trajectories is introduced. This data matrix can served as a substitution of a mathematical model of CS. The use of the data matrix in MPC is demonstrated for linear CS. Different strategies of generating the data matrix and determining an optimal control are analyzed. This analysis is based on the singular value decomposition of the data matrix.