In this document, some novel theoretical and computational techniques for constrained approximation of data-driven systems, are presented. The motivation for the development of these techniques came from structurepreserving matrix approximation problems that appear in the fields of system identification and model predictive control, for data-driven systems and processes. The research reported in this document is focused on finite-state approximation of data-driven systems.Some numerical implementations of the aforementioned techniques in the simulation and model predictive control of some generic data-driven systems, that are related to electrical signal transmission models, are outlined.
Unitary matrices arise in many ways in physics, in particular as a time evolution operator. For a periodically driven system, one frequently wishes to compute a Floquet Hamiltonian that should be a Hermitian operator H such that e−iTH = U(T), where U(T) is the time evolution operator at time corresponding to the period of the system. That is, we want H to be equal to −i times a matrix logarithm of U(T). If the system has a symmetry, such as time reversal symmetry, one can expect H to have a symmetry beyond being Hermitian. We discuss here practical numerical algorithms on computing matrix logarithms that have certain symmetries, which can be used to compute Floquet Hamiltonians that have appropriate symmetries. Along the way, we prove some results on how a symmetry in the Floquet operator U(T) can lead to a symmetry in a basis of Floquet eigenstates.
In this document we study the uniform local path connectivity of sets of m-tuples of pairwise commuting normal matrices with some additional constraints.More specifically, given given ε > 0, a fixed metric ð in Mn(C) m induced by the operator norm · , any collection of r non-
We examine the utility of the quadratic pseudospectrum for understanding and detecting states that are somewhat localized in position and energy, in particular, in the context of condensed matter physics. Specifically, the quadratic pseudospectrum represents a method for approaching systems with incompatible observables { A j∣1 ≤ j ≤ d} as it minimizes collectively the errors ‖ A j v − λ j v‖ while defining a joint approximate spectrum of incompatible observables. Moreover, we derive an important estimate relating the Clifford and quadratic pseudospectra. Finally, we prove that the quadratic pseudospectrum is local and derive the bounds on the errors that are incurred by truncating the system in the vicinity of where the pseudospectrum is being calculated.
We discuss a general method of finding bounds on the norm of a commutator of an operator and a function of a normal operator. As an application we find new bounds on the norm of a commuator with a square root.
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