Utilities

System operators

The struct Operators provides a convenient way to organize and store the full and reduced operators.

LiftAndLearn.Operators — Type

mutable struct Operators

Organize the operators of the system in a structure. The operators currently supported are up to second order.

Fields

A: linear state operator
B: linear input operator
C: linear output operator
D: linear input-output coupling operator
A2: quadratic state operator with redundancy
A3: cubic state operator with redundancy
A4: quartic state operator with redundancy
A2u: quadratic state operator with no redundancy
A3u: cubic state operator with no redundancy
A4u: quartic state operator with no redundancy
A2t: quadratic state operator with redundancy in 3-dim tensor form
N: bilinear (state-input) operator
K: constant operator
f: nonlinear function operator f(x,u)
dims: dimensions of the operators

Note

Currently only supports
- state: up to 4th order
- input: only B matrix
- output: only C and D matrices
- state-input-coupling: bilinear
- constant term: K matrix
- nonlinearity: f(x,u)

source

Math Operations

This package heavily relies on the following math operations

Kronecker Products $\otimes$
Vectorization $\text{vec}(\cdot)$

For Kronecker products we are very thankful to the Kronecker.jl package allowing fast Kronecker product arithmetic. We also, use what is called the unique Kronecker product which eliminates the redundant terms arising from the symmetry of Kronecker products. For details on the unique Kronecker product please refer to the package UniqueKronecker.jl.

Partial Differential Equation Models

Please also refer to the package PolynomialModelReductionDataset.jl providing a suite of polynomial-based systems arising from PDEs. We extensively use these models in our examples.