QuadraticOutputSystems

Documentation for QuadraticOutputSystems.

A package for linear control systems with quadratic output in Julia.

The considered systems are of the form

\[ \Sigma\quad\left\{\quad\begin{aligned} \dot{x}(t) &= Ax(t) + Bu(t),\\ y(t) &= Cx(t) + \left[ \begin{array}{c} x(t)^\top M_1 x(t)\\ \vdots \\ x(t)^\top M_p x(t) \end{array} \right] \end{aligned}\right..\]

The output can be written as

\[ y(t) = Cx(t) + M(x(t)\otimes x(t)), \quad \text{with} \quad M:= \left[\begin{array}{c} \mathrm{vec}(M_1)^\top\\ \vdots\\ \mathrm{vec}(M_p)^\top \end{array}\right] \in \mathbb{R}^{p \times n^2}.\]

This package is based and inspired by ControlSystems.jl and implements some of the results from [1].

Example

using QuadraticOutputSystems, ControlSystems

A = [-2 1; -1 -1]
B = [6; 0]
C = [6 0]
M = 1//2*vec([1 0; 0 1])'

Σ = qoss(A, B, C, M)
# This generates the system:
# QuadraticOutputStateSpace{Rational{Int64}}
# A = 
#  -2//1   1//1
#  -1//1  -1//1
# B = 
#  6//1
#  0//1
# C = 
#  6//1  0//1
# M = 
#  1//2  0//1  0//1  1//2

# Compute the Gramians of the system
gram(Σ, :c)
gram(Σ, :o)

# Compute the H2-norm of the system
h2norm(Σ) # 17.521415467935235

# Given a second system Σr, we can compute the H2-inner product and the H2-error
Σr = qoss(A[1,1], B[1], C[1], M[1])

h2inner(Σ, Σr) # 318.2485207100592
h2norm(Σ - Σr)^2 # 14.75295857988162
h2norm(Σ)^2 + h2norm(Σr)^2 - 2*h2inner(Σ, Σr) # 14.75295857988162

References