PortHamiltonianSystems

Documentation for PortHamiltonianSystems.

A package for port-Hamiltonian Systems in Julia.

The considered systems are of the form

\[ \Sigma\quad\left\{\quad\begin{aligned} \dot{x}(t) &= (J - R)Qx(t) + (G - P)u(t), \\ y(t) &= (G + P)^\top Qx(t) + (S - N)u(t) \end{aligned}\right.,\]

where $J=-J^\top\in \mathbb{R}^{n\times n}$ is skew-symmetric, $R\in \mathbb{R}^{n\times n}$ is positive semi-definite, $Q\in \mathbb{R}^{n\times n}$ is positive definite, $G\in \mathbb{R}^{n\times m}$, $P\in \mathbb{R}^{n\times m}$, $S\in \mathbb{R}^{m\times m}$ and $N\in \mathbb{R}^{m\times m}$.

Example

using LinearAlgebra, ControlSystemsBase
using PortHamiltonianSystems

J = [0 -1; 1 0]
R = [1 -1; -1 2]
Q = I(2)
G = [1; 0;;]

Σph = phss(J, R, Q, G)
# This generates the system:
# PortHamiltonianStateSpace{Float64}
# J = 
#   0.0  -1.0
#  1.0  0.0
# R = 
#   1.0  -1.0
#  -1.0   2.0
# Q = 
#  1.0  0.0
#  0.0  1.0
# G = 
#  1.0
#  0.0
# P = 
#  0.0
#  0.0
# S = 
#  0.0
# N = 
#  0.0

# Compute the observability Gramian
gram(Σph, :o)
# 2×2 Matrix{Float64}:
#  0.5  0.0
#  0.0  0.0

# Compute a numerically minimal realization
Σphr = phminreal(Σph)

# Compute H2 and Hinf errors
norm(Σph - Σphr) # 9.56908750203461e-17
norm(Σph - Σphr, Inf) # 1.8182097724708874e-16

# Computes unstructured state-space realization
Σ = ss(Σph)

# Checking passivity
ispassive(Σ) # true

# Computes pH realization for a given posdef X satisfying the KYP inequality 
X = kyp(Σ)
Σph2 = phss(Σ, X)

norm(Σph - Σph2) # 1.3470909371896273e-16

References