(→Quadratic-Bilinear System: correct notation following https://onlinelibrary.wiley.com/doi/am-pdf/10.1002/nla.2200 ; replace H with Q (to avoid issues with H-norms and transfer functions down the line)) |
(change dimensions to lowercase (reserve uppercase for matrices); make u_i scalar function for QBS (following https://onlinelibrary.wiley.com/doi/am-pdf/10.1002/nla.2200 )) |
||
Line 5: | Line 5: | ||
==Benchmark Model Overview== |
==Benchmark Model Overview== |
||
This page outlines the types of models that are used as benchmark systems. |
This page outlines the types of models that are used as benchmark systems. |
||
− | For this general summary we assume an input <math>u : \mathbb{R} \to \mathbb{R}^ |
+ | For this general summary we assume an input <math>u : \mathbb{R} \to \mathbb{R}^m</math>, |
− | a state <math>x : \mathbb{R} \to \mathbb{R}^ |
+ | a state <math>x : \mathbb{R} \to \mathbb{R}^n</math> and an output <math>y : \mathbb{R} \to \mathbb{R}^q</math>. |
===Linear Time-Invariant System=== |
===Linear Time-Invariant System=== |
||
Line 19: | Line 19: | ||
with: |
with: |
||
− | <math>E \in \mathbb{R}^{ |
+ | <math>E \in \mathbb{R}^{n \times n}</math>, |
− | <math>A \in \mathbb{R}^{ |
+ | <math>A \in \mathbb{R}^{n \times n}</math>, |
− | <math>B \in \mathbb{R}^{ |
+ | <math>B \in \mathbb{R}^{n \times m}</math>, |
− | <math>C \in \mathbb{R}^{ |
+ | <math>C \in \mathbb{R}^{q \times n}</math>. |
Line 36: | Line 36: | ||
with: |
with: |
||
− | <math>E : \mathbb{R} \to \mathbb{R}^{ |
+ | <math>E : \mathbb{R} \to \mathbb{R}^{n \times n}</math>, |
− | <math>A : \mathbb{R} \to \mathbb{R}^{ |
+ | <math>A : \mathbb{R} \to \mathbb{R}^{n \times n}</math>, |
− | <math>B : \mathbb{R} \to \mathbb{R}^{ |
+ | <math>B : \mathbb{R} \to \mathbb{R}^{n \times m}</math>, |
− | <math>C : \mathbb{R} \to \mathbb{R}^{ |
+ | <math>C : \mathbb{R} \to \mathbb{R}^{q \times n}</math>. |
Line 53: | Line 53: | ||
with: |
with: |
||
− | <math>E \in \mathbb{R}^{ |
+ | <math>E \in \mathbb{R}^{n \times n}</math>, |
− | <math>A \in \mathbb{R}^{ |
+ | <math>A \in \mathbb{R}^{n \times n}</math>, |
− | <math> |
+ | <math>Q \in \mathbb{R}^{n \times n^2}</math>, |
− | <math> |
+ | <math>N_i \in \mathbb{R}^{n \times n}</math>, |
− | <math> |
+ | <math>u_i: \mathbb{R} \to \mathbb{R}</math>, |
− | <math> |
+ | <math>B \in \mathbb{R}^{n \times m}</math>, |
+ | <math>C \in \mathbb{R}^{q \times n}</math>. |
||
===Nonlinear Time-Invariant System=== |
===Nonlinear Time-Invariant System=== |
||
Line 71: | Line 72: | ||
with: |
with: |
||
− | <math>E \in \mathbb{R}^{ |
+ | <math>E \in \mathbb{R}^{n \times n}</math>, |
− | <math>A \in \mathbb{R}^{ |
+ | <math>A \in \mathbb{R}^{n \times n}</math>, |
− | <math>B \in \mathbb{R}^{ |
+ | <math>B \in \mathbb{R}^{n \times m}</math>, |
− | <math>C \in \mathbb{R}^{ |
+ | <math>C \in \mathbb{R}^{q \times n}</math>, |
<math>f : \mathbb{R}^N \times \mathbb{R}^M \to \mathbb{R}^N</math>. |
<math>f : \mathbb{R}^N \times \mathbb{R}^M \to \mathbb{R}^N</math>. |
||
Line 89: | Line 90: | ||
with: |
with: |
||
− | <math>E_0 \in \mathbb{R}^{ |
+ | <math>E_0 \in \mathbb{R}^{n \times n}</math>, |
− | <math>E_j \in \mathbb{R}^{ |
+ | <math>E_j \in \mathbb{R}^{n \times n}</math>, |
− | <math>A_0 \in \mathbb{R}^{ |
+ | <math>A_0 \in \mathbb{R}^{n \times n}</math>, |
− | <math>A_i \in \mathbb{R}^{ |
+ | <math>A_i \in \mathbb{R}^{n \times n}</math>, |
− | <math>B \in \mathbb{R}^{ |
+ | <math>B \in \mathbb{R}^{n \times m}</math>, |
− | <math>C \in \mathbb{R}^{ |
+ | <math>C \in \mathbb{R}^{q \times n}</math>. |
===Second-Order System=== |
===Second-Order System=== |
||
Line 107: | Line 108: | ||
with: |
with: |
||
− | <math>M \in \mathbb{R}^{ |
+ | <math>M \in \mathbb{R}^{n \times n}</math>, |
− | <math>E \in \mathbb{R}^{ |
+ | <math>E \in \mathbb{R}^{n \times n}</math>, |
− | <math>K \in \mathbb{R}^{ |
+ | <math>K \in \mathbb{R}^{n \times n}</math>, |
− | <math>B \in \mathbb{R}^{ |
+ | <math>B \in \mathbb{R}^{n \times m}</math>, |
− | <math>C \in \mathbb{R}^{ |
+ | <math>C \in \mathbb{R}^{q \times n}</math>. |
===Nonlinear Second-Order System=== |
===Nonlinear Second-Order System=== |
||
Line 124: | Line 125: | ||
with: |
with: |
||
− | <math>M \in \mathbb{R}^{ |
+ | <math>M \in \mathbb{R}^{n \times n}</math>, |
− | <math>E \in \mathbb{R}^{ |
+ | <math>E \in \mathbb{R}^{n \times n}</math>, |
− | <math>K \in \mathbb{R}^{ |
+ | <math>K \in \mathbb{R}^{n \times n}</math>, |
− | <math>B \in \mathbb{R}^{ |
+ | <math>B \in \mathbb{R}^{n \times m}</math>, |
− | <math>C \in \mathbb{R}^{ |
+ | <math>C \in \mathbb{R}^{q \times n}</math>, |
<math>f : \mathbb{R}^N \times \mathbb{R}^M \to \mathbb{R}^N</math>. |
<math>f : \mathbb{R}^N \times \mathbb{R}^M \to \mathbb{R}^N</math>. |
||
Line 142: | Line 143: | ||
with: |
with: |
||
− | <math>M_0 \in \mathbb{R}^{ |
+ | <math>M_0 \in \mathbb{R}^{n \times n}</math>, |
− | <math>M_i \in \mathbb{R}^{ |
+ | <math>M_i \in \mathbb{R}^{n \times n}</math>, |
− | <math>E_0 \in \mathbb{R}^{ |
+ | <math>E_0 \in \mathbb{R}^{n \times n}</math>, |
− | <math>E_j \in \mathbb{R}^{ |
+ | <math>E_j \in \mathbb{R}^{n \times n}</math>, |
− | <math>K_0 \in \mathbb{R}^{ |
+ | <math>K_0 \in \mathbb{R}^{n \times n}</math>, |
− | <math>K_k \in \mathbb{R}^{ |
+ | <math>K_k \in \mathbb{R}^{n \times n}</math>, |
− | <math>B \in \mathbb{R}^{ |
+ | <math>B \in \mathbb{R}^{n \times m}</math>, |
− | <math>C \in \mathbb{R}^{ |
+ | <math>C \in \mathbb{R}^{q \times n}</math>. |
Revision as of 11:10, 9 August 2022
Note: This page has not been verified by our editors.
Benchmark Model Overview
This page outlines the types of models that are used as benchmark systems.
For this general summary we assume an input ,
a state
and an output
.
Linear Time-Invariant System
with:
,
,
,
.
Linear Time-Varying System
with:
,
,
,
.
Quadratic-Bilinear System
with:
,
,
,
,
,
,
.
Nonlinear Time-Invariant System
with:
,
,
,
,
.
Affine Parametric Linear Time-Invariant System
with:
,
,
,
,
,
.
Second-Order System
with:
,
,
,
,
.
Nonlinear Second-Order System
with:
,
,
,
,
,
.
Affine Parametric Second-Order System
with:
,
,
,
,
,
,
,
.