(add note about constant forcing term) |
(→Affine Parametric Linear Time-Invariant System: add parametrized B and C terms) |
||
Line 83: | Line 83: | ||
:<math> |
:<math> |
||
\begin{align} |
\begin{align} |
||
− | (E + \sum_{i=1}^{\ell} p^E_i E_i)\dot{x}(t) &= (A + \sum_{i=1}^{\ell} p^A_i A_i) x(t) + |
+ | (E + \sum_{i=1}^{\ell} p^E_i E_i)\dot{x}(t) &= (A + \sum_{i=1}^{\ell} p^A_i A_i) x(t) + (B + \sum_{i=1}^{\ell} p^B_i B_i)u(t),\\ |
− | y(t) &= |
+ | y(t) &= (C + \sum_{i=1}^{\ell} p^C_i C_i)x(t), |
\end{align} |
\end{align} |
||
</math> |
</math> |
||
Line 90: | Line 90: | ||
with |
with |
||
− | <math>E \in \mathbb{R}^{n \times n}</math> |
+ | <math>E, E_i \in \mathbb{R}^{n \times n}</math>; |
− | <math> |
+ | <math>A, A_i \in \mathbb{R}^{n \times n}</math>; |
− | <math> |
+ | <math>B, B_i \in \mathbb{R}^{n \times m}</math>; and |
− | <math> |
+ | <math>C, C_i \in \mathbb{R}^{q \times n}</math>, |
− | <math> |
+ | for all <math>i = 1, \ldots, \ell</math>. |
− | <math>C \in \mathbb{R}^{q \times n}</math>. |
||
===Second-Order System=== |
===Second-Order System=== |
Revision as of 15:29, 31 August 2022
Benchmark Model Templates
This page specifies templates for the types of models used as benchmark systems. In particular, the naming schemes established here are used in the corresponding data sets for all benchmarks. For example, always serves as the name of the component matrix applied to the state
in a linear time-invariant system.
For all models we assume an input
, with components
,
a state
,
and an output
.
For all parametric models, we assume each component has
parameters; in cases where a component has fewer than
parameters, the extras are treated as
.
Some benchmarks (e.g., Bone Model) have a constant forcing term, in which case, it is assumed that
is identically
.
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
;
;
; and
,
for all
.
Second-Order System
with
,
,
,
,
.
Nonlinear Second-Order System
with
,
,
,
,
,
,
.
Affine Parametric Second-Order System
with
,
,
,
,
,
,
,
.