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Difference between revisions of "Models"

(→‎Affine Parametric Linear Time-Invariant System: add parametrized B and C terms)
(→‎Affine Parametric Second-Order System: add parametrized B and C, because why not)
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:<math>
 
:<math>
 
\begin{align}
 
\begin{align}
(M + \sum_{i=1}^{\ell} p^M_i M_i)\ddot{x}(t) + (E + \sum_{i=1}^{\ell} p^E_i E_i)\dot{x}(t) + (K + \sum_{i=1}^{\ell} p^K_i K_i)x(t) &= B u(t), \\
+
(M + \sum_{i=1}^{\ell} p^M_i M_i)\ddot{x}(t) + (E + \sum_{i=1}^{\ell} p^E_i E_i)\dot{x}(t) + (K + \sum_{i=1}^{\ell} p^K_i K_i)x(t) &= (B + \sum_{i=1}^{\ell} p^B_i B_i) u(t), \\
y(t) &= C x(t),
+
y(t) &= (C + \sum_{i=1}^{\ell} p^C_i C_i) x(t),
 
\end{align}
 
\end{align}
 
</math>
 
</math>
Line 143: Line 143:
 
with
 
with
   
<math>M \in \mathbb{R}^{n \times n}</math>,
+
<math>M, M_i \in \mathbb{R}^{n \times n}</math>;
<math>M_i \in \mathbb{R}^{n \times n}</math>,
+
<math>E, E_i \in \mathbb{R}^{n \times n}</math>;
<math>E \in \mathbb{R}^{n \times n}</math>,
+
<math>K, K_i \in \mathbb{R}^{n \times n}</math>;
<math>E_i \in \mathbb{R}^{n \times n}</math>,
+
<math>B, B_i \in \mathbb{R}^{n \times m}</math>; and
<math>K \in \mathbb{R}^{n \times n}</math>,
+
<math>C, C_i \in \mathbb{R}^{q \times n}</math>,
<math>K_i \in \mathbb{R}^{n \times n}</math>,
+
for all <math>i = 1, \ldots, \ell</math>.
<math>B \in \mathbb{R}^{n \times m}</math>,
 
<math>C \in \mathbb{R}^{q \times n}</math>.
 

Revision as of 15:32, 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, A always serves as the name of the component matrix applied to the state x(t) in a linear time-invariant system. For all models we assume an input u : \mathbb{R} \to \mathbb{R}^m, with components u_j, j = 1, \ldots, m, a state x : \mathbb{R} \to \mathbb{R}^n, and an output y : \mathbb{R} \to \mathbb{R}^q. For all parametric models, we assume each component has \ell parameters; in cases where a component has fewer than \ell parameters, the extras are treated as 0. Some benchmarks (e.g., Bone Model) have a constant forcing term, in which case, it is assumed that u(t) is identically 1.

Linear Time-Invariant System


\begin{align}
E\dot{x}(t) &= Ax(t) + Bu(t),\\
y(t) &= Cx(t),
\end{align}

with

E \in \mathbb{R}^{n \times n}, A \in \mathbb{R}^{n \times n}, B \in \mathbb{R}^{n \times m}, C \in \mathbb{R}^{q \times n}.


Linear Time-Varying System


\begin{align}
E(t)\dot{x}(t) &= A(t)x(t) + B(t)u(t),\\
y(t) &= C(t)x(t),
\end{align}

with

E : \mathbb{R} \to \mathbb{R}^{n \times n}, A : \mathbb{R} \to \mathbb{R}^{n \times n}, B : \mathbb{R} \to \mathbb{R}^{n \times m}, C : \mathbb{R} \to \mathbb{R}^{q \times n}.


Quadratic-Bilinear System


\begin{align}
 E\dot{x}(t) &= A x(t) + H x(t) \otimes x(t) + \sum_{j=1}^m N_j x(t) u_j(t) + B u(t), \\
 y(t) &= Cx(t),
\end{align}

with

E \in \mathbb{R}^{n \times n}, A \in \mathbb{R}^{n \times n}, H \in \mathbb{R}^{n \times n^2}, N_j \in \mathbb{R}^{n \times n}, B \in \mathbb{R}^{n \times m}, C \in \mathbb{R}^{q \times n}.

Nonlinear Time-Invariant System


\begin{align}
E\dot{x}(t) &= Ax(t) + f(x(t),u(t)) + Bu(t),\\
y(t) &= Cx(t),
\end{align}

with

E \in \mathbb{R}^{n \times n}, A \in \mathbb{R}^{n \times n}, B \in \mathbb{R}^{n \times m}, C \in \mathbb{R}^{q \times n}, f : \mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^n.


Affine Parametric Linear Time-Invariant System


\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) + (B + \sum_{i=1}^{\ell} p^B_i B_i)u(t),\\
y(t) &= (C + \sum_{i=1}^{\ell} p^C_i C_i)x(t),
\end{align}

with

E, E_i \in \mathbb{R}^{n \times n}; A, A_i \in \mathbb{R}^{n \times n}; B, B_i \in \mathbb{R}^{n \times m}; and C, C_i \in \mathbb{R}^{q \times n}, for all i = 1, \ldots, \ell.

Second-Order System


\begin{align}
M \ddot{x}(t) + E \dot{x}(t) + K x(t) &= B u(t), \\
y(t) &= C x(t),
\end{align}

with

M \in \mathbb{R}^{n \times n}, E \in \mathbb{R}^{n \times n}, K \in \mathbb{R}^{n \times n}, B \in \mathbb{R}^{n \times m}, C \in \mathbb{R}^{q \times n}.

Nonlinear Second-Order System


\begin{align}
M \ddot{x}(t) + E \dot{x}(t) + K x(t) &= B u(t) + F f(x(t),u(t)), \\
y(t) &= C x(t),
\end{align}

with

M \in \mathbb{R}^{n \times n}, E \in \mathbb{R}^{n \times n}, K \in \mathbb{R}^{n \times n}, B \in \mathbb{R}^{n \times m}, F \in \mathbb{R}^{n \times n}, C \in \mathbb{R}^{q \times n}, f : \mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^n.

Affine Parametric Second-Order System


\begin{align}
(M + \sum_{i=1}^{\ell} p^M_i M_i)\ddot{x}(t) + (E + \sum_{i=1}^{\ell} p^E_i E_i)\dot{x}(t) + (K + \sum_{i=1}^{\ell} p^K_i K_i)x(t) &= (B + \sum_{i=1}^{\ell} p^B_i B_i) u(t), \\
y(t) &= (C + \sum_{i=1}^{\ell} p^C_i C_i) x(t),
\end{align}

with

M, M_i \in \mathbb{R}^{n \times n}; E, E_i \in \mathbb{R}^{n \times n}; K, K_i \in \mathbb{R}^{n \times n}; B, B_i \in \mathbb{R}^{n \times m}; and C, C_i \in \mathbb{R}^{q \times n}, for all i = 1, \ldots, \ell.