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Difference between revisions of "Models"

 
(8 intermediate revisions by one other user not shown)
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[[Category:Benchmarks]]
+
[[Category:Benchmark]]
   
 
==Benchmark Model Templates==
 
==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, <math>A</math> always serves as the name of the component matrix applied to the state <math>x(t)</math> in a linear time-invariant system.
+
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, <math>A</math> always serves as the name of the component matrix applied to the state <math>x(t)</math> in a linear time-invariant, first-order system.
 
For all models we assume an input <math>u : \mathbb{R} \to \mathbb{R}^m</math>, with components <math>u_j, j = 1, \ldots, m</math>,
 
For all models we assume an input <math>u : \mathbb{R} \to \mathbb{R}^m</math>, with components <math>u_j, j = 1, \ldots, m</math>,
 
a state <math>x : \mathbb{R} \to \mathbb{R}^n</math>,
 
a state <math>x : \mathbb{R} \to \mathbb{R}^n</math>,
Line 25: Line 25:
 
<math>C \in \mathbb{R}^{q \times n}</math>,
 
<math>C \in \mathbb{R}^{q \times n}</math>,
 
<math>D \in \mathbb{R}^{q \times m}</math>.
 
<math>D \in \mathbb{R}^{q \times m}</math>.
  +
  +
By default <math>E = I</math> and <math>D = 0</math>, unless explicitly provided.
   
 
===Linear Time-Varying First-Order System (LTV-FOS)===
 
===Linear Time-Varying First-Order System (LTV-FOS)===
Line 43: Line 45:
 
<math>D : \mathbb{R} \to \mathbb{R}^{q \times m}</math>.
 
<math>D : \mathbb{R} \to \mathbb{R}^{q \times m}</math>.
   
  +
By default <math>E = I</math> and <math>D = 0</math>, unless explicitly provided.
===Quadratic-Bilinear System (QBS)===
 
  +
 
===Affine-Parametric LTI-FOS (AP-LTI-FOS)===
   
 
:<math>
 
:<math>
 
\begin{align}
 
\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), \\
+
(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) &= Cx(t) + Du(t),
+
y(t) &= (C + \sum_{i=1}^{\ell} p^C_i C_i)x(t),
 
\end{align}
 
\end{align}
 
</math>
 
</math>
Line 54: Line 58:
 
with
 
with
   
<math>E \in \mathbb{R}^{n \times n}</math>,
+
<math>E, E_i \in \mathbb{R}^{n \times n}</math>;
<math>A \in \mathbb{R}^{n \times n}</math>,
+
<math>A, A_i \in \mathbb{R}^{n \times n}</math>;
<math>H \in \mathbb{R}^{n \times n^2}</math>,
+
<math>B, B_i \in \mathbb{R}^{n \times m}</math>; and
<math>N_j \in \mathbb{R}^{n \times n}</math>,
+
<math>C, C_i \in \mathbb{R}^{q \times n}</math>,
<math>B \in \mathbb{R}^{n \times m}</math>,
+
for all <math>i = 1, \ldots, \ell</math>.
<math>C \in \mathbb{R}^{q \times n}</math>,
 
<math>D \in \mathbb{R}^{q \times m}</math>.
 
   
  +
By default <math>E = I, E_i = 0</math>, unless explicitly provided. If <math>A_i</math> are provided without <math>A</math>, then it is assumed <math>A = 0</math>. Likewise for <math>B</math>, <math>C</math>, and <math>E</math>.
===Nonlinear Time-Invariant First-Order System (NLTI-FOS)===
 
  +
 
===Linear Time-Invariant Second-Order System (LTI-SOS)===
   
 
:<math>
 
:<math>
 
\begin{align}
 
\begin{align}
E\dot{x}(t) &= Ax(t) + f(x(t),u(t)) + Bu(t),\\
+
M \ddot{x}(t) + E \dot{x}(t) + K x(t) &= B u(t), \\
y(t) &= Cx(t) + Du(t),
+
y(t) &= C_p x(t) + C_v \dot{x}(t) + D u(t),
 
\end{align}
 
\end{align}
 
</math>
 
</math>
Line 73: Line 77:
 
with
 
with
   
 
<math>M \in \mathbb{R}^{n \times n}</math>,
 
<math>E \in \mathbb{R}^{n \times n}</math>,
 
<math>E \in \mathbb{R}^{n \times n}</math>,
<math>A \in \mathbb{R}^{n \times n}</math>,
+
<math>K \in \mathbb{R}^{n \times n}</math>,
 
<math>B \in \mathbb{R}^{n \times m}</math>,
 
<math>B \in \mathbb{R}^{n \times m}</math>,
<math>C \in \mathbb{R}^{q \times n}</math>,
+
<math>C_p, C_v \in \mathbb{R}^{q \times n}</math>,
<math>D \in \mathbb{R}^{q \times m}</math>,
+
<math>D \in \mathbb{R}^{q \times m}</math>.
<math>f : \mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^n</math>.
 
   
 
When <math>C_v = 0</math>, we denote <math>C = C_p</math>. By default <math>E = I</math> and <math>D = 0</math>, unless explicitly provided.
===Affine Parametric LTI-FOS (AP-LTI-FOS)===
 
  +
 
===Affine-Parametric LTI-SOS (AP-LTI-SOS)===
   
 
:<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) + (B + \sum_{i=1}^{\ell} p^B_i B_i)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 + \sum_{i=1}^{\ell} p^C_i C_i)x(t),
+
y(t) &= (C + \sum_{i=1}^{\ell} p^C_i C_i) x(t),
 
\end{align}
 
\end{align}
 
</math>
 
</math>
Line 91: Line 97:
 
with
 
with
   
 
<math>M, M_i \in \mathbb{R}^{n \times n}</math>;
 
<math>E, E_i \in \mathbb{R}^{n \times n}</math>;
 
<math>E, E_i \in \mathbb{R}^{n \times n}</math>;
<math>A, A_i \in \mathbb{R}^{n \times n}</math>;
+
<math>K, K_i \in \mathbb{R}^{n \times n}</math>;
 
<math>B, B_i \in \mathbb{R}^{n \times m}</math>; and
 
<math>B, B_i \in \mathbb{R}^{n \times m}</math>; and
 
<math>C, C_i \in \mathbb{R}^{q \times n}</math>,
 
<math>C, C_i \in \mathbb{R}^{q \times n}</math>,
 
for all <math>i = 1, \ldots, \ell</math>.
 
for all <math>i = 1, \ldots, \ell</math>.
   
  +
By default <math>E = I, E_i = 0</math>, unless explicitly provided. If <math>M_i</math> are provided without <math>M</math>, then it is assumed <math>M = 0</math>. Likewise for <math>K</math>, <math>B</math>, and <math>C</math>.
===Linear Time-Invariant Second-Order System (LTI-SOS)===
 
  +
 
===Quadratic-Bilinear System (QBS)===
   
 
:<math>
 
:<math>
 
\begin{align}
 
\begin{align}
M \ddot{x}(t) + E \dot{x}(t) + K x(t) &= B u(t), \\
+
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) &= C_p x(t) + C_v \dot{x}(t) + D u(t),
+
y(t) &= Cx(t) + Du(t),
 
\end{align}
 
\end{align}
 
</math>
 
</math>
Line 108: Line 117:
 
with
 
with
   
<math>M \in \mathbb{R}^{n \times n}</math>,
 
 
<math>E \in \mathbb{R}^{n \times n}</math>,
 
<math>E \in \mathbb{R}^{n \times n}</math>,
<math>K \in \mathbb{R}^{n \times n}</math>,
+
<math>A \in \mathbb{R}^{n \times n}</math>,
  +
<math>H \in \mathbb{R}^{n \times n^2}</math>,
  +
<math>N_j \in \mathbb{R}^{n \times n}</math>,
 
<math>B \in \mathbb{R}^{n \times m}</math>,
 
<math>B \in \mathbb{R}^{n \times m}</math>,
<math>C_p, C_v \in \mathbb{R}^{q \times n}</math>,
+
<math>C \in \mathbb{R}^{q \times n}</math>,
 
<math>D \in \mathbb{R}^{q \times m}</math>.
 
<math>D \in \mathbb{R}^{q \times m}</math>.
   
 
===Nonlinear Time-Invariant First-Order System (NLTI-FOS)===
When <math>C_v = 0</math>, we denote <math>C = C_p</math>.
 
  +
 
:<math>
 
\begin{align}
  +
E\dot{x}(t) &= Ax(t) + Bu(t) + F f(x(t),u(t)),\\
  +
y(t) &= Cx(t) + Du(t),
 
\end{align}
 
</math>
  +
 
with
  +
  +
<math>E \in \mathbb{R}^{n \times n}</math>,
  +
<math>A \in \mathbb{R}^{n \times n}</math>,
  +
<math>B \in \mathbb{R}^{n \times m}</math>,
 
<math>C \in \mathbb{R}^{q \times n}</math>,
 
<math>D \in \mathbb{R}^{q \times m}</math>,
 
<math>F \in \mathbb{R}^{n \times n}</math>,
 
<math>f : \mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^n</math>.
  +
  +
By default <math>F = I</math>, <math>E = I</math>, <math>D = 0</math>, unless explicitly provided.
   
 
===Nonlinear Time-Invariant Second-Order System (NLTI-SOS)===
 
===Nonlinear Time-Invariant Second-Order System (NLTI-SOS)===
Line 139: Line 168:
 
When <math>C_v = 0</math>, we denote <math>C = C_p</math>.
 
When <math>C_v = 0</math>, we denote <math>C = C_p</math>.
   
  +
By default <math>F = I</math>, <math>E = I</math>, <math>D = 0</math>, unless explicitly provided.
===Affine Parametric LTI-SOS (AP-LTI-SOS)===
 
   
  +
===Other System Classes===
:<math>
 
  +
Affine-parametric and time-varying versions of nonlinear systems are clearly also possible by combining patterns of the above models.
\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}
 
</math>
 
 
with
 
 
<math>M, M_i \in \mathbb{R}^{n \times n}</math>;
 
<math>E, E_i \in \mathbb{R}^{n \times n}</math>;
 
<math>K, K_i \in \mathbb{R}^{n \times n}</math>;
 
<math>B, B_i \in \mathbb{R}^{n \times m}</math>; and
 
<math>C, C_i \in \mathbb{R}^{q \times n}</math>,
 
for all <math>i = 1, \ldots, \ell</math>.
 

Latest revision as of 16:28, 25 March 2024


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, first-order 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 First-Order System (LTI-FOS)


\begin{align}
E\dot{x}(t) &= Ax(t) + Bu(t),\\
y(t) &= Cx(t) + Du(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}, D \in \mathbb{R}^{q \times m}.

By default E = I and D = 0, unless explicitly provided.

Linear Time-Varying First-Order System (LTV-FOS)


\begin{align}
E(t)\dot{x}(t) &= A(t)x(t) + B(t)u(t),\\
y(t) &= C(t)x(t) + D(t)u(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}, D : \mathbb{R} \to \mathbb{R}^{q \times m}.

By default E = I and D = 0, unless explicitly provided.

Affine-Parametric LTI-FOS (AP-LTI-FOS)


\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.

By default E = I, E_i = 0, unless explicitly provided. If A_i are provided without A, then it is assumed A = 0. Likewise for B, C, and E.

Linear Time-Invariant Second-Order System (LTI-SOS)


\begin{align}
M \ddot{x}(t) + E \dot{x}(t) + K x(t) &= B u(t), \\
y(t) &= C_p x(t) + C_v \dot{x}(t) + D u(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_p, C_v \in \mathbb{R}^{q \times n}, D \in \mathbb{R}^{q \times m}.

When C_v = 0, we denote C = C_p. By default E = I and D = 0, unless explicitly provided.

Affine-Parametric LTI-SOS (AP-LTI-SOS)


\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.

By default E = I, E_i = 0, unless explicitly provided. If M_i are provided without M, then it is assumed M = 0. Likewise for K, B, and C.

Quadratic-Bilinear System (QBS)


\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) + Du(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}, D \in \mathbb{R}^{q \times m}.

Nonlinear Time-Invariant First-Order System (NLTI-FOS)


\begin{align}
E\dot{x}(t) &= Ax(t) + Bu(t) + F f(x(t),u(t)),\\
y(t) &= Cx(t) + Du(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}, D \in \mathbb{R}^{q \times m}, F \in \mathbb{R}^{n \times n}, f : \mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^n.

By default F = I, E = I, D = 0, unless explicitly provided.

Nonlinear Time-Invariant Second-Order System (NLTI-SOS)


\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_p x(t) + C_v \dot{x}(t) + D u(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_p, C_v \in \mathbb{R}^{q \times n}, D \in \mathbb{R}^{q \times m}, f : \mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^n.

When C_v = 0, we denote C = C_p.

By default F = I, E = I, D = 0, unless explicitly provided.

Other System Classes

Affine-parametric and time-varying versions of nonlinear systems are clearly also possible by combining patterns of the above models.