Models - Revision history

Saak at 14:28, 25 March 2024

2024-03-25T14:28:03Z

← Older revision		Revision as of 14:28, 25 March 2024
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	[[Category:~~Benchmarks~~]]		[[Category:Benchmark]]

	==Benchmark Model Templates==		==Benchmark Model Templates==

Lund: /* Affine-Parametric LTI-SOS (AP-LTI-SOS) */ fix matrix typos

2023-04-20T19:48:33Z

Affine-Parametric LTI-SOS (AP-LTI-SOS): fix matrix typos

← Older revision		Revision as of 19:48, 20 April 2023
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	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>~~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>.		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>.

	===Quadratic-Bilinear System (QBS)===		===Quadratic-Bilinear System (QBS)===

Lund: update default conditions

2023-04-20T19:45:07Z

update default conditions

← Older revision		Revision as of 19:45, 20 April 2023
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	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.		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>.

	===Linear Time-Invariant Second-Order System (LTI-SOS)===		===Linear Time-Invariant Second-Order System (LTI-SOS)===
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	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>~~E_i~~</math> are provided without <math>E</math>, then it is assumed <math>E = 0</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>.

	===Quadratic-Bilinear System (QBS)===		===Quadratic-Bilinear System (QBS)===

Lund: /* Affine-Parametric LTI-SOS (AP-LTI-SOS) */ update defaults

2023-04-20T19:42:35Z

Affine-Parametric LTI-SOS (AP-LTI-SOS): update defaults

← Older revision		Revision as of 19:42, 20 April 2023
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	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.		By default <math>E = I, E_i = 0</math>, unless explicitly provided. If <math>E_i</math> are provided without <math>E</math>, then it is assumed <math>E = 0</math>.

	===Quadratic-Bilinear System (QBS)===		===Quadratic-Bilinear System (QBS)===

Lund: /* Benchmark Model Templates */ Add comments about defaults

2023-04-20T19:27:41Z

Benchmark Model Templates: Add comments about defaults

← Older revision		Revision as of 19:27, 20 April 2023
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	==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>,
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	<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)===
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	<math>D : \mathbb{R} \to \mathbb{R}^{q \times m}</math>.		<math>D : \mathbb{R} \to \mathbb{R}^{q \times m}</math>.

	===Affine Parametric LTI-FOS (AP-LTI-FOS)===		By default <math>E = I</math> and <math>D = 0</math>, unless explicitly provided.

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

	:<math>		:<math>
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	<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.

	===Linear Time-Invariant Second-Order System (LTI-SOS)===		===Linear Time-Invariant Second-Order System (LTI-SOS)===
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	<math>D \in \mathbb{R}^{q \times m}</math>.		<math>D \in \mathbb{R}^{q \times m}</math>.

	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>E = I</math> and <math>D = 0</math>, unless explicitly provided.

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

	:<math>		:<math>
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	<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.

	===Quadratic-Bilinear System (QBS)===		===Quadratic-Bilinear System (QBS)===
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	<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>,
			<math>F \in \mathbb{R}^{n \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>.

			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)===
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	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.

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

Lund: /* Nonlinear Time-Invariant First-Order System (NLTI-FOS) */ F matrix

2023-04-20T19:14:12Z

Nonlinear Time-Invariant First-Order System (NLTI-FOS): F matrix

← Older revision		Revision as of 19:14, 20 April 2023
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	:<math>		:<math>
	\begin{align}		\begin{align}
	E\dot{x}(t) &= Ax(t) + Bu(t) + f(x(t),u(t)),\\		E\dot{x}(t) &= Ax(t) + Bu(t) + F f(x(t),u(t)),\\
	y(t) &= Cx(t) + Du(t),		y(t) &= Cx(t) + Du(t),
	\end{align}		\end{align}

Lund: /* Nonlinear Time-Invariant First-Order System (NLTI-FOS) */ put f term at end

2023-04-20T19:13:16Z

Nonlinear Time-Invariant First-Order System (NLTI-FOS): put f term at end

← Older revision		Revision as of 19:13, 20 April 2023
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	:<math>		:<math>
	\begin{align}		\begin{align}
	E\dot{x}(t) &= Ax(t) + f(x(t),u(t)~~) + Bu(t~~),\\		E\dot{x}(t) &= Ax(t) + Bu(t) + f(x(t),u(t)),\\
	y(t) &= Cx(t) + Du(t),		y(t) &= Cx(t) + Du(t),
	\end{align}		\end{align}

Lund: /* Nonlinear Time-Invariant Second-Order System (NLTI-SOS) */ add a comment about AP and TV NL systems

2023-04-20T19:12:24Z

Nonlinear Time-Invariant Second-Order System (NLTI-SOS): add a comment about AP and TV NL systems

← Older revision		Revision as of 19:12, 20 April 2023
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	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>.

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

Lund: reorder models

2023-04-20T19:10:44Z

reorder models

Show changes

Lund: /* Affine Parametric Second-Order System */ expand section title

2023-04-20T19:02:02Z

Affine Parametric Second-Order System: expand section title

← Older revision		Revision as of 19:02, 20 April 2023
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	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>.

	===Affine Parametric ~~Second~~-~~Order System~~===		===Affine Parametric LTI-SOS (AP-LTI-SOS)===

	:<math>		:<math>