Loewner Framework - Revision history

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	==Summary==		==Summary==
	In this note we have addressed the Loewner framework for model reduction of some classes of (nonlinear) dynamical systems.
			In this note we have addressed the '''Loewner framework''' for model reduction of some classes of (nonlinear) dynamical systems.
	The underlying philosophy valid for each of the extensions is as follows:		The underlying philosophy valid for each of the extensions is as follows:
	One needs to collect data, i.e. values of high-order transfer functions (either by performing measurements or by direct numerical simulations) and		One needs to collect data, i.e. values of high-order transfer functions (either by performing measurements or by direct numerical simulations) and
	then extract the desired information directly from the model constructed.		then extract the desired information directly from the model constructed.
	The strength of the proposed approaches consists in the fact that, given input-output data, one can construct with basically no computational cost, a model of the data in generalized state-space format.		The strength of the proposed approaches consists in the fact that, given input-output data, one can construct with basically no computational cost, a model of the data in generalized state-space format.

			==Further Variants==

			Additionally, there are further variants of the '''Loewner framework''' for parametric systems <ref name="antoulas12"/>, <ref name="ionita14"/>, singular systems <ref name="antoulas16b"/> and based on time-domain data <ref name="peherstorfer17"/>.

	==References==		==References==
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	<ref name="anderson90"> B.D.O. Anderson, A.C. Antoulas. <span class="plainlinks">[https://doi.org/10.1109/CDC.1990.203941 Rational interpolation and state variable realizations]</span>. Proceedings of the 29th IEEE Conference on Decision and Control: 1865--1870, 1990.</ref>		<ref name="anderson90"> B.D.O. Anderson, A.C. Antoulas. <span class="plainlinks">[https://doi.org/10.1109/CDC.1990.203941 Rational interpolation and state variable realizations]</span>. Proceedings of the 29th IEEE Conference on Decision and Control: 1865--1870, 1990.</ref>

	<~~!--~~ ref name="ionita14"> A.C. Ionita, A.C. Antoulas. <span class="plainlinks">[https://doi.org/10.1137/130914619 Data-Driven Parametrized Model Reduction in the Loewner Framework]</span>. SIAM J. Sci. Comput., 36(3): A984--A1007, 2014.</ref>		<ref name="ionita14"> A.C. Ionita, A.C. Antoulas. <span class="plainlinks">[https://doi.org/10.1137/130914619 Data-Driven Parametrized Model Reduction in the Loewner Framework]</span>. SIAM J. Sci. Comput. 36(3): A984--A1007, 2014.</ref>

			<ref name="antoulas12"> A.C. Antoulas, A.C.Ionita, S.Lefteriu. <span class="plainlinks">[https://doi.org/10.1016/j.laa.2011.07.017 On two-variable rational interpolation]</span>. Linear Algebra and its Applications 436(8): 2889--2915, 2012.</ref>

			<ref name="antoulas16b"> A.C. Antoulas. <span class="plainlinks">[https://doi.org/10.1016/j.aml.2015.10.011 The Loewner framework and transfer functions of singular/rectangular systems]</span>. Applied Mathematics Letters 54: 36--47, 2016.</ref>

	<ref name="~~antoulas12~~"> A.~~C.Antoulas~~, A.~~C.Ionita~~, S.~~Lefteriu~~ <span class="plainlinks">[https://doi.org/10.~~1016~~/~~j.laa~~.~~2011~~.07.~~017 On two-variable rational interpolation]</span>~~. ~~Linear Algebra and its Applications 436~~(8): ~~2889~~--~~2915~~, ~~2012~~.</ref -->		<ref name="peherstorfer17">B. Peherstorfer, S. Gugercin, and K. Willcox <span class="plainlinks">[https://doi.org/10.1137/16M1094750 Data-Driven Reduced Model Construction with Time-Domain Loewner Models]. SIAM J. Sci. Comput. 39(5): A2152--A2178,2017.</ref>

	</references>		</references>

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	</math>		</math>

	==Loewner Framework for Parametric Systems==		<!-- ==Loewner Framework for Parametric Systems== -->




	==Summary==		==Summary==

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 ==Loewner Framework for Linear Switched Systems==
+The restriction of the switching signal <math>\sigma(t)</math> to a finite interval of time <math>[0,T]</math> is given as
 ==Loewner Framework for Parametric Systems==

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	<ref name="anderson90"> B.D.O. Anderson, A.C. Antoulas. <span class="plainlinks">[https://doi.org/10.1109/CDC.1990.203941 Rational interpolation and state variable realizations]</span>. Proceedings of the 29th IEEE Conference on Decision and Control: 1865--1870, 1990.</ref>		<ref name="anderson90"> B.D.O. Anderson, A.C. Antoulas. <span class="plainlinks">[https://doi.org/10.1109/CDC.1990.203941 Rational interpolation and state variable realizations]</span>. Proceedings of the 29th IEEE Conference on Decision and Control: 1865--1870, 1990.</ref>

	<ref name="ionita14"> A.C. Ionita, A.C. Antoulas. <span class="plainlinks">[https://doi.org/10.1137/130914619 Data-Driven Parametrized Model Reduction in the Loewner Framework]</span>. SIAM J. Sci. Comput., 36(3): A984--A1007, 2014.</ref>		<!-- ref name="ionita14"> A.C. Ionita, A.C. Antoulas. <span class="plainlinks">[https://doi.org/10.1137/130914619 Data-Driven Parametrized Model Reduction in the Loewner Framework]</span>. SIAM J. Sci. Comput., 36(3): A984--A1007, 2014.</ref>

	<ref name="antoulas12"> A.C.Antoulas, A.C.Ionita, S.Lefteriu <span class="plainlinks">[https://doi.org/10.1016/j.laa.2011.07.017 On two-variable rational interpolation]</span>. Linear Algebra and its Applications 436(8): 2889--2915, 2012.</ref>		<ref name="antoulas12"> A.C.Antoulas, A.C.Ionita, S.Lefteriu <span class="plainlinks">[https://doi.org/10.1016/j.laa.2011.07.017 On two-variable rational interpolation]</span>. Linear Algebra and its Applications 436(8): 2889--2915, 2012.</ref -->

	</references>		</references>

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 ==Loewner Framework for Quadratic-Bilinear Systems==
+Consider quadratic-bilinear systems <math>\Sigma_{QB}</math> characterized by the following equations:
+:<math>
 ==Loewner Framework for Linear Switched Systems==

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	Next, we turn our attention to bilinear systems <math>\Sigma_B = (A,B,C,0,E,N)</math> characterized by equations:		Next, we turn our attention to bilinear systems <math>\Sigma_B = (A,B,C,0,E,N)</math> characterized by equations:
	:<math>		:<math>
	\Sigma_B : \begin{cases} E\dot{x}(t) &= Ax(t) + Nx(t)u(t) + Bu(t), \\ y(t) = Cx(t), \end{cases} \qquad (11)		\Sigma_B : \begin{cases} E\dot{x}(t) &= Ax(t) + Nx(t)u(t) + Bu(t), \\ y(t) &= Cx(t), \end{cases} \qquad (11)
	</math>		</math>
	where <math>N \in \mathbb{R}^{n \times n}</math> and the other quantities are exactly as in (1).		where <math>N \in \mathbb{R}^{n \times n}</math> and the other quantities are exactly as in (1).
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	</math>		</math>
	The external representation of the bilinear system <math>\Sigma_B</math> can be expressed in terms of [[wikipedia:Volterra_series\|Volterra series]], which describe the nonlinear mapping of admissible inputs <math>u</math> to the output <math>y</math>.		The external representation of the bilinear system <math>\Sigma_B</math> can be expressed in terms of [[wikipedia:Volterra_series\|Volterra series]], which describe the nonlinear mapping of admissible inputs <math>u</math> to the output <math>y</math>.
	The solution of (11) can be written as <math>x(t) = \sum_{i=1}^infty x_i(t)</math>.		The solution of (11) can be written as <math>x(t) = \sum_{i=1}^\infty x_i(t)</math>.
	By considering <math>x_{i-1}(t)</math> as a pseudo-input for the <math>i</math>th equation in (12),		By considering <math>x_{i-1}(t)</math> as a pseudo-input for the <math>i</math>th equation in (12),
	the frequency-domain behavior is described by a series of generalized transfer functions obtained by taking the multivariate Laplace transform of the degree <math>\ell</math> regular kernel.		the frequency-domain behavior is described by a series of generalized transfer functions obtained by taking the [[wikipedia:Multidimensional_transform#Multidimensional_Laplace_transform\|multivariate Laplace transform]] of the degree <math>\ell</math> regular kernel.
	More specifically, introduce the generalized multivariate transfer functions of system <math>\Sigma_B</math>, as follows (for <math>\ell = 1,2 \dots</math>)		More specifically, introduce the generalized multivariate transfer functions of system <math>\Sigma_B</math>, as follows (for <math>\ell = 1,2 \dots</math>)
	:<math>		:<math>
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	The explicit derivation of these types of transfer functions is based on the so-called Volterra series representation.		The explicit derivation of these types of transfer functions is based on the so-called Volterra series representation.
	The characterization of bilinear systems by means of rational functions suggests that reduction of such systems can be performed by means of ''interpolatory methods'',		The characterization of bilinear systems by means of rational functions suggests that reduction of such systems can be performed by means of ''interpolatory methods'',
	i.e. by extending the Loewner framework.		i.e. by extending the '''Loewner framework'''.

	Let <math>\{\mu_1,\mu_2,\dots,\mu_N\}</math> and <math>\{\lambda_1,\lambda_2,\dots,\lambda_N\}</math> be two set of left and respectively, right interpolation		Let <math>\{\mu_1,\mu_2,\dots,\mu_N\}</math> and <math>\{\lambda_1,\lambda_2,\dots,\lambda_N\}</math> be two set of left and respectively, right interpolation
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	\text{two of} \;\; H_1&:& H_1(\mu_1), H_1(\lambda_1) \\		\text{two of} \;\; H_1&:& H_1(\mu_1), H_1(\lambda_1) \\
	\text{three of} \;\; H_2 &:& H_2(\mu_1,\mu_2), H_2(\mu_1,\lambda_1), H_2(\lambda_2,\lambda_1), \\		\text{three of} \;\; H_2 &:& H_2(\mu_1,\mu_2), H_2(\mu_1,\lambda_1), H_2(\lambda_2,\lambda_1), \\
	\text{two of} \;\; H_3 &:& H_3(\mu_1,\mu_2,\lambda_1), H_3(\mu_1,\lambda_2,\~~lambda_2~~) \\		\text{two of} \;\; H_3 &:& H_3(\mu_1,\mu_2,\lambda_1), H_3(\mu_1,\lambda_2,\lambda_1) \\
	\text{one of} \;\; H_4&:& H_4(\mu_1,\mu_2,\lambda_2,\lambda_1),		\text{one of} \;\; H_4&:& H_4(\mu_1,\mu_2,\lambda_2,\lambda_1),
	\end{array}		\end{array}

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 ==Loewner Framework for Bilinear Systems==
+Let <math>\{\mu_1,\mu_2,\dots,\mu_N\}</math> and <math>\{\lambda_1,\lambda_2,\dots,\lambda_N\}</math> be two set of left and respectively, right interpolation
 ==Loewner Framework for Quadratic-Bilinear Systems==

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 We analyze linear time-invariant continuous-time systems described by the following equations:
 :<math>
-\Sigma : \begin{cases} E \dot{x}(t) = Ax(t) + Bu(t) \\ \;\;\; y(t) = Cx(t) + Du(t) \end{cases}
+\Sigma : \begin{cases} E \dot{x}(t) = Ax(t) + Bu(t) \\ \;\;\; y(t) = Cx(t) + Du(t) \end{cases} \quad(1)
 </math>
 where <math>A, E \in \mathbb{R}^{n \times n}</math>, <math>C \in \R^{p \times n}</math>, <math>B \in \mathbb{R}^{n \times m}</math>, <math>D \in \mathbb{R}^{p \times m}</math> and <math>x(t) \in \R^n</math>, <math>u(t) \in R^m</math>, <math>y(t) \in \R^p</math> are state, input and output vector respectively.
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 For simplicity, assume that the data set contains an even number of measurements denoted with <math>2N</math>.
 The next step is to partition the data measurements set into two disjoint data subsets, i.e.  the left subset and the right subset as described in the following
-: Left set: <math>(\mu_i,\ell_i,v_i), i = 1,\dots,N</math>, Right set: <math>(\lambda_j,r_j,w_j), j=1,\dots,N</math>. (2)
+: Left set: <math>(\mu_i,\ell_i,v_i), i = 1,\dots,N</math>, Right set: <math>(\lambda_j,r_j,w_j), j=1,\dots,N. \quad (2)</math>
-More precisely, the data in (2) is composed of <math>N</math> left interpolation points: <math>\{\mu_i\}_{i=1}^N \subset \mathbb{C}</math>, <math>N</math>, left sample values: <math>\{v_i\}_{i=1}^N \subset \mathbb{C}^m</math> and <math>N</math> left tangential directions: <math>\{\ell_i\}_{i=1}^N \subset \mathbb{C}^p</math>.
+More precisely, the data in (2) is composed of <math>N</math> left interpolation points: <math>\{\mu_i\}_{i=1}^N \subset \mathbb{C}</math>, <math>N</math> left sample values: <math>\{v_i\}_{i=1}^N \subset \mathbb{C}^m</math> and <math>N</math> left tangential directions: <math>\{\ell_i\}_{i=1}^N \subset \mathbb{C}^p</math>.
 Additionally, we have <math>N</math> distinct right interpolation points: <math>\{\lambda_i\}_{i=1}^N \subset \mathbb{C}</math> with <math>N</math> right sample values: <math>\{w_i\}_{i=1}^N \subset \mathbb{C}^p</math> and right tangential directions: <math>\{r_i\}_{i=1}^N \subset \mathbb{C}^m</math>.
-We seek a linear system <math>\widehat{Sigma}</math> such that the associated transfer function <math>\widehat{H}(s)</math> is a tangential interpolant to the original one, <math>H(s)</math>.
+We seek a linear system <math>\widehat{\Sigma}</math> such that the associated transfer function <math>\widehat{H}(s)</math> is a tangential interpolant to the original one, <math>H(s)</math>.
 More precisely, the following interpolation conditions should then hold:
 :<math>
 \begin{array}{rcl}
-\ell_i^* \widehat{H}(\mu_i) &= \ell_i^*H(\mu_i) = v_i^* \quad \text{for} \;\; i = 1,\dots,N, \\
+\ell_i^* \widehat{H}(\mu_i) &=& \ell_i^*H(\mu_i) = v_i^* \quad \text{for} \;\; i = 1,\dots,N, \\
-\widehat{H}(\lambda_j)r_j &= H(\lambda_j) r_j = w_j \quad \text{for} \;\; j = 1,\dots,N.
+\widehat{H}(\lambda_j)r_j &=& H(\lambda_j) r_j = w_j \quad \text{for} \;\; j = 1,\dots,N. \qquad(3)
 \end{array}
 </math>
 Next, arrange the measured data into matrix format as follows
 ==Loewner Framework for Bilinear Systems==

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	==Loewner Framework for Linear Systems==		==Loewner Framework for Linear Systems==

			We analyze linear time-invariant continuous-time systems described by the following equations:
			:<math>
			\Sigma : \begin{cases} E \dot{x}(t) = Ax(t) + Bu(t) \\ \;\;\; y(t) = Cx(t) + Du(t) \end{cases}
			</math>
			where <math>A, E \in \mathbb{R}^{n \times n}</math>, <math>C \in \R^{p \times n}</math>, <math>B \in \mathbb{R}^{n \times m}</math>, <math>D \in \mathbb{R}^{p \times m}</math> and <math>x(t) \in \R^n</math>, <math>u(t) \in R^m</math>, <math>y(t) \in \R^p</math> are state, input and output vector respectively.
			The transfer function corresponding to <math>\Sigma:(A,B,C,D,E)</math> is computed as <math>H(s) = C(sE-A)^{-1}B + D</math>.

			The Loewner framework can construct a reduced-order system by means of measured data.
			In this context, data represent sample values of the underlying transfer function <math>H(s)</math> at selected interpolation points (in some cases on the imaginary axis, when approximating the frequency response of the original system).
			A Loewner matrix pencil <math>\lambda\mathbb{L}-\mathbb{L}_s</math> is composed of the Loewner matrix <math>\mathbb{L}</math> and the shifted Loewner matrix <math>\mathbb{L}_s</math>.
			This pencil represents a surrogate of the original matrix pencil <math>\lambda E - A</math>.

			The Loewner framework originates from rational approximation theory (see <ref name="mayo07"/><ref name="antoulas17"/>) and can be used to recover or reduce the original dynamical system by means of interpolating the transfer function.
			For simplicity, assume that the data set contains an even number of measurements denoted with <math>2N</math>.
			The next step is to partition the data measurements set into two disjoint data subsets, i.e. the left subset and the right subset as described in the following
			: Left set: <math>(\mu_i,\ell_i,v_i), i = 1,\dots,N</math>, Right set: <math>(\lambda_j,r_j,w_j), j=1,\dots,N</math>. (2)
			More precisely, the data in (2) is composed of <math>N</math> left interpolation points: <math>\{\mu_i\}_{i=1}^N \subset \mathbb{C}</math>, <math>N</math>, left sample values: <math>\{v_i\}_{i=1}^N \subset \mathbb{C}^m</math> and <math>N</math> left tangential directions: <math>\{\ell_i\}_{i=1}^N \subset \mathbb{C}^p</math>.
			Additionally, we have <math>N</math> distinct right interpolation points: <math>\{\lambda_i\}_{i=1}^N \subset \mathbb{C}</math> with <math>N</math> right sample values: <math>\{w_i\}_{i=1}^N \subset \mathbb{C}^p</math> and right tangential directions: <math>\{r_i\}_{i=1}^N \subset \mathbb{C}^m</math>.
			We seek a linear system <math>\widehat{Sigma}</math> such that the associated transfer function <math>\widehat{H}(s)</math> is a tangential interpolant to the original one, <math>H(s)</math>.
			More precisely, the following interpolation conditions should then hold:
			:<math>
			\begin{array}{rcl}
			\ell_i^* \widehat{H}(\mu_i) &= \ell_i^H(\mu_i) = v_i^ \quad \text{for} \;\; i = 1,\dots,N, \\
			\widehat{H}(\lambda_j)r_j &= H(\lambda_j) r_j = w_j \quad \text{for} \;\; j = 1,\dots,N.
			\end{array}
			</math>
			Next, arrange the measured data into matrix format as follows

	==Loewner Framework for Bilinear Systems==		==Loewner Framework for Bilinear Systems==

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	{{preliminary}} <!-- Do not remove -->		{{preliminary}} <!-- Do not remove -->

	The '''Loewner Framework''' is a frequency-domain system identification and model reduction method, named after [[wikipedia:Charles_Loewner\|Charles Loewner]],		The '''Loewner Framework''' is a frequency-domain [[wikipedia:System_Identification\|system identification]], [[wikipedia:Realization_(systems)\|system realization]] and model reduction method, named after [[wikipedia:Charles_Loewner\|Charles Loewner]],
	who introduced the [[wikipedia:Charles_Loewner#Loewner_matrix_theorem\|Loewner Matrix]] which is essential to this method.		who introduced the [[wikipedia:Charles_Loewner#Loewner_matrix_theorem\|Loewner Matrix]] which is essential to this method based on ideas from <ref name="anderson90"/>.



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	<ref name="carleman32">T. Carleman. <span class="plainlinks">[https://doi.org/10.1007/BF02546499 Application de la théorie des équations intégrales linéaires aux systèmes d'équations différentielles non linéaires]</span>. Acta Math. 59: 63--87, 1932.</ref>		<ref name="carleman32">T. Carleman. <span class="plainlinks">[https://doi.org/10.1007/BF02546499 Application de la théorie des équations intégrales linéaires aux systèmes d'équations différentielles non linéaires]</span>. Acta Math. 59: 63--87, 1932.</ref>

	<ref name="mccormick76"> G.P. McCormick. <span class="plainlinks">[https://doi.org/10.1007/BF01580665 Computability of global solutions to factorable nonconvex programs: Part I -- Convex underestimating problems]</span>. Mathematical Programming 10(1): 147--175 1976.</ref>		<ref name="mccormick76"> G.P. McCormick. <span class="plainlinks">[https://doi.org/10.1007/BF01580665 Computability of global solutions to factorable nonconvex programs: Part I -- Convex underestimating problems]</span>. Mathematical Programming 10(1): 147--175, 1976.</ref>

	<ref name="anderson90"> B.D.O. Anderson, A.C. Antoulas. <span class="plainlinks">[https://doi.org/10.1109/CDC.1990.203941 Rational interpolation and state variable realizations]</span>. Proceedings of the 29th IEEE Conference on Decision and Control: 1865--1870, 1990.</ref>		<ref name="anderson90"> B.D.O. Anderson, A.C. Antoulas. <span class="plainlinks">[https://doi.org/10.1109/CDC.1990.203941 Rational interpolation and state variable realizations]</span>. Proceedings of the 29th IEEE Conference on Decision and Control: 1865--1870, 1990.</ref>