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Intrusive Galerkin Projection for Model Order Reduction of Uncertain Linear Dynamic Systems
Proceedings of the 20th World Congress Proceedings of 20th Proceedings of the the 20th World World Congress The International Federation of Congress Automatic Control Proceedings of the 20th World Congress The Federation of Control The International International Federation of Automatic Automatic Control Available online at www.sciencedirect.com Toulouse, France, July 9-14, 2017 The International Federation of Automatic Control Toulouse, Toulouse, France, France, July July 9-14, 9-14, 2017 2017 Toulouse, France, July 9-14, 2017
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IFAC PapersOnLine 50-1 (2017) 2738–2743
Intrusive Galerkin Projection for Model Intrusive Galerkin Projection for Model Intrusive Galerkin Projection forLinear Model Order Reduction of Uncertain Order Reduction of Uncertain Linear Order Reduction Uncertain Linear DynamicofSystems Dynamic Systems Dynamic Systems L. Nechak ∗∗∗ H-F.Raynaud ∗∗ C. Kulcs´ ar ∗∗ ∗∗ ∗∗ C. Kulcs´ ∗∗ L. Nechak a r ∗∗ L. Nechak H-F.Raynaud a ∗ H-F.Raynaud ∗∗ C. Kulcs´ L. Nechak H-F.Raynaud C. Kulcs´ arr ∗∗ ∗ Laboratoire de Tribologie et Dynamique des Syst`emes, UMR CNRS ∗ ∗ Laboratoire de Tribologie et Dynamique des Syst` emes, UMR CNRS Laboratoire de Tribologie et Syst` UMR CNRS ∗ 5513, ´ Ecole Centrale de Lyon, 36 avenuedes Guy deeemes, Collongue Laboratoire de Tribologie et Dynamique Dynamique des Syst` mes, UMR 69134 CNRS ´ ´ 5513, Ecole Centrale de Lyon, 36 avenue Guy de Collongue 69134 5513, Ecole Centrale de Lyon, 36 avenue Guy de Collongue 69134 ´Ecole ´ Cedex, France (e-mail: [email protected]). 5513, Ecully Centrale de Lyon, 36 avenue Guy de Collongue 69134 ´ ´ Ecully Cedex, France (e-mail: [email protected]). ∗∗ Ecully Cedex, (e-mail: [email protected]). ´ Institut d’Optique Graduate School, Laboratoire Charles Fabry, Ecully Cedex, France France (e-mail: [email protected]). ∗∗ ∗∗ Institut d’Optique Graduate School, Laboratoire Charles Fabry, d’Optique Graduate School, Laboratoire Charles Fabry, ∗∗ Institut CNRS, d’Optique 2 avenue Fresnel, 91127 Palaiseau, France (e-mail: Institut Graduate School, Laboratoire Charles Fabry, CNRS, 2 avenue Fresnel, 91127 Palaiseau, France (e-mail: CNRS, 2 avenue Fresnel, 91127 Palaiseau, France (e-mail: CNRS, [email protected], avenue Fresnel, 91127 Palaiseau, France (e-mail: [email protected], [email protected], [email protected]). [email protected], [email protected]). [email protected]). [email protected]). Abstract: This paper deals with model order reduction of random parameter-dependent (RPD) Abstract: paper deals with model reduction of random parameter-dependent (RPD) Abstract: This paper deals model order reduction of (RPD) linear time This invariant (LTI) systems byorder using the RPD-balanced realization (BR), recently Abstract: This paper (LTI) deals with with modelby order reduction of random random parameter-dependent parameter-dependent (RPD) linear time invariant systems using the RPD-balanced realization (BR), recently linear time invariant (LTI) systems by using the RPD-balanced realization (BR), recently developed byinvariant the same(LTI) authors. A novel way to deal with this crucial issue is(BR), presented. It linear time systems by using the RPD-balanced realization recently developed by the same authors. A novel way to deal with this crucial issue is presented. It developed the same authors. A novel way to deal with this crucial issue is presented. It is based onby the intrusive Galerkin projection of the RPD-BR. Indeed, it is shown, through developed by the same authors. A novel way to deal with this crucial issue is presented. It is based on the intrusive projection of the RPD-BR. it is shown, through is based the Galerkin projection of the Indeed, it is through numerical simulations, thatGalerkin the intrusive Galerkin projection of theIndeed, RPD-BR into the generalized is based on on the intrusive intrusive Galerkin projection ofprojection the RPD-BR. RPD-BR. Indeed, it into is shown, shown, through numerical simulations, that the intrusive Galerkin of the RPD-BR the generalized numerical simulations, that the intrusive Galerkin projection of the RPD-BR into the generalized polynomial chaos (GPC) space enables to generate a deterministic realization preserving numerical simulations, that the intrusive Galerkin projection of the RPD-BR into the generalized polynomial chaos (GPC) (GPC) space enables enables to generate generate deterministic realization preserving polynomial chaos space to aaa the deterministic realization stability properties for the original RPD-model. Moreover, new deterministic statepreserving variables, polynomial chaos (GPC) space enables to generate deterministic realization preserving stability properties for the original RPD-model. Moreover, the new deterministic state variables, stability properties for the original RPD-model. Moreover, the new deterministic state variables, which areproperties the stochastic modes of the RPD-balanced state the variables, are ordered with respect to stability for the original RPD-model. Moreover, new deterministic state variables, which are the stochastic modes of the RPD-balanced state variables, are ordered with respect to which are the stochastic modes of the RPD-balanced state variables, are ordered with respect to their sub-contributions to the RPD-input/output behaviour, measured by deterministic Hankel which are the stochastic modes of the RPD-balanced state variables, are ordered with respect to their sub-contributions to the the RPD-input/output behaviour, measured by deterministic deterministic Hankel their sub-contributions to behaviour, measured by Hankel values. Hence, a deterministic reduced order model is derived by deleting the state variables their sub-contributions to the RPD-input/output RPD-input/output behaviour, measured by deterministic Hankel values. Hence, a deterministic reduced order model is derived by deleting the state variables values. Hence, a deterministic reduced order model is derived by deleting the state variables with small Hankel values and the complete RPD-reduced orderbymodel is then obtained. The values. Hence, a deterministic reduced order model is derived deleting the state variables with small values and the complete RPD-reduced order model is then obtained. The with small Hankel values and the complete RPD-reduced order model is then obtained. The feasibility ofHankel the proposed method is analyzed while its accuracy is compared to the RPDwith small Hankel values and the complete RPD-reduced order model is then obtained. The feasibility of the proposed method is analyzed while its accuracy is compared to the RPDfeasibility of method is truncated balanced realization (RPD-TBR). feasibility of the the proposed proposed method is analyzed analyzed while while its its accuracy accuracy is is compared compared to to the the RPDRPDtruncated balanced balanced realization (RPD-TBR). truncated realization (RPD-TBR). truncated balanced realization (RPD-TBR). © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Uncertain Systems, Random Parameters, Model Reduction, Balanced Truncation, Keywords: Systems, Reduction, Balanced Truncation, Keywords: Uncertain Uncertain Systems, Random Parameters, Model Reduction, Balanced Truncation, Generalized Polynomial Chaos,Random IntrusiveParameters, Projection, Model Non-Intrusive Spectral Projection. Keywords: Uncertain Systems, Random Parameters, Model Reduction, Balanced Truncation, Generalized Polynomial Chaos, Intrusive Projection, Non-Intrusive Spectral Projection. Generalized Polynomial Chaos, Intrusive Projection, Non-Intrusive Spectral Projection. Generalized Polynomial Chaos, Intrusive Projection, Non-Intrusive Spectral Projection. 1. INTRODUCTION points are generated from sparse grid techniques while 1. INTRODUCTION INTRODUCTION points are generated generated from sparse grid techniques while 1. points are techniques while interpolation functionsfrom are sparse chosengrid so the interpolation 1. INTRODUCTION points are generated from sparse grid techniques while interpolation functions are chosen so the interpolation interpolation functions are chosen so the interpolation error be minimized. In (Nechak, L. et al., 2015), a novel interpolation functions are chosen so the interpolation Model reduction has become essential to deal effectively error error betominimized. minimized. In (Nechak, L. et et al., 2015), 2015), a linear novel be (Nechak, L. novel scheme deal with In random parameter-dependent Model reduction has become become essential to deal deal effectively error betominimized. In (Nechak, L. et al., al., 2015), aa linear novel Model reduction has essential to with problems involving simulation, control andeffectively design of scheme scheme deal with random parameter-dependent Model reduction has become essential to deal effectively to deal with random parameter-dependent linear systems to is proposed. Indeed, authors defined, through the with problems involving simulation, control and design of scheme deal with random parameter-dependent linear with problems involving simulation, control and design of high problems dimensional systems. The maincontrol aim is and to efficiently systems is generalized proposed. Indeed, Indeed, authors defined, through the with involving simulation, design of systems proposed. authors defined, the use of theis polynomial chaos (GPC)through formalism, high dimensional systems. The main aim is to efficiently systems is proposed. Indeed, authors defined, through the high dimensional systems. The main aim is to efficiently generate, from a given large scale model, simpler one use of of the the generalized polynomialbalancing chaos formalism, high dimensional systems. Thescale main aim isaa to efficiently polynomial chaos (GPC) (GPC) formalism, a random parameter-dependent transformation generate, from aa the given large model, simpler one use use of the generalized generalized polynomialbalancing chaos (GPC) formalism, generate, from given large scale model, aa simpler one while preserving input/output behaviour and struca(RPD-BT) random parameter-dependent transformation generate, from a given large scale model, simpler one a random parameter-dependent balancing transformation which puts the associated RPD-linear system while preserving of thethe input/output behaviour andinterest struc- a random parameter-dependent balancing transformation while preserving the input/output behaviour and structural properties original model. A great (RPD-BT) which puts puts the associated associated RPD-linear system while preserving of thethe input/output behaviour andinterest struc- (RPD-BT) which the system in an almost-surely balanced form. A RPD-linear statistical analysis tural properties original model. A great (RPD-BT) which puts the associated RPD-linear system tural properties of the original model. A great interest was accorded to of model reduction of linear uncertain dy- in in an almost-surely balanced form. A A statistical analysis tural properties the reduction original model. A great interest form. analysis of an the almost-surely RPD-Hankel balanced singular values is statistical used to define an was accorded to these model of see, linear uncertain dyin an almost-surely balanced form. A statistical analysis was accorded to model reduction linear namical systems last decades,of e.g.,uncertain (Beck, C.dyL of of the RPD-Hankel singular values is used to define an was accorded to model reduction of linear uncertain dythe RPD-Hankel singular values is used to define an almost-sure truncation order for the RPD-BR within the namical systems these last decades, see, e.g., (Beck, C. L of the RPD-Hankel singular values is used to define an namical systems these see, e.g., (Beck, L et al., 1996), (Weile, D.last S. decades, et al., 1999), (Dolgin, Y.C. and almost-sure truncation order for the RPD-BR within the namical systems these last decades, see, e.g., (Beck, C. L almost-sure truncation order for the RPD-BR within the whole probabilistic distribution of random parameters. et al., 1996), 1996), (Weile, D. S. S. etand al.,Hahn, 1999),J., (Dolgin, Y. and and almost-sure truncation order for the RPD-BRparameters. within the et al., (Weile, D. al., 1999), (Dolgin, Y. Zeheb, E., 2005), (Sun, C. et 2006),(Wang, whole probabilistic distribution of random random et al., 1996), (Weile, D. S. etand al.,Hahn, 1999),J., (Dolgin, Y. and whole probabilistic distribution of State variables that are almost-surely weakly parameters. controllable Zeheb, E., 2005), (Sun, C. 2006),(Wang, whole probabilistic distribution of random parameters. Zeheb, E., 2005), (Sun, C. and Hahn, J., 2006),(Wang, Z. Z. etE., al.,2005), 2012). A supplementary issue related to the State State variables that that are almost-surely almost-surely weakly controllable Zeheb, C. and Hahn, J., related 2006),(Wang, variables are weakly and observable are then truncated, giving risecontrollable to a RPDZ. Z. et et al., al., 2012).(Sun, A supplementary supplementary issue to the the State variables that are almost-surely weakly controllable Z. Z. 2012). A to practice of model reduction for this issue class related of systems is and and observable are then truncated, giving rise to aasecond RPDZ. Z. et al., 2012). A supplementary issue related to the observable are then truncated, giving rise to RPDtruncated balanced realization (TBR). First and practice of model reduction for this class of systems is and observable are then truncated, giving rise to asecond RPDpractice of model reduction for this class of systems is to preserve the physical meaning of uncertainty and its truncated balanced realization (TBR). First and practice of model reduction for this class of systems is truncated balanced realization (TBR). First and second order moments of therealization error bound(TBR). relatedFirst to theand truncation to preserve the physical meaning of uncertainty and its truncated balanced second to preserve the physical meaning of uncertainty and its effect on thethedynamical behaviourofafter reduction. Nuorder moments of of thebyerror error bound of related to the the truncation to preserve physical meaning uncertainty andNuits order moments bound related to were characterized moments RPD-Hankel singular effect onstudies the dynamical dynamical behaviour after reduction. order moments of the thebyerror bound of related to the truncation truncation effect the reduction. Numerouson have beenbehaviour proposedafter in this framework. were characterized moments RPD-Hankel singular effect on the dynamical behaviour after reduction. Nuwere characterized by moments of RPD-Hankel singular values associated to the truncated state variables. merous studies have been proposed in this framework. were RPD-Hankel merous studies have proposed in Otherwise, the well-known truncated balanced realization values valuescharacterized associated to toby themoments truncatedofstate state variables.singular merous studies have been been truncated proposed balanced in this this framework. framework. associated truncated Otherwise, the well-known well-known realization values associated to the the truncated stateofvariables. variables. Otherwise, the truncated balanced realization (TBR) technique was extended to linear systems with This paper deals with model reduction RPD-linear sysOtherwise, the well-known truncated balanced realization (TBR) technique was extended to linear systems with This paper deals with model reduction of RPD-linear sys(TBR) technique was extended to linear systems with This paper deals with model reduction of RPD-linear sysuncertain parameters. Baur and to Benner have proposed tems. It proposes the GPC based intrusive Galerkin pro(TBR) technique was extended linear systems with This paper deals with modelbased reduction of RPD-linear sysuncertain parameters. Baur and Benner have proposed tems. It proposes the GPC intrusive Galerkin prouncertain parameters. Baur and Benner have proposed tems. It proposes the GPC based intrusive Galerkin proits combination with interpolation algorithms to proposed generate tems. jectionIttoproposes derive a the RPD-reduced order model. It is shown, uncertain parameters. Baur and Benner haveto GPC basedorder intrusive Galerkin proits combination with interpolation algorithms generate jection to derive a RPD-reduced model. It is shown, its combination with interpolation algorithms to derive order It shown, parametric reduced order models (Baur, U. and Benner, jection throughto numerical simulations, that themodel. intrusive Galerkin its combination withorder interpolation algorithms to generate generate jection tonumerical derive aa RPD-reduced RPD-reduced order It is isGalerkin shown, parametric reduced models (Baur, U. and Benner, through through simulations, that themodel. intrusive parametric reduced order models (Baur, U. and Benner, numerical simulations, that the intrusive Galerkin P., 2009), (Amsallem, D. and Farhat, C., 2011), (Panzer, projection for the RPD-BR, previously introduced by the parametric reduced order models (Baur, U. and Benner, through numerical simulations, that the intrusive Galerkin P., 2009), (Amsallem, D. and and Farhat, C., 2011), 2011), (Panzer, projection for the the RPD-BR, L. previously introduced by the the P., D. Farhat, C., (Panzer, for RPD-BR, previously introduced by H., 2009), 2010). (Amsallem, Reduced-order models are first determined at projection same authors in (Nechak, et al., 2015), produces P., 2009), (Amsallem, D. and Farhat, C., 2011), (Panzer, projection for the RPD-BR, L. previously introduced by theaa H., 2010). Reduced-order models are first determined at same authors in (Nechak, et al., 2015), produces H., 2010). Reduced-order models are first determined at same authors in (Nechak, L. et al., 2015), produces aa some2010). chosen parameter values thenare interpolated, yielding deterministic dynamic system which preserves the asympH., Reduced-order models first determined at same authorsdynamic in (Nechak, L.which et al.,preserves 2015), produces some chosen parameter values values then interpolated, yielding deterministic system the asympasympsome chosen parameter then interpolated, yielding deterministic dynamic system which preserves the a parameter-dependent reduced model. The calculating some chosen parameter values then interpolated, yielding deterministic dynamic system which preserves the asympparameter-dependent reduced model. The calculating calculating aa a parameter-dependent parameter-dependent reduced reduced model. model. The The calculating Copyright © 2017, 2017 IFAC 2793Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright 2017 2793 Copyright ©under 2017 IFAC IFAC 2793Control. Peer review© of International Federation of Automatic Copyright © 2017 responsibility IFAC 2793 10.1016/j.ifacol.2017.08.580
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 L. Nechak et al. / IFAC PapersOnLine 50-1 (2017) 2738–2743
totically stable behaviour of the original RPD-system. Moreover, the deterministic state variables (representing the GPC expansion coefficients of the RPD-balanced state variables) are ordered with respect to their subcontributions to the RPD-input/output behaviour, measured by the deterministic Hankel singular values. These sub-contributions are defined so as to preserve the initial order of the RPD-BR state variables. A reduced-order deterministic model can then be obtained by deleting, in the deterministic system, the state variables with small deterministic Hankel singular values. The truncation order is given by the truncation order of the GPC expansions times the number of almost surely weakly controllable and observable state variables. This method generates a deterministic reduced model which can be more efficiently used in optimal control schemes as defined in (Fisher, J. and Bhattacharya, R., 2009). Hence, the main objective of this study is to analyze and discuss the order reduction of the RPD-BR once projected intrusively into the GPC space. The intrusive and non-intrusive schemes of the GPC expansion are based on Galerkin projections. These are used to deal with numerous problems related to the stability of dynamical systems (Nechak, L. et al., 2011), the prediction of limit cycle oscillation in selfexcited uncertain nonlinear dynamical systems (Nechak, L. et al., 2012), (Nechak, L. et al., 2014), state observer design of uncertain systems (Smith, A. et al., 2007) or optimal control design (Fisher, J. and Bhattacharya, R., 2009). In this paper, the non-intrusive spectral projection is used to determine the RPD-balancing transformation, while the intrusive projection is used to determine the deterministic counterpart of the RPD-BR and thus the associated reduced-order realization. This paper is then organized as follows: the generalized polynomial chaos formalism is recalled in Section 2, followed by a summary of the RPD-BR concept, with the associated projected realization in Section 3; numerical simulations are presented and discussed in Section 4, while a conclusion is given at the end of the paper.
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where x ¯i (for i = 0, 1, ..., P ) are deterministic coefficients, called the stochastic modes of X, and x ¯0 represents its mean value. Otherwise, the number of stochastic modes P +1 is shown to be dependent on the stochastic dimension d and the chaos order p defined by the maximum degree of the used polynomial, that is, P + 1 = (p + d)!/p!d!. The stochastic modes can be obtained via intrusive or nonintrusive schemes exploiting the orthogonality property of GPC, which results in < φi (ξ), φj (ξ) >= δij �φi (ξ)�2 f (ξ)dξ (2) ξ
where δij is the delta Kronecker and f (ξ) = the probability density function of ξ.
d
i=1
fξi is
3. RANDOM PARAMETER-DEPENDENT REDUCED MODELS Let
˙ ξ) = A(ξ)x(t, ξ) + B(ξ)u(t) x(t, y(t, ξ) = C(ξ)x(t, ξ)
(3)
be a linear time invariant system where the state (A), control (B), output (C) matrices are ξ-dependent, as well as the state x(t) ∈ Rn and the output y(t) ∈ Rq vectors. Without loss of generality, the input vector u(t) ∈ Rm is assumed to be independent from the random variable ξ. Moreover, it is assumed that the initial condition does not depend on ξ, that is, x(0, ξ) = x0 . In the following, System (3) is assumed to be almost surely asymptotically stable and almost surely controllable and observable, see (Nechak, L. et al., 2015). Hence, the associated RPD-controllability and observability gramians are then given by the almost surely defined solutions of the RPD versions of Lyapunov equations: A(ξ)W c (ξ) + W c (ξ)A(ξ)T = −B(ξ)B(ξ)T T
T
A(ξ) W o (ξ) + W o (ξ)A(ξ) = −C(ξ) C(ξ)
(4) (5)
3.1 RPD-balanced realization 2. GENERALIZED POLYNOMIAL CHAOS Let (Ω, η, Pr) be a probability space where Ω is the sample space, η the σ-algebra of the subsets of Ω and Pr is the probability measure; let ξ = {ξi }di=1 be a countable set of independent, identically distributed random variables defined from (Ω, η) to (Rd , η d ) where, η d is the σalgebra of Borel subsets of Rd . Let X(ξ(ω)) be a second order stochastic process defined within the Hilbert space L2 (Ω, η, Pr). Otherwise, for the sake of simplicity, the shorthand notation X(ξ) or X will be used instead of X(ξ(ω)). Hence, from the GPC theory, (Xiu, W. and Karniadakis, G. E., 2003) and the Cameron-Martin theorem, (Cameron, H. and Martin, W., 1947), any stochastic process defined as X(ξ) can be approximated by a convergent (in the mean square sense) polynomial series functions φi that are orthogonal with respect to the probability density measures fk , k = 1, ..., d associated to the random variables ξk , k = 1, ..., d. X(ξ) ≈
P i=1
Definition 1: The almost surely minimal and asymptotically stable RPD-LTI system obtained via the RPDtransformation xb (t, ξ) = T −1 b (ξ)x(t, ξ) given by: x˙ b (t, ξ) = Ab (ξ)xb (t, ξ) + B b (ξ)u(t) (6) y(t, ξ) = C b (ξ)xb (t, ξ)
is said to be almost surely balanced if: (7) W c (ξ) = W o (ξ) = Σ(ξ) where the diagonal matrix Σ(ξ) = diag(σ1 (ξ), ..., σn (ξ)) is almost surely solution of the pair of RPD-Lyapunov equations: Ab (ξ)Σ(ξ) + Σ(ξ)Ab (ξ)T = −B b (ξ)B b (ξ)T (8)
Ab (ξ)T Σ(ξ) + Σ(ξ)Ab (ξ) = −C b (ξ)T C b (ξ) (9) th and σi , i = 1, ..., n is the i RPD-Hankel singular value which can be expressed by a truncated GPC expansion σi (ξ) ≈
P
σ ¯i,j φj (ξ)
(10)
j=0
x ¯i φi (ξ)
(1)
where σ ¯i,0 denotes the mean value of the ith Hankel singular value, while the higher-order terms σ ¯i,j define its
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dispersion. The main step is then to determine the RBDbalanced transformation Tb (ξ). The latter is obtained by computing finite order GPC expansions for its elements tij (ξ): tij (ξ) ≈
P �
t¯ij,k φk (ξ)
(11)
k=0
The coefficients t¯ij,k are obtained from the non-intrusive spectral projection of tij (ξ) into the GPC space. They are given by: �tij (ξ), φk (ξ)� t¯ij,k = (12) �φk (ξ)2 � Numerical integration techniques are used to obtain approximations of the coefficients (12), see (Nechak, L. et al., 2015) for more details. The RPD-BR matrices are then obtained as follows: b A (ξ) = T −1 b (ξ)A(ξ)T b (ξ) (13) B b (ξ) = T −1 b (ξ)B(ξ) b C (ξ) = C(ξ)T b (ξ)
In (Nechak, L. et al., 2015), a RPD-TBR is generated by suppressing the state variables with small Hankel singular values which indicate that the corresponding variables are almost surely weakly observable and controllable. To perform order reduction, an almost surely truncation order is determined by a statistical analysis of the RPD-Hankel singular values. Definition 2: An integer 1 < r < n is an almost sure truncation order for the RPD-LTI system (6) if � σr (ξ) < σi (ξ), ∀i ∈ 1, 2, ..., r − 1 (14) σr (ξ) > σj (ξ), ∀j ∈ r + 1, ..., n
Once the almost sure truncation order r has been selected, the RPD-TBR can be generated by deleting the corresponding state variables. In this paper, another way to perform model reduction is proposed and analyzed. It consists in performing an intrusive projection of the RPD-BR into the GPC space. An augmented deterministic state space representation is then obtained where the associated state variables are the stochastic modes of the GPC expansions of the RPD-state variables. A deterministic reduced-order representation is derived from the augmented system and is used to obtain a RPD-reduced model. This method is presented in the following. 3.2 RPD-truncation through intrusive Galerkin projection of the RPD-BR The main idea of the proposed intrusive RPD-TBR is to carry out a Galerkin projection of the RPD-BR into the GPC space after replacing all random quantities by their GPC expansions given as follows: ¯ b (t)T Φ(ξ) xbi (t, ξ) ≈ X i b b T ¯ y (t, ξ) ≈ Y i i (t) Φ(ξ) T u (t, ξ) ≈ U ¯ i (t) Φ(ξ) i (15) b b ¯ aij (ξ) ≈ (Aij )T Φ(ξ) ¯ b )T Φ(ξ) bbij (ξ) ≈ (B ij cb (ξ) ≈ (C ¯ b )T Φ(ξ) ij ij
b T ¯ (t) = [¯ xi,0 (t), ..., x ¯i,P (t)] X i b T yi,0 (t), ..., y¯i,P (t)] Y¯ i (t) = [¯ T ¯ ui,0 (t), ..., u ¯i,P (t)] U i (t) = [¯ T ¯ b = [¯ where A aij,0 , ..., a ¯ij,P ] ij �T � b ¯ = ¯bij,0 , ..., ¯bij,P B ij b T ¯ = [¯ cij,0 , ..., c¯ij,P ] C ij T Φ(ξ) = [φ0 (ξ), ..., φP (ξ)]
Note that since ui (t) is independent on the radom variable ξ, the associated stochastic modes u ¯i,k (t), k = 1, ..., P are null. Carrying out of the two previous steps gives rise to a n × (P + 1)-dimensional linear deterministic state space representation. The new state vector is defined by the coefficients of GPC expansions of the RPD-state variables, (Fisher, J. and Bhattacharya, R., 2009) �
¯ bX ¯ bU ¯ b (t) + B ¯ (t) ¯˙ b (t) = A X b ¯ ¯ ¯ Y b (t) = C b X (t)
(16)
¯ b ∈ Rn(P +1) , A ¯ b ∈ Rn(P +1)×n(P +1) , B ¯b ∈ where X m(P +1)×n(P +1) ¯ R , C b ∈ Rn(P +1)×q(P +1) , Y¯ b ∈ Rq(P +1) , ¯ ∈ Rm(P +1) U After projecting the RPD-BR and since the Galerkin projection does not modify structural properties, the deterministic state variables in the resulting projected system (16), representing the coefficients of the GPC expansions of the RPD-state variables of the RPD-BR, will be ordered so as the order of the original RPD-Hankel singular values is preserved. The deterministic Hankel singular values associated to the resulting deterministic measure the sub-contributions of the new state variables (representing the stochastic modes of the RPD-state variables) to the RPD-input/output behaviour. So, if r is the almost sure truncation order of the RPD-BR, r(P + 1) will be the truncation order of the augmented system (16). So, by ¯b ¯b setting variables X (r+1)(P +1) , ..., X n(P +1) equal to zero, a reduced order system is obtained. It is expressed as follows: � ¯ rX ¯ rU ¯ r (t) + B ¯ (t) ¯˙ r (t) = A X (17) ¯ rX ¯ r (t) Y¯ r (t) = C ¯ r ∈ Rr(P +1)×r(P +1) , B ¯r ∈ ¯ r ∈ Rr(P +1) , A where X ¯ r ∈ Rq(P +1)×r(P +1) while the reduced Rr(P +1)×m(P +1) , C matrices are given by P � r r ¯ ¯ = [A ], A ≈ a ¯rij,k Φk A r ij ij k=0 P � ¯br Φk ¯r ≈ ¯ r = [B r ], B (18) B ij,k ij ij k=0 P � ¯r ≈ ¯ r = [C r ], C c¯rij,k Φk C ij ij k=0
e¯0k0 e¯0k1 where Φk = ...
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e¯0kp
e¯0k1 . . . e¯0kp e¯1k1 . . . e¯1kp with e¯ijk = .. . . .. . . . e¯1kp . . . e¯pkp
�φi ,φj φk � �φ2i �
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 L. Nechak et al. / IFAC PapersOnLine 50-1 (2017) 2738–2743
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Mean value of step response
3
Fig. 1. Mechanical system
2.5
2
1.5
1
0.5
4. APPLICATION AND RESULTS 0
4.1 RPD-Balanced realization The RPD-BR associated to the considered system is generated through a RPD-balancing transformations (11) with p = 3 for Legendre polynomials. The RPD-BR must reproduce the same input-output behaviour as the original RPD-system. This is verified by assessing the accuracy of the first and second order moments of the step response of the RPD-BR. This task is performed by using reference results obtained by the MC method. The instantaneous mean value and variance of the step response of the obtained RPD-BR coincide with the reference results as shown in Fig. 2 and Fig. 3 respectively.
5
10
15
20
Time (sec)
25
30
35
40
−3
9
x 10
Variance of step response
8 7 6 5 4 3 2 1 0
0
5
10
15
20
Time (s)
25
30
35
40
Fig. 3. Instantaneous variance of the step response:—Reference, - - - - RPD-BR is also observed with respect to the reference results. The 1.5
1
4.2 Intrusive RPD-Balanced realization The RPD-BR is projected into the GPC space through a Galerkin projection. The same order p = 3 is used. An augmented deterministic system is obtained. The eigenvalues of the augmented systems derived from projections of original RPD-system and of the associated RPD-BR are plotted in Fig. 4. Eigenvalues of the projected RPDBR coincide with those of the projected original systems. Both projections exhibit the same asymptotically stable dynamics. The instantaneous mean value and variance of the step response of the system obtained from the projected RPD-BR are plotted in Fig. 5. Convenient accuracy
0
Fig. 2. Instantaneous mean value of the step response:—Reference, - - - - RPD-BR
Imaginary parts
In this Section, the proposed method for model reduction of RPD-linear systems is applied to an academic two degrees of freedom mass-spring system (Fig. 1). The control input u = F and the output y are respectively the force applied to and the position of the object of unitary mass m. The damping coefficients d1 = 1 and d2 = 0.65 and the stiffness parameter k2 = 1 are considered deterministic, while the stiffness parameter k1 is assumed to be a random variable with uniform distribution within the interval [0.9, 1.1]. The normalized uniform variable ξ within [-1, 1] is used to describe randomness of k1 through k1 = 1 + 0.1ξ. Hence, for the state vector x = (x1 , x2 , x˙ 1 , x˙ 2 )T the system modelis a RPD-LTI representation in theform (3) with: 0 0 1 0 0 0 0 1 0 k (ξ) k (ξ) d1 d1 0 1 1 A = − , B = 1 , m m1 m1 m1 k1 (ξ)1 k1 (ξ) d1 + d 2 + k2 d1 0 − m2 m2 m2 m2 C = [ 1 0 0 0 ].
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Fig. 4. Eigenvalues within the complex plan: ∗ projection of original RPD-system, + projection of RPD-BR reducibility of the RPD-original system can be analysed through the analysis of the RPD-Hankel singular values.
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Probability density function
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Fig. 5. Instantaneous mean of the step response:—- Reference, - - - - Projected RPD-BR
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Fig. 6. Instantaneous variance of the step response:—Reference, - - - - Projected RPD-BR This enables one to select an almost sure truncation order. GPC expansions can be used in this context, see (Nechak, L. et al., 2015). The probability density functions of the RPD-Hankel singular values are plotted in Fig.7. It can be observed that the two first Hankel singular values are distributed around higher values compared with the two remaining Hankel values. Consequently, r = 2 can be considered as the almost sure truncation order of the RPDBR. In Fig. 8, Hankel singular values of the intrusive projections of the original RPD-system and the associated RPD-BR coincide, showing the accuracy of the built RPDBR and its intrusive projection. Furthermore, the Hankel singular values aggregate into two subsets (Fig. 7): the last eight singular values define the sub-contributions of the coordinates of the projected RPD-state variables to the RPD input/output behaviour. They present small levels which is consistent with the almost surely weak controllability and observability of the two last RPD-state variables. The same remark can be stated about the first eight Hankel singular values. Consequently, r(P + 1) = 8 can be considered as the truncation order of the projected RPD-BR.
Fig. 8. Hankel singular values of the intrusive projections: ∗ RPD-original system, + RPD-BR A reduced RPD-system is generated from the truncated intrusive projection of the RPD-BR (obtained by suppressing the last eight state variables). The instantaneous mean values of its step responses is plotted in figure (9) in comparison with the reference and the instantaneous mean value of the step responses of the RPD-TBR defined in (Nechak, L. et al., 2015). The associated variances are plotted in Fig. 10. While, the average behaviour represented by the instantaneous mean value of the step response is accuraly approximated, a relative error is apparent for the approximated variance. The RPD-TBR gives the best approximation of the variance compared with the RPDreduced model obtained from the intrusive projection of the RPD-BR. This result is consistent with what is expected. The difference in accuracy between the two RPDreduced models is related to the intrusive Galerkin projection which introduces additional numerical errors. 5. CONCLUSION A novel way to generate a reduced uncertain system from the intrusive projection of the RPD-BR is discussed. It is shown through numerical simulations that the organization of the RPD-state variables with respect to their
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6. ACKNOWLEDGMENTS
Mean value of step response
3
This work has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 730890.
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REFERENCES
1.5
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Fig. 9. Instantaneous mean of the step response:—- Reference, - - - - Reduced Projected RPD-BR, -. -. -. -. RPDTBR
0.01
Variance of the step response
0.009 0.008 0.007 0.006 0.005 0.004 0.003 0.002 0.001 0
0
5
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15
20
Time (s)
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Fig. 10. Instantaneous variance of the step response:—Reference, - - - - Reduced Projected RPD-BR, -. -. -. -. RPD-TBR
almost sure controllability and observability is preserved once projected into the GPC space. The generated RPDreduced model exhibits suitable accuracy in the estimation of the first an second order moments of the step response. This study has presented the method with one uncertain parameter. The case of multiple uncertain parameters induces higher chaos orders and thus higher dimension for the intrusive projections. This is the main drawback of methods based on the GPC expansion. However, numerical techniques exist to reduce the chaos order, such as the multi-element generalized polynomial chaos. Otherwise, this study will be completed by the evaluation of the truncation error which consists of two main elements. The first one is related to the truncation order of the GPC expansion while the second is related to truncation of the projected system. Finally, the stability of the proposed reduced model will be investigated in future research.
Amsallem, D. and Farhat, C. (2011). An online method for interpolating linear parametric reduced order models. SIAM Journal on Scientific Computing, 33, 2169–2198. Baur, U. and Benner, P. (2009). Model reduction of parametric systems using balanced truncation and interpolation. Automatisierungstechnik, 57. Beck, C. L, Doyle, J., and Glover, K. (1996). Model reduction of multi-dimensional and uncertain systems. IEEE Transactions on Automatic Control, 41, 1466–477. Cameron, H. and Martin, W. (1947). The orthogonal development of nonlinear functionals in series of fourierhermite functional. Annals of Mathematics, 48, 385. Dolgin, Y. and Zeheb, E. (2005). Model reduction of uncertain systems retaining the uncertainty structure. Systems Control Letters, 54, 771–779. Fisher, J. and Bhattacharya, R. (2009). Linear quadratic regulation of systems with stochastic parameter uncertainties. Automatica, 45, 2831–2841. Nechak, L., Berger, S., and Aubry, E. (2011). A polynomial chaos approach to the robust analysis of the dynamic behaviour of friction systems. European Journal of Mechanics A/Solids, 30, 594–607. Nechak, L., Berger, S., and Aubry, E. (2012). Wienerhaar expansion for the modelling and prediction of the dynamic behaviour of nonlinear uncertain systems. ASME Journal of Dynamic Systems Measurement and Control, 134, 11. Nechak, L., Berger, S., and Aubry, E. (2014). Wiener askey and wiener haar expansions for the analysis and prediction of limit cycle oscillations in uncertain nonlinear dynamic friction systems. ASME Journal of computational and nonlinear dynamics, 9, 12. Nechak, L., Raynaud, H.-F., and Kulcs´ar, C. (2015). Model order reduction of random parameter-dependent linear systems. Automatica, 55, 95–107. Panzer, H. (2010). Parametric model reduction by matrix interpolation. Automatisierungstechnik, 58. Smith, A., Monti, A., and Ponci, F. (2007). Indirect measurement via a polynomial chaos observer. IEEE Transactions on Instrumentation and Measurement, 56, 743–752. Sun, C. and Hahn, J. (2006). Model reduction in the presence of uncertainty in model parameters. Journal of Process Control, 16, 645–649. Wang, Z. Z., Li, L., and Wang, W. F. (2012). Modification algorithm on routh-pade model reduction of interval systems. In Lecture Notes in Computer Science, 6838, 701–704. Weile, D. S., Michielssen, E., and Grimme, E. (1999). A method for generating rational interpolant reduced order models of two parameters linear systems. Applied Mathematics Letters, 12, 93–102. Xiu, W. and Karniadakis, G. E. (2003). Modelling uncertainty in flow simulations via generalized polynomial chaos. Journal of Computational Physics, 187, 137–167.
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Intrusive Galerkin Projection for Model Intrusive Galerkin Projection for Model Intrusive Galerkin Projection forLinear Model Order Reduction of Uncertain Order Reduction of Uncertain Linear Order Reduction Uncertain Linear DynamicofSystems Dynamic Systems Dynamic Systems L. Nechak ∗∗∗ H-F.Raynaud ∗∗ C. Kulcs´ ar ∗∗ ∗∗ ∗∗ C. Kulcs´ ∗∗ L. Nechak a r ∗∗ L. Nechak H-F.Raynaud a ∗ H-F.Raynaud ∗∗ C. Kulcs´ L. Nechak H-F.Raynaud C. Kulcs´ arr ∗∗ ∗ Laboratoire de Tribologie et Dynamique des Syst`emes, UMR CNRS ∗ ∗ Laboratoire de Tribologie et Dynamique des Syst` emes, UMR CNRS Laboratoire de Tribologie et Syst` UMR CNRS ∗ 5513, ´ Ecole Centrale de Lyon, 36 avenuedes Guy deeemes, Collongue Laboratoire de Tribologie et Dynamique Dynamique des Syst` mes, UMR 69134 CNRS ´ ´ 5513, Ecole Centrale de Lyon, 36 avenue Guy de Collongue 69134 5513, Ecole Centrale de Lyon, 36 avenue Guy de Collongue 69134 ´Ecole ´ Cedex, France (e-mail: [email protected]). 5513, Ecully Centrale de Lyon, 36 avenue Guy de Collongue 69134 ´ ´ Ecully Cedex, France (e-mail: [email protected]). ∗∗ Ecully Cedex, (e-mail: [email protected]). ´ Institut d’Optique Graduate School, Laboratoire Charles Fabry, Ecully Cedex, France France (e-mail: [email protected]). ∗∗ ∗∗ Institut d’Optique Graduate School, Laboratoire Charles Fabry, d’Optique Graduate School, Laboratoire Charles Fabry, ∗∗ Institut CNRS, d’Optique 2 avenue Fresnel, 91127 Palaiseau, France (e-mail: Institut Graduate School, Laboratoire Charles Fabry, CNRS, 2 avenue Fresnel, 91127 Palaiseau, France (e-mail: CNRS, 2 avenue Fresnel, 91127 Palaiseau, France (e-mail: CNRS, [email protected], avenue Fresnel, 91127 Palaiseau, France (e-mail: [email protected], [email protected], [email protected]). [email protected], [email protected]). [email protected]). [email protected]). Abstract: This paper deals with model order reduction of random parameter-dependent (RPD) Abstract: paper deals with model reduction of random parameter-dependent (RPD) Abstract: This paper deals model order reduction of (RPD) linear time This invariant (LTI) systems byorder using the RPD-balanced realization (BR), recently Abstract: This paper (LTI) deals with with modelby order reduction of random random parameter-dependent parameter-dependent (RPD) linear time invariant systems using the RPD-balanced realization (BR), recently linear time invariant (LTI) systems by using the RPD-balanced realization (BR), recently developed byinvariant the same(LTI) authors. A novel way to deal with this crucial issue is(BR), presented. It linear time systems by using the RPD-balanced realization recently developed by the same authors. A novel way to deal with this crucial issue is presented. It developed the same authors. A novel way to deal with this crucial issue is presented. It is based onby the intrusive Galerkin projection of the RPD-BR. Indeed, it is shown, through developed by the same authors. A novel way to deal with this crucial issue is presented. It is based on the intrusive projection of the RPD-BR. it is shown, through is based the Galerkin projection of the Indeed, it is through numerical simulations, thatGalerkin the intrusive Galerkin projection of theIndeed, RPD-BR into the generalized is based on on the intrusive intrusive Galerkin projection ofprojection the RPD-BR. RPD-BR. Indeed, it into is shown, shown, through numerical simulations, that the intrusive Galerkin of the RPD-BR the generalized numerical simulations, that the intrusive Galerkin projection of the RPD-BR into the generalized polynomial chaos (GPC) space enables to generate a deterministic realization preserving numerical simulations, that the intrusive Galerkin projection of the RPD-BR into the generalized polynomial chaos (GPC) (GPC) space enables enables to generate generate deterministic realization preserving polynomial chaos space to aaa the deterministic realization stability properties for the original RPD-model. Moreover, new deterministic statepreserving variables, polynomial chaos (GPC) space enables to generate deterministic realization preserving stability properties for the original RPD-model. Moreover, the new deterministic state variables, stability properties for the original RPD-model. Moreover, the new deterministic state variables, which areproperties the stochastic modes of the RPD-balanced state the variables, are ordered with respect to stability for the original RPD-model. Moreover, new deterministic state variables, which are the stochastic modes of the RPD-balanced state variables, are ordered with respect to which are the stochastic modes of the RPD-balanced state variables, are ordered with respect to their sub-contributions to the RPD-input/output behaviour, measured by deterministic Hankel which are the stochastic modes of the RPD-balanced state variables, are ordered with respect to their sub-contributions to the the RPD-input/output behaviour, measured by deterministic deterministic Hankel their sub-contributions to behaviour, measured by Hankel values. Hence, a deterministic reduced order model is derived by deleting the state variables their sub-contributions to the RPD-input/output RPD-input/output behaviour, measured by deterministic Hankel values. Hence, a deterministic reduced order model is derived by deleting the state variables values. Hence, a deterministic reduced order model is derived by deleting the state variables with small Hankel values and the complete RPD-reduced orderbymodel is then obtained. The values. Hence, a deterministic reduced order model is derived deleting the state variables with small values and the complete RPD-reduced order model is then obtained. The with small Hankel values and the complete RPD-reduced order model is then obtained. The feasibility ofHankel the proposed method is analyzed while its accuracy is compared to the RPDwith small Hankel values and the complete RPD-reduced order model is then obtained. The feasibility of the proposed method is analyzed while its accuracy is compared to the RPDfeasibility of method is truncated balanced realization (RPD-TBR). feasibility of the the proposed proposed method is analyzed analyzed while while its its accuracy accuracy is is compared compared to to the the RPDRPDtruncated balanced balanced realization (RPD-TBR). truncated realization (RPD-TBR). truncated balanced realization (RPD-TBR). © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Uncertain Systems, Random Parameters, Model Reduction, Balanced Truncation, Keywords: Systems, Reduction, Balanced Truncation, Keywords: Uncertain Uncertain Systems, Random Parameters, Model Reduction, Balanced Truncation, Generalized Polynomial Chaos,Random IntrusiveParameters, Projection, Model Non-Intrusive Spectral Projection. Keywords: Uncertain Systems, Random Parameters, Model Reduction, Balanced Truncation, Generalized Polynomial Chaos, Intrusive Projection, Non-Intrusive Spectral Projection. Generalized Polynomial Chaos, Intrusive Projection, Non-Intrusive Spectral Projection. Generalized Polynomial Chaos, Intrusive Projection, Non-Intrusive Spectral Projection. 1. INTRODUCTION points are generated from sparse grid techniques while 1. INTRODUCTION INTRODUCTION points are generated generated from sparse grid techniques while 1. points are techniques while interpolation functionsfrom are sparse chosengrid so the interpolation 1. INTRODUCTION points are generated from sparse grid techniques while interpolation functions are chosen so the interpolation interpolation functions are chosen so the interpolation error be minimized. In (Nechak, L. et al., 2015), a novel interpolation functions are chosen so the interpolation Model reduction has become essential to deal effectively error error betominimized. minimized. In (Nechak, L. et et al., 2015), 2015), a linear novel be (Nechak, L. novel scheme deal with In random parameter-dependent Model reduction has become become essential to deal deal effectively error betominimized. In (Nechak, L. et al., al., 2015), aa linear novel Model reduction has essential to with problems involving simulation, control andeffectively design of scheme scheme deal with random parameter-dependent Model reduction has become essential to deal effectively to deal with random parameter-dependent linear systems to is proposed. Indeed, authors defined, through the with problems involving simulation, control and design of scheme deal with random parameter-dependent linear with problems involving simulation, control and design of high problems dimensional systems. The maincontrol aim is and to efficiently systems is generalized proposed. Indeed, Indeed, authors defined, through the with involving simulation, design of systems proposed. authors defined, the use of theis polynomial chaos (GPC)through formalism, high dimensional systems. The main aim is to efficiently systems is proposed. Indeed, authors defined, through the high dimensional systems. The main aim is to efficiently generate, from a given large scale model, simpler one use of of the the generalized polynomialbalancing chaos formalism, high dimensional systems. Thescale main aim isaa to efficiently polynomial chaos (GPC) (GPC) formalism, a random parameter-dependent transformation generate, from aa the given large model, simpler one use use of the generalized generalized polynomialbalancing chaos (GPC) formalism, generate, from given large scale model, aa simpler one while preserving input/output behaviour and struca(RPD-BT) random parameter-dependent transformation generate, from a given large scale model, simpler one a random parameter-dependent balancing transformation which puts the associated RPD-linear system while preserving of thethe input/output behaviour andinterest struc- a random parameter-dependent balancing transformation while preserving the input/output behaviour and structural properties original model. A great (RPD-BT) which puts puts the associated associated RPD-linear system while preserving of thethe input/output behaviour andinterest struc- (RPD-BT) which the system in an almost-surely balanced form. A RPD-linear statistical analysis tural properties original model. A great (RPD-BT) which puts the associated RPD-linear system tural properties of the original model. A great interest was accorded to of model reduction of linear uncertain dy- in in an almost-surely balanced form. A A statistical analysis tural properties the reduction original model. A great interest form. analysis of an the almost-surely RPD-Hankel balanced singular values is statistical used to define an was accorded to these model of see, linear uncertain dyin an almost-surely balanced form. A statistical analysis was accorded to model reduction linear namical systems last decades,of e.g.,uncertain (Beck, C.dyL of of the RPD-Hankel singular values is used to define an was accorded to model reduction of linear uncertain dythe RPD-Hankel singular values is used to define an almost-sure truncation order for the RPD-BR within the namical systems these last decades, see, e.g., (Beck, C. L of the RPD-Hankel singular values is used to define an namical systems these see, e.g., (Beck, L et al., 1996), (Weile, D.last S. decades, et al., 1999), (Dolgin, Y.C. and almost-sure truncation order for the RPD-BR within the namical systems these last decades, see, e.g., (Beck, C. L almost-sure truncation order for the RPD-BR within the whole probabilistic distribution of random parameters. et al., 1996), 1996), (Weile, D. S. S. etand al.,Hahn, 1999),J., (Dolgin, Y. and and almost-sure truncation order for the RPD-BRparameters. within the et al., (Weile, D. al., 1999), (Dolgin, Y. Zeheb, E., 2005), (Sun, C. et 2006),(Wang, whole probabilistic distribution of random random et al., 1996), (Weile, D. S. etand al.,Hahn, 1999),J., (Dolgin, Y. and whole probabilistic distribution of State variables that are almost-surely weakly parameters. controllable Zeheb, E., 2005), (Sun, C. 2006),(Wang, whole probabilistic distribution of random parameters. Zeheb, E., 2005), (Sun, C. and Hahn, J., 2006),(Wang, Z. Z. etE., al.,2005), 2012). A supplementary issue related to the State State variables that that are almost-surely almost-surely weakly controllable Zeheb, C. and Hahn, J., related 2006),(Wang, variables are weakly and observable are then truncated, giving risecontrollable to a RPDZ. Z. et et al., al., 2012).(Sun, A supplementary supplementary issue to the the State variables that are almost-surely weakly controllable Z. Z. 2012). A to practice of model reduction for this issue class related of systems is and and observable are then truncated, giving rise to aasecond RPDZ. Z. et al., 2012). A supplementary issue related to the observable are then truncated, giving rise to RPDtruncated balanced realization (TBR). First and practice of model reduction for this class of systems is and observable are then truncated, giving rise to asecond RPDpractice of model reduction for this class of systems is to preserve the physical meaning of uncertainty and its truncated balanced realization (TBR). First and practice of model reduction for this class of systems is truncated balanced realization (TBR). First and second order moments of therealization error bound(TBR). relatedFirst to theand truncation to preserve the physical meaning of uncertainty and its truncated balanced second to preserve the physical meaning of uncertainty and its effect on thethedynamical behaviourofafter reduction. Nuorder moments of of thebyerror error bound of related to the the truncation to preserve physical meaning uncertainty andNuits order moments bound related to were characterized moments RPD-Hankel singular effect onstudies the dynamical dynamical behaviour after reduction. order moments of the thebyerror bound of related to the truncation truncation effect the reduction. Numerouson have beenbehaviour proposedafter in this framework. were characterized moments RPD-Hankel singular effect on the dynamical behaviour after reduction. Nuwere characterized by moments of RPD-Hankel singular values associated to the truncated state variables. merous studies have been proposed in this framework. were RPD-Hankel merous studies have proposed in Otherwise, the well-known truncated balanced realization values valuescharacterized associated to toby themoments truncatedofstate state variables.singular merous studies have been been truncated proposed balanced in this this framework. framework. associated truncated Otherwise, the well-known well-known realization values associated to the the truncated stateofvariables. variables. Otherwise, the truncated balanced realization (TBR) technique was extended to linear systems with This paper deals with model reduction RPD-linear sysOtherwise, the well-known truncated balanced realization (TBR) technique was extended to linear systems with This paper deals with model reduction of RPD-linear sys(TBR) technique was extended to linear systems with This paper deals with model reduction of RPD-linear sysuncertain parameters. Baur and to Benner have proposed tems. It proposes the GPC based intrusive Galerkin pro(TBR) technique was extended linear systems with This paper deals with modelbased reduction of RPD-linear sysuncertain parameters. Baur and Benner have proposed tems. It proposes the GPC intrusive Galerkin prouncertain parameters. Baur and Benner have proposed tems. It proposes the GPC based intrusive Galerkin proits combination with interpolation algorithms to proposed generate tems. jectionIttoproposes derive a the RPD-reduced order model. It is shown, uncertain parameters. Baur and Benner haveto GPC basedorder intrusive Galerkin proits combination with interpolation algorithms generate jection to derive a RPD-reduced model. It is shown, its combination with interpolation algorithms to derive order It shown, parametric reduced order models (Baur, U. and Benner, jection throughto numerical simulations, that themodel. intrusive Galerkin its combination withorder interpolation algorithms to generate generate jection tonumerical derive aa RPD-reduced RPD-reduced order It is isGalerkin shown, parametric reduced models (Baur, U. and Benner, through through simulations, that themodel. intrusive parametric reduced order models (Baur, U. and Benner, numerical simulations, that the intrusive Galerkin P., 2009), (Amsallem, D. and Farhat, C., 2011), (Panzer, projection for the RPD-BR, previously introduced by the parametric reduced order models (Baur, U. and Benner, through numerical simulations, that the intrusive Galerkin P., 2009), (Amsallem, D. and and Farhat, C., 2011), 2011), (Panzer, projection for the the RPD-BR, L. previously introduced by the the P., D. Farhat, C., (Panzer, for RPD-BR, previously introduced by H., 2009), 2010). (Amsallem, Reduced-order models are first determined at projection same authors in (Nechak, et al., 2015), produces P., 2009), (Amsallem, D. and Farhat, C., 2011), (Panzer, projection for the RPD-BR, L. previously introduced by theaa H., 2010). Reduced-order models are first determined at same authors in (Nechak, et al., 2015), produces H., 2010). Reduced-order models are first determined at same authors in (Nechak, L. et al., 2015), produces aa some2010). chosen parameter values thenare interpolated, yielding deterministic dynamic system which preserves the asympH., Reduced-order models first determined at same authorsdynamic in (Nechak, L.which et al.,preserves 2015), produces some chosen parameter values values then interpolated, yielding deterministic system the asympasympsome chosen parameter then interpolated, yielding deterministic dynamic system which preserves the a parameter-dependent reduced model. The calculating some chosen parameter values then interpolated, yielding deterministic dynamic system which preserves the asympparameter-dependent reduced model. The calculating calculating aa a parameter-dependent parameter-dependent reduced reduced model. model. The The calculating Copyright © 2017, 2017 IFAC 2793Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright 2017 2793 Copyright ©under 2017 IFAC IFAC 2793Control. Peer review© of International Federation of Automatic Copyright © 2017 responsibility IFAC 2793 10.1016/j.ifacol.2017.08.580
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 L. Nechak et al. / IFAC PapersOnLine 50-1 (2017) 2738–2743
totically stable behaviour of the original RPD-system. Moreover, the deterministic state variables (representing the GPC expansion coefficients of the RPD-balanced state variables) are ordered with respect to their subcontributions to the RPD-input/output behaviour, measured by the deterministic Hankel singular values. These sub-contributions are defined so as to preserve the initial order of the RPD-BR state variables. A reduced-order deterministic model can then be obtained by deleting, in the deterministic system, the state variables with small deterministic Hankel singular values. The truncation order is given by the truncation order of the GPC expansions times the number of almost surely weakly controllable and observable state variables. This method generates a deterministic reduced model which can be more efficiently used in optimal control schemes as defined in (Fisher, J. and Bhattacharya, R., 2009). Hence, the main objective of this study is to analyze and discuss the order reduction of the RPD-BR once projected intrusively into the GPC space. The intrusive and non-intrusive schemes of the GPC expansion are based on Galerkin projections. These are used to deal with numerous problems related to the stability of dynamical systems (Nechak, L. et al., 2011), the prediction of limit cycle oscillation in selfexcited uncertain nonlinear dynamical systems (Nechak, L. et al., 2012), (Nechak, L. et al., 2014), state observer design of uncertain systems (Smith, A. et al., 2007) or optimal control design (Fisher, J. and Bhattacharya, R., 2009). In this paper, the non-intrusive spectral projection is used to determine the RPD-balancing transformation, while the intrusive projection is used to determine the deterministic counterpart of the RPD-BR and thus the associated reduced-order realization. This paper is then organized as follows: the generalized polynomial chaos formalism is recalled in Section 2, followed by a summary of the RPD-BR concept, with the associated projected realization in Section 3; numerical simulations are presented and discussed in Section 4, while a conclusion is given at the end of the paper.
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where x ¯i (for i = 0, 1, ..., P ) are deterministic coefficients, called the stochastic modes of X, and x ¯0 represents its mean value. Otherwise, the number of stochastic modes P +1 is shown to be dependent on the stochastic dimension d and the chaos order p defined by the maximum degree of the used polynomial, that is, P + 1 = (p + d)!/p!d!. The stochastic modes can be obtained via intrusive or nonintrusive schemes exploiting the orthogonality property of GPC, which results in < φi (ξ), φj (ξ) >= δij �φi (ξ)�2 f (ξ)dξ (2) ξ
where δij is the delta Kronecker and f (ξ) = the probability density function of ξ.
d
i=1
fξi is
3. RANDOM PARAMETER-DEPENDENT REDUCED MODELS Let
˙ ξ) = A(ξ)x(t, ξ) + B(ξ)u(t) x(t, y(t, ξ) = C(ξ)x(t, ξ)
(3)
be a linear time invariant system where the state (A), control (B), output (C) matrices are ξ-dependent, as well as the state x(t) ∈ Rn and the output y(t) ∈ Rq vectors. Without loss of generality, the input vector u(t) ∈ Rm is assumed to be independent from the random variable ξ. Moreover, it is assumed that the initial condition does not depend on ξ, that is, x(0, ξ) = x0 . In the following, System (3) is assumed to be almost surely asymptotically stable and almost surely controllable and observable, see (Nechak, L. et al., 2015). Hence, the associated RPD-controllability and observability gramians are then given by the almost surely defined solutions of the RPD versions of Lyapunov equations: A(ξ)W c (ξ) + W c (ξ)A(ξ)T = −B(ξ)B(ξ)T T
T
A(ξ) W o (ξ) + W o (ξ)A(ξ) = −C(ξ) C(ξ)
(4) (5)
3.1 RPD-balanced realization 2. GENERALIZED POLYNOMIAL CHAOS Let (Ω, η, Pr) be a probability space where Ω is the sample space, η the σ-algebra of the subsets of Ω and Pr is the probability measure; let ξ = {ξi }di=1 be a countable set of independent, identically distributed random variables defined from (Ω, η) to (Rd , η d ) where, η d is the σalgebra of Borel subsets of Rd . Let X(ξ(ω)) be a second order stochastic process defined within the Hilbert space L2 (Ω, η, Pr). Otherwise, for the sake of simplicity, the shorthand notation X(ξ) or X will be used instead of X(ξ(ω)). Hence, from the GPC theory, (Xiu, W. and Karniadakis, G. E., 2003) and the Cameron-Martin theorem, (Cameron, H. and Martin, W., 1947), any stochastic process defined as X(ξ) can be approximated by a convergent (in the mean square sense) polynomial series functions φi that are orthogonal with respect to the probability density measures fk , k = 1, ..., d associated to the random variables ξk , k = 1, ..., d. X(ξ) ≈
P i=1
Definition 1: The almost surely minimal and asymptotically stable RPD-LTI system obtained via the RPDtransformation xb (t, ξ) = T −1 b (ξ)x(t, ξ) given by: x˙ b (t, ξ) = Ab (ξ)xb (t, ξ) + B b (ξ)u(t) (6) y(t, ξ) = C b (ξ)xb (t, ξ)
is said to be almost surely balanced if: (7) W c (ξ) = W o (ξ) = Σ(ξ) where the diagonal matrix Σ(ξ) = diag(σ1 (ξ), ..., σn (ξ)) is almost surely solution of the pair of RPD-Lyapunov equations: Ab (ξ)Σ(ξ) + Σ(ξ)Ab (ξ)T = −B b (ξ)B b (ξ)T (8)
Ab (ξ)T Σ(ξ) + Σ(ξ)Ab (ξ) = −C b (ξ)T C b (ξ) (9) th and σi , i = 1, ..., n is the i RPD-Hankel singular value which can be expressed by a truncated GPC expansion σi (ξ) ≈
P
σ ¯i,j φj (ξ)
(10)
j=0
x ¯i φi (ξ)
(1)
where σ ¯i,0 denotes the mean value of the ith Hankel singular value, while the higher-order terms σ ¯i,j define its
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dispersion. The main step is then to determine the RBDbalanced transformation Tb (ξ). The latter is obtained by computing finite order GPC expansions for its elements tij (ξ): tij (ξ) ≈
P �
t¯ij,k φk (ξ)
(11)
k=0
The coefficients t¯ij,k are obtained from the non-intrusive spectral projection of tij (ξ) into the GPC space. They are given by: �tij (ξ), φk (ξ)� t¯ij,k = (12) �φk (ξ)2 � Numerical integration techniques are used to obtain approximations of the coefficients (12), see (Nechak, L. et al., 2015) for more details. The RPD-BR matrices are then obtained as follows: b A (ξ) = T −1 b (ξ)A(ξ)T b (ξ) (13) B b (ξ) = T −1 b (ξ)B(ξ) b C (ξ) = C(ξ)T b (ξ)
In (Nechak, L. et al., 2015), a RPD-TBR is generated by suppressing the state variables with small Hankel singular values which indicate that the corresponding variables are almost surely weakly observable and controllable. To perform order reduction, an almost surely truncation order is determined by a statistical analysis of the RPD-Hankel singular values. Definition 2: An integer 1 < r < n is an almost sure truncation order for the RPD-LTI system (6) if � σr (ξ) < σi (ξ), ∀i ∈ 1, 2, ..., r − 1 (14) σr (ξ) > σj (ξ), ∀j ∈ r + 1, ..., n
Once the almost sure truncation order r has been selected, the RPD-TBR can be generated by deleting the corresponding state variables. In this paper, another way to perform model reduction is proposed and analyzed. It consists in performing an intrusive projection of the RPD-BR into the GPC space. An augmented deterministic state space representation is then obtained where the associated state variables are the stochastic modes of the GPC expansions of the RPD-state variables. A deterministic reduced-order representation is derived from the augmented system and is used to obtain a RPD-reduced model. This method is presented in the following. 3.2 RPD-truncation through intrusive Galerkin projection of the RPD-BR The main idea of the proposed intrusive RPD-TBR is to carry out a Galerkin projection of the RPD-BR into the GPC space after replacing all random quantities by their GPC expansions given as follows: ¯ b (t)T Φ(ξ) xbi (t, ξ) ≈ X i b b T ¯ y (t, ξ) ≈ Y i i (t) Φ(ξ) T u (t, ξ) ≈ U ¯ i (t) Φ(ξ) i (15) b b ¯ aij (ξ) ≈ (Aij )T Φ(ξ) ¯ b )T Φ(ξ) bbij (ξ) ≈ (B ij cb (ξ) ≈ (C ¯ b )T Φ(ξ) ij ij
b T ¯ (t) = [¯ xi,0 (t), ..., x ¯i,P (t)] X i b T yi,0 (t), ..., y¯i,P (t)] Y¯ i (t) = [¯ T ¯ ui,0 (t), ..., u ¯i,P (t)] U i (t) = [¯ T ¯ b = [¯ where A aij,0 , ..., a ¯ij,P ] ij �T � b ¯ = ¯bij,0 , ..., ¯bij,P B ij b T ¯ = [¯ cij,0 , ..., c¯ij,P ] C ij T Φ(ξ) = [φ0 (ξ), ..., φP (ξ)]
Note that since ui (t) is independent on the radom variable ξ, the associated stochastic modes u ¯i,k (t), k = 1, ..., P are null. Carrying out of the two previous steps gives rise to a n × (P + 1)-dimensional linear deterministic state space representation. The new state vector is defined by the coefficients of GPC expansions of the RPD-state variables, (Fisher, J. and Bhattacharya, R., 2009) �
¯ bX ¯ bU ¯ b (t) + B ¯ (t) ¯˙ b (t) = A X b ¯ ¯ ¯ Y b (t) = C b X (t)
(16)
¯ b ∈ Rn(P +1) , A ¯ b ∈ Rn(P +1)×n(P +1) , B ¯b ∈ where X m(P +1)×n(P +1) ¯ R , C b ∈ Rn(P +1)×q(P +1) , Y¯ b ∈ Rq(P +1) , ¯ ∈ Rm(P +1) U After projecting the RPD-BR and since the Galerkin projection does not modify structural properties, the deterministic state variables in the resulting projected system (16), representing the coefficients of the GPC expansions of the RPD-state variables of the RPD-BR, will be ordered so as the order of the original RPD-Hankel singular values is preserved. The deterministic Hankel singular values associated to the resulting deterministic measure the sub-contributions of the new state variables (representing the stochastic modes of the RPD-state variables) to the RPD-input/output behaviour. So, if r is the almost sure truncation order of the RPD-BR, r(P + 1) will be the truncation order of the augmented system (16). So, by ¯b ¯b setting variables X (r+1)(P +1) , ..., X n(P +1) equal to zero, a reduced order system is obtained. It is expressed as follows: � ¯ rX ¯ rU ¯ r (t) + B ¯ (t) ¯˙ r (t) = A X (17) ¯ rX ¯ r (t) Y¯ r (t) = C ¯ r ∈ Rr(P +1)×r(P +1) , B ¯r ∈ ¯ r ∈ Rr(P +1) , A where X ¯ r ∈ Rq(P +1)×r(P +1) while the reduced Rr(P +1)×m(P +1) , C matrices are given by P � r r ¯ ¯ = [A ], A ≈ a ¯rij,k Φk A r ij ij k=0 P � ¯br Φk ¯r ≈ ¯ r = [B r ], B (18) B ij,k ij ij k=0 P � ¯r ≈ ¯ r = [C r ], C c¯rij,k Φk C ij ij k=0
e¯0k0 e¯0k1 where Φk = ...
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e¯0kp
e¯0k1 . . . e¯0kp e¯1k1 . . . e¯1kp with e¯ijk = .. . . .. . . . e¯1kp . . . e¯pkp
�φi ,φj φk � �φ2i �
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Mean value of step response
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Fig. 1. Mechanical system
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4.1 RPD-Balanced realization The RPD-BR associated to the considered system is generated through a RPD-balancing transformations (11) with p = 3 for Legendre polynomials. The RPD-BR must reproduce the same input-output behaviour as the original RPD-system. This is verified by assessing the accuracy of the first and second order moments of the step response of the RPD-BR. This task is performed by using reference results obtained by the MC method. The instantaneous mean value and variance of the step response of the obtained RPD-BR coincide with the reference results as shown in Fig. 2 and Fig. 3 respectively.
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Fig. 3. Instantaneous variance of the step response:—Reference, - - - - RPD-BR is also observed with respect to the reference results. The 1.5
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4.2 Intrusive RPD-Balanced realization The RPD-BR is projected into the GPC space through a Galerkin projection. The same order p = 3 is used. An augmented deterministic system is obtained. The eigenvalues of the augmented systems derived from projections of original RPD-system and of the associated RPD-BR are plotted in Fig. 4. Eigenvalues of the projected RPDBR coincide with those of the projected original systems. Both projections exhibit the same asymptotically stable dynamics. The instantaneous mean value and variance of the step response of the system obtained from the projected RPD-BR are plotted in Fig. 5. Convenient accuracy
0
Fig. 2. Instantaneous mean value of the step response:—Reference, - - - - RPD-BR
Imaginary parts
In this Section, the proposed method for model reduction of RPD-linear systems is applied to an academic two degrees of freedom mass-spring system (Fig. 1). The control input u = F and the output y are respectively the force applied to and the position of the object of unitary mass m. The damping coefficients d1 = 1 and d2 = 0.65 and the stiffness parameter k2 = 1 are considered deterministic, while the stiffness parameter k1 is assumed to be a random variable with uniform distribution within the interval [0.9, 1.1]. The normalized uniform variable ξ within [-1, 1] is used to describe randomness of k1 through k1 = 1 + 0.1ξ. Hence, for the state vector x = (x1 , x2 , x˙ 1 , x˙ 2 )T the system modelis a RPD-LTI representation in theform (3) with: 0 0 1 0 0 0 0 1 0 k (ξ) k (ξ) d1 d1 0 1 1 A = − , B = 1 , m m1 m1 m1 k1 (ξ)1 k1 (ξ) d1 + d 2 + k2 d1 0 − m2 m2 m2 m2 C = [ 1 0 0 0 ].
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Fig. 4. Eigenvalues within the complex plan: ∗ projection of original RPD-system, + projection of RPD-BR reducibility of the RPD-original system can be analysed through the analysis of the RPD-Hankel singular values.
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Probability density function
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Fig. 5. Instantaneous mean of the step response:—- Reference, - - - - Projected RPD-BR
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Fig. 6. Instantaneous variance of the step response:—Reference, - - - - Projected RPD-BR This enables one to select an almost sure truncation order. GPC expansions can be used in this context, see (Nechak, L. et al., 2015). The probability density functions of the RPD-Hankel singular values are plotted in Fig.7. It can be observed that the two first Hankel singular values are distributed around higher values compared with the two remaining Hankel values. Consequently, r = 2 can be considered as the almost sure truncation order of the RPDBR. In Fig. 8, Hankel singular values of the intrusive projections of the original RPD-system and the associated RPD-BR coincide, showing the accuracy of the built RPDBR and its intrusive projection. Furthermore, the Hankel singular values aggregate into two subsets (Fig. 7): the last eight singular values define the sub-contributions of the coordinates of the projected RPD-state variables to the RPD input/output behaviour. They present small levels which is consistent with the almost surely weak controllability and observability of the two last RPD-state variables. The same remark can be stated about the first eight Hankel singular values. Consequently, r(P + 1) = 8 can be considered as the truncation order of the projected RPD-BR.
Fig. 8. Hankel singular values of the intrusive projections: ∗ RPD-original system, + RPD-BR A reduced RPD-system is generated from the truncated intrusive projection of the RPD-BR (obtained by suppressing the last eight state variables). The instantaneous mean values of its step responses is plotted in figure (9) in comparison with the reference and the instantaneous mean value of the step responses of the RPD-TBR defined in (Nechak, L. et al., 2015). The associated variances are plotted in Fig. 10. While, the average behaviour represented by the instantaneous mean value of the step response is accuraly approximated, a relative error is apparent for the approximated variance. The RPD-TBR gives the best approximation of the variance compared with the RPDreduced model obtained from the intrusive projection of the RPD-BR. This result is consistent with what is expected. The difference in accuracy between the two RPDreduced models is related to the intrusive Galerkin projection which introduces additional numerical errors. 5. CONCLUSION A novel way to generate a reduced uncertain system from the intrusive projection of the RPD-BR is discussed. It is shown through numerical simulations that the organization of the RPD-state variables with respect to their
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Mean value of step response
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This work has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 730890.
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REFERENCES
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Fig. 9. Instantaneous mean of the step response:—- Reference, - - - - Reduced Projected RPD-BR, -. -. -. -. RPDTBR
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Fig. 10. Instantaneous variance of the step response:—Reference, - - - - Reduced Projected RPD-BR, -. -. -. -. RPD-TBR
almost sure controllability and observability is preserved once projected into the GPC space. The generated RPDreduced model exhibits suitable accuracy in the estimation of the first an second order moments of the step response. This study has presented the method with one uncertain parameter. The case of multiple uncertain parameters induces higher chaos orders and thus higher dimension for the intrusive projections. This is the main drawback of methods based on the GPC expansion. However, numerical techniques exist to reduce the chaos order, such as the multi-element generalized polynomial chaos. Otherwise, this study will be completed by the evaluation of the truncation error which consists of two main elements. The first one is related to the truncation order of the GPC expansion while the second is related to truncation of the projected system. Finally, the stability of the proposed reduced model will be investigated in future research.
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