Hindawi Publishing CorporationJournal of MathematicsVolume 2013 Article ID 365909 6 pageshttpdxdoiorg1011552013365909

Research ArticleA Unified Software Framework for Empirical Gramians

Christian Himpe and Mario Ohlberger

Institute for Computational and Applied Mathematics University of Munster Einsteinstraszlige 62 48149 Munster Germany

Correspondence should be addressed to Christian Himpe christianhimpewwude

Received 11 July 2013 Accepted 2 August 2013

Academic Editor Beny Neta

Copyright copy 2013 C Himpe and M OhlbergerThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

A common approach in model reduction is balanced truncation which is based on Gramian matrices classifying certain attributesof states or parameters of a given dynamic system Initially restricted to linear systems the empirical Gramians not only extendedthis concept to nonlinear systems but also provided a uniform computational method This work introduces a unified softwareframework supplying routines for six types of empirical Gramians The Gramian types will be discussed and applied in a modelreduction framework for multiple-input multiple-output systems

1 Introduction

In a control system setting balanced truncation is a well-known technique for model reduction Introduced by [1]Gramian matrices were employed to determine controllabil-ity and observability of linear systems From these Gramiansa balancing transformation can be computed enabling thetruncation for example of states that are neither controllablenor observable

With [2] empirical (controllability and observability)Gramians were introduced which correspond to the analyti-cal Gramians for linear systems while extending the conceptof systemGramians to nonlinear systems which are generallygiven by

(119905) = 119891 (119909 119906 119901)

119910 (119905) = 119892 (119909 119906 119901)

(1)

with the system function 119891 and output function 119892 of states119909 input 119906 and parameters 119901 in the special case of anunparameterized linear system 119891 = 119860119909(119905) + 119861119906(119905) and 119892 =

119862119909(119905) These empirical Gramians are computed by averagingsimulations or experimental datawith perturbations in inputsand initial states

The emgr framework presented here encompasses sixempiricalGramians namely the controllability observabilitycross sensitivity identifiability and joint Gramians To adapt

the computation of empirical Gramians to the operating set-ting of the system the initial state and the input are the mainvariables which are perturbed by rotations and scaling Thesets of rotations provided are 1 (unit matrix) and minus1 1

(negative unit matrix and unit matrix)Though these are verybasic sets and thus might not reflect all dynamics especiallywith interrelated states and parameters they allow a veryefficient Gramian assembly Scales may be freely chosenThe subdivision of the scales may be linear logarithmic orgeometric Finally there are several options to average againstthe arithmetic average [2] the median a steady state [3] andadditionally the principal components of the simulations ordata via a proper orthogonal decomposition (POD)

2 Empirical Gramians

Concerned with the reduction of states the controllabilityobservability and cross Gramians are presented next fol-lowed by the sensitivity identifiability and joint Gramianswhich are used for parameter and combined reduction Forthe purpose of defining the Gramians a linear time-invariantcontrol system is assumed

(119905) = 119860119909 (119905) + 119861119906 (119905)

119910 (119905) = 119862119909 (119905)

(2)

2 Journal of Mathematics

with the states 119909 isin R119899 control or input 119906 isin R119898 output 119910 isin

R119900 system matrix 119860 isin R119899times119899 input matrix 119861 isin R119899times119898 andoutput matrix 119862 isin R119900times119899

The necessary perturbations are given by six sets ofwhich 119864

119906 119877

119906 119876

119906 define the input perturbations while sets

119864

119909 119877

119909 119876

119909 define the initial state perturbations

119864

119906= 119890

119894isin R119895

1003817

1003817

1003817

1003817

119890

119894

1003817

1003817

1003817

1003817

= 1 119890

119894119890

119895 = 119894= 0 119894 = 1 119898

119864

119909= 119891

119894isin R119899

1003817

1003817

1003817

1003817

119891

119894

1003817

1003817

1003817

1003817

= 1 119891

119894119891

119895 = 119894= 0 119894 = 1 119899

119877

119906= 119878

119894isin R119895times119895

119878

lowast

119894119878

119894= 1 119894 = 1 119904

119877

119909= 119879

119894isin R119899times119899

119879

lowast

119894119879

119894= 1 119894 = 1 119905

119876

119906= 119888

119894isin R 119888

119894gt 0 119894 = 1 119902

119876

119909= 119889

119894isin R 119889

119894gt 0 119894 = 1 119903

(3)

These sets should correspond to the ranges in inputs andinitial states the system is operating in

21 Controllability Gramian Controllability is a quantifica-tion of how well a state can be driven by input Analyticallythe controllability Gramian is given by the smallest semi-positive definite solution of the Lyapunov equation 119860119882

119862+

119882

119862119860

119879

= minus119861119861

119879 If the underlying system is asymptoticallystable the controllability Gramian can also be defined usingthe linear input-to-state map

119882

119862= int

infin

0

119890

119860120591

119861119861

119879

119890

119860119879120591

119889120591 (4)

Following [3] the empirical controllability Gramian isdefined by the following

Definition 1 For sets 119864

119906 119877119906 and 119876

119906 and input 119906(119905) and

input during the steady state119909(119906) the empirical controllabilityGramian is given by

119882

119862=

1

1003816

1003816

1003816

1003816

119876

119906

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119877

119906

1003816

1003816

1003816

1003816

|119876119906|

sum

ℎ=1

|119877119906|

sum

119894=1

119898

sum

119895=1

1

119888

2

ℎ

int

infin

0

Ψ

ℎ119894119895

(119905) 119889119905

Ψ

ℎ119894119895

(119905) = (119909

ℎ119894119895

(119905) minus 119909) (119909

ℎ119894119895

(119905) minus 119909)

lowast

isin R119899times119899

(5)

with 119909

ℎ119894119895 being the states for the input configuration 119906

ℎ119894119895

(119905) =

119888

ℎ119878

119894119890

119895119906(119905) + 119906

Originally in [2] 119906(119905) was restricted to 120575(119905) but extendedin [3] to arbitrary input under the name of empirical covari-ance matrix 119909 can be the arithmetic average the median thesteady state or the principal components Restricting 119877

119906to

minus1 1 simplifies the input perturbation to

119906

ℎ119894119895

(119905) = minus1

119894

119902

ℎ119890

119895119906 (119905) + 119906 (6)

22 Observability Gramian Observability quantifies howwell a change in a state is reflected by the output Theanalytical observability Gramian is given by the smallest

semipositive definite solution of the Lyapunov equation119860

119879

119882

119874+ 119882

119874119860 = minus119862

119879

119862 Given an asymptotically stableunderlying system the observability Gramian can also bedefined using the state-to-output map

119882

119874= int

infin

0

119890

119860119879120591

119862

119879

119862119890

119860120591

119889120591 (7)

The empirical observability Gramian is defined as describedin [2 3]

Definition 2 For sets 119864119909 119877119909 and119876

119909and output 119910 during the

steady state 119909(119910) the empirical observability Gramian is givenby

119882

119874=

1

1003816

1003816

1003816

1003816

119876

119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119877

119909

1003816

1003816

1003816

1003816

|119876119909|

sum

119896=1

|119877119909|

sum

119897=1

1

119889

2

119896

119879

119897int

infin

0

Ψ

119896119897

(119905) 119889119905 119879

lowast

119897

Ψ

119896119897

119886119887= (119910

119896119897119886

(119905) minus 119910)

lowast

(119910

119896119897119887

(119905) minus 119910) isin R

(8)

with 119910

119896119897119886 being the systems output for the initial state conf-iguration 119909

119896119897119886

0= 119889

119896119878

119897119891

119886+ 119909

119910 can be the arithmetic average the median the steady-state output or the principal components Restricting 119877

119909to

minus1 1 simplifies (8) to

119882

119874=

1

1003816

1003816

1003816

1003816

119876

119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119877

119909

1003816

1003816

1003816

1003816

|119876119909|

sum

119896=1

|119877119909|

sum

119897=1

1

119889

2

119896

int

infin

0

Ψ

119896119897

(119905) 119889119905(9)

and simplifies the initial state perturbation to

119909

119896119897119886

0= minus1

119897

119889

119896119891

119886+ 119909

(10)

23 Cross Gramian The cross Gramian [4] makes a com-bined statement about the controllability and observabilitygiven that the system has the same number of inputs andoutputs If the system is also symmetric meaning that thesystem transfer function is symmetric then the absolute valueof these Gramiansrsquo eigenvalues equals the Hankel singularvalues It is originally computed as the smallest solution of theSylvester equation 119860119882

119883+119882

119883119860 = minus119861119862 The cross Gramian

can also be defined using the input-to-state and state-to-output maps if the underlying system is asymptoticallystable

119882

119883= int

infin

0

119890

119860120591

119861119862119890

119860120591

119889120591 (11)

The empirical cross Gramian has been introduced in [5] forSISO systems and was extended to MIMO systems in [6]

Journal of Mathematics 3

Definition 3 For sets 119864119906 119864119909 119877119906 119877119909 119876119906 and 119876

119909and input

119906 during steady state 119909 with output 119910 the empirical crossGramian is given by

119882

119883=

1

1003816

1003816

1003816

1003816

119876

119906

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119877

119906

1003816

1003816

1003816

1003816

119898

1003816

1003816

1003816

1003816

119876

119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119877

119909

1003816

1003816

1003816

1003816

sdot

|119876119906|

sum

ℎ=1

|119877119906|

sum

119894=1

119898

sum

119895=1

|119876119909|

sum

119896=1

|119877119909|

sum

119897=1

1

119888

ℎ119889

119896

int

infin

0

119879

119897Ψ

ℎ119894119895119896119897

(119905) 119879

lowast

119897119889119905

Ψ

ℎ119894119895119896119897

119886119887(119905) = 119891

lowast

119887119879

lowast

119896Δ119909

ℎ119894119895

(119905) 119890

lowast

119894119878

lowast

ℎΔ119910

119896119897119886

(119905)

Δ119909

ℎ119894119895

(119905) = (119909

ℎ119894119895

(119905) minus 119909)

Δ119910

119896119897119886

(119905) = (119910

119896119897119886

(119905) minus 119910)

(12)

with 119909

ℎ119894119895 and 119910

119896119897119886 being the states and output for the input119906

ℎ119894119895

(119905) = 119888

ℎ119878

119894119890

119895119906(119905) + 119906 and initial state 119909119896119897119886

0= 119889

119896119879

119897119891

119886+ 119909

respectively

119909 and 119910 can be the arithmetic average the median thesteady state or the principal components of the outputAgain restricting 119877

119906and 119877

119909to minus1 1 simplifies (12) to

Ψ

ℎ119894119895119896119897

119886119887= (minus1)

119894+119897

119891

lowast

119887Δ119909

ℎ119894119895

(119905) 119890

lowast

119895Δ119910

119896119897119886

(119905) (13)

as well as simplifying input and initial state perturbation to

119906

ℎ119894119895

(119905) = minus1

119894

119888

ℎ119890

119895119906 (119905) + 119906

119909

119896119897119886

0= minus1

119897

119889

119896119891

119886+ 119909

(14)

24 Sensitivity Gramian The sensitivity Gramian allowscontrollability-based parameter reduction and identificationIt is based on [7] and aimed formodels that can be partitionedas follows

= 119891 (119909 119906 119901) = 119891 (119909 119906) +

119875

sum

119896=1

119891 (119909 119901

119896) (15)

The parameters 119901 isin R119875 are handled here as additionalinputs All summands of the partitioned system are treated asindependent subsystems and thus a controllability Gramianfor each subsystem can be computed Each parameter con-trollability is encoded in the sum of singular values of theassociated subcontrollability Gramian 119882

119862119896 The sensitivity

Gramian is now given by the diagonal matrix with eachdiagonal element being the trace of a subcontrollabilityGramian

119882

119878=

119875

sum

119896=1

tr (119882119862119896

) (120575

119894119895)

119894=119895=119896

(16)

The controllability and thus identifiability of each parameterare then given by the corresponding diagonal entry of thesensitivity Gramian119882

119878 For partitionable linear systems the

sum of all subsystems controllability Gramians119882119862119896

and the

parameter-free subsystems Gramian 119882

1198620equals the usual

controllability Gramian [7]

119882

119862= 119882

1198620+

119875

sum

119896=1

119882

119862119896 (17)

The sensitivity Gramian can be applied to nonpartitionablemodels with reduced accuracy

25 Identifiability Gramian The identifiability Gramian ena-bles observability-based parameter identification and conse-quently parameter reduction As described in [8] the dyna-mic system states are augmented with as many states as para-meters that are constant over time and have the initial valueof the (prior) parameter value

One has

= (

119901

) = (

119891 (119909 119906 119901)

0

)

119910 = 119892 (119909 119906 119901)

(18)

The observability Gramian of this augmented system holdsthe observability information of states and parameters Toextract the parameter specific observability the Schur com-plement can be applied to the augmented observability Gra-mian

119882 = (

119882

119874119882

119876

119882

lowast

119876119882

119875

)

997904rArr 119882

119868= 119882

119875minus119882

lowast

119876119882

minus1

119874119882

119876

(19)

26 Joint Gramian Based on the identifiability Gramianprocedure the cross Gramian can be employed for a con-current state and parameter reduction (see [6]) As for theidentifiability gramian the system is augmented by constantparameter states Additionally as many inputs and outputs asparameters are augmented as well The augmented inputs actvia identity on the augmented states Likewise the augmentedstates are mapped by identity to the augmented outputs topreserve symmetry

= (

119901

) = (

119891 (119909 119906 119901)

V)

119910 = (

119892 (119909 119906 119901)

1)

(20)

to preserve symmetry The cross Gramian of this specialaugmented system similar to the identifiability Gramianholds the cross Gramian of the original system as well as across-identifiability Gramian119882 119868 which can be extracted witha Schur complement from the joint Gramian

119882

119869= (

119882

119883119882

119876

119882

119902119882

119875

)

997904rArr 119882 119868= 119882

119875minus119882

lowast

119876119882

minus1

119883119882

119876

(21)

4 Journal of Mathematics

Table 1 Empirical gramian application matrix

Gramian type Statereduction

Parameterreduction

Combinedreduction

119882

119862() times times

119882

119874() times times

119882

119862amp 119882

119874 times times

119882

119883 () times

119882

119878() times

119882

119868() ()

119882

119869()

3 Implementation

The emgr software framework presented here provides auniform interface to compute all six empirical Gramians andis given by

W = emgr (f g q t w p vcfg u us xs um xm yd) (22)

with f and g being handles to the system and the outputfunction both requiring the signature f(xup) andg(xup) q is a vector defining the systems number ofinputs states and outputs t is a three-component vectorcontaining start time time step and stop time w is a charactersetting the Gramian type for an overview on the applicabilityof Gramian types see Table 1

The following arguments are optional p holds anyparameters The ten-component vector vcfg configures theavailable options including averaging types input and statescale subdivisions and perturbation rotations u provides theinput to119891 and119892 while us and xs set steady input and steadystate um and xm define the scales of the perturbation Las-tly yd allows passing experimental data to be used insteadof generated snapshots

The parameters reducing empirical Gramians (sensitivityidentifiability and joint) are an encapsulation of the statereducing empirical Gramians (controllability observabilityand cross) Computation of the latter is extensively vector-ized exploiting the Gramian matrix assembly format Forexample the empirical observability Gramian assembly from(8) can be computationally simplified to

Ψ

119896119897

119886119887= (119910

119896119897119886

(119905) minus 119910)

lowast

(119910

119896119897119887

(119905) minus 119910)

997888rarr

120595

119896119897

119886(119905) = (119910

119896119897119886

(119905) minus 119910)

Ψ

119896119897

119886119887= vec (120595119896119897

119886)

lowast

vec (120595119896119897119887)

(23)

Computation of empirical Gramians using emgr is veryportable since only basic vector and matrix operations arerequired Necessary integrations meaning simulations forgiven inputs or initial states are accomplished either by thefirst-order Eulerrsquos method second-order Adams-Bashforthmethod or second-order Leapfrog method The empiricalGramian framework emgr as well as the following exper-iments is released under an open source license is com-patible with OCTAVE and MATLAB and can be found

Time

Rela

tive e

rror

10 20 30 40 50 60 70 80 90 100123456789

10

0002

0004

0006

0008

0010

0012

0014

Figure 1 Relative error in reduced system output by balanced trun-cation using the empirical controllability Gramian and the empiricalobservabilityGramian WC = emgr(fg[1010010][00011]ldquocrdquop) WO = emgr(fg[1010010][00011]ldquoordquop)

at httpgramiande or at the Oberwolfach References onMathematical Software

4 Numerical Experiments

To demonstrate the various empirical Gramians computedby the emgr framework a symmetric nonlinear MIMOsystem with one hundred states ten inputs and ten outputsis employed The system matrix is generated randomly withensured stability and symmetry the input matrix 119861 is alsoa random matrix and the output matrix is given by 119862 =

119861

119879 Furthermore a random but element-wise parametrizedsource term 119864

119901of dimension 119899 = 100 parameters is added

Input is applied through a delta impulse

119891 (119909 119906 119901) = = 119860 arsinh (119909) + 119861119906 + 119864

119901

119892 (119909) = 119910 = 119862119909

(24)

First a state reduction using the empirical controllabilityGramian and the empirical observability Gramian throughbalanced truncation is performed in Figure 1 reducingthe number of states to the number of outputs Balancedtruncation as a classic approach inmodel order reductionwillbe used as a baseline to which the following methods will becompared

Next a state reduction by direct truncation employing theempirical cross Gramian is demonstrated in Figure 2 againreducing the number of states to ten The state reduction viadirect truncation of the cross Gramian has about the sameerror but requires only one Gramian and no balancing

The empirical sensitivity Gramian can be applied if theunderlying system can be partitioned such that 119891(119909 119906 119901) =

119891(119909 119906) + 119891(119909 119901) To be able to use it in this setting theparametrized source term is reduced to the number ofoutputs in Figure 3 The sensitivity Gramian is the fastestparameter reduction method but has a high relative error inoutputs Since the parameters of the source term are reducedthe cumulative effects in the original system are the origin ofthe increasing error over time Next the parametrized source

Journal of Mathematics 5

0005

001

0015

002

0025

003

Time

Rela

tive e

rror

10 20 30 40 50 60 70 80 90 100123456789

10

Figure 2 Relative error in reduced system output by truncationusing the empirical cross Gramian WX = emgr(fg[1010010]

[00011]ldquoxrdquop)

01

02

03

04

05

06

07

08

Time

Rela

tive e

rror

10 20 30 40 50 60 70 80 90 100123456789

10

Figure 3 Relative error in reduced system output with reducedsource term reduction by truncation using the empirical sensitivityGramian WS = emgr(fg[1010010][00011]ldquosrdquop)

term is reduced by the empirical identifiability Gramian inFigure 4

Taking five times as long for the parameter reduction asthe sensitivity Gramian the identifiability Gramian is abouttwo orders of magnitude more accurate

Finally in Figure 5 the same system undergoes a com-bined state and parameter reduction using the empiricaljoint Gramian The joint Gramian is the only Gramianallowing direct balancing-free combined reduction of stateand parameter space with a comparable relative error and yettakes about the same time for the reduction as the identifiabil-ity Gramian This combined reduction generates a reduced-order model of which the relative error is comparable to theother reduced system output models

5 Future Work

The emgr framework already allows a wide range of com-putations of empirical Gramians for state or parameterreduction Apart from model order reduction the empiricalGramians can be employed for system identification taskslike parameter identification or sensitivity analysis as wellas decentralized control nonlinearity measurement anduncertainty quantification

00001

00002

00003

00004

00005

00006

Time

Rela

tive e

rror

10 20 30 40 50 60 70 80 90 100123456789

10

Figure 4 Relative error in reduced system output with reducedsource term reduction by truncation using the empiricalidentifiability Gramian WI = emgr(fg [1010010]

[00011]ldquoirdquop)

03

005

01

015

02

025

Time

Rela

tive e

rror

10 20 30 40 50 60 70 80 90 100123456789

10

Figure 5 Relative error in system output with combinedstate and parameter reduction reduction by truncation ofparameters and states using the empirical joint GramianWJ = emgr(fg[1010010][00011]ldquojrdquop)

Further work will enhance the flexibility while keepingthe interface as simple as possible Following [8] allowingfactorial designs will greatly enlarge the field of applicationFinally extending the use of the cross Gramian (and thus thejoint Gramian) to nonsymmetric systems [4] will enable acombined state and parameter reduction for general linearand nonlinear models without balancing

Acknowledgments

The authors acknowledge the support by the DeutscheForschungsgemeinschaft and the Open Access PublicationFund of the University of Munster

References

[1] B C Moore ldquoPrincipal component analysis in linear systemscontrollability observability andmodel reductionrdquo IEEETrans-actions on Automatic Control vol 26 no 1 pp 17ndash32 1981

[2] S Lall J E Marsden and S Glavaski ldquoEmpirical modelreduction of controlled nonlinear systemsrdquo in Proceedings of theInternational Federation of Automatic Control World Congress(IFAC rsquo99) pp 473ndash478 1999

6 Journal of Mathematics

[3] J Hahn and T F Edgar ldquoBalancing approach to minimalrealization and model reduction of stable nonlinear systemsrdquoIndustrial and Engineering Chemistry Research vol 41 no 9 pp2204ndash2212 2002

[4] A C Antoulas Approximation of Large-Scale Dynamical Sys-tems Advances in Design and Control SIAM Philadelphia PaUSA 2005

[5] S Streif R Findeisen andE Bullinger ldquoRelating cross gramiansand sensitivity analysis in systems biologyrdquo in Proceedings ofthe 17th International Symposium on Mathematical Theory ofNetworks and Systems (MTNS rsquo06) pp 437ndash442 2006

[6] C Himpe and M Ohlberger ldquoCross-Gramian based combinedstate and parameter reductionrdquo httparxivorgabs13020634

[7] C Sun and J Hahn ldquoParameter reduction for stable dynamicalsystems based on Hankel singular values and sensitivity analy-sisrdquoChemical Engineering Science vol 61 no 16 pp 5393ndash54032006

[8] D Geffen R Findeisen M Schliemann F Allgower and MGuay ldquoObservability based parameter identifiability for bio-chemical reaction networksrdquo in Proceedings of the AmericanControl Conference (ACC rsquo08) pp 2130ndash2135 June 2008

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2 Journal of Mathematics

with the states 119909 isin R119899 control or input 119906 isin R119898 output 119910 isin

R119900 system matrix 119860 isin R119899times119899 input matrix 119861 isin R119899times119898 andoutput matrix 119862 isin R119900times119899

The necessary perturbations are given by six sets ofwhich 119864

119906 119877

119906 119876

119906 define the input perturbations while sets

119864

119909 119877

119909 119876

119909 define the initial state perturbations

119864

119906= 119890

119894isin R119895

1003817

1003817

1003817

1003817

119890

119894

1003817

1003817

1003817

1003817

= 1 119890

119894119890

119895 = 119894= 0 119894 = 1 119898

119864

119909= 119891

119894isin R119899

1003817

1003817

1003817

1003817

119891

119894

1003817

1003817

1003817

1003817

= 1 119891

119894119891

119895 = 119894= 0 119894 = 1 119899

119877

119906= 119878

119894isin R119895times119895

119878

lowast

119894119878

119894= 1 119894 = 1 119904

119877

119909= 119879

119894isin R119899times119899

119879

lowast

119894119879

119894= 1 119894 = 1 119905

119876

119906= 119888

119894isin R 119888

119894gt 0 119894 = 1 119902

119876

119909= 119889

119894isin R 119889

119894gt 0 119894 = 1 119903

(3)

These sets should correspond to the ranges in inputs andinitial states the system is operating in

21 Controllability Gramian Controllability is a quantifica-tion of how well a state can be driven by input Analyticallythe controllability Gramian is given by the smallest semi-positive definite solution of the Lyapunov equation 119860119882

119862+

119882

119862119860

119879

= minus119861119861

119879 If the underlying system is asymptoticallystable the controllability Gramian can also be defined usingthe linear input-to-state map

119882

119862= int

infin

0

119890

119860120591

119861119861

119879

119890

119860119879120591

119889120591 (4)

Following [3] the empirical controllability Gramian isdefined by the following

Definition 1 For sets 119864

119906 119877119906 and 119876

119906 and input 119906(119905) and

input during the steady state119909(119906) the empirical controllabilityGramian is given by

119882

119862=

1

1003816

1003816

1003816

1003816

119876

119906

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119877

119906

1003816

1003816

1003816

1003816

|119876119906|

sum

ℎ=1

|119877119906|

sum

119894=1

119898

sum

119895=1

1

119888

2

ℎ

int

infin

0

Ψ

ℎ119894119895

(119905) 119889119905

Ψ

ℎ119894119895

(119905) = (119909

ℎ119894119895

(119905) minus 119909) (119909

ℎ119894119895

(119905) minus 119909)

lowast

isin R119899times119899

(5)

with 119909

ℎ119894119895 being the states for the input configuration 119906

ℎ119894119895

(119905) =

119888

ℎ119878

119894119890

119895119906(119905) + 119906

Originally in [2] 119906(119905) was restricted to 120575(119905) but extendedin [3] to arbitrary input under the name of empirical covari-ance matrix 119909 can be the arithmetic average the median thesteady state or the principal components Restricting 119877

119906to

minus1 1 simplifies the input perturbation to

119906

ℎ119894119895

(119905) = minus1

119894

119902

ℎ119890

119895119906 (119905) + 119906 (6)

22 Observability Gramian Observability quantifies howwell a change in a state is reflected by the output Theanalytical observability Gramian is given by the smallest

semipositive definite solution of the Lyapunov equation119860

119879

119882

119874+ 119882

119874119860 = minus119862

119879

119862 Given an asymptotically stableunderlying system the observability Gramian can also bedefined using the state-to-output map

119882

119874= int

infin

0

119890

119860119879120591

119862

119879

119862119890

119860120591

119889120591 (7)

The empirical observability Gramian is defined as describedin [2 3]

Definition 2 For sets 119864119909 119877119909 and119876

119909and output 119910 during the

steady state 119909(119910) the empirical observability Gramian is givenby

119882

119874=

1

1003816

1003816

1003816

1003816

119876

119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119877

119909

1003816

1003816

1003816

1003816

|119876119909|

sum

119896=1

|119877119909|

sum

119897=1

1

119889

2

119896

119879

119897int

infin

0

Ψ

119896119897

(119905) 119889119905 119879

lowast

119897

Ψ

119896119897

119886119887= (119910

119896119897119886

(119905) minus 119910)

lowast

(119910

119896119897119887

(119905) minus 119910) isin R

(8)

with 119910

119896119897119886 being the systems output for the initial state conf-iguration 119909

119896119897119886

0= 119889

119896119878

119897119891

119886+ 119909

119910 can be the arithmetic average the median the steady-state output or the principal components Restricting 119877

119909to

minus1 1 simplifies (8) to

119882

119874=

1

1003816

1003816

1003816

1003816

119876

119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119877

119909

1003816

1003816

1003816

1003816

|119876119909|

sum

119896=1

|119877119909|

sum

119897=1

1

119889

2

119896

int

infin

0

Ψ

119896119897

(119905) 119889119905(9)

and simplifies the initial state perturbation to

119909

119896119897119886

0= minus1

119897

119889

119896119891

119886+ 119909

(10)

23 Cross Gramian The cross Gramian [4] makes a com-bined statement about the controllability and observabilitygiven that the system has the same number of inputs andoutputs If the system is also symmetric meaning that thesystem transfer function is symmetric then the absolute valueof these Gramiansrsquo eigenvalues equals the Hankel singularvalues It is originally computed as the smallest solution of theSylvester equation 119860119882

119883+119882

119883119860 = minus119861119862 The cross Gramian

can also be defined using the input-to-state and state-to-output maps if the underlying system is asymptoticallystable

119882

119883= int

infin

0

119890

119860120591

119861119862119890

119860120591

119889120591 (11)

The empirical cross Gramian has been introduced in [5] forSISO systems and was extended to MIMO systems in [6]

Journal of Mathematics 3

Definition 3 For sets 119864119906 119864119909 119877119906 119877119909 119876119906 and 119876

119909and input

119906 during steady state 119909 with output 119910 the empirical crossGramian is given by

119882

119883=

1

1003816

1003816

1003816

1003816

119876

119906

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119877

119906

1003816

1003816

1003816

1003816

119898

1003816

1003816

1003816

1003816

119876

119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119877

119909

1003816

1003816

1003816

1003816

sdot

|119876119906|

sum

ℎ=1

|119877119906|

sum

119894=1

119898

sum

119895=1

|119876119909|

sum

119896=1

|119877119909|

sum

119897=1

1

119888

ℎ119889

119896

int

infin

0

119879

119897Ψ

ℎ119894119895119896119897

(119905) 119879

lowast

119897119889119905

Ψ

ℎ119894119895119896119897

119886119887(119905) = 119891

lowast

119887119879

lowast

119896Δ119909

ℎ119894119895

(119905) 119890

lowast

119894119878

lowast

ℎΔ119910

119896119897119886

(119905)

Δ119909

ℎ119894119895

(119905) = (119909

ℎ119894119895

(119905) minus 119909)

Δ119910

119896119897119886

(119905) = (119910

119896119897119886

(119905) minus 119910)

(12)

with 119909

ℎ119894119895 and 119910

119896119897119886 being the states and output for the input119906

ℎ119894119895

(119905) = 119888

ℎ119878

119894119890

119895119906(119905) + 119906 and initial state 119909119896119897119886

0= 119889

119896119879

119897119891

119886+ 119909

respectively

119909 and 119910 can be the arithmetic average the median thesteady state or the principal components of the outputAgain restricting 119877

119906and 119877

119909to minus1 1 simplifies (12) to

Ψ

ℎ119894119895119896119897

119886119887= (minus1)

119894+119897

119891

lowast

119887Δ119909

ℎ119894119895

(119905) 119890

lowast

119895Δ119910

119896119897119886

(119905) (13)

as well as simplifying input and initial state perturbation to

119906

ℎ119894119895

(119905) = minus1

119894

119888

ℎ119890

119895119906 (119905) + 119906

119909

119896119897119886

0= minus1

119897

119889

119896119891

119886+ 119909

(14)

24 Sensitivity Gramian The sensitivity Gramian allowscontrollability-based parameter reduction and identificationIt is based on [7] and aimed formodels that can be partitionedas follows

= 119891 (119909 119906 119901) = 119891 (119909 119906) +

119875

sum

119896=1

119891 (119909 119901

119896) (15)

The parameters 119901 isin R119875 are handled here as additionalinputs All summands of the partitioned system are treated asindependent subsystems and thus a controllability Gramianfor each subsystem can be computed Each parameter con-trollability is encoded in the sum of singular values of theassociated subcontrollability Gramian 119882

119862119896 The sensitivity

Gramian is now given by the diagonal matrix with eachdiagonal element being the trace of a subcontrollabilityGramian

119882

119878=

119875

sum

119896=1

tr (119882119862119896

) (120575

119894119895)

119894=119895=119896

(16)

The controllability and thus identifiability of each parameterare then given by the corresponding diagonal entry of thesensitivity Gramian119882

119878 For partitionable linear systems the

sum of all subsystems controllability Gramians119882119862119896

and the

parameter-free subsystems Gramian 119882

1198620equals the usual

controllability Gramian [7]

119882

119862= 119882

1198620+

119875

sum

119896=1

119882

119862119896 (17)

The sensitivity Gramian can be applied to nonpartitionablemodels with reduced accuracy

25 Identifiability Gramian The identifiability Gramian ena-bles observability-based parameter identification and conse-quently parameter reduction As described in [8] the dyna-mic system states are augmented with as many states as para-meters that are constant over time and have the initial valueof the (prior) parameter value

One has

= (

119901

) = (

119891 (119909 119906 119901)

0

)

119910 = 119892 (119909 119906 119901)

(18)

The observability Gramian of this augmented system holdsthe observability information of states and parameters Toextract the parameter specific observability the Schur com-plement can be applied to the augmented observability Gra-mian

119882 = (

119882

119874119882

119876

119882

lowast

119876119882

119875

)

997904rArr 119882

119868= 119882

119875minus119882

lowast

119876119882

minus1

119874119882

119876

(19)

26 Joint Gramian Based on the identifiability Gramianprocedure the cross Gramian can be employed for a con-current state and parameter reduction (see [6]) As for theidentifiability gramian the system is augmented by constantparameter states Additionally as many inputs and outputs asparameters are augmented as well The augmented inputs actvia identity on the augmented states Likewise the augmentedstates are mapped by identity to the augmented outputs topreserve symmetry

= (

119901

) = (

119891 (119909 119906 119901)

V)

119910 = (

119892 (119909 119906 119901)

1)

(20)

to preserve symmetry The cross Gramian of this specialaugmented system similar to the identifiability Gramianholds the cross Gramian of the original system as well as across-identifiability Gramian119882 119868 which can be extracted witha Schur complement from the joint Gramian

119882

119869= (

119882

119883119882

119876

119882

119902119882

119875

)

997904rArr 119882 119868= 119882

119875minus119882

lowast

119876119882

minus1

119883119882

119876

(21)

4 Journal of Mathematics

Table 1 Empirical gramian application matrix

Gramian type Statereduction

Parameterreduction

Combinedreduction

119882

119862() times times

119882

119874() times times

119882

119862amp 119882

119874 times times

119882

119883 () times

119882

119878() times

119882

119868() ()

119882

119869()

3 Implementation

The emgr software framework presented here provides auniform interface to compute all six empirical Gramians andis given by

W = emgr (f g q t w p vcfg u us xs um xm yd) (22)

with f and g being handles to the system and the outputfunction both requiring the signature f(xup) andg(xup) q is a vector defining the systems number ofinputs states and outputs t is a three-component vectorcontaining start time time step and stop time w is a charactersetting the Gramian type for an overview on the applicabilityof Gramian types see Table 1

The following arguments are optional p holds anyparameters The ten-component vector vcfg configures theavailable options including averaging types input and statescale subdivisions and perturbation rotations u provides theinput to119891 and119892 while us and xs set steady input and steadystate um and xm define the scales of the perturbation Las-tly yd allows passing experimental data to be used insteadof generated snapshots

The parameters reducing empirical Gramians (sensitivityidentifiability and joint) are an encapsulation of the statereducing empirical Gramians (controllability observabilityand cross) Computation of the latter is extensively vector-ized exploiting the Gramian matrix assembly format Forexample the empirical observability Gramian assembly from(8) can be computationally simplified to

Ψ

119896119897

119886119887= (119910

119896119897119886

(119905) minus 119910)

lowast

(119910

119896119897119887

(119905) minus 119910)

997888rarr

120595

119896119897

119886(119905) = (119910

119896119897119886

(119905) minus 119910)

Ψ

119896119897

119886119887= vec (120595119896119897

119886)

lowast

vec (120595119896119897119887)

(23)

Computation of empirical Gramians using emgr is veryportable since only basic vector and matrix operations arerequired Necessary integrations meaning simulations forgiven inputs or initial states are accomplished either by thefirst-order Eulerrsquos method second-order Adams-Bashforthmethod or second-order Leapfrog method The empiricalGramian framework emgr as well as the following exper-iments is released under an open source license is com-patible with OCTAVE and MATLAB and can be found

Time

Rela

tive e

rror

10 20 30 40 50 60 70 80 90 100123456789

10

0002

0004

0006

0008

0010

0012

0014

Figure 1 Relative error in reduced system output by balanced trun-cation using the empirical controllability Gramian and the empiricalobservabilityGramian WC = emgr(fg[1010010][00011]ldquocrdquop) WO = emgr(fg[1010010][00011]ldquoordquop)

at httpgramiande or at the Oberwolfach References onMathematical Software

4 Numerical Experiments

To demonstrate the various empirical Gramians computedby the emgr framework a symmetric nonlinear MIMOsystem with one hundred states ten inputs and ten outputsis employed The system matrix is generated randomly withensured stability and symmetry the input matrix 119861 is alsoa random matrix and the output matrix is given by 119862 =

119861

119879 Furthermore a random but element-wise parametrizedsource term 119864

119901of dimension 119899 = 100 parameters is added

Input is applied through a delta impulse

119891 (119909 119906 119901) = = 119860 arsinh (119909) + 119861119906 + 119864

119901

119892 (119909) = 119910 = 119862119909

(24)

First a state reduction using the empirical controllabilityGramian and the empirical observability Gramian throughbalanced truncation is performed in Figure 1 reducingthe number of states to the number of outputs Balancedtruncation as a classic approach inmodel order reductionwillbe used as a baseline to which the following methods will becompared

Next a state reduction by direct truncation employing theempirical cross Gramian is demonstrated in Figure 2 againreducing the number of states to ten The state reduction viadirect truncation of the cross Gramian has about the sameerror but requires only one Gramian and no balancing

The empirical sensitivity Gramian can be applied if theunderlying system can be partitioned such that 119891(119909 119906 119901) =

119891(119909 119906) + 119891(119909 119901) To be able to use it in this setting theparametrized source term is reduced to the number ofoutputs in Figure 3 The sensitivity Gramian is the fastestparameter reduction method but has a high relative error inoutputs Since the parameters of the source term are reducedthe cumulative effects in the original system are the origin ofthe increasing error over time Next the parametrized source

Journal of Mathematics 5

0005

001

0015

002

0025

003

Time

Rela

tive e

rror

10 20 30 40 50 60 70 80 90 100123456789

10

Figure 2 Relative error in reduced system output by truncationusing the empirical cross Gramian WX = emgr(fg[1010010]

[00011]ldquoxrdquop)

01

02

03

04

05

06

07

08

Time

Rela

tive e

rror

10 20 30 40 50 60 70 80 90 100123456789

10

Figure 3 Relative error in reduced system output with reducedsource term reduction by truncation using the empirical sensitivityGramian WS = emgr(fg[1010010][00011]ldquosrdquop)

term is reduced by the empirical identifiability Gramian inFigure 4

Taking five times as long for the parameter reduction asthe sensitivity Gramian the identifiability Gramian is abouttwo orders of magnitude more accurate

Finally in Figure 5 the same system undergoes a com-bined state and parameter reduction using the empiricaljoint Gramian The joint Gramian is the only Gramianallowing direct balancing-free combined reduction of stateand parameter space with a comparable relative error and yettakes about the same time for the reduction as the identifiabil-ity Gramian This combined reduction generates a reduced-order model of which the relative error is comparable to theother reduced system output models

5 Future Work

The emgr framework already allows a wide range of com-putations of empirical Gramians for state or parameterreduction Apart from model order reduction the empiricalGramians can be employed for system identification taskslike parameter identification or sensitivity analysis as wellas decentralized control nonlinearity measurement anduncertainty quantification

00001

00002

00003

00004

00005

00006

Time

Rela

tive e

rror

10 20 30 40 50 60 70 80 90 100123456789

10

Figure 4 Relative error in reduced system output with reducedsource term reduction by truncation using the empiricalidentifiability Gramian WI = emgr(fg [1010010]

[00011]ldquoirdquop)

03

005

01

015

02

025

Time

Rela

tive e

rror

10 20 30 40 50 60 70 80 90 100123456789

10

Figure 5 Relative error in system output with combinedstate and parameter reduction reduction by truncation ofparameters and states using the empirical joint GramianWJ = emgr(fg[1010010][00011]ldquojrdquop)

Further work will enhance the flexibility while keepingthe interface as simple as possible Following [8] allowingfactorial designs will greatly enlarge the field of applicationFinally extending the use of the cross Gramian (and thus thejoint Gramian) to nonsymmetric systems [4] will enable acombined state and parameter reduction for general linearand nonlinear models without balancing

Acknowledgments

The authors acknowledge the support by the DeutscheForschungsgemeinschaft and the Open Access PublicationFund of the University of Munster

References

[1] B C Moore ldquoPrincipal component analysis in linear systemscontrollability observability andmodel reductionrdquo IEEETrans-actions on Automatic Control vol 26 no 1 pp 17ndash32 1981

[2] S Lall J E Marsden and S Glavaski ldquoEmpirical modelreduction of controlled nonlinear systemsrdquo in Proceedings of theInternational Federation of Automatic Control World Congress(IFAC rsquo99) pp 473ndash478 1999

6 Journal of Mathematics

[3] J Hahn and T F Edgar ldquoBalancing approach to minimalrealization and model reduction of stable nonlinear systemsrdquoIndustrial and Engineering Chemistry Research vol 41 no 9 pp2204ndash2212 2002

[4] A C Antoulas Approximation of Large-Scale Dynamical Sys-tems Advances in Design and Control SIAM Philadelphia PaUSA 2005

[5] S Streif R Findeisen andE Bullinger ldquoRelating cross gramiansand sensitivity analysis in systems biologyrdquo in Proceedings ofthe 17th International Symposium on Mathematical Theory ofNetworks and Systems (MTNS rsquo06) pp 437ndash442 2006

[6] C Himpe and M Ohlberger ldquoCross-Gramian based combinedstate and parameter reductionrdquo httparxivorgabs13020634

[7] C Sun and J Hahn ldquoParameter reduction for stable dynamicalsystems based on Hankel singular values and sensitivity analy-sisrdquoChemical Engineering Science vol 61 no 16 pp 5393ndash54032006

[8] D Geffen R Findeisen M Schliemann F Allgower and MGuay ldquoObservability based parameter identifiability for bio-chemical reaction networksrdquo in Proceedings of the AmericanControl Conference (ACC rsquo08) pp 2130ndash2135 June 2008

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Mathematics 3

Definition 3 For sets 119864119906 119864119909 119877119906 119877119909 119876119906 and 119876

119909and input

119906 during steady state 119909 with output 119910 the empirical crossGramian is given by

119882

119883=

1

1003816

1003816

1003816

1003816

119876

119906

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119877

119906

1003816

1003816

1003816

1003816

119898

1003816

1003816

1003816

1003816

119876

119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119877

119909

1003816

1003816

1003816

1003816

sdot

|119876119906|

sum

ℎ=1

|119877119906|

sum

119894=1

119898

sum

119895=1

|119876119909|

sum

119896=1

|119877119909|

sum

119897=1

1

119888

ℎ119889

119896

int

infin

0

119879

119897Ψ

ℎ119894119895119896119897

(119905) 119879

lowast

119897119889119905

Ψ

ℎ119894119895119896119897

119886119887(119905) = 119891

lowast

119887119879

lowast

119896Δ119909

ℎ119894119895

(119905) 119890

lowast

119894119878

lowast

ℎΔ119910

119896119897119886

(119905)

Δ119909

ℎ119894119895

(119905) = (119909

ℎ119894119895

(119905) minus 119909)

Δ119910

119896119897119886

(119905) = (119910

119896119897119886

(119905) minus 119910)

(12)

with 119909

ℎ119894119895 and 119910

119896119897119886 being the states and output for the input119906

ℎ119894119895

(119905) = 119888

ℎ119878

119894119890

119895119906(119905) + 119906 and initial state 119909119896119897119886

0= 119889

119896119879

119897119891

119886+ 119909

respectively

119909 and 119910 can be the arithmetic average the median thesteady state or the principal components of the outputAgain restricting 119877

119906and 119877

119909to minus1 1 simplifies (12) to

Ψ

ℎ119894119895119896119897

119886119887= (minus1)

119894+119897

119891

lowast

119887Δ119909

ℎ119894119895

(119905) 119890

lowast

119895Δ119910

119896119897119886

(119905) (13)

as well as simplifying input and initial state perturbation to

119906

ℎ119894119895

(119905) = minus1

119894

119888

ℎ119890

119895119906 (119905) + 119906

119909

119896119897119886

0= minus1

119897

119889

119896119891

119886+ 119909

(14)

24 Sensitivity Gramian The sensitivity Gramian allowscontrollability-based parameter reduction and identificationIt is based on [7] and aimed formodels that can be partitionedas follows

= 119891 (119909 119906 119901) = 119891 (119909 119906) +

119875

sum

119896=1

119891 (119909 119901

119896) (15)

The parameters 119901 isin R119875 are handled here as additionalinputs All summands of the partitioned system are treated asindependent subsystems and thus a controllability Gramianfor each subsystem can be computed Each parameter con-trollability is encoded in the sum of singular values of theassociated subcontrollability Gramian 119882

119862119896 The sensitivity

Gramian is now given by the diagonal matrix with eachdiagonal element being the trace of a subcontrollabilityGramian

119882

119878=

119875

sum

119896=1

tr (119882119862119896

) (120575

119894119895)

119894=119895=119896

(16)

The controllability and thus identifiability of each parameterare then given by the corresponding diagonal entry of thesensitivity Gramian119882

119878 For partitionable linear systems the

sum of all subsystems controllability Gramians119882119862119896

and the

parameter-free subsystems Gramian 119882

1198620equals the usual

controllability Gramian [7]

119882

119862= 119882

1198620+

119875

sum

119896=1

119882

119862119896 (17)

The sensitivity Gramian can be applied to nonpartitionablemodels with reduced accuracy

25 Identifiability Gramian The identifiability Gramian ena-bles observability-based parameter identification and conse-quently parameter reduction As described in [8] the dyna-mic system states are augmented with as many states as para-meters that are constant over time and have the initial valueof the (prior) parameter value

One has

= (

119901

) = (

119891 (119909 119906 119901)

0

)

119910 = 119892 (119909 119906 119901)

(18)

The observability Gramian of this augmented system holdsthe observability information of states and parameters Toextract the parameter specific observability the Schur com-plement can be applied to the augmented observability Gra-mian

119882 = (

119882

119874119882

119876

119882

lowast

119876119882

119875

)

997904rArr 119882

119868= 119882

119875minus119882

lowast

119876119882

minus1

119874119882

119876

(19)

26 Joint Gramian Based on the identifiability Gramianprocedure the cross Gramian can be employed for a con-current state and parameter reduction (see [6]) As for theidentifiability gramian the system is augmented by constantparameter states Additionally as many inputs and outputs asparameters are augmented as well The augmented inputs actvia identity on the augmented states Likewise the augmentedstates are mapped by identity to the augmented outputs topreserve symmetry

= (

119901

) = (

119891 (119909 119906 119901)

V)

119910 = (

119892 (119909 119906 119901)

1)

(20)

to preserve symmetry The cross Gramian of this specialaugmented system similar to the identifiability Gramianholds the cross Gramian of the original system as well as across-identifiability Gramian119882 119868 which can be extracted witha Schur complement from the joint Gramian

119882

119869= (

119882

119883119882

119876

119882

119902119882

119875

)

997904rArr 119882 119868= 119882

119875minus119882

lowast

119876119882

minus1

119883119882

119876

(21)

4 Journal of Mathematics

Table 1 Empirical gramian application matrix

Gramian type Statereduction

Parameterreduction

Combinedreduction

119882

119862() times times

119882

119874() times times

119882

119862amp 119882

119874 times times

119882

119883 () times

119882

119878() times

119882

119868() ()

119882

119869()

3 Implementation

The emgr software framework presented here provides auniform interface to compute all six empirical Gramians andis given by

W = emgr (f g q t w p vcfg u us xs um xm yd) (22)

with f and g being handles to the system and the outputfunction both requiring the signature f(xup) andg(xup) q is a vector defining the systems number ofinputs states and outputs t is a three-component vectorcontaining start time time step and stop time w is a charactersetting the Gramian type for an overview on the applicabilityof Gramian types see Table 1

The following arguments are optional p holds anyparameters The ten-component vector vcfg configures theavailable options including averaging types input and statescale subdivisions and perturbation rotations u provides theinput to119891 and119892 while us and xs set steady input and steadystate um and xm define the scales of the perturbation Las-tly yd allows passing experimental data to be used insteadof generated snapshots

The parameters reducing empirical Gramians (sensitivityidentifiability and joint) are an encapsulation of the statereducing empirical Gramians (controllability observabilityand cross) Computation of the latter is extensively vector-ized exploiting the Gramian matrix assembly format Forexample the empirical observability Gramian assembly from(8) can be computationally simplified to

Ψ

119896119897

119886119887= (119910

119896119897119886

(119905) minus 119910)

lowast

(119910

119896119897119887

(119905) minus 119910)

997888rarr

120595

119896119897

119886(119905) = (119910

119896119897119886

(119905) minus 119910)

Ψ

119896119897

119886119887= vec (120595119896119897

119886)

lowast

vec (120595119896119897119887)

(23)

Computation of empirical Gramians using emgr is veryportable since only basic vector and matrix operations arerequired Necessary integrations meaning simulations forgiven inputs or initial states are accomplished either by thefirst-order Eulerrsquos method second-order Adams-Bashforthmethod or second-order Leapfrog method The empiricalGramian framework emgr as well as the following exper-iments is released under an open source license is com-patible with OCTAVE and MATLAB and can be found

Time

Rela

tive e

rror

10 20 30 40 50 60 70 80 90 100123456789

10

0002

0004

0006

0008

0010

0012

0014

Figure 1 Relative error in reduced system output by balanced trun-cation using the empirical controllability Gramian and the empiricalobservabilityGramian WC = emgr(fg[1010010][00011]ldquocrdquop) WO = emgr(fg[1010010][00011]ldquoordquop)

at httpgramiande or at the Oberwolfach References onMathematical Software

4 Numerical Experiments

To demonstrate the various empirical Gramians computedby the emgr framework a symmetric nonlinear MIMOsystem with one hundred states ten inputs and ten outputsis employed The system matrix is generated randomly withensured stability and symmetry the input matrix 119861 is alsoa random matrix and the output matrix is given by 119862 =

119861

119879 Furthermore a random but element-wise parametrizedsource term 119864

119901of dimension 119899 = 100 parameters is added

Input is applied through a delta impulse

119891 (119909 119906 119901) = = 119860 arsinh (119909) + 119861119906 + 119864

119901

119892 (119909) = 119910 = 119862119909

(24)

First a state reduction using the empirical controllabilityGramian and the empirical observability Gramian throughbalanced truncation is performed in Figure 1 reducingthe number of states to the number of outputs Balancedtruncation as a classic approach inmodel order reductionwillbe used as a baseline to which the following methods will becompared

Next a state reduction by direct truncation employing theempirical cross Gramian is demonstrated in Figure 2 againreducing the number of states to ten The state reduction viadirect truncation of the cross Gramian has about the sameerror but requires only one Gramian and no balancing

The empirical sensitivity Gramian can be applied if theunderlying system can be partitioned such that 119891(119909 119906 119901) =

119891(119909 119906) + 119891(119909 119901) To be able to use it in this setting theparametrized source term is reduced to the number ofoutputs in Figure 3 The sensitivity Gramian is the fastestparameter reduction method but has a high relative error inoutputs Since the parameters of the source term are reducedthe cumulative effects in the original system are the origin ofthe increasing error over time Next the parametrized source

Journal of Mathematics 5

0005

001

0015

002

0025

003

Time

Rela

tive e

rror

10 20 30 40 50 60 70 80 90 100123456789

10

Figure 2 Relative error in reduced system output by truncationusing the empirical cross Gramian WX = emgr(fg[1010010]

[00011]ldquoxrdquop)

01

02

03

04

05

06

07

08

Time

Rela

tive e

rror

10 20 30 40 50 60 70 80 90 100123456789

10

Figure 3 Relative error in reduced system output with reducedsource term reduction by truncation using the empirical sensitivityGramian WS = emgr(fg[1010010][00011]ldquosrdquop)

term is reduced by the empirical identifiability Gramian inFigure 4

Taking five times as long for the parameter reduction asthe sensitivity Gramian the identifiability Gramian is abouttwo orders of magnitude more accurate

Finally in Figure 5 the same system undergoes a com-bined state and parameter reduction using the empiricaljoint Gramian The joint Gramian is the only Gramianallowing direct balancing-free combined reduction of stateand parameter space with a comparable relative error and yettakes about the same time for the reduction as the identifiabil-ity Gramian This combined reduction generates a reduced-order model of which the relative error is comparable to theother reduced system output models

5 Future Work

The emgr framework already allows a wide range of com-putations of empirical Gramians for state or parameterreduction Apart from model order reduction the empiricalGramians can be employed for system identification taskslike parameter identification or sensitivity analysis as wellas decentralized control nonlinearity measurement anduncertainty quantification

00001

00002

00003

00004

00005

00006

Time

Rela

tive e

rror

10 20 30 40 50 60 70 80 90 100123456789

10

Figure 4 Relative error in reduced system output with reducedsource term reduction by truncation using the empiricalidentifiability Gramian WI = emgr(fg [1010010]

[00011]ldquoirdquop)

03

005

01

015

02

025

Time

Rela

tive e

rror

10 20 30 40 50 60 70 80 90 100123456789

10

Figure 5 Relative error in system output with combinedstate and parameter reduction reduction by truncation ofparameters and states using the empirical joint GramianWJ = emgr(fg[1010010][00011]ldquojrdquop)

Further work will enhance the flexibility while keepingthe interface as simple as possible Following [8] allowingfactorial designs will greatly enlarge the field of applicationFinally extending the use of the cross Gramian (and thus thejoint Gramian) to nonsymmetric systems [4] will enable acombined state and parameter reduction for general linearand nonlinear models without balancing

Acknowledgments

The authors acknowledge the support by the DeutscheForschungsgemeinschaft and the Open Access PublicationFund of the University of Munster

References

[1] B C Moore ldquoPrincipal component analysis in linear systemscontrollability observability andmodel reductionrdquo IEEETrans-actions on Automatic Control vol 26 no 1 pp 17ndash32 1981

[2] S Lall J E Marsden and S Glavaski ldquoEmpirical modelreduction of controlled nonlinear systemsrdquo in Proceedings of theInternational Federation of Automatic Control World Congress(IFAC rsquo99) pp 473ndash478 1999

6 Journal of Mathematics

[3] J Hahn and T F Edgar ldquoBalancing approach to minimalrealization and model reduction of stable nonlinear systemsrdquoIndustrial and Engineering Chemistry Research vol 41 no 9 pp2204ndash2212 2002

[4] A C Antoulas Approximation of Large-Scale Dynamical Sys-tems Advances in Design and Control SIAM Philadelphia PaUSA 2005

[5] S Streif R Findeisen andE Bullinger ldquoRelating cross gramiansand sensitivity analysis in systems biologyrdquo in Proceedings ofthe 17th International Symposium on Mathematical Theory ofNetworks and Systems (MTNS rsquo06) pp 437ndash442 2006

[6] C Himpe and M Ohlberger ldquoCross-Gramian based combinedstate and parameter reductionrdquo httparxivorgabs13020634

[7] C Sun and J Hahn ldquoParameter reduction for stable dynamicalsystems based on Hankel singular values and sensitivity analy-sisrdquoChemical Engineering Science vol 61 no 16 pp 5393ndash54032006

[8] D Geffen R Findeisen M Schliemann F Allgower and MGuay ldquoObservability based parameter identifiability for bio-chemical reaction networksrdquo in Proceedings of the AmericanControl Conference (ACC rsquo08) pp 2130ndash2135 June 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Mathematics

Table 1 Empirical gramian application matrix

Gramian type Statereduction

Parameterreduction

Combinedreduction

119882

119862() times times

119882

119874() times times

119882

119862amp 119882

119874 times times

119882

119883 () times

119882

119878() times

119882

119868() ()

119882

119869()

3 Implementation

The emgr software framework presented here provides auniform interface to compute all six empirical Gramians andis given by

W = emgr (f g q t w p vcfg u us xs um xm yd) (22)

with f and g being handles to the system and the outputfunction both requiring the signature f(xup) andg(xup) q is a vector defining the systems number ofinputs states and outputs t is a three-component vectorcontaining start time time step and stop time w is a charactersetting the Gramian type for an overview on the applicabilityof Gramian types see Table 1

The following arguments are optional p holds anyparameters The ten-component vector vcfg configures theavailable options including averaging types input and statescale subdivisions and perturbation rotations u provides theinput to119891 and119892 while us and xs set steady input and steadystate um and xm define the scales of the perturbation Las-tly yd allows passing experimental data to be used insteadof generated snapshots

The parameters reducing empirical Gramians (sensitivityidentifiability and joint) are an encapsulation of the statereducing empirical Gramians (controllability observabilityand cross) Computation of the latter is extensively vector-ized exploiting the Gramian matrix assembly format Forexample the empirical observability Gramian assembly from(8) can be computationally simplified to

Ψ

119896119897

119886119887= (119910

119896119897119886

(119905) minus 119910)

lowast

(119910

119896119897119887

(119905) minus 119910)

997888rarr

120595

119896119897

119886(119905) = (119910

119896119897119886

(119905) minus 119910)

Ψ

119896119897

119886119887= vec (120595119896119897

119886)

lowast

vec (120595119896119897119887)

(23)

Computation of empirical Gramians using emgr is veryportable since only basic vector and matrix operations arerequired Necessary integrations meaning simulations forgiven inputs or initial states are accomplished either by thefirst-order Eulerrsquos method second-order Adams-Bashforthmethod or second-order Leapfrog method The empiricalGramian framework emgr as well as the following exper-iments is released under an open source license is com-patible with OCTAVE and MATLAB and can be found

Time

Rela

tive e

rror

10 20 30 40 50 60 70 80 90 100123456789

10

0002

0004

0006

0008

0010

0012

0014

Figure 1 Relative error in reduced system output by balanced trun-cation using the empirical controllability Gramian and the empiricalobservabilityGramian WC = emgr(fg[1010010][00011]ldquocrdquop) WO = emgr(fg[1010010][00011]ldquoordquop)

at httpgramiande or at the Oberwolfach References onMathematical Software

4 Numerical Experiments

To demonstrate the various empirical Gramians computedby the emgr framework a symmetric nonlinear MIMOsystem with one hundred states ten inputs and ten outputsis employed The system matrix is generated randomly withensured stability and symmetry the input matrix 119861 is alsoa random matrix and the output matrix is given by 119862 =

119861

119879 Furthermore a random but element-wise parametrizedsource term 119864

119901of dimension 119899 = 100 parameters is added

Input is applied through a delta impulse

119891 (119909 119906 119901) = = 119860 arsinh (119909) + 119861119906 + 119864

119901

119892 (119909) = 119910 = 119862119909

(24)

First a state reduction using the empirical controllabilityGramian and the empirical observability Gramian throughbalanced truncation is performed in Figure 1 reducingthe number of states to the number of outputs Balancedtruncation as a classic approach inmodel order reductionwillbe used as a baseline to which the following methods will becompared

Next a state reduction by direct truncation employing theempirical cross Gramian is demonstrated in Figure 2 againreducing the number of states to ten The state reduction viadirect truncation of the cross Gramian has about the sameerror but requires only one Gramian and no balancing

The empirical sensitivity Gramian can be applied if theunderlying system can be partitioned such that 119891(119909 119906 119901) =

119891(119909 119906) + 119891(119909 119901) To be able to use it in this setting theparametrized source term is reduced to the number ofoutputs in Figure 3 The sensitivity Gramian is the fastestparameter reduction method but has a high relative error inoutputs Since the parameters of the source term are reducedthe cumulative effects in the original system are the origin ofthe increasing error over time Next the parametrized source

Journal of Mathematics 5

0005

001

0015

002

0025

003

Time

Rela

tive e

rror

10 20 30 40 50 60 70 80 90 100123456789

10

Figure 2 Relative error in reduced system output by truncationusing the empirical cross Gramian WX = emgr(fg[1010010]

[00011]ldquoxrdquop)

01

02

03

04

05

06

07

08

Time

Rela

tive e

rror

10 20 30 40 50 60 70 80 90 100123456789

10

Figure 3 Relative error in reduced system output with reducedsource term reduction by truncation using the empirical sensitivityGramian WS = emgr(fg[1010010][00011]ldquosrdquop)

term is reduced by the empirical identifiability Gramian inFigure 4

Taking five times as long for the parameter reduction asthe sensitivity Gramian the identifiability Gramian is abouttwo orders of magnitude more accurate

Finally in Figure 5 the same system undergoes a com-bined state and parameter reduction using the empiricaljoint Gramian The joint Gramian is the only Gramianallowing direct balancing-free combined reduction of stateand parameter space with a comparable relative error and yettakes about the same time for the reduction as the identifiabil-ity Gramian This combined reduction generates a reduced-order model of which the relative error is comparable to theother reduced system output models

5 Future Work

The emgr framework already allows a wide range of com-putations of empirical Gramians for state or parameterreduction Apart from model order reduction the empiricalGramians can be employed for system identification taskslike parameter identification or sensitivity analysis as wellas decentralized control nonlinearity measurement anduncertainty quantification

00001

00002

00003

00004

00005

00006

Time

Rela

tive e

rror

10 20 30 40 50 60 70 80 90 100123456789

10

Figure 4 Relative error in reduced system output with reducedsource term reduction by truncation using the empiricalidentifiability Gramian WI = emgr(fg [1010010]

[00011]ldquoirdquop)

03

005

01

015

02

025

Time

Rela

tive e

rror

10 20 30 40 50 60 70 80 90 100123456789

10

Figure 5 Relative error in system output with combinedstate and parameter reduction reduction by truncation ofparameters and states using the empirical joint GramianWJ = emgr(fg[1010010][00011]ldquojrdquop)

Further work will enhance the flexibility while keepingthe interface as simple as possible Following [8] allowingfactorial designs will greatly enlarge the field of applicationFinally extending the use of the cross Gramian (and thus thejoint Gramian) to nonsymmetric systems [4] will enable acombined state and parameter reduction for general linearand nonlinear models without balancing

Acknowledgments

The authors acknowledge the support by the DeutscheForschungsgemeinschaft and the Open Access PublicationFund of the University of Munster

References

[1] B C Moore ldquoPrincipal component analysis in linear systemscontrollability observability andmodel reductionrdquo IEEETrans-actions on Automatic Control vol 26 no 1 pp 17ndash32 1981

[2] S Lall J E Marsden and S Glavaski ldquoEmpirical modelreduction of controlled nonlinear systemsrdquo in Proceedings of theInternational Federation of Automatic Control World Congress(IFAC rsquo99) pp 473ndash478 1999

6 Journal of Mathematics

[3] J Hahn and T F Edgar ldquoBalancing approach to minimalrealization and model reduction of stable nonlinear systemsrdquoIndustrial and Engineering Chemistry Research vol 41 no 9 pp2204ndash2212 2002

[4] A C Antoulas Approximation of Large-Scale Dynamical Sys-tems Advances in Design and Control SIAM Philadelphia PaUSA 2005

[5] S Streif R Findeisen andE Bullinger ldquoRelating cross gramiansand sensitivity analysis in systems biologyrdquo in Proceedings ofthe 17th International Symposium on Mathematical Theory ofNetworks and Systems (MTNS rsquo06) pp 437ndash442 2006

[6] C Himpe and M Ohlberger ldquoCross-Gramian based combinedstate and parameter reductionrdquo httparxivorgabs13020634

[7] C Sun and J Hahn ldquoParameter reduction for stable dynamicalsystems based on Hankel singular values and sensitivity analy-sisrdquoChemical Engineering Science vol 61 no 16 pp 5393ndash54032006

[8] D Geffen R Findeisen M Schliemann F Allgower and MGuay ldquoObservability based parameter identifiability for bio-chemical reaction networksrdquo in Proceedings of the AmericanControl Conference (ACC rsquo08) pp 2130ndash2135 June 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Mathematics 5

0005

001

0015

002

0025

003

Time

Rela

tive e

rror

10 20 30 40 50 60 70 80 90 100123456789

10

Figure 2 Relative error in reduced system output by truncationusing the empirical cross Gramian WX = emgr(fg[1010010]

[00011]ldquoxrdquop)

01

02

03

04

05

06

07

08

Time

Rela

tive e

rror

10 20 30 40 50 60 70 80 90 100123456789

10

Figure 3 Relative error in reduced system output with reducedsource term reduction by truncation using the empirical sensitivityGramian WS = emgr(fg[1010010][00011]ldquosrdquop)

term is reduced by the empirical identifiability Gramian inFigure 4

Taking five times as long for the parameter reduction asthe sensitivity Gramian the identifiability Gramian is abouttwo orders of magnitude more accurate

Finally in Figure 5 the same system undergoes a com-bined state and parameter reduction using the empiricaljoint Gramian The joint Gramian is the only Gramianallowing direct balancing-free combined reduction of stateand parameter space with a comparable relative error and yettakes about the same time for the reduction as the identifiabil-ity Gramian This combined reduction generates a reduced-order model of which the relative error is comparable to theother reduced system output models

5 Future Work

The emgr framework already allows a wide range of com-putations of empirical Gramians for state or parameterreduction Apart from model order reduction the empiricalGramians can be employed for system identification taskslike parameter identification or sensitivity analysis as wellas decentralized control nonlinearity measurement anduncertainty quantification

00001

00002

00003

00004

00005

00006

Time

Rela

tive e

rror

10 20 30 40 50 60 70 80 90 100123456789

10

Figure 4 Relative error in reduced system output with reducedsource term reduction by truncation using the empiricalidentifiability Gramian WI = emgr(fg [1010010]

[00011]ldquoirdquop)

03

005

01

015

02

025

Time

Rela

tive e

rror

10 20 30 40 50 60 70 80 90 100123456789

10

Figure 5 Relative error in system output with combinedstate and parameter reduction reduction by truncation ofparameters and states using the empirical joint GramianWJ = emgr(fg[1010010][00011]ldquojrdquop)

Further work will enhance the flexibility while keepingthe interface as simple as possible Following [8] allowingfactorial designs will greatly enlarge the field of applicationFinally extending the use of the cross Gramian (and thus thejoint Gramian) to nonsymmetric systems [4] will enable acombined state and parameter reduction for general linearand nonlinear models without balancing

Acknowledgments

The authors acknowledge the support by the DeutscheForschungsgemeinschaft and the Open Access PublicationFund of the University of Munster

References

[1] B C Moore ldquoPrincipal component analysis in linear systemscontrollability observability andmodel reductionrdquo IEEETrans-actions on Automatic Control vol 26 no 1 pp 17ndash32 1981

[2] S Lall J E Marsden and S Glavaski ldquoEmpirical modelreduction of controlled nonlinear systemsrdquo in Proceedings of theInternational Federation of Automatic Control World Congress(IFAC rsquo99) pp 473ndash478 1999

6 Journal of Mathematics

[3] J Hahn and T F Edgar ldquoBalancing approach to minimalrealization and model reduction of stable nonlinear systemsrdquoIndustrial and Engineering Chemistry Research vol 41 no 9 pp2204ndash2212 2002

[4] A C Antoulas Approximation of Large-Scale Dynamical Sys-tems Advances in Design and Control SIAM Philadelphia PaUSA 2005

[5] S Streif R Findeisen andE Bullinger ldquoRelating cross gramiansand sensitivity analysis in systems biologyrdquo in Proceedings ofthe 17th International Symposium on Mathematical Theory ofNetworks and Systems (MTNS rsquo06) pp 437ndash442 2006

[6] C Himpe and M Ohlberger ldquoCross-Gramian based combinedstate and parameter reductionrdquo httparxivorgabs13020634

[7] C Sun and J Hahn ldquoParameter reduction for stable dynamicalsystems based on Hankel singular values and sensitivity analy-sisrdquoChemical Engineering Science vol 61 no 16 pp 5393ndash54032006

[8] D Geffen R Findeisen M Schliemann F Allgower and MGuay ldquoObservability based parameter identifiability for bio-chemical reaction networksrdquo in Proceedings of the AmericanControl Conference (ACC rsquo08) pp 2130ndash2135 June 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Mathematics

[3] J Hahn and T F Edgar ldquoBalancing approach to minimalrealization and model reduction of stable nonlinear systemsrdquoIndustrial and Engineering Chemistry Research vol 41 no 9 pp2204ndash2212 2002

[4] A C Antoulas Approximation of Large-Scale Dynamical Sys-tems Advances in Design and Control SIAM Philadelphia PaUSA 2005

[5] S Streif R Findeisen andE Bullinger ldquoRelating cross gramiansand sensitivity analysis in systems biologyrdquo in Proceedings ofthe 17th International Symposium on Mathematical Theory ofNetworks and Systems (MTNS rsquo06) pp 437ndash442 2006

[6] C Himpe and M Ohlberger ldquoCross-Gramian based combinedstate and parameter reductionrdquo httparxivorgabs13020634

[7] C Sun and J Hahn ldquoParameter reduction for stable dynamicalsystems based on Hankel singular values and sensitivity analy-sisrdquoChemical Engineering Science vol 61 no 16 pp 5393ndash54032006

[8] D Geffen R Findeisen M Schliemann F Allgower and MGuay ldquoObservability based parameter identifiability for bio-chemical reaction networksrdquo in Proceedings of the AmericanControl Conference (ACC rsquo08) pp 2130ndash2135 June 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of