research article cross-gramian-based combined...

Research ArticleCross-Gramian-Based Combined State and Parameter Reductionfor Large-Scale Control Systems

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 christianhimpeuni-muensterde

Received 14 April 2014 Accepted 1 June 2014 Published 2 July 2014

Academic Editor Victor Sreeram

Copyright copy 2014 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

This work introduces the empirical cross-gramian for multiple-input-multiple-output systems The cross-gramian is a tool forreducing the state space of control systems by conjoining controllability and observability information into a single matrixand does not require balancing Its empirical gramian variant extends the applicability of the cross-gramian to nonlinearsystems Furthermore for parametrized systems the empirical gramians can also be utilized for sensitivity analysis or parameteridentification and thus for parameter reduction This work also introduces the empirical joint gramian which is derived from theempirical cross-gramian The joint gramian allows not only a reduction of the parameter space but also the combined state andparameter space reduction which is tested on a linear and a nonlinear control system Controllability- and observability-basedcombined reduction methods are also presented which are benchmarked against the joint gramian

1 Introduction

The evaluation of large-scale dynamical systems which arisefor example from complex networks or discretized partialdifferential equations may require model reduction due tolimitations in computing power or memory A reductionof the state space generates a surrogate model resemblingthe same dynamics up to a small error For parametrizedsystems the model order reduction has to take into accountthe associated parameter space to ensure the validity ofthe reduced order model If the parameter space is of highdimension a repeated evaluation at various locations of theparameter space for example during optimization of inverseproblems may also necessitate a model reduction yet forthe parameter space This contribution is concerned withcombined state and parameter reduction targeting modelswith high-dimensional state and parameter spaces

The efficient reduction of large-scale nonlinear con-trol systems is a challenging task especially in the caseof parametrized systems with high-dimensional state andparameter spaces where a combined reduction of parametersand states may be required to allow repeated evaluation Forinstance an inverse problem on a neural network with many

nodes and unknown connectivitymodeled as a parametrizednonlinear control system requires long runtimes duringparameter estimation due to system size and parametercount Large-scale neural networks havewidespreaduse suchas forward control problems on artificial neural networks orinverse problems on biological neural networks A real-lifeexample is the reconstruction of connectivity between brainregions from activity measurements like EEG or fMRI (seeeg [1])

To lower computational complexity the parameter andstate spaces are to be confined to low-dimensional sub-spaces without affecting the systems dynamics significantlyProjection-based model order reduction techniques are con-cerned with determining projections to such subspacesmapping the high-dimensionalmodel to a reduced order low-dimensional surrogate model

The methods presented in this work are rooted in bal-anced truncation [2] and proper orthogonal decomposition(POD) [3] As an alternative to the here presented methodusing empirical gramians another class of balancing-relatedapproaches focuses on solving Lyapunov and Sylvester equa-tions (see eg [4]) For parametrized systems the reduced-basis also method [5] should be noted here

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 843869 13 pageshttpdxdoiorg1011552014843869

2 Mathematical Problems in Engineering

Since the number of a systems inputs and outputs usuallyremains fixed the maps to and from the intermediary statescharacterize the reducibility of a system [6] The balancedtruncation approach introduced in [2] balances a systemin terms of controllability and observability where control-lability quantifies how well a state is driven by the inputand observability quantifies how well changes in a stateare reflected in the output Excluding the least controllableand observable states by truncating the balanced system areduced order mapping from inputs to outputs is approxi-mated

A linear time-invariant control system is composed of alinear dynamic system and a linear output transformation

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

119910 (119905) = 119862119909 (119905)

(1)

with states 119909(119905) isin R119899 input or control 119906(119905) isin R119898 and outputs119910(119905) isin R119900 The system matrix 119860 isin R119899times119899 transforms thestates the input matrix 119861 isin R119899times119898 introduces external inputor control and the output matrix 119862 isin R119900times119899 transforms thestates to the outputs

Controllability and observability can be assessed throughthe associated controllability gramian119882

119862= int

infin

0

119890

119860119905

119861119861

119879

119890

119860119879119905

119889119905 and observability gramian 119882

119874= int

infin

0

119890

119860119879119905

119862

119879

119862119890

119860119905

119889119905Classically119882

119862and119882

119874are computed as the smallest positive

semi-definite solutions of the Lyapunov equations 119860119882119862+

119882

119862119860

119879

= minus119861119861

119879 and 119860

119879

119882

119874+ 119882

119874119860 = minus119862

119879

119862 respectivelyTo make a compound statement about controllability andobservability 119882

119862and 119882

119874have to be balanced [7] The

singular values of the resulting balanced gramian correspondto the Hankel singular values of the system with theirmagnitude describing how controllable and observable theassociated state is

This work focuses on cross-gramian-based methods formodel reduction which combine controllability and observ-ability information into one gramian and is elaboratelydescribed in [8] The cross-gramian 119882

119883= int

infin

0

119890

119860119905

119861119862119890

119860119905

119889119905

was introduced in [9] and corresponds to a solution of theSylvester equation 119860119882

119883+119882

119883119860 = minus119861119862

An alternative to solving the Lyapunov or Sylvestermatrixequations apart from the analytic approaches for examplein [10] is the method of empirical gramians which wasintroduced in the works [11 12] and enables the compu-tation of gramian matrices also for nonlinear systems bymere basic vector and matrix operations This concept wasextended among others in [13] providing more general inputsignals Particularly noted should be [14 15] for developingthe empirical cross-gramian for single-input-single-output(SISO) systems in the context of sensitivity analysis

In this paper the empirical cross-gramian is generalizedto be applicable to multiple-input-multiple-output (MIMO)systems For the gramian-based parameter reduction thegroundwork has been laid by [16] from the observabilityand by [17] from the controllability point of view From thecross-gramian perspective of parameter reduction a newgramian namely the joint gramian is introduced in thiswork Furthermore the concept of gramian-based combined

state and parameter reduction is established Using empiricalgramians it is shown that combined reduction allows efficientmodel order reduction of linear and nonlinear control sys-tems

To begin the cross-gramian and its properties arereviewed in Section 2 Next the empirical cross-gramianfor MIMO systems is developed in Section 3 Section 4introduces combined state and parameter reduction in twovariants first an observability- and second a controllability-based approach the former is enhanced to a cross-gramian-based combined reduction which is presented in Section 5Finally numerical experiments are conducted in Section 6comparing the newly presented methods for a linear andnonlinear neural network as well as a nonlinear benchmarkproblem

2 Review of the Cross-Gramian

A brief review of the cross-gramian along with its applicationto model reduction of linear time-invariant control systemsis given next The cross-gramian 119882

119883(also known by the

symbol119882119862119874

) was introduced in a sequence of works [9 18ndash23] and encodes controllability and observability into a singlegramian matrix defined as the product of controllability andobservability operator it can only be computed for square(a system with the same number of inputs and outputs) andasymptotically stable systems

119882

119883= int

infin

0

119890

119860119905

119861119862119890

119860119905

119889119905 (2)

Equivalently the cross-gramian is given as a solution to theSylvester equation 119860119882

119883+ 119882

119883119860 = minus119861119862 Approximate solu-

tions for the Sylvester equation were discussed in [24ndash26] Ifthe system is also symmetric the following relation betweenthe cross the controllability and observability gramian holds[9]

119882

2

119883= 119882

119862119882

119874997904rArr

1003816

1003816

1003816

1003816

120582 (119882

119883)

1003816

1003816

1003816

1003816

=radic120582 (119882

119862119882

119874)

(3)

While a SISO system is always symmetric [9] a linearMIMOsystem not only requires the same number of inputs andoutputs but also the system gain 119866 = minus119862119860

minus1

119861 has to besymmetric [24] then a symmetric transformation 119869 with119860119869 = 119869119860

119879 and 119861 = 119862

119879

119869 exist Trivially for 119869 = 1 the systemwould be restricted by 119860 = 119860

119879 and 119861 = 119862

119879 such a system iscalled state space symmetric

As presented in [9] the trace of the cross-gramian equalshalf the gain of a SISO system (119860 119887 119888) where now 119887 isin R119899times1

and 119888 isin R1times119899

tr (119882119883) = minus

1

2

119888119860

minus1

119887 (4)

Because the trace equals the sum of eigenvalues the cross-gramians eigenvalues are associated with the system gain (4)This result was used in [14 15] for parameter identificationpurposes using the system gain as a sensitivity measure Anextension of (4) from [9 Theorem 3] for MIMO systems isdeveloped (see also [27]) next

Mathematical Problems in Engineering 3

Corollary 1 Given a linear square asymptotically stableMIMO system then the trace of the cross-gramian relates tothe system gain as follows

tr (119882119883) = minus

1

2

tr (119862119860minus1

119861) (5)

Proof For an asymptotically stable system the trace of thecross-gramian in the form of (2) is given by

tr (119882119883) = tr(int

infin

0

119890

119860119905

119861119862119890

119860119905

119889119905)

= int

infin

0

tr (119890119860119905119861119862119890119860119905) 119889119905

= int

infin

0

tr (119862119890119860119905119890119860119905119861) 119889119905

= tr(intinfin

0

119862119890

2119860119905

119861119889119905)

= tr(119862intinfin

0

119890

2119860119905

119889119905119861)

= tr(119862(minus12

119860

minus1

)119861)

= minus

1

2

tr (119862119860minus1

119861)

(6)

Employing the cross-gramian instead of controllabilityand observability gramian means only a single gramian hasto be computed And since no balancing is required thetruncation procedure can be simplified to a direct truncation([28] [8 Ch 123]) A balancing transformation can beapproximated by the singular value decomposition (SVD) ofthe cross-gramianThe approximated Hankel singular valuesof the diagonal matrix119863 are sorted by the controllability andobservability of the states A projection to a subspace of thestate space is then given by truncation of 119880 and 119881

119882

119883

SVD= 119880119863119881 = (119880

1119880

2) (

119863

10

0 119863

2

)(

119881

1

119881

2

) (7)

The matrices 119880119881 isin R119899times119899 are partitioned based on athreshold 120598 le 2sum

119899

119896=119903+1119863

2119896119896into 119880

1isin R119899times119903 119880

2isin

R119899times(119899minus119903) and 119881

1isin R119903times119899 119881

2isin R(119899minus119903)times119899 This leads to

the following reduced order model (here the (one-sided)Galerkin projection is used since the (two-sided) Petrov-Galerkin projection may produce unstable reduced ordermodels)

119860 = 119880

119879

1119860119880

1

119861 = 119880

119879

1119861

119862 = 119862119880

1 119909

0= 119880

119879

1119909

0

997904rArr

(119905) =

119860119909 (119905) +

119861119906 (119905)

119910 (119905) =

119862119909 (119905)

(8)

Apart from truncation-basedmodel reduction the cross-gramian has applications for example in system identifica-tion [10] and decentralized control [29ndash31] by computing a

participation matrix (see [32]) based on the cross-gramianLastly the cross-gramian also has the benefit of convey-ing more information than controllability and observabilitygramian since the systemrsquos Cauchy index is given by thecross-gramianrsquos signature [19]

3 Empirical Cross-Gramian

In this section the empirical cross-gramian for MIMO sys-tems is introduced For general possibly nonlinear controlsystems of the form

(119905) = 119891 (119909 (119905) 119906 (119905))

119910 (119905) = 119892 (119909 (119905) 119906 (119905))

(9)

with states 119909(119905) isin R119899 input or control 119906(119905) isin R119898 outputs119910(119905) isin R119900 a vector field 119891 R119899

times R119898

rarr R119899 and anoutput function 119892 R119899

times R119898

rarr R119900 the procedure fromSection 2 is not viable In [11 12 33] the concept of empirical(controllability and observability) gramians was introducedThis is a PODmethod based solely on state space simulationsof the system [34] These empirical gramians correspondto the classic gramians for linear systems as shown in [11]Subsequently this approach and its field of application wereadvanced by [32 35ndash37] Because the empirical gramians canbe aligned to the operating region of the underlying systemin terms of initial states and input or control the empiricalgramians carry more detailed information on the system [38]than the gramians computed as solutions ofmatrix equations

Empirical gramians are based on averaging the responseof a system that is perturbed in inputs and initial statesInitially the perturbed input was restricted to a delta impulse119906(119905) = 120575(119905) which was broadened to more general input con-figurations in [13] under the name of empirical controllabilitycovariance matrix and empirical observability covariancematrix

Thenecessary perturbation sets are systematically definednext these should reflect the operating range of the under-lying system 119864

119906and 119864

119909are sets of standard directions for

the inputs and initial states Sets 119877119906and 119877

119909are orthogonal

transformations (rotations) to these standard directions ofinputs and initial states respectively while 119876

119906and 119876

119909hold

scales to these directions

119864

119906= 119890

119894isin R

119895

1003817

1003817

1003817

1003817

119890

119894

1003817

1003817

1003817

1003817

= 1 119890

119894119890

119895 = 119894= 0 119894 = 1 119898

119864

119909= 119891

119894isin R

119899

1003817

1003817

1003817

1003817

119891

119894

1003817

1003817

1003817

1003817

= 1 119891

119894119891

119895 = 119894= 0 119894 = 1 119899

119877

119906= 119878

119894isin R

119895times119895

119878

119879

119894119878

119894= 1 119894 = 1 119904

119877

119909= 119879

119894isin R

119899times119899

119879

119879

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

(10)

Along the lines of the empirical controllability gramianand empirical observability gramian [11] the empirical cross-gramian for SISO systems was introduced in [14] In this


work as a new contribution the empirical cross-gramian isgeneralized to squareMIMO systems Hence the scope of thecross-gramian is extended to nonlinear control systems andprovides an alternative nonlinear cross-gramian to [10] Fora general (possibly nonlinear) MIMO system with dim(119906) =dim(119909) the empirical cross-gramian is defined as follows

Definition 2 (empirical cross-gramian) For sets 119864119906 119864

119909 119877

119906

119877

119909 119876

119906 and 119876

119909 input 119906 during steady state 119909 with output

119910 the empirical cross-gramian

119882

119883relating the states 119909ℎ119894119895 of

input119906ℎ119894119895(119905) = 119888

ℎ119878

119894119890

119895120575(119905)+119906 to output119910119896119897119887 of119909119896119897119887

0= 119889

119896119879

119897119891

119887+119909

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

119879

119897int

infin

0

Ψ

ℎ119894119895119896119897

(119905) 119889119905 119879

119879

119897

Ψ

ℎ119894119895119896119897

119886119887(119905) = 119891

119879

119886119879

119879

119897Δ119909

ℎ119894119895

(119905) 119890

119879

119895119878

119879

119894Δ119910

119896119897119887

(119905) isin R

Δ119909

ℎ119894119895

(119905) = (119909

ℎ119894119895

(119905) minus 119909)

Δ119910

119896119897119887

(119905) = (119910

119896119897119887

(119905) minus 119910)

(11)

Essentially the empirical cross-gramian is an averagedcross-gramian over snapshots with the specified perturba-tions in input and initial states around steady state input 119909(119906)and steady state output 119910(119909)

Next similar to [11 14] the equality of the cross-gramianand the empirical cross-gramian for linear MIMO controlsystems are shown next

Lemma 3 (empirical cross-gramian) For any nonempty sets119877

119906 119877

119909 119876

119906 and 119876

119909the empirical cross-gramian

119882

119883of an

asymptotically stable linear control system is equal to the cross-gramian

Proof For an asymptotically stable linear control system theinput-to-state and state-to-output maps are given by

Δ119909 (119905) = 119909 (119905) = 119890

119860119905

119861119906 (119905)

Δ119910 (119905) = 119910 (119905) = 119862119890

119860119905

119909

0

(12)

thus

Ψ

ℎ119894119895119896119897

119886119887= 119891

119879

119886119879

119879

119897(119890

119860119905

119861119888

ℎ119878

119894119890

119895) 119890

119879

119895119878

119879

119894(119862119890

119860119905

119889

119896119879

119897119891

119887)

= 119888

ℎ119889

119896119891

119879

119886119879

119879

119897119890

119860119905

119861119862119890

119860119905

119879

119897119891

119887

997904rArr Ψ

ℎ119894119895119896119897

= 119888

ℎ119889

119896119879

119879

119897119890

119860119905

119861119862119890

119860119905

119879

119897

997904rArr

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

int

infin

0

119890

119860119905

119861119862119890

119860119905

119889119905

= int

infin

0

119890

119860119905

119861119862119890

119860119905

119889119905

= 119882

119883

(13)

As for the other empirical gramians this proof is onlyvalid for impulse input yet a similar approach to [13] canbe used to extend the empirical cross-gramian yieldingan empirical cross-covariance matrix by allowing generaldiscrete input signals The snapshots 119909ℎ119894119895 and 119910

119896119897119887 can notonly be computed as simulations on demand but also beincluded from observed experimental data In case the data iscollected at discrete times 119905 in regular intervals Δ119905 following[39] a discrete representation of the empirical cross-gramianis given here too

Definition 4 (discrete empirical cross-gramian) For sets 119864119906

119864

119909 119877

119906 119877

119909 119876

119906 and 119876

119909 input 119906 during steady state 119909 with

output 119910 the discrete empirical cross-gramian W119883relating

the states 119909ℎ119894119895 of input 119906ℎ119894119895(119905) = 119888

ℎ119878

119894119890

119895120575(119905) + 119906 to output 119910119896119897119887

of 1199091198961198971198870

= 119889

119896119879

119897119891

119887+ 119909 is given by

W119883=

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

Δ119905

119888

ℎ119889

119896

119879

119897

T

sum

119905=0

Ψ

ℎ119894119895119896119897

119905119879

119879

119897

Ψ

ℎ119894119895119896119897

119886119887119905= 119891

119879

119886119879

119879

119897Δ119909

ℎ119894119895

119905119890

119879

119895119878

119879

119894Δ119910

119896119897119887

119905isin R

Δ119909

ℎ119894119895

119905= (119909

ℎ119894119895

119905minus 119909)

Δ119910

119896119897119887

119905= (119910

119896119897119887

119905minus 119910)

(14)

Computational complexity depends largely on the number ofscales and rotations of perturbations as well as the order ofintegration used to generate the snapshots 119909ℎ119894119895 and 119910119896119897119887

The empirical cross-gramian enables state reduction forsquare nonlinear control systems without an additional bal-ancing procedure using direct truncation where the approx-imately balancing projection 119880 is computed by an SVD andtruncated to 119880

1 analogous to (7)

(119905) = 119880

119879

1119891 (119880

1119909 (119905) 119906 (119905))

119910 (119905) = 119892 (119880

1119909 (119905) 119906 (119905))

(15)

Like POD to quantify how close the subspace obtainedby reduction is approximating the state space a measure oftotal preserved energy [11 40] can also be employed here

119864 =

sum

119896

119894=1120590

119894

sum

119899

119894=1120590

119894

(16)

for 119896 retained states of an 119899-dimensional model with 119899 minus 119896

truncated states related to the 119899 minus 119896 lowest singular values ofthe empirical cross-gramian 120590

119894(119882

119883)


4 Combined State and Parameter Reduction

Two methods for combined reduction allowing simultane-ous reduction of state and parameter spaces are proposedan observability-based and a controllability-based ansatz Forparametrized general control systems with parameters 120579 isin

R119901 (119905) = 119891 (119909 (119905) 119906 (119905) 120579)

119910 (119905) = 119892 (119909 (119905) 119906 (119905) 120579)

(17)

in [16] the identifiability gramian was introduced (in [41]a similar concept is presented) which extends the conceptof observability from states to parameters Augmenting thestates of a given system by its parameters 120579 as constantcomponents the parameters are treated like states

(119905) = (

(119905)

120579 (119905)

) = (

119891 (119909 (119905) 119906 (119905) 120579)

0

)

119910 (119905) = 119892 (119909 (119905) 119906 (119905) 120579)

0= (

119909

0

120579

)

(18)

The initial states 0are also augmented by the given param-

eter value 120579 yielding (119905) isin R119899+119901 The identifiability ofthe parameters is obtained through the observability of theseparameter-states by the augmented observability gramian

119882

119874= (

119882

119874119882

119872

119882

119879

119872119882

119875

) isin R(119899+119901)times(119899+119901)

(19)

with the state observability gramian 119882

119874isin R119899times119899 the

parameter observability gramian119882119875isin R119901times119901 and a mixture

matrix 119882119872

isin R119899times119901 From the observability gramian of thisaugmented system the identifiability gramian119882

119868isin R119901times119901 can

be extracted via the Schur complement

119882

119868= 119882

119875minus119882

119879

119872119882

119874

minus1

119882

119872 (20)

For an approximation of the identifiability gramian 119882

119868 the

parameter observability gramian 119882

119875asymp 119882

119868itself is often

sufficient The identifiability is then given as the observ-ability of the parameters through the singular values of119882

119868or approximately by 119882

119875 respectively Instead of using

the identifiability information for parameter identificationas in [16 42] a projection to the dominant parametersubspace is computed from119882

119868 Similar to the cross-gramian

approach a singular value decomposition of the approximateidentifiability gramian yields the reduced parameters 120579

119882

119868

SVD= ΠΔΛ = (

Π

1

Π

2

)(

Δ

10

0 Δ

2

) (Λ

1Λ

2)

997904rArr

120579 = Π

119879

1120579 Π

1

120579 asymp 120579

(21)

The partitioning depends on the singular values in Δ A trun-cation of the projection Π results in the reduced parameters

120579 and the associated parameter reduced order model

(119905) = 119891 (119909 (119905) 119906 (119905) Π

1

120579)

119910 (119905) = 119892 (119909 (119905) 119906 (119905) Π

1

120579)

(22)

Next a combined reduction of state and parameterspace is introduced With the identifiability gramian-basedparameter reduction first the parameter space of the systemis reduced The observability of the states is encoded in theaugmented observability gramian too it can be extracted asthe upper-left 119899times119899matrix from

119882

119874 Then after computation

of a controllability gramian 119882

119862 the state space is reduced

by balanced truncation (balanced truncation provides two-sided truncated projection matrices 119880

1and 119881

1 see [43])

This results in an observability-based combined state andparameter reduction

(119905) = 119881

1119891 (119880

1119909 (119905) 119906 (119905) Π

1

120579)

119910 (119905) = 119892 (119880

1119909 (119905) 119906 (119905) Π

1

120579)

(23)

Similarly a controllability-based combined reduction canbe achieved by a parameter reduction using the sensitivitygramian from [44] for additive partitionable systems

119891 (119909 119906 120579) = 119891 (119909 119906) +

119901

sum

119896=1

119891 (119909 120579

119896)

997904rArr 119882

119862= 119882

1198620+

119901

sum

119896=1

119882

119862119896

119882

119878= (

tr (1198821198621) 0

d0 tr (119882

119862119901)

) isin R119901times119901

(24)

which is based on [17 45] and treats the parameters asadditional inputs By the sensitivity gramian controllabilityinformation on the parameters is provided which also allowsa parameter reduction again by a singular value decompo-sition of 119882

119878 Since an approximate controllability gramian

119882

119862is also computed in process (24) after computation of

an observability gramian119882119874 the parameter reduced system

is reduced in states by balanced truncation This resultsin a controllability-based combined state and parameterreductionThe controllability-based combined reduced ordermodel has the same form as the observability-based reducedmodel (23)

5 Joint Gramian and Combined Reduction

In addition to controllability- and observability-based com-bined reduction a cross-gramian-based combined reductionis proposed next which for symmetric control systems isenabled by the empirical cross-gramian for MIMO systemsfrom Section 3 Aggregating the computation of controlla-bility and observability not only for states like the cross-gramian but also for identifiability of parameters leads to areduction and identification method requiring a new singlegramian Here the same augmented system (18) is used Thesystems symmetry is not affected by the augmentation withthe constant (parameter-) states since in terms of a linearsystem the system components 119860 119861 119862 are expanded withzeros This leads to the following new gramian matrix which


utilizes the cross-gramian and thus unifies controllability andobservability of states and parameters

Definition 5 (joint gramian) The joint gramian 119882

119869is the

cross-gramian of a square augmented system see (18)

The joint gramian also has a 2 times 2 block structure

119882

119869= (

119882

119883119882

119872

119882

119898119882

119875

) isin R(119899+119901)times(119899+119901)

(25)

with the state cross-gramian 119882

119883isin R119899times119899 the parameter

cross-gramian 119882119875isin R119901times119901 and the mixture matrices 119882

119872isin

R119899times119901

119882

119898isin R119901times119899 The parameter-states are uncontrollable

yielding 119882

119875= 0 and 119882

119898= 0 since no inputs affect the

augmented states Thus a Schur complement of the jointgramian to extract the parameter associated lower right blockmatrix will always be zero

119882 119868= 119882

119875minus119882

119898119882

minus1

119883119882

119872= 0 minus 0 119882

minus1

119883119882

119872= 0 (26)

Yet the identifiability information on the parameters isencoded in the nonzero mixture matrix 119882

119872 By taking the

symmetric part of the joint gramian 119882

119869= 05 (119882

119869+ 119882

119879

119869)

one obtains the cross-identifiability gramian119882 119868

119882 119868= 0 minus

1

4

119882

119879

119872119882

minus1

119883119882

119872 (27)

Taking the inverse of the symmetric part of the cross-gramian is too costly in a large-scale setting But the Schurcomplement can be approximated by using

119863 = diag (119882119883)

119882

minus1

119883asymp 119908

minus1

119883= 119863

minus1

minus 119863

minus1

(119882

119883minus 119863)119863

minus1

(28)

as a coarse approximation (this approximation of the inverseis of complexity 1198992) to the inverse from [46] Thus a moreefficient cross-identifiability gramian is given by

119882 119868= minus

1

4

119882

119879

119872119908

minus1

119883119882

119872 (29)

A reduced set of parameters

120579 is again computed bya truncated projection obtained from the singular valuedecomposition of the cross-identifiability gramian

119882 119868asymp 119882

119875

SVD= ΠΔΛ = (

Π

1

Π

2

)(

Δ

10

0 Δ

2

) (Λ

1Λ

2)

997904rArr

120579 = Π

119879

1120579 Π

1

120579 asymp 120579

(30)

After a parameter reduction

(119905) = 119891 (119909 (119905) 119906 (119905) Π

1

120579)

119910 (119905) = 119892 (119909 (119905) 119906 (119905) Π

1

120579)

(31)

the states can be reducedwith a state reduction by direct trun-cation employing the usual cross-gramian119882

119883 a byproduct

of the joint gramian 119882119869 which is the upper-left 119899 times 119899 block

matrix of119882119869

(119905) = 119881

1119891 (119880

1119909 (119905) 119906 (119905) Π

1

120579)

119910 (119905) = 119892 (119880

1119909 (119905) 119906 (119905) Π

1

120579)

(32)

For this combined reduction of states and parametersno further gramians have to be computed and no balancingtransformation is required

6 Implementation and Numerical Results

For an efficient implementation the structure of the gramiancomputation is exploited First the empirical gramians allowextensive parallelization Each combination of directiontransformation and scale can be processed separately yield-ing a sub-gramian Second the assembly of each sub-gramiancan be comprehensively vectorized since it consists of vectoradditions inner products and outer products In the specialcase of the empirical cross-gramian and thus the empiricaljoint gramian which is an encapsulation of the empiricalcross-gramian organizing the observability snapshots into athird-order-tensor and exploiting generalized transpositionsresult in a very efficient gramian assembly (see emgrm) Thefinal resulting gramian is the normalized accumulation overall sub-gramians For further details about the implementa-tion see [44]

All gramians for the numerical results are computedby the empirical gramian framework introduced in [44]The empirical gramian framework emgr [47] can be foundat httpgramiande and is compatible with Octave [48]and Matlab [49] The source code used for the followingexperiments is provided as supplementary material

The error measure employed in the following experi-ments is the relative 1198712-error for a vector valued time series[50] and is defined as follows

Definition 6 The relative 1198712-error for two vector valued timeseries 119910 and 119910 is given by

120598 =

1003817

1003817

1003817

1003817

119910 minus 119910

1003817

1003817

1003817

10038172

1003817

1003817

1003817

1003817

119910

1003817

1003817

1003817

10038172

(33)

with the 1198712-norm of a (discrete) time series

1003817

1003817

1003817

1003817

119910

1003817

1003817

1003817

10038172=

radic

sum

119905

1003817

1003817

1003817

1003817

119910(119905)

1003817

1003817

1003817

1003817

2

2 (34)

Numerical results for three models are presented nextFirst state reduction is applied to a nonlinear benchmarkproblem to validate the applicability of the empirical cross-gramian Then a linear and nonlinear parametrized controlsystem is considered for the state parameter and combinedreduction

61 Nonlinear Benchmark Introduced in [51] this nonlinearbenchmark (this benchmark is also listed in the MORwiki


[52]) has been used in [37 40 53] as a test problem forthe assessment of the empirical controllability gramian andempirical observability gramian in a balanced truncationmodel order reduction settingThis benchmark systemmod-els a circuit consisting of capacitors and nonlinear resistors(a resistor parallel-connected to a diode) Its mathematicalmodel is given by the following nonlinear SISO controlsystem

(119905) =

(

(

(

(

minus119892(119909

1(119905)) minus 119892 (119909

1(119905) minus 119909

2(119905))

119892 (119909

1(119905) minus 119909

2(119905)) minus 119892 (119909

2(119905) minus 119909

3(119905))

119892 (119909

119896minus1(119905) minus 119909

119896(119905)) minus 119892 (119909

119896(119905) minus 119909

119909+1(119905))

119892 (119909

119873minus1(119905) minus 119909

119873(119905))

)

)

)

)

+

(

(

(

(

119906(119905)

0

0

0

)

)

)

)

119910 (119905) = 119909

1(119905)

(35)

with the nonlinear function 119892 R rarr R

119892 (119909) = exp (119909) + 119909 minus 1 (36)

In this setting with dim(119909) = 100 a zero initial state 1199090=

0 and a decaying exponential input 119906(119905) = 119890

minus119905 are employedFigure 1 shows the relative 119871

2-error in the reduced ordermodels outputs reduced by balanced truncation of empiricalcontrollability and observability gramian and direct trunca-tion of the empirical cross-gramian

After a steep initial drop in the error the relative outputerror remains near the machine precision Both methodsbalanced truncation and direct truncation perform verysimilarly and require less than 10 of the full order modelrsquosstate dimension to reach the error plateau The empiricalcross-gramian-based direct truncation exhibits a slightlylower output error

62 State Reduction Next a parametrized linear controlsystem and a parametrized nonlinear control system arereduced in states and parameters and combined in states andparameters The parametrized linear control system model[17] is given by

= 119860119909 + 119861119906 + 120579

119910 = 119862119909

(37)

with an additional parametrized source term 120579 isin R119873The system is ensured to be asymptotically stable thus

Re(1205821119899

(119860)) lt 0 which is a central requirement forall empirical gramians to be computable Furthermore

100

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

10minus16

100

101

Balanced truncationDirect truncation

Figure 1 Relative 1198712 output error in the reduced order nonlinearbenchmark after the state reduction using balanced truncation anddirect truncation of the empirical cross-gramian

the system is chosen to be state space symmetric 119860 = 119860

119879

and 119862 = 119861

119879 A state space symmetric system providesthat all system gramians the controllability gramian theobservability gramian and the cross-gramian are equal [54]This allows the verification of the empirical cross-gramian

The parametrized nonlinear control system model isgiven by

= 119860 tanh(14

119909) + 119861119906 + 120579

119910 = 119862119909

(38)

with a hyperbolic tangent nonlinearity using the same systemmatrices 119860 119861 119862 as in (37) This model is related to thehyperbolic network model from [55]

The linear and nonlinear model are subject to zero initialstates 119909

0= 0 and impulse input 119906(119905) = 120575(119905) For the following

experiments a state space dimension 119899 = dim(119909(119905)) = 256thus 119901 = dim(120579) = 256 is assumed as well as an input andoutput dimension of 119898 = dim(119906(119905)) = 119900 = dim(119910(119905)) =

16 The parameters are drawn from a uniform distribution120579

119894= 119880(0 01) the system matrix 119860 is generated as a sparse

uniformly random matrix with ensured stability and theinput matrix 119861 is a dense uniformly random matrix yieldingthe output matrix 119862 = 119861

119879The linear model in (37) and the nonlinear model in

(38) are first reduced in states using the following methodsbalanced truncation utilizing the empirical controllabilitygramian and empirical observability gramian direct trunca-tion of the empirical cross-gramian presented in Section 3Furthermore a method closely related to balanced POD [56


50 100 150 200 250

Balanced PODBalanced truncationDirect truncation

100

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

(a)

50 100 150 200 250


100

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

(b)

Figure 2 Relative 1198712 output error in the reduced order linear and nonlinear models after the state reduction using balanced POD balancedtruncation and direct truncation of the empirical cross-gramian

57] is tested too For the linear model an approximate cross-gramian 119882

119884is computed using the approach from [22 58]

which obtains the cross-gramian by computing the control-lability gramian of the system augmented with its adjointsystem (this approach is also implemented in the empiricalgramian framework as empirical linear cross gramian) Forthe nonlinear model an approximate cross-gramian is com-puted following [56] by the product119882

119884= 119882

119862119882

119874 In Figure 2

the relative 1198712-error in the outputs 119910 is plotted for reducedorders dim(119909) = 1 119899 minus 1

The state reduced models in the linear setting generatedby balanced POD balanced truncation and direct trunca-tion are of similar quality yet balanced POD and directtruncation perform slightly better than balanced truncationIn the nonlinear setting balanced truncation outperformsbalanced POD and direct truncation for which the outputerror flattens above a reduced order of about 40 of theoriginal model order

The better performance of balanced truncation for thenonlinear model is due to use of two-sided projections asopposed to the one-sided projections used for balanced PODand direct truncation here

63 Parameter Reduction The linear model in (37) and thenonlinear model in (38) are reduced in parameters usingthe empirical sensitivity gramian the empirical identifiabilitygramian and empirical cross identifiability gramian from theempirical joint gramian Figure 3 depicts the relative 1198712-errorin the outputs 119910 for the reduced orders dim(120579) = 1 119899 minus 1

For the linear and the nonlinear model thecontrollability-based sensitivity gramian performs worstwith the slowest decline in output error The observability-based identifiability gramian and the cross-gramian-basedcross-identifiability gramian show a sharper descent of theoutput error of similar quality in the linear and nonlinearsetting yet the observability-based parameter reductionexhibits a steeper drop of the output error for reduced ordersup to 40 of the original parameter space

Since the sensitivity gramian is a diagonal matrix theassociated projections reorder the parameters and thusexclude all effects of the truncated parameters while theidentifiability and cross-identifiability gramian use linearcombinations of parameters to be truncated

64 Combined Reduction The linear model in (37) and thenonlinearmodel in (38) are next reduced in states and param-eters employing the methods presented in Section 4 andSection 5 Controllability-based combined reduction uses theempirical sensitivity gramian for parameter reduction andbalanced truncation for the state reduction Observability-based combined reduction uses the empirical identifiabil-ity gramian for parameter reduction and also balancedtruncation for the state reduction The cross-gramian-basedcombined reduction utilizes the empirical joint gramianthe cross-identifiability gramian is used for the parameterreduction and the direct truncation of the cross-gramian isused for the state reduction In Figures 4 and 5 the relative 1198712-error in the outputs 119910 is plotted for reduced orders dim(119909) =1 119899 minus 1 and dim(120579) = 1 119899 minus 1


50 100 150 200 250

Sensitivity gramianIdentifiability gramianCross-identifiability gramian

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

(a)

50 100 150 200 250


100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

(b)

Figure 3 Relative 1198712 output error in the reduced order linear and nonlinear models after the parameter reduction using the sensitivitygramian identifiability gramian cross-identifiability gramian

128

128

256 256

States

Param

eters

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

WS + WO

(a)

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

128

128

256 256

States

Param

eters

WC +WI

(b)

WJ

128

128

256 256

States

Param

eters

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

(c)Figure 4 Relative 1198712 output error in the reduced order linear model after the combined reduction using controllability-based observability-based and cross-gramian-based combined reduction

For a better comparison a cross-section of the surfacesin Figures 4 and 5 along the diagonals are plotted in Figure 6showing the reduced order modelrsquos output error for the samereduced order in states and parameters dim(119909) = dim(120579)

For all reduced order models obtained by combinedstate and parameter reduction the parameter reduction errordominates the output error As for the parameter reduction

the controllability-based combined reduction by sensitivitygramian and balanced truncation performs worst with a slowdescent in the output error In the linear setting the cross-gramian-based joint gramian performs significantly betterthan the combination of identifiability gramian and balancedtruncation In the nonlinear setting the reduced ordermodelsof these methods exhibit similar behavior which is due to


128

128

256 256

States

Param

eters

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

WS + WO

(a)

128

128

256 256

States

Param

eters

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

WC + WI

(b)

128

128

256 256

States

Param

eters

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

WJ

(c)

Figure 5 Relative 119871

2 output error in the reduced order nonlinear model after the combined reduction using controllability-basedobservability-based and cross-gramian-based combined reduction

50 100 150 200 250

Controllability-basedObservability-basedCross-gramian-based

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

(a)

50 100 150 200 250


100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

(b)

Figure 6 Relative 1198712 output error in the reduced order linear and nonlinearmodels after the combined reduction using controllability-basedobservability-based and cross-gramian-based combined reduction for the same reduced order of states and parameters

the higher state reduction error (Figure 2) in the cross-gramian-based approach

To assess the efficiency of the presented methods theoffline times (the time required to assemble the necessaryempirical gramian matrices) are compared in Figure 7

The state reduction of the linear model is accomplishedfastest by the balanced-POD-related method since the com-putational effort required for computation is equivalent toa single empirical controllability gramian For the nonlinearmodel this advantage does not exist since no adjoint system


Linear

state

redu

ction

WYWC +WO

WX

Non

linear

Non

linear

Non

linear

state

redu

ction

WYWC +WO

WX

parameter

Linear

parameter

redu

ction

WSWIWJ

WSWIWJre

duction

Linear

combined

redu

ction

combined

redu

ction

WS + WO

WJ

WC + WI

WS + WO

WJ

WC + WI

0 10 20 30 40 50 60

t (s)

Figure 7 Offline time for the empirical gramians and the associated decompositions for the state parameter and combined reduction in thelinear and nonlinear setting

for the nonlinear model is provided Notably the cross-gramian for the direct truncation is computed slightly fasterthan controllability observability gramian and balancingtransformation for the balanced truncation

Among the empirical gramians for parameter reductionthe sensitivity gramian is computed fastest yet due to the higherror this is the least applicable Between the identifiabilityand cross-identifiability gramian which exhibit a comparableoutput error the cross-gramian-based approach is consider-ably faster

For the combined reduction the offline times are similarto the offline times of the parameter reduction with theexception of the controllability-based combined reductionwhich now takes longer than the cross-gramian-based jointgramian because of the additional observability gramianrequired for the balanced truncation state reductionThus theempirical joint gramian is the fastest of the testedmethods forcombined reduction and provides a competitive output error

7 Conclusion

In this paper the empirical cross-gramian for MIMO systemsand the empirical joint gramian for parameter and combinedstate and parameter reduction have been introduced andbenchmarked The empirical cross-gramian allows a statereduction of linear and nonlinear systems (for a transfer ofthe symmetry classification to nonlinear systems see [10])The empirical joint gramian enables not only cross-gramian-based parameter reduction but also an efficient combinedstate and parameter reduction Both the empirical cross-gramian and the empirical joint gramian have been shown tobe a viable alternative to balanced truncation and balancing-based combined reduction approaches

Further research has to be conducted on two-sidedprojections and error bounds for the reduced order modelsand on applying the empirical cross-gramian to nonlinearor nonsymmetric systems Existing extensions for nonsym-metric systems are generalizing the symmetry constraint toorthogonal symmetry [59] or embedding into a symmetricsystem [24]

The empirical cross-gramian for MIMO systems com-pletes the set of empirical gramians for state reductionwhile the joint gramian completes the body of parameteridentification gramians and enables combined reductionwithout balancing

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the Deutsche Forschungsge-meinschaft the Open Access Publication Fund of the Uni-versity of Munster DFG EXC 1003 Cells in Motion Clusterof Excellence Munster Germany as well as the Center forDeveloping Mathematics in Interaction DEMAIN MunsterGermany

References

[1] R J Moran S J Kiebel K E Stephan R B Reilly J Daunizeauand K J Friston ldquoA neural mass model of spectral responses inelectrophysiologyrdquoNeuroImage vol 37 no 3 pp 706ndash720 2007


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

[3] K Kunisch and S Volkwein ldquoControl of the Burgers equationby a reduced-order approach using proper orthogonal decom-positionrdquo Journal of Optimization Theory and Applications vol102 no 2 pp 345ndash371 1999

[4] P Benner ldquoSolving large-scale control problemsrdquo IEEE ControlSystems vol 24 no 1 pp 44ndash59 2004

[5] B Haasdonk and M Ohlberger ldquoReduced basis method forfinite volume approximations of parametrized linear evolutionequationsrdquo Mathematical Modelling and Numerical Analysisvol 42 no 2 pp 277ndash302 2008

[6] P M van Dooren ldquoGramian based model reduction of large-scale dynamical systemsrdquo in Numerical Analysis 1999 vol 420of Research Notes in Mathematics pp 231ndash247 2000

[7] A Laub M Heath C Paige and R Ward ldquoComputation ofsystem balancing transformations and other applications ofsimultaneous diagonalization algorithmsrdquo IEEE Transactionson Automatic Control vol 32 no 2 pp 115ndash122 1987

[8] A C Antoulas Approximation of Large-Scale Dynamical Sys-tems vol 6 of Advances in Design and Control Society forIndustrial and Applied Mathematics 2005

[9] K V Fernando andHNicholson ldquoOn the structure of balancedand other principal representations of SISO systemsrdquo IEEETransactions on Automatic Control vol 28 no 2 pp 228ndash2311983

[10] T C Ionescu K Fujimoto and J M A Scherpen ldquoSingularvalue analysis of nonlinear symmetric systemsrdquo IEEE Transac-tions on Automatic Control vol 56 no 9 pp 2073ndash2086 2011

[11] S Lall J E Marsden and S Glavaski ldquoEmpirical modelreduction of controlled nonlinear systemsrdquo in Proceedings of theIFACWorld Congress pp 473ndash478 1999

[12] S Lall J E Marsden and S Glavaski ldquoA subspace approach tobalanced truncation for model reduction of nonlinear controlsystemsrdquo International Journal of Robust and Nonlinear Controlvol 12 no 6 pp 519ndash535 2002

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

[14] S Streif R Findeisen andE Bullinger ldquoRelating cross gramiansand sensitivity analysis in systems biologyrdquoTheory of Networksand Systems vol 10 no 4 pp 437ndash442 2006

[15] S Streif S Waldherr F Allgower and R Findeisen ldquoSteadystate sensitivity analysis of biochemical reaction networks abrief review and new methodsrdquo in Methods in BioengineeringA Jayaraman and J Hahn Eds Systems Analysis of BiologicalNetworks pp 129ndash148 Artech House MIT Press 2009

[16] D Geffen R Findeisen M Schliemann F Allgower andM Guay ldquoObservability based parameter identifiability forbiochemical reaction networksrdquo in Proceedings of the AmericanControl Conference (ACC 08) pp 2130ndash2135 Seattle WashUSA June 2008

[17] 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

[18] K V Fernando and H Nicholson ldquoMinimality of SISO linearsystemsrdquo Proceedings of the IEEE vol 70 no 10 pp 1241ndash12421982

[19] K V Fernando and H Nicholson ldquoOn the Cauchy index oflinear systemsrdquo IEEE Transactions on Automatic Control vol28 no 2 pp 222ndash224 1983

[20] K V Fernando and H Nicholson ldquoOn a fundamental propertyof the cross-Gramianmatrixrdquo IEEETransactions onCircuits andSystems vol 31 no 5 pp 504ndash505 1984

[21] K V Fernando and H Nicholson ldquoReachability observabilitandminimality ofMIMO systemsrdquo Proceedings of the IEEE vol72 no 12 pp 1820ndash1821 1984

[22] K V Fernando and H Nicholson ldquoOn the cross-Gramian forsymmetric MIMO systemsrdquo IEEE Transactions on Circuits andSystems vol 32 no 5 pp 487ndash489 1985

[23] A J Laub L M Silverman and M Verma ldquoA note on cross-Grammians for symmetric realizationsrdquo Proceedings of theIEEE vol 71 no 7 pp 904ndash905 1983

[24] D C Sorensen and A C Antoulas ldquoProjection methods forbalanced model reductionrdquo Technical Report 2001

[25] D C Sorensen and A C Antoulas ldquoThe Sylvester equationand approximate balanced reductionrdquo Linear Algebra and itsApplications vol 351-352 pp 671ndash700 2002

[26] U Baur and P Benner ldquoCross-gramian based model reductionfor data-sparse systemsrdquo Electronic Transactions on NumericalAnalysis vol 31 pp 256ndash270 2008

[27] A Gheondea and R J Ober ldquoA trace formula for Hankeloperatorsrdquo Proceedings of the American Mathematical Societyvol 127 no 7 pp 2007ndash2012 1999

[28] R W Aldhaheri ldquoModel order reduction via real Schur-formdecompositionrdquo International Journal of Control vol 53 no 3pp 709ndash716 1991

[29] B Moaveni and A Khaki-Sedigh ldquoInput-output pairing basedon cross-gramian matrixrdquo in Proceedings of the SICE-ICASEInternational Joint Conference pp 2378ndash2380 Busan Republicof Korea October 2006

[30] B Moaveni and A Khaki-Sedigh ldquoA new approach to computethe cross-Gramian matrix and its application in input-outputpairing of linear multivariable plantsrdquo Journal of Applied Sci-ences vol 8 no 4 pp 608ndash614 2008

[31] B Moaveni and A K Sedigh ldquoInput-output pairing analysis foruncertain multivariable processesrdquo Journal of Process Controlvol 18 no 6 pp 527ndash532 2008

[32] H R Shaker andM Komareji ldquoControl configuration selectionformultivariable nonlinear systemsrdquo Industrial and EngineeringChemistry Research vol 51 no 25 pp 8583ndash8587 2012

[33] J Hahn and T F Edgar ldquoReduction of nonlinear models usingbalancing of empirical gramians and Galerkin projectionsrdquo inProceedings of the American Control Conference vol 4 pp2864ndash2868 June 2000

[34] A C Antoulas and D C Sorensen ldquoApproximation of large-scale dynamical systems an overviewrdquo International Journal ofApplied Mathematics and Computer Science vol 11 no 5 pp1093ndash1121 2001

[35] J Hahn and T F Edgar ldquoAn improved method for nonlinearmodel reduction using balancing of empirical gramiansrdquo Com-puters and Chemical Engineering vol 26 no 10 pp 1379ndash13972002

[36] J Hahn T F Edgar and W Marquardt ldquoControllability andobservability covariance matrices for the analysis and orderreduction of stable nonlinear systemsrdquo Journal of Process Con-trol vol 13 no 2 pp 115ndash127 2003

[37] M Condon and R Ivanov ldquoEmpirical balanced truncation ofnonlinear systemsrdquo Journal of Nonlinear Science vol 14 no 5pp 405ndash414 (2005) 2004


[38] A K Singh and J Hahn ldquoOn the use of empirical gramians forcontrollability and observability analysisrdquo in Proceedings of theAmerican Control Conference (ACC 05) pp 140ndash146 June 2005

[39] J Hahn and T F Edgar ldquoA gramian based approach tononlinearity quantification and model classificationrdquo Industrialand Engineering Chemistry Research vol 40 no 24 pp 5724ndash5731 2001

[40] M Condon and R Ivanov ldquoModel reduction of nonlinearsystemsrdquo COMPEL The International Journal for Computationand Mathematics in Electrical and Electronic Engineering vol23 no 2 pp 547ndash557 2004

[41] A K Singh and J Hahn ldquoDetermining optimal sensor locationsfor state and parameter estimation for stable nonlinear systemsrdquoIndustrial amp Engineering Chemistry Research vol 44 no 15 pp5645ndash5659 2005

[42] C Eberle and C Ament ldquoIdentifiability and online estimationof diagnostic parameters with in the glucose insulin homeosta-sisrdquo BioSystems vol 107 no 3 pp 135ndash141 2012

[43] A Keil and J L Gouze ldquoModel reduction of modular systemsusing balancing methodsrdquo Tech Rep Munich University ofTechnology 2003

[44] C Himpe andM Ohlberger ldquoA unified software framework forempirical Gramiansrdquo Journal of Mathematics vol 2013 ArticleID 365909 6 pages 2013

[45] C Sun and J Hahn ldquoModel reduction in the presence ofuncertainty in model parametersrdquo Journal of Process Controlvol 16 no 6 pp 645ndash649 2006

[46] M Wu B Yin A Vosoughi C Studer J R Cavallaro and CDick ldquoApproximate matrix inversion for high-throughput datadetection in the large-scaleMIMO uplinkrdquo in Proceedings of theIEEE International Symposium on Circuits and Systems (ISCASrsquo13) pp 2155ndash2158 Beijing China May 2013

[47] C Himpe ldquoemgrmdashEmpirical Gramian Frameworkrdquo httpgramiande

[48] Octave Community GNU Octave 38 2014 httpgnuorgsoftwareoctave

[49] MATLAB Version 83 (R2014a) 2014[50] L Fortuna and M Fransca Optimal and Robust Control

Advanced Topics with MATLAB Taylor amp Francis 2012[51] Y Chen Model order reduction for nonlinear systems [MS

thesis] Massachusetts Institute of Technology 1999[52] MORwiki Community MORwikimdashModel Order Reduction

Wiki 2014 httpmodelreductionorg[53] M Condon ldquoModel reduction of nonlinear systemsrdquo Proceed-

ings in Applied Mathematics and Mechanics vol 7 no 1 pp2130011ndash2130012

[54] WQ Liu V Sreeram and K L Teo ldquoModel reduction for state-space symmetric systemsrdquo SystemsampControl Letters vol 34 no4 pp 209ndash215 1998

[55] Y Quan H Zhang and L Cai ldquoModeling and control based ona new neural network modelrdquo in Proceedings of the AmericanControl Conference vol 3 pp 1928ndash1929 Arlington Va USAJune 2001

[56] K Willcox and J Peraire ldquoBalanced model reduction via theproper orthogonal decompositionrdquo AIAA Journal vol 40 no11 pp 2323ndash2330 2002

[57] A Barbagallo V F de Felice and K K Nagarajan ldquoReducedorder modeling of a couette flow using balanced proper orthog-onal decompositionrdquo ERCOFTAC 2008

[58] H R Shaker ldquoGeneralized cross-gramian for linear systemsrdquo inProceedings of the 7th IEEE Conference on Industrial Electronicsand Applications (ICIEA rsquo12) pp 749ndash751 July 2012

[59] J A De Abreu-Garcia and F W Fairman ldquoA note on crossGrammians for orthogonally symmetric realizationsrdquo IEEETransactions on Automatic Control vol 31 no 9 pp 866ndash8681986

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Since the number of a systems inputs and outputs usuallyremains fixed the maps to and from the intermediary statescharacterize the reducibility of a system [6] The balancedtruncation approach introduced in [2] balances a systemin terms of controllability and observability where control-lability quantifies how well a state is driven by the inputand observability quantifies how well changes in a stateare reflected in the output Excluding the least controllableand observable states by truncating the balanced system areduced order mapping from inputs to outputs is approxi-mated

A linear time-invariant control system is composed of alinear dynamic system and a linear output transformation

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

119910 (119905) = 119862119909 (119905)

(1)

with states 119909(119905) isin R119899 input or control 119906(119905) isin R119898 and outputs119910(119905) isin R119900 The system matrix 119860 isin R119899times119899 transforms thestates the input matrix 119861 isin R119899times119898 introduces external inputor control and the output matrix 119862 isin R119900times119899 transforms thestates to the outputs

Controllability and observability can be assessed throughthe associated controllability gramian119882

119862= int

infin

0

119890

119860119905

119861119861

119879

119890

119860119879119905

119889119905 and observability gramian 119882

119874= int

infin

0

119890

119860119879119905

119862

119879

119862119890

119860119905

119889119905Classically119882

119862and119882

119874are computed as the smallest positive

semi-definite solutions of the Lyapunov equations 119860119882119862+

119882

119862119860

119879

= minus119861119861

119879 and 119860

119879

119882

119874+ 119882

119874119860 = minus119862

119879

119862 respectivelyTo make a compound statement about controllability andobservability 119882

119862and 119882

119874have to be balanced [7] The

singular values of the resulting balanced gramian correspondto the Hankel singular values of the system with theirmagnitude describing how controllable and observable theassociated state is

This work focuses on cross-gramian-based methods formodel reduction which combine controllability and observ-ability information into one gramian and is elaboratelydescribed in [8] The cross-gramian 119882

119883= int

infin

0

119890

119860119905

119861119862119890

119860119905

119889119905

was introduced in [9] and corresponds to a solution of theSylvester equation 119860119882

119883+119882

119883119860 = minus119861119862

An alternative to solving the Lyapunov or Sylvestermatrixequations apart from the analytic approaches for examplein [10] is the method of empirical gramians which wasintroduced in the works [11 12] and enables the compu-tation of gramian matrices also for nonlinear systems bymere basic vector and matrix operations This concept wasextended among others in [13] providing more general inputsignals Particularly noted should be [14 15] for developingthe empirical cross-gramian for single-input-single-output(SISO) systems in the context of sensitivity analysis

In this paper the empirical cross-gramian is generalizedto be applicable to multiple-input-multiple-output (MIMO)systems For the gramian-based parameter reduction thegroundwork has been laid by [16] from the observabilityand by [17] from the controllability point of view From thecross-gramian perspective of parameter reduction a newgramian namely the joint gramian is introduced in thiswork Furthermore the concept of gramian-based combined

state and parameter reduction is established Using empiricalgramians it is shown that combined reduction allows efficientmodel order reduction of linear and nonlinear control sys-tems

To begin the cross-gramian and its properties arereviewed in Section 2 Next the empirical cross-gramianfor MIMO systems is developed in Section 3 Section 4introduces combined state and parameter reduction in twovariants first an observability- and second a controllability-based approach the former is enhanced to a cross-gramian-based combined reduction which is presented in Section 5Finally numerical experiments are conducted in Section 6comparing the newly presented methods for a linear andnonlinear neural network as well as a nonlinear benchmarkproblem

2 Review of the Cross-Gramian

A brief review of the cross-gramian along with its applicationto model reduction of linear time-invariant control systemsis given next The cross-gramian 119882

119883(also known by the

symbol119882119862119874

) was introduced in a sequence of works [9 18ndash23] and encodes controllability and observability into a singlegramian matrix defined as the product of controllability andobservability operator it can only be computed for square(a system with the same number of inputs and outputs) andasymptotically stable systems

119882

119883= int

infin

0

119890

119860119905

119861119862119890

119860119905

119889119905 (2)

Equivalently the cross-gramian is given as a solution to theSylvester equation 119860119882

119883+ 119882

119883119860 = minus119861119862 Approximate solu-

tions for the Sylvester equation were discussed in [24ndash26] Ifthe system is also symmetric the following relation betweenthe cross the controllability and observability gramian holds[9]

119882

2

119883= 119882

119862119882

119874997904rArr

1003816

1003816

1003816

1003816

120582 (119882

119883)

1003816

1003816

1003816

1003816

=radic120582 (119882

119862119882

119874)

(3)

While a SISO system is always symmetric [9] a linearMIMOsystem not only requires the same number of inputs andoutputs but also the system gain 119866 = minus119862119860

minus1

119861 has to besymmetric [24] then a symmetric transformation 119869 with119860119869 = 119869119860

119879 and 119861 = 119862

119879

119869 exist Trivially for 119869 = 1 the systemwould be restricted by 119860 = 119860

119879 and 119861 = 119862

119879 such a system iscalled state space symmetric

As presented in [9] the trace of the cross-gramian equalshalf the gain of a SISO system (119860 119887 119888) where now 119887 isin R119899times1

and 119888 isin R1times119899

tr (119882119883) = minus

1

2

119888119860

minus1

119887 (4)

Because the trace equals the sum of eigenvalues the cross-gramians eigenvalues are associated with the system gain (4)This result was used in [14 15] for parameter identificationpurposes using the system gain as a sensitivity measure Anextension of (4) from [9 Theorem 3] for MIMO systems isdeveloped (see also [27]) next



tr (119882119883) = minus

1

2

tr (119862119860minus1

119861) (5)


tr (119882119883) = tr(int

infin

0

119890

119860119905

119861119862119890

119860119905

119889119905)

= int

infin

0

tr (119890119860119905119861119862119890119860119905) 119889119905

= int

infin

0

tr (119862119890119860119905119890119860119905119861) 119889119905

= tr(intinfin

0

119862119890

2119860119905

119861119889119905)

= tr(119862intinfin

0

119890

2119860119905

119889119905119861)

= tr(119862(minus12

119860

minus1

)119861)

= minus

1

2

tr (119862119860minus1

119861)

(6)


119882

119883

SVD= 119880119863119881 = (119880

1119880

2) (

119863

10

0 119863

2

)(

119881

1

119881

2

) (7)


119899

119896=119903+1119863

2119896119896into 119880

1isin R119899times119903 119880

2isin

R119899times(119899minus119903) and 119881

1isin R119903times119899 119881



119860 = 119880

119879

1119860119880

1

119861 = 119880

119879

1119861

119862 = 119862119880

1 119909

0= 119880

119879

1119909

0

997904rArr

(119905) =

119860119909 (119905) +

119861119906 (119905)

119910 (119905) =

119862119909 (119905)

(8)





(119905) = 119891 (119909 (119905) 119906 (119905))

119910 (119905) = 119892 (119909 (119905) 119906 (119905))

(9)


times R119898


times R119898




119906and 119864





119906and 119876

119909hold


119864

119906= 119890

119894isin R

119895

1003817

1003817

1003817

1003817

119890

119894

1003817

1003817

1003817

1003817

= 1 119890

119894119890

119895 = 119894= 0 119894 = 1 119898

119864

119909= 119891

119894isin R

119899

1003817

1003817

1003817

1003817

119891

119894

1003817

1003817

1003817

1003817

= 1 119891

119894119891

119895 = 119894= 0 119894 = 1 119899

119877

119906= 119878

119894isin R

119895times119895

119878

119879

119894119878

119894= 1 119894 = 1 119904

119877

119909= 119879

119894isin R

119899times119899

119879

119879

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

(10)





119909 119877

119906

119877

119909 119876

119906 and 119876



119882


input119906ℎ119894119895(119905) = 119888

ℎ119878

119894119890

119895120575(119905)+119906 to output119910119896119897119887 of119909119896119897119887

0= 119889

119896119879

119897119891

119887+119909

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

119879

119897int

infin

0

Ψ

ℎ119894119895119896119897

(119905) 119889119905 119879

119879

119897

Ψ

ℎ119894119895119896119897

119886119887(119905) = 119891

119879

119886119879

119879

119897Δ119909

ℎ119894119895

(119905) 119890

119879

119895119878

119879

119894Δ119910

119896119897119887

(119905) isin R

Δ119909

ℎ119894119895

(119905) = (119909

ℎ119894119895

(119905) minus 119909)

Δ119910

119896119897119887

(119905) = (119910

119896119897119887

(119905) minus 119910)

(11)




119906 119877

119909 119876

119906 and 119876


119882

119883of an



Δ119909 (119905) = 119909 (119905) = 119890

119860119905

119861119906 (119905)

Δ119910 (119905) = 119910 (119905) = 119862119890

119860119905

119909

0

(12)

thus

Ψ

ℎ119894119895119896119897

119886119887= 119891

119879

119886119879

119879

119897(119890

119860119905

119861119888

ℎ119878

119894119890

119895) 119890

119879

119895119878

119879

119894(119862119890

119860119905

119889

119896119879

119897119891

119887)

= 119888

ℎ119889

119896119891

119879

119886119879

119879

119897119890

119860119905

119861119862119890

119860119905

119879

119897119891

119887

997904rArr Ψ

ℎ119894119895119896119897

= 119888

ℎ119889

119896119879

119879

119897119890

119860119905

119861119862119890

119860119905

119879

119897

997904rArr

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

int

infin

0

119890

119860119905

119861119862119890

119860119905

119889119905

= int

infin

0

119890

119860119905

119861119862119890

119860119905

119889119905

= 119882

119883

(13)




119864

119909 119877

119906 119877

119909 119876

119906 and 119876




ℎ119878

119894119890

119895120575(119905) + 119906 to output 119910119896119897119887

of 1199091198961198971198870

= 119889

119896119879

119897119891

119887+ 119909 is given by

W119883=

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

Δ119905

119888

ℎ119889

119896

119879

119897

T

sum

119905=0

Ψ

ℎ119894119895119896119897

119905119879

119879

119897

Ψ

ℎ119894119895119896119897

119886119887119905= 119891

119879

119886119879

119879

119897Δ119909

ℎ119894119895

119905119890

119879

119895119878

119879

119894Δ119910

119896119897119887

119905isin R

Δ119909

ℎ119894119895

119905= (119909

ℎ119894119895

119905minus 119909)

Δ119910

119896119897119887

119905= (119910

119896119897119887

119905minus 119910)

(14)



1 analogous to (7)

(119905) = 119880

119879

1119891 (119880

1119909 (119905) 119906 (119905))

119910 (119905) = 119892 (119880

1119909 (119905) 119906 (119905))

(15)


119864 =

sum

119896

119894=1120590

119894

sum

119899

119894=1120590

119894

(16)



119894(119882

119883)




R119901 (119905) = 119891 (119909 (119905) 119906 (119905) 120579)

119910 (119905) = 119892 (119909 (119905) 119906 (119905) 120579)

(17)


(119905) = (

(119905)

120579 (119905)

) = (

119891 (119909 (119905) 119906 (119905) 120579)

0

)

119910 (119905) = 119892 (119909 (119905) 119906 (119905) 120579)

0= (

119909

0

120579

)

(18)



119882

119874= (

119882

119874119882

119872

119882

119879

119872119882

119875

) isin R(119899+119901)times(119899+119901)

(19)


119874isin R119899times119899 the


matrix 119882119872


119868isin R119901times119901 can


119882

119868= 119882

119875minus119882

119879

119872119882

119874

minus1

119882

119872 (20)


119868 the


119875asymp 119882








119882

119868

SVD= ΠΔΛ = (

Π

1

Π

2

)(

Δ

10

0 Δ

2

) (Λ

1Λ

2)

997904rArr

120579 = Π

119879

1120579 Π

1

120579 asymp 120579

(21)



(119905) = 119891 (119909 (119905) 119906 (119905) Π

1

120579)

119910 (119905) = 119892 (119909 (119905) 119906 (119905) Π

1

120579)

(22)


119882





1and 119881

1 see [43])


(119905) = 119881

1119891 (119880

1119909 (119905) 119906 (119905) Π

1

120579)

119910 (119905) = 119892 (119880

1119909 (119905) 119906 (119905) Π

1

120579)

(23)


119891 (119909 119906 120579) = 119891 (119909 119906) +

119901

sum

119896=1

119891 (119909 120579

119896)

997904rArr 119882

119862= 119882

1198620+

119901

sum

119896=1

119882

119862119896

119882

119878= (

tr (1198821198621) 0

d0 tr (119882

119862119901)


(24)



119882









119869is the



119882

119869= (

119882

119883119882

119872

119882

119898119882

119875

) isin R(119899+119901)times(119899+119901)

(25)




119872isin

R119899times119901

119882


yielding 119882

119875= 0 and 119882



119882 119868= 119882

119875minus119882

119898119882

minus1

119883119882

119872= 0 minus 0 119882

minus1

119883119882

119872= 0 (26)




119869= 05 (119882

119869+ 119882

119879

119869)


119882 119868= 0 minus

1

4

119882

119879

119872119882

minus1

119883119882

119872 (27)


119863 = diag (119882119883)

119882

minus1

119883asymp 119908

minus1

119883= 119863

minus1

minus 119863

minus1

(119882

119883minus 119863)119863

minus1

(28)


119882 119868= minus

1

4

119882

119879

119872119908

minus1

119883119882

119872 (29)



119882 119868asymp 119882

119875

SVD= ΠΔΛ = (

Π

1

Π

2

)(

Δ

10

0 Δ

2

) (Λ

1Λ

2)

997904rArr

120579 = Π

119879

1120579 Π

1

120579 asymp 120579

(30)


(119905) = 119891 (119909 (119905) 119906 (119905) Π

1

120579)

119910 (119905) = 119892 (119909 (119905) 119906 (119905) Π

1

120579)

(31)


119883 a byproduct


matrix of119882119869

(119905) = 119881

1119891 (119880

1119909 (119905) 119906 (119905) Π

1

120579)

119910 (119905) = 119892 (119880

1119909 (119905) 119906 (119905) Π

1

120579)

(32)







120598 =

1003817

1003817

1003817

1003817

119910 minus 119910

1003817

1003817

1003817

10038172

1003817

1003817

1003817

1003817

119910

1003817

1003817

1003817

10038172

(33)


1003817

1003817

1003817

1003817

119910

1003817

1003817

1003817

10038172=

radic

sum

119905

1003817

1003817

1003817

1003817

119910(119905)

1003817

1003817

1003817

1003817

2

2 (34)





(119905) =

(

(

(

(

minus119892(119909

1(119905)) minus 119892 (119909

1(119905) minus 119909

2(119905))

119892 (119909

1(119905) minus 119909

2(119905)) minus 119892 (119909

2(119905) minus 119909

3(119905))

119892 (119909

119896minus1(119905) minus 119909

119896(119905)) minus 119892 (119909

119896(119905) minus 119909

119909+1(119905))

119892 (119909

119873minus1(119905) minus 119909

119873(119905))

)

)

)

)

+

(

(

(

(

119906(119905)

0

0

0

)

)

)

)

119910 (119905) = 119909

1(119905)

(35)


119892 (119909) = exp (119909) + 119909 minus 1 (36)







= 119860119909 + 119861119906 + 120579

119910 = 119862119909

(37)


Re(1205821119899


100

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

10minus16

100

101




119879

and 119862 = 119861



= 119860 tanh(14

119909) + 119861119906 + 120579

119910 = 119862119909

(38)











50 100 150 200 250


100

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

(a)

50 100 150 200 250


100

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

(b)





119884= 119882

119862119882

119874 In Figure 2









50 100 150 200 250


100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

(a)

50 100 150 200 250


100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

(b)


128

128

256 256

States

Param

eters

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

WS + WO

(a)

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

128

128

256 256

States

Param

eters

WC +WI

(b)

WJ

128

128

256 256

States

Param

eters

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7






128

128

256 256

States

Param

eters

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

WS + WO

(a)

128

128

256 256

States

Param

eters

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

WC + WI

(b)

128

128

256 256

States

Param

eters

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

WJ

(c)



50 100 150 200 250


100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

(a)

50 100 150 200 250


100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

(b)






Linear

state

redu

ction

WYWC +WO

WX

Non

linear

Non

linear

Non

linear

state

redu

ction

WYWC +WO

WX

parameter

Linear

parameter

redu

ction

WSWIWJ

WSWIWJre

duction

Linear

combined

redu

ction

combined

redu

ction

WS + WO

WJ

WC + WI

WS + WO

WJ

WC + WI

0 10 20 30 40 50 60

t (s)





7 Conclusion






Acknowledgments


References





































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











tr (119882119883) = minus

1

2

tr (119862119860minus1

119861) (5)


tr (119882119883) = tr(int

infin

0

119890

119860119905

119861119862119890

119860119905

119889119905)

= int

infin

0

tr (119890119860119905119861119862119890119860119905) 119889119905

= int

infin

0

tr (119862119890119860119905119890119860119905119861) 119889119905

= tr(intinfin

0

119862119890

2119860119905

119861119889119905)

= tr(119862intinfin

0

119890

2119860119905

119889119905119861)

= tr(119862(minus12

119860

minus1

)119861)

= minus

1

2

tr (119862119860minus1

119861)

(6)


119882

119883

SVD= 119880119863119881 = (119880

1119880

2) (

119863

10

0 119863

2

)(

119881

1

119881

2

) (7)


119899

119896=119903+1119863

2119896119896into 119880

1isin R119899times119903 119880

2isin

R119899times(119899minus119903) and 119881

1isin R119903times119899 119881



119860 = 119880

119879

1119860119880

1

119861 = 119880

119879

1119861

119862 = 119862119880

1 119909

0= 119880

119879

1119909

0

997904rArr

(119905) =

119860119909 (119905) +

119861119906 (119905)

119910 (119905) =

119862119909 (119905)

(8)





(119905) = 119891 (119909 (119905) 119906 (119905))

119910 (119905) = 119892 (119909 (119905) 119906 (119905))

(9)


times R119898


times R119898




119906and 119864





119906and 119876

119909hold


119864

119906= 119890

119894isin R

119895

1003817

1003817

1003817

1003817

119890

119894

1003817

1003817

1003817

1003817

= 1 119890

119894119890

119895 = 119894= 0 119894 = 1 119898

119864

119909= 119891

119894isin R

119899

1003817

1003817

1003817

1003817

119891

119894

1003817

1003817

1003817

1003817

= 1 119891

119894119891

119895 = 119894= 0 119894 = 1 119899

119877

119906= 119878

119894isin R

119895times119895

119878

119879

119894119878

119894= 1 119894 = 1 119904

119877

119909= 119879

119894isin R

119899times119899

119879

119879

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

(10)





119909 119877

119906

119877

119909 119876

119906 and 119876



119882


input119906ℎ119894119895(119905) = 119888

ℎ119878

119894119890

119895120575(119905)+119906 to output119910119896119897119887 of119909119896119897119887

0= 119889

119896119879

119897119891

119887+119909

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

119879

119897int

infin

0

Ψ

ℎ119894119895119896119897

(119905) 119889119905 119879

119879

119897

Ψ

ℎ119894119895119896119897

119886119887(119905) = 119891

119879

119886119879

119879

119897Δ119909

ℎ119894119895

(119905) 119890

119879

119895119878

119879

119894Δ119910

119896119897119887

(119905) isin R

Δ119909

ℎ119894119895

(119905) = (119909

ℎ119894119895

(119905) minus 119909)

Δ119910

119896119897119887

(119905) = (119910

119896119897119887

(119905) minus 119910)

(11)




119906 119877

119909 119876

119906 and 119876


119882

119883of an



Δ119909 (119905) = 119909 (119905) = 119890

119860119905

119861119906 (119905)

Δ119910 (119905) = 119910 (119905) = 119862119890

119860119905

119909

0

(12)

thus

Ψ

ℎ119894119895119896119897

119886119887= 119891

119879

119886119879

119879

119897(119890

119860119905

119861119888

ℎ119878

119894119890

119895) 119890

119879

119895119878

119879

119894(119862119890

119860119905

119889

119896119879

119897119891

119887)

= 119888

ℎ119889

119896119891

119879

119886119879

119879

119897119890

119860119905

119861119862119890

119860119905

119879

119897119891

119887

997904rArr Ψ

ℎ119894119895119896119897

= 119888

ℎ119889

119896119879

119879

119897119890

119860119905

119861119862119890

119860119905

119879

119897

997904rArr

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

int

infin

0

119890

119860119905

119861119862119890

119860119905

119889119905

= int

infin

0

119890

119860119905

119861119862119890

119860119905

119889119905

= 119882

119883

(13)




119864

119909 119877

119906 119877

119909 119876

119906 and 119876




ℎ119878

119894119890

119895120575(119905) + 119906 to output 119910119896119897119887

of 1199091198961198971198870

= 119889

119896119879

119897119891

119887+ 119909 is given by

W119883=

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

Δ119905

119888

ℎ119889

119896

119879

119897

T

sum

119905=0

Ψ

ℎ119894119895119896119897

119905119879

119879

119897

Ψ

ℎ119894119895119896119897

119886119887119905= 119891

119879

119886119879

119879

119897Δ119909

ℎ119894119895

119905119890

119879

119895119878

119879

119894Δ119910

119896119897119887

119905isin R

Δ119909

ℎ119894119895

119905= (119909

ℎ119894119895

119905minus 119909)

Δ119910

119896119897119887

119905= (119910

119896119897119887

119905minus 119910)

(14)



1 analogous to (7)

(119905) = 119880

119879

1119891 (119880

1119909 (119905) 119906 (119905))

119910 (119905) = 119892 (119880

1119909 (119905) 119906 (119905))

(15)


119864 =

sum

119896

119894=1120590

119894

sum

119899

119894=1120590

119894

(16)



119894(119882

119883)




R119901 (119905) = 119891 (119909 (119905) 119906 (119905) 120579)

119910 (119905) = 119892 (119909 (119905) 119906 (119905) 120579)

(17)


(119905) = (

(119905)

120579 (119905)

) = (

119891 (119909 (119905) 119906 (119905) 120579)

0

)

119910 (119905) = 119892 (119909 (119905) 119906 (119905) 120579)

0= (

119909

0

120579

)

(18)



119882

119874= (

119882

119874119882

119872

119882

119879

119872119882

119875

) isin R(119899+119901)times(119899+119901)

(19)


119874isin R119899times119899 the


matrix 119882119872


119868isin R119901times119901 can


119882

119868= 119882

119875minus119882

119879

119872119882

119874

minus1

119882

119872 (20)


119868 the


119875asymp 119882








119882

119868

SVD= ΠΔΛ = (

Π

1

Π

2

)(

Δ

10

0 Δ

2

) (Λ

1Λ

2)

997904rArr

120579 = Π

119879

1120579 Π

1

120579 asymp 120579

(21)



(119905) = 119891 (119909 (119905) 119906 (119905) Π

1

120579)

119910 (119905) = 119892 (119909 (119905) 119906 (119905) Π

1

120579)

(22)


119882





1and 119881

1 see [43])


(119905) = 119881

1119891 (119880

1119909 (119905) 119906 (119905) Π

1

120579)

119910 (119905) = 119892 (119880

1119909 (119905) 119906 (119905) Π

1

120579)

(23)


119891 (119909 119906 120579) = 119891 (119909 119906) +

119901

sum

119896=1

119891 (119909 120579

119896)

997904rArr 119882

119862= 119882

1198620+

119901

sum

119896=1

119882

119862119896

119882

119878= (

tr (1198821198621) 0

d0 tr (119882

119862119901)


(24)



119882









119869is the



119882

119869= (

119882

119883119882

119872

119882

119898119882

119875

) isin R(119899+119901)times(119899+119901)

(25)




119872isin

R119899times119901

119882


yielding 119882

119875= 0 and 119882



119882 119868= 119882

119875minus119882

119898119882

minus1

119883119882

119872= 0 minus 0 119882

minus1

119883119882

119872= 0 (26)




119869= 05 (119882

119869+ 119882

119879

119869)


119882 119868= 0 minus

1

4

119882

119879

119872119882

minus1

119883119882

119872 (27)


119863 = diag (119882119883)

119882

minus1

119883asymp 119908

minus1

119883= 119863

minus1

minus 119863

minus1

(119882

119883minus 119863)119863

minus1

(28)


119882 119868= minus

1

4

119882

119879

119872119908

minus1

119883119882

119872 (29)



119882 119868asymp 119882

119875

SVD= ΠΔΛ = (

Π

1

Π

2

)(

Δ

10

0 Δ

2

) (Λ

1Λ

2)

997904rArr

120579 = Π

119879

1120579 Π

1

120579 asymp 120579

(30)


(119905) = 119891 (119909 (119905) 119906 (119905) Π

1

120579)

119910 (119905) = 119892 (119909 (119905) 119906 (119905) Π

1

120579)

(31)


119883 a byproduct


matrix of119882119869

(119905) = 119881

1119891 (119880

1119909 (119905) 119906 (119905) Π

1

120579)

119910 (119905) = 119892 (119880

1119909 (119905) 119906 (119905) Π

1

120579)

(32)







120598 =

1003817

1003817

1003817

1003817

119910 minus 119910

1003817

1003817

1003817

10038172

1003817

1003817

1003817

1003817

119910

1003817

1003817

1003817

10038172

(33)


1003817

1003817

1003817

1003817

119910

1003817

1003817

1003817

10038172=

radic

sum

119905

1003817

1003817

1003817

1003817

119910(119905)

1003817

1003817

1003817

1003817

2

2 (34)





(119905) =

(

(

(

(

minus119892(119909

1(119905)) minus 119892 (119909

1(119905) minus 119909

2(119905))

119892 (119909

1(119905) minus 119909

2(119905)) minus 119892 (119909

2(119905) minus 119909

3(119905))

119892 (119909

119896minus1(119905) minus 119909

119896(119905)) minus 119892 (119909

119896(119905) minus 119909

119909+1(119905))

119892 (119909

119873minus1(119905) minus 119909

119873(119905))

)

)

)

)

+

(

(

(

(

119906(119905)

0

0

0

)

)

)

)

119910 (119905) = 119909

1(119905)

(35)


119892 (119909) = exp (119909) + 119909 minus 1 (36)







= 119860119909 + 119861119906 + 120579

119910 = 119862119909

(37)


Re(1205821119899


100

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

10minus16

100

101




119879

and 119862 = 119861



= 119860 tanh(14

119909) + 119861119906 + 120579

119910 = 119862119909

(38)











50 100 150 200 250


100

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

(a)

50 100 150 200 250


100

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

(b)





119884= 119882

119862119882

119874 In Figure 2









50 100 150 200 250


100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

(a)

50 100 150 200 250


100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

(b)


128

128

256 256

States

Param

eters

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

WS + WO

(a)

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

128

128

256 256

States

Param

eters

WC +WI

(b)

WJ

128

128

256 256

States

Param

eters

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7






128

128

256 256

States

Param

eters

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

WS + WO

(a)

128

128

256 256

States

Param

eters

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

WC + WI

(b)

128

128

256 256

States

Param

eters

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

WJ

(c)



50 100 150 200 250


100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

(a)

50 100 150 200 250


100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

(b)






Linear

state

redu

ction

WYWC +WO

WX

Non

linear

Non

linear

Non

linear

state

redu

ction

WYWC +WO

WX

parameter

Linear

parameter

redu

ction

WSWIWJ

WSWIWJre

duction

Linear

combined

redu

ction

combined

redu

ction

WS + WO

WJ

WC + WI

WS + WO

WJ

WC + WI

0 10 20 30 40 50 60

t (s)





7 Conclusion






Acknowledgments


References





































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119909 119877

119906

119877

119909 119876

119906 and 119876



119882


input119906ℎ119894119895(119905) = 119888

ℎ119878

119894119890

119895120575(119905)+119906 to output119910119896119897119887 of119909119896119897119887

0= 119889

119896119879

119897119891

119887+119909

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

119879

119897int

infin

0

Ψ

ℎ119894119895119896119897

(119905) 119889119905 119879

119879

119897

Ψ

ℎ119894119895119896119897

119886119887(119905) = 119891

119879

119886119879

119879

119897Δ119909

ℎ119894119895

(119905) 119890

119879

119895119878

119879

119894Δ119910

119896119897119887

(119905) isin R

Δ119909

ℎ119894119895

(119905) = (119909

ℎ119894119895

(119905) minus 119909)

Δ119910

119896119897119887

(119905) = (119910

119896119897119887

(119905) minus 119910)

(11)




119906 119877

119909 119876

119906 and 119876


119882

119883of an



Δ119909 (119905) = 119909 (119905) = 119890

119860119905

119861119906 (119905)

Δ119910 (119905) = 119910 (119905) = 119862119890

119860119905

119909

0

(12)

thus

Ψ

ℎ119894119895119896119897

119886119887= 119891

119879

119886119879

119879

119897(119890

119860119905

119861119888

ℎ119878

119894119890

119895) 119890

119879

119895119878

119879

119894(119862119890

119860119905

119889

119896119879

119897119891

119887)

= 119888

ℎ119889

119896119891

119879

119886119879

119879

119897119890

119860119905

119861119862119890

119860119905

119879

119897119891

119887

997904rArr Ψ

ℎ119894119895119896119897

= 119888

ℎ119889

119896119879

119879

119897119890

119860119905

119861119862119890

119860119905

119879

119897

997904rArr

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

int

infin

0

119890

119860119905

119861119862119890

119860119905

119889119905

= int

infin

0

119890

119860119905

119861119862119890

119860119905

119889119905

= 119882

119883

(13)




119864

119909 119877

119906 119877

119909 119876

119906 and 119876




ℎ119878

119894119890

119895120575(119905) + 119906 to output 119910119896119897119887

of 1199091198961198971198870

= 119889

119896119879

119897119891

119887+ 119909 is given by

W119883=

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

Δ119905

119888

ℎ119889

119896

119879

119897

T

sum

119905=0

Ψ

ℎ119894119895119896119897

119905119879

119879

119897

Ψ

ℎ119894119895119896119897

119886119887119905= 119891

119879

119886119879

119879

119897Δ119909

ℎ119894119895

119905119890

119879

119895119878

119879

119894Δ119910

119896119897119887

119905isin R

Δ119909

ℎ119894119895

119905= (119909

ℎ119894119895

119905minus 119909)

Δ119910

119896119897119887

119905= (119910

119896119897119887

119905minus 119910)

(14)



1 analogous to (7)

(119905) = 119880

119879

1119891 (119880

1119909 (119905) 119906 (119905))

119910 (119905) = 119892 (119880

1119909 (119905) 119906 (119905))

(15)


119864 =

sum

119896

119894=1120590

119894

sum

119899

119894=1120590

119894

(16)



119894(119882

119883)




R119901 (119905) = 119891 (119909 (119905) 119906 (119905) 120579)

119910 (119905) = 119892 (119909 (119905) 119906 (119905) 120579)

(17)


(119905) = (

(119905)

120579 (119905)

) = (

119891 (119909 (119905) 119906 (119905) 120579)

0

)

119910 (119905) = 119892 (119909 (119905) 119906 (119905) 120579)

0= (

119909

0

120579

)

(18)



119882

119874= (

119882

119874119882

119872

119882

119879

119872119882

119875

) isin R(119899+119901)times(119899+119901)

(19)


119874isin R119899times119899 the


matrix 119882119872


119868isin R119901times119901 can


119882

119868= 119882

119875minus119882

119879

119872119882

119874

minus1

119882

119872 (20)


119868 the


119875asymp 119882








119882

119868

SVD= ΠΔΛ = (

Π

1

Π

2

)(

Δ

10

0 Δ

2

) (Λ

1Λ

2)

997904rArr

120579 = Π

119879

1120579 Π

1

120579 asymp 120579

(21)



(119905) = 119891 (119909 (119905) 119906 (119905) Π

1

120579)

119910 (119905) = 119892 (119909 (119905) 119906 (119905) Π

1

120579)

(22)


119882





1and 119881

1 see [43])


(119905) = 119881

1119891 (119880

1119909 (119905) 119906 (119905) Π

1

120579)

119910 (119905) = 119892 (119880

1119909 (119905) 119906 (119905) Π

1

120579)

(23)


119891 (119909 119906 120579) = 119891 (119909 119906) +

119901

sum

119896=1

119891 (119909 120579

119896)

997904rArr 119882

119862= 119882

1198620+

119901

sum

119896=1

119882

119862119896

119882

119878= (

tr (1198821198621) 0

d0 tr (119882

119862119901)


(24)



119882









119869is the



119882

119869= (

119882

119883119882

119872

119882

119898119882

119875

) isin R(119899+119901)times(119899+119901)

(25)




119872isin

R119899times119901

119882


yielding 119882

119875= 0 and 119882



119882 119868= 119882

119875minus119882

119898119882

minus1

119883119882

119872= 0 minus 0 119882

minus1

119883119882

119872= 0 (26)




119869= 05 (119882

119869+ 119882

119879

119869)


119882 119868= 0 minus

1

4

119882

119879

119872119882

minus1

119883119882

119872 (27)


119863 = diag (119882119883)

119882

minus1

119883asymp 119908

minus1

119883= 119863

minus1

minus 119863

minus1

(119882

119883minus 119863)119863

minus1

(28)


119882 119868= minus

1

4

119882

119879

119872119908

minus1

119883119882

119872 (29)



119882 119868asymp 119882

119875

SVD= ΠΔΛ = (

Π

1

Π

2

)(

Δ

10

0 Δ

2

) (Λ

1Λ

2)

997904rArr

120579 = Π

119879

1120579 Π

1

120579 asymp 120579

(30)


(119905) = 119891 (119909 (119905) 119906 (119905) Π

1

120579)

119910 (119905) = 119892 (119909 (119905) 119906 (119905) Π

1

120579)

(31)


119883 a byproduct


matrix of119882119869

(119905) = 119881

1119891 (119880

1119909 (119905) 119906 (119905) Π

1

120579)

119910 (119905) = 119892 (119880

1119909 (119905) 119906 (119905) Π

1

120579)

(32)







120598 =

1003817

1003817

1003817

1003817

119910 minus 119910

1003817

1003817

1003817

10038172

1003817

1003817

1003817

1003817

119910

1003817

1003817

1003817

10038172

(33)


1003817

1003817

1003817

1003817

119910

1003817

1003817

1003817

10038172=

radic

sum

119905

1003817

1003817

1003817

1003817

119910(119905)

1003817

1003817

1003817

1003817

2

2 (34)





(119905) =

(

(

(

(

minus119892(119909

1(119905)) minus 119892 (119909

1(119905) minus 119909

2(119905))

119892 (119909

1(119905) minus 119909

2(119905)) minus 119892 (119909

2(119905) minus 119909

3(119905))

119892 (119909

119896minus1(119905) minus 119909

119896(119905)) minus 119892 (119909

119896(119905) minus 119909

119909+1(119905))

119892 (119909

119873minus1(119905) minus 119909

119873(119905))

)

)

)

)

+

(

(

(

(

119906(119905)

0

0

0

)

)

)

)

119910 (119905) = 119909

1(119905)

(35)


119892 (119909) = exp (119909) + 119909 minus 1 (36)







= 119860119909 + 119861119906 + 120579

119910 = 119862119909

(37)


Re(1205821119899


100

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

10minus16

100

101




119879

and 119862 = 119861



= 119860 tanh(14

119909) + 119861119906 + 120579

119910 = 119862119909

(38)











50 100 150 200 250


100

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

(a)

50 100 150 200 250


100

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

(b)





119884= 119882

119862119882

119874 In Figure 2









50 100 150 200 250


100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

(a)

50 100 150 200 250


100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

(b)


128

128

256 256

States

Param

eters

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

WS + WO

(a)

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

128

128

256 256

States

Param

eters

WC +WI

(b)

WJ

128

128

256 256

States

Param

eters

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7






128

128

256 256

States

Param

eters

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

WS + WO

(a)

128

128

256 256

States

Param

eters

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

WC + WI

(b)

128

128

256 256

States

Param

eters

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

WJ

(c)



50 100 150 200 250


100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

(a)

50 100 150 200 250


100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

(b)






Linear

state

redu

ction

WYWC +WO

WX

Non

linear

Non

linear

Non

linear

state

redu

ction

WYWC +WO

WX

parameter

Linear

parameter

redu

ction

WSWIWJ

WSWIWJre

duction

Linear

combined

redu

ction

combined

redu

ction

WS + WO

WJ

WC + WI

WS + WO

WJ

WC + WI

0 10 20 30 40 50 60

t (s)





7 Conclusion






Acknowledgments


References





































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












R119901 (119905) = 119891 (119909 (119905) 119906 (119905) 120579)

119910 (119905) = 119892 (119909 (119905) 119906 (119905) 120579)

(17)


(119905) = (

(119905)

120579 (119905)

) = (

119891 (119909 (119905) 119906 (119905) 120579)

0

)

119910 (119905) = 119892 (119909 (119905) 119906 (119905) 120579)

0= (

119909

0

120579

)

(18)



119882

119874= (

119882

119874119882

119872

119882

119879

119872119882

119875

) isin R(119899+119901)times(119899+119901)

(19)


119874isin R119899times119899 the


matrix 119882119872


119868isin R119901times119901 can


119882

119868= 119882

119875minus119882

119879

119872119882

119874

minus1

119882

119872 (20)


119868 the


119875asymp 119882








119882

119868

SVD= ΠΔΛ = (

Π

1

Π

2

)(

Δ

10

0 Δ

2

) (Λ

1Λ

2)

997904rArr

120579 = Π

119879

1120579 Π

1

120579 asymp 120579

(21)



(119905) = 119891 (119909 (119905) 119906 (119905) Π

1

120579)

119910 (119905) = 119892 (119909 (119905) 119906 (119905) Π

1

120579)

(22)


119882





1and 119881

1 see [43])


(119905) = 119881

1119891 (119880

1119909 (119905) 119906 (119905) Π

1

120579)

119910 (119905) = 119892 (119880

1119909 (119905) 119906 (119905) Π

1

120579)

(23)


119891 (119909 119906 120579) = 119891 (119909 119906) +

119901

sum

119896=1

119891 (119909 120579

119896)

997904rArr 119882

119862= 119882

1198620+

119901

sum

119896=1

119882

119862119896

119882

119878= (

tr (1198821198621) 0

d0 tr (119882

119862119901)


(24)



119882









119869is the



119882

119869= (

119882

119883119882

119872

119882

119898119882

119875

) isin R(119899+119901)times(119899+119901)

(25)




119872isin

R119899times119901

119882


yielding 119882

119875= 0 and 119882



119882 119868= 119882

119875minus119882

119898119882

minus1

119883119882

119872= 0 minus 0 119882

minus1

119883119882

119872= 0 (26)




119869= 05 (119882

119869+ 119882

119879

119869)


119882 119868= 0 minus

1

4

119882

119879

119872119882

minus1

119883119882

119872 (27)


119863 = diag (119882119883)

119882

minus1

119883asymp 119908

minus1

119883= 119863

minus1

minus 119863

minus1

(119882

119883minus 119863)119863

minus1

(28)


119882 119868= minus

1

4

119882

119879

119872119908

minus1

119883119882

119872 (29)



119882 119868asymp 119882

119875

SVD= ΠΔΛ = (

Π

1

Π

2

)(

Δ

10

0 Δ

2

) (Λ

1Λ

2)

997904rArr

120579 = Π

119879

1120579 Π

1

120579 asymp 120579

(30)


(119905) = 119891 (119909 (119905) 119906 (119905) Π

1

120579)

119910 (119905) = 119892 (119909 (119905) 119906 (119905) Π

1

120579)

(31)


119883 a byproduct


matrix of119882119869

(119905) = 119881

1119891 (119880

1119909 (119905) 119906 (119905) Π

1

120579)

119910 (119905) = 119892 (119880

1119909 (119905) 119906 (119905) Π

1

120579)

(32)







120598 =

1003817

1003817

1003817

1003817

119910 minus 119910

1003817

1003817

1003817

10038172

1003817

1003817

1003817

1003817

119910

1003817

1003817

1003817

10038172

(33)


1003817

1003817

1003817

1003817

119910

1003817

1003817

1003817

10038172=

radic

sum

119905

1003817

1003817

1003817

1003817

119910(119905)

1003817

1003817

1003817

1003817

2

2 (34)





(119905) =

(

(

(

(

minus119892(119909

1(119905)) minus 119892 (119909

1(119905) minus 119909

2(119905))

119892 (119909

1(119905) minus 119909

2(119905)) minus 119892 (119909

2(119905) minus 119909

3(119905))

119892 (119909

119896minus1(119905) minus 119909

119896(119905)) minus 119892 (119909

119896(119905) minus 119909

119909+1(119905))

119892 (119909

119873minus1(119905) minus 119909

119873(119905))

)

)

)

)

+

(

(

(

(

119906(119905)

0

0

0

)

)

)

)

119910 (119905) = 119909

1(119905)

(35)


119892 (119909) = exp (119909) + 119909 minus 1 (36)







= 119860119909 + 119861119906 + 120579

119910 = 119862119909

(37)


Re(1205821119899


100

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

10minus16

100

101




119879

and 119862 = 119861



= 119860 tanh(14

119909) + 119861119906 + 120579

119910 = 119862119909

(38)











50 100 150 200 250


100

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

(a)

50 100 150 200 250


100

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

(b)





119884= 119882

119862119882

119874 In Figure 2









50 100 150 200 250


100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

(a)

50 100 150 200 250


100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

(b)


128

128

256 256

States

Param

eters

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

WS + WO

(a)

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

128

128

256 256

States

Param

eters

WC +WI

(b)

WJ

128

128

256 256

States

Param

eters

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7






128

128

256 256

States

Param

eters

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

WS + WO

(a)

128

128

256 256

States

Param

eters

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

WC + WI

(b)

128

128

256 256

States

Param

eters

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

WJ

(c)



50 100 150 200 250


100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

(a)

50 100 150 200 250


100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

(b)






Linear

state

redu

ction

WYWC +WO

WX

Non

linear

Non

linear

Non

linear

state

redu

ction

WYWC +WO

WX

parameter

Linear

parameter

redu

ction

WSWIWJ

WSWIWJre

duction

Linear

combined

redu

ction

combined

redu

ction

WS + WO

WJ

WC + WI

WS + WO

WJ

WC + WI

0 10 20 30 40 50 60

t (s)





7 Conclusion






Acknowledgments


References





































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119869is the



119882

119869= (

119882

119883119882

119872

119882

119898119882

119875

) isin R(119899+119901)times(119899+119901)

(25)




119872isin

R119899times119901

119882


yielding 119882

119875= 0 and 119882



119882 119868= 119882

119875minus119882

119898119882

minus1

119883119882

119872= 0 minus 0 119882

minus1

119883119882

119872= 0 (26)




119869= 05 (119882

119869+ 119882

119879

119869)


119882 119868= 0 minus

1

4

119882

119879

119872119882

minus1

119883119882

119872 (27)


119863 = diag (119882119883)

119882

minus1

119883asymp 119908

minus1

119883= 119863

minus1

minus 119863

minus1

(119882

119883minus 119863)119863

minus1

(28)


119882 119868= minus

1

4

119882

119879

119872119908

minus1

119883119882

119872 (29)



119882 119868asymp 119882

119875

SVD= ΠΔΛ = (

Π

1

Π

2

)(

Δ

10

0 Δ

2

) (Λ

1Λ

2)

997904rArr

120579 = Π

119879

1120579 Π

1

120579 asymp 120579

(30)


(119905) = 119891 (119909 (119905) 119906 (119905) Π

1

120579)

119910 (119905) = 119892 (119909 (119905) 119906 (119905) Π

1

120579)

(31)


119883 a byproduct


matrix of119882119869

(119905) = 119881

1119891 (119880

1119909 (119905) 119906 (119905) Π

1

120579)

119910 (119905) = 119892 (119880

1119909 (119905) 119906 (119905) Π

1

120579)

(32)







120598 =

1003817

1003817

1003817

1003817

119910 minus 119910

1003817

1003817

1003817

10038172

1003817

1003817

1003817

1003817

119910

1003817

1003817

1003817

10038172

(33)


1003817

1003817

1003817

1003817

119910

1003817

1003817

1003817

10038172=

radic

sum

119905

1003817

1003817

1003817

1003817

119910(119905)

1003817

1003817

1003817

1003817

2

2 (34)





(119905) =

(

(

(

(

minus119892(119909

1(119905)) minus 119892 (119909

1(119905) minus 119909

2(119905))

119892 (119909

1(119905) minus 119909

2(119905)) minus 119892 (119909

2(119905) minus 119909

3(119905))

119892 (119909

119896minus1(119905) minus 119909

119896(119905)) minus 119892 (119909

119896(119905) minus 119909

119909+1(119905))

119892 (119909

119873minus1(119905) minus 119909

119873(119905))

)

)

)

)

+

(

(

(

(

119906(119905)

0

0

0

)

)

)

)

119910 (119905) = 119909

1(119905)

(35)


119892 (119909) = exp (119909) + 119909 minus 1 (36)







= 119860119909 + 119861119906 + 120579

119910 = 119862119909

(37)


Re(1205821119899


100

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

10minus16

100

101




119879

and 119862 = 119861



= 119860 tanh(14

119909) + 119861119906 + 120579

119910 = 119862119909

(38)











50 100 150 200 250


100

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

(a)

50 100 150 200 250


100

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

(b)





119884= 119882

119862119882

119874 In Figure 2









50 100 150 200 250


100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

(a)

50 100 150 200 250


100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

(b)


128

128

256 256

States

Param

eters

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

WS + WO

(a)

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

128

128

256 256

States

Param

eters

WC +WI

(b)

WJ

128

128

256 256

States

Param

eters

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7






128

128

256 256

States

Param

eters

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

WS + WO

(a)

128

128

256 256

States

Param

eters

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

WC + WI

(b)

128

128

256 256

States

Param

eters

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

WJ

(c)



50 100 150 200 250


100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

(a)

50 100 150 200 250


100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

(b)






Linear

state

redu

ction

WYWC +WO

WX

Non

linear

Non

linear

Non

linear

state

redu

ction

WYWC +WO

WX

parameter

Linear

parameter

redu

ction

WSWIWJ

WSWIWJre

duction

Linear

combined

redu

ction

combined

redu

ction

WS + WO

WJ

WC + WI

WS + WO

WJ

WC + WI

0 10 20 30 40 50 60

t (s)





7 Conclusion






Acknowledgments


References





































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(119905) =

(

(

(

(

minus119892(119909

1(119905)) minus 119892 (119909

1(119905) minus 119909

2(119905))

119892 (119909

1(119905) minus 119909

2(119905)) minus 119892 (119909

2(119905) minus 119909

3(119905))

119892 (119909

119896minus1(119905) minus 119909

119896(119905)) minus 119892 (119909

119896(119905) minus 119909

119909+1(119905))

119892 (119909

119873minus1(119905) minus 119909

119873(119905))

)

)

)

)

+

(

(

(

(

119906(119905)

0

0

0

)

)

)

)

119910 (119905) = 119909

1(119905)

(35)


119892 (119909) = exp (119909) + 119909 minus 1 (36)







= 119860119909 + 119861119906 + 120579

119910 = 119862119909

(37)


Re(1205821119899


100

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

10minus16

100

101




119879

and 119862 = 119861



= 119860 tanh(14

119909) + 119861119906 + 120579

119910 = 119862119909

(38)











50 100 150 200 250


100

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

(a)

50 100 150 200 250


100

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

(b)





119884= 119882

119862119882

119874 In Figure 2









50 100 150 200 250


100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

(a)

50 100 150 200 250


100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

(b)


128

128

256 256

States

Param

eters

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

WS + WO

(a)

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

128

128

256 256

States

Param

eters

WC +WI

(b)

WJ

128

128

256 256

States

Param

eters

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7






128

128

256 256

States

Param

eters

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

WS + WO

(a)

128

128

256 256

States

Param

eters

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

WC + WI

(b)

128

128

256 256

States

Param

eters

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

WJ

(c)



50 100 150 200 250


100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

(a)

50 100 150 200 250


100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

(b)






Linear

state

redu

ction

WYWC +WO

WX

Non

linear

Non

linear

Non

linear

state

redu

ction

WYWC +WO

WX

parameter

Linear

parameter

redu

ction

WSWIWJ

WSWIWJre

duction

Linear

combined

redu

ction

combined

redu

ction

WS + WO

WJ

WC + WI

WS + WO

WJ

WC + WI

0 10 20 30 40 50 60

t (s)





7 Conclusion






Acknowledgments


References





































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










50 100 150 200 250


100

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

(a)

50 100 150 200 250


100

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

(b)





119884= 119882

119862119882

119874 In Figure 2









50 100 150 200 250


100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

(a)

50 100 150 200 250


100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

(b)


128

128

256 256

States

Param

eters

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

WS + WO

(a)

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

128

128

256 256

States

Param

eters

WC +WI

(b)

WJ

128

128

256 256

States

Param

eters

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7






128

128

256 256

States

Param

eters

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

WS + WO

(a)

128

128

256 256

States

Param

eters

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

WC + WI

(b)

128

128

256 256

States

Param

eters

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

WJ

(c)



50 100 150 200 250


100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

(a)

50 100 150 200 250


100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

(b)






Linear

state

redu

ction

WYWC +WO

WX

Non

linear

Non

linear

Non

linear

state

redu

ction

WYWC +WO

WX

parameter

Linear

parameter

redu

ction

WSWIWJ

WSWIWJre

duction

Linear

combined

redu

ction

combined

redu

ction

WS + WO

WJ

WC + WI

WS + WO

WJ

WC + WI

0 10 20 30 40 50 60

t (s)





7 Conclusion






Acknowledgments


References





































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










50 100 150 200 250


100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

(a)

50 100 150 200 250


100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

(b)


128

128

256 256

States

Param

eters

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

WS + WO

(a)

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

128

128

256 256

States

Param

eters

WC +WI

(b)

WJ

128

128

256 256

States

Param

eters

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7






128

128

256 256

States

Param

eters

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

WS + WO

(a)

128

128

256 256

States

Param

eters

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

WC + WI

(b)

128

128

256 256

States

Param

eters

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

WJ

(c)



50 100 150 200 250


100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

(a)

50 100 150 200 250


100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

(b)






Linear

state

redu

ction

WYWC +WO

WX

Non

linear

Non

linear

Non

linear

state

redu

ction

WYWC +WO

WX

parameter

Linear

parameter

redu

ction

WSWIWJ

WSWIWJre

duction

Linear

combined

redu

ction

combined

redu

ction

WS + WO

WJ

WC + WI

WS + WO

WJ

WC + WI

0 10 20 30 40 50 60

t (s)





7 Conclusion






Acknowledgments


References





































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










128

128

256 256

States

Param

eters

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

WS + WO

(a)

128

128

256 256

States

Param

eters

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

WC + WI

(b)

128

128

256 256

States

Param

eters

100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

WJ

(c)



50 100 150 200 250


100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

(a)

50 100 150 200 250


100

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

(b)






Linear

state

redu

ction

WYWC +WO

WX

Non

linear

Non

linear

Non

linear

state

redu

ction

WYWC +WO

WX

parameter

Linear

parameter

redu

ction

WSWIWJ

WSWIWJre

duction

Linear

combined

redu

ction

combined

redu

ction

WS + WO

WJ

WC + WI

WS + WO

WJ

WC + WI

0 10 20 30 40 50 60

t (s)





7 Conclusion






Acknowledgments


References





































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Linear

state

redu

ction

WYWC +WO

WX

Non

linear

Non

linear

Non

linear

state

redu

ction

WYWC +WO

WX

parameter

Linear

parameter

redu

ction

WSWIWJ

WSWIWJre

duction

Linear

combined

redu

ction

combined

redu

ction

WS + WO

WJ

WC + WI

WS + WO

WJ

WC + WI

0 10 20 30 40 50 60

t (s)





7 Conclusion






Acknowledgments


References





































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article cross-gramian-based combined...

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