Research Article Recursive Reduced-Order Algor ithm...

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Research Article Recursive Reduced-Order Algorithm for Singularly Perturbed Cross Grammian Algebraic Sylvester Equation Intessar Al-Iedani 1,2 and Zoran Gajic 1 1 Electrical and Computer Engineering Department, Rutgers University, New Brunswick, NJ, USA 2 Electrical Engineering Department, College of Engineering, University of Basrah, Basrah, Iraq Correspondence should be addressed to Intessar Al-Iedani; [email protected] Received 10 July 2016; Accepted 24 October 2016 Academic Editor: Guangming Xie Copyright © 2016 I. Al-Iedani and Z. Gajic. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A new recursive algorithm is developed for solving the algebraic Sylvester equation that defines the cross Grammian of singularly perturbed linear systems. e cross Grammian matrix provides aggregate information about controllability and observability of a linear system. e solution is obtained in terms of reduced-order algebraic Sylvester equations that correspond to slow and fast subsystems of a singularly perturbed system. e rate of convergence of the proposed algorithm is (), where is a small singular perturbation parameter that indicates separation of slow and fast state variables. Several real physical system examples are solved to demonstrate efficiency of the proposed algorithm. 1. Introduction Singularly perturbed systems have multiple time scales cor- responding to fast and slow state space variables. For a system with two time scales, the slow time scale is related to the eigenvalues that are close to the imaginary axis and that represent the slow state space variables (slow modes) of the system, while the fast time scale is related to those that are far from the imaginary axis and that represent the fast state space variables (fast modes) of the system. Many algorithms exist in the literature for solving diverse problems related to analysis and control of singularly perturbed linear systems. Fixed point recursive numerical methods were first proposed in [1] and used in [2–4] to solve the closed and open loop optimal control problems. ose methods led thereaſter to the Hamiltonian approach, which solves the linear-quadratic optimal control and filtering problem by decomposing the algebraic Riccati equations into pure- slow and pure-fast reduced-order algebraic Riccati equations [5]. e exact decomposition into pure-slow and pure-fast subsystems led to the use of parallel algorithms [6, 7] to solve the algebraic Riccati equation of the linear-quadratic optimal control problem. Moreover, some iterative methods were also used to solve this problem (see, e.g., [8] and the references therein). Most of the previous studies consider solving the algebraic Riccati equation, as it represents the most important equation of the optimal control and filtering problems. e system under investigation in this paper must be asymptotically stable, controllable, and observable. e test for controllability and observability of the system is usually done separately using the controllability and observability Grammians. In many applications, the reduced-order system is welcome to lower computational complexity. Model order reduction retains only state space variables that are both strongly controllable and strongly observable. is requires investigating the behavior of state space variables and bal- ancing the controllability and observability Grammians, such that they are diagonal and identical. It has been shown in [9] that studying the controllability and observability of the system, separately, can be misleading; a method that directly assesses the combination of the two properties is preferred. erefore, the cross Grammian matrix was defined in [10] as an alternative approach to the existing controllability and observability Grammian matrices. Unlike the controllability and observability Grammians, the cross Grammian contains Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2016, Article ID 2452746, 9 pages http://dx.doi.org/10.1155/2016/2452746

Transcript of Research Article Recursive Reduced-Order Algor ithm...

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Research ArticleRecursive Reduced-Order Algorithm for Singularly PerturbedCross Grammian Algebraic Sylvester Equation

Intessar Al-Iedani12 and Zoran Gajic1

1Electrical and Computer Engineering Department Rutgers University New Brunswick NJ USA2Electrical Engineering Department College of Engineering University of Basrah Basrah Iraq

Correspondence should be addressed to Intessar Al-Iedani intessaraliedanirutgersedu

Received 10 July 2016 Accepted 24 October 2016

Academic Editor Guangming Xie

Copyright copy 2016 I Al-Iedani and Z Gajic This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

A new recursive algorithm is developed for solving the algebraic Sylvester equation that defines the cross Grammian of singularlyperturbed linear systems The cross Grammian matrix provides aggregate information about controllability and observability ofa linear system The solution is obtained in terms of reduced-order algebraic Sylvester equations that correspond to slow and fastsubsystems of a singularly perturbed systemThe rate of convergence of the proposed algorithm is 119874(120576) where 120576 is a small singularperturbation parameter that indicates separation of slow and fast state variables Several real physical system examples are solvedto demonstrate efficiency of the proposed algorithm

1 Introduction

Singularly perturbed systems have multiple time scales cor-responding to fast and slow state space variables For asystem with two time scales the slow time scale is relatedto the eigenvalues that are close to the imaginary axis andthat represent the slow state space variables (slow modes)of the system while the fast time scale is related to thosethat are far from the imaginary axis and that represent thefast state space variables (fast modes) of the system Manyalgorithms exist in the literature for solving diverse problemsrelated to analysis and control of singularly perturbed linearsystems Fixed point recursive numerical methods were firstproposed in [1] and used in [2ndash4] to solve the closedand open loop optimal control problems Those methodsled thereafter to the Hamiltonian approach which solvesthe linear-quadratic optimal control and filtering problemby decomposing the algebraic Riccati equations into pure-slow and pure-fast reduced-order algebraic Riccati equations[5] The exact decomposition into pure-slow and pure-fastsubsystems led to the use of parallel algorithms [6 7] tosolve the algebraic Riccati equation of the linear-quadraticoptimal control problem Moreover some iterative methods

were also used to solve this problem (see eg [8] and thereferences therein) Most of the previous studies considersolving the algebraic Riccati equation as it represents themost important equation of the optimal control and filteringproblems

The system under investigation in this paper must beasymptotically stable controllable and observable The testfor controllability and observability of the system is usuallydone separately using the controllability and observabilityGrammians In many applications the reduced-order systemis welcome to lower computational complexity Model orderreduction retains only state space variables that are bothstrongly controllable and strongly observable This requiresinvestigating the behavior of state space variables and bal-ancing the controllability and observability Grammians suchthat they are diagonal and identical It has been shown in[9] that studying the controllability and observability of thesystem separately can be misleading a method that directlyassesses the combination of the two properties is preferredTherefore the cross Grammian matrix was defined in [10]as an alternative approach to the existing controllability andobservability Grammian matrices Unlike the controllabilityand observability Grammians the cross Grammian contains

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 2452746 9 pageshttpdxdoiorg10115520162452746

2 Mathematical Problems in Engineering

information about both controllability and observability ofthe system

In this paper a new recursive algorithm is proposed tosolve the algebraic Sylvester equation of linear singularlyperturbed systems whose solution defines the cross Gram-mian matrix The algorithm is obtained in terms of reduced-order algebraic Sylvester equations corresponding to slowand fast subsystems The solutions of full-order algebraicSylvester equations for finding the cross Grammian matrixwere considered in [11 12]

The remainder of the paper is organized as followsSection 2 reviews the controllability observability and thecross Grammian matrices The proposed recursive algorithmis then described in Section 3 In Section 4 several casestudies are considered to demonstrate the performance ofthe proposed algorithm Then the conclusions follow inSection 5

2 The Cross Grammian Matrix

Consider a linear dynamic system

119889119909119889119905 = 119860119909 (119905) + 119861119906 (119905) 119910 (119905) = 119862119909 (119905) + 119863119906 (119905)

(1)

where 119909(119905) isin R119899 are state variables 119906(119905) isin R119898 are controlinputs and 119910(119905) isin R119901 are measured outputs Assume thatsystem (1) is asymptotically stable controllable and observ-able Controllability and observability of the system canbe measured using the controllability and the observabilityGrammians defined respectively as

119882119862 = intinfin0

119890119860119905119861119861119879119890119860119879119905 (2)

119882119874 = intinfin0

119890119860119879119905119862119879119862119890119860119905 (3)

For asymptotically stable controllable and observablesystems Grammians (2) and (3) are positive definite andrepresented the solutions of the algebraic Lyapunov equations

119860119882119862 + 119882119862119860119879 = minus119861119861119879119860119879119882119874 + 119882119874119860 = minus119862119879119862 (4)

Assuming system (1) is square that is the number ofinputs equals the number of outputs 119898 = 119901 the crossGrammian matrix was defined in [10] for single-input single-output (SISO) systems as

119882119883 = intinfin0

119890119860119905119861119862119890119860119905 (5)

and represented by the solution to the algebraic Sylvesterequation

119860119882119883 + 119882119883119860 + 119861119862 = 0 (6)

In this context the Sylvester algebraic equation (6) hasa unique solution if and only if 119860 and minus119860 have distincteigenvalues [13] Several numerical solutions for the Sylvesterequation were proposed in the literature see for example[14ndash16] and the references therein The definition in (5)was extended in [17ndash19] to include multi-input multioutput(MIMO) systems

Furthermore for MIMO symmetric systems the relationbetween controllability and observability on the one handand the cross Grammian on the other hand is given by [20]

1198822119883 = 119882119862119882119874 (7)

3 A Recursive Algorithm for FindingCross Grammians for Singularly PerturbedLinear Systems

The singularly perturbed structure can be obtained by parti-tioning the system matrices in (1) as follows [6 21]

119860 = [[

1198601 119860211986031205761198604120576

]]

119861 = [[

11986111198612120576]]

119862 = [1198621 1198622]

(8)

where 120576 is a small positive singular perturbation parameter119860 119861 and 119862 are constant matrices of appropriate dimensionsBased on the singular perturbation theory [6 21] a singularlyperturbed linear system in the explicit state variable standardform is given by

1198891199091 (119905)119889119905 = 11986011199091 (119905) + 11986021199092 (119905) + 1198611119906 (119905) 1205761198891199092 (119905)119889119905 = 11986031199091 (119905) + 11986041199092 (119905) + 1198612119906 (119905)

119910 (119905) = 11986211199091 (119905) + 11986221199092 (119905) + 119863119906 (119905) (9)

where 1199091(119905) isin R1198991 are the slow state variables and 1199092(119905) isin R1198992

are the fast state variables Assuming that 1198604 is nonsingularthe eigenvalues of matrix 119860 consist of two disjoint groupsone corresponds to the slow subsystem 120582119904(119860) and the othercorresponds to the fast subsystem 120582119891(119860) If the two sub-systems have a mixture of slow and fast eigenvalues then atechnique has to be applied to convert the system into itsstandard singularly perturbed form defined in (9) We willgive examples on this case in Sections 42 and 43

The nature of the cross Grammian matrix 119882119883 defined in(6) corresponding to the system singularly perturbed formdefined in (9) is

119882119883 = [1198821 12057611988221198823 1198824 ] (10)

Mathematical Problems in Engineering 3

Using (8) and (10) in (6) we get the partitioned form of thealgebraic Sylvester equation as follows

11986011198821 + 11986021198823 + 11988211198601 + 11988221198603 + 11986111198621 = 012057611986011198822 + 11986021198824 + 11988211198602 + 11988221198604 + 11986111198622 = 011986031198821 + 11986041198823 + 12057611988231198601 + 11988241198603 + 11986121198621 = 0

12057611986031198822 + 11986041198824 + 12057611988231198602 + 11988241198604 + 11986121198622 = 0(11)

Setting 120576 = 0 we get the following approximate algebraicequations

1198601119882(0)1 + 1198602119882(0)3 + 119882(0)1 1198601 + 119882(0)2 1198603 + 11986111198621 = 01198602119882(0)4 + 119882(0)1 1198602 + 119882(0)2 1198604 + 11986111198622 = 01198603119882(0)1 + 1198604119882(0)3 + 119882(0)4 1198603 + 11986121198621 = 0

1198604119882(0)4 + 119882(0)4 1198604 + 11986121198622 = 0(12)

The solution of equations (12) is given in terms ofthe following reduced-order algebraic Sylvester equationscorresponding to the slow and fast subsystems

1198604119882(0)4 + 119882(0)4 1198604 + 11986121198622 = 01198600119882(0)1 + 119882(0)1 1198600 + 1198660 = 0 (13)

In addition we have from (12)

119882(0)2 = minus (11986111198622 + 1198602119882(0)4 + 119882(0)1 1198602) 119860minus14 119882(0)3 = minus119860minus14 (11986121198621 + 1198603119882(0)1 + 119882(0)4 1198603) (14)

where

1198600 = 1198601 minus 1198602119860minus14 1198603 (15)

1198660 = minus1198602119860minus14 (11986121198621 + 119882(0)4 1198603)minus (1198602119882(0)4 + 11986111198622) 119860minus14 1198603 + 11986111198621

(16)

To find a unique solution of (13) we impose the followingassumption

Assumption 1 Matrices 1198600 and 1198604 are asymptotically stableIn consequence unique solutions of (13)-(14) exist

Defining the approximation error as

1198821 = 119882(0)1 + 12057611986411198822 = 119882(0)2 + 12057611986421198823 = 119882(0)3 + 12057611986431198824 = 119882(0)4 + 1205761198644

(17)

and subtracting (12) from (11) we get the following errorequations after some algebra

11986041198644 + 11986441198604= minus1198603 (119882(0)2 + 1205761198642) minus (119882(0)3 + 1205761198643) 1198602

11986031198641 + 11986041198643 + 11986441198603 = minus (119882(0)3 + 1205761198643) 119860111986021198644 + 11986411198602 + 11986421198604 = minus1198601 (119882(0)2 + 1205761198642) 11986011198641 + 11986021198643 + 11986411198601 + 11986421198603 = 0

(18)

From the first equation in (18) we can observe thatthe unknown errors 1198642 and 1198643 are multiplied by a smallparameter 120576 A similar situation is in the second and thethird equations of (18) Therefore we propose the followingalgorithm for solving error equations (18)

31 The Proposed Algorithm Start with 119864(0)2 = 0 and 119864(0)3 = 0and recursively evaluate

1198604119864(119894+1)4 + 119864(119894+1)4 1198604= minus1198603 (119882(0)2 + 120576119864(119894)2 ) minus (119882(0)3 + 120576119864(119894)3 ) 1198602

1198600119864(119894+1)1 + 119864(119894+1)1 1198600= 1198602119860minus14 (119882(0)3 + 120576119864(119894)3 ) 1198601

+ 1198601 (119882(0)2 + 120576119864(119894)2 ) 119860minus14 1198603 + 1198602119860minus14 119864(119894+1)4 1198603+ 1198602119864(119894+1)4 119860minus14 1198603

119864(119894+1)2= minus (1198601 (119882(0)2 + 120576119864(119894)2 ) + 1198602119864(119894+1)4 + 119864(119894+1)1 1198602) 119860minus14

119864(119894+1)3= minus119860minus14 ((119882(0)3 + 120576119864(119894)3 ) 1198601 + 1198603119864(119894+1)1 + 119864(119894+1)4 1198603)

for 119894 = 0 1 2

(19)

Theorem 2 Assuming that matrices 1198600 and 1198604 are asymp-totically stable algorithm (19) converges to the exact solutionof (18) with a rate of convergence 119874(120576) that is

10038171003817100381710038171003817119864(119894+1)119895 minus 119864(119894)119895 10038171003817100381710038171003817 = 119874 (120576) 10038171003817100381710038171003817119864(119894)119895 minus 11986411989510038171003817100381710038171003817 = 119874 (120576119894) (20)

for 119895 = 1 2 3 4 and 119894 = 1 2 Therefore the exact solution 119882119883 can be obtained with

an accuracy of 119874(120576119894) after performing 119894 iterations on theproposed algorithm (19) as follows

119882(119894)119895 = 119882119895 + 120576119864(119894)119895 for 119895 = 1 2 3 4 119894 = 1 2 (21)

4 Mathematical Problems in Engineering

Proof Using Assumption 1 that is 1198600 and 1198604 are asymptoti-cally stable it can be shown that (19) represents a contractionmapping [22] that is

10038171003817100381710038171003817119864(119894)119895 minus 11986411989510038171003817100381710038171003817 = 119874 (120576119894) for 119895 = 1 2 3 4 119894 = 1 2 (22)

Formula (22) will be also valid if

119864(119894+1)2 = minus (1198601 (119882(0)2 + 120576119864(119894)2 ) + 1198602119864(119894)4 + 119864(119894)1 1198602) 119860minus14 119864(119894+1)3 = minus119860minus14 ((119882(0)3 + 120576119864(119894)3 ) 1198601 + 1198603119864(119894)1 + 119864(119894)4 1198603)

for 119894 = 0 1 2 119864(0)1 = 0 119864(0)4 = 0(23)

Formula (22) implies that algorithm (19) is convergent Using119864(infin)119895 for 119895 = 1 2 3 4 in (19) and comparing it to (18) it canbe seen that algorithm (19) converges to the unique solutionof (18)

4 Case Studies

Three case studies are considered to demonstrate the pro-posed algorithm a fourth-order aircraft example whosemathematical model is in the explicit singularly perturbedform defined in (9) in which with accuracy of 119874(120576) theslow eigenvalues are all contained in the approximate slowsubsystem represented by 1198600 and all fast eigenvalues arecontained in the approximate fast subsystem representedby 1198604 a fifth-order chemical plant model given in implicitsingularly perturbed form (it has two slow and three fasteigenvalues but the state variables have to be reordered toachieve explicit singularly perturbed form defined in (9)) a

tenth-order hydrogen gas reformer used to provide hydrogento a fuel cell fromhydrogen rich fuels (natural gasmethanol)

41 L-1011 Aircraft Here we consider the lateral axis equa-tions of the rigid body model of L-1011 aircraft at cruisecondition [23]The state variables are the bank angle roll rateyaw rate and sideslip angle which are represented in the statevector 119909(119905) = [1199091 1199092 1199093 1199094]119879 in the same order The inputvector consists of two variables the rudder deflection 120575119903 andthe aileron deflection 120575119886 and is given as 119906(119905) = [120575119903 120575119886]119879 Thesystem matrices are given as

119860 =[[[[[[

0 1 0 00 minus189 039 minus5550 minus0034 minus298 243

0034 minus00011 minus099 minus021

]]]]]]

119861 =[[[[[[

0 0036 minus16

minus095 minus0032003 0

]]]]]]

119862 = [1 0 0 00 1 0 0]

(24)

The eigenvalues of thematrix119860 areminus01016minus14811plusmn06292119894and minus20162The system is asymptotically stable (all eigenval-ues are in the left half plane) controllable and observableMoreover there is only one slow mode with eigenvalueminus00899 and there are three fast modes with eigenvaluesminus14891 plusmn 07686119894 and minus21017 The singular perturbationparameter 120576 = 007 which is the ratio between the fastestslow eigenvalue and the slowest fast eigenvalue Solving thealgebraic Sylvesterrsquos equation (6) the cross Grammian matrixcan be obtained as follows

119882119883 =[[[[[[

minus377168467243 minus225320146563 minus460285451131 1268652326564minus043134179103 minus062940797540 minus027809983914 1332572206696minus014313989569 000048741612 minus001060057667 minus002122054103020967036557 006408641105 minus003861451004 000250243909

]]]]]]

(25)

Using the proposed algorithm the initial cross Grammi-an matrix 119882(0) (first-order approximate solution) is obtained

as follows

119882(0) =[[[[[[

minus398315817315 minus225882783809 minus446229986109 1232553354098minus041906814039 minus042418654922 minus021443252362 0883962065453minus014908955443 minus000284907928 000235249833 minus0005866460180202330975578 0001553980682 0000836843461 minus000419856136

]]]]]]

(26)

Mathematical Problems in Engineering 5

Table 1 Error norm values for each iteration for L-1011 aircraftsystem

Number of iterations 119894 10038171003817100381710038171003817119882119883 minus 119882(119894)119883 1003817100381710038171003817100381722 7503328625983908119890 minus 043 2611125417476592119890 minus 054 9085062505535487119890 minus 075 3161097999142058119890 minus 086 1099918110580769119890 minus 097 3830668858621788119890 minus 118 1368874658220410119890 minus 129 9598118726733237119890 minus 1410 6474407682370149119890 minus 14

Comparing the exact solution 119882119883 to the first-orderapproximate solution of the cross Grammian matrix bycalculating the error norm we get

10038171003817100381710038171003817119882119883 minus 119882(0)100381710038171003817100381710038172 = 0624920763816596 (27)

Then the cross Grammian matrix is calculated using the pro-posed recursive algorithm The error norm at each iterationis shown in Table 1 By taking the error norm it can be seenthat the algorithm converges rapidly to the exact solution

42 Chemical Plant In this section the linearized chemicalplant considered in [24] is chosen to explain the behavior ofthe proposed algorithm when the linear singularly perturbedsystem is not in the explicit standard form (9) The systemmatrices are given as follows

119860 =[[[[[[[[[

minus01094 00628 0 0 01306 minus2136 09807 0 0

0 1595 minus3149 1547 00 00355 2632 minus4257 18550 00027 0 01636 minus01625

]]]]]]]]]

119861 =[[[[[[[[[

0 000638 000838 minus0139601004 minus020600063 minus00128

]]]]]]]]]

119862 = [1 0 0 0 00 0 0 0 1]

(28)

The eigenvalues of matrix 119860 are minus59822 minus28408 minus08954minus00774 and minus00141 which indicates that this system hastwo slow modes (eigenvalues) The singular perturbationparameter 120576 is chosen as the ratio of the fastest sloweigenvalue to the slowest fast eigenvalue and is equal to

120576 = 0086 = 0077408954 Introducing the followingpermutation matrix that exchanges the second row of matrix119860 with its fifth row that is

119875 =[[[[[[[[[

1 0 0 0 00 0 0 0 10 0 1 0 00 0 0 1 00 1 0 0 0

]]]]]]]]]

(29)

the explicit singularly perturbed form of the system matricescan be obtained as follows

119860SP = 119875119860119875119861SP = 119875119861119862SP = 119862119875

(30)

Thereby they are calculated as

119860SP

=[[[[[[[[[[

minus01094 0 0 0 006280 minus01625 0 01636 000270 0 minus3149 1547 159500 18550 2632 minus4257 00355

13060 0 09807 0 minus21320

]]]]]]]]]]

119861SP =[[[[[[[[[

0 000063 minus0012800838 minus0139601004 minus020600638 0

]]]]]]]]]

119862SP = [1 0 0 0 00 1 0 0 0]

(31)

Matrices 119860SP and 119860 have the same eigenvalues and the samenumber of slow and fast modes However the slow and fasteigenvalues are now correctly separated into two disjointgroups The slow approximate subsystem (see (15)) has theeigenvalues minus00788 and minus00160 and the fast approximatesubsystem has the fast eigenvalues minus59521 27925 andminus07933 (eigenvalues of 1198604)

Solving the algebraic Sylvesterrsquos equation (6) the crossGrammian matrix can be obtained as follows

6 Mathematical Problems in Engineering

119882119883 =[[[[[[[[[

minus007530866 minus011561658 minus000985227 minus000806596 minus000985946minus023194836 minus049463876 minus003897379 minus003313413 minus003676743minus007341430 minus042776115 minus002785663 minus002552098 minus002384797minus012954567 minus052436133 minus003510447 minus003163354 minus003102749minus005734203 minus026232141 minus001916962 minus001692499 minus001682689

]]]]]]]]]

(32)

Calculating the initial cross Grammian matrix 119882(0) using theproposed algorithm and comparing the result to the exact

solution of the cross Grammian using the error norm we getthe following results

119882(0) =[[[[[[[[[

minus007894318 minus013640657 minus001124158 minus000932742 minus00110635minus025878609 minus056732132 minus004537161 minus003829073 minus004292241minus005767992 minus050330195 0 0 0minus012521917 minus060940965 0 0 0minus004496552 minus031507279 0 0 0

]]]]]]]]]

10038171003817100381710038171003817119882119883 minus 119882(0)100381710038171003817100381710038172 = 0160161520985517

(33)

The cross Grammian matrix is then calculated using theproposed recursive algorithm The error norm for eachiteration is shown in Table 2

It can be seen from Table 2 that the proposed algorithmconverges to the exact solution according to the convergenceresult stated in Theorem 2

43 Natural Gas Hydrogen Reformer In this section weinvestigate the behavior of the proposed algorithm in caseof higher order singularly perturbed systems The linearized10th-ordermathematical model of the gas hydrogen reformerintroduced and studied in [25ndash27] is chosen The systemmatrices are given as follows [25]

119860 =

[[[[[[[[[[[[[[[[[[[[[[[

minus0074 0 0 0 0 0 minus353 10748 0 00 minus1468 minus253 0 0 0 0 0 25582 139110 0 minus156 0 0 0 0 0 0 335860 0 0 minus1245 21263 0 11269 11269 0 00 0 0 0 minus3333 0 0 0 0 00 0 0 0 0 minus3243 32304 32304 0 00 0 0 0 0 3318 minus344 minus341 0 990420 0 0 22197 0 0 minus2532 minus2549 0 325260 0 20354 0 0 0 18309 1214 minus0358 minus3304

00188 0 81642 0 0 0 56043 53994 0 minus1361

]]]]]]]]]]]]]]]]]]]]]]]

119861 = [0 0 0 0 012 0 0 0 0 00 0 0 0 0 01834 0 0 0 0]

119879

119862 = [1 0 0 0 0 0 0 0 0 00 0994 minus0088 0 0 0 0 0 0 0]

(34)

Mathematical Problems in Engineering 7

Table 2 Error norm values for each iteration for the chemical plant

Number of iterations 119894 10038171003817100381710038171003817119882119883 minus 119882(119894)119883 1003817100381710038171003817100381722 6549205248602119890 minus 34 3305462486403183119890 minus 046 1644553914568401119890 minus 058 7961048437784984119890 minus 0710 3791658707506841119890 minus 0812 1790737974576206119890 minus 0914 8421250159940589119890 minus 1116 3944852245039569119890 minus 1218 1781054537942190119890 minus 1320 1746372245754009119890 minus 14

The eigenvalues of the system matrix 119860 are as followsminus66068minus1579minus89137minus12175minus333minus277plusmn0547119894minus1468minus0358 and minus00861 All real parts of those eigenvalues liein the left part of the complex plane hence the systemis asymptotically stable Moreover the system has multipletime scales (slow and fast) since there are three eigenvalueslocated very close to the imaginary axis while the otherseven eigenvalues are located far from that axis The singularperturbation parameter 120576 is chosen as the ratio of the fastestslow eigenvalue to the slowest fast eigenvalue and is equal to120576 = 052 = 1468277

What are supposed to be slowmode eigenvalues obtainedvia (15) minus1468 minus00838 and minus1284495 and what aresupposed to be the fastmode eigenvalues (eigenvalues of1198604)minus0358minus66068minus89144minus13954minus2829plusmn0886 andminus3333

Table 3 Error norm values for each iteration for the gas reformersystem

Number of iterations 119894 10038171003817100381710038171003817119882119883 minus 119882(119894)119883 1003817100381710038171003817100381722 566707478610707585 1326046318575198610 038648979991069112 50958037951316119890 minus 0215 5949048480951119890 minus 0320 6505261253303921119890 minus 0423 6630466299301564119890 minus 0525 7912793405030453119890 minus 0630 7210694103862186119890 minus 0735 3050169378906093119890 minus 0840 8638446874677160119890 minus 10

do not display the slow-fast time scale separation Similarto the chemical plant example in the previous section wecan notice that the slow and fast eigenvalues are not wellseparated into two disjoint groups However since the gasformer system is of higher order it will be a little cumbersometo come up with a permutation matrix that would convert thesystem into its explicit singularly perturbed form Thereforethe algorithm presented in [21 28] and used by [27] to studythe slow and fast dynamics of the gas reformer system isalso considered here to convert the system from an implicitsingularly perturbed form to the explicit singularly perturbedform The algorithm in [21 28] is based on introducing asimilarity transformation 119879 that transforms the general linearsystem in implicit singularly perturbed form into the explicitsingularly perturbed form defined in (9) The similaritytransformation 119879 is given by [27]

119879 =

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

48557 21797 minus03535 17829 11374 minus52659 minus51468 1 0 00 0 0 0 0 0 0 0 1 0

minus03794 07980 minus00771 0 0 52667 05148 0 0 10 0 1 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 00 0 0 0 1 0 0 0 0 00 0 0 0 0 1 0 0 0 00 0 0 0 0 0 1 0 0 00 0 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 0 1

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

(35)

8 Mathematical Problems in Engineering

The new transformed system matrices are defined as

119860SP = 119879119860119879minus1119861SP = 119879119861119862SP = 119862119879minus1

(36)

Thereby they are calculated as

119860SP =

[[[[[[[[[[[[[[[[[[[[[[[

minus03192 55761 minus3138 minus0353 05681 363094 minus0259 minus0050 55156 313790 minus0358 0 20354 0 0 0 1831 12140 minus3304

minus00866 20413 minus12315 minus0125 01544 98473 19252 01901 minus03195 123140 0 0 minus1245 0 0 0 0 0 335860 0 0 0 minus1245 21263 0 11269 11269 00 0 0 0 0 minus3333 0 0 0 00 0 0 0 0 0 minus3243 32304 32304 00 0 0 22197 0 0 3318 minus344 minus341 990420 0 20354 0 22197 0 0 minus2532 minus2549 32526

00188 0 81642 0 minus00057 minus03629 0213 5625 53962 minus1361

]]]]]]]]]]]]]]]]]]]]]]]

119861SP = [136488 0 0 0 0 012 0 0 0 0minus96577 0 09659 0 0 0 01834 0 0 0]

119879

119862SP = [01697 0 minus04636 00243 minus0302 minus19303 113788 1112 minus01697 0463600802 0 10266 00195 minus0143 minus91238 minus11825 minus0115 minus00802 minus10266]

(37)

The matrices 119860SP and 119860 have the same eigenvalues sincethey are preserved under the similarity transformation Theslow mode eigenvalues are minus1468 minus0358 and minus008552while the fast mode eigenvalues are minus660682 minus15789 89137minus12174 minus3333 and minus27697 plusmn 060087 They are clearlyseparated now into two disjoint groups

Using our proposed algorithm to calculate the initial crossGrammian matrix 119882(0) and compare the result to the exactsolution of the algebraic Sylvester equation we get the errornorm as 10038171003817100381710038171003817119882119883 minus 119882(0)100381710038171003817100381710038172 = 53489147248362102 (38)

The cross Grammian matrix is then calculated usingthe proposed recursive algorithm The error norm for eachiteration is shown in Table 3

5 Conclusions

The algorithm was developed to solve the algebraic Sylvesterequation whose solution defines the cross Grammian ofsingularly perturbed linear systems The algorithm is veryefficient defined in terms of reduced-order subproblemscorresponding to slow and fast subsystems and convergesrapidly to the required solution The efficacy of the algorithmis demonstrated on three real physical examples The algo-rithm can be directly applied to singularly perturbed systems

in the explicit standard forms A similarity transformationneeds to be applied to singularly perturbed systems in implicitforms to convert them into their explicit form before theproposed algorithm can be applied to this class of systems

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

References

[1] Z Gajic Multimodel control and estimation of linear stochastic[PhD Dissertation] Michigan State University 1984

[2] T Grodt and Z Gajic ldquoThe recursive reduced-order numericalsolution of the singularly perturbed matrix differential Riccatiequationrdquo IEEE Transactions on Automatic Control vol 33 no8 pp 751ndash754 1988

[3] W Su Z Gajic and X Shen ldquoThe recursive reduced-ordersolution of an open-loop control problem of linear singularlyperturbed systemsrdquo IEEE Transactions on Automatic Controlvol 37 no 2 pp 279ndash281 1992

[4] Z Gajic andM-T LimOptimal Control of Singularly PerturbedLinear Systems and Applications High-Accuracy TechniquesMarcel Dekker New York NY USA 2001

[5] W C Su Z Gajic and X M Shen ldquoThe exact slow-fastdecomposition of the algebraic Ricatti equation of singularly

Mathematical Problems in Engineering 9

perturbed systemsrdquo IEEE Transactions on Automatic Controlvol 37 no 9 pp 1456ndash1459 1992

[6] Z Gajic and X Shen Parallel Algorithms for Optimal Control ofLarge Scale Linear Systems Springer London UK 1993

[7] I Borno andZGajic ldquoParallel algorithms for optimal control ofweakly coupled and singularly perturbed jump linear systemsrdquoAutomatica vol 31 no 7 pp 985ndash988 1995

[8] H Mukaidani H Xu and K Mizukami ldquoNew iterative algo-rithm for algebraic riccati equation related to 119867infin controlproblem of singularly perturbed systemsrdquo IEEE Transactions onAutomatic Control vol 46 no 10 pp 1659ndash1666 2001

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

[10] 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

[11] V Sreeram ldquoRecursive technique for computation of Grami-ansrdquo IEE Proceedings D Control Theory and Applications vol140 no 3 pp 160ndash166 1993

[12] K CWan andV Sreeram ldquoSolution of the bilinearmatrix equa-tion using Astrom-Jury-Agniel algorithmrdquo IEE Proceedings-Dvol 140 pp 21160ndash2166 1993

[13] P Lancaster and M Tismenetsky The Theory of MatricesComputer Science and Applied Mathematics Academic PressOrlando Fla USA 2nd edition 1985

[14] U Baur ldquoLow rank solution of data-sparse Sylvester equationsrdquoNumerical Linear Algebra with Applications vol 15 no 9 pp837ndash851 2008

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

[16] P Benner E S Quintana-Ortı and G Quintana-Ortı ldquoSolvingstable Sylvester equations via rational iterative schemesrdquo Journalof Scientific Computing vol 28 no 1 pp 51ndash83 2006

[17] A J Laub L M Silverman and M Verma ldquoNote on cross-grammians for symmetric realizationsrdquo Proceedings of the IEEEvol 71 no 7 pp 904ndash905 1983

[18] K Fernando and H Nicholson ldquoOn a fundamental property ofthe cross- Gramian matrixrdquo IEEE Transactions on Circuits andSystems vol 31 no 5 pp 504ndash505 1984

[19] K V Fernando and H Nicholson ldquoReachabilty observabilityand minimality of mimo systemsrdquo Proceedings of the IEEE vol72 no 12 pp 1820ndash1821 1984

[20] 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

[21] P Kokotovic H Khalil and J OrsquoReilly Singular PerturbationMethods in Control Analysis and Design Academic PressLondon UK 1986

[22] DG LuenbergerOptimization byVector SpaceMethodsWiley-Interscience New York NY USA 1969

[23] A N Andry Jr J C Chung and E Y Shapiro ldquoModalizedobserversrdquo IEEE Transactions on Automatic Control vol 29 no7 pp 669ndash672 1984

[24] V G Shankar and K Ramar ldquoPole assignment with minimumeigenvalue sensitivity to plant parameter variationsrdquo Interna-tiona Jornal of Control vol 23 no 4 pp 493ndash504 1976

[25] J Pukrushpan A Stefanopoulou and H Peng Control of FuelCell Power Systems Principles Modeling Analysis and FeedbackDesign Springer Berlin Germany 2004

[26] J Pukrushpan A Stefanopoulou S Varigonda J Eborn andC Haugstetter ldquoControl-oriented model of fuel processor forhydrogen generation in fuel cell applicationsrdquoControl Engineer-ing Practice vol 14 no 3 pp 277ndash293 2006

[27] M Skataric and Z Gajic ldquoSlow and fast dynamics of a naturalgas hydrogen reformerrdquo International Journal of HydrogenEnergy vol 38 no 35 pp 15173ndash15179 2013

[28] H K Khalil ldquoTime scale decomposition of linear implicitsingularly perturbed systemsrdquo IEEE Transactions on AutomaticControl vol 29 no 11 pp 1054ndash1056 1984

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Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

Page 2: Research Article Recursive Reduced-Order Algor ithm …downloads.hindawi.com/journals/mpe/2016/2452746.pdf · Research Article Recursive Reduced-Order Algor ithm for Singularly Perturbed

2 Mathematical Problems in Engineering

information about both controllability and observability ofthe system

In this paper a new recursive algorithm is proposed tosolve the algebraic Sylvester equation of linear singularlyperturbed systems whose solution defines the cross Gram-mian matrix The algorithm is obtained in terms of reduced-order algebraic Sylvester equations corresponding to slowand fast subsystems The solutions of full-order algebraicSylvester equations for finding the cross Grammian matrixwere considered in [11 12]

The remainder of the paper is organized as followsSection 2 reviews the controllability observability and thecross Grammian matrices The proposed recursive algorithmis then described in Section 3 In Section 4 several casestudies are considered to demonstrate the performance ofthe proposed algorithm Then the conclusions follow inSection 5

2 The Cross Grammian Matrix

Consider a linear dynamic system

119889119909119889119905 = 119860119909 (119905) + 119861119906 (119905) 119910 (119905) = 119862119909 (119905) + 119863119906 (119905)

(1)

where 119909(119905) isin R119899 are state variables 119906(119905) isin R119898 are controlinputs and 119910(119905) isin R119901 are measured outputs Assume thatsystem (1) is asymptotically stable controllable and observ-able Controllability and observability of the system canbe measured using the controllability and the observabilityGrammians defined respectively as

119882119862 = intinfin0

119890119860119905119861119861119879119890119860119879119905 (2)

119882119874 = intinfin0

119890119860119879119905119862119879119862119890119860119905 (3)

For asymptotically stable controllable and observablesystems Grammians (2) and (3) are positive definite andrepresented the solutions of the algebraic Lyapunov equations

119860119882119862 + 119882119862119860119879 = minus119861119861119879119860119879119882119874 + 119882119874119860 = minus119862119879119862 (4)

Assuming system (1) is square that is the number ofinputs equals the number of outputs 119898 = 119901 the crossGrammian matrix was defined in [10] for single-input single-output (SISO) systems as

119882119883 = intinfin0

119890119860119905119861119862119890119860119905 (5)

and represented by the solution to the algebraic Sylvesterequation

119860119882119883 + 119882119883119860 + 119861119862 = 0 (6)

In this context the Sylvester algebraic equation (6) hasa unique solution if and only if 119860 and minus119860 have distincteigenvalues [13] Several numerical solutions for the Sylvesterequation were proposed in the literature see for example[14ndash16] and the references therein The definition in (5)was extended in [17ndash19] to include multi-input multioutput(MIMO) systems

Furthermore for MIMO symmetric systems the relationbetween controllability and observability on the one handand the cross Grammian on the other hand is given by [20]

1198822119883 = 119882119862119882119874 (7)

3 A Recursive Algorithm for FindingCross Grammians for Singularly PerturbedLinear Systems

The singularly perturbed structure can be obtained by parti-tioning the system matrices in (1) as follows [6 21]

119860 = [[

1198601 119860211986031205761198604120576

]]

119861 = [[

11986111198612120576]]

119862 = [1198621 1198622]

(8)

where 120576 is a small positive singular perturbation parameter119860 119861 and 119862 are constant matrices of appropriate dimensionsBased on the singular perturbation theory [6 21] a singularlyperturbed linear system in the explicit state variable standardform is given by

1198891199091 (119905)119889119905 = 11986011199091 (119905) + 11986021199092 (119905) + 1198611119906 (119905) 1205761198891199092 (119905)119889119905 = 11986031199091 (119905) + 11986041199092 (119905) + 1198612119906 (119905)

119910 (119905) = 11986211199091 (119905) + 11986221199092 (119905) + 119863119906 (119905) (9)

where 1199091(119905) isin R1198991 are the slow state variables and 1199092(119905) isin R1198992

are the fast state variables Assuming that 1198604 is nonsingularthe eigenvalues of matrix 119860 consist of two disjoint groupsone corresponds to the slow subsystem 120582119904(119860) and the othercorresponds to the fast subsystem 120582119891(119860) If the two sub-systems have a mixture of slow and fast eigenvalues then atechnique has to be applied to convert the system into itsstandard singularly perturbed form defined in (9) We willgive examples on this case in Sections 42 and 43

The nature of the cross Grammian matrix 119882119883 defined in(6) corresponding to the system singularly perturbed formdefined in (9) is

119882119883 = [1198821 12057611988221198823 1198824 ] (10)

Mathematical Problems in Engineering 3

Using (8) and (10) in (6) we get the partitioned form of thealgebraic Sylvester equation as follows

11986011198821 + 11986021198823 + 11988211198601 + 11988221198603 + 11986111198621 = 012057611986011198822 + 11986021198824 + 11988211198602 + 11988221198604 + 11986111198622 = 011986031198821 + 11986041198823 + 12057611988231198601 + 11988241198603 + 11986121198621 = 0

12057611986031198822 + 11986041198824 + 12057611988231198602 + 11988241198604 + 11986121198622 = 0(11)

Setting 120576 = 0 we get the following approximate algebraicequations

1198601119882(0)1 + 1198602119882(0)3 + 119882(0)1 1198601 + 119882(0)2 1198603 + 11986111198621 = 01198602119882(0)4 + 119882(0)1 1198602 + 119882(0)2 1198604 + 11986111198622 = 01198603119882(0)1 + 1198604119882(0)3 + 119882(0)4 1198603 + 11986121198621 = 0

1198604119882(0)4 + 119882(0)4 1198604 + 11986121198622 = 0(12)

The solution of equations (12) is given in terms ofthe following reduced-order algebraic Sylvester equationscorresponding to the slow and fast subsystems

1198604119882(0)4 + 119882(0)4 1198604 + 11986121198622 = 01198600119882(0)1 + 119882(0)1 1198600 + 1198660 = 0 (13)

In addition we have from (12)

119882(0)2 = minus (11986111198622 + 1198602119882(0)4 + 119882(0)1 1198602) 119860minus14 119882(0)3 = minus119860minus14 (11986121198621 + 1198603119882(0)1 + 119882(0)4 1198603) (14)

where

1198600 = 1198601 minus 1198602119860minus14 1198603 (15)

1198660 = minus1198602119860minus14 (11986121198621 + 119882(0)4 1198603)minus (1198602119882(0)4 + 11986111198622) 119860minus14 1198603 + 11986111198621

(16)

To find a unique solution of (13) we impose the followingassumption

Assumption 1 Matrices 1198600 and 1198604 are asymptotically stableIn consequence unique solutions of (13)-(14) exist

Defining the approximation error as

1198821 = 119882(0)1 + 12057611986411198822 = 119882(0)2 + 12057611986421198823 = 119882(0)3 + 12057611986431198824 = 119882(0)4 + 1205761198644

(17)

and subtracting (12) from (11) we get the following errorequations after some algebra

11986041198644 + 11986441198604= minus1198603 (119882(0)2 + 1205761198642) minus (119882(0)3 + 1205761198643) 1198602

11986031198641 + 11986041198643 + 11986441198603 = minus (119882(0)3 + 1205761198643) 119860111986021198644 + 11986411198602 + 11986421198604 = minus1198601 (119882(0)2 + 1205761198642) 11986011198641 + 11986021198643 + 11986411198601 + 11986421198603 = 0

(18)

From the first equation in (18) we can observe thatthe unknown errors 1198642 and 1198643 are multiplied by a smallparameter 120576 A similar situation is in the second and thethird equations of (18) Therefore we propose the followingalgorithm for solving error equations (18)

31 The Proposed Algorithm Start with 119864(0)2 = 0 and 119864(0)3 = 0and recursively evaluate

1198604119864(119894+1)4 + 119864(119894+1)4 1198604= minus1198603 (119882(0)2 + 120576119864(119894)2 ) minus (119882(0)3 + 120576119864(119894)3 ) 1198602

1198600119864(119894+1)1 + 119864(119894+1)1 1198600= 1198602119860minus14 (119882(0)3 + 120576119864(119894)3 ) 1198601

+ 1198601 (119882(0)2 + 120576119864(119894)2 ) 119860minus14 1198603 + 1198602119860minus14 119864(119894+1)4 1198603+ 1198602119864(119894+1)4 119860minus14 1198603

119864(119894+1)2= minus (1198601 (119882(0)2 + 120576119864(119894)2 ) + 1198602119864(119894+1)4 + 119864(119894+1)1 1198602) 119860minus14

119864(119894+1)3= minus119860minus14 ((119882(0)3 + 120576119864(119894)3 ) 1198601 + 1198603119864(119894+1)1 + 119864(119894+1)4 1198603)

for 119894 = 0 1 2

(19)

Theorem 2 Assuming that matrices 1198600 and 1198604 are asymp-totically stable algorithm (19) converges to the exact solutionof (18) with a rate of convergence 119874(120576) that is

10038171003817100381710038171003817119864(119894+1)119895 minus 119864(119894)119895 10038171003817100381710038171003817 = 119874 (120576) 10038171003817100381710038171003817119864(119894)119895 minus 11986411989510038171003817100381710038171003817 = 119874 (120576119894) (20)

for 119895 = 1 2 3 4 and 119894 = 1 2 Therefore the exact solution 119882119883 can be obtained with

an accuracy of 119874(120576119894) after performing 119894 iterations on theproposed algorithm (19) as follows

119882(119894)119895 = 119882119895 + 120576119864(119894)119895 for 119895 = 1 2 3 4 119894 = 1 2 (21)

4 Mathematical Problems in Engineering

Proof Using Assumption 1 that is 1198600 and 1198604 are asymptoti-cally stable it can be shown that (19) represents a contractionmapping [22] that is

10038171003817100381710038171003817119864(119894)119895 minus 11986411989510038171003817100381710038171003817 = 119874 (120576119894) for 119895 = 1 2 3 4 119894 = 1 2 (22)

Formula (22) will be also valid if

119864(119894+1)2 = minus (1198601 (119882(0)2 + 120576119864(119894)2 ) + 1198602119864(119894)4 + 119864(119894)1 1198602) 119860minus14 119864(119894+1)3 = minus119860minus14 ((119882(0)3 + 120576119864(119894)3 ) 1198601 + 1198603119864(119894)1 + 119864(119894)4 1198603)

for 119894 = 0 1 2 119864(0)1 = 0 119864(0)4 = 0(23)

Formula (22) implies that algorithm (19) is convergent Using119864(infin)119895 for 119895 = 1 2 3 4 in (19) and comparing it to (18) it canbe seen that algorithm (19) converges to the unique solutionof (18)

4 Case Studies

Three case studies are considered to demonstrate the pro-posed algorithm a fourth-order aircraft example whosemathematical model is in the explicit singularly perturbedform defined in (9) in which with accuracy of 119874(120576) theslow eigenvalues are all contained in the approximate slowsubsystem represented by 1198600 and all fast eigenvalues arecontained in the approximate fast subsystem representedby 1198604 a fifth-order chemical plant model given in implicitsingularly perturbed form (it has two slow and three fasteigenvalues but the state variables have to be reordered toachieve explicit singularly perturbed form defined in (9)) a

tenth-order hydrogen gas reformer used to provide hydrogento a fuel cell fromhydrogen rich fuels (natural gasmethanol)

41 L-1011 Aircraft Here we consider the lateral axis equa-tions of the rigid body model of L-1011 aircraft at cruisecondition [23]The state variables are the bank angle roll rateyaw rate and sideslip angle which are represented in the statevector 119909(119905) = [1199091 1199092 1199093 1199094]119879 in the same order The inputvector consists of two variables the rudder deflection 120575119903 andthe aileron deflection 120575119886 and is given as 119906(119905) = [120575119903 120575119886]119879 Thesystem matrices are given as

119860 =[[[[[[

0 1 0 00 minus189 039 minus5550 minus0034 minus298 243

0034 minus00011 minus099 minus021

]]]]]]

119861 =[[[[[[

0 0036 minus16

minus095 minus0032003 0

]]]]]]

119862 = [1 0 0 00 1 0 0]

(24)

The eigenvalues of thematrix119860 areminus01016minus14811plusmn06292119894and minus20162The system is asymptotically stable (all eigenval-ues are in the left half plane) controllable and observableMoreover there is only one slow mode with eigenvalueminus00899 and there are three fast modes with eigenvaluesminus14891 plusmn 07686119894 and minus21017 The singular perturbationparameter 120576 = 007 which is the ratio between the fastestslow eigenvalue and the slowest fast eigenvalue Solving thealgebraic Sylvesterrsquos equation (6) the cross Grammian matrixcan be obtained as follows

119882119883 =[[[[[[

minus377168467243 minus225320146563 minus460285451131 1268652326564minus043134179103 minus062940797540 minus027809983914 1332572206696minus014313989569 000048741612 minus001060057667 minus002122054103020967036557 006408641105 minus003861451004 000250243909

]]]]]]

(25)

Using the proposed algorithm the initial cross Grammi-an matrix 119882(0) (first-order approximate solution) is obtained

as follows

119882(0) =[[[[[[

minus398315817315 minus225882783809 minus446229986109 1232553354098minus041906814039 minus042418654922 minus021443252362 0883962065453minus014908955443 minus000284907928 000235249833 minus0005866460180202330975578 0001553980682 0000836843461 minus000419856136

]]]]]]

(26)

Mathematical Problems in Engineering 5

Table 1 Error norm values for each iteration for L-1011 aircraftsystem

Number of iterations 119894 10038171003817100381710038171003817119882119883 minus 119882(119894)119883 1003817100381710038171003817100381722 7503328625983908119890 minus 043 2611125417476592119890 minus 054 9085062505535487119890 minus 075 3161097999142058119890 minus 086 1099918110580769119890 minus 097 3830668858621788119890 minus 118 1368874658220410119890 minus 129 9598118726733237119890 minus 1410 6474407682370149119890 minus 14

Comparing the exact solution 119882119883 to the first-orderapproximate solution of the cross Grammian matrix bycalculating the error norm we get

10038171003817100381710038171003817119882119883 minus 119882(0)100381710038171003817100381710038172 = 0624920763816596 (27)

Then the cross Grammian matrix is calculated using the pro-posed recursive algorithm The error norm at each iterationis shown in Table 1 By taking the error norm it can be seenthat the algorithm converges rapidly to the exact solution

42 Chemical Plant In this section the linearized chemicalplant considered in [24] is chosen to explain the behavior ofthe proposed algorithm when the linear singularly perturbedsystem is not in the explicit standard form (9) The systemmatrices are given as follows

119860 =[[[[[[[[[

minus01094 00628 0 0 01306 minus2136 09807 0 0

0 1595 minus3149 1547 00 00355 2632 minus4257 18550 00027 0 01636 minus01625

]]]]]]]]]

119861 =[[[[[[[[[

0 000638 000838 minus0139601004 minus020600063 minus00128

]]]]]]]]]

119862 = [1 0 0 0 00 0 0 0 1]

(28)

The eigenvalues of matrix 119860 are minus59822 minus28408 minus08954minus00774 and minus00141 which indicates that this system hastwo slow modes (eigenvalues) The singular perturbationparameter 120576 is chosen as the ratio of the fastest sloweigenvalue to the slowest fast eigenvalue and is equal to

120576 = 0086 = 0077408954 Introducing the followingpermutation matrix that exchanges the second row of matrix119860 with its fifth row that is

119875 =[[[[[[[[[

1 0 0 0 00 0 0 0 10 0 1 0 00 0 0 1 00 1 0 0 0

]]]]]]]]]

(29)

the explicit singularly perturbed form of the system matricescan be obtained as follows

119860SP = 119875119860119875119861SP = 119875119861119862SP = 119862119875

(30)

Thereby they are calculated as

119860SP

=[[[[[[[[[[

minus01094 0 0 0 006280 minus01625 0 01636 000270 0 minus3149 1547 159500 18550 2632 minus4257 00355

13060 0 09807 0 minus21320

]]]]]]]]]]

119861SP =[[[[[[[[[

0 000063 minus0012800838 minus0139601004 minus020600638 0

]]]]]]]]]

119862SP = [1 0 0 0 00 1 0 0 0]

(31)

Matrices 119860SP and 119860 have the same eigenvalues and the samenumber of slow and fast modes However the slow and fasteigenvalues are now correctly separated into two disjointgroups The slow approximate subsystem (see (15)) has theeigenvalues minus00788 and minus00160 and the fast approximatesubsystem has the fast eigenvalues minus59521 27925 andminus07933 (eigenvalues of 1198604)

Solving the algebraic Sylvesterrsquos equation (6) the crossGrammian matrix can be obtained as follows

6 Mathematical Problems in Engineering

119882119883 =[[[[[[[[[

minus007530866 minus011561658 minus000985227 minus000806596 minus000985946minus023194836 minus049463876 minus003897379 minus003313413 minus003676743minus007341430 minus042776115 minus002785663 minus002552098 minus002384797minus012954567 minus052436133 minus003510447 minus003163354 minus003102749minus005734203 minus026232141 minus001916962 minus001692499 minus001682689

]]]]]]]]]

(32)

Calculating the initial cross Grammian matrix 119882(0) using theproposed algorithm and comparing the result to the exact

solution of the cross Grammian using the error norm we getthe following results

119882(0) =[[[[[[[[[

minus007894318 minus013640657 minus001124158 minus000932742 minus00110635minus025878609 minus056732132 minus004537161 minus003829073 minus004292241minus005767992 minus050330195 0 0 0minus012521917 minus060940965 0 0 0minus004496552 minus031507279 0 0 0

]]]]]]]]]

10038171003817100381710038171003817119882119883 minus 119882(0)100381710038171003817100381710038172 = 0160161520985517

(33)

The cross Grammian matrix is then calculated using theproposed recursive algorithm The error norm for eachiteration is shown in Table 2

It can be seen from Table 2 that the proposed algorithmconverges to the exact solution according to the convergenceresult stated in Theorem 2

43 Natural Gas Hydrogen Reformer In this section weinvestigate the behavior of the proposed algorithm in caseof higher order singularly perturbed systems The linearized10th-ordermathematical model of the gas hydrogen reformerintroduced and studied in [25ndash27] is chosen The systemmatrices are given as follows [25]

119860 =

[[[[[[[[[[[[[[[[[[[[[[[

minus0074 0 0 0 0 0 minus353 10748 0 00 minus1468 minus253 0 0 0 0 0 25582 139110 0 minus156 0 0 0 0 0 0 335860 0 0 minus1245 21263 0 11269 11269 0 00 0 0 0 minus3333 0 0 0 0 00 0 0 0 0 minus3243 32304 32304 0 00 0 0 0 0 3318 minus344 minus341 0 990420 0 0 22197 0 0 minus2532 minus2549 0 325260 0 20354 0 0 0 18309 1214 minus0358 minus3304

00188 0 81642 0 0 0 56043 53994 0 minus1361

]]]]]]]]]]]]]]]]]]]]]]]

119861 = [0 0 0 0 012 0 0 0 0 00 0 0 0 0 01834 0 0 0 0]

119879

119862 = [1 0 0 0 0 0 0 0 0 00 0994 minus0088 0 0 0 0 0 0 0]

(34)

Mathematical Problems in Engineering 7

Table 2 Error norm values for each iteration for the chemical plant

Number of iterations 119894 10038171003817100381710038171003817119882119883 minus 119882(119894)119883 1003817100381710038171003817100381722 6549205248602119890 minus 34 3305462486403183119890 minus 046 1644553914568401119890 minus 058 7961048437784984119890 minus 0710 3791658707506841119890 minus 0812 1790737974576206119890 minus 0914 8421250159940589119890 minus 1116 3944852245039569119890 minus 1218 1781054537942190119890 minus 1320 1746372245754009119890 minus 14

The eigenvalues of the system matrix 119860 are as followsminus66068minus1579minus89137minus12175minus333minus277plusmn0547119894minus1468minus0358 and minus00861 All real parts of those eigenvalues liein the left part of the complex plane hence the systemis asymptotically stable Moreover the system has multipletime scales (slow and fast) since there are three eigenvalueslocated very close to the imaginary axis while the otherseven eigenvalues are located far from that axis The singularperturbation parameter 120576 is chosen as the ratio of the fastestslow eigenvalue to the slowest fast eigenvalue and is equal to120576 = 052 = 1468277

What are supposed to be slowmode eigenvalues obtainedvia (15) minus1468 minus00838 and minus1284495 and what aresupposed to be the fastmode eigenvalues (eigenvalues of1198604)minus0358minus66068minus89144minus13954minus2829plusmn0886 andminus3333

Table 3 Error norm values for each iteration for the gas reformersystem

Number of iterations 119894 10038171003817100381710038171003817119882119883 minus 119882(119894)119883 1003817100381710038171003817100381722 566707478610707585 1326046318575198610 038648979991069112 50958037951316119890 minus 0215 5949048480951119890 minus 0320 6505261253303921119890 minus 0423 6630466299301564119890 minus 0525 7912793405030453119890 minus 0630 7210694103862186119890 minus 0735 3050169378906093119890 minus 0840 8638446874677160119890 minus 10

do not display the slow-fast time scale separation Similarto the chemical plant example in the previous section wecan notice that the slow and fast eigenvalues are not wellseparated into two disjoint groups However since the gasformer system is of higher order it will be a little cumbersometo come up with a permutation matrix that would convert thesystem into its explicit singularly perturbed form Thereforethe algorithm presented in [21 28] and used by [27] to studythe slow and fast dynamics of the gas reformer system isalso considered here to convert the system from an implicitsingularly perturbed form to the explicit singularly perturbedform The algorithm in [21 28] is based on introducing asimilarity transformation 119879 that transforms the general linearsystem in implicit singularly perturbed form into the explicitsingularly perturbed form defined in (9) The similaritytransformation 119879 is given by [27]

119879 =

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

48557 21797 minus03535 17829 11374 minus52659 minus51468 1 0 00 0 0 0 0 0 0 0 1 0

minus03794 07980 minus00771 0 0 52667 05148 0 0 10 0 1 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 00 0 0 0 1 0 0 0 0 00 0 0 0 0 1 0 0 0 00 0 0 0 0 0 1 0 0 00 0 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 0 1

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

(35)

8 Mathematical Problems in Engineering

The new transformed system matrices are defined as

119860SP = 119879119860119879minus1119861SP = 119879119861119862SP = 119862119879minus1

(36)

Thereby they are calculated as

119860SP =

[[[[[[[[[[[[[[[[[[[[[[[

minus03192 55761 minus3138 minus0353 05681 363094 minus0259 minus0050 55156 313790 minus0358 0 20354 0 0 0 1831 12140 minus3304

minus00866 20413 minus12315 minus0125 01544 98473 19252 01901 minus03195 123140 0 0 minus1245 0 0 0 0 0 335860 0 0 0 minus1245 21263 0 11269 11269 00 0 0 0 0 minus3333 0 0 0 00 0 0 0 0 0 minus3243 32304 32304 00 0 0 22197 0 0 3318 minus344 minus341 990420 0 20354 0 22197 0 0 minus2532 minus2549 32526

00188 0 81642 0 minus00057 minus03629 0213 5625 53962 minus1361

]]]]]]]]]]]]]]]]]]]]]]]

119861SP = [136488 0 0 0 0 012 0 0 0 0minus96577 0 09659 0 0 0 01834 0 0 0]

119879

119862SP = [01697 0 minus04636 00243 minus0302 minus19303 113788 1112 minus01697 0463600802 0 10266 00195 minus0143 minus91238 minus11825 minus0115 minus00802 minus10266]

(37)

The matrices 119860SP and 119860 have the same eigenvalues sincethey are preserved under the similarity transformation Theslow mode eigenvalues are minus1468 minus0358 and minus008552while the fast mode eigenvalues are minus660682 minus15789 89137minus12174 minus3333 and minus27697 plusmn 060087 They are clearlyseparated now into two disjoint groups

Using our proposed algorithm to calculate the initial crossGrammian matrix 119882(0) and compare the result to the exactsolution of the algebraic Sylvester equation we get the errornorm as 10038171003817100381710038171003817119882119883 minus 119882(0)100381710038171003817100381710038172 = 53489147248362102 (38)

The cross Grammian matrix is then calculated usingthe proposed recursive algorithm The error norm for eachiteration is shown in Table 3

5 Conclusions

The algorithm was developed to solve the algebraic Sylvesterequation whose solution defines the cross Grammian ofsingularly perturbed linear systems The algorithm is veryefficient defined in terms of reduced-order subproblemscorresponding to slow and fast subsystems and convergesrapidly to the required solution The efficacy of the algorithmis demonstrated on three real physical examples The algo-rithm can be directly applied to singularly perturbed systems

in the explicit standard forms A similarity transformationneeds to be applied to singularly perturbed systems in implicitforms to convert them into their explicit form before theproposed algorithm can be applied to this class of systems

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

References

[1] Z Gajic Multimodel control and estimation of linear stochastic[PhD Dissertation] Michigan State University 1984

[2] T Grodt and Z Gajic ldquoThe recursive reduced-order numericalsolution of the singularly perturbed matrix differential Riccatiequationrdquo IEEE Transactions on Automatic Control vol 33 no8 pp 751ndash754 1988

[3] W Su Z Gajic and X Shen ldquoThe recursive reduced-ordersolution of an open-loop control problem of linear singularlyperturbed systemsrdquo IEEE Transactions on Automatic Controlvol 37 no 2 pp 279ndash281 1992

[4] Z Gajic andM-T LimOptimal Control of Singularly PerturbedLinear Systems and Applications High-Accuracy TechniquesMarcel Dekker New York NY USA 2001

[5] W C Su Z Gajic and X M Shen ldquoThe exact slow-fastdecomposition of the algebraic Ricatti equation of singularly

Mathematical Problems in Engineering 9

perturbed systemsrdquo IEEE Transactions on Automatic Controlvol 37 no 9 pp 1456ndash1459 1992

[6] Z Gajic and X Shen Parallel Algorithms for Optimal Control ofLarge Scale Linear Systems Springer London UK 1993

[7] I Borno andZGajic ldquoParallel algorithms for optimal control ofweakly coupled and singularly perturbed jump linear systemsrdquoAutomatica vol 31 no 7 pp 985ndash988 1995

[8] H Mukaidani H Xu and K Mizukami ldquoNew iterative algo-rithm for algebraic riccati equation related to 119867infin controlproblem of singularly perturbed systemsrdquo IEEE Transactions onAutomatic Control vol 46 no 10 pp 1659ndash1666 2001

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

[10] 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

[11] V Sreeram ldquoRecursive technique for computation of Grami-ansrdquo IEE Proceedings D Control Theory and Applications vol140 no 3 pp 160ndash166 1993

[12] K CWan andV Sreeram ldquoSolution of the bilinearmatrix equa-tion using Astrom-Jury-Agniel algorithmrdquo IEE Proceedings-Dvol 140 pp 21160ndash2166 1993

[13] P Lancaster and M Tismenetsky The Theory of MatricesComputer Science and Applied Mathematics Academic PressOrlando Fla USA 2nd edition 1985

[14] U Baur ldquoLow rank solution of data-sparse Sylvester equationsrdquoNumerical Linear Algebra with Applications vol 15 no 9 pp837ndash851 2008

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

[16] P Benner E S Quintana-Ortı and G Quintana-Ortı ldquoSolvingstable Sylvester equations via rational iterative schemesrdquo Journalof Scientific Computing vol 28 no 1 pp 51ndash83 2006

[17] A J Laub L M Silverman and M Verma ldquoNote on cross-grammians for symmetric realizationsrdquo Proceedings of the IEEEvol 71 no 7 pp 904ndash905 1983

[18] K Fernando and H Nicholson ldquoOn a fundamental property ofthe cross- Gramian matrixrdquo IEEE Transactions on Circuits andSystems vol 31 no 5 pp 504ndash505 1984

[19] K V Fernando and H Nicholson ldquoReachabilty observabilityand minimality of mimo systemsrdquo Proceedings of the IEEE vol72 no 12 pp 1820ndash1821 1984

[20] 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

[21] P Kokotovic H Khalil and J OrsquoReilly Singular PerturbationMethods in Control Analysis and Design Academic PressLondon UK 1986

[22] DG LuenbergerOptimization byVector SpaceMethodsWiley-Interscience New York NY USA 1969

[23] A N Andry Jr J C Chung and E Y Shapiro ldquoModalizedobserversrdquo IEEE Transactions on Automatic Control vol 29 no7 pp 669ndash672 1984

[24] V G Shankar and K Ramar ldquoPole assignment with minimumeigenvalue sensitivity to plant parameter variationsrdquo Interna-tiona Jornal of Control vol 23 no 4 pp 493ndash504 1976

[25] J Pukrushpan A Stefanopoulou and H Peng Control of FuelCell Power Systems Principles Modeling Analysis and FeedbackDesign Springer Berlin Germany 2004

[26] J Pukrushpan A Stefanopoulou S Varigonda J Eborn andC Haugstetter ldquoControl-oriented model of fuel processor forhydrogen generation in fuel cell applicationsrdquoControl Engineer-ing Practice vol 14 no 3 pp 277ndash293 2006

[27] M Skataric and Z Gajic ldquoSlow and fast dynamics of a naturalgas hydrogen reformerrdquo International Journal of HydrogenEnergy vol 38 no 35 pp 15173ndash15179 2013

[28] H K Khalil ldquoTime scale decomposition of linear implicitsingularly perturbed systemsrdquo IEEE Transactions on AutomaticControl vol 29 no 11 pp 1054ndash1056 1984

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Algebra

Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 3: Research Article Recursive Reduced-Order Algor ithm …downloads.hindawi.com/journals/mpe/2016/2452746.pdf · Research Article Recursive Reduced-Order Algor ithm for Singularly Perturbed

Mathematical Problems in Engineering 3

Using (8) and (10) in (6) we get the partitioned form of thealgebraic Sylvester equation as follows

11986011198821 + 11986021198823 + 11988211198601 + 11988221198603 + 11986111198621 = 012057611986011198822 + 11986021198824 + 11988211198602 + 11988221198604 + 11986111198622 = 011986031198821 + 11986041198823 + 12057611988231198601 + 11988241198603 + 11986121198621 = 0

12057611986031198822 + 11986041198824 + 12057611988231198602 + 11988241198604 + 11986121198622 = 0(11)

Setting 120576 = 0 we get the following approximate algebraicequations

1198601119882(0)1 + 1198602119882(0)3 + 119882(0)1 1198601 + 119882(0)2 1198603 + 11986111198621 = 01198602119882(0)4 + 119882(0)1 1198602 + 119882(0)2 1198604 + 11986111198622 = 01198603119882(0)1 + 1198604119882(0)3 + 119882(0)4 1198603 + 11986121198621 = 0

1198604119882(0)4 + 119882(0)4 1198604 + 11986121198622 = 0(12)

The solution of equations (12) is given in terms ofthe following reduced-order algebraic Sylvester equationscorresponding to the slow and fast subsystems

1198604119882(0)4 + 119882(0)4 1198604 + 11986121198622 = 01198600119882(0)1 + 119882(0)1 1198600 + 1198660 = 0 (13)

In addition we have from (12)

119882(0)2 = minus (11986111198622 + 1198602119882(0)4 + 119882(0)1 1198602) 119860minus14 119882(0)3 = minus119860minus14 (11986121198621 + 1198603119882(0)1 + 119882(0)4 1198603) (14)

where

1198600 = 1198601 minus 1198602119860minus14 1198603 (15)

1198660 = minus1198602119860minus14 (11986121198621 + 119882(0)4 1198603)minus (1198602119882(0)4 + 11986111198622) 119860minus14 1198603 + 11986111198621

(16)

To find a unique solution of (13) we impose the followingassumption

Assumption 1 Matrices 1198600 and 1198604 are asymptotically stableIn consequence unique solutions of (13)-(14) exist

Defining the approximation error as

1198821 = 119882(0)1 + 12057611986411198822 = 119882(0)2 + 12057611986421198823 = 119882(0)3 + 12057611986431198824 = 119882(0)4 + 1205761198644

(17)

and subtracting (12) from (11) we get the following errorequations after some algebra

11986041198644 + 11986441198604= minus1198603 (119882(0)2 + 1205761198642) minus (119882(0)3 + 1205761198643) 1198602

11986031198641 + 11986041198643 + 11986441198603 = minus (119882(0)3 + 1205761198643) 119860111986021198644 + 11986411198602 + 11986421198604 = minus1198601 (119882(0)2 + 1205761198642) 11986011198641 + 11986021198643 + 11986411198601 + 11986421198603 = 0

(18)

From the first equation in (18) we can observe thatthe unknown errors 1198642 and 1198643 are multiplied by a smallparameter 120576 A similar situation is in the second and thethird equations of (18) Therefore we propose the followingalgorithm for solving error equations (18)

31 The Proposed Algorithm Start with 119864(0)2 = 0 and 119864(0)3 = 0and recursively evaluate

1198604119864(119894+1)4 + 119864(119894+1)4 1198604= minus1198603 (119882(0)2 + 120576119864(119894)2 ) minus (119882(0)3 + 120576119864(119894)3 ) 1198602

1198600119864(119894+1)1 + 119864(119894+1)1 1198600= 1198602119860minus14 (119882(0)3 + 120576119864(119894)3 ) 1198601

+ 1198601 (119882(0)2 + 120576119864(119894)2 ) 119860minus14 1198603 + 1198602119860minus14 119864(119894+1)4 1198603+ 1198602119864(119894+1)4 119860minus14 1198603

119864(119894+1)2= minus (1198601 (119882(0)2 + 120576119864(119894)2 ) + 1198602119864(119894+1)4 + 119864(119894+1)1 1198602) 119860minus14

119864(119894+1)3= minus119860minus14 ((119882(0)3 + 120576119864(119894)3 ) 1198601 + 1198603119864(119894+1)1 + 119864(119894+1)4 1198603)

for 119894 = 0 1 2

(19)

Theorem 2 Assuming that matrices 1198600 and 1198604 are asymp-totically stable algorithm (19) converges to the exact solutionof (18) with a rate of convergence 119874(120576) that is

10038171003817100381710038171003817119864(119894+1)119895 minus 119864(119894)119895 10038171003817100381710038171003817 = 119874 (120576) 10038171003817100381710038171003817119864(119894)119895 minus 11986411989510038171003817100381710038171003817 = 119874 (120576119894) (20)

for 119895 = 1 2 3 4 and 119894 = 1 2 Therefore the exact solution 119882119883 can be obtained with

an accuracy of 119874(120576119894) after performing 119894 iterations on theproposed algorithm (19) as follows

119882(119894)119895 = 119882119895 + 120576119864(119894)119895 for 119895 = 1 2 3 4 119894 = 1 2 (21)

4 Mathematical Problems in Engineering

Proof Using Assumption 1 that is 1198600 and 1198604 are asymptoti-cally stable it can be shown that (19) represents a contractionmapping [22] that is

10038171003817100381710038171003817119864(119894)119895 minus 11986411989510038171003817100381710038171003817 = 119874 (120576119894) for 119895 = 1 2 3 4 119894 = 1 2 (22)

Formula (22) will be also valid if

119864(119894+1)2 = minus (1198601 (119882(0)2 + 120576119864(119894)2 ) + 1198602119864(119894)4 + 119864(119894)1 1198602) 119860minus14 119864(119894+1)3 = minus119860minus14 ((119882(0)3 + 120576119864(119894)3 ) 1198601 + 1198603119864(119894)1 + 119864(119894)4 1198603)

for 119894 = 0 1 2 119864(0)1 = 0 119864(0)4 = 0(23)

Formula (22) implies that algorithm (19) is convergent Using119864(infin)119895 for 119895 = 1 2 3 4 in (19) and comparing it to (18) it canbe seen that algorithm (19) converges to the unique solutionof (18)

4 Case Studies

Three case studies are considered to demonstrate the pro-posed algorithm a fourth-order aircraft example whosemathematical model is in the explicit singularly perturbedform defined in (9) in which with accuracy of 119874(120576) theslow eigenvalues are all contained in the approximate slowsubsystem represented by 1198600 and all fast eigenvalues arecontained in the approximate fast subsystem representedby 1198604 a fifth-order chemical plant model given in implicitsingularly perturbed form (it has two slow and three fasteigenvalues but the state variables have to be reordered toachieve explicit singularly perturbed form defined in (9)) a

tenth-order hydrogen gas reformer used to provide hydrogento a fuel cell fromhydrogen rich fuels (natural gasmethanol)

41 L-1011 Aircraft Here we consider the lateral axis equa-tions of the rigid body model of L-1011 aircraft at cruisecondition [23]The state variables are the bank angle roll rateyaw rate and sideslip angle which are represented in the statevector 119909(119905) = [1199091 1199092 1199093 1199094]119879 in the same order The inputvector consists of two variables the rudder deflection 120575119903 andthe aileron deflection 120575119886 and is given as 119906(119905) = [120575119903 120575119886]119879 Thesystem matrices are given as

119860 =[[[[[[

0 1 0 00 minus189 039 minus5550 minus0034 minus298 243

0034 minus00011 minus099 minus021

]]]]]]

119861 =[[[[[[

0 0036 minus16

minus095 minus0032003 0

]]]]]]

119862 = [1 0 0 00 1 0 0]

(24)

The eigenvalues of thematrix119860 areminus01016minus14811plusmn06292119894and minus20162The system is asymptotically stable (all eigenval-ues are in the left half plane) controllable and observableMoreover there is only one slow mode with eigenvalueminus00899 and there are three fast modes with eigenvaluesminus14891 plusmn 07686119894 and minus21017 The singular perturbationparameter 120576 = 007 which is the ratio between the fastestslow eigenvalue and the slowest fast eigenvalue Solving thealgebraic Sylvesterrsquos equation (6) the cross Grammian matrixcan be obtained as follows

119882119883 =[[[[[[

minus377168467243 minus225320146563 minus460285451131 1268652326564minus043134179103 minus062940797540 minus027809983914 1332572206696minus014313989569 000048741612 minus001060057667 minus002122054103020967036557 006408641105 minus003861451004 000250243909

]]]]]]

(25)

Using the proposed algorithm the initial cross Grammi-an matrix 119882(0) (first-order approximate solution) is obtained

as follows

119882(0) =[[[[[[

minus398315817315 minus225882783809 minus446229986109 1232553354098minus041906814039 minus042418654922 minus021443252362 0883962065453minus014908955443 minus000284907928 000235249833 minus0005866460180202330975578 0001553980682 0000836843461 minus000419856136

]]]]]]

(26)

Mathematical Problems in Engineering 5

Table 1 Error norm values for each iteration for L-1011 aircraftsystem

Number of iterations 119894 10038171003817100381710038171003817119882119883 minus 119882(119894)119883 1003817100381710038171003817100381722 7503328625983908119890 minus 043 2611125417476592119890 minus 054 9085062505535487119890 minus 075 3161097999142058119890 minus 086 1099918110580769119890 minus 097 3830668858621788119890 minus 118 1368874658220410119890 minus 129 9598118726733237119890 minus 1410 6474407682370149119890 minus 14

Comparing the exact solution 119882119883 to the first-orderapproximate solution of the cross Grammian matrix bycalculating the error norm we get

10038171003817100381710038171003817119882119883 minus 119882(0)100381710038171003817100381710038172 = 0624920763816596 (27)

Then the cross Grammian matrix is calculated using the pro-posed recursive algorithm The error norm at each iterationis shown in Table 1 By taking the error norm it can be seenthat the algorithm converges rapidly to the exact solution

42 Chemical Plant In this section the linearized chemicalplant considered in [24] is chosen to explain the behavior ofthe proposed algorithm when the linear singularly perturbedsystem is not in the explicit standard form (9) The systemmatrices are given as follows

119860 =[[[[[[[[[

minus01094 00628 0 0 01306 minus2136 09807 0 0

0 1595 minus3149 1547 00 00355 2632 minus4257 18550 00027 0 01636 minus01625

]]]]]]]]]

119861 =[[[[[[[[[

0 000638 000838 minus0139601004 minus020600063 minus00128

]]]]]]]]]

119862 = [1 0 0 0 00 0 0 0 1]

(28)

The eigenvalues of matrix 119860 are minus59822 minus28408 minus08954minus00774 and minus00141 which indicates that this system hastwo slow modes (eigenvalues) The singular perturbationparameter 120576 is chosen as the ratio of the fastest sloweigenvalue to the slowest fast eigenvalue and is equal to

120576 = 0086 = 0077408954 Introducing the followingpermutation matrix that exchanges the second row of matrix119860 with its fifth row that is

119875 =[[[[[[[[[

1 0 0 0 00 0 0 0 10 0 1 0 00 0 0 1 00 1 0 0 0

]]]]]]]]]

(29)

the explicit singularly perturbed form of the system matricescan be obtained as follows

119860SP = 119875119860119875119861SP = 119875119861119862SP = 119862119875

(30)

Thereby they are calculated as

119860SP

=[[[[[[[[[[

minus01094 0 0 0 006280 minus01625 0 01636 000270 0 minus3149 1547 159500 18550 2632 minus4257 00355

13060 0 09807 0 minus21320

]]]]]]]]]]

119861SP =[[[[[[[[[

0 000063 minus0012800838 minus0139601004 minus020600638 0

]]]]]]]]]

119862SP = [1 0 0 0 00 1 0 0 0]

(31)

Matrices 119860SP and 119860 have the same eigenvalues and the samenumber of slow and fast modes However the slow and fasteigenvalues are now correctly separated into two disjointgroups The slow approximate subsystem (see (15)) has theeigenvalues minus00788 and minus00160 and the fast approximatesubsystem has the fast eigenvalues minus59521 27925 andminus07933 (eigenvalues of 1198604)

Solving the algebraic Sylvesterrsquos equation (6) the crossGrammian matrix can be obtained as follows

6 Mathematical Problems in Engineering

119882119883 =[[[[[[[[[

minus007530866 minus011561658 minus000985227 minus000806596 minus000985946minus023194836 minus049463876 minus003897379 minus003313413 minus003676743minus007341430 minus042776115 minus002785663 minus002552098 minus002384797minus012954567 minus052436133 minus003510447 minus003163354 minus003102749minus005734203 minus026232141 minus001916962 minus001692499 minus001682689

]]]]]]]]]

(32)

Calculating the initial cross Grammian matrix 119882(0) using theproposed algorithm and comparing the result to the exact

solution of the cross Grammian using the error norm we getthe following results

119882(0) =[[[[[[[[[

minus007894318 minus013640657 minus001124158 minus000932742 minus00110635minus025878609 minus056732132 minus004537161 minus003829073 minus004292241minus005767992 minus050330195 0 0 0minus012521917 minus060940965 0 0 0minus004496552 minus031507279 0 0 0

]]]]]]]]]

10038171003817100381710038171003817119882119883 minus 119882(0)100381710038171003817100381710038172 = 0160161520985517

(33)

The cross Grammian matrix is then calculated using theproposed recursive algorithm The error norm for eachiteration is shown in Table 2

It can be seen from Table 2 that the proposed algorithmconverges to the exact solution according to the convergenceresult stated in Theorem 2

43 Natural Gas Hydrogen Reformer In this section weinvestigate the behavior of the proposed algorithm in caseof higher order singularly perturbed systems The linearized10th-ordermathematical model of the gas hydrogen reformerintroduced and studied in [25ndash27] is chosen The systemmatrices are given as follows [25]

119860 =

[[[[[[[[[[[[[[[[[[[[[[[

minus0074 0 0 0 0 0 minus353 10748 0 00 minus1468 minus253 0 0 0 0 0 25582 139110 0 minus156 0 0 0 0 0 0 335860 0 0 minus1245 21263 0 11269 11269 0 00 0 0 0 minus3333 0 0 0 0 00 0 0 0 0 minus3243 32304 32304 0 00 0 0 0 0 3318 minus344 minus341 0 990420 0 0 22197 0 0 minus2532 minus2549 0 325260 0 20354 0 0 0 18309 1214 minus0358 minus3304

00188 0 81642 0 0 0 56043 53994 0 minus1361

]]]]]]]]]]]]]]]]]]]]]]]

119861 = [0 0 0 0 012 0 0 0 0 00 0 0 0 0 01834 0 0 0 0]

119879

119862 = [1 0 0 0 0 0 0 0 0 00 0994 minus0088 0 0 0 0 0 0 0]

(34)

Mathematical Problems in Engineering 7

Table 2 Error norm values for each iteration for the chemical plant

Number of iterations 119894 10038171003817100381710038171003817119882119883 minus 119882(119894)119883 1003817100381710038171003817100381722 6549205248602119890 minus 34 3305462486403183119890 minus 046 1644553914568401119890 minus 058 7961048437784984119890 minus 0710 3791658707506841119890 minus 0812 1790737974576206119890 minus 0914 8421250159940589119890 minus 1116 3944852245039569119890 minus 1218 1781054537942190119890 minus 1320 1746372245754009119890 minus 14

The eigenvalues of the system matrix 119860 are as followsminus66068minus1579minus89137minus12175minus333minus277plusmn0547119894minus1468minus0358 and minus00861 All real parts of those eigenvalues liein the left part of the complex plane hence the systemis asymptotically stable Moreover the system has multipletime scales (slow and fast) since there are three eigenvalueslocated very close to the imaginary axis while the otherseven eigenvalues are located far from that axis The singularperturbation parameter 120576 is chosen as the ratio of the fastestslow eigenvalue to the slowest fast eigenvalue and is equal to120576 = 052 = 1468277

What are supposed to be slowmode eigenvalues obtainedvia (15) minus1468 minus00838 and minus1284495 and what aresupposed to be the fastmode eigenvalues (eigenvalues of1198604)minus0358minus66068minus89144minus13954minus2829plusmn0886 andminus3333

Table 3 Error norm values for each iteration for the gas reformersystem

Number of iterations 119894 10038171003817100381710038171003817119882119883 minus 119882(119894)119883 1003817100381710038171003817100381722 566707478610707585 1326046318575198610 038648979991069112 50958037951316119890 minus 0215 5949048480951119890 minus 0320 6505261253303921119890 minus 0423 6630466299301564119890 minus 0525 7912793405030453119890 minus 0630 7210694103862186119890 minus 0735 3050169378906093119890 minus 0840 8638446874677160119890 minus 10

do not display the slow-fast time scale separation Similarto the chemical plant example in the previous section wecan notice that the slow and fast eigenvalues are not wellseparated into two disjoint groups However since the gasformer system is of higher order it will be a little cumbersometo come up with a permutation matrix that would convert thesystem into its explicit singularly perturbed form Thereforethe algorithm presented in [21 28] and used by [27] to studythe slow and fast dynamics of the gas reformer system isalso considered here to convert the system from an implicitsingularly perturbed form to the explicit singularly perturbedform The algorithm in [21 28] is based on introducing asimilarity transformation 119879 that transforms the general linearsystem in implicit singularly perturbed form into the explicitsingularly perturbed form defined in (9) The similaritytransformation 119879 is given by [27]

119879 =

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

48557 21797 minus03535 17829 11374 minus52659 minus51468 1 0 00 0 0 0 0 0 0 0 1 0

minus03794 07980 minus00771 0 0 52667 05148 0 0 10 0 1 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 00 0 0 0 1 0 0 0 0 00 0 0 0 0 1 0 0 0 00 0 0 0 0 0 1 0 0 00 0 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 0 1

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

(35)

8 Mathematical Problems in Engineering

The new transformed system matrices are defined as

119860SP = 119879119860119879minus1119861SP = 119879119861119862SP = 119862119879minus1

(36)

Thereby they are calculated as

119860SP =

[[[[[[[[[[[[[[[[[[[[[[[

minus03192 55761 minus3138 minus0353 05681 363094 minus0259 minus0050 55156 313790 minus0358 0 20354 0 0 0 1831 12140 minus3304

minus00866 20413 minus12315 minus0125 01544 98473 19252 01901 minus03195 123140 0 0 minus1245 0 0 0 0 0 335860 0 0 0 minus1245 21263 0 11269 11269 00 0 0 0 0 minus3333 0 0 0 00 0 0 0 0 0 minus3243 32304 32304 00 0 0 22197 0 0 3318 minus344 minus341 990420 0 20354 0 22197 0 0 minus2532 minus2549 32526

00188 0 81642 0 minus00057 minus03629 0213 5625 53962 minus1361

]]]]]]]]]]]]]]]]]]]]]]]

119861SP = [136488 0 0 0 0 012 0 0 0 0minus96577 0 09659 0 0 0 01834 0 0 0]

119879

119862SP = [01697 0 minus04636 00243 minus0302 minus19303 113788 1112 minus01697 0463600802 0 10266 00195 minus0143 minus91238 minus11825 minus0115 minus00802 minus10266]

(37)

The matrices 119860SP and 119860 have the same eigenvalues sincethey are preserved under the similarity transformation Theslow mode eigenvalues are minus1468 minus0358 and minus008552while the fast mode eigenvalues are minus660682 minus15789 89137minus12174 minus3333 and minus27697 plusmn 060087 They are clearlyseparated now into two disjoint groups

Using our proposed algorithm to calculate the initial crossGrammian matrix 119882(0) and compare the result to the exactsolution of the algebraic Sylvester equation we get the errornorm as 10038171003817100381710038171003817119882119883 minus 119882(0)100381710038171003817100381710038172 = 53489147248362102 (38)

The cross Grammian matrix is then calculated usingthe proposed recursive algorithm The error norm for eachiteration is shown in Table 3

5 Conclusions

The algorithm was developed to solve the algebraic Sylvesterequation whose solution defines the cross Grammian ofsingularly perturbed linear systems The algorithm is veryefficient defined in terms of reduced-order subproblemscorresponding to slow and fast subsystems and convergesrapidly to the required solution The efficacy of the algorithmis demonstrated on three real physical examples The algo-rithm can be directly applied to singularly perturbed systems

in the explicit standard forms A similarity transformationneeds to be applied to singularly perturbed systems in implicitforms to convert them into their explicit form before theproposed algorithm can be applied to this class of systems

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

References

[1] Z Gajic Multimodel control and estimation of linear stochastic[PhD Dissertation] Michigan State University 1984

[2] T Grodt and Z Gajic ldquoThe recursive reduced-order numericalsolution of the singularly perturbed matrix differential Riccatiequationrdquo IEEE Transactions on Automatic Control vol 33 no8 pp 751ndash754 1988

[3] W Su Z Gajic and X Shen ldquoThe recursive reduced-ordersolution of an open-loop control problem of linear singularlyperturbed systemsrdquo IEEE Transactions on Automatic Controlvol 37 no 2 pp 279ndash281 1992

[4] Z Gajic andM-T LimOptimal Control of Singularly PerturbedLinear Systems and Applications High-Accuracy TechniquesMarcel Dekker New York NY USA 2001

[5] W C Su Z Gajic and X M Shen ldquoThe exact slow-fastdecomposition of the algebraic Ricatti equation of singularly

Mathematical Problems in Engineering 9

perturbed systemsrdquo IEEE Transactions on Automatic Controlvol 37 no 9 pp 1456ndash1459 1992

[6] Z Gajic and X Shen Parallel Algorithms for Optimal Control ofLarge Scale Linear Systems Springer London UK 1993

[7] I Borno andZGajic ldquoParallel algorithms for optimal control ofweakly coupled and singularly perturbed jump linear systemsrdquoAutomatica vol 31 no 7 pp 985ndash988 1995

[8] H Mukaidani H Xu and K Mizukami ldquoNew iterative algo-rithm for algebraic riccati equation related to 119867infin controlproblem of singularly perturbed systemsrdquo IEEE Transactions onAutomatic Control vol 46 no 10 pp 1659ndash1666 2001

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

[10] 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

[11] V Sreeram ldquoRecursive technique for computation of Grami-ansrdquo IEE Proceedings D Control Theory and Applications vol140 no 3 pp 160ndash166 1993

[12] K CWan andV Sreeram ldquoSolution of the bilinearmatrix equa-tion using Astrom-Jury-Agniel algorithmrdquo IEE Proceedings-Dvol 140 pp 21160ndash2166 1993

[13] P Lancaster and M Tismenetsky The Theory of MatricesComputer Science and Applied Mathematics Academic PressOrlando Fla USA 2nd edition 1985

[14] U Baur ldquoLow rank solution of data-sparse Sylvester equationsrdquoNumerical Linear Algebra with Applications vol 15 no 9 pp837ndash851 2008

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

[16] P Benner E S Quintana-Ortı and G Quintana-Ortı ldquoSolvingstable Sylvester equations via rational iterative schemesrdquo Journalof Scientific Computing vol 28 no 1 pp 51ndash83 2006

[17] A J Laub L M Silverman and M Verma ldquoNote on cross-grammians for symmetric realizationsrdquo Proceedings of the IEEEvol 71 no 7 pp 904ndash905 1983

[18] K Fernando and H Nicholson ldquoOn a fundamental property ofthe cross- Gramian matrixrdquo IEEE Transactions on Circuits andSystems vol 31 no 5 pp 504ndash505 1984

[19] K V Fernando and H Nicholson ldquoReachabilty observabilityand minimality of mimo systemsrdquo Proceedings of the IEEE vol72 no 12 pp 1820ndash1821 1984

[20] 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

[21] P Kokotovic H Khalil and J OrsquoReilly Singular PerturbationMethods in Control Analysis and Design Academic PressLondon UK 1986

[22] DG LuenbergerOptimization byVector SpaceMethodsWiley-Interscience New York NY USA 1969

[23] A N Andry Jr J C Chung and E Y Shapiro ldquoModalizedobserversrdquo IEEE Transactions on Automatic Control vol 29 no7 pp 669ndash672 1984

[24] V G Shankar and K Ramar ldquoPole assignment with minimumeigenvalue sensitivity to plant parameter variationsrdquo Interna-tiona Jornal of Control vol 23 no 4 pp 493ndash504 1976

[25] J Pukrushpan A Stefanopoulou and H Peng Control of FuelCell Power Systems Principles Modeling Analysis and FeedbackDesign Springer Berlin Germany 2004

[26] J Pukrushpan A Stefanopoulou S Varigonda J Eborn andC Haugstetter ldquoControl-oriented model of fuel processor forhydrogen generation in fuel cell applicationsrdquoControl Engineer-ing Practice vol 14 no 3 pp 277ndash293 2006

[27] M Skataric and Z Gajic ldquoSlow and fast dynamics of a naturalgas hydrogen reformerrdquo International Journal of HydrogenEnergy vol 38 no 35 pp 15173ndash15179 2013

[28] H K Khalil ldquoTime scale decomposition of linear implicitsingularly perturbed systemsrdquo IEEE Transactions on AutomaticControl vol 29 no 11 pp 1054ndash1056 1984

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Algebra

Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 4: Research Article Recursive Reduced-Order Algor ithm …downloads.hindawi.com/journals/mpe/2016/2452746.pdf · Research Article Recursive Reduced-Order Algor ithm for Singularly Perturbed

4 Mathematical Problems in Engineering

Proof Using Assumption 1 that is 1198600 and 1198604 are asymptoti-cally stable it can be shown that (19) represents a contractionmapping [22] that is

10038171003817100381710038171003817119864(119894)119895 minus 11986411989510038171003817100381710038171003817 = 119874 (120576119894) for 119895 = 1 2 3 4 119894 = 1 2 (22)

Formula (22) will be also valid if

119864(119894+1)2 = minus (1198601 (119882(0)2 + 120576119864(119894)2 ) + 1198602119864(119894)4 + 119864(119894)1 1198602) 119860minus14 119864(119894+1)3 = minus119860minus14 ((119882(0)3 + 120576119864(119894)3 ) 1198601 + 1198603119864(119894)1 + 119864(119894)4 1198603)

for 119894 = 0 1 2 119864(0)1 = 0 119864(0)4 = 0(23)

Formula (22) implies that algorithm (19) is convergent Using119864(infin)119895 for 119895 = 1 2 3 4 in (19) and comparing it to (18) it canbe seen that algorithm (19) converges to the unique solutionof (18)

4 Case Studies

Three case studies are considered to demonstrate the pro-posed algorithm a fourth-order aircraft example whosemathematical model is in the explicit singularly perturbedform defined in (9) in which with accuracy of 119874(120576) theslow eigenvalues are all contained in the approximate slowsubsystem represented by 1198600 and all fast eigenvalues arecontained in the approximate fast subsystem representedby 1198604 a fifth-order chemical plant model given in implicitsingularly perturbed form (it has two slow and three fasteigenvalues but the state variables have to be reordered toachieve explicit singularly perturbed form defined in (9)) a

tenth-order hydrogen gas reformer used to provide hydrogento a fuel cell fromhydrogen rich fuels (natural gasmethanol)

41 L-1011 Aircraft Here we consider the lateral axis equa-tions of the rigid body model of L-1011 aircraft at cruisecondition [23]The state variables are the bank angle roll rateyaw rate and sideslip angle which are represented in the statevector 119909(119905) = [1199091 1199092 1199093 1199094]119879 in the same order The inputvector consists of two variables the rudder deflection 120575119903 andthe aileron deflection 120575119886 and is given as 119906(119905) = [120575119903 120575119886]119879 Thesystem matrices are given as

119860 =[[[[[[

0 1 0 00 minus189 039 minus5550 minus0034 minus298 243

0034 minus00011 minus099 minus021

]]]]]]

119861 =[[[[[[

0 0036 minus16

minus095 minus0032003 0

]]]]]]

119862 = [1 0 0 00 1 0 0]

(24)

The eigenvalues of thematrix119860 areminus01016minus14811plusmn06292119894and minus20162The system is asymptotically stable (all eigenval-ues are in the left half plane) controllable and observableMoreover there is only one slow mode with eigenvalueminus00899 and there are three fast modes with eigenvaluesminus14891 plusmn 07686119894 and minus21017 The singular perturbationparameter 120576 = 007 which is the ratio between the fastestslow eigenvalue and the slowest fast eigenvalue Solving thealgebraic Sylvesterrsquos equation (6) the cross Grammian matrixcan be obtained as follows

119882119883 =[[[[[[

minus377168467243 minus225320146563 minus460285451131 1268652326564minus043134179103 minus062940797540 minus027809983914 1332572206696minus014313989569 000048741612 minus001060057667 minus002122054103020967036557 006408641105 minus003861451004 000250243909

]]]]]]

(25)

Using the proposed algorithm the initial cross Grammi-an matrix 119882(0) (first-order approximate solution) is obtained

as follows

119882(0) =[[[[[[

minus398315817315 minus225882783809 minus446229986109 1232553354098minus041906814039 minus042418654922 minus021443252362 0883962065453minus014908955443 minus000284907928 000235249833 minus0005866460180202330975578 0001553980682 0000836843461 minus000419856136

]]]]]]

(26)

Mathematical Problems in Engineering 5

Table 1 Error norm values for each iteration for L-1011 aircraftsystem

Number of iterations 119894 10038171003817100381710038171003817119882119883 minus 119882(119894)119883 1003817100381710038171003817100381722 7503328625983908119890 minus 043 2611125417476592119890 minus 054 9085062505535487119890 minus 075 3161097999142058119890 minus 086 1099918110580769119890 minus 097 3830668858621788119890 minus 118 1368874658220410119890 minus 129 9598118726733237119890 minus 1410 6474407682370149119890 minus 14

Comparing the exact solution 119882119883 to the first-orderapproximate solution of the cross Grammian matrix bycalculating the error norm we get

10038171003817100381710038171003817119882119883 minus 119882(0)100381710038171003817100381710038172 = 0624920763816596 (27)

Then the cross Grammian matrix is calculated using the pro-posed recursive algorithm The error norm at each iterationis shown in Table 1 By taking the error norm it can be seenthat the algorithm converges rapidly to the exact solution

42 Chemical Plant In this section the linearized chemicalplant considered in [24] is chosen to explain the behavior ofthe proposed algorithm when the linear singularly perturbedsystem is not in the explicit standard form (9) The systemmatrices are given as follows

119860 =[[[[[[[[[

minus01094 00628 0 0 01306 minus2136 09807 0 0

0 1595 minus3149 1547 00 00355 2632 minus4257 18550 00027 0 01636 minus01625

]]]]]]]]]

119861 =[[[[[[[[[

0 000638 000838 minus0139601004 minus020600063 minus00128

]]]]]]]]]

119862 = [1 0 0 0 00 0 0 0 1]

(28)

The eigenvalues of matrix 119860 are minus59822 minus28408 minus08954minus00774 and minus00141 which indicates that this system hastwo slow modes (eigenvalues) The singular perturbationparameter 120576 is chosen as the ratio of the fastest sloweigenvalue to the slowest fast eigenvalue and is equal to

120576 = 0086 = 0077408954 Introducing the followingpermutation matrix that exchanges the second row of matrix119860 with its fifth row that is

119875 =[[[[[[[[[

1 0 0 0 00 0 0 0 10 0 1 0 00 0 0 1 00 1 0 0 0

]]]]]]]]]

(29)

the explicit singularly perturbed form of the system matricescan be obtained as follows

119860SP = 119875119860119875119861SP = 119875119861119862SP = 119862119875

(30)

Thereby they are calculated as

119860SP

=[[[[[[[[[[

minus01094 0 0 0 006280 minus01625 0 01636 000270 0 minus3149 1547 159500 18550 2632 minus4257 00355

13060 0 09807 0 minus21320

]]]]]]]]]]

119861SP =[[[[[[[[[

0 000063 minus0012800838 minus0139601004 minus020600638 0

]]]]]]]]]

119862SP = [1 0 0 0 00 1 0 0 0]

(31)

Matrices 119860SP and 119860 have the same eigenvalues and the samenumber of slow and fast modes However the slow and fasteigenvalues are now correctly separated into two disjointgroups The slow approximate subsystem (see (15)) has theeigenvalues minus00788 and minus00160 and the fast approximatesubsystem has the fast eigenvalues minus59521 27925 andminus07933 (eigenvalues of 1198604)

Solving the algebraic Sylvesterrsquos equation (6) the crossGrammian matrix can be obtained as follows

6 Mathematical Problems in Engineering

119882119883 =[[[[[[[[[

minus007530866 minus011561658 minus000985227 minus000806596 minus000985946minus023194836 minus049463876 minus003897379 minus003313413 minus003676743minus007341430 minus042776115 minus002785663 minus002552098 minus002384797minus012954567 minus052436133 minus003510447 minus003163354 minus003102749minus005734203 minus026232141 minus001916962 minus001692499 minus001682689

]]]]]]]]]

(32)

Calculating the initial cross Grammian matrix 119882(0) using theproposed algorithm and comparing the result to the exact

solution of the cross Grammian using the error norm we getthe following results

119882(0) =[[[[[[[[[

minus007894318 minus013640657 minus001124158 minus000932742 minus00110635minus025878609 minus056732132 minus004537161 minus003829073 minus004292241minus005767992 minus050330195 0 0 0minus012521917 minus060940965 0 0 0minus004496552 minus031507279 0 0 0

]]]]]]]]]

10038171003817100381710038171003817119882119883 minus 119882(0)100381710038171003817100381710038172 = 0160161520985517

(33)

The cross Grammian matrix is then calculated using theproposed recursive algorithm The error norm for eachiteration is shown in Table 2

It can be seen from Table 2 that the proposed algorithmconverges to the exact solution according to the convergenceresult stated in Theorem 2

43 Natural Gas Hydrogen Reformer In this section weinvestigate the behavior of the proposed algorithm in caseof higher order singularly perturbed systems The linearized10th-ordermathematical model of the gas hydrogen reformerintroduced and studied in [25ndash27] is chosen The systemmatrices are given as follows [25]

119860 =

[[[[[[[[[[[[[[[[[[[[[[[

minus0074 0 0 0 0 0 minus353 10748 0 00 minus1468 minus253 0 0 0 0 0 25582 139110 0 minus156 0 0 0 0 0 0 335860 0 0 minus1245 21263 0 11269 11269 0 00 0 0 0 minus3333 0 0 0 0 00 0 0 0 0 minus3243 32304 32304 0 00 0 0 0 0 3318 minus344 minus341 0 990420 0 0 22197 0 0 minus2532 minus2549 0 325260 0 20354 0 0 0 18309 1214 minus0358 minus3304

00188 0 81642 0 0 0 56043 53994 0 minus1361

]]]]]]]]]]]]]]]]]]]]]]]

119861 = [0 0 0 0 012 0 0 0 0 00 0 0 0 0 01834 0 0 0 0]

119879

119862 = [1 0 0 0 0 0 0 0 0 00 0994 minus0088 0 0 0 0 0 0 0]

(34)

Mathematical Problems in Engineering 7

Table 2 Error norm values for each iteration for the chemical plant

Number of iterations 119894 10038171003817100381710038171003817119882119883 minus 119882(119894)119883 1003817100381710038171003817100381722 6549205248602119890 minus 34 3305462486403183119890 minus 046 1644553914568401119890 minus 058 7961048437784984119890 minus 0710 3791658707506841119890 minus 0812 1790737974576206119890 minus 0914 8421250159940589119890 minus 1116 3944852245039569119890 minus 1218 1781054537942190119890 minus 1320 1746372245754009119890 minus 14

The eigenvalues of the system matrix 119860 are as followsminus66068minus1579minus89137minus12175minus333minus277plusmn0547119894minus1468minus0358 and minus00861 All real parts of those eigenvalues liein the left part of the complex plane hence the systemis asymptotically stable Moreover the system has multipletime scales (slow and fast) since there are three eigenvalueslocated very close to the imaginary axis while the otherseven eigenvalues are located far from that axis The singularperturbation parameter 120576 is chosen as the ratio of the fastestslow eigenvalue to the slowest fast eigenvalue and is equal to120576 = 052 = 1468277

What are supposed to be slowmode eigenvalues obtainedvia (15) minus1468 minus00838 and minus1284495 and what aresupposed to be the fastmode eigenvalues (eigenvalues of1198604)minus0358minus66068minus89144minus13954minus2829plusmn0886 andminus3333

Table 3 Error norm values for each iteration for the gas reformersystem

Number of iterations 119894 10038171003817100381710038171003817119882119883 minus 119882(119894)119883 1003817100381710038171003817100381722 566707478610707585 1326046318575198610 038648979991069112 50958037951316119890 minus 0215 5949048480951119890 minus 0320 6505261253303921119890 minus 0423 6630466299301564119890 minus 0525 7912793405030453119890 minus 0630 7210694103862186119890 minus 0735 3050169378906093119890 minus 0840 8638446874677160119890 minus 10

do not display the slow-fast time scale separation Similarto the chemical plant example in the previous section wecan notice that the slow and fast eigenvalues are not wellseparated into two disjoint groups However since the gasformer system is of higher order it will be a little cumbersometo come up with a permutation matrix that would convert thesystem into its explicit singularly perturbed form Thereforethe algorithm presented in [21 28] and used by [27] to studythe slow and fast dynamics of the gas reformer system isalso considered here to convert the system from an implicitsingularly perturbed form to the explicit singularly perturbedform The algorithm in [21 28] is based on introducing asimilarity transformation 119879 that transforms the general linearsystem in implicit singularly perturbed form into the explicitsingularly perturbed form defined in (9) The similaritytransformation 119879 is given by [27]

119879 =

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

48557 21797 minus03535 17829 11374 minus52659 minus51468 1 0 00 0 0 0 0 0 0 0 1 0

minus03794 07980 minus00771 0 0 52667 05148 0 0 10 0 1 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 00 0 0 0 1 0 0 0 0 00 0 0 0 0 1 0 0 0 00 0 0 0 0 0 1 0 0 00 0 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 0 1

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

(35)

8 Mathematical Problems in Engineering

The new transformed system matrices are defined as

119860SP = 119879119860119879minus1119861SP = 119879119861119862SP = 119862119879minus1

(36)

Thereby they are calculated as

119860SP =

[[[[[[[[[[[[[[[[[[[[[[[

minus03192 55761 minus3138 minus0353 05681 363094 minus0259 minus0050 55156 313790 minus0358 0 20354 0 0 0 1831 12140 minus3304

minus00866 20413 minus12315 minus0125 01544 98473 19252 01901 minus03195 123140 0 0 minus1245 0 0 0 0 0 335860 0 0 0 minus1245 21263 0 11269 11269 00 0 0 0 0 minus3333 0 0 0 00 0 0 0 0 0 minus3243 32304 32304 00 0 0 22197 0 0 3318 minus344 minus341 990420 0 20354 0 22197 0 0 minus2532 minus2549 32526

00188 0 81642 0 minus00057 minus03629 0213 5625 53962 minus1361

]]]]]]]]]]]]]]]]]]]]]]]

119861SP = [136488 0 0 0 0 012 0 0 0 0minus96577 0 09659 0 0 0 01834 0 0 0]

119879

119862SP = [01697 0 minus04636 00243 minus0302 minus19303 113788 1112 minus01697 0463600802 0 10266 00195 minus0143 minus91238 minus11825 minus0115 minus00802 minus10266]

(37)

The matrices 119860SP and 119860 have the same eigenvalues sincethey are preserved under the similarity transformation Theslow mode eigenvalues are minus1468 minus0358 and minus008552while the fast mode eigenvalues are minus660682 minus15789 89137minus12174 minus3333 and minus27697 plusmn 060087 They are clearlyseparated now into two disjoint groups

Using our proposed algorithm to calculate the initial crossGrammian matrix 119882(0) and compare the result to the exactsolution of the algebraic Sylvester equation we get the errornorm as 10038171003817100381710038171003817119882119883 minus 119882(0)100381710038171003817100381710038172 = 53489147248362102 (38)

The cross Grammian matrix is then calculated usingthe proposed recursive algorithm The error norm for eachiteration is shown in Table 3

5 Conclusions

The algorithm was developed to solve the algebraic Sylvesterequation whose solution defines the cross Grammian ofsingularly perturbed linear systems The algorithm is veryefficient defined in terms of reduced-order subproblemscorresponding to slow and fast subsystems and convergesrapidly to the required solution The efficacy of the algorithmis demonstrated on three real physical examples The algo-rithm can be directly applied to singularly perturbed systems

in the explicit standard forms A similarity transformationneeds to be applied to singularly perturbed systems in implicitforms to convert them into their explicit form before theproposed algorithm can be applied to this class of systems

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

References

[1] Z Gajic Multimodel control and estimation of linear stochastic[PhD Dissertation] Michigan State University 1984

[2] T Grodt and Z Gajic ldquoThe recursive reduced-order numericalsolution of the singularly perturbed matrix differential Riccatiequationrdquo IEEE Transactions on Automatic Control vol 33 no8 pp 751ndash754 1988

[3] W Su Z Gajic and X Shen ldquoThe recursive reduced-ordersolution of an open-loop control problem of linear singularlyperturbed systemsrdquo IEEE Transactions on Automatic Controlvol 37 no 2 pp 279ndash281 1992

[4] Z Gajic andM-T LimOptimal Control of Singularly PerturbedLinear Systems and Applications High-Accuracy TechniquesMarcel Dekker New York NY USA 2001

[5] W C Su Z Gajic and X M Shen ldquoThe exact slow-fastdecomposition of the algebraic Ricatti equation of singularly

Mathematical Problems in Engineering 9

perturbed systemsrdquo IEEE Transactions on Automatic Controlvol 37 no 9 pp 1456ndash1459 1992

[6] Z Gajic and X Shen Parallel Algorithms for Optimal Control ofLarge Scale Linear Systems Springer London UK 1993

[7] I Borno andZGajic ldquoParallel algorithms for optimal control ofweakly coupled and singularly perturbed jump linear systemsrdquoAutomatica vol 31 no 7 pp 985ndash988 1995

[8] H Mukaidani H Xu and K Mizukami ldquoNew iterative algo-rithm for algebraic riccati equation related to 119867infin controlproblem of singularly perturbed systemsrdquo IEEE Transactions onAutomatic Control vol 46 no 10 pp 1659ndash1666 2001

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

[10] 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

[11] V Sreeram ldquoRecursive technique for computation of Grami-ansrdquo IEE Proceedings D Control Theory and Applications vol140 no 3 pp 160ndash166 1993

[12] K CWan andV Sreeram ldquoSolution of the bilinearmatrix equa-tion using Astrom-Jury-Agniel algorithmrdquo IEE Proceedings-Dvol 140 pp 21160ndash2166 1993

[13] P Lancaster and M Tismenetsky The Theory of MatricesComputer Science and Applied Mathematics Academic PressOrlando Fla USA 2nd edition 1985

[14] U Baur ldquoLow rank solution of data-sparse Sylvester equationsrdquoNumerical Linear Algebra with Applications vol 15 no 9 pp837ndash851 2008

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

[16] P Benner E S Quintana-Ortı and G Quintana-Ortı ldquoSolvingstable Sylvester equations via rational iterative schemesrdquo Journalof Scientific Computing vol 28 no 1 pp 51ndash83 2006

[17] A J Laub L M Silverman and M Verma ldquoNote on cross-grammians for symmetric realizationsrdquo Proceedings of the IEEEvol 71 no 7 pp 904ndash905 1983

[18] K Fernando and H Nicholson ldquoOn a fundamental property ofthe cross- Gramian matrixrdquo IEEE Transactions on Circuits andSystems vol 31 no 5 pp 504ndash505 1984

[19] K V Fernando and H Nicholson ldquoReachabilty observabilityand minimality of mimo systemsrdquo Proceedings of the IEEE vol72 no 12 pp 1820ndash1821 1984

[20] 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

[21] P Kokotovic H Khalil and J OrsquoReilly Singular PerturbationMethods in Control Analysis and Design Academic PressLondon UK 1986

[22] DG LuenbergerOptimization byVector SpaceMethodsWiley-Interscience New York NY USA 1969

[23] A N Andry Jr J C Chung and E Y Shapiro ldquoModalizedobserversrdquo IEEE Transactions on Automatic Control vol 29 no7 pp 669ndash672 1984

[24] V G Shankar and K Ramar ldquoPole assignment with minimumeigenvalue sensitivity to plant parameter variationsrdquo Interna-tiona Jornal of Control vol 23 no 4 pp 493ndash504 1976

[25] J Pukrushpan A Stefanopoulou and H Peng Control of FuelCell Power Systems Principles Modeling Analysis and FeedbackDesign Springer Berlin Germany 2004

[26] J Pukrushpan A Stefanopoulou S Varigonda J Eborn andC Haugstetter ldquoControl-oriented model of fuel processor forhydrogen generation in fuel cell applicationsrdquoControl Engineer-ing Practice vol 14 no 3 pp 277ndash293 2006

[27] M Skataric and Z Gajic ldquoSlow and fast dynamics of a naturalgas hydrogen reformerrdquo International Journal of HydrogenEnergy vol 38 no 35 pp 15173ndash15179 2013

[28] H K Khalil ldquoTime scale decomposition of linear implicitsingularly perturbed systemsrdquo IEEE Transactions on AutomaticControl vol 29 no 11 pp 1054ndash1056 1984

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 5: Research Article Recursive Reduced-Order Algor ithm …downloads.hindawi.com/journals/mpe/2016/2452746.pdf · Research Article Recursive Reduced-Order Algor ithm for Singularly Perturbed

Mathematical Problems in Engineering 5

Table 1 Error norm values for each iteration for L-1011 aircraftsystem

Number of iterations 119894 10038171003817100381710038171003817119882119883 minus 119882(119894)119883 1003817100381710038171003817100381722 7503328625983908119890 minus 043 2611125417476592119890 minus 054 9085062505535487119890 minus 075 3161097999142058119890 minus 086 1099918110580769119890 minus 097 3830668858621788119890 minus 118 1368874658220410119890 minus 129 9598118726733237119890 minus 1410 6474407682370149119890 minus 14

Comparing the exact solution 119882119883 to the first-orderapproximate solution of the cross Grammian matrix bycalculating the error norm we get

10038171003817100381710038171003817119882119883 minus 119882(0)100381710038171003817100381710038172 = 0624920763816596 (27)

Then the cross Grammian matrix is calculated using the pro-posed recursive algorithm The error norm at each iterationis shown in Table 1 By taking the error norm it can be seenthat the algorithm converges rapidly to the exact solution

42 Chemical Plant In this section the linearized chemicalplant considered in [24] is chosen to explain the behavior ofthe proposed algorithm when the linear singularly perturbedsystem is not in the explicit standard form (9) The systemmatrices are given as follows

119860 =[[[[[[[[[

minus01094 00628 0 0 01306 minus2136 09807 0 0

0 1595 minus3149 1547 00 00355 2632 minus4257 18550 00027 0 01636 minus01625

]]]]]]]]]

119861 =[[[[[[[[[

0 000638 000838 minus0139601004 minus020600063 minus00128

]]]]]]]]]

119862 = [1 0 0 0 00 0 0 0 1]

(28)

The eigenvalues of matrix 119860 are minus59822 minus28408 minus08954minus00774 and minus00141 which indicates that this system hastwo slow modes (eigenvalues) The singular perturbationparameter 120576 is chosen as the ratio of the fastest sloweigenvalue to the slowest fast eigenvalue and is equal to

120576 = 0086 = 0077408954 Introducing the followingpermutation matrix that exchanges the second row of matrix119860 with its fifth row that is

119875 =[[[[[[[[[

1 0 0 0 00 0 0 0 10 0 1 0 00 0 0 1 00 1 0 0 0

]]]]]]]]]

(29)

the explicit singularly perturbed form of the system matricescan be obtained as follows

119860SP = 119875119860119875119861SP = 119875119861119862SP = 119862119875

(30)

Thereby they are calculated as

119860SP

=[[[[[[[[[[

minus01094 0 0 0 006280 minus01625 0 01636 000270 0 minus3149 1547 159500 18550 2632 minus4257 00355

13060 0 09807 0 minus21320

]]]]]]]]]]

119861SP =[[[[[[[[[

0 000063 minus0012800838 minus0139601004 minus020600638 0

]]]]]]]]]

119862SP = [1 0 0 0 00 1 0 0 0]

(31)

Matrices 119860SP and 119860 have the same eigenvalues and the samenumber of slow and fast modes However the slow and fasteigenvalues are now correctly separated into two disjointgroups The slow approximate subsystem (see (15)) has theeigenvalues minus00788 and minus00160 and the fast approximatesubsystem has the fast eigenvalues minus59521 27925 andminus07933 (eigenvalues of 1198604)

Solving the algebraic Sylvesterrsquos equation (6) the crossGrammian matrix can be obtained as follows

6 Mathematical Problems in Engineering

119882119883 =[[[[[[[[[

minus007530866 minus011561658 minus000985227 minus000806596 minus000985946minus023194836 minus049463876 minus003897379 minus003313413 minus003676743minus007341430 minus042776115 minus002785663 minus002552098 minus002384797minus012954567 minus052436133 minus003510447 minus003163354 minus003102749minus005734203 minus026232141 minus001916962 minus001692499 minus001682689

]]]]]]]]]

(32)

Calculating the initial cross Grammian matrix 119882(0) using theproposed algorithm and comparing the result to the exact

solution of the cross Grammian using the error norm we getthe following results

119882(0) =[[[[[[[[[

minus007894318 minus013640657 minus001124158 minus000932742 minus00110635minus025878609 minus056732132 minus004537161 minus003829073 minus004292241minus005767992 minus050330195 0 0 0minus012521917 minus060940965 0 0 0minus004496552 minus031507279 0 0 0

]]]]]]]]]

10038171003817100381710038171003817119882119883 minus 119882(0)100381710038171003817100381710038172 = 0160161520985517

(33)

The cross Grammian matrix is then calculated using theproposed recursive algorithm The error norm for eachiteration is shown in Table 2

It can be seen from Table 2 that the proposed algorithmconverges to the exact solution according to the convergenceresult stated in Theorem 2

43 Natural Gas Hydrogen Reformer In this section weinvestigate the behavior of the proposed algorithm in caseof higher order singularly perturbed systems The linearized10th-ordermathematical model of the gas hydrogen reformerintroduced and studied in [25ndash27] is chosen The systemmatrices are given as follows [25]

119860 =

[[[[[[[[[[[[[[[[[[[[[[[

minus0074 0 0 0 0 0 minus353 10748 0 00 minus1468 minus253 0 0 0 0 0 25582 139110 0 minus156 0 0 0 0 0 0 335860 0 0 minus1245 21263 0 11269 11269 0 00 0 0 0 minus3333 0 0 0 0 00 0 0 0 0 minus3243 32304 32304 0 00 0 0 0 0 3318 minus344 minus341 0 990420 0 0 22197 0 0 minus2532 minus2549 0 325260 0 20354 0 0 0 18309 1214 minus0358 minus3304

00188 0 81642 0 0 0 56043 53994 0 minus1361

]]]]]]]]]]]]]]]]]]]]]]]

119861 = [0 0 0 0 012 0 0 0 0 00 0 0 0 0 01834 0 0 0 0]

119879

119862 = [1 0 0 0 0 0 0 0 0 00 0994 minus0088 0 0 0 0 0 0 0]

(34)

Mathematical Problems in Engineering 7

Table 2 Error norm values for each iteration for the chemical plant

Number of iterations 119894 10038171003817100381710038171003817119882119883 minus 119882(119894)119883 1003817100381710038171003817100381722 6549205248602119890 minus 34 3305462486403183119890 minus 046 1644553914568401119890 minus 058 7961048437784984119890 minus 0710 3791658707506841119890 minus 0812 1790737974576206119890 minus 0914 8421250159940589119890 minus 1116 3944852245039569119890 minus 1218 1781054537942190119890 minus 1320 1746372245754009119890 minus 14

The eigenvalues of the system matrix 119860 are as followsminus66068minus1579minus89137minus12175minus333minus277plusmn0547119894minus1468minus0358 and minus00861 All real parts of those eigenvalues liein the left part of the complex plane hence the systemis asymptotically stable Moreover the system has multipletime scales (slow and fast) since there are three eigenvalueslocated very close to the imaginary axis while the otherseven eigenvalues are located far from that axis The singularperturbation parameter 120576 is chosen as the ratio of the fastestslow eigenvalue to the slowest fast eigenvalue and is equal to120576 = 052 = 1468277

What are supposed to be slowmode eigenvalues obtainedvia (15) minus1468 minus00838 and minus1284495 and what aresupposed to be the fastmode eigenvalues (eigenvalues of1198604)minus0358minus66068minus89144minus13954minus2829plusmn0886 andminus3333

Table 3 Error norm values for each iteration for the gas reformersystem

Number of iterations 119894 10038171003817100381710038171003817119882119883 minus 119882(119894)119883 1003817100381710038171003817100381722 566707478610707585 1326046318575198610 038648979991069112 50958037951316119890 minus 0215 5949048480951119890 minus 0320 6505261253303921119890 minus 0423 6630466299301564119890 minus 0525 7912793405030453119890 minus 0630 7210694103862186119890 minus 0735 3050169378906093119890 minus 0840 8638446874677160119890 minus 10

do not display the slow-fast time scale separation Similarto the chemical plant example in the previous section wecan notice that the slow and fast eigenvalues are not wellseparated into two disjoint groups However since the gasformer system is of higher order it will be a little cumbersometo come up with a permutation matrix that would convert thesystem into its explicit singularly perturbed form Thereforethe algorithm presented in [21 28] and used by [27] to studythe slow and fast dynamics of the gas reformer system isalso considered here to convert the system from an implicitsingularly perturbed form to the explicit singularly perturbedform The algorithm in [21 28] is based on introducing asimilarity transformation 119879 that transforms the general linearsystem in implicit singularly perturbed form into the explicitsingularly perturbed form defined in (9) The similaritytransformation 119879 is given by [27]

119879 =

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

48557 21797 minus03535 17829 11374 minus52659 minus51468 1 0 00 0 0 0 0 0 0 0 1 0

minus03794 07980 minus00771 0 0 52667 05148 0 0 10 0 1 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 00 0 0 0 1 0 0 0 0 00 0 0 0 0 1 0 0 0 00 0 0 0 0 0 1 0 0 00 0 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 0 1

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

(35)

8 Mathematical Problems in Engineering

The new transformed system matrices are defined as

119860SP = 119879119860119879minus1119861SP = 119879119861119862SP = 119862119879minus1

(36)

Thereby they are calculated as

119860SP =

[[[[[[[[[[[[[[[[[[[[[[[

minus03192 55761 minus3138 minus0353 05681 363094 minus0259 minus0050 55156 313790 minus0358 0 20354 0 0 0 1831 12140 minus3304

minus00866 20413 minus12315 minus0125 01544 98473 19252 01901 minus03195 123140 0 0 minus1245 0 0 0 0 0 335860 0 0 0 minus1245 21263 0 11269 11269 00 0 0 0 0 minus3333 0 0 0 00 0 0 0 0 0 minus3243 32304 32304 00 0 0 22197 0 0 3318 minus344 minus341 990420 0 20354 0 22197 0 0 minus2532 minus2549 32526

00188 0 81642 0 minus00057 minus03629 0213 5625 53962 minus1361

]]]]]]]]]]]]]]]]]]]]]]]

119861SP = [136488 0 0 0 0 012 0 0 0 0minus96577 0 09659 0 0 0 01834 0 0 0]

119879

119862SP = [01697 0 minus04636 00243 minus0302 minus19303 113788 1112 minus01697 0463600802 0 10266 00195 minus0143 minus91238 minus11825 minus0115 minus00802 minus10266]

(37)

The matrices 119860SP and 119860 have the same eigenvalues sincethey are preserved under the similarity transformation Theslow mode eigenvalues are minus1468 minus0358 and minus008552while the fast mode eigenvalues are minus660682 minus15789 89137minus12174 minus3333 and minus27697 plusmn 060087 They are clearlyseparated now into two disjoint groups

Using our proposed algorithm to calculate the initial crossGrammian matrix 119882(0) and compare the result to the exactsolution of the algebraic Sylvester equation we get the errornorm as 10038171003817100381710038171003817119882119883 minus 119882(0)100381710038171003817100381710038172 = 53489147248362102 (38)

The cross Grammian matrix is then calculated usingthe proposed recursive algorithm The error norm for eachiteration is shown in Table 3

5 Conclusions

The algorithm was developed to solve the algebraic Sylvesterequation whose solution defines the cross Grammian ofsingularly perturbed linear systems The algorithm is veryefficient defined in terms of reduced-order subproblemscorresponding to slow and fast subsystems and convergesrapidly to the required solution The efficacy of the algorithmis demonstrated on three real physical examples The algo-rithm can be directly applied to singularly perturbed systems

in the explicit standard forms A similarity transformationneeds to be applied to singularly perturbed systems in implicitforms to convert them into their explicit form before theproposed algorithm can be applied to this class of systems

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

References

[1] Z Gajic Multimodel control and estimation of linear stochastic[PhD Dissertation] Michigan State University 1984

[2] T Grodt and Z Gajic ldquoThe recursive reduced-order numericalsolution of the singularly perturbed matrix differential Riccatiequationrdquo IEEE Transactions on Automatic Control vol 33 no8 pp 751ndash754 1988

[3] W Su Z Gajic and X Shen ldquoThe recursive reduced-ordersolution of an open-loop control problem of linear singularlyperturbed systemsrdquo IEEE Transactions on Automatic Controlvol 37 no 2 pp 279ndash281 1992

[4] Z Gajic andM-T LimOptimal Control of Singularly PerturbedLinear Systems and Applications High-Accuracy TechniquesMarcel Dekker New York NY USA 2001

[5] W C Su Z Gajic and X M Shen ldquoThe exact slow-fastdecomposition of the algebraic Ricatti equation of singularly

Mathematical Problems in Engineering 9

perturbed systemsrdquo IEEE Transactions on Automatic Controlvol 37 no 9 pp 1456ndash1459 1992

[6] Z Gajic and X Shen Parallel Algorithms for Optimal Control ofLarge Scale Linear Systems Springer London UK 1993

[7] I Borno andZGajic ldquoParallel algorithms for optimal control ofweakly coupled and singularly perturbed jump linear systemsrdquoAutomatica vol 31 no 7 pp 985ndash988 1995

[8] H Mukaidani H Xu and K Mizukami ldquoNew iterative algo-rithm for algebraic riccati equation related to 119867infin controlproblem of singularly perturbed systemsrdquo IEEE Transactions onAutomatic Control vol 46 no 10 pp 1659ndash1666 2001

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

[10] 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

[11] V Sreeram ldquoRecursive technique for computation of Grami-ansrdquo IEE Proceedings D Control Theory and Applications vol140 no 3 pp 160ndash166 1993

[12] K CWan andV Sreeram ldquoSolution of the bilinearmatrix equa-tion using Astrom-Jury-Agniel algorithmrdquo IEE Proceedings-Dvol 140 pp 21160ndash2166 1993

[13] P Lancaster and M Tismenetsky The Theory of MatricesComputer Science and Applied Mathematics Academic PressOrlando Fla USA 2nd edition 1985

[14] U Baur ldquoLow rank solution of data-sparse Sylvester equationsrdquoNumerical Linear Algebra with Applications vol 15 no 9 pp837ndash851 2008

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

[16] P Benner E S Quintana-Ortı and G Quintana-Ortı ldquoSolvingstable Sylvester equations via rational iterative schemesrdquo Journalof Scientific Computing vol 28 no 1 pp 51ndash83 2006

[17] A J Laub L M Silverman and M Verma ldquoNote on cross-grammians for symmetric realizationsrdquo Proceedings of the IEEEvol 71 no 7 pp 904ndash905 1983

[18] K Fernando and H Nicholson ldquoOn a fundamental property ofthe cross- Gramian matrixrdquo IEEE Transactions on Circuits andSystems vol 31 no 5 pp 504ndash505 1984

[19] K V Fernando and H Nicholson ldquoReachabilty observabilityand minimality of mimo systemsrdquo Proceedings of the IEEE vol72 no 12 pp 1820ndash1821 1984

[20] 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

[21] P Kokotovic H Khalil and J OrsquoReilly Singular PerturbationMethods in Control Analysis and Design Academic PressLondon UK 1986

[22] DG LuenbergerOptimization byVector SpaceMethodsWiley-Interscience New York NY USA 1969

[23] A N Andry Jr J C Chung and E Y Shapiro ldquoModalizedobserversrdquo IEEE Transactions on Automatic Control vol 29 no7 pp 669ndash672 1984

[24] V G Shankar and K Ramar ldquoPole assignment with minimumeigenvalue sensitivity to plant parameter variationsrdquo Interna-tiona Jornal of Control vol 23 no 4 pp 493ndash504 1976

[25] J Pukrushpan A Stefanopoulou and H Peng Control of FuelCell Power Systems Principles Modeling Analysis and FeedbackDesign Springer Berlin Germany 2004

[26] J Pukrushpan A Stefanopoulou S Varigonda J Eborn andC Haugstetter ldquoControl-oriented model of fuel processor forhydrogen generation in fuel cell applicationsrdquoControl Engineer-ing Practice vol 14 no 3 pp 277ndash293 2006

[27] M Skataric and Z Gajic ldquoSlow and fast dynamics of a naturalgas hydrogen reformerrdquo International Journal of HydrogenEnergy vol 38 no 35 pp 15173ndash15179 2013

[28] H K Khalil ldquoTime scale decomposition of linear implicitsingularly perturbed systemsrdquo IEEE Transactions on AutomaticControl vol 29 no 11 pp 1054ndash1056 1984

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 6: Research Article Recursive Reduced-Order Algor ithm …downloads.hindawi.com/journals/mpe/2016/2452746.pdf · Research Article Recursive Reduced-Order Algor ithm for Singularly Perturbed

6 Mathematical Problems in Engineering

119882119883 =[[[[[[[[[

minus007530866 minus011561658 minus000985227 minus000806596 minus000985946minus023194836 minus049463876 minus003897379 minus003313413 minus003676743minus007341430 minus042776115 minus002785663 minus002552098 minus002384797minus012954567 minus052436133 minus003510447 minus003163354 minus003102749minus005734203 minus026232141 minus001916962 minus001692499 minus001682689

]]]]]]]]]

(32)

Calculating the initial cross Grammian matrix 119882(0) using theproposed algorithm and comparing the result to the exact

solution of the cross Grammian using the error norm we getthe following results

119882(0) =[[[[[[[[[

minus007894318 minus013640657 minus001124158 minus000932742 minus00110635minus025878609 minus056732132 minus004537161 minus003829073 minus004292241minus005767992 minus050330195 0 0 0minus012521917 minus060940965 0 0 0minus004496552 minus031507279 0 0 0

]]]]]]]]]

10038171003817100381710038171003817119882119883 minus 119882(0)100381710038171003817100381710038172 = 0160161520985517

(33)

The cross Grammian matrix is then calculated using theproposed recursive algorithm The error norm for eachiteration is shown in Table 2

It can be seen from Table 2 that the proposed algorithmconverges to the exact solution according to the convergenceresult stated in Theorem 2

43 Natural Gas Hydrogen Reformer In this section weinvestigate the behavior of the proposed algorithm in caseof higher order singularly perturbed systems The linearized10th-ordermathematical model of the gas hydrogen reformerintroduced and studied in [25ndash27] is chosen The systemmatrices are given as follows [25]

119860 =

[[[[[[[[[[[[[[[[[[[[[[[

minus0074 0 0 0 0 0 minus353 10748 0 00 minus1468 minus253 0 0 0 0 0 25582 139110 0 minus156 0 0 0 0 0 0 335860 0 0 minus1245 21263 0 11269 11269 0 00 0 0 0 minus3333 0 0 0 0 00 0 0 0 0 minus3243 32304 32304 0 00 0 0 0 0 3318 minus344 minus341 0 990420 0 0 22197 0 0 minus2532 minus2549 0 325260 0 20354 0 0 0 18309 1214 minus0358 minus3304

00188 0 81642 0 0 0 56043 53994 0 minus1361

]]]]]]]]]]]]]]]]]]]]]]]

119861 = [0 0 0 0 012 0 0 0 0 00 0 0 0 0 01834 0 0 0 0]

119879

119862 = [1 0 0 0 0 0 0 0 0 00 0994 minus0088 0 0 0 0 0 0 0]

(34)

Mathematical Problems in Engineering 7

Table 2 Error norm values for each iteration for the chemical plant

Number of iterations 119894 10038171003817100381710038171003817119882119883 minus 119882(119894)119883 1003817100381710038171003817100381722 6549205248602119890 minus 34 3305462486403183119890 minus 046 1644553914568401119890 minus 058 7961048437784984119890 minus 0710 3791658707506841119890 minus 0812 1790737974576206119890 minus 0914 8421250159940589119890 minus 1116 3944852245039569119890 minus 1218 1781054537942190119890 minus 1320 1746372245754009119890 minus 14

The eigenvalues of the system matrix 119860 are as followsminus66068minus1579minus89137minus12175minus333minus277plusmn0547119894minus1468minus0358 and minus00861 All real parts of those eigenvalues liein the left part of the complex plane hence the systemis asymptotically stable Moreover the system has multipletime scales (slow and fast) since there are three eigenvalueslocated very close to the imaginary axis while the otherseven eigenvalues are located far from that axis The singularperturbation parameter 120576 is chosen as the ratio of the fastestslow eigenvalue to the slowest fast eigenvalue and is equal to120576 = 052 = 1468277

What are supposed to be slowmode eigenvalues obtainedvia (15) minus1468 minus00838 and minus1284495 and what aresupposed to be the fastmode eigenvalues (eigenvalues of1198604)minus0358minus66068minus89144minus13954minus2829plusmn0886 andminus3333

Table 3 Error norm values for each iteration for the gas reformersystem

Number of iterations 119894 10038171003817100381710038171003817119882119883 minus 119882(119894)119883 1003817100381710038171003817100381722 566707478610707585 1326046318575198610 038648979991069112 50958037951316119890 minus 0215 5949048480951119890 minus 0320 6505261253303921119890 minus 0423 6630466299301564119890 minus 0525 7912793405030453119890 minus 0630 7210694103862186119890 minus 0735 3050169378906093119890 minus 0840 8638446874677160119890 minus 10

do not display the slow-fast time scale separation Similarto the chemical plant example in the previous section wecan notice that the slow and fast eigenvalues are not wellseparated into two disjoint groups However since the gasformer system is of higher order it will be a little cumbersometo come up with a permutation matrix that would convert thesystem into its explicit singularly perturbed form Thereforethe algorithm presented in [21 28] and used by [27] to studythe slow and fast dynamics of the gas reformer system isalso considered here to convert the system from an implicitsingularly perturbed form to the explicit singularly perturbedform The algorithm in [21 28] is based on introducing asimilarity transformation 119879 that transforms the general linearsystem in implicit singularly perturbed form into the explicitsingularly perturbed form defined in (9) The similaritytransformation 119879 is given by [27]

119879 =

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

48557 21797 minus03535 17829 11374 minus52659 minus51468 1 0 00 0 0 0 0 0 0 0 1 0

minus03794 07980 minus00771 0 0 52667 05148 0 0 10 0 1 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 00 0 0 0 1 0 0 0 0 00 0 0 0 0 1 0 0 0 00 0 0 0 0 0 1 0 0 00 0 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 0 1

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

(35)

8 Mathematical Problems in Engineering

The new transformed system matrices are defined as

119860SP = 119879119860119879minus1119861SP = 119879119861119862SP = 119862119879minus1

(36)

Thereby they are calculated as

119860SP =

[[[[[[[[[[[[[[[[[[[[[[[

minus03192 55761 minus3138 minus0353 05681 363094 minus0259 minus0050 55156 313790 minus0358 0 20354 0 0 0 1831 12140 minus3304

minus00866 20413 minus12315 minus0125 01544 98473 19252 01901 minus03195 123140 0 0 minus1245 0 0 0 0 0 335860 0 0 0 minus1245 21263 0 11269 11269 00 0 0 0 0 minus3333 0 0 0 00 0 0 0 0 0 minus3243 32304 32304 00 0 0 22197 0 0 3318 minus344 minus341 990420 0 20354 0 22197 0 0 minus2532 minus2549 32526

00188 0 81642 0 minus00057 minus03629 0213 5625 53962 minus1361

]]]]]]]]]]]]]]]]]]]]]]]

119861SP = [136488 0 0 0 0 012 0 0 0 0minus96577 0 09659 0 0 0 01834 0 0 0]

119879

119862SP = [01697 0 minus04636 00243 minus0302 minus19303 113788 1112 minus01697 0463600802 0 10266 00195 minus0143 minus91238 minus11825 minus0115 minus00802 minus10266]

(37)

The matrices 119860SP and 119860 have the same eigenvalues sincethey are preserved under the similarity transformation Theslow mode eigenvalues are minus1468 minus0358 and minus008552while the fast mode eigenvalues are minus660682 minus15789 89137minus12174 minus3333 and minus27697 plusmn 060087 They are clearlyseparated now into two disjoint groups

Using our proposed algorithm to calculate the initial crossGrammian matrix 119882(0) and compare the result to the exactsolution of the algebraic Sylvester equation we get the errornorm as 10038171003817100381710038171003817119882119883 minus 119882(0)100381710038171003817100381710038172 = 53489147248362102 (38)

The cross Grammian matrix is then calculated usingthe proposed recursive algorithm The error norm for eachiteration is shown in Table 3

5 Conclusions

The algorithm was developed to solve the algebraic Sylvesterequation whose solution defines the cross Grammian ofsingularly perturbed linear systems The algorithm is veryefficient defined in terms of reduced-order subproblemscorresponding to slow and fast subsystems and convergesrapidly to the required solution The efficacy of the algorithmis demonstrated on three real physical examples The algo-rithm can be directly applied to singularly perturbed systems

in the explicit standard forms A similarity transformationneeds to be applied to singularly perturbed systems in implicitforms to convert them into their explicit form before theproposed algorithm can be applied to this class of systems

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

References

[1] Z Gajic Multimodel control and estimation of linear stochastic[PhD Dissertation] Michigan State University 1984

[2] T Grodt and Z Gajic ldquoThe recursive reduced-order numericalsolution of the singularly perturbed matrix differential Riccatiequationrdquo IEEE Transactions on Automatic Control vol 33 no8 pp 751ndash754 1988

[3] W Su Z Gajic and X Shen ldquoThe recursive reduced-ordersolution of an open-loop control problem of linear singularlyperturbed systemsrdquo IEEE Transactions on Automatic Controlvol 37 no 2 pp 279ndash281 1992

[4] Z Gajic andM-T LimOptimal Control of Singularly PerturbedLinear Systems and Applications High-Accuracy TechniquesMarcel Dekker New York NY USA 2001

[5] W C Su Z Gajic and X M Shen ldquoThe exact slow-fastdecomposition of the algebraic Ricatti equation of singularly

Mathematical Problems in Engineering 9

perturbed systemsrdquo IEEE Transactions on Automatic Controlvol 37 no 9 pp 1456ndash1459 1992

[6] Z Gajic and X Shen Parallel Algorithms for Optimal Control ofLarge Scale Linear Systems Springer London UK 1993

[7] I Borno andZGajic ldquoParallel algorithms for optimal control ofweakly coupled and singularly perturbed jump linear systemsrdquoAutomatica vol 31 no 7 pp 985ndash988 1995

[8] H Mukaidani H Xu and K Mizukami ldquoNew iterative algo-rithm for algebraic riccati equation related to 119867infin controlproblem of singularly perturbed systemsrdquo IEEE Transactions onAutomatic Control vol 46 no 10 pp 1659ndash1666 2001

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

[10] 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

[11] V Sreeram ldquoRecursive technique for computation of Grami-ansrdquo IEE Proceedings D Control Theory and Applications vol140 no 3 pp 160ndash166 1993

[12] K CWan andV Sreeram ldquoSolution of the bilinearmatrix equa-tion using Astrom-Jury-Agniel algorithmrdquo IEE Proceedings-Dvol 140 pp 21160ndash2166 1993

[13] P Lancaster and M Tismenetsky The Theory of MatricesComputer Science and Applied Mathematics Academic PressOrlando Fla USA 2nd edition 1985

[14] U Baur ldquoLow rank solution of data-sparse Sylvester equationsrdquoNumerical Linear Algebra with Applications vol 15 no 9 pp837ndash851 2008

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

[16] P Benner E S Quintana-Ortı and G Quintana-Ortı ldquoSolvingstable Sylvester equations via rational iterative schemesrdquo Journalof Scientific Computing vol 28 no 1 pp 51ndash83 2006

[17] A J Laub L M Silverman and M Verma ldquoNote on cross-grammians for symmetric realizationsrdquo Proceedings of the IEEEvol 71 no 7 pp 904ndash905 1983

[18] K Fernando and H Nicholson ldquoOn a fundamental property ofthe cross- Gramian matrixrdquo IEEE Transactions on Circuits andSystems vol 31 no 5 pp 504ndash505 1984

[19] K V Fernando and H Nicholson ldquoReachabilty observabilityand minimality of mimo systemsrdquo Proceedings of the IEEE vol72 no 12 pp 1820ndash1821 1984

[20] 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

[21] P Kokotovic H Khalil and J OrsquoReilly Singular PerturbationMethods in Control Analysis and Design Academic PressLondon UK 1986

[22] DG LuenbergerOptimization byVector SpaceMethodsWiley-Interscience New York NY USA 1969

[23] A N Andry Jr J C Chung and E Y Shapiro ldquoModalizedobserversrdquo IEEE Transactions on Automatic Control vol 29 no7 pp 669ndash672 1984

[24] V G Shankar and K Ramar ldquoPole assignment with minimumeigenvalue sensitivity to plant parameter variationsrdquo Interna-tiona Jornal of Control vol 23 no 4 pp 493ndash504 1976

[25] J Pukrushpan A Stefanopoulou and H Peng Control of FuelCell Power Systems Principles Modeling Analysis and FeedbackDesign Springer Berlin Germany 2004

[26] J Pukrushpan A Stefanopoulou S Varigonda J Eborn andC Haugstetter ldquoControl-oriented model of fuel processor forhydrogen generation in fuel cell applicationsrdquoControl Engineer-ing Practice vol 14 no 3 pp 277ndash293 2006

[27] M Skataric and Z Gajic ldquoSlow and fast dynamics of a naturalgas hydrogen reformerrdquo International Journal of HydrogenEnergy vol 38 no 35 pp 15173ndash15179 2013

[28] H K Khalil ldquoTime scale decomposition of linear implicitsingularly perturbed systemsrdquo IEEE Transactions on AutomaticControl vol 29 no 11 pp 1054ndash1056 1984

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 7: Research Article Recursive Reduced-Order Algor ithm …downloads.hindawi.com/journals/mpe/2016/2452746.pdf · Research Article Recursive Reduced-Order Algor ithm for Singularly Perturbed

Mathematical Problems in Engineering 7

Table 2 Error norm values for each iteration for the chemical plant

Number of iterations 119894 10038171003817100381710038171003817119882119883 minus 119882(119894)119883 1003817100381710038171003817100381722 6549205248602119890 minus 34 3305462486403183119890 minus 046 1644553914568401119890 minus 058 7961048437784984119890 minus 0710 3791658707506841119890 minus 0812 1790737974576206119890 minus 0914 8421250159940589119890 minus 1116 3944852245039569119890 minus 1218 1781054537942190119890 minus 1320 1746372245754009119890 minus 14

The eigenvalues of the system matrix 119860 are as followsminus66068minus1579minus89137minus12175minus333minus277plusmn0547119894minus1468minus0358 and minus00861 All real parts of those eigenvalues liein the left part of the complex plane hence the systemis asymptotically stable Moreover the system has multipletime scales (slow and fast) since there are three eigenvalueslocated very close to the imaginary axis while the otherseven eigenvalues are located far from that axis The singularperturbation parameter 120576 is chosen as the ratio of the fastestslow eigenvalue to the slowest fast eigenvalue and is equal to120576 = 052 = 1468277

What are supposed to be slowmode eigenvalues obtainedvia (15) minus1468 minus00838 and minus1284495 and what aresupposed to be the fastmode eigenvalues (eigenvalues of1198604)minus0358minus66068minus89144minus13954minus2829plusmn0886 andminus3333

Table 3 Error norm values for each iteration for the gas reformersystem

Number of iterations 119894 10038171003817100381710038171003817119882119883 minus 119882(119894)119883 1003817100381710038171003817100381722 566707478610707585 1326046318575198610 038648979991069112 50958037951316119890 minus 0215 5949048480951119890 minus 0320 6505261253303921119890 minus 0423 6630466299301564119890 minus 0525 7912793405030453119890 minus 0630 7210694103862186119890 minus 0735 3050169378906093119890 minus 0840 8638446874677160119890 minus 10

do not display the slow-fast time scale separation Similarto the chemical plant example in the previous section wecan notice that the slow and fast eigenvalues are not wellseparated into two disjoint groups However since the gasformer system is of higher order it will be a little cumbersometo come up with a permutation matrix that would convert thesystem into its explicit singularly perturbed form Thereforethe algorithm presented in [21 28] and used by [27] to studythe slow and fast dynamics of the gas reformer system isalso considered here to convert the system from an implicitsingularly perturbed form to the explicit singularly perturbedform The algorithm in [21 28] is based on introducing asimilarity transformation 119879 that transforms the general linearsystem in implicit singularly perturbed form into the explicitsingularly perturbed form defined in (9) The similaritytransformation 119879 is given by [27]

119879 =

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

48557 21797 minus03535 17829 11374 minus52659 minus51468 1 0 00 0 0 0 0 0 0 0 1 0

minus03794 07980 minus00771 0 0 52667 05148 0 0 10 0 1 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 00 0 0 0 1 0 0 0 0 00 0 0 0 0 1 0 0 0 00 0 0 0 0 0 1 0 0 00 0 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 0 1

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

(35)

8 Mathematical Problems in Engineering

The new transformed system matrices are defined as

119860SP = 119879119860119879minus1119861SP = 119879119861119862SP = 119862119879minus1

(36)

Thereby they are calculated as

119860SP =

[[[[[[[[[[[[[[[[[[[[[[[

minus03192 55761 minus3138 minus0353 05681 363094 minus0259 minus0050 55156 313790 minus0358 0 20354 0 0 0 1831 12140 minus3304

minus00866 20413 minus12315 minus0125 01544 98473 19252 01901 minus03195 123140 0 0 minus1245 0 0 0 0 0 335860 0 0 0 minus1245 21263 0 11269 11269 00 0 0 0 0 minus3333 0 0 0 00 0 0 0 0 0 minus3243 32304 32304 00 0 0 22197 0 0 3318 minus344 minus341 990420 0 20354 0 22197 0 0 minus2532 minus2549 32526

00188 0 81642 0 minus00057 minus03629 0213 5625 53962 minus1361

]]]]]]]]]]]]]]]]]]]]]]]

119861SP = [136488 0 0 0 0 012 0 0 0 0minus96577 0 09659 0 0 0 01834 0 0 0]

119879

119862SP = [01697 0 minus04636 00243 minus0302 minus19303 113788 1112 minus01697 0463600802 0 10266 00195 minus0143 minus91238 minus11825 minus0115 minus00802 minus10266]

(37)

The matrices 119860SP and 119860 have the same eigenvalues sincethey are preserved under the similarity transformation Theslow mode eigenvalues are minus1468 minus0358 and minus008552while the fast mode eigenvalues are minus660682 minus15789 89137minus12174 minus3333 and minus27697 plusmn 060087 They are clearlyseparated now into two disjoint groups

Using our proposed algorithm to calculate the initial crossGrammian matrix 119882(0) and compare the result to the exactsolution of the algebraic Sylvester equation we get the errornorm as 10038171003817100381710038171003817119882119883 minus 119882(0)100381710038171003817100381710038172 = 53489147248362102 (38)

The cross Grammian matrix is then calculated usingthe proposed recursive algorithm The error norm for eachiteration is shown in Table 3

5 Conclusions

The algorithm was developed to solve the algebraic Sylvesterequation whose solution defines the cross Grammian ofsingularly perturbed linear systems The algorithm is veryefficient defined in terms of reduced-order subproblemscorresponding to slow and fast subsystems and convergesrapidly to the required solution The efficacy of the algorithmis demonstrated on three real physical examples The algo-rithm can be directly applied to singularly perturbed systems

in the explicit standard forms A similarity transformationneeds to be applied to singularly perturbed systems in implicitforms to convert them into their explicit form before theproposed algorithm can be applied to this class of systems

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

References

[1] Z Gajic Multimodel control and estimation of linear stochastic[PhD Dissertation] Michigan State University 1984

[2] T Grodt and Z Gajic ldquoThe recursive reduced-order numericalsolution of the singularly perturbed matrix differential Riccatiequationrdquo IEEE Transactions on Automatic Control vol 33 no8 pp 751ndash754 1988

[3] W Su Z Gajic and X Shen ldquoThe recursive reduced-ordersolution of an open-loop control problem of linear singularlyperturbed systemsrdquo IEEE Transactions on Automatic Controlvol 37 no 2 pp 279ndash281 1992

[4] Z Gajic andM-T LimOptimal Control of Singularly PerturbedLinear Systems and Applications High-Accuracy TechniquesMarcel Dekker New York NY USA 2001

[5] W C Su Z Gajic and X M Shen ldquoThe exact slow-fastdecomposition of the algebraic Ricatti equation of singularly

Mathematical Problems in Engineering 9

perturbed systemsrdquo IEEE Transactions on Automatic Controlvol 37 no 9 pp 1456ndash1459 1992

[6] Z Gajic and X Shen Parallel Algorithms for Optimal Control ofLarge Scale Linear Systems Springer London UK 1993

[7] I Borno andZGajic ldquoParallel algorithms for optimal control ofweakly coupled and singularly perturbed jump linear systemsrdquoAutomatica vol 31 no 7 pp 985ndash988 1995

[8] H Mukaidani H Xu and K Mizukami ldquoNew iterative algo-rithm for algebraic riccati equation related to 119867infin controlproblem of singularly perturbed systemsrdquo IEEE Transactions onAutomatic Control vol 46 no 10 pp 1659ndash1666 2001

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

[10] 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

[11] V Sreeram ldquoRecursive technique for computation of Grami-ansrdquo IEE Proceedings D Control Theory and Applications vol140 no 3 pp 160ndash166 1993

[12] K CWan andV Sreeram ldquoSolution of the bilinearmatrix equa-tion using Astrom-Jury-Agniel algorithmrdquo IEE Proceedings-Dvol 140 pp 21160ndash2166 1993

[13] P Lancaster and M Tismenetsky The Theory of MatricesComputer Science and Applied Mathematics Academic PressOrlando Fla USA 2nd edition 1985

[14] U Baur ldquoLow rank solution of data-sparse Sylvester equationsrdquoNumerical Linear Algebra with Applications vol 15 no 9 pp837ndash851 2008

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

[16] P Benner E S Quintana-Ortı and G Quintana-Ortı ldquoSolvingstable Sylvester equations via rational iterative schemesrdquo Journalof Scientific Computing vol 28 no 1 pp 51ndash83 2006

[17] A J Laub L M Silverman and M Verma ldquoNote on cross-grammians for symmetric realizationsrdquo Proceedings of the IEEEvol 71 no 7 pp 904ndash905 1983

[18] K Fernando and H Nicholson ldquoOn a fundamental property ofthe cross- Gramian matrixrdquo IEEE Transactions on Circuits andSystems vol 31 no 5 pp 504ndash505 1984

[19] K V Fernando and H Nicholson ldquoReachabilty observabilityand minimality of mimo systemsrdquo Proceedings of the IEEE vol72 no 12 pp 1820ndash1821 1984

[20] 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

[21] P Kokotovic H Khalil and J OrsquoReilly Singular PerturbationMethods in Control Analysis and Design Academic PressLondon UK 1986

[22] DG LuenbergerOptimization byVector SpaceMethodsWiley-Interscience New York NY USA 1969

[23] A N Andry Jr J C Chung and E Y Shapiro ldquoModalizedobserversrdquo IEEE Transactions on Automatic Control vol 29 no7 pp 669ndash672 1984

[24] V G Shankar and K Ramar ldquoPole assignment with minimumeigenvalue sensitivity to plant parameter variationsrdquo Interna-tiona Jornal of Control vol 23 no 4 pp 493ndash504 1976

[25] J Pukrushpan A Stefanopoulou and H Peng Control of FuelCell Power Systems Principles Modeling Analysis and FeedbackDesign Springer Berlin Germany 2004

[26] J Pukrushpan A Stefanopoulou S Varigonda J Eborn andC Haugstetter ldquoControl-oriented model of fuel processor forhydrogen generation in fuel cell applicationsrdquoControl Engineer-ing Practice vol 14 no 3 pp 277ndash293 2006

[27] M Skataric and Z Gajic ldquoSlow and fast dynamics of a naturalgas hydrogen reformerrdquo International Journal of HydrogenEnergy vol 38 no 35 pp 15173ndash15179 2013

[28] H K Khalil ldquoTime scale decomposition of linear implicitsingularly perturbed systemsrdquo IEEE Transactions on AutomaticControl vol 29 no 11 pp 1054ndash1056 1984

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 8: Research Article Recursive Reduced-Order Algor ithm …downloads.hindawi.com/journals/mpe/2016/2452746.pdf · Research Article Recursive Reduced-Order Algor ithm for Singularly Perturbed

8 Mathematical Problems in Engineering

The new transformed system matrices are defined as

119860SP = 119879119860119879minus1119861SP = 119879119861119862SP = 119862119879minus1

(36)

Thereby they are calculated as

119860SP =

[[[[[[[[[[[[[[[[[[[[[[[

minus03192 55761 minus3138 minus0353 05681 363094 minus0259 minus0050 55156 313790 minus0358 0 20354 0 0 0 1831 12140 minus3304

minus00866 20413 minus12315 minus0125 01544 98473 19252 01901 minus03195 123140 0 0 minus1245 0 0 0 0 0 335860 0 0 0 minus1245 21263 0 11269 11269 00 0 0 0 0 minus3333 0 0 0 00 0 0 0 0 0 minus3243 32304 32304 00 0 0 22197 0 0 3318 minus344 minus341 990420 0 20354 0 22197 0 0 minus2532 minus2549 32526

00188 0 81642 0 minus00057 minus03629 0213 5625 53962 minus1361

]]]]]]]]]]]]]]]]]]]]]]]

119861SP = [136488 0 0 0 0 012 0 0 0 0minus96577 0 09659 0 0 0 01834 0 0 0]

119879

119862SP = [01697 0 minus04636 00243 minus0302 minus19303 113788 1112 minus01697 0463600802 0 10266 00195 minus0143 minus91238 minus11825 minus0115 minus00802 minus10266]

(37)

The matrices 119860SP and 119860 have the same eigenvalues sincethey are preserved under the similarity transformation Theslow mode eigenvalues are minus1468 minus0358 and minus008552while the fast mode eigenvalues are minus660682 minus15789 89137minus12174 minus3333 and minus27697 plusmn 060087 They are clearlyseparated now into two disjoint groups

Using our proposed algorithm to calculate the initial crossGrammian matrix 119882(0) and compare the result to the exactsolution of the algebraic Sylvester equation we get the errornorm as 10038171003817100381710038171003817119882119883 minus 119882(0)100381710038171003817100381710038172 = 53489147248362102 (38)

The cross Grammian matrix is then calculated usingthe proposed recursive algorithm The error norm for eachiteration is shown in Table 3

5 Conclusions

The algorithm was developed to solve the algebraic Sylvesterequation whose solution defines the cross Grammian ofsingularly perturbed linear systems The algorithm is veryefficient defined in terms of reduced-order subproblemscorresponding to slow and fast subsystems and convergesrapidly to the required solution The efficacy of the algorithmis demonstrated on three real physical examples The algo-rithm can be directly applied to singularly perturbed systems

in the explicit standard forms A similarity transformationneeds to be applied to singularly perturbed systems in implicitforms to convert them into their explicit form before theproposed algorithm can be applied to this class of systems

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

References

[1] Z Gajic Multimodel control and estimation of linear stochastic[PhD Dissertation] Michigan State University 1984

[2] T Grodt and Z Gajic ldquoThe recursive reduced-order numericalsolution of the singularly perturbed matrix differential Riccatiequationrdquo IEEE Transactions on Automatic Control vol 33 no8 pp 751ndash754 1988

[3] W Su Z Gajic and X Shen ldquoThe recursive reduced-ordersolution of an open-loop control problem of linear singularlyperturbed systemsrdquo IEEE Transactions on Automatic Controlvol 37 no 2 pp 279ndash281 1992

[4] Z Gajic andM-T LimOptimal Control of Singularly PerturbedLinear Systems and Applications High-Accuracy TechniquesMarcel Dekker New York NY USA 2001

[5] W C Su Z Gajic and X M Shen ldquoThe exact slow-fastdecomposition of the algebraic Ricatti equation of singularly

Mathematical Problems in Engineering 9

perturbed systemsrdquo IEEE Transactions on Automatic Controlvol 37 no 9 pp 1456ndash1459 1992

[6] Z Gajic and X Shen Parallel Algorithms for Optimal Control ofLarge Scale Linear Systems Springer London UK 1993

[7] I Borno andZGajic ldquoParallel algorithms for optimal control ofweakly coupled and singularly perturbed jump linear systemsrdquoAutomatica vol 31 no 7 pp 985ndash988 1995

[8] H Mukaidani H Xu and K Mizukami ldquoNew iterative algo-rithm for algebraic riccati equation related to 119867infin controlproblem of singularly perturbed systemsrdquo IEEE Transactions onAutomatic Control vol 46 no 10 pp 1659ndash1666 2001

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

[10] 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

[11] V Sreeram ldquoRecursive technique for computation of Grami-ansrdquo IEE Proceedings D Control Theory and Applications vol140 no 3 pp 160ndash166 1993

[12] K CWan andV Sreeram ldquoSolution of the bilinearmatrix equa-tion using Astrom-Jury-Agniel algorithmrdquo IEE Proceedings-Dvol 140 pp 21160ndash2166 1993

[13] P Lancaster and M Tismenetsky The Theory of MatricesComputer Science and Applied Mathematics Academic PressOrlando Fla USA 2nd edition 1985

[14] U Baur ldquoLow rank solution of data-sparse Sylvester equationsrdquoNumerical Linear Algebra with Applications vol 15 no 9 pp837ndash851 2008

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

[16] P Benner E S Quintana-Ortı and G Quintana-Ortı ldquoSolvingstable Sylvester equations via rational iterative schemesrdquo Journalof Scientific Computing vol 28 no 1 pp 51ndash83 2006

[17] A J Laub L M Silverman and M Verma ldquoNote on cross-grammians for symmetric realizationsrdquo Proceedings of the IEEEvol 71 no 7 pp 904ndash905 1983

[18] K Fernando and H Nicholson ldquoOn a fundamental property ofthe cross- Gramian matrixrdquo IEEE Transactions on Circuits andSystems vol 31 no 5 pp 504ndash505 1984

[19] K V Fernando and H Nicholson ldquoReachabilty observabilityand minimality of mimo systemsrdquo Proceedings of the IEEE vol72 no 12 pp 1820ndash1821 1984

[20] 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

[21] P Kokotovic H Khalil and J OrsquoReilly Singular PerturbationMethods in Control Analysis and Design Academic PressLondon UK 1986

[22] DG LuenbergerOptimization byVector SpaceMethodsWiley-Interscience New York NY USA 1969

[23] A N Andry Jr J C Chung and E Y Shapiro ldquoModalizedobserversrdquo IEEE Transactions on Automatic Control vol 29 no7 pp 669ndash672 1984

[24] V G Shankar and K Ramar ldquoPole assignment with minimumeigenvalue sensitivity to plant parameter variationsrdquo Interna-tiona Jornal of Control vol 23 no 4 pp 493ndash504 1976

[25] J Pukrushpan A Stefanopoulou and H Peng Control of FuelCell Power Systems Principles Modeling Analysis and FeedbackDesign Springer Berlin Germany 2004

[26] J Pukrushpan A Stefanopoulou S Varigonda J Eborn andC Haugstetter ldquoControl-oriented model of fuel processor forhydrogen generation in fuel cell applicationsrdquoControl Engineer-ing Practice vol 14 no 3 pp 277ndash293 2006

[27] M Skataric and Z Gajic ldquoSlow and fast dynamics of a naturalgas hydrogen reformerrdquo International Journal of HydrogenEnergy vol 38 no 35 pp 15173ndash15179 2013

[28] H K Khalil ldquoTime scale decomposition of linear implicitsingularly perturbed systemsrdquo IEEE Transactions on AutomaticControl vol 29 no 11 pp 1054ndash1056 1984

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 9: Research Article Recursive Reduced-Order Algor ithm …downloads.hindawi.com/journals/mpe/2016/2452746.pdf · Research Article Recursive Reduced-Order Algor ithm for Singularly Perturbed

Mathematical Problems in Engineering 9

perturbed systemsrdquo IEEE Transactions on Automatic Controlvol 37 no 9 pp 1456ndash1459 1992

[6] Z Gajic and X Shen Parallel Algorithms for Optimal Control ofLarge Scale Linear Systems Springer London UK 1993

[7] I Borno andZGajic ldquoParallel algorithms for optimal control ofweakly coupled and singularly perturbed jump linear systemsrdquoAutomatica vol 31 no 7 pp 985ndash988 1995

[8] H Mukaidani H Xu and K Mizukami ldquoNew iterative algo-rithm for algebraic riccati equation related to 119867infin controlproblem of singularly perturbed systemsrdquo IEEE Transactions onAutomatic Control vol 46 no 10 pp 1659ndash1666 2001

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

[10] 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

[11] V Sreeram ldquoRecursive technique for computation of Grami-ansrdquo IEE Proceedings D Control Theory and Applications vol140 no 3 pp 160ndash166 1993

[12] K CWan andV Sreeram ldquoSolution of the bilinearmatrix equa-tion using Astrom-Jury-Agniel algorithmrdquo IEE Proceedings-Dvol 140 pp 21160ndash2166 1993

[13] P Lancaster and M Tismenetsky The Theory of MatricesComputer Science and Applied Mathematics Academic PressOrlando Fla USA 2nd edition 1985

[14] U Baur ldquoLow rank solution of data-sparse Sylvester equationsrdquoNumerical Linear Algebra with Applications vol 15 no 9 pp837ndash851 2008

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

[16] P Benner E S Quintana-Ortı and G Quintana-Ortı ldquoSolvingstable Sylvester equations via rational iterative schemesrdquo Journalof Scientific Computing vol 28 no 1 pp 51ndash83 2006

[17] A J Laub L M Silverman and M Verma ldquoNote on cross-grammians for symmetric realizationsrdquo Proceedings of the IEEEvol 71 no 7 pp 904ndash905 1983

[18] K Fernando and H Nicholson ldquoOn a fundamental property ofthe cross- Gramian matrixrdquo IEEE Transactions on Circuits andSystems vol 31 no 5 pp 504ndash505 1984

[19] K V Fernando and H Nicholson ldquoReachabilty observabilityand minimality of mimo systemsrdquo Proceedings of the IEEE vol72 no 12 pp 1820ndash1821 1984

[20] 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

[21] P Kokotovic H Khalil and J OrsquoReilly Singular PerturbationMethods in Control Analysis and Design Academic PressLondon UK 1986

[22] DG LuenbergerOptimization byVector SpaceMethodsWiley-Interscience New York NY USA 1969

[23] A N Andry Jr J C Chung and E Y Shapiro ldquoModalizedobserversrdquo IEEE Transactions on Automatic Control vol 29 no7 pp 669ndash672 1984

[24] V G Shankar and K Ramar ldquoPole assignment with minimumeigenvalue sensitivity to plant parameter variationsrdquo Interna-tiona Jornal of Control vol 23 no 4 pp 493ndash504 1976

[25] J Pukrushpan A Stefanopoulou and H Peng Control of FuelCell Power Systems Principles Modeling Analysis and FeedbackDesign Springer Berlin Germany 2004

[26] J Pukrushpan A Stefanopoulou S Varigonda J Eborn andC Haugstetter ldquoControl-oriented model of fuel processor forhydrogen generation in fuel cell applicationsrdquoControl Engineer-ing Practice vol 14 no 3 pp 277ndash293 2006

[27] M Skataric and Z Gajic ldquoSlow and fast dynamics of a naturalgas hydrogen reformerrdquo International Journal of HydrogenEnergy vol 38 no 35 pp 15173ndash15179 2013

[28] H K Khalil ldquoTime scale decomposition of linear implicitsingularly perturbed systemsrdquo IEEE Transactions on AutomaticControl vol 29 no 11 pp 1054ndash1056 1984

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 10: Research Article Recursive Reduced-Order Algor ithm …downloads.hindawi.com/journals/mpe/2016/2452746.pdf · Research Article Recursive Reduced-Order Algor ithm for Singularly Perturbed

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


A GENERALIZED CONJUGATEGRADIENTALGOR ITHM FOR …i.stanford.edu/pub/cstr/reports/cs/tr/77/638/CS-TR-77-638.pdf · A GENERALIZED CONJUGATEGRADIENTALGOR ITHM FOR SOLVINGA CLASS Ok-QUADRATIC

A GENERALIZED CONJUGATEGRADIENTALGOR ITHM FOR …i.stanford.edu/pub/cstr/reports/cs/tr/77/638/CS-TR-77-638.pdf · A GENERALIZED CONJUGATEGRADIENTALGOR ITHM FOR SOLVINGA CLASS Ok-QUADRATIC

Recursive Dynamic CS: Recursive Recovery of Sparse Signal ...home.engineering.iastate.edu/~namrata/TSP_RecRecon_Paper_4.pdf · Recursive Dynamic CS: Recursive Recovery of Sparse Signal

Recursive Dynamic CS: Recursive Recovery of Sparse Signal ...home.engineering.iastate.edu/~namrata/TSP_RecRecon_Paper_4.pdf · Recursive Dynamic CS: Recursive Recovery of Sparse Signal

[Marc Moonen] SVD and Signal Processing III Algor(BookFi.org)

[Marc Moonen] SVD and Signal Processing III Algor(BookFi.org)

Char t Parsing Reading - staff.um.edu.mtstaff.um.edu.mt/mros1/hlt/PDF/Chart-Parsing_lavie09.pdfThe Main Algor ithm: parsing input 1 1. 0 2. If Agenda is empty and then set 1, find

Char t Parsing Reading - staff.um.edu.mtstaff.um.edu.mt/mros1/hlt/PDF/Chart-Parsing_lavie09.pdfThe Main Algor ithm: parsing input 1 1. 0 2. If Agenda is empty and then set 1, find

ITHM 101 INTRODUCTION TO THE HOSPITALITY INDUSTRY … · 2014-06-26 · ITHM 101 Introduction to the Hospitality Industry 3 Prep. 08-09-03. Prof. Manuel A. Rivera Ramirez Rev. 02-04-06

ITHM 101 INTRODUCTION TO THE HOSPITALITY INDUSTRY … · 2014-06-26 · ITHM 101 Introduction to the Hospitality Industry 3 Prep. 08-09-03. Prof. Manuel A. Rivera Ramirez Rev. 02-04-06

ALGOR mc-tutorial

ALGOR mc-tutorial

Chapter 11 – Recursion Recursive Processes Writing a Recursive Method A Recursive Factorial Method Comparison of Recursive and Iterative Solutions Recursive.

Chapter 11 – Recursion Recursive Processes Writing a Recursive Method A Recursive Factorial Method Comparison of Recursive and Iterative Solutions Recursive.

ITHM 370 Hospitality Sales and Marketing 1...ITHM 370 Hospitality Sales and Marketing 5 Updated, 06/09/2011 6. Identificar los elementos de publicidad, relaciones públicas, venta

ITHM 370 Hospitality Sales and Marketing 1...ITHM 370 Hospitality Sales and Marketing 5 Updated, 06/09/2011 6. Identificar los elementos de publicidad, relaciones públicas, venta

Applications Mining: and Cursebobd/ECML_Tutorial/ECML_handouts.… · ithm Sample S of N in R d; m mixture components; , . min: considered. theoretical ithm.) 1 a k the space R d

Applications Mining: and Cursebobd/ECML_Tutorial/ECML_handouts.… · ithm Sample S of N in R d; m mixture components; , . min: considered. theoretical ithm.) 1 a k the space R d

Curso de ALGOR - astroscu.unam.mxfarah/FEA/Curso FEA ALGOR/Clase 5/ALGOR.pdf · SolidWorks CAD prt asm sat igs step ALGOR CAE esd dxf - prt sat igs* step* WorkingModel CAE Normalizados

Curso de ALGOR - astroscu.unam.mxfarah/FEA/Curso FEA ALGOR/Clase 5/ALGOR.pdf · SolidWorks CAD prt asm sat igs step ALGOR CAE esd dxf - prt sat igs* step* WorkingModel CAE Normalizados

Tutorial Algor

Tutorial Algor

Recursive Static Route · Information About Recursive Static Route Recursive Static Routes Arecursivestaticrouteisaroutewhosenexthopandthedestinationnetworkarecoveredbyanotherlearned

Recursive Static Route · Information About Recursive Static Route Recursive Static Routes Arecursivestaticrouteisaroutewhosenexthopandthedestinationnetworkarecoveredbyanotherlearned

Princeton)University)–)ComputerScience) COS226:*Data ...€¦ · ithm read ithm ithm ifyo ithm ithm ithm ohpl inga ohpl sort ohpl lgor ingt ohpl ohpl ohpl sort ingm ease uare idea

Princeton)University)–)ComputerScience) COS226:*Data ...€¦ · ithm read ithm ithm ifyo ithm ithm ithm ohpl inga ohpl sort ohpl lgor ingt ohpl ohpl ohpl sort ingm ease uare idea

Algor ithm for Flicker Pr ediction caused by EAFapic/uploads/Forum/poster11.pdf · 2013-11-07 · spectrum of EAF 1 for the melting period. Figure 1 – EAF construction and layout.

Algor ithm for Flicker Pr ediction caused by EAFapic/uploads/Forum/poster11.pdf · 2013-11-07 · spectrum of EAF 1 for the melting period. Figure 1 – EAF construction and layout.

CO NV E R GE NCE OF TH E DOMINANT P O LE ALGOR ITHM A …sleij101/Preprints/RSl07preprint.pdf · CO NV E R GE NCE OF TH E DOMINANT P O LE ALGOR ITHM A ND R A YLEIGH QUO TI E NT ITER

CO NV E R GE NCE OF TH E DOMINANT P O LE ALGOR ITHM A …sleij101/Preprints/RSl07preprint.pdf · CO NV E R GE NCE OF TH E DOMINANT P O LE ALGOR ITHM A ND R A YLEIGH QUO TI E NT ITER

ithm Bijections - su18.eecs70.org · ithm) If p then ( Z = p Z ) = f 1 ;p 1 g . ithm Algorithm : I If b = then egcd( a ; 0 ( a ; 1 ; 0 ) . I let ( d 0; x 0; y 0 egcd( b ; a d b )

ithm Bijections - su18.eecs70.org · ithm) If p then ( Z = p Z ) = f 1 ;p 1 g . ithm Algorithm : I If b = then egcd( a ; 0 ( a ; 1 ; 0 ) . I let ( d 0; x 0; y 0 egcd( b ; a d b )

ALGOR Software Offers a Cost-Effective, Quality and Feature-Rich

ALGOR Software Offers a Cost-Effective, Quality and Feature-Rich

Algor Simulation Detailed Brochure Us0

Algor Simulation Detailed Brochure Us0

An Algor ithm to Estimate the Two-Way Fixed EffectsModelweb.stanford.edu/group/fwolak/cgi-bin/sites/default/files/jem-2014... · An Algor ithm to Estimate the Two-Way Fixed EffectsModel

An Algor ithm to Estimate the Two-Way Fixed EffectsModelweb.stanford.edu/group/fwolak/cgi-bin/sites/default/files/jem-2014... · An Algor ithm to Estimate the Two-Way Fixed EffectsModel

Parallel Algor

Parallel Algor