research article suboptimal control using model order...

Research ArticleSuboptimal Control Using Model Order Reduction

Avadh Pati Awadhesh Kumar and Dinesh Chandra

Electrical Engineering Department Motilal Nehru National Institute of Technology Allahabad India

Correspondence should be addressed to Avadh Pati eravadhmnnitgmailcom

Received 27 September 2013 Accepted 19 December 2013 Published 3 February 2014

Academic Editors X Ma and J Yu

Copyright copy 2014 Avadh Pati et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A Pade approximation based technique for designing a suboptimal controller is presented The technique uses matching of bothtime moments and Markov parameters for model order reduction In this method the suboptimal controller is first derived forreduced order model and then implemented for higher order plant by partial feedback of measurable states

1 Introduction

The design of an optimal control law for large scale systemsis inconvenient due to one of the main drawbacks of optimalcontrol theory that requires feedback from all state variablesthat are defined to describe the dynamics of the plant It israre that all the states are available for measurement [1] andif one tries to design optimal controller Kalman filter or stateobserver is required to reconstruct the missing state of theplantThis introduces the high order dynamics in the controlfunction and leads to a complicated and costly controller Inorder tominimize this problem an attempt has beenmade forthe design of incomplete state feedback suboptimal controllaw using measurable states [2ndash4] In this paper a methodfor suboptimal controller design is proposedThe suboptimalcontrol can be achieved by using model order reductiontechnique for higher order system

The plant (higher order system) can be reduced by Padeapproximation technique (matching of both time momentsand Markov parameters) into reduced order model andthen with the help of reduced order controller one candesign the suboptimal controller for plant (higher ordersystems) The usefulness of techniques for deriving reducedorder approximations of high order system has been alreadyaccepted due to the advantages of reduced computationaleffort and increased understanding of original system [5 6]Consequently a large number of time-domain (Hankel normbalance truncation and Krylov subspace projection) andfrequency-domain (Moment matching Pade approximationRouth-Pade approximation and pole clustering methods)

system simplification techniques have been developed tosuit the different requirements and a variety of numericalapproximation are applied to obtain the reduced order model[7] In this paper the plant (higher order system) is reducedby Pade approximation technique and suboptimal controlleris derived with the reduced order model Then the subop-timal controller is implemented for original plant (higherorder system) This reduces the computational complexity ofcontroller design Two numerical examples are included toillustrate the procedure

2 The Design Procedure

Consider a single-input single-output linear (119899th-order)dynamic system described by

= 119860119909 (119905) + 119861119906 (119905)

119910 = 119862119909 (119905)

(1)

with the quadratic cost function

119895 = int

infin

0

119909119879

(119905) 119876119909 (119905) + 1198771199062

(119905) (2)

where 119876 is the (119899 times 119899) positive semidefinite matrix and 119877is a positive weight 119860 119861 and 119862 are matrices of appropriatedimensions It is well know that the optimal feedback controllaw is linear combination of the state variables A blockdiagram of the plant using control law has been shown inFigure 1

Hindawi Publishing CorporationChinese Journal of EngineeringVolume 2014 Article ID 797581 5 pageshttpdxdoiorg1011552014797581

2 Chinese Journal of Engineering

Plantminus

+

k

r(t)

Xlowast

ylowast(t)

kTX

lowast

Figure 1 Block diagram of controlled system

Higher order Model order Reduced order Suboptimal systems reduction model controller

Figure 2 Flow diagram of suboptimal control design

The control action of the plant is given as

119906 (119905) = minus119877minus1

119861119879

119875119909 = minus119896119879

119909 (3)

where 119875 is the symmetric positive definite matrix whoseelements may be obtained by solving the following matrixRiccati equation

119860119879

119875 + 119875119860 minus 119875119861119877minus1

119861119879

119875 + 119876 = 0 (4)

The closed loop transfer function with controller is obtainedas

119879 lowast (119904) = 119862119879

[119904119868 minus 119860 + 119861119896119879

]

minus1

119861

=

119904119899minus1

+ 1198861119904119899minus2

+ sdot sdot sdot + 119886119899minus21199042

+ 119886119899minus1119904 + 119886119899

119904119899+ 1198871119904119899minus1+ sdot sdot sdot + 119887

119899minus21199042+ 119887119899minus1119904 + 119887119899

(5)

21Model Order Reduction Procedure by Pade ApproximationConsider a 119899th order system [5] as

119866119899(119904) =

1198861119904119899minus1

+ 1198862119904119899minus2

+ sdot sdot sdot + 119886119899


119899

= 1199050+ 1199051119904 + 11990521199042

+ sdot sdot sdot

(expansion around 119904 = 0)

= 1198721119904minus1

+1198722119904minus2

+ sdot sdot sdot

(expansion around 119904 = infin)

(6)

It is desired to obtain a stable 119903th order approximant as

119866119903(119904) =

1198861015840

1119904119899minus1

+ 1198861015840

2119904119899minus2

+ sdot sdot sdot + 1198861015840

119903

119904119899+ 1198871015840

1119904119899minus1+ sdot sdot sdot + 119887

1015840

119903

= 1199051015840

0+ 1199051015840

1119904 + 1199051015840

21199042

+ sdot sdot sdot


= 1198721015840

1119904minus1

+1198721015840

2119904minus2

+ sdot sdot sdot


(7)

It is easy to verify that the following equations hold true

1199051015840

1=

1198861015840

119903

1198871015840

119903

119894 = 1

1199051015840

119894= 1198861015840

119903+1minus119894minus

119894minus1

sum

119895

(1199051015840

1198941198871015840

119903+119895minus119894) 1198871015840

119903

minus1

119894 = 2 3

(8)

and 1198861015840119894= 0 for 119894 le 0 1198871015840

0= 1 1198871015840119894= 0 for 119894 le minus1

1198721015840

119894=

1198861015840

119894 119894 = 1

1198861015840

119894minus

119894minus1

sum

119895=1

1198721015840

1198951198871015840

119894minus119895 119894 = 2 3

119886119894= 119887119894= 0 for 119894 = 119903 + 1 119903 + 2

(9)

We seek a reduced order model so as to satisfy

1199051015840

0= 1199050997904rArr 1198861015840

119903+1minus119894=

119894

sum

119895=1

1199051198951198871015840

119894minus119895 119894 isin 1 2 120582

1198721015840

119894= 119872119894997904rArr 1198861015840

119894=

119894

sum

119895=1

1198721198951198871015840

119894minus119895 119894 isin 1 2 120574

(10)

where 120582 + 120574 = 2119903Using (8)ndash(16) the reduced order model is obtained

22 Suboptimal Controller Design Procedure The suboptimalcontroller [4] is designed with the help of reduced ordermodel by choosing the optimal gainmatrix in such a way that

119870sub = [119870red 0 sdot sdot sdot 0] (11)

where 119870red is optimal gain matrix of reduced order modelobtained using (3) and (4) Finally this 119870sub is applied tohigher order system in order to control the plant (higherorder system)

23 Flow Diagram of Suboptimal Controller Design SeeFigure 2

24 Comparison of Responses of Optimal and SuboptimalController Using (5) one can try to determine the closed looptransfer functions with optimal controller and suboptimalcontroller with the gain matrices 119870opt and 119870sub respectivelybeing derived for comparison

3 Numerical Examples

The step-by-step procedure to obtain reduced the ordermodel is explained with the help of the example presentedbelow and procedure is derived for suboptimal controller

Example 1 Consider the following 4th order system [3]

1198664(119904) =

1199043

+ 71199042

+ 24119904 + 24

1199044+ 101199043+ 351199042+ 50119904 + 24

(12)

Chinese Journal of Engineering 3

76543210

0

01

02

03

04

05

06

07

08

09

1

Time (s)

High order systemReduced order model

Am

plitu

de

Figure 3 Step response of higher order system and reduced ordermodel

Suppose that reduced order model of the form

1198662(119904) =

1198861015840

1119904 + 1198861015840

2

1199042+ 1198871015840

1119904 + 1198871015840

2

(13)

is required

Reduction Steps The approximant can systematically bederived in the following steps for second order approximant

Step 1 Expand (12) in power series around 119904 = 0 and 119904 = infin

1198664(119904) =

1199043

+ 71199042

+ 24119904 + 24

1199044+ 101199043+ 351199042+ 50119904 + 24

= 1 minus 10833119904 + 109021199042

minus 106641199043

+ sdot sdot sdot

= 1119904minus1

minus 3119904minus2

+ 19119904minus3

minus 111119904minus4

+ 571119904minus5

+ sdot sdot sdot

(14)


1198662(119904) =

1198861015840

1119904 + 1198861015840

2

1199042+ 1198871015840

1119904 + 1198871015840

2

= 1199051015840

1+ 1199051015840

2119904 + 1199051015840

31199042

+ sdot sdot sdot

= 1198721015840

1119904minus1

+1198721015840

2119904minus2

+1198721015840

3119904minus3

+ sdot sdot sdot

(15)

The time moment and Markov parameter expansion canobtain about 119904 = 0 and 119904 = infin

Step 3 For perfect matching we have to match the (2119903) term

1199051= 1199051015840

1

1198721= 1198721015840

1

1199052= 1199051015840

2

1198722= 1198721015840

2

(16)

Step 4 By considering the above assumptions we can get thesecond order reduced model by solving the following and getthe variables of reduced order model (13)

1199051015840

1=

1198861015840

2

1198871015840

2

1199051015840

2=

1198861015840

1minus 1199051015840

11198871015840

1

1198871015840

2

1198721015840

1= 1198861015840

1

1198721015840

2= 1198861015840

2minus1198721015840

11198871015840

1

(17)

But for better overall time response approximation equalemphasis is given to both time moments andMarkov param-eters For second order reduced model we have to match2119903 terms and satisfy 120582 + 120574 = 2119903 here 119903 = 2 and consider120582 = 120574 = 2

By solving (17) we get

1198861015840

1= 1

1198861015840

2= 240096

1198871015840

1= 270096

1198871015840

2= 240096

(18)

and require the reduced order model obtained as

1198662(119904) =

119904 + 240096

1199042+ 270096119904 + 240096

(19)

The step response of higher order system (12) and reducedorder model (19) is plotted as shown in Figure 3

Model (19) is seen to be a closed approximant to (12)The optimal controller and suboptimal controller can

be designed for a particular performance index by solvingRiccati equation (4)

The optimal controller for actual higher order system (12)is obtained by using (3)-(4) as

119870opt = [00626 01275 00857 00208] (20)

and the transfer function with optimal controller is obtainedby putting the value of gain matrix (20) in (5) as

119879119870opt(119904) =

1199043

+ 71199042

+ 24119904 + 24

1199044+ 1006119904

3+ 3513119904

2+ 5086119904 + 2402

(21)

Similarly the optimal gain matrix for reduced order model(19) by using (3) and (4) is as follows

119870red = [00193 00208] (22)

The closed loop transfer function of reduced ordermodel (19)is obtained by putting (22) in (5) as

119879119870red(119904) =

119904 + 2401

1199042+ 2703119904 + 2403

(23)

4 Chinese Journal of EngineeringA

mpl

itude

87654321

0

0

01

02

03

04

05

06

07

08

09

1

Time (s)

High order system with optimal controllerReduced order model with optimal controller

Figure 4 Step response of higher order and reduced order modelwith optimal controller

Am

plitu

de

8765432100

01

02

03

04

05

06

07

08

09

1

Time (s)

High order system with optimal controllerHigh order system with suboptimal controller

Figure 5The step response of higher order systemwith optimal andwith suboptimal controller

The step response of higher order system with optimal con-troller (21) and reduced order model with optimal controller(23) is shown in Figure 4

It is found that (23) is a closed approximation to (21)The suboptimal controller (11) is obtained with the help

of reduced order model using 119870sub = [119870red 0 sdot sdot sdot 0]Thus 119870sub takes the following form

119870sub = [00193 00208 0 0] (24)

The transfer function with suboptimal controller is obtainedby putting the value of (24) in (5) as

119879119870sub (119904) =

1199043

+ 71199042

+ 24119904 + 24

1199044+ 1002119904

3+ 3502119904

2+ 50119904 + 24

(25)

The step response of higher order systemwith optimal control(21) and with suboptimal controller (25) is shown in Figure 5

The step responses of higher order system with optimalcontroller (21) and with suboptimal controller (25) are shown

25

2

15

1

05

0

Am

plitu

de

0 1 2 3 4 5 6 7 8 9 10

Time (s)


Figure 6 Step response of higher order system and reduced ordermodel with optimal controller

25

2

15

1

05

0

Am

plitu

de

0 1 2 3 4 5 6 7 8 9 10

Time (s)

High order system with optimal controllerHigh order model with suboptimal controller

Figure 7The step response of higher order systemwith optimal andsuboptimal controller

in Figure 5The response of suboptimal controller (25) is seenidentical to (21) This shows that we can control the plantwith the help of only measurable states which reduces thecomplexity and cost

Example 2 Consider the higher order system [6]

1198664(119904) =

91199043

+ 421199042

+ 31119904 + 10

1199044+ 81199043+ 211199042+ 22119904 + 8

(26)

The optimal controller for (26) is obtained using (3) and (4)as

119870opt = [00821 01603 01405 00623] (27)

The closed loop transfer function using (5) is expressed as

119879opt (119904) =91199043

+ 421199042

+ 31119904 + 10

1199044+ 8082119904

3+ 2116119904

2+ 2214119904 + 8062

(28)


It is assumed that one state is not available for measurementThus a reduced order model (119903 = 3) is desired which takesthe following form

1198663(119904) =

91199042

+ 7145119904 + 2735

1199043+ 4127119904

2+ 495119904 + 2188

(29)

And the optimal gainmatrix for (29) is obtainedwith the helpof (3) and (4) as

119870red = [02053 03686 02177] (30)

The closed loop transfer function of (29) with optimalcontroller is obtained by putting (30) into (5) as

119879red (119904) =91199042

+ 7145119904 + 2735

1199043+ 4333119904

2+ 5319119904 + 2406

(31)

The comparison of the result of step responses of higher ordersystemwith optimal controller (28) and reduced ordermodelwith optimal controller (31) is shown in Figure 6

It is clear that the step response of higher order systemwith optimal controller (28) and reduced order model withoptimal controller (31) is very close to identical

The suboptimal gain matrix is obtained for the system(26) with the help of (11) and (30) and is expressed as

119870sub = [02053 03686 02177 0] (32)

The closed loop transfer function of higher order system(26) with suboptimal controller is obtained using (32) and (5)as

119879119870sub(119904) =

91199043

+ 421199042

+ 31119904 + 10

1199044+ 8205119904

3+ 2137119904

2+ 2222119904 + 8

(33)

The step responses of (28) and (33) are shown in Figure 7 andare found identical in transient as well as steady state region

The results of the two examples signify that the subop-timal control provides a good platform for controlling thehigher order system and reduces the complexity and cost ofcontrol action

For purpose of comparison the suboptimal controller isalso derived using the technique in [4] The result obtainedin [4] shows remarkable deviation in the transient part whichconfirms the observation of [4] that for better overall timeresponse approximation matching of both time momentsand Markov parameters is required

4 Conclusion

A design technique of suboptimal controller using partialfeedback is proposed The method is based on the Padeapproximation technique (matching of both time momentsand Markov parameters) for model order reduction whichresults in better overall time response approximation In thismethod the suboptimal controller is first derived for reducedorder model and then implemented for higher order plantThis reduces the design complexity and cost of control action

The method also avoids use of observer or Kalman filter forreconstruction of missing states

In this paper equal emphasis is given for matchingof time moments and Markov parameters however theoptimal choice of Markov parameters and time moments tobe matched remains unresolved and it is open to furtherinvestigation

Conflict of Interests

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

References

[1] G Bengtsson and S Lindahl ldquoA design scheme for incompletestate or output feedback with applications to boiler and powersystem controlrdquo Automatica vol 10 no 1 pp 15ndash30 1974

[2] T C Hsia ldquoAn approach for incomplete state feedback controlsystems designrdquo IEEE Transactions on Automatic Control volAC-17 pp 383ndash386 1972

[3] V P Singh P Chaubey and D Chandra ldquoModel order reduc-tion of continuous time systems using pole clustering andChebyshev polynomialsrdquo in Proceedings of the IEEE StudentsConference on Engineering and Systems (SCES rsquo12) 2012

[4] J Pal ldquoSuboptimal control using Pade approximation tech-niquerdquo IEEE Transactions on Automatic Control vol AC-25 no5 1980

[5] V Singh D Chandra and H Kar ldquoImproved Routh-Padeapproximants a computer-aided approachrdquo IEEE Transactionson Automatic Control vol 49 no 2 pp 292ndash296 2004

[6] D Chandra An investigation into Routhrsquos Pade approximants[PhD thesis] Department of Electrical Engineering MotilalNehru National Institute of Technology Allahabad India

[7] D S Janardhanan ldquoModel Order Reduction and ControllerDesignrdquo httpwebiitdacinsimjanascoursesmaterialeel879sp topics 01pdf

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2 Chinese Journal of Engineering

Plantminus

+

k

r(t)

Xlowast

ylowast(t)

kTX

lowast

Figure 1 Block diagram of controlled system

Higher order Model order Reduced order Suboptimal systems reduction model controller

Figure 2 Flow diagram of suboptimal control design

The control action of the plant is given as

119906 (119905) = minus119877minus1

119861119879

119875119909 = minus119896119879

119909 (3)

where 119875 is the symmetric positive definite matrix whoseelements may be obtained by solving the following matrixRiccati equation

119860119879

119875 + 119875119860 minus 119875119861119877minus1

119861119879

119875 + 119876 = 0 (4)

The closed loop transfer function with controller is obtainedas

119879 lowast (119904) = 119862119879

[119904119868 minus 119860 + 119861119896119879

]

minus1

119861

=

119904119899minus1

+ 1198861119904119899minus2

+ sdot sdot sdot + 119886119899minus21199042

+ 119886119899minus1119904 + 119886119899


119899minus21199042+ 119887119899minus1119904 + 119887119899

(5)

21Model Order Reduction Procedure by Pade ApproximationConsider a 119899th order system [5] as

119866119899(119904) =

1198861119904119899minus1

+ 1198862119904119899minus2

+ sdot sdot sdot + 119886119899


119899

= 1199050+ 1199051119904 + 11990521199042

+ sdot sdot sdot


= 1198721119904minus1

+1198722119904minus2

+ sdot sdot sdot


(6)

It is desired to obtain a stable 119903th order approximant as

119866119903(119904) =

1198861015840

1119904119899minus1

+ 1198861015840

2119904119899minus2

+ sdot sdot sdot + 1198861015840

119903

119904119899+ 1198871015840

1119904119899minus1+ sdot sdot sdot + 119887

1015840

119903

= 1199051015840

0+ 1199051015840

1119904 + 1199051015840

21199042

+ sdot sdot sdot


= 1198721015840

1119904minus1

+1198721015840

2119904minus2

+ sdot sdot sdot


(7)

It is easy to verify that the following equations hold true

1199051015840

1=

1198861015840

119903

1198871015840

119903

119894 = 1

1199051015840

119894= 1198861015840

119903+1minus119894minus

119894minus1

sum

119895

(1199051015840

1198941198871015840

119903+119895minus119894) 1198871015840

119903

minus1

119894 = 2 3

(8)

and 1198861015840119894= 0 for 119894 le 0 1198871015840

0= 1 1198871015840119894= 0 for 119894 le minus1

1198721015840

119894=

1198861015840

119894 119894 = 1

1198861015840

119894minus

119894minus1

sum

119895=1

1198721015840

1198951198871015840

119894minus119895 119894 = 2 3

119886119894= 119887119894= 0 for 119894 = 119903 + 1 119903 + 2

(9)

We seek a reduced order model so as to satisfy

1199051015840

0= 1199050997904rArr 1198861015840

119903+1minus119894=

119894

sum

119895=1

1199051198951198871015840

119894minus119895 119894 isin 1 2 120582

1198721015840

119894= 119872119894997904rArr 1198861015840

119894=

119894

sum

119895=1

1198721198951198871015840

119894minus119895 119894 isin 1 2 120574

(10)

where 120582 + 120574 = 2119903Using (8)ndash(16) the reduced order model is obtained

22 Suboptimal Controller Design Procedure The suboptimalcontroller [4] is designed with the help of reduced ordermodel by choosing the optimal gainmatrix in such a way that

119870sub = [119870red 0 sdot sdot sdot 0] (11)

where 119870red is optimal gain matrix of reduced order modelobtained using (3) and (4) Finally this 119870sub is applied tohigher order system in order to control the plant (higherorder system)

23 Flow Diagram of Suboptimal Controller Design SeeFigure 2

24 Comparison of Responses of Optimal and SuboptimalController Using (5) one can try to determine the closed looptransfer functions with optimal controller and suboptimalcontroller with the gain matrices 119870opt and 119870sub respectivelybeing derived for comparison

3 Numerical Examples

The step-by-step procedure to obtain reduced the ordermodel is explained with the help of the example presentedbelow and procedure is derived for suboptimal controller

Example 1 Consider the following 4th order system [3]

1198664(119904) =

1199043

+ 71199042

+ 24119904 + 24

1199044+ 101199043+ 351199042+ 50119904 + 24

(12)


76543210

0

01

02

03

04

05

06

07

08

09

1

Time (s)


Am

plitu

de



1198662(119904) =

1198861015840

1119904 + 1198861015840

2

1199042+ 1198871015840

1119904 + 1198871015840

2

(13)

is required



1198664(119904) =

1199043

+ 71199042

+ 24119904 + 24

1199044+ 101199043+ 351199042+ 50119904 + 24

= 1 minus 10833119904 + 109021199042

minus 106641199043

+ sdot sdot sdot

= 1119904minus1

minus 3119904minus2

+ 19119904minus3


+ 571119904minus5

+ sdot sdot sdot

(14)


1198662(119904) =

1198861015840

1119904 + 1198861015840

2

1199042+ 1198871015840

1119904 + 1198871015840

2

= 1199051015840

1+ 1199051015840

2119904 + 1199051015840

31199042

+ sdot sdot sdot

= 1198721015840

1119904minus1

+1198721015840

2119904minus2

+1198721015840

3119904minus3

+ sdot sdot sdot

(15)



1199051= 1199051015840

1

1198721= 1198721015840

1

1199052= 1199051015840

2

1198722= 1198721015840

2

(16)


1199051015840

1=

1198861015840

2

1198871015840

2

1199051015840

2=

1198861015840

1minus 1199051015840

11198871015840

1

1198871015840

2

1198721015840

1= 1198861015840

1

1198721015840

2= 1198861015840

2minus1198721015840

11198871015840

1

(17)



1198861015840

1= 1

1198861015840

2= 240096

1198871015840

1= 270096

1198871015840

2= 240096

(18)


1198662(119904) =

119904 + 240096

1199042+ 270096119904 + 240096

(19)





119870opt = [00626 01275 00857 00208] (20)


119879119870opt(119904) =

1199043

+ 71199042

+ 24119904 + 24

1199044+ 1006119904

3+ 3513119904

2+ 5086119904 + 2402

(21)


119870red = [00193 00208] (22)


119879119870red(119904) =

119904 + 2401

1199042+ 2703119904 + 2403

(23)


mpl

itude

87654321

0

0

01

02

03

04

05

06

07

08

09

1

Time (s)



Am

plitu

de

8765432100

01

02

03

04

05

06

07

08

09

1

Time (s)






119870sub = [00193 00208 0 0] (24)


119879119870sub (119904) =

1199043

+ 71199042

+ 24119904 + 24

1199044+ 1002119904

3+ 3502119904

2+ 50119904 + 24

(25)



25

2

15

1

05

0

Am

plitu

de

0 1 2 3 4 5 6 7 8 9 10

Time (s)



25

2

15

1

05

0

Am

plitu

de

0 1 2 3 4 5 6 7 8 9 10

Time (s)





1198664(119904) =

91199043

+ 421199042

+ 31119904 + 10

1199044+ 81199043+ 211199042+ 22119904 + 8

(26)


119870opt = [00821 01603 01405 00623] (27)


119879opt (119904) =91199043

+ 421199042

+ 31119904 + 10

1199044+ 8082119904

3+ 2116119904

2+ 2214119904 + 8062

(28)



1198663(119904) =

91199042

+ 7145119904 + 2735

1199043+ 4127119904

2+ 495119904 + 2188

(29)


119870red = [02053 03686 02177] (30)


119879red (119904) =91199042

+ 7145119904 + 2735

1199043+ 4333119904

2+ 5319119904 + 2406

(31)




119870sub = [02053 03686 02177 0] (32)


119879119870sub(119904) =

91199043

+ 421199042

+ 31119904 + 10

1199044+ 8205119904

3+ 2137119904

2+ 2222119904 + 8

(33)




4 Conclusion






References










RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










76543210

0

01

02

03

04

05

06

07

08

09

1

Time (s)


Am

plitu

de



1198662(119904) =

1198861015840

1119904 + 1198861015840

2

1199042+ 1198871015840

1119904 + 1198871015840

2

(13)

is required



1198664(119904) =

1199043

+ 71199042

+ 24119904 + 24

1199044+ 101199043+ 351199042+ 50119904 + 24

= 1 minus 10833119904 + 109021199042

minus 106641199043

+ sdot sdot sdot

= 1119904minus1

minus 3119904minus2

+ 19119904minus3


+ 571119904minus5

+ sdot sdot sdot

(14)


1198662(119904) =

1198861015840

1119904 + 1198861015840

2

1199042+ 1198871015840

1119904 + 1198871015840

2

= 1199051015840

1+ 1199051015840

2119904 + 1199051015840

31199042

+ sdot sdot sdot

= 1198721015840

1119904minus1

+1198721015840

2119904minus2

+1198721015840

3119904minus3

+ sdot sdot sdot

(15)



1199051= 1199051015840

1

1198721= 1198721015840

1

1199052= 1199051015840

2

1198722= 1198721015840

2

(16)


1199051015840

1=

1198861015840

2

1198871015840

2

1199051015840

2=

1198861015840

1minus 1199051015840

11198871015840

1

1198871015840

2

1198721015840

1= 1198861015840

1

1198721015840

2= 1198861015840

2minus1198721015840

11198871015840

1

(17)



1198861015840

1= 1

1198861015840

2= 240096

1198871015840

1= 270096

1198871015840

2= 240096

(18)


1198662(119904) =

119904 + 240096

1199042+ 270096119904 + 240096

(19)





119870opt = [00626 01275 00857 00208] (20)


119879119870opt(119904) =

1199043

+ 71199042

+ 24119904 + 24

1199044+ 1006119904

3+ 3513119904

2+ 5086119904 + 2402

(21)


119870red = [00193 00208] (22)


119879119870red(119904) =

119904 + 2401

1199042+ 2703119904 + 2403

(23)


mpl

itude

87654321

0

0

01

02

03

04

05

06

07

08

09

1

Time (s)



Am

plitu

de

8765432100

01

02

03

04

05

06

07

08

09

1

Time (s)






119870sub = [00193 00208 0 0] (24)


119879119870sub (119904) =

1199043

+ 71199042

+ 24119904 + 24

1199044+ 1002119904

3+ 3502119904

2+ 50119904 + 24

(25)



25

2

15

1

05

0

Am

plitu

de

0 1 2 3 4 5 6 7 8 9 10

Time (s)



25

2

15

1

05

0

Am

plitu

de

0 1 2 3 4 5 6 7 8 9 10

Time (s)





1198664(119904) =

91199043

+ 421199042

+ 31119904 + 10

1199044+ 81199043+ 211199042+ 22119904 + 8

(26)


119870opt = [00821 01603 01405 00623] (27)


119879opt (119904) =91199043

+ 421199042

+ 31119904 + 10

1199044+ 8082119904

3+ 2116119904

2+ 2214119904 + 8062

(28)



1198663(119904) =

91199042

+ 7145119904 + 2735

1199043+ 4127119904

2+ 495119904 + 2188

(29)


119870red = [02053 03686 02177] (30)


119879red (119904) =91199042

+ 7145119904 + 2735

1199043+ 4333119904

2+ 5319119904 + 2406

(31)




119870sub = [02053 03686 02177 0] (32)


119879119870sub(119904) =

91199043

+ 421199042

+ 31119904 + 10

1199044+ 8205119904

3+ 2137119904

2+ 2222119904 + 8

(33)




4 Conclusion






References










RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










mpl

itude

87654321

0

0

01

02

03

04

05

06

07

08

09

1

Time (s)



Am

plitu

de

8765432100

01

02

03

04

05

06

07

08

09

1

Time (s)






119870sub = [00193 00208 0 0] (24)


119879119870sub (119904) =

1199043

+ 71199042

+ 24119904 + 24

1199044+ 1002119904

3+ 3502119904

2+ 50119904 + 24

(25)



25

2

15

1

05

0

Am

plitu

de

0 1 2 3 4 5 6 7 8 9 10

Time (s)



25

2

15

1

05

0

Am

plitu

de

0 1 2 3 4 5 6 7 8 9 10

Time (s)





1198664(119904) =

91199043

+ 421199042

+ 31119904 + 10

1199044+ 81199043+ 211199042+ 22119904 + 8

(26)


119870opt = [00821 01603 01405 00623] (27)


119879opt (119904) =91199043

+ 421199042

+ 31119904 + 10

1199044+ 8082119904

3+ 2116119904

2+ 2214119904 + 8062

(28)



1198663(119904) =

91199042

+ 7145119904 + 2735

1199043+ 4127119904

2+ 495119904 + 2188

(29)


119870red = [02053 03686 02177] (30)


119879red (119904) =91199042

+ 7145119904 + 2735

1199043+ 4333119904

2+ 5319119904 + 2406

(31)




119870sub = [02053 03686 02177 0] (32)


119879119870sub(119904) =

91199043

+ 421199042

+ 31119904 + 10

1199044+ 8205119904

3+ 2137119904

2+ 2222119904 + 8

(33)




4 Conclusion






References










RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











1198663(119904) =

91199042

+ 7145119904 + 2735

1199043+ 4127119904

2+ 495119904 + 2188

(29)


119870red = [02053 03686 02177] (30)


119879red (119904) =91199042

+ 7145119904 + 2735

1199043+ 4333119904

2+ 5319119904 + 2406

(31)




119870sub = [02053 03686 02177 0] (32)


119879119870sub(119904) =

91199043

+ 421199042

+ 31119904 + 10

1199044+ 8205119904

3+ 2137119904

2+ 2222119904 + 8

(33)




4 Conclusion






References










RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article suboptimal control using model order...

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