Enhancing the Classical Closed-Loop Algorithm in terms of Power

Research ArticleEnhancing the Classical Closed-Loop Algorithm in terms ofPower Consumption

Rahman Mirzaei12 and Seyed Sina Kourehli3

1Department of Civil Engineering Bonab Branch Islamic Azad University Bonab Iran2The Research Center of Optimization and Engineering Tabriz Iran3Department of Civil Engineering Ahar Branch Islamic Azad University Ahar Iran

Correspondence should be addressed to Seyed Sina Kourehli s-kourehliiau-aharacir

Received 25 September 2014 Accepted 1 February 2015

Academic Editor Amir Hossein Gandomi

Copyright copy R Mirzaei and S S Kourehli 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

An approach is suggested to reduce the peak and average control forces of actively controlled structures In this method responsesof an actively controlled building should be much smaller than the responses of the same building controlled by the similar passivecontrol mechanismThis approach leads to a time varying gain matrix which is not restricted by external excitation but it is relatedto the selection of a scalar function Extensive numerical analyses by using various scalar functions show that the proposed strategyeffectively can reduce the need of the required control force consumptions

1 Introduction

Significant efforts have been directed toward the possibility ofemploying protective systems in civil engineering structuresincluding passive semiactive active and hybrid control sys-tems [1ndash5] Among these active control systems are the mostadvanced technique to achieve this issue It makes a controlmechanism act in a manner that automatically supplies theforces to control the responses of a structure depending uponthe severity of the external forces and the instant state of thestructure during vibrations The direction and magnitude ofthe controlling forces in an active system are estimated by acompetent algorithm General concept of themodern controlmethods to reduce structural vibrations due to externalexcitations was proposed in the 1960s [6] Thereafter a vastamount of efforts has been devoted to develop a varietyof control algorithms based on different control designcriteria like the classical optimal control method [7] thepole assignment method [8] the predictive control method[9] and the instantaneous optimal control method [10 11]Although the linear quadratic regulator algorithm LQR [6]is known as the one of the most famous and historic methodswith widespread applications in different fields (because of

its simple procedure and ease of implementation on actuallarge-scale systems) only the classical closed-loop methodscontrol is feasible for structural control applications Thismeans that there are some other methods which performbetter than the closed-loop control algorithm however sincethey need foreknowledge of entire external disturbance timehistory they are not applicable On the other hand sincethe Riccati equation is obtained by ignoring the earthquakeexcitation term classical closed-loop control algorithms areapproximately optimal methods and do not entirely satisfythe optimality conditions [12] To overcome the shortcomingsof the classical optimal control algorithms several methodshave been proposed until now such as the instantaneousoptimal control method [10 11] the generalized optimalactive control (GOAC) [13] and the instantaneous opti-mal Wilson-120579 method [11] These algorithms in spite oftheir suitable performances are sensitive to change of timeincrement Also they could not significantly improve theperformance of the controlled structure with respect to theclassical optimal control methods Basu and Nagarajaiahproposed a wavelet-based adaptive linear quadratic regulatorformulation for optimal control problems [14] They alsopresented a method for the control of time varying systems

Hindawi Publishing CorporationAdvances in Civil EngineeringVolume 2015 Article ID 291076 9 pageshttpdxdoiorg1011552015291076

2 Advances in Civil Engineering

based on wavelet transformation [15] Bagheri and Aminiproposed a procedure based on pattern search method andwavelet to acquire a more efficient control scheme than theLQR [16]

In this paper an approach called new linear quadraticregulator (NLQR) is proposed in which a relationshipis established between the active and passive structuralresponses via a proper transformation matrix Then thisrelationship is linked to a scalar function (considering anadditional assumption) which causes control force compu-tations to be quite simple In order to evaluate effectivenessand performance of the proposed method it is appliedto an 8-story shear type building structure to reduce itsseismic vibration responses during the El-Centro earthquakeexcitation A few proper scalar functions are derived usingthe El-Centro earthquake ground acceleration and then theperformance of this building is evaluated during some otherearthquake excitations

2 Classical Optimal Closed-LoopAlgorithm (CCLQR)

Consider a building equipped with an active control systemexcited by a strong ground motion The governing dynamicequation of motion may be written as the following matrixform

119872 + 119862 + 119870119909 = minus119872119864119892 + 119863119906 (119905) (1)

where 119909 is the 119899-dimensional displacement vector and dotsstate the derivative of 119909 with respect to time as the velocityand acceleration vectors 119872 119862 and 119870 are the 119899 times 119899 massdamping and stiffness matrices of the structure respectively119864 is the 119899 times 1 influence vector of the ground acceleration onthe building masses 119863 is the 119899 times 119898 location matrix of thecontrol forces affecting the structure and 119906(119905) is the 119898 times 1

control force vector applied by the119898 actuatorsWith somemanipulations the equation ofmotionmay be

rewritten in terms of the state-space variables 119885 as follows

= 119860119885 (119905) + 119861119906 (119905) + 119867119891 (119905) 119885 (1199050) = 1198850 (2)

in which 1199050 is the initial time instant 119885(119905) is the vector ofstate variables and 119860 depicts the system matrix respectivelyVector 119885 and matrix 119860 are defined as follows

119885 (119905) = [119909 (119905) (119905)] 119860 = [0 119868

minus119872minus1119870 minus119872

minus1119862] (3)

In addition matrix 119861 and vector119867 are given as

119861 = [0

119872minus1119863] 119867 = [

0

minus119864] (4)

In the classical linear optimal control a performance index119869(119905) is defined in order to minimize control forces to achievethe best structural performances as

119869 = int

119905119891

0

[119885119879(119905) 119876119885 (119905) + 119906

119879(119905) 119877119906 (119905)] 119889119905 (5)

where 119876 is a 2119899 times 2119899 positive semidefinite weighting matrixrelated to the structural response119877 is an 119903times119903 positive definiteweighting matrix related to the active control force and 119905119891indicates the terminal time that should be longer than theearthquake duration

In classical closed-loop quadratic regulator (CCLQR)optimal control force is achieved by minimizing a perfor-mance index 119869(119905) with respect to the vectors 119885(119905) and 119906(119905) asfollows

119906 (119905) = minus1

2119877minus1119861119879119875 (119905) 119885 (119905) (6)

where 119875(119905) is the Riccati matrix obtained by solving thefollowing nonlinear matrix equation

[ (119905) + 119875 (119905) 119860 minus1

2119875 (119905) 119861119877

minus1119861119879119875 (119905) + 119860

119879119875 (119905) + 2119876]119885 (119905)

+ 119875 (119905)119867119891 (119905) = 0 119875 (119905119891) = 0

(7)

There are two assumptions to solve (7) first the externaldisturbance 119891(119905) is set to zero (it is assumed as a stochasticwhite noise process) second the Riccati matrix is a constantone over the time Although the second assumption isalmost acceptable the first assumption is not fulfilled in allsituations Anyway (7) can be rewritten as the following one

119875119860 minus1

2119875119861119877minus1119861119879119875 + 119860

119879119875 + 2119876 = 0 (8)

By selecting appropriate 119876 and 119877 weighting matrices thisequation can simply be solved and according to (6) theconstant gain matrix is determined as

119866 = minus1

2119877minus1119861119879119875 (9)

3 Proposed Linear QuadraticRegulator (NLQR)

Aswe know by removing control forces of activemass drivers(AMDs) an active control system may passively perform astuned mass dampers (TMDs) Hence from control mech-anisms point of view first we assume that the building iscontrolled by a virtual passive controller which includes afew single degree of freedom systems as TMDs tuned to thebuilding Then by employing the similar active control sys-tem in which TMDs were replaced by AMDs control forcesare determined in the way that the responses of the activecontrolled building are a portion of the similar responses inthe passively controlled building The new proposed methodor the new linear quadratic regulator (NLQR) is formed basedon this idea In the first case we assume that the dynamicequation of motion of the passively controlled building interms of the state-space variables shown by 119885 is

119885 = 119860119885 (119905) + 119867119891 (119905) 119885 (1199050) = 0 (10)

Now suppose that in practice the building is actively con-trolled byAMDswith the same location and characteristics of

Advances in Civil Engineering 3

the passive control system AMDs apply active control forcesto the building such that the building responses are muchreduced with respect to the structural responses of the firstcase as

119885 (119905) = 119879 (119905) 119885 (119905) (11)

where 119879(119905) is an admissible transformation matrix It shouldbe mentioned that the passive control system is a virtualsystem therefore there is no need to know the responsesof the passively controlled building In other words thetransformation matrix is not tuned for a specific externalexcitation but it is considered based on the building prop-erties in a way where the building performs better againstexternal disturbances By using first derivative of (11) withrespect to time we have

(119905) = (119905) 119885 (119905) + 119879 (119905)

119885 (119905) (12)

By combining (2) and (10) in (12) and using the transformrelation of (11) and (6) after some manipulations (13) isobtained

( (119905) minus 119860119879 (119905) + 119879 (119905) 119860 +1

2119861119877minus1119861119879119875119879 (119905))119885 (119905)

= (119868 minus 119879 (119905))119867119891 (119905)

(13)

For further convenience assume that the transformmatrix of(11) is chosen as an identity matrix premultiplied by a scalartime dependent function as follows

119879 (119905) = 120572 (119905) sdot 119868 (119905) = (119905) sdot 119868 (14)

Then the external excitation term in (13) is directly related tothe passive state-space variable through a simple form

( (119905) sdot 119868 +120572 (119905)

2119861119877minus1119861119879119875)119885 (119905) = (1 minus 120572 (119905))119867119891 (119905) or

( (119905)

(1 minus 120572 (119905))sdot 119868 +

120572 (119905)

2 (1 minus 120572 (119905))119861119877minus1119861119879119875)119885 (119905) = 119867119891 (119905)

(15)

This equation represents the relation between external exci-tation and passively controlled responses of the structure Byinserting (15) in optimization process of (7) and using (11) aswell as ignoring the first derivative of 119875matrix the followingequation is obtained

119875(119860 + ( (119905)

2120572 (119905) (1 minus 120572 (119905))) 119868)

minus1

2119875119861(minus(

1 minus 120572 (119905)

120572 (119905)) sdot 119877)

minus1

119861119879119875

+ (119860 + ( (119905)

2120572 (119905) (1 minus 120572 (119905))) 119868)

119879

119875 + 2119876 = 0

(16)

This equation is a modified form of (8) in which matrices119860 and 119877 are changed based on the selected scalar function120572(119905) Now by proper adjustment of the scalar function 120572(119905)and solving (16) time dependent gain matrix is achieved as

119866 (119905) = minus1

2119877minus1119861119879119875 (119905) (17)

4 Limitations of the Scalar Function 120572(119905)

With a glance at (16) we find out that the proposed methodmodifies definition of matrices 119860 and 119877 in (8) to the newforms as follows

119860 997888rarr 119860 + 120582 (119905) 119868 = 119860 + ( (119905)

2120572 (119905) (1 minus 120572 (119905))) 119868

119877 997888rarr 120574 (119905) 119877 = minus(1 minus 120572 (119905)

120572 (119905)) sdot 119877

(18)

Equation (18) dictates that the scalar function 120572(119905) has tosatisfy the following two important criteria (1) we know thatbased on the Lyapunov stability criteria in the continuous-time state-space formulation an active controlled buildingwill be stable if the eigenvalues of its system matrix havenegative real parts On the other hand according to the 120572-shift method by using 119860 + 120582(119905)119868 we forced our system tohave eigenvalues with real parts less than minus120582(119905) Hence ifthe coefficient of identity matrix 120582(119905) added to 119860 in (18)is positive stability of the whole system will be ensuredduring control time (2) 119877matrix is a positive definite matrixNecessarily therefore the coefficient of 120574(119905) in (18) should bepositive over the time to satisfy this criterion

5 Numerical Example

In order to compare performance of the proposed method(NLQR) in comparisonwith the classical closed-loop optimalalgorithm (CCLQR) an eight-story shear type building isconsidered All the properties of the floors are identical thefloor mass of each story is equal to 3455 tons the elasticstiffness of each story is 3404 times 105 kNm and the internaldamping capacity of each story is 2937 tons-secmThe activecontrol system includes an activemass damperdriver (AMD)mechanism which is installed at the roof The characteristicsof AMD are as follows its mass is 2963 tons its tunedfrequency is 98 of the predominant vibration frequencyof the building without control and its damping is 25 tons-secm

51 Ground Accelerations Performance of the active con-trolled building is considered during four different groundmotions which include the El-Centro earthquake theHyogoken-Nanbu (Kobe) earthquake the Landers earth-quake and the Parkfield earthquake These acceleration timehistories are shown in Figures 1 to 4

The active control system is designed such that thecontrolled building presents the best performance duringthe El-Centro earthquake excitation Then behavior of thecontrolled building during other earthquake excitations isevaluated and their performances are compared together

52 Selection of a Proper Scalar Function 120572(119905) Based onSection 4 three different functions are presented here assuitable options for the scalar function The first scalarfunction is as follows

1205721 (119905) = 1 + 119890120573119905 120573 isin 119877 (19)

4 Advances in Civil EngineeringAc

c (g

)

Time (s)0 5 10 15 20 25 30 35 40 45 50 55

minus03

minus02

minus01

001020304

Figure 1 Time history of El-Centro earthquake

Acc

(g)

0 5 10 15 20 25 30 35 40 45 50minus1

minus05

0

05

1

Time (s)

Figure 2 Time history of Hyogoken-Nanbu earthquake

This function is arbitrarily selected to achieve desirable per-formances against the El-Centro groundmotionAppropriatevalue for variable 120573 is equal to 2

As a second case another scalar function is proposed asfollows

1205722 (119905) = 1 + 119890minus120573(119905minus119905

119900)2

119905119900 ≫ 119905119891 120573 isin 119877 (20)

Appropriate value for variable 120573 is set to 0001 Parameter 1199050 isan auxiliary variable to set the necessary conditions presentedin Section 3 which should be much longer than the length ofthe control time duration As a rule 1199050 may be almost two orthree times greater than the control time

Different than the above functions the last scalar func-tion is proposed as an acceleration time history dependentfunction This function uses time history of the recordedearthquake until that time to find the instantaneous Riccatimatrix in each time instant The proposed function is asfollows

1205723 (119905) = 119890minus120578119905

(1 + 119890minus119891(119905)

)

where 119891 (119905) = int

119905

0

100381610038161003816100381610038161003816100381610038161003816

119892

119892

100381610038161003816100381610038161003816100381610038161003816

119889119905

119899

120578 isin 119877+ 119899 isin 119877

(21)

After extensive analysis the appropriate values for variables120578 and 119899 in (21) are set to be 01 and 15 respectively Thisfunction satisfies the necessary conditions of Section 4 butas it was previously mentioned the rates of change of thetwo variables 120582(119905) and 120574(119905) are very rapid Hence the Riccati

0 5 10 15 20 25 30 35 40 45 50minus015

minus01

minus005

0005

01015

02

Time (s)

Acc

(g)

Figure 3 Time history of Landers earthquake

Acc

(g)

0 5 10 15 20 25 30 35minus03

minus02

minus01

001020304

Time (s)

Figure 4 Time history of Parkfield earthquake

matrix independent of the previous and the next time instantshould be found

53 Evaluation Criteria To evaluate the efficiency of controlalgorithms researchers have employed various indices suchas the maximum displacement velocity or acceleration ofthe stories the drift ratios of the adjacent floors and themaximumbase shear of the structures In this paper differentcriteria are developed in order to painstakingly comparethe efficiency of different algorithms Table 1 summarizedthese categories of criteria in which indices 1198691 through1198693 represent the criteria for the maximum displacementvelocity and acceleration responses of the top story which arenormalized to their corresponding uncontrolled values thatis the structure without any active or passive control systems

The performance index 1198694 represents the normalizedmaximum base shear of the controlled building with respectto the uncontrolled one Indices 1198695 through 1198698 show theroot mean square (RMS) of the maximum story responsessuch as the displacement velocity acceleration and thestory drifts with respect to the corresponding responsequantities in the uncontrolled case Finally indices 1198699 and11986910 represent the maximum and average amounts of therequired control forces with respect to the classical closed-loop optimal control algorithm Meanwhile an additionalparameter called efficiency index (EI) is defined as theaverage of the indices 1198691 through 1198698 All of these indices helpto show an overall insight on the performances of the variouscontrol systems


Table 1 Performance indices

1198691=

max 1003816100381610038161003816Dists1003816100381610038161003816con

max 1003816100381610038161003816Dists1003816100381610038161003816uncon

1198692 =max 1003816100381610038161003816Velts

1003816100381610038161003816con

max 1003816100381610038161003816Velts1003816100381610038161003816uncon

1198693=

max 1003816100381610038161003816Accts1003816100381610038161003816con

max 1003816100381610038161003816Accts1003816100381610038161003816uncon

1198694=

max 10038161003816100381610038161198811198871003816100381610038161003816con

max 10038161003816100381610038161198811198871003816100381610038161003816uncon

Top story peak Dis Top story peak Vel Top story peak Acc Peak base shear

1198695=

RMS (max |Dis|con)RMS (max |Dis|uncon)

1198696=

RMS (max |Vel|con)RMS (max |Vel|uncon)

1198697=

RMS (max |Acc|con)RMS (max |Acc|uncon)

1198698=

RMS (max |Dri|con)RMS (max |Dri|uncon)

RMS of the stories RMS of the stories RMS of the stories RMS of the storiesPeak Dis Peak Vel Peak Acc Peak drift

1198699=

max |Cf|conminus119894max |Cf|conminus119895

11986910

=ave |Cf|conminus119894ave |Cf|conminus119895

EI =8

sum

119894=1

119869119894

8

Peak control force Average control force Efficiency index

54 Selection of Weighting Matrices In many researches [17]variety of 119876 weighting matrix has been suggested Properarrangement of the weightingmatrix elements and increasingor decreasing its values may significantly affect the stability ofthe controlled structure and the performance of the controlsystem to achieve the objectives In order to investigate theperformance of the proposed method using various scalarfunctions it is assumed that the arrangements of the 119876

weighting matrix are the same for the proposed method andthe classical optimal algorithm

119876 = 1 times 104[119870 0

0 119872] (22)

where119870 and119872 are thematrices with dimensionless numeri-cal values corresponding to the stiffness andmass matrices ofthe controlled building omitting the stiffness andmass valuesof the active mass damperdriver Weighting matrix relatedto the control force 119877 is assigned to be equal to 1 for allalgorithms Notice that matrix 119877 is a scalar quantity becauseonly one AMD is installed at the roof level

55 Stability Diagram of the Actively Controlled BuildingSince the proposed method results in a time varying gainmatrix stability of the whole controlled building may bechanged over the time In order to inspect this issue stabilitydiagrams of the building equipped with active and passivesystems are depicted together in Figures 5 to 7 Stabilitydiagram of the building using scalar function1205721(119905) is plottedin Figure 5 At the beginning of ground motion the structureis more stable while as time goes away poles of actively con-trolled building move toward passively controlled buildingpoles On the other hand in Figure 6 the stability diagramfor the case corresponding to scalar function 1205722(119905) shows areverse treatment in comparison with scalar function 1205721(119905)Finally in Figure 7 the stability diagram of the controlledbuilding using 1205723(119905) is depicted In this figure a mannersimilar to case one can be seen

56 Response of the Controlled Building during the El-CentroEarthquake Excitation In order to examine performance ofthe proposed control method linear time history analysis iscarried out for all mentioned cases Maximum accelerationvelocity and displacement responses of all floors including

minus18minus60

minus40

minus20

minus16 minus14 minus12 minus10 minus8 minus6 minus4 minus2 0Real part of eigenvalues

0

20

40

60

Imag

inar

y pa

rt o

f eig

enva

lues

Passive controlNLQR-start time

NLQR-mid timeNLQR-end time

Figure 5 Stability diagram of the controlled building using 1205721(119905)

scalar function the El-Centro earthquake

passive and active control systems using CCLQR and NLQRwith different scenarios during the El-Centro earthquakeexcitation are compared with the responses of the uncon-trolled building as shown in Figure 8 in which the proposedmethod using various scenarios can significantly decrease allresponses of the floorsThe best responses belong to the thirdscalar function 1205723(119905) which is almost the same as the CCLQRalgorithm

Numerical results of the responses of the controlledbuilding are briefly tabulated in Table 2 The maximum andthe root mean square of the top story responses maximumand average required control force maximum values of thebase shear and the root mean square of the floor driftsfor different control systems are presented Comparing theresults shows that the best performance belongs to CCLQRand the results of the new proposed method using thethird scalar function 1205723(119905) are also very close to it Theonly difference is in consuming control forces which maybe observed in columns 8 and 9 of Table 2 Decreasing inmaximum control force is only about 12 but decreasing inthe average (or total) required control force is about 28Thisis the major capability of the proposed method It is noted


Table 2 Structural peak response of the controlled building during the El-Centro earthquake excitation

Case Top storyDis (cm)

Top storyVel (ms)

Top storyAcc (ms2)

RMS ofDis (cm)

RMS ofVel (ms)

RMS ofAcc (ms2)

Max Cof(KN)

Ave Cof(KN)

Max Bshear (KN)

RMS ofdrifts (cm)

No control 193 106 79 3916 223 1568 mdash mdash 1107 734Passive control 152 089 64 3109 184 1313 mdash mdash 924 587CCLQR 670 063 360 1374 141 892 108282 11984 408 274NLQR-1205721(119905) 890 068 460 1834 152 1009 76315 8325 549 342NLQR-1205722(119905) 780 068 440 1610 149 940 73979 9917 486 312NLQR-1205723(119905) 680 063 410 1407 144 925 106947 8600 417 277

minus18minus60

minus40

minus20


0

20

40

60

Imag

inar

y pa

rt o

f eig

enva

lues





that selecting proper scalar function plays an important rolein decreasing the need of the total power supply of the system

57 Active Control Forces against the El-Centro Earth-quake Excitation Active control force time histories of theproposed method using three various scalar functions arecompared to that of CCLQR as shown in Figures 9 and 10In Figure 9 it is observed that the control system using thefirst scalar function 1205721(119905) totally determines active controlforces lower than CCLQR Figure 10 shows that at the starttime of control duration the system using the second scalarfunction 1205722(119905) determines active control forces lower thanCCLQR but after a while by changing its poles control forcesare increased Therefore it is expected that the performanceof the control system using 1205722(119905) is better than the previousone but not as good as CCLQR

Figure 11 shows that the control system using the thirdscalar function requires the same control forces as CCLQRwhen the acceleration groundmotion is large Additionally asthe severity of the earthquake is decreased the control forcedemand decreases too

Briefly regarding the above discussion we can say NLQRwith 1205723(119905) may present the best performance of the control

minus18minus60

minus40

minus20


0

20

40

60

Imag

inar

y pa

rt o

f eig

enva

lues





system since instant control force follows the severity of theground excitations

58 Response of the Controlled Building during the OtherEarthquake Excitations Performances of various controlsystems which are designed to achieve the high efficiencyduring the El-Centro earthquake are examined under afew other earthquakes excitations including the Hyogoken-Nanbu earthquake the Landers earthquake and the Parkfieldearthquake After computing performance indices of thecontrolled building for each earthquake are listed in Tables3 to 5 The indices of controlled building using NLQR-1205721(119905)show the lowest efficiency in comparisonwith the other casesHigh values of its separate indices or its averaged index EIconfirm this statement

Generally the control system using 1205722(119905) presents themiddle performance between the three scalar functionsexcept the Landers earthquake in which the performance isbetter than NLQR-1205723(119905) By noticing the earthquake recordsin Figures 1 to 4 it is seen that the strong part of the Landersrecord is distributed over the whole duration of the signalwhile for the other earthquakes it is concentrated at a shortlength beyond the beginning Since NLQR-1205722(119905) increases


Displacement (cm)5 10 15 20

Stor

y

1

2

3

4

5

6

7

8

Velocity (ms)02 04 06 08 10

Stor

y

1

2

3

4

5

6

7

8

3 4 5 6 7 8

Stor

y

1

2

3

4

5

6

7

8

Without controlPassive control

CCLQR

Acceleration (ms2)

NLQR-120572(t) = 1205721

NLQR-120572(t) = 1205722NLQR-120572(t) = 1205723


CCLQRNLQR-120572(t) = 1205721

NLQR-120572(t) = 1205722NLQR-120572(t) = 1205723


CCLQRNLQR-120572(t) = 1205721

NLQR-120572(t) = 1205722NLQR-120572(t) = 1205723

Figure 8 Maximum responses of the floors comparing active passive and uncontrolled cases when the structure is subjected to the El-Centro earthquake excitation

Time (s)0 10 20 30 40 50 60

Con

trol f

orce

(kN

)

minus1000

minus500

0

500

1000

CCLQR NLQR-120572(t) = 1205721

Figure 9 Control force time history of NLQR using 1205721(119905) scalar

function during the El-Centro earthquake excitation

control force with time it is expected to act more effectivethan the other scalar functions in this case

NLQR-1205723(119905) in almost all cases gives satisfactory per-formances close to the CCLQR with lower maximum andaverage control forces demand This may play a crucial rolewhen severe earthquakes happen

The average values of the results obtained from allearthquakes are summarized in Table 6 As can be seen theproposed control method using various scalar functions maygenerally reduce responses as well as both the maximumand the average required control forces of the building muchbetter than a similar control system Extensive results during

Time (s)0 10 20 30 40 50 60

Con

trol f

orce

(kN

)

0

500

1000

CCLQR

minus1000

minus500

NLQR-120572(t) = 1205722



different strong shakings show that high efficiency of NLQRis obtained when a scalar function related to the externalexcitation or combination of two dissimilar functions like thesecond and the third scalar functions is employed

6 Conclusion

In this paper an approach called NLQR is suggestedfor reducing the peak and average required control forcesin actively controlled structures employing optimal con-trol methods In order to evaluate the effectiveness of theproposed method the responses of an 8-story shear type


Time (s)0 10 20 30 40 50 60

0

500

1000

CCLQR

minus1000

minus500

Con

trol f

orce

(kN

)

NLQR-120572(t) = 1205723



Table 3 Performance indices for building subjected to Hyogoken-Nanbu earthquake excitation

Index Passivecontrol CCLQR NLQR-1205721(119905) NLQR-1205722(119905) NLQR-1205723(119905)

1198691 094 050 074 057 0551198692 096 060 075 066 0651198693 096 055 088 068 0661198694 094 051 084 062 0561198695 094 051 074 057 0561198696 096 061 076 067 0661198697 095 073 097 083 0791198698 094 054 081 063 058EI mdash 057 081 065 0631198699 mdash 1 066 094 08911986910 mdash 1 095 092 080

Table 4 Performance indices for building subjected to Landersground excitation



building structure subjected to different ground motions areevaluated Different types of scalar functions are also utilizedfor this purpose The extensive analyses show that NLQRcan decrease all responses of the building similar to CCLQR

Table 5 Performance indices for building subjected to Parkfieldground excitation



Table 6 Averaged performance indices of the controlled buildingsubjected to the El-Centro Hyogoken-Nanbu Landers and Park-field earthquakes



but almost in all cases it effectively reduces the need for therequired control force consumptions

Conflict of Interests

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

References

[1] N M Rahbari and S Talatahari ldquoOn the efficiency of semi-active smart structures self-regulating MR dampers controlsystem for tall buildingsrdquo The Structural Design of Tall andSpecial Buildings vol 23 no 13 pp 1027ndash1044 2014

[2] X Zhang and F Y Cheng ldquoControl force estimation in seismicbuilding designrdquo in Proceedings of the Structures Congress pp1510ndash1522 Orlando Fla USA May 2010

[3] F Y Cheng H Jiang and K Lou Smart Structures InnovativeSystems for Seismic Response Control Taylor amp Francis BocaRaton Fla USA 2008


[4] F Yang R Sedaghati and E Esmailzadeh ldquoOptimal design ofdistributed tunedmass dampers for passive vibration control ofstructuresrdquo Structural Control and Health Monitoring vol 22no 2 pp 221ndash236 2015

[5] C C Patel and R S Jangid ldquoSeismic response of dynamicallysimilar adjacent structures connected with viscous dampersrdquoThe IES Journal Part A Civil and Structural Engineering vol 3no 1 pp 1ndash13 2010

[6] J T P Yao ldquoConcept of structural controlrdquo Structural Divisionvol 98 no 7 pp 1567ndash1574 1972

[7] N S Xu and Z H Yang ldquoPredictive structural control based ondominant internal model approachrdquo Automatica vol 35 no 1pp 59ndash67 1999

[8] H P Gavin and U Aldemir ldquoOptimal control of earthquakeresponse using semiactive isolationrdquo Journal of EngineeringMechanics vol 131 no 8 pp 769ndash776 2005

[9] J N Yang ldquoControl of tall buildings under earthquake excita-tionrdquo Journal of the EngineeringMechanics Division vol 108 noEM5 pp 883ndash849 1982

[10] J N Yang A Akbarpour and P Ghaemmaghami ldquoOptimalcontrol algorithms for earthquake excited buildingsrdquo in Struc-tural Control Proceedings of the Second International Sympo-sium on Structural Control University of Waterloo OntarioCanada July 15ndash17 1985 pp 748ndash761 Springer Berlin Ger-many 1987

[11] O Bahar M R Banan M Mahzoon and Y KitagawaldquoInstantaneous optimal Wilson-Θ control methodrdquo Journal ofEngineering Mechanics vol 129 no 11 pp 1268ndash1276 2003

[12] T T SoongActive Structural ControlTheory and Practice JohnWiley amp Sons New York NY USA 1990

[13] F Y Cheng and P Tian ldquoGeneralized optimal active controlalgorithm for nonlinear seismic structuresrdquo in Proceedings ofthe 10thWorld Conference on Earthquake Engineering pp 3677ndash3682 Madrid Spain 1992

[14] B Basu and S Nagarajaiah ldquoA wavelet-based time-varyingadaptive LQR algorithm for structural controlrdquo EngineeringStructures vol 30 no 9 pp 2470ndash2477 2008

[15] B Basu and S Nagarajaiah ldquoMulti scale wavelet-LQR controllerfor linear time varying systemsrdquo Journal of EngineeringMechan-ics vol 136 no 9 pp 1143ndash1151 2010

[16] A Bagheri and F Amini ldquoControl of structures under uniformhazard earthquake excitation via wavelet analysis and patternsearch methodrdquo Structural Control and Health Monitoring vol20 no 5 pp 671ndash685 2013

[17] R Mirzaei and O Bahar ldquoA new view on optimal controlalgorithmsrdquo Seismology and Earthquake Engineering vol 13 no3 pp 195ndash207 2011

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



based on wavelet transformation [15] Bagheri and Aminiproposed a procedure based on pattern search method andwavelet to acquire a more efficient control scheme than theLQR [16]

In this paper an approach called new linear quadraticregulator (NLQR) is proposed in which a relationshipis established between the active and passive structuralresponses via a proper transformation matrix Then thisrelationship is linked to a scalar function (considering anadditional assumption) which causes control force compu-tations to be quite simple In order to evaluate effectivenessand performance of the proposed method it is appliedto an 8-story shear type building structure to reduce itsseismic vibration responses during the El-Centro earthquakeexcitation A few proper scalar functions are derived usingthe El-Centro earthquake ground acceleration and then theperformance of this building is evaluated during some otherearthquake excitations

2 Classical Optimal Closed-LoopAlgorithm (CCLQR)

Consider a building equipped with an active control systemexcited by a strong ground motion The governing dynamicequation of motion may be written as the following matrixform

119872 + 119862 + 119870119909 = minus119872119864119892 + 119863119906 (119905) (1)

where 119909 is the 119899-dimensional displacement vector and dotsstate the derivative of 119909 with respect to time as the velocityand acceleration vectors 119872 119862 and 119870 are the 119899 times 119899 massdamping and stiffness matrices of the structure respectively119864 is the 119899 times 1 influence vector of the ground acceleration onthe building masses 119863 is the 119899 times 119898 location matrix of thecontrol forces affecting the structure and 119906(119905) is the 119898 times 1

control force vector applied by the119898 actuatorsWith somemanipulations the equation ofmotionmay be

rewritten in terms of the state-space variables 119885 as follows

= 119860119885 (119905) + 119861119906 (119905) + 119867119891 (119905) 119885 (1199050) = 1198850 (2)

in which 1199050 is the initial time instant 119885(119905) is the vector ofstate variables and 119860 depicts the system matrix respectivelyVector 119885 and matrix 119860 are defined as follows

119885 (119905) = [119909 (119905) (119905)] 119860 = [0 119868

minus119872minus1119870 minus119872

minus1119862] (3)

In addition matrix 119861 and vector119867 are given as

119861 = [0

119872minus1119863] 119867 = [

0

minus119864] (4)

In the classical linear optimal control a performance index119869(119905) is defined in order to minimize control forces to achievethe best structural performances as

119869 = int

119905119891

0

[119885119879(119905) 119876119885 (119905) + 119906

119879(119905) 119877119906 (119905)] 119889119905 (5)

where 119876 is a 2119899 times 2119899 positive semidefinite weighting matrixrelated to the structural response119877 is an 119903times119903 positive definiteweighting matrix related to the active control force and 119905119891indicates the terminal time that should be longer than theearthquake duration

In classical closed-loop quadratic regulator (CCLQR)optimal control force is achieved by minimizing a perfor-mance index 119869(119905) with respect to the vectors 119885(119905) and 119906(119905) asfollows

119906 (119905) = minus1

2119877minus1119861119879119875 (119905) 119885 (119905) (6)

where 119875(119905) is the Riccati matrix obtained by solving thefollowing nonlinear matrix equation

[ (119905) + 119875 (119905) 119860 minus1

2119875 (119905) 119861119877

minus1119861119879119875 (119905) + 119860

119879119875 (119905) + 2119876]119885 (119905)

+ 119875 (119905)119867119891 (119905) = 0 119875 (119905119891) = 0

(7)

There are two assumptions to solve (7) first the externaldisturbance 119891(119905) is set to zero (it is assumed as a stochasticwhite noise process) second the Riccati matrix is a constantone over the time Although the second assumption isalmost acceptable the first assumption is not fulfilled in allsituations Anyway (7) can be rewritten as the following one

119875119860 minus1

2119875119861119877minus1119861119879119875 + 119860

119879119875 + 2119876 = 0 (8)

By selecting appropriate 119876 and 119877 weighting matrices thisequation can simply be solved and according to (6) theconstant gain matrix is determined as

119866 = minus1

2119877minus1119861119879119875 (9)

3 Proposed Linear QuadraticRegulator (NLQR)

Aswe know by removing control forces of activemass drivers(AMDs) an active control system may passively perform astuned mass dampers (TMDs) Hence from control mech-anisms point of view first we assume that the building iscontrolled by a virtual passive controller which includes afew single degree of freedom systems as TMDs tuned to thebuilding Then by employing the similar active control sys-tem in which TMDs were replaced by AMDs control forcesare determined in the way that the responses of the activecontrolled building are a portion of the similar responses inthe passively controlled building The new proposed methodor the new linear quadratic regulator (NLQR) is formed basedon this idea In the first case we assume that the dynamicequation of motion of the passively controlled building interms of the state-space variables shown by 119885 is

119885 = 119860119885 (119905) + 119867119891 (119905) 119885 (1199050) = 0 (10)

Now suppose that in practice the building is actively con-trolled byAMDswith the same location and characteristics of



119885 (119905) = 119879 (119905) 119885 (119905) (11)


(119905) = (119905) 119885 (119905) + 119879 (119905)

119885 (119905) (12)


( (119905) minus 119860119879 (119905) + 119879 (119905) 119860 +1

2119861119877minus1119861119879119875119879 (119905))119885 (119905)

= (119868 minus 119879 (119905))119867119891 (119905)

(13)


119879 (119905) = 120572 (119905) sdot 119868 (119905) = (119905) sdot 119868 (14)


( (119905) sdot 119868 +120572 (119905)

2119861119877minus1119861119879119875)119885 (119905) = (1 minus 120572 (119905))119867119891 (119905) or

( (119905)

(1 minus 120572 (119905))sdot 119868 +

120572 (119905)

2 (1 minus 120572 (119905))119861119877minus1119861119879119875)119885 (119905) = 119867119891 (119905)

(15)


119875(119860 + ( (119905)

2120572 (119905) (1 minus 120572 (119905))) 119868)

minus1

2119875119861(minus(

1 minus 120572 (119905)

120572 (119905)) sdot 119877)

minus1

119861119879119875

+ (119860 + ( (119905)

2120572 (119905) (1 minus 120572 (119905))) 119868)

119879

119875 + 2119876 = 0

(16)


119866 (119905) = minus1

2119877minus1119861119879119875 (119905) (17)



119860 997888rarr 119860 + 120582 (119905) 119868 = 119860 + ( (119905)

2120572 (119905) (1 minus 120572 (119905))) 119868

119877 997888rarr 120574 (119905) 119877 = minus(1 minus 120572 (119905)

120572 (119905)) sdot 119877

(18)


5 Numerical Example





1205721 (119905) = 1 + 119890120573119905 120573 isin 119877 (19)


c (g

)

Time (s)0 5 10 15 20 25 30 35 40 45 50 55

minus03

minus02

minus01

001020304


Acc

(g)

0 5 10 15 20 25 30 35 40 45 50minus1

minus05

0

05

1

Time (s)




1205722 (119905) = 1 + 119890minus120573(119905minus119905

119900)2

119905119900 ≫ 119905119891 120573 isin 119877 (20)



1205723 (119905) = 119890minus120578119905

(1 + 119890minus119891(119905)

)

where 119891 (119905) = int

119905

0

100381610038161003816100381610038161003816100381610038161003816

119892

119892

100381610038161003816100381610038161003816100381610038161003816

119889119905

119899

120578 isin 119877+ 119899 isin 119877

(21)


0 5 10 15 20 25 30 35 40 45 50minus015

minus01

minus005

0005

01015

02

Time (s)

Acc

(g)


Acc

(g)

0 5 10 15 20 25 30 35minus03

minus02

minus01

001020304

Time (s)







1198691=

max 1003816100381610038161003816Dists1003816100381610038161003816con

max 1003816100381610038161003816Dists1003816100381610038161003816uncon

1198692 =max 1003816100381610038161003816Velts

1003816100381610038161003816con

max 1003816100381610038161003816Velts1003816100381610038161003816uncon

1198693=

max 1003816100381610038161003816Accts1003816100381610038161003816con

max 1003816100381610038161003816Accts1003816100381610038161003816uncon

1198694=

max 10038161003816100381610038161198811198871003816100381610038161003816con

max 10038161003816100381610038161198811198871003816100381610038161003816uncon


1198695=


1198696=


1198697=


1198698=



1198699=


11986910


EI =8

sum

119894=1

119869119894

8




119876 = 1 times 104[119870 0

0 119872] (22)




minus18minus60

minus40

minus20


0

20

40

60

Imag

inar

y pa

rt o

f eig

enva

lues










Top storyVel (ms)

Top storyAcc (ms2)

RMS ofDis (cm)

RMS ofVel (ms)

RMS ofAcc (ms2)

Max Cof(KN)

Ave Cof(KN)

Max Bshear (KN)

RMS ofdrifts (cm)


minus18minus60

minus40

minus20


0

20

40

60

Imag

inar

y pa

rt o

f eig

enva

lues









minus18minus60

minus40

minus20


0

20

40

60

Imag

inar

y pa

rt o

f eig

enva

lues










Stor

y

1

2

3

4

5

6

7

8

Velocity (ms)02 04 06 08 10

Stor

y

1

2

3

4

5

6

7

8

3 4 5 6 7 8

Stor

y

1

2

3

4

5

6

7

8


CCLQR

Acceleration (ms2)

NLQR-120572(t) = 1205721

NLQR-120572(t) = 1205722NLQR-120572(t) = 1205723


CCLQRNLQR-120572(t) = 1205721

NLQR-120572(t) = 1205722NLQR-120572(t) = 1205723


CCLQRNLQR-120572(t) = 1205721

NLQR-120572(t) = 1205722NLQR-120572(t) = 1205723


Time (s)0 10 20 30 40 50 60

Con

trol f

orce

(kN

)

minus1000

minus500

0

500

1000

CCLQR NLQR-120572(t) = 1205721






Time (s)0 10 20 30 40 50 60

Con

trol f

orce

(kN

)

0

500

1000

CCLQR

minus1000

minus500

NLQR-120572(t) = 1205722




6 Conclusion



Time (s)0 10 20 30 40 50 60

0

500

1000

CCLQR

minus1000

minus500

Con

trol f

orce

(kN

)

NLQR-120572(t) = 1205723



















References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119885 (119905) = 119879 (119905) 119885 (119905) (11)


(119905) = (119905) 119885 (119905) + 119879 (119905)

119885 (119905) (12)


( (119905) minus 119860119879 (119905) + 119879 (119905) 119860 +1

2119861119877minus1119861119879119875119879 (119905))119885 (119905)

= (119868 minus 119879 (119905))119867119891 (119905)

(13)


119879 (119905) = 120572 (119905) sdot 119868 (119905) = (119905) sdot 119868 (14)


( (119905) sdot 119868 +120572 (119905)

2119861119877minus1119861119879119875)119885 (119905) = (1 minus 120572 (119905))119867119891 (119905) or

( (119905)

(1 minus 120572 (119905))sdot 119868 +

120572 (119905)

2 (1 minus 120572 (119905))119861119877minus1119861119879119875)119885 (119905) = 119867119891 (119905)

(15)


119875(119860 + ( (119905)

2120572 (119905) (1 minus 120572 (119905))) 119868)

minus1

2119875119861(minus(

1 minus 120572 (119905)

120572 (119905)) sdot 119877)

minus1

119861119879119875

+ (119860 + ( (119905)

2120572 (119905) (1 minus 120572 (119905))) 119868)

119879

119875 + 2119876 = 0

(16)


119866 (119905) = minus1

2119877minus1119861119879119875 (119905) (17)



119860 997888rarr 119860 + 120582 (119905) 119868 = 119860 + ( (119905)

2120572 (119905) (1 minus 120572 (119905))) 119868

119877 997888rarr 120574 (119905) 119877 = minus(1 minus 120572 (119905)

120572 (119905)) sdot 119877

(18)


5 Numerical Example





1205721 (119905) = 1 + 119890120573119905 120573 isin 119877 (19)


c (g

)

Time (s)0 5 10 15 20 25 30 35 40 45 50 55

minus03

minus02

minus01

001020304


Acc

(g)

0 5 10 15 20 25 30 35 40 45 50minus1

minus05

0

05

1

Time (s)




1205722 (119905) = 1 + 119890minus120573(119905minus119905

119900)2

119905119900 ≫ 119905119891 120573 isin 119877 (20)



1205723 (119905) = 119890minus120578119905

(1 + 119890minus119891(119905)

)

where 119891 (119905) = int

119905

0

100381610038161003816100381610038161003816100381610038161003816

119892

119892

100381610038161003816100381610038161003816100381610038161003816

119889119905

119899

120578 isin 119877+ 119899 isin 119877

(21)


0 5 10 15 20 25 30 35 40 45 50minus015

minus01

minus005

0005

01015

02

Time (s)

Acc

(g)


Acc

(g)

0 5 10 15 20 25 30 35minus03

minus02

minus01

001020304

Time (s)







1198691=

max 1003816100381610038161003816Dists1003816100381610038161003816con

max 1003816100381610038161003816Dists1003816100381610038161003816uncon

1198692 =max 1003816100381610038161003816Velts

1003816100381610038161003816con

max 1003816100381610038161003816Velts1003816100381610038161003816uncon

1198693=

max 1003816100381610038161003816Accts1003816100381610038161003816con

max 1003816100381610038161003816Accts1003816100381610038161003816uncon

1198694=

max 10038161003816100381610038161198811198871003816100381610038161003816con

max 10038161003816100381610038161198811198871003816100381610038161003816uncon


1198695=


1198696=


1198697=


1198698=



1198699=


11986910


EI =8

sum

119894=1

119869119894

8




119876 = 1 times 104[119870 0

0 119872] (22)




minus18minus60

minus40

minus20


0

20

40

60

Imag

inar

y pa

rt o

f eig

enva

lues










Top storyVel (ms)

Top storyAcc (ms2)

RMS ofDis (cm)

RMS ofVel (ms)

RMS ofAcc (ms2)

Max Cof(KN)

Ave Cof(KN)

Max Bshear (KN)

RMS ofdrifts (cm)


minus18minus60

minus40

minus20


0

20

40

60

Imag

inar

y pa

rt o

f eig

enva

lues









minus18minus60

minus40

minus20


0

20

40

60

Imag

inar

y pa

rt o

f eig

enva

lues










Stor

y

1

2

3

4

5

6

7

8

Velocity (ms)02 04 06 08 10

Stor

y

1

2

3

4

5

6

7

8

3 4 5 6 7 8

Stor

y

1

2

3

4

5

6

7

8


CCLQR

Acceleration (ms2)

NLQR-120572(t) = 1205721

NLQR-120572(t) = 1205722NLQR-120572(t) = 1205723


CCLQRNLQR-120572(t) = 1205721

NLQR-120572(t) = 1205722NLQR-120572(t) = 1205723


CCLQRNLQR-120572(t) = 1205721

NLQR-120572(t) = 1205722NLQR-120572(t) = 1205723


Time (s)0 10 20 30 40 50 60

Con

trol f

orce

(kN

)

minus1000

minus500

0

500

1000

CCLQR NLQR-120572(t) = 1205721






Time (s)0 10 20 30 40 50 60

Con

trol f

orce

(kN

)

0

500

1000

CCLQR

minus1000

minus500

NLQR-120572(t) = 1205722




6 Conclusion



Time (s)0 10 20 30 40 50 60

0

500

1000

CCLQR

minus1000

minus500

Con

trol f

orce

(kN

)

NLQR-120572(t) = 1205723



















References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










c (g

)

Time (s)0 5 10 15 20 25 30 35 40 45 50 55

minus03

minus02

minus01

001020304


Acc

(g)

0 5 10 15 20 25 30 35 40 45 50minus1

minus05

0

05

1

Time (s)




1205722 (119905) = 1 + 119890minus120573(119905minus119905

119900)2

119905119900 ≫ 119905119891 120573 isin 119877 (20)



1205723 (119905) = 119890minus120578119905

(1 + 119890minus119891(119905)

)

where 119891 (119905) = int

119905

0

100381610038161003816100381610038161003816100381610038161003816

119892

119892

100381610038161003816100381610038161003816100381610038161003816

119889119905

119899

120578 isin 119877+ 119899 isin 119877

(21)


0 5 10 15 20 25 30 35 40 45 50minus015

minus01

minus005

0005

01015

02

Time (s)

Acc

(g)


Acc

(g)

0 5 10 15 20 25 30 35minus03

minus02

minus01

001020304

Time (s)







1198691=

max 1003816100381610038161003816Dists1003816100381610038161003816con

max 1003816100381610038161003816Dists1003816100381610038161003816uncon

1198692 =max 1003816100381610038161003816Velts

1003816100381610038161003816con

max 1003816100381610038161003816Velts1003816100381610038161003816uncon

1198693=

max 1003816100381610038161003816Accts1003816100381610038161003816con

max 1003816100381610038161003816Accts1003816100381610038161003816uncon

1198694=

max 10038161003816100381610038161198811198871003816100381610038161003816con

max 10038161003816100381610038161198811198871003816100381610038161003816uncon


1198695=


1198696=


1198697=


1198698=



1198699=


11986910


EI =8

sum

119894=1

119869119894

8




119876 = 1 times 104[119870 0

0 119872] (22)




minus18minus60

minus40

minus20


0

20

40

60

Imag

inar

y pa

rt o

f eig

enva

lues










Top storyVel (ms)

Top storyAcc (ms2)

RMS ofDis (cm)

RMS ofVel (ms)

RMS ofAcc (ms2)

Max Cof(KN)

Ave Cof(KN)

Max Bshear (KN)

RMS ofdrifts (cm)


minus18minus60

minus40

minus20


0

20

40

60

Imag

inar

y pa

rt o

f eig

enva

lues









minus18minus60

minus40

minus20


0

20

40

60

Imag

inar

y pa

rt o

f eig

enva

lues










Stor

y

1

2

3

4

5

6

7

8

Velocity (ms)02 04 06 08 10

Stor

y

1

2

3

4

5

6

7

8

3 4 5 6 7 8

Stor

y

1

2

3

4

5

6

7

8


CCLQR

Acceleration (ms2)

NLQR-120572(t) = 1205721

NLQR-120572(t) = 1205722NLQR-120572(t) = 1205723


CCLQRNLQR-120572(t) = 1205721

NLQR-120572(t) = 1205722NLQR-120572(t) = 1205723


CCLQRNLQR-120572(t) = 1205721

NLQR-120572(t) = 1205722NLQR-120572(t) = 1205723


Time (s)0 10 20 30 40 50 60

Con

trol f

orce

(kN

)

minus1000

minus500

0

500

1000

CCLQR NLQR-120572(t) = 1205721






Time (s)0 10 20 30 40 50 60

Con

trol f

orce

(kN

)

0

500

1000

CCLQR

minus1000

minus500

NLQR-120572(t) = 1205722




6 Conclusion



Time (s)0 10 20 30 40 50 60

0

500

1000

CCLQR

minus1000

minus500

Con

trol f

orce

(kN

)

NLQR-120572(t) = 1205723



















References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











1198691=

max 1003816100381610038161003816Dists1003816100381610038161003816con

max 1003816100381610038161003816Dists1003816100381610038161003816uncon

1198692 =max 1003816100381610038161003816Velts

1003816100381610038161003816con

max 1003816100381610038161003816Velts1003816100381610038161003816uncon

1198693=

max 1003816100381610038161003816Accts1003816100381610038161003816con

max 1003816100381610038161003816Accts1003816100381610038161003816uncon

1198694=

max 10038161003816100381610038161198811198871003816100381610038161003816con

max 10038161003816100381610038161198811198871003816100381610038161003816uncon


1198695=


1198696=


1198697=


1198698=



1198699=


11986910


EI =8

sum

119894=1

119869119894

8




119876 = 1 times 104[119870 0

0 119872] (22)




minus18minus60

minus40

minus20


0

20

40

60

Imag

inar

y pa

rt o

f eig

enva

lues










Top storyVel (ms)

Top storyAcc (ms2)

RMS ofDis (cm)

RMS ofVel (ms)

RMS ofAcc (ms2)

Max Cof(KN)

Ave Cof(KN)

Max Bshear (KN)

RMS ofdrifts (cm)


minus18minus60

minus40

minus20


0

20

40

60

Imag

inar

y pa

rt o

f eig

enva

lues









minus18minus60

minus40

minus20


0

20

40

60

Imag

inar

y pa

rt o

f eig

enva

lues










Stor

y

1

2

3

4

5

6

7

8

Velocity (ms)02 04 06 08 10

Stor

y

1

2

3

4

5

6

7

8

3 4 5 6 7 8

Stor

y

1

2

3

4

5

6

7

8


CCLQR

Acceleration (ms2)

NLQR-120572(t) = 1205721

NLQR-120572(t) = 1205722NLQR-120572(t) = 1205723


CCLQRNLQR-120572(t) = 1205721

NLQR-120572(t) = 1205722NLQR-120572(t) = 1205723


CCLQRNLQR-120572(t) = 1205721

NLQR-120572(t) = 1205722NLQR-120572(t) = 1205723


Time (s)0 10 20 30 40 50 60

Con

trol f

orce

(kN

)

minus1000

minus500

0

500

1000

CCLQR NLQR-120572(t) = 1205721






Time (s)0 10 20 30 40 50 60

Con

trol f

orce

(kN

)

0

500

1000

CCLQR

minus1000

minus500

NLQR-120572(t) = 1205722




6 Conclusion



Time (s)0 10 20 30 40 50 60

0

500

1000

CCLQR

minus1000

minus500

Con

trol f

orce

(kN

)

NLQR-120572(t) = 1205723



















References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Top storyVel (ms)

Top storyAcc (ms2)

RMS ofDis (cm)

RMS ofVel (ms)

RMS ofAcc (ms2)

Max Cof(KN)

Ave Cof(KN)

Max Bshear (KN)

RMS ofdrifts (cm)


minus18minus60

minus40

minus20


0

20

40

60

Imag

inar

y pa

rt o

f eig

enva

lues









minus18minus60

minus40

minus20


0

20

40

60

Imag

inar

y pa

rt o

f eig

enva

lues










Stor

y

1

2

3

4

5

6

7

8

Velocity (ms)02 04 06 08 10

Stor

y

1

2

3

4

5

6

7

8

3 4 5 6 7 8

Stor

y

1

2

3

4

5

6

7

8


CCLQR

Acceleration (ms2)

NLQR-120572(t) = 1205721

NLQR-120572(t) = 1205722NLQR-120572(t) = 1205723


CCLQRNLQR-120572(t) = 1205721

NLQR-120572(t) = 1205722NLQR-120572(t) = 1205723


CCLQRNLQR-120572(t) = 1205721

NLQR-120572(t) = 1205722NLQR-120572(t) = 1205723


Time (s)0 10 20 30 40 50 60

Con

trol f

orce

(kN

)

minus1000

minus500

0

500

1000

CCLQR NLQR-120572(t) = 1205721






Time (s)0 10 20 30 40 50 60

Con

trol f

orce

(kN

)

0

500

1000

CCLQR

minus1000

minus500

NLQR-120572(t) = 1205722




6 Conclusion



Time (s)0 10 20 30 40 50 60

0

500

1000

CCLQR

minus1000

minus500

Con

trol f

orce

(kN

)

NLQR-120572(t) = 1205723



















References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Stor

y

1

2

3

4

5

6

7

8

Velocity (ms)02 04 06 08 10

Stor

y

1

2

3

4

5

6

7

8

3 4 5 6 7 8

Stor

y

1

2

3

4

5

6

7

8


CCLQR

Acceleration (ms2)

NLQR-120572(t) = 1205721

NLQR-120572(t) = 1205722NLQR-120572(t) = 1205723


CCLQRNLQR-120572(t) = 1205721

NLQR-120572(t) = 1205722NLQR-120572(t) = 1205723


CCLQRNLQR-120572(t) = 1205721

NLQR-120572(t) = 1205722NLQR-120572(t) = 1205723


Time (s)0 10 20 30 40 50 60

Con

trol f

orce

(kN

)

minus1000

minus500

0

500

1000

CCLQR NLQR-120572(t) = 1205721






Time (s)0 10 20 30 40 50 60

Con

trol f

orce

(kN

)

0

500

1000

CCLQR

minus1000

minus500

NLQR-120572(t) = 1205722




6 Conclusion



Time (s)0 10 20 30 40 50 60

0

500

1000

CCLQR

minus1000

minus500

Con

trol f

orce

(kN

)

NLQR-120572(t) = 1205723



















References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Time (s)0 10 20 30 40 50 60

0

500

1000

CCLQR

minus1000

minus500

Con

trol f

orce

(kN

)

NLQR-120572(t) = 1205723



















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











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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