Nature of seismic control force in acceleration feedback

STRUCTURAL CONTROL AND HEALTH MONITORINGStruct. Control Health Monit. (2012)Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/stc.1496

Nature of seismic control force in acceleration feedback

Takehiko Asai1, B. F. Spencer Jr.1,*,†, Hirokazu Iemura2 and Chia-Ming Chang1

1Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801-2352, USA2Kinki Polytechnic College, Osaka 596-0103, Japan

SUMMARY

Active control methods have been applied to numerous civil structures in recent years. A significant amount of theresearch on active control methods has been based on full-state feedback using the linear quadratic regulator(LQR) control algorithm; because measurement of the full state (i.e., the displacements and velocities of all DOFs)is difficult, such algorithms are impractical in full-scale implementations. Output feedback strategies based onmeasured acceleration at a limited number of points has been proposed and validated. However, a thorough under-standing of the dissipative nature of the associated control forces and the way in which these forces protect thestructure have been elusive. This paper considers the hysteric behavior of the control forces produced by thewidely employed linear quadratic Gaussian (LQG)-based acceleration feedback control strategies and exploresparallels with the pseudo-negative stiffness (PNS) semi-active damping systems. Numerical simulation studiescarried out on one-story and three-story buildings with active bracing show that the LQG-based algorithms arequite versatile and can produce controllers with a variety of behaviors, including the PNS, depending upon thecontrol objectives chosen. Copyright © 2012 John Wiley & Sons, Ltd.

Received 17 May 2011; Revised 10 November 2011; Accepted 16 March 2012

KEYWORDS: active control; hysteretic loop; pseudo-negative stiffness; linear quadratic Gaussian; acceleration feedback

1. INTRODUCTION

The control of civil structures excited by severe earthquakes or strong wind has attracted much atten-tion since the concept of active control system was first suggested by Yao [1]. More than 20 years haspassed since the first full-scale application of an actively controlled building, the Kyobashi Centerbuilding, was achieved in 1989 in Japan by the Kajima Corporation [2–4]. Subsequently, active controlstrategies have been applied to many buildings and bridges, particularly in Asia. To date, many studiesrelated to active control methods have been performed, and significant progress has been made towardprotecting civil structures from severe environmental loads using these advances [5].

Linear quadratic regulator (LQR) control theory has been studied widely for active control in earthquakeengineering applications, because it is easy to understand. Despite this intense activity, only a few studieshave been carried out to understand the nature of the forces prescribed by active control strategies. Iemuraand Pradono [6] investigated the hysteretic loops produced by the LQR controllers and demonstrated thatthey have negative stiffness. However, employing the LQR algorithms for control of civil structures isimpracticable because the LQR control is based on full-state feedback (i.e., all displacement and velocitymeasurements for each of the structural DOFs). Direct measurement of such displacements and velocitiesat arbitrary locations on large-scale structures is difficult, if not impossible, to achieve.

To avoid this problem, output feedback control based on the linear quadratic Gaussian (LQG) controlwas proposed by Spencer et al. [7,8] and Yang et al. [9] using acceleration measurements. In contrast to

*Correspondence to: B. F. Spencer Jr., Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801–2352 USA.†E-mail: [email protected]

Copyright © 2012 John Wiley & Sons, Ltd.

T. ASAI ET AL.

displacements and velocities, accelerations can be readily obtained using inexpensive and reliable sensorsplaced at arbitrary points on the structure. In the LQG control algorithm, displacements and velocities areestimated from measured accelerations using a Kalman filter; these values are then used to determine thecontrol force. Therefore, fewer sensors are needed than when implementing the LQR controller directly.

In this paper, the behavior of the control forces produced by the LQG-based acceleration feedback controlstrategies are investigated, and parallels are explored with the negative stiffness approaches shown, recentlyto be effective, by Iemura et al. [10], Iemura and Pradono [11,12], Iemura et al. [12,13], Viti et al. [14], Liet al. [15], Nagarajaiah et al. [16], andWebber and Boston [17]. First, the formulation of the problem is pro-vided, including the equations of motion and the associated state space representation. Then, LQG controltheory is briefly overviewed, along with the pseudo-negative stiffness (PNS) algorithms proposed by Iemuraet al. [10], Iemura and Pradono [11,12], Iemura et al. [12,13]. Subsequently, numerical results are presentedfor scaled one-story and three-story building models. Conclusions obtained from this study then follow.

2. ACCELERATION FEEDBACK CONTROL: PROBLEM FORMULATION

Consider a structural system, excited by a one-dimensional earthquake loading, with an equation ofmotion given by

Ms€x þ Cs _xþKsx ¼ Gsf �MsLs€xg (1)

where f is the vector of control forces, €xgis the ground acceleration, Ms, Cs, and Ks are the mass, lineardamping and stiffness matrices of the structure, respectively, and Gs and Ls are the influence coefficientvectors of the control force and structural mass, respectively. x is the displacement vector, which is com-posed of the displacement of each floor relative to the ground. An upper dot represents the time derivative.

A state space representation of the equation of motion can be written as

_z ¼ Azþ Bf þ E€xg (2)

ym ¼ Cmzþ Dmf þ Fm€xg þ v (3)

ye ¼ Cezþ Def þ Fe€xg (4)

where z is the state variable of the story displacements and velocities relative to the base, that is,

z ¼ xT _xT� �T

(5)

ym is the vector corresponding to the measured outputs including absolute accelerations, v is an obser-vation noise vector, ye is the vector corresponding to the regulated outputs that are used for evaluationof the system. The matrices A, B, and E are given as

A ¼ 0 I�Ms

�1Ks �Ms�1Cs

� �; B ¼ 0

Ms�1Gs

� �; E ¼ 0

�Ls

� �(6)

And Cm, Ce, Dm, De, Fm, and Fe are appropriately chosen matrices corresponding to the associatedoutput vectors. Acceleration feedback control strategies for the structure described in Equations (2–4)can be derived based on LQG methods. For the control design, the absolute acceleration of the ground,€xg, is taken to be a stationary filtered white noise, and an infinite horizon performance index is chosenthat weights the regulated output vector, ye, that is,

J ¼ limt!1

1tE

Z t

0Cezþ Defð ÞTQ Cezþ Defð Þ þ fTRf

n odt

� �(7)

where Q and R are called weighting matrices. Further, the observation noise is assumed to be identi-cally distributed, statistically independent Gaussian white noise processes. The separation principle isinvoked to allow the control and estimation problems to be considered independently. The resultingcontroller is of the form [18,19]

f ¼�Kz (8)

where z is the estimated state vector obtained from the Kalman filter, and K is the full state feedbackgain matrix for the deterministic regulator problem given by

Copyright © 2012 John Wiley & Sons, Ltd. Struct. Control Health Monit. (2012)DOI: 10.1002/stc

NATURE OF SEISMIC CONTROL FORCE IN ACCELERATION FEEDBACK

K ¼ ~R�1 ~Nþ BTP� �

(9)

P is the solution of the algebraic Riccati equation given by

P~Aþ ~ATP� PBR�1BTPþ ~Q ¼ 0 (10)

and

~Q ¼ CeTQCe � ~N~R

�1 ~NT

(11)

~R ¼ Rþ DeTQDe (12)

~A ¼ A� B~R�1 ~N

T(13)

The Kalman filter optimal estimator is given by

_z ¼Az þ Bf þ Lðym �Cmz �Dmf � Fm€xgÞ (14)

L ¼ SCm þWEFmT

� �~R�1 (15)

where S is the solution of the algebraic Ricatti equation given by

S~Aþ ~ATS� S~GSþ ~H ¼ 0 (16)

and

~A ¼ AT � CmR�1 WFmEð Þ (17)

~G ¼ CmT~R�1Cm (18)

~H ¼ WEET �W2EFm ~R�1FmET (19)

~R ¼ VþWFmFmT (20)

where W and V are magnitude of the constant two-sided spectral densities for the white noises used inthe LQG control design. The controller given in Equation (8) has been shown to be effective in protect-ing structural systems from seismic loading [20].

3. HYSTERETIC CONTROL FORCE LOOPS

To investigate the nature of the energy dissipation capabilities of acceleration feedback controlstrategies clearly, first, a one-story building model shown in Figure 1(a) is considered. Subse-quently, the three-story model shown in Figure 1(b) is investigated to determine if the trendsfound in the one-story model are also seen in multi-DOFs structures. The three-story buildingmodel was previously investigated by Dyke et al. [21,22]. Because the actuator is installedbetween the first floor and the ground, its displacement is equal to the displacement of the firstfloor of the structure relative to the ground. Here, xi; _xi;€xai and di represent relative displacement,relative velocity, absolute accelerations of ith floor, and inter-story drift between ith and (i�1)thfloor (i.e., xi� xi� 1), respectively. f is the force in the actuator installed between the ground andthe first floor.

The models are subjected to 1940 El Centro NS earthquake record using numerical simulation. Tosatisfy scaling laws, the earthquake must be reproduced at five times the recorded rate. The time historyof the employed earthquake record is shown in Figure 2.

To explore how the weighting matrices affect the natural frequency, the damping ratio, and thehysteretic loops produced by the LQG controller, various values of the weighting matrices Q and Rin Equation (7) are employed. The earthquake input energy is defined as [23]


Figure 1. Schematic diagram. (a) One-story building model; and (b) three-story building model.

0 1 2 3 4 5 6 7 8 9 10-4

-2

0

2

4

Time (sec)

Acc

eler

atio

n (m

/sec

2 )

Figure 2. Earthquake record of the 1940 El Centro NS.

T. ASAI ET AL.

Ei ¼ �Z t

0_xTMsLs€xgdt (21)

and investigated for each case. The LQR controller is applied as well by assuming that all state vectorsare measurable and compared with the analogous LQG controller.

3.1. One-story model

For the one-story structure as shown in Figure 1(a), the given in Equation (1) are

Ms ¼ Ms ¼ 98:3 ðkgÞ; Cs ¼ Cs ¼ 125 ðN sec m-1 Þ; Ks ¼ Ks ¼ 5:16� 105 ðN m-1Þf ¼ f ; Gs ¼ Gs ¼ �1; Ls ¼ Ls ¼ 1:

(22)

The state vector z in Equations (2–5) is then

z ¼ x1 _x1½ �T: (23)

Also, the measurement vector ym in Equation (3) and the evaluation vector ye in Equation (4) aredefined by

ym ¼ x1 €xa1½ �T (24)

ye ¼ x1 €xa1½ �T (25)

respectively. Therefore, in this case, the matrices Cm, Ce, Dm, De, Em, and Ee can be written as

Cm ¼ Ce ¼1 0

� Ks

Ms� Cs

Ms

" #; Dm ¼ De ¼

0Gs

Ms

" #; Fm ¼ 02�1; Fe ¼ 03�1 (26)

Hence, the weighting matrices Q and R in Equation. (7) should be a 2� 2 matrix and a scalar, R,respectively. Here, the weighting matrix Q is set as follows: all of the elements of the displacementweighting matrix Qd are zero, except for Qd11 = 1, and all of the elements of the acceleration weighting



matrix Qa are zero, except for Qa22 = 1. And we assume that power spectral densities W and V are5� 104 and I2� 2, respectively. These values are determined by trial and error so that the LQG showsgood performance. Calculation to determine K in Equation (9) and L in Equation (15) are performedusing the control toolbox in MATLAB (MathWorks, Natick, MA, USA) [24].

Figures 3 and 4 show how natural frequencies and damping ratios are affected by changing thevalue of R in the case of both displacement and acceleration weightings, respectively. From theseparation theorem, the poles (i.e., natural frequencies and damping ratios) obtained by the LQG arethe sum of the poles from the LQR and the Kalman filter. From Figure 3, the actuator controlled bydisplacement weighted LQR and LQG results in the natural frequency of the building model increasingas the control authority increases (i.e., R decreases), which means that the controller added positivestiffness to the structure. In contrast, Figure 4 shows that acceleration weighting leads to smallernatural frequency as the control authority is increased (i.e., R decreases), which indicates that thecontroller adds a negative stiffness on the structure. Figures 3 and 4 also show that the damping ratiosbecome larger as the control force increases in both cases.

Figure 5 shows hysteretic loops of the actuator produced by the LQR and the LQG controllers when1940 El Centro NS is input. Here, four cases are considered for the LQR and the LQG controllers: (a)Q =Qd, R= 10�10 (displacement weighting with small control force); (b) Q =Qd, R= 10�12

(displacement weighting with large control force): (c) Q =Qa, R= 10�3 (acceleration weighting withsmall control force); and (d) Q=Qa, R= 10�5 (acceleration weighting with large control force). Peakvalues of relative displacement, absolute acceleration, and actuator force are summarized in Table I.Plots of energy input to the structure by the earthquake, that is, Equation (21), for the four cases areshown in Figure 6.

The results shown in Figures 3 and 4 are confirmed in Figure 5, where the hysteretic loops of theactuators controlled by the LQG and the LQR are seen to produce both positive and negative stiffness,depending on the weighting matrices employed. Moreover, the LQG and the LQR produced quitesimilar hysteresis loops, as well as earthquake input energy (see Figure 6), although the LQR controllershowed slightly better performance. Finally, Table I confirms that the displacement responses werereduced by the controller using displacement weighting, and the acceleration responses were reducedby the controllers using acceleration weighting.

10-1410-1210-10

100

200

300

R

Nat

ural

Fre

quen

cy (

rad/

s)

(a)(b)

LQR/LQG

Uncontrolled

10-1410-1210-100

0.5

1

R

Dam

ping

Rat

io

(a)

(b)

Figure 3. Natural frequency and damping ratio of a one-story building model for displacement weighting: (a)Q =Qd, R= 10�10; (b) Q=Qd, R = 10�12.

10-610-410-2

20

40

60

80

R

Nat

ural

Fre

quen

cy (

rad/

s)

(c)

(d)

LQR/LQG

Uncontrolled

10-610-410-20

0.5

1

R

Dam

ping

Rat

io

(c)

(d)

Figure 4. Natural frequency and damping ratio of a one-story building model for acceleration weighting: (c)Q=Qa, R= 10�3; (d) Q=Qa, R= 10�5.


-0.1 -0.05 0 0.05 0.1 0.15

-0.1 -0.05 0 0.05 0.1 0.15

-500

0

500

1st Story Displacement (cm)

Act

uato

r Fo

rce

(N)

(a)

LQG

LQR


Act

uato

r Fo

rce

(N)

(b)

-500

0

500

-0.1 -0.05 0 0.05 0.1 0.15

-0.1 -0.05 0 0.05 0.1 0.15

-500

0

500

-500

0

500


Act

uato

r Fo

rce

(N)

(c)


Act

uato

r Fo

rce

(N)

(d)

Figure 5. Hysteretic control force loops for a one-story model produced by LQR and LQG to 1940 El Centro NS:(a) Q =Qd, R= 10�10; (b) Q =Qd, R= 10�12; (c) Q=Qa, R= 10�3; and (d) Q=Qa, R= 10�5.

Table I. Peak values for a one-story model to 1940 El Centro NS: (a) Q=Qd, R= 10�10; (b)Q=Qd, R = 10�12; (cQ=Qa, R= 10�3; (d) Q=Qa, R= 10�5.

Parameter Uncontrolled

(a) (b) (c) (d)

LQG LQR LQG LQR LQG LQR LQG LQR

x1 (cm) 0.173 0.099 0.098 0.032 0.029 0.099 0.097 0.102 0.097€xa1(m/s2) 9.098 5.389 5.310 3.798 3.728 4.995 4.957 3.404 3.333f (N) 93.3 93.7 274.1 261.4 152.5 152.6 422.6 408.3

LQG, linear quadratic Gaussian; LQR, linear quadratic regulator.

T. ASAI ET AL.

Copyright © 2012 John Wiley & Sons, Ltd. Struct. Control Health Monit. (2012DOI: 10.1002/stc

)

The next section investigates active control of a three-story model to see if the trends found in theone-story controlled structure are also found in multi-DOF structures.

3.2. Three-story model

For the three-story structure as shown in Figure 1(b),

Ms ¼98:3 0 00 98:3 00 0 98:3

24

35 ðkgÞ; Cs ¼

175 �50 0�50 100 �500 �50 50

24

35 ðN sec m-1 Þ

Ks ¼ 10512:0 �6:84 0�6:84 13:7 �6:84

0 �6:84 6:84

24

35 ðN m-1Þ; f ¼ f ; Gs ¼

�100

24

35; Ls ¼

111

24

35

(27)

The state vector z in Equations (2–5) is then

z ¼ x1 x2 x3 _x1 _x2 _x3½ �T (28)

Also, the measurement vector ym in Equation (3) and the evaluation vector ye in Equation (4) aredefined by

ym ¼ x1 €xa1 €xa2 €xa3½ �T (29)

ye ¼ x1 d2 d3 €xa1 €xa2 €xa3½ � (30)

Therefore, the matrices Cm, Ce, Dm, De, Em, and Ee become

)

0 2 4 6 8 100

1

2

3

Time (sec)

Ene

rgy

(N*m

)

(a)

0 2 4 6 8 100

Time (sec)

Ene

rgy

(N*m

)

(c)

0 2 4 6 8 100

1

2

3

Time (sec)

Ene

rgy

(N*m

)

(b)

0 2 4 6 8 100

1

2

3

1

2

3

Time (sec)

Ene

rgy

(N*m

)

(d)

LQG

LQR

Figure 6. Earthquake input energy of a one-story model by LQG and LQR to 1940 El Centro NS: (a) Q=Qd,R= 1010; (b) Q=Qd, R = 10�12; (c) Q=Qa, R= 10�3; (d) Q=Qa, R = 10�5.


Cm ¼ 1 0 0 01�3

�Ms�1Ks �Ms

�1Cs

� �; Ce ¼

1 0Δ 02�2

�Ms�1Ks �Ms

�1C

24

35; Δ ¼ �1 1 0

0 �1 1

� �

Dm ¼ 0Ms

�1Gs

� �; De ¼ 03�1

Ms�1Gs

� �; Fm ¼ 04�1; Fe ¼ 06�1

(31)

Hence, the weighting matrices Q and R in Equation (7) should be a 6� 6 matrix and a scalar R,respectively. Here, we consider two cases. The first case weights the second and third inter-story driftsequally, whereas the weighting on the first floor relative displacement is taken as a controldesign parameter a, that is Qd = diag([a 1 1 0 0 0]). The second case places equal weighting on theacceleration of each floor of the structure, that is, Qa = diag([0 0 0 1 1 1]). The power spectral densitiesof the disturbanceW and the measurement noise vectorV are chosen to be 5� 104 and I4� 4, respectively,so as to achieve good control performance.

Figure. 7 shows contours of the natural frequencies and damping ratios of first, second, and thirdmodes for the displacement weighted LQR/LQG controllers as a function of R and a. The natural fre-quencies of the uncontrolled model for three modes are 34.2, 99.3, and 148.5 rad/s, and noted in thisfigure by a heavier line weight. As can be seen here, the natural frequency of the first mode increasesas a increases, regardless of the value of R. This result indicates that the controller can produce bothpositive and negative stiffness contributions to the first mode. Additionally, Figure 7 shows that thenatural frequencies of the second and third modes are not substantially affected by a, and that thedamping ratio increases generally as the control force gets larger, as was the case of the one-storybuilding model.

Figure 8 shows the natural frequencies and damping ratios of the first, second, and third modes as afunction of R for acceleration weighting. For this case, the natural frequencies of all modes decrease asthe control force increases (i.e., R decreases); additionally, the damping ratio of the first mode increasesas the control force is increased, which is again similar to the one-story model.

Six specific cases are considered for more investigation: (a) Q=Qd (a= 1), R= 10�10 (equal inter-story displacement weighting, with small control force); (b)Q =Qd (a= 1), R = 10�12 (equal inter-storydisplacement weighting, with large control force); (c) Q =Qd (a= 0.01), R= 10�10.5 (inter-story dis-placement weighting mainly on the second and third stories, with small control force); (d) Q =Qd

(a= 0.01), R= 10�13.5 (inter-story displacement weighting mainly on the second and third stories, withlarge control force); (e)Q =Qa, R= 10�3.5 (equal acceleration weighting, with small control force); and


2530

35 40

45

50

Uncontrolled (34.2)

(a) (b)

(c) (d)

R

Natural Frequency (rad/s) for 1st mode

10-1410-1210-1010-2

100

R10-1410-1210-1010-2

100

R10-1410-1210-1010-2

100

R10-1410-1210-1010-2

100

R10-1410-1210-1010-2

100

R10-1410-1210-1010-2

100

0.60.50.4

0.3

0.2

0.20.1

(a) (b)

(c) (d)

Damping ratio for 1st mode

130

120110100

(a) (b)

(c) (d)

Natural Frequency (rad/s) for 2nd mode

0.2

0.2

0.3

0.1

(a) (b)

(c) (d)

Damping ratio for 2nd mode

400

350

300250

200150

Uncontrolled (148.5)(a) (b)

(c) (d)

Natural Frequency (rad/s) for 3rd mode

0.60.5

0.40.30.2

0.1

(a) (b)

(c) (d)

Damping ratio for 3rd mode

Figure 7. Natural frequency and damping ratio of a three-story model for displacement weighting: (a)Q=Qd (a=1),R=10�10; (b) Q=Qd (a=1), R=10�12; (c) Q=Qd (a=0.01), R=10�10.5; (d) Q=Qd (a=0.01), R=10�13.5.

T. ASAI ET AL.

(f) Q=Qa, R= 10�5.5 (equal acceleration weighting, with large control force). These controllers are markedwith an ‘x’ in Figures 7 and 8. Figure 7 shows that controller (a) produces positive stiffness in all modes,controller (b) produces positive stiffness in the first and second modes and negative stiffness in third mode,and controllers (c), and (d) produce negative stiffness in first mode and positive stiffness in second and thirdmodes. Figure 8 shows that controllers (e) and (f) produce negative stiffness in all three modes.

The hysteretic loops (i.e., control force versus displacement) for these six cases are shown inFigure 9 for the structure subjected by the NS component of the 1940 El Centro earthquake. Thedominance of the first mode in the response is seen in Figure 9, that is, controllers (b) and (d) producepositive and negative stiffness, respectively. We can also see that controller (f) has negative stiffness asin the case of one-story building model. Thus, the LQR/LQG can produce both positive and negativestiffness, depending on the control weightings chosen. Moreover, the LQR and LQG hysteretic loopsand earthquake input energies are nearly identical.

Peak values of relative displacement, inter-story drift, and absolute acceleration for each floor, alongwith actuator force, are summarized in Table II. When the control force is small (i.e., controllers (a), (c),and (e)), the controller has little impact on the responses. However, when the control force is large, thenature of the controllers is clearer; here, controllers (d) and (f) reduce absolute acceleration for each floormore than controller (b). Again, the LQG controller has comparable performance to its LQR counterpart.

The plots of earthquake energy input to the structure as given by Equation (21) are shown inFigure 10 for cases (a)–(f). Controllers producing negative stiffness (i.e., controllers (d) and (f)) aremore effective at limiting the input earthquake energy than the controller producing positive stiffness(i.e., controller (b)), even though the peak actuator forces of the cases (d) and (f) are smaller than thecase (b) as shown in Table II.


10-610-4

10

20

30

40

R

10-610-4

R

10-610-4

R

Nat

ural

Fre

quen

cy (

rad/

s)

1st mode

(e)(f)

LQR/LQG

Unontrolled

10-610-40

0.5

1

R

Dam

ping

Rat

io

10-610-40

0.5

1

R

Dam

ping

Rat

io

10-610-40

0.5

1

R

Dam

ping

Rat

io

1st mode

(e)

(f)

70

80

90

100

Nat

ural

Fre

quen

cy (

rad/

s)

2nd mode

(e)

(f)

2nd mode

(e)

(f)

140

145

150

Nat

ural

Fre

quen

cy (

rad/

s)

3rd mode

(e)

(f)

3rd mode

(e) (f)

Figure 8. Natural frequency and damping ratio of a three-story model for acceleration weighting: (e) Q=Qa,R= 10�3.5; (f) Q=Qa, R= 10�5.5.


4. PSEUDO-NEGATIVE STIFFNESS METHOD

The PNS method proposed by Iemura et al. [10], Iemura and Pradono [11,12], Iemura et al. [12,13] hasprovided insight into the manner in which actively, semi-actively, and even passively controlled struc-tures reduce structural responses. This section briefly discusses the PNS method and compares theresults with those obtained from the acceleration-based LQG feedback controller. The basic conceptof PNS method is producing control force Fd expressed by

Fd ¼ Kduþ Cd _u (32)

where Kd is a selected negative stiffness value, Cd is a selected damping coefficient and u is a pistondisplacement (stroke). The PNS method was used to realize controllers for various devices, such asa variable-orifice oil damper [6], a magnetorheological (MR) damper [25], a skyhook damper [26],and a negative stiffness friction damper [26,27]. As an example, an MR damper is used in this paper.

Input voltage to an MR damper can also be controlled based on displacement and motion directionsonly. Figure 11 shows the voltage schemes. In the PNS methods, the force always changes from alarger value to a smaller one, depending on the direction. The input voltage V follows the solid linewhen the direction of motion is positive (from left to right) and follows the dashed line when the direc-tion is negative (from right to left). A mathematical expression for this mechanism is shown as

V x; _xð Þ ¼ � sgn _xð Þ � S� xþ Vm (33)

where S is the absolute slope of the line, Vm is the input voltage at the central position, and x is thestroke of the MR damper.


-0.2 -0.1 0 0.1 0.2 0.3-1000

0

1000


Act

uato

r Fo

rce

(N)

(a)

-0.2 -0.1 0 0.1 0.2 0.3-1000

0

1000


Act

uato

r Fo

rce

(N)

(b)

-0.2 -0.1 0 0.1 0.2 0.3-1000

0

1000


Act

uato

r Fo

rce

(N)

(c)

-0.2 -0.1 0 0.1 0.2 0.3-1000

0

1000


Act

uato

r Fo

rce

(N)

(d)

-0.2 -0.1 0 0.1 0.2 0.3-1000

0

1000


Act

uato

r Fo

rce

(N)

(e)

-0.2 -0.1 0 0.1 0.2 0.3-1000

0

1000


Act

uato

r Fo

rce

(N)

(f)

LQG

LQR

Figure 9. Hysteretic control force loops of a three-story model produced by LQR and LQG to 1940 El Centro NS:(a)Q=Qd (a= 1), R = 10�10; (b)Q =Qd (a= 1), R= 10�12; (c)Q=Qd (a= 0.01), R = 10�10.5; (d)Q=Qd (a= 0.01),

R= 10�13.5; (e) Q=Qa, R= 10�3.5; (f) Q=Qa, R= 10�5.5.

T. ASAI ET AL.

In this paper, theMR damper is modeled by using the Bouc–Wenmodel [28] as depicted in Figure 12[29,30]. Here, the forces on either side of the rigid bar are equivalent; therefore,

c1 _y ¼ azþ k0 xd � yð Þ þ c0 _xd � _yð Þ (34)

where the evolutionary variable z is governed by

_z ¼ �g _xd � _yj jz zj jn�1 � b _xd � _yð Þ zj jn þ A _xd � _yð Þ: (35)

The total force can be written as

f ¼ c1 _yþ k1 xd � xoð Þ: (36)

where k1 is the accumulator stiffness, c0 is the viscous damping observed at larger velocities. Then, c1 isincluded in the model to produce the roll-off that was observed in the experimental data at low velocities,k0 is present to control the stiffness at large velocities and x0 is the initial displacements of spring k1associated with the nominal damper force caused by the accumulator. Also, the viscous damping constantsalso vary linearly with the applied voltage. Therefore, the following relations are proposed.

a ¼ a uð Þ ¼ aa þ abu; c1 ¼ c1 uð Þ ¼ c1a þ c1bu and c0 ¼ c0 uð Þ ¼ c0a þ c0bu (37)

where the dynamics involved in the MR fluid reaching rheological equilibrium are accounted for throughthe first order filter

_u ¼ �� u� vð Þ (38)

and v is the voltage applied to the current driver.


Table

II.Peakvalues

forathree-storymodel

to19

40ElCentroNS:(a)Q=Q

d(a

=1),R=10

�10;(b)Q=Q

d(a

=1),R=10

�12;(c)Q=Q

d(a

=0.01),R=10

�10.5;(d)Q=Q

d

(a=0.01),R=10

�13.5;(e)Q=Q

a,R=10

�3.5;(f)Q=Q

a,R=10

�5.5.

Param

eter

Uncontrolled

(a)

(b)

(c)

(d)

(e)

(f)

LQG

LQR

LQG

LQR

LQG

LQR

LQG

LQR

LQG

LQR

LQG

LQR

x 1(cm)

0.541

0.230

0.228

0.099

0.096

0.240

0.242

0.196

0.192

0.212

0.210

0.221

0.216

x 2(cm)

0.826

0.365

0.362

0.162

0.157

0.375

0.377

0.220

0.215

0.332

0.327

0.258

0.253

x 3(cm)

0.964

0.458

0.454

0.249

0.242

0.461

0.465

0.237

0.231

0.401

0.396

0.277

0.272

d 2(cm)

0.320

0.158

0.157

0.131

0.127

0.152

0.154

0.041

0.041

0.129

0.127

0.056

0.056

d 3(cm)

0.202

0.099

0.097

0.087

0.085

0.093

0.095

0.031

0.031

0.077

0.075

0.039

0.038

€ x a1(m

s-2)

8.664

4.684

4.563

4.069

3.928

4.179

4.285

2.690

2.653

3.402

3.358

1.601

1.583

€ x a2(m

s-2)

10.450

4.905

4.819

4.239

4.075

4.615

4.691

1.533

1.514

4.107

4.034

1.664

1.642

€ x a3(m

s-2)

14.061

6.879

6.782

6.084

5.938

6.484

6.578

2.167

2.136

5.351

5.240

2.686

2.647

f(N

)276.6

275.5

944.0

946.9

277.2

278.7

887.9

872.4

318.2

312.7

872.4

859.3

LQG,lin

earquadratic

Gaussian;

LQR,linearquadratic

regulator.



0 2 4 6 8 100

5

10

Time (sec)

Ene

rgy

(N*m

)

(a)

0 2 4 6 8 100

5

10

Time (sec)

Ene

rgy

(N*m

)

(b)

0 2 4 6 8 100

5

10

Time (sec)

Ene

rgy

(N*m

)

(c)

0 2 4 6 8 100

5

10

Time (sec)

Ene

rgy

(N*m

)

(d)

0 2 4 6 8 100

5

10

Time (sec)

Ene

rgy

(N*m

)

(e)

0 2 4 6 8 100

5

10

Time (sec)

Ene

rgy

(N*m

)(f)

LQG

LQR

Figure 10. Earthquake input energy of a three-story model by LQG and LQR to 1940 El Centro NS: (a) Q=Qd (a=1),R=10�10; (b)Q=Qd (a=1), R=10

�12; (c)Q=Qd (a=0.01), R=10�10.5; (d)Q=Qd (a=0.01), R=10

�13.5; (e)Q=Qa,R=10�3.5; (f) Q=Qa, R=10

�5.5.

Figure 11. Input voltage (function of displacement and motion direction).

T. ASAI ET AL.

To design controllers, two values of the slope S are used: (A) S = 5 (V/cm), and (B) S = 10 (V/cm),whereas the maximum voltage is set 2.25V [30] and the Vm is fixed at 1.125V that is, half of the max-imum. Also, Table III shows the values of parameters for the MR damper model used in this paper [30].

The hysteretic loops produced by the PNS controller using the MR damper are investigated.Figure 13 shows the results obtained from the same three-story model as used in the active control case.From Figure 13, we can confirm that the hysteretic loops of the MR damper controlled by the PNS


Figure 12. Proposed mechanical model of the magnetorheological damper.

Table III. Parameters for the magnetorheological damper model.

Parameter Value Parameter Value Parameter Value

c0a 21.0 (N sec cm�1) k1 5.00 (N cm�1) b 363 (cm�2)c0b 3.50 (N sec cm�1 V�1) x0 14.3 (cm) A 301k0 46.9 (N cm�1) aa 140 (N cm�1) n 2c1a 283 (N sec cm�1) ab 695 (N cm�1 V�1) Z 190 (sec�1)c1b 2.95 (N sec cm�1 V�1) g 363 (cm�2)

-0.2 -0.1 0 0.1 0.2 0.3

-1000

-500

0

500

1000


MR

Dam

per

Forc

e (N

)

-0.2 -0.1 0 0.1 0.2 0.3

-1000

-500

0

500

1000


MR

Dam

per

Forc

e (N

)

(A) (B)

Figure 13. Hysteretic loops of A three-story model produced by pseudo-negative stiffness to 1940 El Centro NS:(A) S = 5V/cm; and (B) S = 10V/cm.


have negative stiffness. The relationships between the input voltage to the MR damper and the relativedisplacement of the first floor are depicted in Figure 14.

Figure 15 shows the earthquake input energies. For comparisons of the three methods producing neg-ative stiffness, Figure 15 includes not only the two PNS controllers but also case (d) of the displacementweighted LQG (i.e., Q=Qd (a=0.01), R=10�13.5), which is redefined as case (C) and case (f) of theacceleration weighted LQG (i.e.,Q=Qa, R=10�5.5) which is redefined as case (D) here. The peak valuesof structural responses, such as relative displacements, inter-story drifts, and absolute accelerations of eachfloor, from the 1940 El Centro NS earthquake record for these four cases, are also presented in Table IV.

Figures 13 and 15 and Table IV show that the differences in the performance of the two PNS/MRdamper controllers (i.e., cases (A) and (B)) are small in terms of the shapes of hysteretic loops, earthquakeinput energies, and peak response values. From Table IV, the displacements of the second and third floorsrelative to the ground (i,e., x1, and x2) are seen to be reduced better by the PNS/MR damper controllers(i.e., cases (A) and (B)) than by the LQG active controllers (i.e., cases (C) and (D)). On the other hand,the performance of the PNS semi-active controller is inferior to the LQG active controllers in terms ofthe inter-story drifts (i.e., d2 and d3) and absolute accelerations (i.e., €xa1; €xa2; and €xa3 ). This result isexpected, as the LQG controllers explicitly seek to minimize inter-story drifts and acceleration responses.Finally, Figure 15 shows that the PNS/MR damper controllers have reasonable performance; however,active controller cases (C) and (D) show superior performance in terms of the earthquake input energy.


0 2 4 6 8 100

2

4

6

8

10

12

Time (sec)

Ene

rgy

(N*m

)

(A)(B)

(C)(D)

Figure 15. Earthquake input energy of a three-story model by pseudo-negative stiffness to 1940 El Centro NS: (AS= 5V/cm; (B) S = 10V/cm; (C) linear quadratic Gaussian (LQG) (Q=Qd (a= 0.01), R= 10�13.5); (D) LQG

(Q=Qa, R= 10�5.5).

Table IV. Comparisons of peak values of a three-story model obtained by pseudo-negative stiffness to 1940 ECentro NS: (A) S = 5V/cm; (B) S = 10V/cm; (C) linear quadratic Gaussian (LQG) (Q=Qd), (a= 0.01)

R = 10�13.5); (D) LQG (Q=Qa, R = 10�5.5).

Parameter Uncontrolled (A) (B) (C) (D)

x1(cm) 0.541 0.106 0.113 0.196 0.221x2(cm) 0.826 0.196 0.177 0.220 0.258x3(cm) 0.964 0.295 0.272 0.237 0.277d2(cm) 0.320 0.138 0.138 0.041 0.056d3(cm) 0.202 0.103 0.104 0.031 0.039€xa1(m s�2) 8.664 2.983 3.742 2.690 1.601€xa2(m s�2) 10.450 4.337 4.854 1.533 1.664€xa3(m s�2) 14.061 7.200 7.212 2.167 2.686f (N) 804.9 921.3 887.9 872.4

-0.2 -0.1 0 0.1 0.2 0.30

0.5

1

1.5

2


App

lied

Vol

tage

(V

)

(A)

-0.2 -0.1 0 0.1 0.2 0.30

0.5

1

1.5

2


App

lied

Vol

tage

(V

)

(B)

Figure 14. Applied voltage to magnetorheological damper of a three-story model to 1940 El Centro NS: (A)S = 5V/cm; and (B) S = 10V/cm.

T. ASAI ET AL.

Copyright © 2012 John Wiley & Sons, Ltd. Struct. Control Health Monit. (2012DOI: 10.1002/stc

)

l,

5. CONCLUSIONS

This paper investigated the nature of the hysteric behavior of the control forces produced by thewidely employed LQG-based acceleration feedback control strategies and explored parallels withthe PNS semi-active damping systems. Numerical simulation studies carried out on one-story andthree-story buildings with active bracing show that the LQG-based algorithms are quite versatileand can produce controllers with a variety of behaviors, including the PNS, depending upon thecontrol objectives chosen. Additionally, the numerical results demonstrated that the presentedLQG-based acceleration feedback control had performance comparable to the LQR in the pre-sented SDOF and 3DOF building models.

)


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Nature of seismic control force in acceleration feedback

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