ro

Mex

) aplictionr bgape.

erablenancct thetural chis undthe buy add[15].

sied as passive control which does not require an external power

which uses a mass without spring and dashpot [5]. In this paper,we use AMD type actuator for the active vibration control.

In order to achieve a good performance, it is essential to designan effective control strategy, which should be simple, robust, andfault tolerant. Many attempts have been made to introduceadvanced controllers for the active vibration control of building

structures. Instead of changing the structure stiffness, a pole-place-

time applications [4]. The great advantages of PID control overr physical mean-lgorithmsuctural viapplied to

the building displacement due to wind excitation. In [12,1and PID controllers were used in the numerical simulati[31], a Proportional-integral (PI) controller with an AMD is usedto attenuate the structural motion due to earthquake. However,these control results are not satisfactory, because it is difcult totune PID gains to guarantee good performances such as rise-time,overshoot, settling time, and steady-state error [13]. Moreover,these works do not discuss the stability analysis of these activecontrol systems.

Corresponding author. Tel.: +52 55 57473734.E-mail address: yuw@ctrl.cinvestav.mx (W. Yu).

Engineering Structures 81 (2014) 208218

Contents lists availab

g

lsesource [28], and active control which uses sensors and active actu-ators to control the unwanted vibrations [27]. There are manyactive control devices designed for structural control applications[7]. The active mass damper (AMD) is the most popular actuator,

the others are that they are simple and have cleaings. Although theory research in PID control aestablished, it is still not well developed in strcontrol. In [19], a simple proportional control ishttp://dx.doi.org/10.1016/j.engstruct.2014.09.0420141-0296/ 2014 Elsevier Ltd. All rights reserved.is wellbrationreduce3], PDons. InControl device and controller design are the main focus of thetraditional active vibration control systems [9,10]. Since the forceexerted by the earthquake and wind on the structures are veryhuge and uncertain, these large civil structures require a largeamount of energy to control it. The structural control can be clas-

the building structure. Some model-free controllers, such as slidingmode control (SMC) [33], neural network control [17], and fuzzylogic control [25] are still complex.

PID control is widely used in industrial applications. Withoutmodel knowledge, PID control may be the best controller in real-1. Introduction

The building structures are vulnmade hazards, which may result inhuman losses. It is essential to protethe human occupants and non-structhreats. One approach to mitigate talter the dynamic characteristics ofgiven load, which can be achieved bdampers or actuators to the buildingto natural and man-ial, environmental, andse structures, includingomponents from theseesirable behavior is toilding with respect to aing control devices like

ment H1 control corresponding to a target damping ratio is pro-posed in [20]. In order to avoid the higher order problem in H1control, the balanced truncation is applied in [23]. In [8], thegenetic algorithm is used to determine the feedback control. Thereare several optimal control algorithms applied for the active vibra-tion control of building structures, for example ltered linear qua-dratic control (LQ) [24], linear quadratic regulator (LQR) [1], andlinear quadratic Gaussian (LQG) [14]. All these controllers aremodel-based, that are complex and demands the exact model ofStabilityStability analysis of active vibration contPD/PID control

Suresh Thenozhi, Wen Yu Departamento de Control Automtico, CINVESTAV-IPN (National Polytechnic Institute),

a r t i c l e i n f o

Article history:Received 12 August 2013Revised 23 September 2014Accepted 24 September 2014Available online 17 October 2014

Keywords:Active vibration controlBuilding structuresPID control

a b s t r a c t

Proportional-derivative (PDalgorithms in industrial aplers on the structural vibravibration control system fotions for choosing the PIDtwo-story building prototy

Engineerin

journal homepage: www.el of building structures using

ico City, Mexico

nd proportional-integral-derivative (PID) controllers are the most popularations. However, there are few published theory results of PD/PID control-control applications. In this paper, we analyze the stability of the active

oth the linear and nonlinear structures. We give explicit sufcient condi-ins. The theory conclusions are veried via numerical simulations and aThese results give validation of our theory analysis.

2014 Elsevier Ltd. All rights reserved.

le at ScienceDirect

Structures

vier .com/locate /engstruct

While there is no doubt about the advances in the structuralcontrol eld, there still exist some areas which need more explora-tion [30] . The active devices have the ability to add force onto thebuilding structure. A poorly designed controller will lead to anundesirable control performance, which can even damage thebuilding. So it is desired to study the stability of the closed-loopsystem. Only a few structural controllers such as H1 and SMC con-sider the stability in their design, whereas the other control strat-egies do not. However, these designs have concerned only thelinear stiffness, since it represents a simple and efcient model atleast for a small operational range. In practice, these building struc-tures possess nonlinear behavior like the hysteresis phenomenon[6]. Also, there is a lack of experimental verication of these con-trollers. The practical implementation of a controller will be chal-lenging if these issues were not addressed.

In this paper, we use standard industrial PD and PID controllersfor the active vibration control. The main contribution is that wegive theory analysis of these PD/PID controllers. Both the linearand nonlinear cases for structural stiffness are considered in the

Mx C _x fs fe 2For unidirectional motion, the parameters can be simplied as

[19]:

M

m1 0 00 m2 ..

.

..

. ... . .

. ...

0 0 mn

2666664

3777775 2 R

nn;

C

c1 c2 c2 0 0c2 c2 c3 ..

. ...

..

. ... . .

. ... ..

.

. .

266666666

3777777772 Rnn;

3

S. Thenozhi, W. Yu / Engineering Structures 81 (2014) 208218 209analysis. BoucWen model is used to model the nonlinear hyster-esis phenomenon. The sufcient conditions for asymptotic stabilityare derived, which are simple and explicit. The controller gains canbe decided directly from these conditions. Numerical simulationsare given to compare with SMC. An active vibration control systemfor a two-story building structure equipped with an AMD is con-structed for the experimental study. The experimental results viaPD and PID controllers are discussed and the effectiveness of ourtheory results is demonstrated.

2. Model and active control of building structures

Consider a simplebuilding structure,which canbemodeledby [6],

mx c _x kx f e 1where m is the mass, c is the damping coefcient, k is the stiffness,f e is an external force applied to the structure, and x; _x, and x are thedisplacement, velocity, and acceleration, respectively.

A model for a linear multi-story structure with n-degree-of-freedom (n-DOF) is shown in Fig. 1. Here it is assumed that themass of the structure is concentrated at each oor. Neglectinggravity force and assuming that a horizontal force is acting onthe structure base, the equation of motion of the n-oor structurecan be expressed as [19],

1k

2k

1x

2x

1c

2c

1m

2m

nk

n1m

nm

n1x

nx

ncgx

Fig. 1. Mechanical model of a n-DOF building structure... .. cn1 cn cn0 0 cn cn

4 5x 2 Rn; fs f s;1; . . . ; f s;n

2 Rn is the structure stiffness force vector,and fe 2 Rn is the external force vector applied to the structure,such as earthquake and wind excitations.

If the relationship between the lateral force fs and the resultingdeformation x is linear, then fs is

fs Kx; where K

k1k2 k2 0 0k2 k2k3 ..

. ...

..

. ... . .

. ... ..

.

..

. ... kn1kn kn

0 0 kn kn

2666666664

37777777752Rnn

4If the relationship between the lateral force fs and the resultingdeformation x is nonlinear, then the stiffness component is saidto be inelastic [6]. This happens when the structure is excited bya very strong force, that deforms the structure beyond its limit oflinear elastic behavior. BoucWen model gives a realistic represen-tation of the structural behavior under strong earthquake excita-tions. The force-displacement relationship of each of the stiffnesselements (ignoring any coupling effects) agrees the following rela-tionship [32]:

f s;i kixi 1 kigui; i 1; . . . ;n 5where the rst part is the elastic stiffness and the second part is theinelastic stiffness, ki is the linear stiffness dened in Eq. (4), and gare positive numbers, and ui is the nonlinear restoring force whichsatises

_ui g1 d _xi bj _xijjuijp1ui c _xijuijph i

6

x

Fig. 2. Hysteresis loop of BoucWen model.

where d;b; c, and p are positive numbers. The BoucWen model hashysteresis property. Its input displacement and the output force isshown in Fig. 2. The dynamic properties of the BoucWen modelhas been analyzed in [16].

In the case of closed-loop control systems, its input and outputvariables may respond to a few nonlinearities. From the controlpoint of view, it is crucial to investigate the effects of the nonlin-earities on the structural dynamics.

2.1. PD control

PD control may be the simplest controller for the structuralvibration control system, which provides high robustness withrespect to uncertainties. PD control has the following form

k1

k2

x1

x2 + xd

c2

m1

m2

u

cd

md

x2

Amplifier

PD/PID Controller

x1

x2

xg

m1

m2

umd

210 S. Thenozhi, W. Yu / Engineering Structures 81 (2014) 208218The BoucWen model represented in Eqs. (5) and (6) is said tobe bounded input-bounded output (BIBO) stable, if and only if theset Xbw with initial conditions u0 is non-empty. The set Xbw isdened as: u0 2 R such that f s is bounded for all C1 input signal,and x with xed values of parameters d; b; c, and p;ua and ub aredened as

ua d

b cp

s; ub

d

c bp

s7

For any bounded input signal x, the corresponding hysteresisoutput f s is also bounded. On the other hand if u0 2 Xbw ;,then the model output f s is unbounded. Table 1 shows how theparameter d; b; c, affect the stability property of the BoucWenmodel.

Passivity is the property stating that the system storage energyis always lesser than its supply energy. On the other hand, theactive systems generate energy. In [16], it is shown that theBoucWenmodel is passive with respect to its storage energy. Case1 in Table 1 describes the physical system sufciently well and pre-serves both the BIBO stability and passivity properties.

The nonlinear differential equation Eq. (6) is continuos depen-dence on time. It is locally Lipschitz. For the case p > 1, we can con-clude that Eq. (6) has a unique solution on a time interval 0; t0 .This property will be used later during the stability analysis.

The main objective of structural control is to reduce the acceler-ation response of buildings to a comfortable level. In order toattenuate the vibrations caused by the external force, an AMD isinstalled on the structure, see Fig. 3. The closed-loop system withthe control force u is dened as

Mx C _x fs fe Cu d 8where u 2 Rn is the control signals applied to the dampers, d 2 Rnis the damping and friction force vector of the dampers, andC 2 Rnn is the location matrix of the dampers, dened as follows.

Ci;j 1 if i j s0 otherwise

; 8i; j 2 f1; . . . ; ng; s# f1; . . . ;ng 9

where s are the oors on which the dampers are installed. In the

case of a two-story building C C1;1 C1;2C2;1 C2;2

, if the damper is

placed on second oor, s f2g, C 0 00 1

. If the damper is placed

on both rst and second oor, then s f1;2g;C 1 00 1

.

The damper force f d, exerted by the q-th damper on the struc-ture is

Table 1Stability of BoucWen model with different d; b; c.

Case Conditions Xbw Upper bound of utj j1 d > 0;b c > 0 and b cP 0 R max u0j j;ua 2 d > 0;b c < 0 and bP 0 ub;ub max u0j j;ua 3 d < 0;b c > 0 and b cP 0 R max u0j j;ub 4 d < 0;b c < 0 and bP 0 u ;u max u0j j;u a a b5 d 0;b c > 0 and b cP 0 R u0j j6 All other conditions ; Unboundedf d;q md;qxs xd;q uq dq 10where md;q is the mass of the q-th damper, xs is the acceleration ofs-th oor on which the damper is installed, xd;q is the acceleration ofq-th damper, uq is the control signal applied to the q-th damper, and

dq cd;q _xd;q qmd;qg tanh bh _xd;q 11

where cd;q and _xd;q are the damping coefcient and velocity of theq-th damper respectively and the second term is the Columb fric-tion represented using a hyperbolic tangent dependent on a largepositive constant bh where q is the friction coefcient betweenthe q-th damper and the oor on which it is attached and g is thegravity constant [22].

Since AMD controller adds force to the building structure, thisforce may stabilize or destabilize the building structure. If the con-trol algorithm generates unstable signal, the dampers will producea force which can make the building structure unstable. This ismore crucial for nonlinear devices, because even for a boundedinput signal, nonlinear devices may produce unstable output.

Obviously, the building structures in open-loop are asymptoti-cally stable when there is no external force, fe 0. This is also truein the case of inelastic stiffness, due to the BIBO stability and pas-sivity properties. During excitation, the ideal active control forcerequired for cancelling out the vibration completely is Cu fe.However, it is impossible because fe is not always measurableand is much bigger than any control device force. Hence, the objec-tive of the active control is to maintain the vibration as small aspossible by minimizing the relative movement between the struc-tural oors. In the next section, we will discuss the simple PD andPID controllers and their stability analysis.

xg

c1

Fig. 3. Building structure equipped with AMD.Fig. 4. PD/PID control for a two-story building.

x C Kd x x f 25

_x f 2

_x f 2f _x 6 _x Kf _x f Kf f 27

ng Structures 81 (2014) 208218 211u Kpx xd Kd _x _xd 12where Kp and Kd are positive-denite constant matrices, which cor-respond to the proportional and derivative gains, respectively andxd is the desired position. In active vibration control of buildingstructures, the references are xd _xd 0, hence Eq. (12) becomesu Kpx Kd _x 13

The aim of the controller design is to choose the suitable gainsKp and Kd in (13), such that the closed-loop system is stable. With-out loss of generality, we use a two-story building structure asshown in Fig. 4.

When the structural parameters in Eq. (8) are completelyknown, i.e. there are no uncertainties and fs is linear as in Eq. (4)then the building structure is a linear determinant system. Manypapers have used this model for the structure control design, suchas PID control [12], H2 control [23], and optimal control [1]. How-ever, they did not discuss the stability problem.

Assuming d 0, the closed-loop system with the PD control inEq. (13) is

Mx C _x Kx fe CKpx Kd _x 14

where M m1 00 m2

>0;C c1c2 c2c2 c2

>0;K k1k2 k2k2 k2

>0;

x x1x2

;fe m1xg

m2xg

;Kp kp1 00 kp2

>0, and Kd kd1 00 kd2

>0. The

damper is installed on the second oor, then C 0 00 1

. Now we

are in a position to study the system represented in (14) using linear

techniques. Eq. (14) can be written in the state-space form

_z Aclz fcl 15where z x_x

2R4;Acl 022 I22M1 KCKp

M1 CCKd

2R44,and fcl 012 fTe

h iT2R4.

The stability of the closed-loop system in Eq. (15) depends onthe system matrix Acl. Its characteristic polynomial is

detsI Acl s4 a1s3 a2s2 a3s a4 16where

a1 1m1 c1 c2 1m2

c2 kd2

a2 1m1m2 c1kd2 c2kd2 m1kp2 c1c2 k1m2 k2m1 k2m2

a3 1m1m2 k1kd2 k2kd2 c1kp2 c2kp2 c1k2 c2k1

a4 1m1m2 k1kp2 k2kp2 k1k2

17

Using LienardChipart criterion [21], the closed-loop system Acl isstable if and only if

ai > 0; i 1;2;3;4 and a1a2a3 a21a4 a23 > 0 18

Now the designer can directly choose the controller gains,which can satisfy the ve inequalities given by Eq. (18).

In practice, the parameters of the building structure are partlyknown and the structure model might have nonlinearities suchas the hysteresis phenomenon. It is convenient to express Eq.

S. Thenozhi, W. Yu / Engineeri(8) as

Mx C _x f Cu 19where Kf is any positive denite matrix. In this paper, we select Kfas

C > Kf > 0 28So

_V 6 _xT C Kd Kf _x fTK1f f 29If we choose Kd > 0, then

_V 6 _xTQ _x lf 6 km Q _xk k2 fTK1f f 30where Q Kd C Kf > 0. V is therefore an ISS-Lyapunov function.Using Theorem 1 from [26], the boundedness of fTK1f f 6 lf impliesthat the regulation error _xk k is bounded. It is noted that when_xk k2Q > lf ; 8t 2 0; T 31

_V < 0. Now we prove that the total time during which _xk k2Q > lf isnite. Let Tk denotes the time interval during which _xk k2Q > lf .Using the matrix inequality

XTY YTX 6 XTKX YTK1Y 26which is valid for any X; Y 2 Rnm and any 0 < K KT 2 Rnn, wecan write the scalar variable _xTf as

T 1 T 1 T T T 1where

f fs fe d 20The building structure with the PD control in Eq. (13) can be

now written as

Mx C _x f C Kpx Kd _x 21

Since Eq. (21) is a nonlinear system and M;C, and f areunknown, RouthHurwitz stability criterion in Eq. (16) cannotbe applied here. The following theorem gives the stability anal-ysis of the PD control in Eq. (13). In order to simplify the proof,we rst consider Cnn Inn, i.e., each oor has an actuatorinstalled on it.

Theorem 1. Consider the structural system as Eq. (19) controlled bythe PD controller as Eq. (13), the closed-loop system as Eq. (21) isstable, provided that the control gains satisfy1

Kp > 0; Kd > 0 22The derivative of the regulation error x converges to the residual set

D _x _xj _xk k2Q 6 lfn o

23

where lf P fTK1f f and C > Kf > 0.

Proof. We select the systems energy as the Lyapunov candidate V.

V 12

_xTM _x 12xTKpx 24

The rst term of Eq. (24) represents the kinetic energy and thesecond term is the virtual elastic potential energy. Since M and Kpare positive denite matrices, V P 0. The derivative of Eq. (24) is

_V _xTMx _xTKpx _xT C _x f Kpx Kd _x _xTKpx

_ T _ _ T1 K > 0 means K is a positive denite matrix, i.e., with any vector x; xTKx > 0, all ofits eigenvalues are positive.

In order to analyze the stability of PID controller, Eq. (35) isexpres

u n K

Nocan be

Mx

In2

g Sddt

n

x_x

64 75 Kix_x

M1 C _x f Kpx Kd _x n 64 75 38

The equilibrium of Eq. (38) is n;x; _x n;0;0 . Since at equilib-rium point x 0 and _x 0, the equilibrium is f 0 ;0;0 . In order to

move

n nsed by

Kpx Kd _x nix; n0 0

36

w substituting Eq. (36) into Eq. (19), the closed-loop systemwritten as

C _x f Kpx Kd _x n 37

matrix form, the closed-loop system is3 2 3(1) If only nite times that _xk k2Q > lf stay outside the circle ofradius lf (and then reenter), _xk k2Q > lf will eventually stayinside of this circle.

(2) If _xk k2Q > lf leave the circle innite times, since the totaltime _xk k2Q > lf leave the circle is nite, thenX1k1

Tk < 1; limk!1

Tk 0 32

So _xk k2Q is bounded via an invariant set argument. From Eq.(30) _xk k is also bounded. Let _xk k2Q denotes the largest track-ing error during the Tk interval. Then Eq. (32) and bounded_xk k2Q imply that

limk!1

_xk k2Q lfh i

0 33

So _xk k2Q will converge to lf , hence Eq. (23) is achieved.

Since V P 0;V decreases until _xk k2Q 6 lf . Total time of_xk k2Q > lf being nite means that V 12 _xTM _x 12xTKpx is bounded,

hence the regulation error _x is bounded. h

It is well known that the regulation error becomes smaller whileincreasing the gain Kd. The cost of large Kd is that the transient per-formance becomes slow. Only when Kd !1, the regulation errorconverges to zero [18]. However, it would seem better to use asmaller Kd if the system contains high-frequency noise signals.

2.2. PID control

From the above section it is clear that any increase in the deriv-ative gain Kd can decrease the regulation error, but causes a slowresponse. In the control viewpoint, the regulation error can beremoved by introducing an integral component to the PD control,i.e., modify the PD control into PID control. The PID control lawcan be expressed as

u Kpx xd KiZ t0x xdds Kd _x _xd 34

where Ki > 0 correspond to the integration gain. For the regulationcase xd _xd 0, Eq. (34) becomes

u Kpx KiZ t0xds Kd _x 35

212 S. Thenozhi, W. Yu / Engineerinthe equilibrium to origin, we dene

f 0 39The nal closed-loop equation becomes

Mx C _x f Kpx Kd _x n f 0 n Kix

40

In order to analyze the stability of Eq. (40), we rst give the follow-ing properties.

P1. The positive denite matrix M satises the followingcondition.

0 < kmM 6 Mk k 6 kMM 6 m 41where kmM and kMM are the minimum and maximum eigen-values of the matrix M, respectively and m > 0 is the upper bound.

P2. The term f is Lipschitz over ~x and ~y

f ~x f ~y k k 6 kf ~x ~yk k 42Most of the uncertainties are rst-order continuous functions.

Since fs; fe, and d are rst-order continuous functions and satisfyLipschitz condition, P2 can be established using Eq. (20). Now wecalculate the lower bound of

Rf dx.Z t

0fdx

Z t0fsdx

Z t0fedx

Z t0d dx 43

We dene the lower bound ofR t0 fsdx is f s and for

R t0 d dx is d.

Compared with fs and d; fe is much bigger in the case of earth-quake. We dene the lower bound of

R t0 fedx is f e. Finally, the

lower bound kf is

kf f s f e d 44

The following theorem gives the stability analysis of PID con-troller (36).

Theorem 2. Consider the structural system as Eq. (19) controlled bythe PID controller as Eq. (36), the closed-loop system as Eq. (40) isasymptotically stable at the equilibrium n f 0 ;x; _x T 0, providedthat the control gains satisfy

km Kp

P32kf kc

kM Ki 6 /km Kp

kM M km Kd P / 1 kckM M

kmC

45

where / 13 km M km Kp

q.

Proof. Here, the Lyapunov function is dened as

V 12

_xTM _x 12xTKpx a2 n

TK1i n xTn axTM _xa2xTKdx

Z t0fdx kf 46

where kf is dened in Eq. (46) such that V 0 0. In order to showthat V P 0, it is separated into three parts, such that V P3i1ViV1 16x

TKpx a2 xTKdx

Z t0fdx kf P 0 47

V2 16xTKpx a2 n

TK1i n xTn

P1 1

kmKp xk k2 akmK1i nk k2 xk k nk k 48

tructures 81 (2014) 2082182 6 2

When aP 3kmK1i kmKp

,

R t" #(

ng SV2 P12

kmKp

3

rxk k

3

kmKp

snk k

!2P 0 49

and

V3 16xTKpx 12 _x

TM _x axTM _x 50

~yTA~xP ~yk k A~xk kP ~yk k Ak k ~xk kP kMAj j ~yk k ~xk k 51when a 6

13kmMkmKp

pkM M

V3 P12

13kmKp xk k2 kmM _xk k2 2akMM xk k _xk k

12

kmKp

3

rxk k

kmM

p_xk k

!2P 0 52

If 13

rkmK1i k

32mKpk

12mMP kMM 53

there exists13 kmMkmKp

qkMM P aP

3kmK1i kmKp

54

The derivative of Eq. (46) is

_V _xTMx _xTKpx anTK1i n _xTn xTn a _xTM _x axTMx a _xTKdx _xT

f _xT C _x f Kpx Kd _x n f 0 _xTKpx anTK1i n _xTn xTn a _xTM _x axT C _x f Kpx Kd _x n f 0

axTKd _x _xTf 55

From Eq. (42)

axT f 0 f 6 akf xk k2 56Using Eq. (26) we can write

axTC _x 6 akc xTx _xT _x 57

where Ck k 6 kc .Since n Kix; nTK1i n becomes axTn and xTn becomes xTKix,

then

_V _xT C Kd aM akc _x xT aKp Ki akf akc

x 58Using Eqs. (41) and (58) becomes,

_V 6 _xT kmC km Kd akMM akc _x xT akmKp kMKi akf akc

x 59

If kmC km Kd P a kM M kc and km Kp

P 1a kM Ki kf kc ,then _V 0; xk k decreases. From (54) and km K1i

1kM Ki , if

km Kd P13km M km Kp

r1 kc

kM M

kmC

km Kp

P32kf kc

60

then Eq. (45) is established.Finally, we prove the asymptotic stability of the closed-loop

system as Eq. (40). There exists a ball R of radius q > 0 centered atthe origin of the state-space on which _V 0. The origin of theclosed-loop equation as Eq. (40) is a stable equilibrium. Since theclosed-loop equation is autonomous, we use La Salles theorem.

S. Thenozhi, W. Yu / EngineeriDene X asCu 0 00 1

kp1 00 kp2

x1x2

ki1 00 ki2

0 x1dsR t0 x2ds

kd1 00 kd2

_x1_x2

63

Cu 0kp2x2 ki2R t0 x2ds kd2 _x2

64

where the scalars kp2; ki2, and kd2 are the proportional, integral, andderivative gains, respectively. In this case Eq. (45) becomes,

kp2 P32kf kc

ki2 6 ~/minfkp2gkM M

k 65X z t xT ; _xT ; nT T 2 R3n : _V 0n o n 2 Rn; x 0 2 Rn; _x 0 2 Rnf g 61

From Eq. (55), _V 0 if and only if x _x 0. For a solution z t tobelong to X for all t P 0, it is necessary and sufcient thatx _x 0 for all t P 0. Therefore, it must also hold that x 0 forall t P 0. We conclude that from the closed-loop system as Eq.(40), if z t 2 X for all t P 0, then f x f 0 n f 0 and _n 0.It implies that n 0 for all t P 0. So z t 0 is the only initial con-dition in X for which z t 2 X for all t P 0. We conclude from theabove discussions that the origin of the closed-loop system as Eq.(40) is asymptotically stable. It establishes the stability of the pro-posed controller, in the sense that the domain of attraction can beenlarged with a suitable choice of the gains. Namely, increasingKp the basin of attraction will grow. h

Remark 1. Since the stiffness of the building structure has hyster-esis property, the hysteresis output depends on both the instanta-neous and the history of the deformation. This deformation beforeapplying the force (loading) and after removing the force (unload-ing) is not the same, i.e, the equilibrium position before the earth-quake and after the vibration dies out is not the same. After theearthquake, the stable point is moved. This corresponds to theterm f 0 . So we cannot conclude that the closed-loop system isglobally stable.

It is well known that, in the absence of the uncertainties andexternal force, f 0, the PD control as Eq. (13) with any positivegains can drive the closed-loop system asymptotically stable. Themain objective of the integral action can be regarded to cancel f.In order to decrease integral gain, an estimated f is applied tothe PID control as Eq. (36). The PID control with an approximateforce compensation f^ is

u Kpx Kd _x n f^; n Kix 62The above theorem is also applicable for the PID controllers

with an approximate f compensation as in Eq. (62). The condition

for PID gains in Eq. (45) becomes km Kp

P 32~kf kch i

and

kM Ki 6 3/2~kfkckM M ;

~kf kf .If the number of dampers installed on the buildings is less than

the number of the building oors (n), then the resulting system istermed as under-actuated system. In that case, the location matrixC should be included along with the gain matrices. In our experi-ment, there is only one damper installed (second oor) on thestructure. The PID controller becomes

tructures 81 (2014) 208218 213kd2 P ~/ 1 ckM M kmC

(

easy to verify that the closed-loop system with PID control(Acl 2 R55; kp2 350; ki2 2200, and kd2 45 is stable.

If the SMC switching gain g is greater than the system uncer-tainty bound, then the r z converges to zero. We consider theswitching gain of SMC g 1:3.

Here the structure is excited by a step input and the corre-sponding vibration response is reduced by applying the above con-trollers. The control objective is to bring the structural vibration asclose to zero as possible. The control signal is directly applied asthe force without applying any constraint by neglecting the dam-per dynamics.

Fig. 5 shows the time response of the second oor displacementfor both controlled and uncontrolled cases, the unite is centimetre.It shows that all the three controller reduces the structure motion.The PD controller reduces the structure oscillations but has a bigsteady-state error. This error can be reduced by introducing anintegral term, hence a PID controller, which can achieve a zerosteady-state error. The control signals are shown in Figs. 68, theunites are volt.

The performance of the SMC lies between the PD and PID con-troller but its control signal has many high frequency switching,which may not be acceptable for some mechanical dampers. How-ever in practice, the active system cannot achieve this much atten-uation due to the actuator limitations.

0 5 10 15 20 25 30 35 40 45 50-0.5

0

0.5

1

1.5

2

Time (s)

Dis

plac

emen

t

No controlPDPIDSMC

Fig. 5. The displacements of the second oor using PD, PID, and SMC control.

g Structures 81 (2014) 208218controller, SMC is a popular robust controller which is often seenin the structural vibration control applications [30]. A switchingcontrol law is used to drive the state trajectory onto a pre-speciedsurface. In the case of structural vibration control, this surface cor-responds to a desired system dynamics.

A general class of discontinuous structural control is dened bythe following relationships [34].

u ueq gsignr g if r > 00 if r 0g if r < 0

8>: ; g > 0 67

where the linear term ueq is the equivalent control force, r is thesliding surface and sign r sign r1 ; . . . ; sign r2n T . The slidingsurface can be a function of the regulation error, thenr xT ; _xT T z 2 R2n. The equivalent control can be estimatedusing a low pass lter or neglected, if the system parameters areunknown.ureal sat utheory utheory if utheory < mmax

mmax if utheory P mmax 66

where utheory is the theory force, ureal is the actual control force, mmaxis the maximum torque of the AMD actuator. Now the linear PIDcontroller becomes nonlinear PID. The asymptotic stability of Theo-rem 2 becomes stable as Theorem 1, see [2].

3. Simulations and experimental results

3.1. Numerical simulations

Consider the systemdescribed by Eq. (2)with linear stiffness, hasthe following set of parameters: the matrix M is m1 3:3 kg andm2 6:1 kg;C is given by c1 2:5 N s=m and c2 1:4 N s=m, andK is given by k1 4080 N=m and k2 4260 N=m. These parametersare obtained by identifying the two-story lab prototype [11].

We compare the performances of PD, PID, and SMC. Like PIDwhere ~/ 13 km M minfkp2g

q.

Remark 2. The PID tuning methods are different for the systemwith and without prior knowledge. If the system parameters areunknown, then auto-tuning techniques are employed to choose thegains either on-line or off-line. These techniques are broadlyclassied into direct and indirect methods [3]. In direct method,the closed-loop response of the system is observed and thecontroller gains are tuned directly based on the past experienceand heuristic rules. In the case of indirect method, the structureparameters are identied rst from the measured output andbased on these identied parameters the controller is then tunedto achieve a desired system dynamics. This paper provides a tuningmethod that ensures a stable closed-loop performance. For thatpurpose, the structural parameters kM M , kmC; kf , and kc aredetermined from the identied parameters.

Remark 3. The PID control as Eq. (34) does not need exact infor-mation about the building structure as Eq. (8). It uses only the dis-placements of the building and upper bound estimation of thebuilding parameters. If the actual control force to the buildingstructure satises Eq. (45), the closed-loop system is stable. Andthis condition is easy to be satised from the above remark. Sowe does not require the theory force (34) to match actual controlforce for PID structure control. However, in many cases the actualcontrol force cannot reach the theory force as Eq. (34) due to theactuator limitations, which causes saturation.

214 S. Thenozhi, W. Yu / EngineerinIf the PD control as Eq. (13) has kp2 350 and kd2 45, they sat-isfy the condition as Eq. (18), hence the closed-loop is stable. It is0 5 10 15 20 25 30 35 40 45 50-1.5

-1

-0.5

0

Con

trol s

igna

l

0 5 10 15 20 25 30 35 40 45 50-1.5

-1

-0.5

0

Time (s)

Con

trol s

igna

l

Fig. 6. Control signal of PD control for the simulation model.Time (s)

Fig. 7. Control signal of PID control for the simulation model.

3.2. Experimental results

To illustrate the theory analysis results, a two-story buildingprototype is constructed which is mounted on a shaking table,see Fig. 11. The building structure is constructed of aluminum.The shaking table is actuated using a hydraulic control system(FEEDBACK EHS 160), which is used to generate earthquake sig-nals. The AMD is a linear servo actuator (STB1108, Copley ControlsCorp.), which is mounted on the second oor. The moving mass ofthe damper weights 5% (0:45 kg) of the total building mass. Thelinear servo mechanism is driven by a digital servo drive (AccelnetMicro Panel, Copley Controls Corp). ServoToGo II I/O board is usedfor the data acquisition purpose.

The PD/PID control needs the structure position and velocitydata. During the seismic excitation, the reference where the dis-placement and velocity sensors are attached will also move, as aresult the absolute value of the above parameters cannot besensed. Alternatively, accelerometers can provide inexpensiveand reliable measurement of the acceleration at strategic pointson the structure. Three accelerometers (Summit Instruments13203B) were used to measure the absolute accelerations on theground and each oor. The ground acceleration is then subtractedfrom the each oor accelerations to get the relative oor move-

sampling frequency of 1:0 kHz. The control signal generated bythe control algorithm is fed as voltage input to the amplier. Thecurrent control loop is used to control the AMD operation. Theamplier converts its voltage input to a respective current outputwith a gain of 0.5. The AMD have a force constant of 6:26 N=A or3:13 N=V.

Now, we show the procedure for selecting the gains for a stableoperation. The theorems in this paper give sufcient conditions forthe minimal values of the proportional and derivative gains andmaximal values of the integral gains. In order to do a fair compar-

AMD

HydraulicShaker

Accelerometer

DataAcquisition

Unit

Fig. 11. Two-story building prototype with the shaking table.

0 5 10 15 20 25 30-3

-2

-1

0

1

2

3

Time (s)

Dis

plac

emen

t (cm

) UncontrolledPD

Fig. 12. The displacements of the rst oor using PD control.

0 5 10 15Frequency (Hz)

Fig. 10. Comparison of the measured and estimated position data using Fourierspectra.

S. Thenozhi, W. Yu / Engineering Sment. The relative velocity and position data are then estimatedusing the numerical integrator proposed in [29].

The position estimation with respect to the Loma Prieta East-West earthquake signal is shown in Fig. 9. The effect of the pro-posed numerical integrator on frequency characteristics is studiedby plotting Fourier spectra. A sinusoidal signal composed with6 Hz;7 Hz, and 8 Hz is used here to excite the linear actuator.The linear actuator has a position and acceleration sensor. Theacceleration of the actuator is measured, which is then integratedtwice to obtain the position estimation. The FFT diagram of themeasured and estimated position is generated, see Fig. 10. As canbe seen from the gure that the frequency information is notaffected except in the low frequency range. This low-frequencyerror is caused due to the presence of bias and noise in the accel-erometer output.

The control programs were operated in Windows XP withMatlab 6.5/Simulink. All the control actions were employed at a

0 5 10 15 20 25 30-15

-10

-5

0

5

10

15

Dis

plac

emen

t (m

m)

24.8 25 25.2 25.4 25.6 25.8 26

-202 Measured

Estimated

0 5 10 15 20 25 30 35 40 45 50-1.5

-1-0.5

00.5

11.5

Time (s)

Con

trol s

igna

l

Fig. 8. Control signal of SMC control for the simulation model.Time (sec)

Fig. 9. Comparison of the measured and estimated position data.0

0.5

1

1.5x 10-4

Mag

nitu

de

EstimatedMeasured

tructures 81 (2014) 208218 215ison, both the PD and PID controller uses the same proportionaland derivative gains. We rst design the PID controller basedon the identied parameters of the two-story lab prototype.

chieved using the controllers. Figs. 1215 show the time responsef the rst and second oor displacements for both controlled andncontrolled cases. The control algorithm outputs are shown inigs. 16 and 17.From Table 2 one can observe that the controllers effectively

ecrease the vibration. The controlled response using the PD con-oller is reduced signicantly by applying a damping providedy the derivative gain. Figs. 13 and 15 show the vibration attenu-tion achieved by adding an integral action to the above PD con-oller. The results demonstrate that PID controller performs

better than PD controller.

Remark 4. It is worth to note the frequency characteristics of anintegrator. An ideal integrator acts like a low-pass lter. The bodemagnitude plot of an ideal integrator is shown in Fig. 18. At 1.6 Hzthe integrator attenuates the input power by 20 dB and at 16 Hz itreaches to 40 dB. During earthquake the structure oscillates at itsnatural frequencies. If the natural frequency is very small then theintegrator produces a larger output. The structure prototype weused for the experiments have natural frequencies 2:1 Hz and8:9 Hz. Since these frequencies have an attenuation more than20 dB a larger value can be used for Ki. On the other hand, if thebuilding has a natural frequency less than 1:6 Hz, then the integralgain should be reduced accordingly. The error input to theintegrator is the position data. From Figs. 1215 we can see that

g S ctures 81 (2014) 2082180 5 10 15 20 25 30-3-2-10123

Time (s)

Dis

plac

emen

t (cm

)

UncontrolledPID

Fig. 13. The displacements of the rst oor using PID control.

0 5 10 15 20 25 30-5

-3

-1

1

3

5

Time (s)

Dis

plac

emen

t (cm

) UncontrolledPD

Fig. 14. The displacements of the second oor using PD control.

216 S. Thenozhi, W. Yu / EngineerinThe following set of parameters were used for the control design:kM M 6:1; kmC 0:6; kf 365, and kc 5:8. Applying thesevalues in Theorem 2 we get

km Kp

P 556; kM Ki 3066; km Kd P 65 68In order to evaluate the performance, these controllers are

implemented to control the vibration on the excited lab prototype.The control performance is evaluated in terms of their ability toreduce the relative displacement of each oor of the building.The proportional, derivative, and integral gains are furtheradjusted to obtain a higher attenuation. Finally, the PID controllergains are chosen to be

kp 635; ki 3000; kd 65 69and the PD controller gains are

kp 635; kd 65 70Theorem 1 requires that the PD controller gains need to be posi-

tive. In the experiments, the negative gains resulted in an unstableclosed-loop operation, which satises the conditions in Theorem 1.Since Theorem 2 provides sufcient conditions, violating it doesnot mean instability. We have found that, when kM Ki is morethan 4200, the system becomes unstable. This satises the condi-tion as Eq. (68).

Table-2 shows the mean squared error, MSE 1NPN

i1e2i of the

displacement with proposed controllers, here N is the number ofdata samples and e xd x x, where x is the position

the position data for the most part takes successive positive andnegative values. Hence, the integrator output for high frequencyinput signal is small due to the rapid cancellation between thesepositive and negative values.

Table 2Comparison of vibration attenuation obtained using PD and PID controller.

0 5 10 15 20 25 30-5

-3

-1

1

3

5

Time (s)

Dis

plac

emen

t (cm

) UncontrolledPID

Fig. 15. The displacements of the second oor using PID control.Control action PD control PID control No control

Floor-1 displacement 0.1699 0.1281 1.0688Floor-2 displacement 0.5141 0.3386 3.3051

0 5 10 15 20 25 30-1.5

-1

-0.5

0

0.5

1

1.5

Time (s)

Con

trol s

igna

l (V)

Fig. 16. Control signal of PD control for the prototype.

0 5 10 15 20 25 30

-1

-0.5

0

0.5

1

Con

trol s

igna

l (V)aouF

dtrbatr

truTime (s)

Fig. 17. Control signal of PID control for the prototype.

ng S0 2 4 6 8 10 12 14 16 18 20

-40

-30

-20

-10

0

10

20

Mag

nitu

de (d

B)

Frequency (Hz)

Fig. 18. Bode magnitude plot of an ideal integrator.

S. Thenozhi, W. Yu / EngineeriSometimes, the integral control results in an actuator satura-tion. But as discussed in Remark 2, the output of the integrator issmall in our case. Fig. 19 shows the magnitude spectrum of controlsignals of the PD and PID controllers. As the building structure isexcited mainly in its natural frequency (2:1 Hz), the major controlaction occurs in this zone. Even though the Ki gain is large, PID con-troller produces less control effort than the PD controller, but stillachieves a better vibration attenuation.

Remark 5. From our experience, the classic SMC performs poorwhile x starts damping from a large to a small value for the shakingtable. In Fig. 20, after 22 s we can see that the vibration levelincreases. This is due to the fact that SMC switches aggressivelywith a gain of g, even though the actual vibration is considerablysmall, see Fig. 21.

Technol 2003;40:117.

of the linear quadratic Gaussian and input estimation approaches. J Sound Vib

0 5 10 15 20 25 30-5

-3

-1

1

3

5

Time (s)

Dis

plac

emen

t (cm

)

UncontrolledSMC

Fig. 20. The displacements of the second oor using SMC control.

0 5 10 15 20 25 30

-1-0.5

00.5

1

Time (s)

Con

trol s

igna

l (V)

Fig. 21. Control signal of SMC control for the prototype.

Fig. 19. Fourier spectrums of PD and PID control signals.2007;301:42949.[15] Housner GW et al. Present and future. J Eng Mech 1997;123:897974.[16] Ikhouane F, Maosa V, Rodellar J. Dynamic properties of the hysteretic Bouc

Wen model. Syst Control Lett 2007;56:197205.[17] Kim JT, Jung HJ, Lee IW. Optimal structural control using neural networks. J

Eng Mech 2000;126:2015.[18] Lewis FL, Dawson DM, Abdallah CT. Robot manipulator control: theory and

practice. 2nd ed. Marcel Dekker, Inc; 2004.[19] Nerves AC, Krishnan R. Active control strategies for tall civil structures. Proc

IEEE Int Conf Ind Electron Control Instrum 1995;2:9627.[20] Park W, Park KS, Koh HM. Active control of large structures using a bilinear

pole-shifting transform with H1 control method. Eng Struct 2008;30:333644.

[21] Poznyak AS. Advanced mathematical tools for automatic control engineers.Deterministic systems, Vol. I. Springer; 2009.

[22] Roldn C, Campa FJ, Altuzarra O, Amezua E. Automatic identication of theinertia and friction of an electromechanical actuator. New advances inmechanisms, transmissions and applications, vol. 17. Netherlands: Springer;[8] Du H, Zhang N. H1 control for buildings with time delay in control via linearmatrix inequalities and genetic algorithms. Eng Struct 2008;30:8192.

[9] Fisco NR, Adeli H. Smart structures: part Iactive and semi-active control.Scientia Iran 2011;18:27584.

[10] Fisco NR, Adeli H. Smart structures: part IIhybrid control systems and controlstrategies. Scientia Iran 2011;18:28595.

[11] Garrido-Moctezuma RA, Concha SA. Estimation of the parameters of structuresusing acceleration measurements. In: 16th IFAC symposium on systemidentication, vol. 16. Brussels, Belgium; 2012. p. 179196.

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[13] Guclu R. Sliding mode and PID control of a structural system againstearthquake. Math Comput Modell 2006;44:2107.

[14] Ho CC, Ma CK. Active vibration control of structural systems by a combination4. Conclusion

In this paper, the model of building structures with an activevibration control is analyzed. The theory contribution of this paperis that the stability of the AMD PD/PID control for building struc-tures is proven. By using Lyapunov theory, sufcient conditionsof stability are derived to tune the PD/PID gains. The technicaladvance of this paper is that a systematic tuning method of PIDis proposed based on the stability analysis. The theory results aresuccessfully applied to a numerical example and a two-story build-ing prototype. The results show that even though the chosen gainsare not optimal, the controllers guarantee stable controlperformances.

Acknowledgments

The authors are grateful to Dr. Antonio Concha for his help inidentifying the building parameters. The authors would like tothank Mr. Jess Meza and Mr. Gerardo Castro for their assistanceto complete the experiments. The rst author would like to thankConsejo Nacional de Ciencia y Tecnologa (CONACyT) of Mexico forthe nancial support. This work was partially supported by StateKey Laboratory of Synthetical Automation for Process Industriesand the Project 111 (No. B08015) of China.

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218 S. Thenozhi, W. Yu / Engineering Structures 81 (2014) 208218

Stability analysis of active vibration control of building structures using PD/PID control1 Introduction2 Model andactive control of building structures2.1 PD control2.2 PID control

3 Simulations and experimental results3.1 Numerical simulations3.2 Experimental results

4 ConclusionAcknowledgmentsReferences