]{ }

} g

u&

e

= stiffness matrix, { }

e

u&

PARAMETRIC STUDY OF MODIFIED CONTINUOUS BANG-BANG CONTROLLER 
(SC-051) 

Yoyong Arfiadi1* 

1Department of Civil Engineering, Atma Jaya Yogyakarta University, Yogyakarta, Indonesia 
	*e-mail of corresponding author: yoyong@mail.uajy.ac.id or yoyong@staff.uajy.ac.id 

ABSTRACT 
Active control systems have been used to reduce structural response due to earthquake excitations. 
Several control algorithms have been proposed including linear quadratic regulator and bang-bang control 
law. While bang-bang control algorithm outperforms linear quadratic regulator in terms of the control 
energy’s imitation, it switches the controller from maximum to maximum capacity in one direction to 
another direction, which may not be appropriate to some devices. In this paper a modified bang-bang type 
control law which is continuous but bounded in control capacity is investigated. The control law considered 
in this paper utilizes a hyperbolic tangent function to imitate a bang-bang type controller, thence it is a 
continuous but bounded controller. A parametric study of weighting matrix that influences the control 
effectiveness is investigated in this paper. The choice of the weighting matrix is important such that the 
control energy approaches the maximum energy but avoid the switching effect. A simulation to a three 
story building utilizing active bracing system is carried out. From the simulation it can be shown that the 
effective controllers are the ones that the value of the hyperbolic tangent of the argument approaches 1. 
Keywords: Response reduction, active control, modified bang-bang control, continuous control, bonded controller 

1.  INTRODUCTION 
Reducing response of structures due to earthquake is the main concern of structural engineers. One of the methods to reduce structural response is by using active vibration control. Several control algorithms have been proposed in order to obtain the optimum control force. The base line in modern control theory is the linear quadratic regulator (LQR) control law [1]. In LQR method the control force is regulated by multiplying the structural responses by the gain. The resulting control force is a linear function and therefore the control force is unlimited and depends on the magnitude of the gain and structural responses. However, the capacity of the device is not unlimited. In order to limit the maximum control force, a bang-bang control law is usually utilized where the control forces always reach the actuators capacity [2]. However, bangbang control algorithms tend to have switching effects that are not suitable for many actuators especially those are used for civil engineering structures. 
In order to alleviate the switching effects Arfiadi and Hadi [3] utilize a continuous function to regulate the control forces. The function to be used in this case is a hyperbolic tangent function (tanh(.)). By using this continuous function the actuator’s capacity may not necessarily reaches the maximum yet it bounds by its capacity. The resulting control law is a continuous but bounded control algorithm. In order the control algorithms perform well as expected, the variables involved in control law should be designed such that the control law mimics the bang-bang control law. 

2.  STRUCTURES WITH ACTIVE CONTROL 
Consider a multi degree of freedom structure with active control, where the equations of motion can be written in the form:

[ M	]{U&}+[ C	]{U }+[ K U

= [ b	]{ f } − [ M	]{ ˆ 	&	.(1)

s	s	s	s
where  [ M s] = mass matrix,  [ C s ] = damping matrix,  [ K s ]
respect to the ground,  [ b	] 	= control force location matrix,

c	s	o
U  = displacement vector with
{c }f= control force vector,  { ˆ 	} = effect of 

s	o

ground acceleration to the structure and	&	g	= ground acceleration. The super dot in the equation indicates

derivative with respect to time.
Equation (1) can be written in the state space form as 

The 3rd International Conference of EACEF (European Asian Civil Engineering Forum)	S - 211
Universitas Atma Jaya Yogyakarta, INDONESIA, September 20-22, 2011 

{ }  [A ]{Z}

Z

{ }

{ }

{}

{ }  [  ]{ }

{c }

[ ][

[ ][

{ }  [A ]  {S }

S

{ }

{ }

Structural and Construction Engineering

where

{Z} =

⎧ U ⎫ 	⎡ 	[0] n × n
⎨ 	⎬ , [ A ] = ⎢ 	− 1

&	=	+ [ B ]{ f c } + {E}w

[I] n × n	⎤ 	⎡ 	[0] n × n a	⎤ 	⎧
− 1	, [ B ] = ⎢ 	− 1	, [ E ] = ⎨

... 2)

{0}	n × 1	⎫ 	⎧ {0} n × 1 ⎫
− 1	⎬ = ⎨ 	⎬ 	.(3a-d) 

⎩	& U	⎭	⎣−	[ Ms	]	[ Ks	]	−	[ Ms	]	[Cs	] ⎦	⎣ [M]	[ bs	]⎥⎦	⎩[Ms	]	{ es}⎭	⎩−	1	n× 1	⎭

3.  CONTROL ALGORITHMS 
Control force in equation (1) or (2) shall be optimized according to an appropriate control law. One of the most common control algorithms is the linear quadratic regulator (LQR). In LQR control law the control force is optimized such that minimizes the performance index: 
∞

1
J = 2

∫ (Z
0

T	T
Q	Z	+	f	[ R ]{ f c}	)dt	..(4)

where [Q ] and [ R

]

are positive definite weighting matrices.   The resulting control law is of the form:

−1	T

{fc } = − [ R ]	[ B ]	[P]{Z}
where matrix [P ] is obtained from the Ricatti’s equation [1]:

T	−1	T

.(5) 

P	A]+	[ A]	[P]−	P	B][ R]	[ B]	[P]+	[Q]=	0	...(6)
Control law in equation (5) is a linear feedback control law, where the maximum control force is not limited. 
In practice the control force is bounded due to the capacity of the actuators to supply the control force. 
In order to bound the control force at the capacity of the actuator a performance index as taken as follow 
[2] : 

J	=	∫({Z}T [Qo	]{Z})dt	..(7)

where [ Q o 	] is positive definite matrix, with the bound in control energy: 

fci	(t) ≤	fci	,max	..(8)

in which	fci	is the ith element of control forces in {fc}
The resulting control force is
{fc }=	− {f ci	} max	sgn([ B] T	[ S])	...(9)
The costate vector	{S}can be obtained from a differential equation:
&	T
=	+	[ Qo	]{Z}	..(10)
Note also that the signum function in equation (9) switches the actuator from minimum and maximum
capacity, i.e.,
{fc }=	− {fci	}	max	if	sgn([B]T	[ S]) >	0	...(11a)
{fc }=	{ fci	} max	if	sgn([ B] T	[ S])	<	0	.(11b)

The computation of costate vector	S  is done online and therefore, will result in time delay. To alleviate
the online computation of the costate vector	S  , Wu and Soong  [2] utilize Lyapunov function as the
performance objective, i.e.,
V	=	{Z}T  [Pc	]{Z}	.(12)
The derivative of Lyapunov function is: 

S - 212	The 3rd International Conference of EACEF (European Asian Civil Engineering Forum)
Universitas Atma Jaya Yogyakarta, INDONESIA, September 20-22, 2011 

{ } [

]{ }

V

Z

Z

{ }

ˆ o

Structural and Construction Engineering

&
=

& T	&
P	c ]{Z} + {Z}T [ P	c	...(13) 

Substituting equation (2) into equation (13) results in:
&	T	T	T
V	=	{Z}	( [ A]	[ P	c ]+	[ Pc	][ A]) {Z}+	2([ B] {fc	} +	E	w )	[ P	c ]{Z}	.(14)
In order to make the derivative of Lyapunov function negative, all terms in equation  (14) has to be
negative. In order to make the first term negative:

[ A ] T

[ P
c

] + [ P ][ A ] = − [ Q	]	...(15)
c	c 

where [Qc	]is a positive semi-definite matrix. While in order to make the second term negative can be

achieved through:
T 
{fc }=	−	{ fci	} max	sgn([ B]	[ Pc	]{Z})	.(16)
Because of the signum function, the controller swithes from maximum to maximum capacity in one to another direction, which may not be apropriate for some devices. To alleviate the controller switches from its  maximum  capacity  a  continuous  control  law  is  preferrable.  Arfiadi  and  Hadi  [3]  has  proposed  a continuous bounded control law utilizing a hyperbolic tangent function of the form

{fc } = − { f	} tanh( α [ B ] T

[ P ]{Z})	..(17) 

c, max	c

The effectiveness of the control law depends on the argument in the tanh(.) function. The order of {Z}may

be predicted from the uncontrolled structures, while matrix  [P c	]depends on the weighting matrix  [Qc	]in

equation (15) and α is used to scale the argument such that the value of tanh approaches ≈ 1.

4.  PARAMETRIC STUDY 
A three-storey shear building utilizing an active bracing system as

m3

m2

shown in Figure  1 and has been discussed in  [3] is used as the
example. The active bracing is installed at the first floor to control 
the vibration of the structure. The structural parameters are mi = 1 t, 
ci =  1.407  kN-s/m  and  ki  =  980  kN/m,  where  i  =  1,  2,  3.  The
equations of motion can be represented as in equation (1) where the matrices in equation (1) are
⎡1	0	0⎤ 	⎡ 2 814	− 1 407	0	⎤
⎢ 	⎥ 	⎢ 	⎥

m1	[ M	] =	0	1

0	[ C	] =	− 1 407	2 814	− 1 407

s	⎢ 	⎥ 	s	⎢
⎣ 0	0	1 ⎦ 	⎣
⎡ 1960	− 980	0
⎢

⎥
0	− 1 407	1 407 ⎦
⎤ 	⎧ 1 ⎫ 	⎧ − 1 ⎫
⎥ 	⎪ 	⎪ 	⎪ 	⎪ 

[ K	]	=	−	980	1960	−	980	{ b	}=	0	{ e	}=	−	1

s	⎢

Figure 1.   A three story building

⎥ 	s	⎨ 	⎬ 	⎨ 	⎬
⎣ 	⎦ 	⎪ 	⎪ 	⎪ 	⎪

with active bracing
In order to investigate the effect of matrix [ Q

c

]

0	− 980	980	⎩ 0 ⎭ 	⎩ − 1 ⎭
in equation (15) to the response reduction and control force 

history, the values of  [Qc	]	are varied to represent low and high authority of the controller in the continuous

modified bang-bang- control law, while α is set constant equals to 10. Since the objective is to minimize

the displacement of the third (top) floor with respect to the ground, the elements of  [Qc	]are taken as

follows:	Qc	(i,j) = 0 for I = 1, 2, . . ., 6 and j = 1,2, . . ., 6,, except for i = j = 3 , i.e.,	Qc	(3,3) = varies. The

structure is then subjected to El Centro 1940 NS ground acceleration scaled to 0.1 g, while the control

force is bound to a maximum capacity = 1.5 kN. The results of simulations are depicted in Figures 2 and 3.

Table 1 shows the simulation results, where |Z| max is taken as the uncontrolled case.

The 3rd International Conference of EACEF (European Asian Civil Engineering Forum)	S - 213
Universitas Atma Jaya Yogyakarta, INDONESIA, September 20-22, 2011 

Structural and Construction Engineering 

Figure 2. Qc(3,3) vs. top floor response reduction 

Figure 3. Qc(3,3) vs. tanh	(α	[ B] T	[ P	]{Z}	)
c 	max

From Table 1, it can be seen that when tanh

( α 	T P B	) ]	]{Z} [ [	approaches 1, the response reduction is 

c 	max 
maximum. The time history response for Qc(3,3) = 500 can be seen in Figure 4. 
A sample controller is then simulated subject to different earthquake excitations. The results are shown in 
Table 2. Due to space limitations time history responses subject to Kobe 1995 only is depicted in Figure 5. 

S - 214	The 3rd International Conference of EACEF (European Asian Civil Engineering Forum)
Universitas Atma Jaya Yogyakarta, INDONESIA, September 20-22, 2011 

Structural and Construction Engineering 

Table 1. Structural response due to El Centro 1940 NS ground acceleration 0.1 g

Q(3,3)

1
10
50
100
180
200
250
300
400
500
1000
2000
3000
5000
10000
100000

Uncontrolled	Controlled
top floor	top floor
displacement	displacement
(m)	(m)
0.02593	0.02051
0.02593	0.01975
0.02593	0.01137
0.02593	0.00935
0.02593	0.00813
0.02593	0.00804
0.02593	0.00791
0.02593	0.00784
0.02593	0.00777
0.02593	0.00775
0.02593	0.00773
0.02593	0.00760
0.02593	0.00752
0.02593	0.00755
0.02593	0.00749
0.02593	0.00727

%	α
reduction	[B]T[P]|{Z}|max

0.61%	0.008
5.79%	0.077
23.57%	0.383
37.54%	0.766
50.08%	1.380
51.98%	1.533
55.82%	1.916
58.76%	2.299
61.62%	3.066
63.73%	3.832
68.81%	7.664
69.89%	15.328
70.02%	22.991
70.12%	38.319
70.56%	76.638
68.21%	766.380

tanh(αBT P |{Z}|max)	fcmax

0.008	0.01
0.076	0.11
0.365	0.42
0.645	0.65
0.881	0.87
0.911	0.93
0.958	1.06
0.980	1.16
0.996	1.30
0.999	1.38
1.000	1.49
1.000	1.50
1.000	1.50
1.000	1.50
1.000	1.50
1.000	1.50 

Table 2. Results for Qc(3,3) = 500 subject to different earthquake excitations scaled to 0.1g

Earthquake

El Centro
Kobe
Hachinohe 
Northridge

α B]T[P]|{Z}|max	tanh(αBT P |{Z}|max

3.8319	0.99906
4.9437	0.9999
3.4048	0.9978
3.2999	0.99728

Uncontrolled	Controlled
top floor	top floor
displacement	displacement
(m)	(m)
0.02593	0.00940
0.03216	0.01202
0.02314	0.00907
0.02064	0.01446

%
fcmax
reduction
63.7%	1.38
62.6%	1.36
60.8%	1.34
29.9%	1.46 

5.  CONCLUSSIONS 
The continuous but bounded control law is considered in this paper. The control law utilizes hyperbolic tangent function to represent continuous but bounded control law. A parametric study to the weighting matrix show that the effectiveness of the control law is achieved when the value of the hyperbolic tangent function of the argument (tanh(.)) approaches 1. 

6.  REFERENCES 
[1]  Ogata, K. (1990). Modern Control Engineering. Prentice-Hall, Inc., Englewood Cliffs 
[2]  Wu, Z. and Soong, TT. (1996). Modified Bang-Bang Control Law for Structural Control Implementation. 
	Journal of Engineering Mechanics,122(8), 771-777 
[3]  Arfiadi, Y. and Hadi, MNS. (2006). Continuous Bounded Controller for Active Control of Structures. 
	Computers and Structures, 84, 798-807. 

The 3rd International Conference of EACEF (European Asian Civil Engineering Forum)	S - 215
Universitas Atma Jaya Yogyakarta, INDONESIA, September 20-22, 2011 

Structural and Construction Engineering 

El Centro 0.1 g, Q(3,3) = 500 
0.03 
uncontrolled 
0.02	controlled
0.01 
	0 
-0.01 
-0.02 
-0.03 
0	5	10	15
1.5 
1 
0.5 
0 
-0.5 
-1 
-1.5 
0	5	10	15
Time (s) 

Figure 4. Time history resposes for Qc(3,3) = 500 subject to El Centro 0.1g. 

Kobe 0.1 g, Q(3,3) = 500 
0.04 
uncontrolled 
0.02	controlled

0 

-0.02 

-0.040	5	10	15	20	25	30	35

1.5 
1 
0.5 
0 
-0.5 
-1 
-1.5 
0	5	10	15	20	25	30	35
Time (s) 

Figure 5. Time history resposes for Qc(3,3) = 500 subject to Kobe 0.1g. 

S - 216	The 3rd International Conference of EACEF (European Asian Civil Engineering Forum)
Universitas Atma Jaya Yogyakarta, INDONESIA, September 20-22, 2011