DSpace at VNU: A study on the application of hedge algebras to active fuzzy control of a seism-excited structure

Article

A study on the application of hedge
algebras to active fuzzy control
of a seism-excited structure

Journal of Vibration and Control
18(14) 2186–2200
! The Author(s) 2011
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DOI: 10.1177/1077546311429057
jvc.sagepub.com

Nguyen Dinh Duc1, Nhu-Lan Vu2, Duc-Trung Tran3
and Hai-Le Bui3

Abstract
The active control problem of seism-excited civil structures has attracted considerable attention in recent years. In this
paper, conventional, hedge-algebras-based and optimal hedge-algebras-based fuzzy controllers, respectively denoted by
HAFCs and OHAFCs, are designed to suppress vibrations of a structure against earthquake. The interested structure is a
building modeled as a four-degrees-of-freedom structure system with one actuator, which is an active tendon, installed on
the first floor. The structural system is simulated against the ground motion, acting on the base, of the El Centro
earthquake (Mw ¼ 7.1) in the USA on 18 May 1940. The control effects of FC, HAFC and OHAFC are compared via
the time history of the floor displacements and velocities, control error and control force of the structure.

Keywords
Active control, building, earthquake, fuzzy control, hedge algebras
Received: 18 October 2010; accepted: 26 August 2011

1. Introduction

Vibration occurs in most structures, machines and
dynamic systems. Vibration can be found in daily life
as well as in engineering structures. Undesired vibration results in structural fatigue, lowering the strength
and safety of the structure, and reducing the accuracy
and reliability of the equipment in the system. The
problem of undesired vibration reduction has been
established for many years and solving it has become
more attractive nowadays in order to ensure the safety
of the structure, and increase the reliability and durability of the equipment (Teng et al., 2000; Anh et al.,
2007).
A critical aspect in the design of civil engineering
structures is the reduction of response quantities, such
as velocities, deflections and forces, induced by environmental dynamic loadings (i.e. wind and earthquake).
In recent years, the reduction of structural response,
caused by dynamic effects, has become a subject of
research, and many structural control concepts have
been implemented in practice (Yan et al., 1998; Park
et al., 2002; Guclu, 2006; Pourzeynali et al., 2007;
Guclu and Yazici, 2008).

Depending on the control methods, vibration control
in the structure can be divided into two categories, namely
passive control and active control. Passive structural control uses energy absorption, so as to reduce displacement
in the structure. Passive vibration control devices have
traditionally been used, because they do not require an
energy feed and therefore do not run the risk of generating
unstable states. However, passive vibration control
devices have no sensors and cannot respond to variations
in the parameters of the object being controlled or the
controlling device. Recent development of control

theory and technique has brought vibration control
from passive to active and the active control method has
1

University of Engineering and Technology, Vietnam National University,
Hanoi, Hanoi, Vietnam
2
Institute of Information Technology, Vietnam Academy of Science and
Technology, Hanoi, Vietnam
3
School of Mechanical Engineering, Hanoi University of Science and
Technology, Hanoi, Vietnam
Corresponding author:
Nguyen Dinh Duc, University of Engineering and Technology, Vietnam
National University, Hanoi, 144 Xuan Thuy Street, Hanoi, Vietnam
Email:

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Duc et al.

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become more effective in use. An active vibration controller is equipped with sensors and actuators, and it requires
power (Teng et al., 2000; Preumont and Suto, 2008).
Fuzzy set theory, introduced by Zadeh (1965), has
provided a mathematical tool that is useful for modeling
uncertain (imprecise) and vague data and has been
presented in many real situations. Recently, many

researches on active fuzzy control of vibrating structures
have been done. In Teng et al. (2000), fuzzy theory was
applied to active control of a cantilever beam. The
optimal control method was also applied to process
structural control for comparison. The fuzzy supervisory technique for the active control of earthquakeexcited building structures was studied by Park et al.
(2002). Pourzeynali et al. (2007) designed and optimized
different parameters of an active-tuned mass damper
control scheme to obtain the best results in the reduction
of the building response under earthquake excitations
using genetic algorithms (GAs) and fuzzy logic.
In Guclu and Yazici (2008), fuzzy and proportional–
derivative (PD) controllers were designed for active
control of a real building against earthquake. Battaini
et al. (1999) studied the response of a three-story frame,
subjected to earthquake excitation, controlled by an
active mass driver located on the top floor. Li et al.
(2010) developed a fuzzy logic-based control algorithm
to control a nonlinear high-rise structure under earthquake excitation using an active mass damper device.
Wang and Lin (2007) developed variable structure and
fuzzy sliding mode controllers for the active control of a
building with an active-tuned mass damper.
Although a fuzzy controller (FC) is flexible and easy
to use, its semantic order of linguistic values is not
closely guaranteed and its fuzzification and defuzzification methods are quite complicated.
Hedge algebras (HAs) were introduced in 1990 and
have been investigated since (Ho and Wechler, 1990,
1992; Ho et al., 1999; Ho and Nam, 2002; Ho et al.,
2006; Ho, 2007; Ho and Long, 2007; Ho et al., 2008).
The authors of HAs discovered that linguistic values can
formulate an algebraic structure (Ho and Wechler,

1990, 1992) and, in the Complete Hedge Algebras
Structure (Ho, 2007; Ho and Long, 2007), the main
property is that the semantic order of linguistic values
is always guaranteed. It is even a rich enough algebraic
structure (Ho and Nam, 2002) to completely describe
reasoning processes. HAs can be considered as a
mathematical order-based structure of term-domains,
the ordering relation of which is induced by the meaning
of linguistic terms in these domains. It is shown that
each term-domain has its own order relation induced
by the meaning of terms, called the semantically ordering relation. Many interesting semantic properties of
terms can be formulated in terms of this relation and
some of these can be taken to form an axioms system of

HAs. These algebras form an algebraic foundation to
study a type of fuzzy logic, called linguistic-valued logic,
and provide a good mathematical tool to define and
investigate the concept of fuzziness of vague terms and
the quantification problem and some approximate reasoning methods. In Ho et al. (2008), HA theory was first
applied to fuzzy control and it provided very much
better results than FC. The studied object in Ho et al.
(2008), which is a single-undeformable pendulum without external loads, where its state equations are solved
by the Euler method with a sample time of 1 second, is
too simple to evaluate completely its control effect.
This suggests to us, in this paper, applying HAs in
active fuzzy control of a structure, which is a building
modeled as a four-degrees-of-freedom structure system
against earthquakes with three controllers (FC, hedgealgebras-based fuzzy controller (HAFC) and optimal
hedge-algebras-based fuzzy controllers (OHAFC)) in
order to compare their control effect, where the state

equations are solved by the Newmark method and the
sample time is 0.01 second.
This paper is organized as follows. In Section 2, the
dynamic model of the structural system is given. The
idea and basic formulas of HAs are summarized in
Section 3. In Section 4, the FCs of the structural
system are presented. Results and discussion are given
in Section 5. Conclusions are presented in Section 6.

2. Dynamic model of the structural
system
Consider an earthquake-excited four-floor one-way
shear building structure equipped with an active
tendon on the first floor (u2), as shown in Figure 1.

m4
x4
k4

c4

m3
k3

c3

x3

m2
k2

c2

x2

u2
m1
k1

c1

x¨ 0

Figure 1. The structural system.

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Journal of Vibration and Control 18(14)
Table 1. The system parameters

x¨ 0 , m/s2

4
2

Floor i

Mass
mi (103 kg)

Damping
ci (102 Ns/m)

Stiffness
ki (105 N/m)

0

1
2
3
4

450
345
345
345

261.7
4670
4100
3500

180.5
3260

2850
2500

–2
–4
0

10

20

30

40

50

Time, s

Reproduced with kind permission from Elsevier (Guclu, 2006).

Figure 2. The north-south acceleration component of the 1940
El Centro earthquake.

3. Hedge algebras
The equations of motion of the system subjected to the
north-south acceleration component of the 1940 El
Centro earthquake x€ 0 (see Figure 2), with control force
vector {u}, can be written as
€ þ ½CŠfxg

_ þ ½KŠfxg ¼ fug À ½MŠfrgx€ 0
½MŠfxg

ð1Þ

where {x} ¼ [x1 x2 x3 x4]T, {u} ¼ [Àu2 u2 0 0]T represents
the horizontal component of the active tendon force
and the 4 Â 1 vector {r} is the influence vector representing the displacement of each degree of freedom resulting
from static application of a unit ground displacement.
The 4 Â 4 matrices [M], [C] and [K] represent the structural mass, damping and stiffness matrices, respectively.
The mass matrix for a building structure, with the
assumption of masses lumped at floor levels, is a diagonal matrix in which the mass of each story is sorted on
its diagonal, as given in the following:
2
3
m1 0
0
0
6 0 m2 0
0 7
7
½MŠ ¼ 6
ð2Þ
4 0
0 m3 0 5
0
0
0 m4
where mi is the ith floor mass.
The structural stiffness matrix [K] is developed based

on the individual stiffness, ki, of each floor is given in
Equation (3):
2
3
Àk2
0
0
k1
6 Àk2 k1 þ k2
Àk3
0 7
7
ð3Þ
½KŠ ¼ 6
4 0
Àk3
k2 þ k3 Àk4 5
0
0
Àk4
k4
The structural damping matrix [C] is given as
2
3
c1
Àc2
0
0
6 Àc2 c1 þ c2
Àc3

0 7
7
½CŠ ¼ 6
4 0
Àc3
c2 þ c3 Àc4 5
0
0
Àc4
c4

ð4Þ

The system parameters are given in Table 1 (Guclu,
2006).

In this section, the idea and basic formulas of HAs are
summarized based on definitions, theorems and propositions in Ho and Wechler (1990, 1992), Ho et al. (1999),
Ho and Nam (2002), Ho et al. (2006), Ho (2007), Ho
and Long (2007) and Ho et al. (2008).
By the meaning of the term we can observe that
extremely small < very small < small < approximately
small < little small < big < very big < extremely big. . .
So, we have a new viewpoint: term-domains can be
modeled by a poset (partially ordered set), a semantics-based order structure.
Next, we explain how we can find this structure.
Consider TRUTH as a linguistic variable and let X
be its term-set. Assume that its linguistic hedges used to
express the TRUTH are Extremely, Very, Approximately,
Little, which for short are denoted correspondingly by E,

V, A and L, and its primary terms are false and true. Then,
X ¼ {true, V true, E true, EA true, A true, LA true, L true,
L false, false, A false, V false, E false . . .} [ {0, W, 1} is a
term-domain of TRUTH, where 0, W and 1 are specific
constants called absolutely false, neutral and absolutely
true, respectively.
A term-domain X can be ordered based on the
following observations.
– Each primary term has a sign that expresses a semantic tendency. For instance, true has a tendency of
‘going up’, called positive one, and it is denoted by
cþ, while false has a tendency of ‘going down’, called
negative one, denoted by cÀ. In general, we always
have cþ ! cÀ, semantically.
– Each hedge also has a sign. It is positive if it increases
the semantic tendency of the primary terms and
negative if it decreases this tendency. For instance,
V is positive with respect to both primary terms,
while L has the reverse effect and hence it is negative.
Denote by HÀ the set of all negative hedges and by
Hþ the set of all positive ones of TRUTH.
The term-set X can be considered as an abstract
algebra AX ¼ (X, G, C, H, ), where G ¼ {cÀ, cþ},

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C ¼ {0, W, 1}, H ¼ Hþ [ HÀ and is a partially ordering relation on X. It is assumed that HÀ ¼ {h–1, . . . , h–q},
where h–1 < h–2 < . . . < h–q, Hþ ¼ {h1, . . . , hp}, where
h1 < h 2 < . . . < h p.
The fuzziness measure of vague terms and hedges of
term-domains is defined as follow (Ho et al., 2008:
Definition 2): a fm: X ! [0, 1] is said to be a fuzziness
measure of terms in X if:
P
– fm(cÀ) þ fm(cþ) ¼ 1 and
h2H fm(hu) ¼ fm(u), for
8u 2 X;
– for
the
constants
0,
W
and
1,
fm(0) ¼ fm(W) ¼ fm(1) ¼ 0;
fmðhyÞ
– for 8x, y 2 X, 8h 2 H, fmðhxÞ
fmðxÞ ¼ fmð yÞ : this proportion
does not depend on specific elements, called fuzziness
measure of the hedge h and denoted by m(h).
For each fuzziness measure fm on X, we have (Ho
et al., 2008: Proposition 1):
–
–
–
–

–

fm(hx) ¼ m(h)fm(x), for every x 2 X;
fm(cÀ) þ fm(cþ) ¼ 1;
P
À þ
P Àq i p, i6¼0 fm ðhi cÞ ¼ fmðcÞ, c 2 {c , c };
P Àq i p, i6¼ 0 fmðhi xÞ ¼ fmðxÞ;
P
Àq i À1 ðhi Þ ¼  and
1 i p  ðhi Þ ¼
where
,
> 0 and  þ
¼ 1.

A function Sign, X ! {À1, 0, 1}, is a mapping that
is defined recursively as follows, for h, h’ 2 H and
c 2 {cÀ, cþ} (Ho et al., 2008: Definition 3):
– Sign(cÀ) ¼ À1, Sign(cþ) ¼ þ1;
– Sign(hc) ¼ ÀSign(c), if h is negative with regard to c;
Sign(hc) ¼ þ Sign(c), if h is positive with regard to c;
– Sign(h’hx) ¼ ÀSign(hx), if h’hx 6¼ hx and h’ is negative with regard to h; Sign(h’hx) ¼ þ Sign(hx), if
h’hx 6¼ hx and h’ is positive with regard to h;
– Sign(h’hx) ¼ 0 if h’hx ¼ hx.
Let fm be a fuzziness measure on X. A semantically
quantifying mapping (SQM) ’: X ! [0, 1], which is
induced by fm on X, is defined as follows (Ho et al.,
2008: Definition 4):
(i) ’(W) ¼  ¼ fm(cÀ), ’(cÀ) ¼  À fm(cÀ) ¼

fm(cÀ),
’(cþ) ¼  þ fm(cþ);
P
(ii) ’(hjx) ¼ ’(x) þ Sign(hjx)f ji¼Signð j Þ fmðhi xÞ À !
ðhj xÞ fmðhj xÞg, where j 2 {j: À q j p & j 6¼ 0} ¼
[–q^p] and !(hjx) ¼ 12[1 þ Sign(hjx)Sign(hphjx)(
À )].
It can be seen that the mapping ’ is completely
defined by (p þ q) free parameters: one parameter
of the fuzziness measure of a primary term and
(p þ q À 1) parameters of the fuzziness measure of
hedges.

Example: Consider a HA AX ¼ (X, G, C, H, ),
where
G ¼ {small,
large};
C ¼ {0,
W,
1};
HÀ ¼ {Little} ¼ {h–1}; q ¼ 1; Hþ ¼ {Very} ¼ {h1}; p ¼ 1;
 ¼ 0.5;  ¼ 0.5;
¼ 0.5 ( þ
¼ 1). Hence:
m(Very) ¼ 0.5; m(Little) ¼ 0.5;
fm(small ) ¼ 0.5; fm(large) ¼ 0.5;
’(small ) ¼  À fm(small ) ¼ 0.5 À 0.5 Â 0.5 ¼ 0.25;
’(Very small ) ¼ ’(small ) þ Sign(Very small ) Â (fm(Very
small ) À 0.5fm(Very small )) ¼ 0.25 þ (–1) Â 0.5 Â 0.5
Â0.5 ¼ 0.125;

’(Little
small ) ¼ ’(small ) þ Sign(Little
small ) Â
(fm(Little small ) À 0.5fm(Little small )) ¼ 0.25 þ
(þ1) Â 0.5 Â 0.5 Â 0.5 ¼ 0.375;
’(large) ¼  þ fm(large) ¼ 0.5 þ 0.5 Â 0.5 ¼ 0.75;
’(Very large) ¼ ’(large) þ Sign(Very large) Â (fm(Very
large) À 0.5fm(Very
large)) ¼ 0.75 þ (þ1) Â 0.5 Â 0.5 Â 0.5 ¼ 0.875;
’(Little
large) ¼ ’(large) þ Sign(Little
large) Â
(fm(Little large) À 0.5fm(Little large)) ¼ 0.75 þ (–1)
 0.5  0.5  0.5 ¼ 0.625.
’(Very Very small ) ¼ ’(Very small ) þ Sign(Very Very
small ) Â (fm(Very Very small ) À 0.5fm(Very Very
small )) ¼ 0.125 þ (–1) Â 0.5 Â 0.5 Â 0.5 Â 0.5 ¼
0.0625;
’(Little Very small ) ¼ ’(Very small ) þ Sign(Little Very
small ) Â (fm(Little Very small ) À 0.5fm(Little Very
small )) ¼ 0.125 þ (þ1) Â 0.5 Â 0.5 Â 0.5 Â 0.5 ¼
0.1875;
’(Very Little small ) ¼ ’(Little small ) þ Sign(Very Little
small ) Â (fm(Very Little small ) À 0.5fm(Very Little
small )) ¼ 0.375 þ (–1) Â 0.5 Â 0.5 Â 0.5 Â 0.5 ¼ 0.3125;
’(Little Little small ) ¼ ’(Little small ) þ Sign(Little
Little small ) Â (fm(Little Little small ) À 0.5fm
(Little Little small ))
¼ 0.375 þ (þ1) Â 0.5 Â
0.5 Â 0.5 Â 0.5 ¼ 0.4375;

’(Little Little large) ¼ ’(Little large) þ Sign(Little
Little large) Â (fm(Little Little large) À 0.5fm
(Little Little large)) ¼ 0.625 þ (–1) Â 0.5 Â0.5 Â
0.5 Â 0.5 ¼ 0.5625;
’(Very Little large) ¼ ’(Little large) þ Sign(Very Little
large) Â (fm(Very Little large) À 0.5fm(Very Little
large)) ¼ 0.625 þ (þ1) Â 0.5 Â 0.5 Â 0.5 Â 0.5 ¼
0.6875;
]’(Little Very large) ¼ ’(Very large) þ Sign(Little Very
large) Â (fm(Little Very large) À 0.5fm(Little
Very large)) ¼ 0.875 þ (–1) Â 0.5 Â 0.5 Â 0.5 Â 0.5 ¼
0.8125;
’(Very Very large) ¼ ’(Very large) þ Sign(Very
Very large) Â (fm(Very Very large) À 0.5fm(Very
Very large)) ¼ 0.875 þ (þ1) Â 0.5 Â 0.5 Â 0.5 Â 0.5 ¼
0.9375.
The above mappings ’ could be arranged based on
their semantic order, as shown in Figure 3.

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Journal of Vibration and Control 18(14)

j (Very Very large) = 0.9375

j (Very large) = 0.875

j (Little Very large) = 0.8125

j (large) = 0.75

j (Very Little large) = 0.6875

j (Little large) = 0.625

1
j (Little Little large) = 0.5625

j (W) = 0.5

j (Little Little small) = 0.4375

j (Little small) = 0.375

j (Very Little small) = 0.3125

j (small) = 0.25

j (Little Very small) = 0.1875

j (Very small) = 0.125

0 .5
j (Very Very small ) = 0.0625

0

Figure 3. Semantically quantifying mappings ’.

4. Fuzzy controllers of the structural
system

x2
FUZZY
CONTROLLERS

.

The FCs are based on the closed-loop fuzzy system
shown in Figure 4, where u2 is determined by the
above-mentioned controllers (FC, HAFC and
OHAFC) and x2 and x_ 2 are determined from Equation
(1) by using the Newmark method with sample time
Át ¼ 0.01 s. The goal of controllers is to reduce displacement in the second floor, so as to reduce displacements
in the structure.
It is assumed that the universes of discourse of two
state variables are ÀxÃ2 x2 xÃ2 (xÃ2 ¼ 0.2 m) and
Àx_ Ã2 x_ 2 x_ Ã2 (x_ Ã2 ¼ 0.6 m/s), and of the control force
it is À6 Â 106 u2 6 Â 106 (N). In the following parts
of this section, the establishing steps of the controllers
will be presented.

4.1. Conventional fuzzy controller
of the structure

x2

u2
m2
k2
m1
.

x2

u2

c2

x2

x1
x2

Figure 4. Fuzzy controllers of the structural system.

4.1.2. Fuzzy rule base. The fuzzy associative memory

In this section, the FC of the structure is established (for
establishing steps of a FC, see Mandal, 2006) using
Mamdani’s inference and centroid defuzzification
method with 15 control rules. The configuration of the
FC is shown in Figure 5.

4.1.1. Fuzzifier. Five membership functions for x2 in
its interval are established with values negative big
(NB), negative (N), zero (Z), positive (P) and positive

big (PB), as shown in Figure 6. Three membership
functions for x_ 2 in its interval are established with
values N, Z and P, as shown in Figure 7 (Guclu and
Yazici, 2008).
Then, seven membership functions for u2 in its
interval are established with values negative very
big (NVB), NB, N, Z, P, PB and positive very big
(PVB), as shown in Figure 8 (Guclu and Yazici,
2008).

table (FAM table) is established as shown in Table 2
for the actuator on the first floor (Guclu and Yazici,
2008).

4.2. Hedge-algebras-based fuzzy controller
of the structure
In FC, the FAM table is formulated in Table 2. The
linguistic labels in Table 2 have to be transformed into
the new ones, given in Tables 3 and 4, that are suitable
to describe linguistically reference domains of [0, 1] and
can be modeled by suitable HAs. The HAs of the state
variables x2 and x_ 2 are AX ¼ (X, G, C, H, ), where
X ¼ x2 or x_ 2 , G ¼ {small, large}, C ¼ {0, W, 1},
H ¼ {H–, Hþ} ¼ {Little, Very}, and the HAs of the
control variable AU ¼ (U, G, C, H, ), where U ¼ u2,
with the same sets G, C and H as for x2 and x_ 2 , however,
their terms describe different quantitative semantics
based on different real reference domains.

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Centroid Method
Fuzzy Rule Base
(FAM table)

Fuzzifier

State
variables

Defuzzifier

Control
voltage

Fuzzy Inference Engine
(Mamdani Method)

Figure 5. The configuration of the fuzzy controller. FAM: fuzzy associative memory.

NB

N

Z

1

PB

P

x_ 2
x 2 (m)

−x *2

Table 2. Fuzzy associative memory table for the actuator on the
first floor

x *2

0

Figure 6. Membership functions for x2. NB: negative big, N:
negative, Z: zero, P: positive, PB: positive big.

x2

N

Z

P

NB
N
Z
P
PB

PVB
PB
P
Z
N

PB
P
Z
N
NB

P
Z
N
NB
NVB

NB: negative big, N: negative, Z: zero, P: positive, PB: positive big, PVB:
positive very big, NVB: negative very big.

N

P

Z

1

Table 3. Linguistic transformation for x2 and x_ 2

.

x 2 (m/s)
.

−x *2

.

Figure 7. Membership functions for x_ 2. N: negative, Z: zero, P:
positive.

NVB

NB

N

Z

1

P

N

Z

P

PB

Small

Little small

W

Little large

Large

NB: negative big, N: negative, Z: zero, P: positive, PB: positive big.

−x *2

0

NB

PB

PVB

Table 4. Linguistic transformation for u2
NVB

NB

N

Z

P

PB

PVB

Very
Very
small

Little
Very
small

Very
Little
small

W

Very
Little
large

Little
Very
large

Very
Very
large

NVB: negative very big, NB: negative big, N: negative, Z: zero, P: positive,
PB: positive big, PVB: positive very big.

u2 (N)
−6×106

0

6×106

Figure 8. Membership functions for u2. NVB: negative very big,
NB: negative big, N: negative, Z: zero, P: positive, PB: positive big,
PVB: positive very big.

The SQMs ’ are determined and are shown in Tables
5 and 6 (see Section 3).
The configuration of the HAFC is shown in Figure 9.

4.2.1. Semantization and desemantization. Note
that, for convenience in presenting the quantitative

Table 5. Parameters of semantically quantifying mapping s for x2
and x_ 2
Small

Little small

W

Little large

Large

0.25

0.375

0.5

0.625

0.75

semantics formalism in studying the meaning of
vague terms, we have assumed that the common reference domain of the linguistic variables is the interval [0,
1], called the semantic domain of the linguistic variables.
In applications, we need use the values in the reference
domains, for example, the interval [a, b], of the linguistic

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Journal of Vibration and Control 18(14)

variables and, therefore, we have to transform the interval [a, b] into [0, 1] and vice versa. The transformation
(linear interpolation) of the interval [a, b] into [0, 1] is
called a semantization and its converse transformation
from [0, 1] into [a, b] is called a desemantization. The new
terminology ‘semantization’ was defined and accepted
in Ho et al. (2008).
The semantizations for each state variable are defined
by the transformations given in Figures 10 and 11. The
semantization and desemantization for the control
variable are defined by the transformation given in
Figure 12 (x2 , x_ 2 and u2 are replaced with x2s , x_ 2s and
u2s when transforming from the real domain to the
semantic one, respectively).

sets plays an important role; however, in the HAs
approach, the algebraic structure is essential and, hence,
so are the SQMs. So, the meaning of terms or the fuzziness
measure of terms and hedges, which are the parameters of
SQMs or parameters of the fuzziness measure of primary
terms and hedges, are very important.
In the OHAFC, the parameters of the fuzziness
measure of primary terms and hedges of u2 are now

considered as design variables and their intervals are
determined as follows:
 ¼ ½0:4 Ä 0:6Š;  ¼ ½0:4 Ä 0:6Š

4.2.2. HAs rule base. We have the SAM (semantic
associative memory) table based on the FAM table
(Table 2) with SQMs as shown in Table 7 for the
actuator on the first floor.

Domain of x 2

0.0625

0.1875

0.3125

0

.
x *2

Domain of x 2

.

0.375

W

Very
Little
large

Little
Very
large

Very
Very
large

0.5

0.6875

0.8125

0.9375

Semantization

0.75

.

0

0.625

Figure 11. Transformation: x_ 2 to x_ 2s .

–6×106

0

–6×106

0.0625

0

0.9375

Domain of u2

Domain of u2s

Figure 12. Transformation: u2 to u2s.

Linear Interpolation
State
variables

0

.
−x *2

Table 6. Parameters of semantically quantifying mapping s for u2

Very
Little
small

0.25

Domain of x 2s

In this section, the OHAFC of the structure is established, where a GA is used as the search algorithm,
based on the code of Chipperfield et al. (1994).
Note that in the fuzzy sets approach, linguistic terms
are merely labels of fuzzy sets, that is, the shape of fuzzy

Little
Very
small

x2

Figure 10. Transformation: x2 to x2s .

4.3. Optimal hedge-algebras-based fuzzy
controller of the structure

Very
Very
small

0

Domain of x 2s

4.2.3. HAs inference. We propose a HAs inference
method described by Quantifying Semantic Surface
established through the points that present the control
rules occurring in Table 7, as shown in Figure 13.
Hence, u2s is determined by linear interpolations
through x2s and x_ 2s . For example, if x2s ¼ 0.7 (point
X21) and x_ 2s ¼ 0.6 (point X22), then u2s ¼ 0.8625
(point U2).

−x*2

Linear Interpolation
HAs Rule Base
(SAM table)

Desemantization

Control
voltage

HAs Inference Engine
(Linear Interpolation)

Figure 9. The configuration of the hedge-algebras-based fuzzy controller. HA: hedge algebra, SAM: semantic associative memory.

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Duc et al.

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Table 7. Semantic associative memory table for the actuator on the first floor
x_ 2s
x2s

Little small: 0.375

W: 0.5

Small: 0.25
Little small: 0.375
W: 0.5
Little large: 0.625
large: 0.75

Very Very large: 0.9375
Little Very large: 0.8125
Very Little large: 0.6875
W: 0.5
Very Little small: 0.3125

Little Very
Very Little
W: 0.5
Very Little
Little Very

Little large: 0.625
large: 0.8125
large: 0.6875

Very Little large: 0.6875
W: 0.5
Very Little small: 0.3125
Little Very small: 0.1875
Very Very small: 0.0625

small: 0.3125
small: 0.1875

U2
1

u2s 0.5

0
0.8

X22
X21
0.7

0.6

0.5
x 2s

0.4

0.3

0.2

0.7
0.6
0.5 x.
2s
0.4
0.3

Figure 13. Quantifying the semantic surface.

The goal function g is defined as follows:

g¼

n
X
i¼0

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x22 ðiÞ x_ 22 ðiÞ
þ
¼ min
ðxÃ2 Þ2 ðx_ Ã2 Þ2

ð5Þ

where n is the number of control cycles.
The parameters using the GA are determined as
follows (Chipperfield et al., 1994): number of individuals per subpopulations: 10; number of generations:
300; recombination probability: 0.8; number of
variables: 6; fidelity of solution: 10.

5. Results and discussion
The results include the time history of the floor displacements and velocities, control error e and control force of
the structure for both controlled and uncontrolled cases
in order to compare the control effect of the FC, HAFC
and OHAFC, where the error e, which measures the
performance of the controllers, is defined as follows:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x23
x_ 23
x22
x24
x_ 22
x_ 24
e¼
þ
þ
þ
þ
þ
ð6Þ
ðxÃ2 Þ2 ðx_ Ã2 Þ2 ðxÃ2 Þ2 ðx_ Ã2 Þ2 ðxÃ2 Þ2 ðx_ Ã2 Þ2

Figures 14–17 show the time responses of the first,

second, third and fourth floor displacements, respectively. The maximum floor drift is shown in Figure 18.
A comparison of the effectiveness of the three controllers used in this study is presented in Table 8. Figures
19–22 show the time responses of the first, second, third
and fourth floor velocities, respectively. The control
error e is shown in Figure 23. Figure 24 presents the
time response of the control force u2.
As shown in above-mentioned figures and tables,
vibration amplitudes of the floors are decreased successfully with the FC, HAFC and OHAFC. The HAFC
provides better results and an easier implementation in
comparison with the FC.
From Figure 3 and Tables 3–6 it can be conceded
that the semantic order of the HAFC is always
guaranteed.
The semantization method of the HAFC, executed by
linear interpolations (see Figures 10–12), is simpler than
the fuzzification method of the FC, executed by determining the shape, number and density distribution of
the membership functions (see Figures 6–8).
The desemantization method of the HAFC, executed
by linear interpolations (see Figure 12), is much simpler
than the defuzzification method (centroid method À in
this paper) of the FC.

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Journal of Vibration and Control 18(14)

0.3

Uncontrolled

FC

0.2
0.1

The first floor displacements, m

0
–0.1
–0.2
HAFC
–0.3
0
0.3

OHAFC

5

10

15

20

21

22

23

24

25

30

35

40

45

50

26

27

28

29

30

0.2
0.1
0

–0.1
–0.2
–0.3
20

25
Time, s

Figure 14. Displacement x1 (m) versus time (s). FC: fuzzy controller, HAFC: hedge-algebras-based fuzzy controller, OHAFC: optimal
hedge-algebras-based fuzzy controller.

0.3
Uncontrolled

FC

0.2

The second floor displacements, m

0.1
0
–0.1
–0.2
–0.3

OHAFC

HAFC
0

5

10

15

20

–0.3
20

21

22

23

24

25

30

35

40

45

50

26

27

28

29

30

0.3
0.2
0.1
0
–0.1
–0.2
25
Time, s

Figure 15. Displacement x2 (m) versus time (s). FC: fuzzy controller, HAFC: hedge-algebras-based fuzzy controller, OHAFC: optimal
hedge-algebras-based fuzzy controller.

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0.3

Uncontrolled

FC

0.2
0.1

The third floor displacement, m

0
–0.1
–0.2
OHAFC

HAFC

0

5

10

15

20

20

21

22

23

24

25

30

35

40

45

50

26

27

28

29

30

0.3
0.2
0.1
0
–0.1
–0.2
25
Time, s

Figure 16. Displacement x3 (m) versus time (s). FC: fuzzy controller, HAFC: hedge-algebras-based fuzzy controller, OHAFC: optimal
hedge-algebras-based fuzzy controller.

0.3
Uncontrolled

FC

0.2
0.1

The fourth floor displacement, m

0
–0.1
–0.2
–0.3

OHAFC

HAFC

0

5

10

15

20

–0.3
20

21

22

23

24

25

30

35

40

45

50

26

27

28

29

30

0.3
0.2
0.1
0
–0.1
–0.2
25
Time, s

Figure 17. Displacement x4 (m) versus time (s). FC: fuzzy controller, HAFC: hedge-algebras-based fuzzy controller, OHAFC: optimal
hedge-algebras-based fuzzy controller.

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Journal of Vibration and Control 18(14)

4

Uncontrolled

3
Floor

FC
HAFC
OHAFC

2

1
0.18

0.2

0.22

0.24

0.26

0.28

0.3

0.32

Max. Floor Drift, m

Figure 18. The maximum floor drift. FC: fuzzy controller, HAFC: hedge-algebras-based fuzzy controller, OHAFC: optimal hedgealgebras-based fuzzy controller.
Table 8. Comparison of the effectiveness of the three controllers
Controlled to uncontrolled
displacement ratio (reduction ratio)
Building floor

Maximum uncontrolled
displacement, m

FC

HAFC

OHAFC

1
2
3
4

0.278
0.289
0.297
0.302

0.818
0.785
0.784
0.779

0.718
0.673
0.672
0.671

0.692
0.647
0.654
0.662

FC: fuzzy controller, HAFC: hedge-algebras-based fuzzy controller, OHAFC: optimal hedge-algebras-based fuzzy controller

1
Uncontrolled

FC

0.5

The first floor velocity, m/s

0
–0.5
HAFC

–1

OHAFC

0

5

10

15

20

–1
20

21

22

23

24

25

30

35

40

45

50

26

27

28

29

30

1
0.5
0
–0.5

25
Time, s

Figure 19. Velocity x_ 1 (m/s) versus time (s). FC: fuzzy controller, HAFC: hedge-algebras-based fuzzy controller, OHAFC: optimal
hedge-algebras-based fuzzy controller.

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1
Uncontrolle

FC

0.5

The second floor velocity, m/s

0
–0.5
OHAFC

HAFC

–1

0

5

10

15

20

–1
20

21

22

23

24

25

30

35

40

45

50

26

27

28

29

30

1
0.5
0
–0.5

25
Time, s

Figure 20. Velocity x_ 2 (m/s) versus time (s). FC: fuzzy controller, HAFC: hedge-algebras-based fuzzy controller, OHAFC: optimal
hedge-algebras-based fuzzy controller.

1.2
Uncontrolle

FC

0.8
0.4
0

The third floor velocity, m/s

–0.4
–0.8

HAFC

–1.2

OHAFC

0

5

10

15

20

–1.2
20

21

22

23

24

25

30

35

40

45

50

26

27

28

29

30

1.2
0.8
0.4
0
–0.4
–0.8
25
Time, s

Figure 21. Velocity x_ 3 (m/s) versus time (s). FC: fuzzy controller, HAFC: hedge-algebras-based fuzzy controller, OHAFC: optimal
hedge-algebras-based fuzzy controller.

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Journal of Vibration and Control 18(14)

1.2
Uncontrolled

FC

0.8
0.4
0
The fourth floor velocity, m/s

–0.4
–0.8
HAFC

–1.2

0

OHAFC

5

10

15

20

25

30

35

40

45

50

21

22

23

24

25
Time, s

26

27

28

29

30

1.2
0.8
0.4
0
–0.4
–0.8
–1.2
20

Figure 22. Velocity x_ 4 (m/s) versus time (s). FC: fuzzy controller, HAFC: hedge-algebras-based fuzzy controller, OHAFC: optimal
hedge-algebras-based fuzzy controller.

3.5
3
2.5

Uncontrolled

HAFC

FC

OHAFC

2
1.5

Control error e

1
0.5
0

0

5

10

15

20

0
20

21

22

23

24

25

30

35

40

45

50

26

27

28

29

30

3.5
3
2.5
2
1.5

1
0.5
25
Time, s

Figure 23. Control error e versus time (s). FC: fuzzy controller, HAFC: hedge-algebras-based fuzzy controller, OHAFC: optimal
hedge-algebras-based fuzzy controller.

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5

x 106
FC

Control force u2, N

0

OHAFC

HAFC

–5

5

0
x 106

5

10

15

20

16

17

18

19

25

30

35

40

45

50

21

22

23

24

25

0

–5
15

20
Time, s

Figure 24. Control force u2 (N) versus time (s). FC: fuzzy controller, HAFC: hedge-algebras-based fuzzy controller, OHAFC: optimal
hedge-algebras-based fuzzy controller.

The inference method of the HAFC, executed by
linear interpolations (see Figure 13), is also much simpler than that of the FC (Mamdani method À in this
paper).
In order to describe three, five, seven, . . . , n linguistic
labels by HAs, only two independent parameters ( and
, see Section 3) are needed. Thus, there are two design

variables to establish an optimal HAFC. For an optimal
FC based on n linguistic labels, there are (n  3) design
variables (each triangular membership function needs
three design variables). Hence, an optimal HAFC is
simpler and more efficient than an optimal FC when
designing and implementing.
The HAFC, a new fuzzy control algorithm, does not
require fuzzy sets to provide the semantics of the
linguistic terms used in the fuzzy rule system, rather
the semantics are obtained through the SQMs. In the
algebraic approach, the design of a HAFC leads to
the determination of the parameter of SQMs, which
are the fuzziness measure of primary terms and linguistic hedges occurring in the fuzzy model.

6. Conclusions
In the present work, new FCs based on HAs are applied
for active control of a structure against earthquake. The
main results are summarized as follows.

The algebraic approach to term-domains of linguistic
variables is quite different from the fuzzy sets one in the
representation of the meaning of linguistic terms and
the methodology of solving the fuzzy multiple conditional reasoning problems.
It is clear that the HAFC is simpler, more effective
and more understandable in comparison with the FC
for actively controlling the above-mentioned seismexcited civil structure.
The proposed HAs inference method allows directly
establishing an inference engine from SAM tables.
In fuzzy logic, many important concepts, such as
fuzzy set, T-norm, S-norm, intersection, union, complement, composition, etc., are used in approximate

reasoning. This is an advantage for the process of flexible reasoning, but there are too many factors, such as
the shape and number of membership functions, defuzzification method, etc., influencing the precision of
the reasoning process and it is difficult to optimize.
Those are subjective factors that cause error in determining the values of control process. Meanwhile,
approximate reasoning based on HAs, from the beginning, does not use the fuzzy set concept and its precision
is obviously not influenced by this concept. Therefore,
the method based on HAs does not need to determine
the shape and number of the membership function,
neither does it need to solve defuzzification problem.

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Journal of Vibration and Control 18(14)

In addition, in calculation, while there is a large number
of membership functions, the volume of calculation
based on fuzzy control increases quickly; meanwhile,
the volume of calculation based on HAs does not
increase much with very simple calculation. With these
above advantages, it is definitely possible to use HA
theory for many different controlling problems.
Funding
This paper was supported by the National Foundation for
Science and Technology Development of Vietnam–
NAFOSTED.

References

Anh ND, Matsuhisa H, Viet LD and Yasuda M (2007)
Vibration control of an inverted pendulum type structure
by passive mass-spring-pendulum dynamic vibration
absorber. Journal of Sound and Vibration 307: 187–201.
Battaini M, Casciati F and Faravelli L (1999) Fuzzy control of
structural vibration. An active mass system driven by a
fuzzy controller. Earthquake Engineering & Structural
Dynamics 27: 1267–1276.
ChipperEeld A, Fleming P, Pohlheim H and Fonseca C (1994)
Genetic Algorithm Toolbox. Sheffield: Department of
Automatic Control and Systems Engineering, The
University of Sheffield.
Guclu R (2006) Sliding mode and PID control of a structural
system against earthquake. Mathematical and Computer
Modelling 44: 210–217.
Guclu R and Yazici H (2008) Vibration control of a structure
with ATMD against earthquake using fuzzy logic controllers. Journal of Sound and Vibration 318: 36–49.
Ho NC (2007) A topological completion of reEned hedge algebras and a model of fuzziness of linguistic terms and
hedges. Fuzzy Sets and Systems 158: 436–451.
Ho NC, Lan VN and Viet LX (2006) An interpolative reasoning method based on hedge algebras and its application to
problem of fuzzy control. In: Proceedings of the 10th
WSEAS International on Computers, Vouliagmeni,
Athens, Greece, July, pp. 526–534.
Ho NC, Lan VN and Viet LX (2008) Optimal hedge-algebrasbased controller: design and application. Fuzzy Sets and
Systems 159: 968–989.

Ho NC and Long NV (2007) Fuzziness measure on complete
hedge algebras and quantifying semantics of terms in linear
hedge algebras. Fuzzy Sets and Systems 158: 452–471.
Ho NC and Nam HV (2002) An algebraic approach to linguistic hedges in Zadeh’s fuzzy logic. Fuzzy Sets and

Systems 129: 229–254.
Ho NC, Nam HV, Khang TD and Chau NH (1999) Hedge
algebras, linguistic-valued logic and their application to
fuzzy reasoning. International Journal of Uncertainty
Fuzziness of Knowledge-based Systems 7: 347–361.
Ho NC and Wechler W (1990) Hedge algebras: an algebraic
approach to structure of sets of linguistic truth values.
Fuzzy Sets and Systems 35: 281–293.
Ho NC and Wechler W (1992) Extended hedge algebras and
their application to fuzzy logic. Fuzzy Sets and Systems 52:
259–281.
Li L, Song G and Ou J (2010) Hybrid active mass damper
(AMD) vibration suppression of nonlinear high-rise structure using fuzzy logic control algorithm under earthquake
excitations. Structural Control and Health Monitoring 18:
698–709.
Mandal AK (2006) Introduction to Control Engineering. New
Delhi: New Age International Publishers.
Park KS, Koh HM and Ok SY (2002) Active control of earthquake excited structures using fuzzy supervisory technique.
Advances in Engineering Software 33: 761–768.
Pourzeynali S, Lavasani HH and Modarayi AH (2007) Active
control of high rise building structures using fuzzy logic
and genetic algorithms. Engineering Structures 29:
346–357.
Preumont A and Suto K (2008) Active Control of Structures.
West Sussex, UK: John Wiley & Sons.
Teng TL, Peng CP and Chuang C (2000) A study on the application of fuzzy theory to structural active control.
Computer Methods in Applied Mechanics and Engineering
189: 439–448.
Wang AP and Lin YH (2007) Vibration control of a tall building subjected to earthquake excitation. Journal of Sound
and Vibration 299: 757–773.

Yan N, Wang CM and Balendra T (1998) Composite mass
dampers for vibration control of wind-excited towers.
Journal of Sound and Vibration 213: 301–316.
Zadeh LA (1965) Fuzzy sets. Information and Control 8:
338–353.

Downloaded from jvc.sagepub.com at UNIV OF PITTSBURGH on February 12, 2015