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

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.. This suggests to us, in this pap

A study on the application of hedge

algebras to active fuzzy control

of a seism-excited structure

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

vibra-tion 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

dura-bility 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

environ-mental 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 con-trol 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: ducnd@vnu.edu.vn

18(14) 2186–2200

! The Author(s) 2011 Reprints and permissions:

sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/1077546311429057 jvc.sagepub.com

become more effective in use An active vibration

control-ler 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

supervi-sory technique for the active control of

earthquake-excited 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

earth-quake 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

defuzzifica-tion 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

order-ing relation Many interestorder-ing 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 rea-soning 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 with-out 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, hedge-algebras-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

x1

k1

m1

m2

m3

m4

x2

x3

x4

k2

k4

k3

u2

c1

x¨0

c2

c4

c3

Figure 1 The structural system

The equations of motion of the system subjected to the

north-south acceleration component of the 1940 El

Centro earthquake €x0 (see Figure 2), with control force

vector {u}, can be written as

½Mf €xg þ ½Cf _xg þ ½Kfxg ¼ fug  ½Mfrg €x0 ð1Þ

where {x} ¼ [x1x2x3x4]T, {u} ¼ [u2u20 0]Trepresents

the horizontal component of the active tendon force

and the 4  1 vector {r} is the influence vector

represent-ing the displacement of each degree of freedom resultrepresent-ing

from static application of a unit ground displacement

The 4  4 matrices [M], [C] and [K] represent the

struc-tural mass, damping and stiffness matrices, respectively

The mass matrix for a building structure, with the

assumption of masses lumped at floor levels, is a

diag-onal matrix in which the mass of each story is sorted on

its diagonal, as given in the following:

½M ¼

2 6 4

3 7

where miis 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):

½K ¼

k2 k1þk2 k3 0

0 k3 k2þk3 k4

2

6

4

3 7

The structural damping matrix [C] is given as

½C ¼

c2 c1þc2 c3 0

0 c3 c2þc3 c4

2

6

4

3 7

The system parameters are given in Table 1 (Guclu,

2006)

3 Hedge algebras

In this section, the idea and basic formulas of HAs are summarized based on definitions, theorems and propo-sitions 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 seman-tics-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 seman-tic 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 c

In general, we always have cþ

c

, 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 Hthe set of all negative hedges and by

Hþthe 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 {c, cþ},

Time, s

–4

–2

0

2

4

x¨0

Figure 2 The north-south acceleration component of the 1940

El Centro earthquake

Table 1 The system parameters

Floor i

Mass

mi(103kg)

Damping

ci(102Ns/m)

Stiffness

ki(105N/m)

1 450 261.7 180.5

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

C ¼{0, W, 1}, H ¼ Hþ

[H

and  is a partially order-ing relation on X It is assumed that H

¼{h– , , h–q}, where h– < h– < < h–q, Hþ¼{h1, , hp}, where

h1< h2< < hp

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:

– fm(c) þ fm(cþ) ¼ 1 and P

h2H fm(hu) ¼ fm(u), for 8u 2 X;

fm(0) ¼ fm(W) ¼ fm(1) ¼ 0;

– for 8x, y 2 X, 8h 2 H, fmðhxÞfmðxÞ ¼fmðhyÞfmð yÞ: this proportion

does not depend on specific elements, called fuzziness

measure of the hedge hand 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(c) þ fm(cþ) ¼ 1;

q  i  p, i6¼0fm ðhicÞ ¼ fmðcÞ, c 2 {c, cþ};

q  i  p, i6¼ 0fmðhixÞ ¼ fmðxÞ;

q  i  1ðhiÞ ¼ and P

1  i  p ðhiÞ ¼ where

,  > 0 and  þ  ¼ 1

A function Sign, X ! {1, 0, 1}, is a mapping that

is defined recursively as follows, for h, h’ 2 H and

c 2{c

, cþ

} (Ho et al., 2008: Definition 3):

– Sign(c) ¼ 1, Sign(cþ) ¼ þ1;

– Sign(hc) ¼ Sign(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 Sign(hx), if h’hx 6¼ hx and h’ is

nega-tive with regard to h; Sign(h’hx) ¼ þ Sign(hx), if

h’hx 6¼ hxand 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(c

), ’(c

) ¼   fm(c

) ¼ fm(c

),

’(cþ) ¼  þ fm(cþ);

(ii) ’(hjx) ¼ ’(x) þ Sign(hjx)fPj

i¼Signð j ÞfmðhixÞ  !

ðhjxÞ fmðhjxÞg, where j 2{j:  q  j  p & j 6¼ 0} ¼

[–q^p] and !(hjx) ¼1

2[1 þ Sign(hjx)Sign(hphjx)(  )]

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  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};

H¼{Little} ¼ {h–1}; q ¼ 1; Hþ¼{Very} ¼ {h1}; p ¼ 1;

 ¼0.5;  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   fm(small ) ¼ 0.5  0.5  0.5 ¼ 0.25;

’(Very small ) ¼ ’(small ) þ Sign(Very small )  (fm(Very small)  0.5fm(Very small )) ¼ 0.25 þ (–1)  0.5  0.5

0.5 ¼ 0.125;

’(Little small) ¼ ’(small ) þ Sign(Little small)  (fm(Little small)  0.5fm(Little small)) ¼ 0.25 þ (þ1)  0.5  0.5  0.5 ¼ 0.375;

’(large) ¼  þ fm(large) ¼ 0.5 þ 0.5  0.5 ¼ 0.75;

’(Very large) ¼ ’(large) þ Sign(Very large)  (fm(Very large)  0.5fm(Very

large)) ¼ 0.75 þ (þ1)  0.5  0.5  0.5 ¼ 0.875;

’(Little large) ¼ ’(large) þ Sign(Little large)  (fm(Little large)  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)  (fm(Very Very small )  0.5fm(Very Very small)) ¼ 0.125 þ (–1)  0.5  0.5 0.5  0.5 ¼ 0.0625;

’(Little Very small ) ¼ ’(Very small ) þ Sign(Little Very small)  (fm(Little Very small )  0.5fm(Little Very small)) ¼0.125 þ (þ1)  0.5  0.5  0.5  0.5 ¼ 0.1875;

’(Very Little small ) ¼ ’(Little small ) þ Sign(Very Little small)  (fm(Very Little small )  0.5fm(Very Little small)) ¼ 0.375 þ (–1)  0.5  0.5  0.5  0.5 ¼ 0.3125;

’(Little Little small ) ¼ ’(Little small ) þ Sign(Little Little small)  (fm(Little Little small )  0.5fm (Little Little small)) ¼0.375 þ (þ1)  0.5  0.5  0.5  0.5 ¼ 0.4375;

’(Little Little large) ¼ ’(Little large) þ Sign(Little Little large) (fm(Little Little large) 0.5fm (Little Little large)) ¼0.625 þ (–1)  0.5 0.5  0.5  0.5 ¼ 0.5625;

’(Very Little large) ¼ ’(Little large) þ Sign(Very Little large)  (fm(Very Little large)  0.5fm(Very Little large)) ¼ 0.625 þ (þ1)  0.5  0.5  0.5  0.5 ¼ 0.6875;

]’(Little Very large) ¼ ’(Very large) þ Sign(Little Very large) (fm(Little Very large) 0.5fm(Little Very large)) ¼ 0.875 þ (–1)  0.5  0.5  0.5  0.5 ¼ 0.8125;

’(Very Very large) ¼’(Very large) þSign(Very Very large)  (fm(Very Very large)  0.5fm(Very Very large)) ¼ 0.875 þ (þ1)  0.5  0.5  0.5  0.5 ¼ 0.9375

The above mappings ’ could be arranged based on their semantic order, as shown in Figure 3

4 Fuzzy controllers of the structural

system

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 x2and _x2 are determined from Equation

(1) by using the Newmark method with sample time

t ¼ 0.01 s The goal of controllers is to reduce

displace-ment in the second floor, so as to reduce displacedisplace-ments

in the structure

It is assumed that the universes of discourse of two

state variables are x

2x2x

2 (x

2¼0.2 m) and

x_

2 x_2x_

2 ( _x

2¼0.6 m/s), and of the control force

it is 6  106 u2 6  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

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 _x2 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)

4.1.2 Fuzzy rule base. The fuzzy associative memory 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 _x2 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 x2and _x2, however, their terms describe different quantitative semantics based on different real reference domains

0

1 5

0

Figure 3 Semantically quantifying mappings ’

FUZZY CONTROLLERS

x2

x2

x1

x2

m2

m1

x .2

x .2

c2

u2

u2

k2

Figure 4 Fuzzy controllers of the structural system

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

semantics formalism in studying the meaning of vague terms, we have assumed that the common refer-ence 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

Fuzzy Rule Base (FAM table)

Fuzzy Inference Engine (Mamdani Method)

State variables

Centroid Method

Control voltage

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

N

2

x (m) 0

NB

Figure 6 Membership functions for x2 NB: negative big, N:

negative, Z: zero, P: positive, PB: positive big

Z

(m/s) 0

N

x .2

Figure 7 Membership functions for _x2 N: negative, Z: zero, P:

positive

N

u2 (N) 0

NVB

PVB NB

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

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

x2

_x2

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

Table 3 Linguistic transformation for x2and _x2

Small Little small W Little large Large

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

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.

Table 5 Parameters of semantically quantifying mapping s for x2 and _x2

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

variables and, therefore, we have to transform the

inter-val [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, _x2 and u2are replaced with x2s, _x2s and

u2s when transforming from the real domain to the

semantic one, respectively)

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

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 _x2s For example, if x2s¼0.7 (point

X21) and _x2s¼0.6 (point X22), then u2s¼0.8625

(point U2)

4.3 Optimal hedge-algebras-based fuzzy

controller of the structure

In this section, the OHAFC of the structure is

estab-lished, 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

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:

 ¼½0:4  0:6;  ¼ 0:4  0:6½ 

Table 6 Parameters of semantically quantifying mapping s for u2

Very

Very

small

Little

Very

small

Very Little small W

Very Little large

Little Very large

Very Very large 0.0625 0.1875 0.3125 0.5 0.6875 0.8125 0.9375

HAs Rule Base (SAM table)

HAs Inference Engine (Linear Interpolation)

Semantization State

variables

Linear Interpolation

Control voltage Desemantization

Linear Interpolation

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

0

0

0.75 0.25

Figure 10 Transformation: x2to x2s

0

0

0.625 0.375

Figure 11 Transformation: _x2to _x2s

0

0 0.0625

0.9375

Figure 12 Transformation: u2to u2s

The goal function g is defined as follows:

g ¼Xn

i¼0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

x2ðiÞ

ðx

2Þ2þ

_

x2ðiÞ

ðx_

2Þ2

s

where n is the number of control cycles

The parameters using the GA are determined as

follows (Chipperfield et al., 1994): number of

individ-uals 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

displace-ments 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:

e ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

x2

2

ðx

2Þ2þ

_

x2 2

ðx_

2Þ2þ

x2 3

ðx

2Þ2þ

_

x2 3

ðx_

2Þ2þ

x2 4

ðx

2Þ2þ

_

x2 4

ðx_

2Þ2

s

ð6Þ

Figures 14–17 show the time responses of the first, second, third and fourth floor displacements, respec-tively The maximum floor drift is shown in Figure 18

A comparison of the effectiveness of the three control-lers 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 success-fully 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 deter-mining 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  in this paper) of the FC

Table 7 Semantic associative memory table for the actuator on the first floor

x2s

_x2s

Little small: 0.375 W: 0.5 Little large: 0.625 Small: 0.25 Very Very large: 0.9375 Little Very large: 0.8125 Very Little large: 0.6875 Little small: 0.375 Little Very large: 0.8125 Very Little large: 0.6875 W: 0.5

W: 0.5 Very Little large: 0.6875 W: 0.5 Very Little small: 0.3125 Little large: 0.625 W: 0.5 Very Little small: 0.3125 Little Very small: 0.1875 large: 0.75 Very Little small: 0.3125 Little Very small: 0.1875 Very Very small: 0.0625

0.3 0.4 0.5 0.6 0.7

0.2 0.3

0.4 0.5

0.6 0.7

0.8 0 0.5 1

X21

X22

U2

u2s

x .2s

x2s

Figure 13 Quantifying the semantic surface

20 21 22 23 24 25 26 27 28 29 30 –0.3

–0.2 –0.1 0 0.1 0.2 0.3

–0.3 –0.2 –0.1 0 0.1 0.2 0.3

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 –0.2 –0.1 0 0.1 0.2 0.3

–0.3 –0.2 –0.1 0 0.1 0.2 0.3

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

0 5 10 15 20 25 30 35 40 45 50 –0.2

–0.1 0 0.1 0.2 0.3

Time, s

–0.2 –0.1 0 0.1 0.2 0.3

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 –0.2 –0.1 0 0.1 0.2 0.3

Time, s

–0.3 –0.2 –0.1 0 0.1 0.2 0.3

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