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 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav 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 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 not require an energy feed and therefore 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 University of Engineering and Technology, Vietnam National University, Hanoi, Hanoi, Vietnam Institute of Information Technology, Vietnam Academy of Science and Technology, Hanoi, Vietnam 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 Downloaded from jvc.sagepub.com at UNIV OF PITTSBURGH on February 12, 2015 Duc et al 2187 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 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 In Section 4, the FCs of the structural system are presented Results and discussion are given in Section Conclusions are presented in Section 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 m4 x4 k4 c4 m3 k3 c3 x3 m2 k2 c2 x2 u2 m1 k1 c1 xă Figure The structural system Downloaded from jvc.sagepub.com at UNIV OF PITTSBURGH on February 12, 2015 x1 2188 Journal of Vibration and Control 18(14) Table The system parameters xă , m/s2 Floor i Mass mi (103 kg) Damping ci (102 Ns/m) Stiffness ki (105 N/m) 450 345 345 345 261.7 4670 4100 3500 180.5 3260 2850 2500 –2 –4 10 20 30 40 50 Time, s Reproduced with kind permission from Elsevier (Guclu, 2006) Figure The north-south acceleration component of the 1940 El Centro earthquake Hedge algebras The equations of motion of the system subjected to the north-south acceleration component of the 1940 El Centro earthquake x€ (see Figure 2), with control force vector {u}, can be written as ỵ ẵCfxg _ ỵ ẵKfxg ẳ fug ẵMfrgx ẵMfxg 1ị where {x} ẳ [x1 x2 x3 x4]T, {u} ¼ [Àu2 u2 0]T represents the horizontal component of the active tendon force and the  vector {r} is the influence vector representing the displacement of each degree of freedom resulting from static application of a unit ground displacement The  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: m1 0 m2 0 7 ẵM ẳ 2ị 0 m3 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): Àk2 0 k1 k2 k1 ỵ k2 k3 7 3ị ẵK ẳ k3 k2 ỵ k3 k4 0 Àk4 k4 The structural damping matrix [C] is given as c1 Àc2 0 Àc2 c1 ỵ c2 c3 7 ẵC ẳ c3 c2 ỵ c3 c4 0 Àc4 c4 ð4Þ The system parameters are given in Table (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 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 cỵ, while false has a tendency of going down’, called negative one, denoted by cÀ In general, we always have cỵ ! 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 HÀ the set of all negative hedges and by Hỵ 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 ẳ {c, cỵ}, Downloaded from jvc.sagepub.com at UNIV OF PITTSBURGH on February 12, 2015 Duc et al 2189 C ẳ {0, W, 1}, H ẳ Hỵ [ HÀ and is a partially ordering relation on X It is assumed that HÀ ¼ {h–1, , h–q}, where h–1 < h–2 < < hq, Hỵ ẳ {h1, , hp}, where h1 < h < < 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(c) ỵ fm(cỵ) ẳ and h2H fm(hu) ¼ fm(u), for 8u X; – for the constants 0, W and 1, fm(0) ¼ fm(W) ¼ fm(1) ¼ 0; fmðhyÞ – for 8x, y X, 8h H, fmhxị fmxị ẳ fm yị : 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 X; fm(c) ỵ fm(cỵ) ẳ 1; P ỵ P q i p, i6ẳ0 fm hi cị ¼ fmðcÞ, c {c , c }; P Àq i p, i6ẳ fmhi xị ẳ fmxị; P q i hi ị ẳ  and i p  hi ị ẳ where , > and  þ ¼ A function Sign, X ! {À1, 0, 1}, is a mapping that is defined recursively as follows, for h, h H and c {c, cỵ} (Ho et al., 2008: Definition 3): – Sign(cÀ) ¼ À1, Sign(cỵ) ẳ ỵ1; Sign(hc) ẳ Sign(c), if h is negative with regard to c; Sign(hc) ẳ ỵ 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(hhx) ẳ ỵ Sign(hx), if hhx 6ẳ hx and h’ is positive with regard to h; – Sign(h’hx) ¼ 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  fm(c) ẳ fm(c), (cỵ) ẳ  ỵ fm(cỵ); P (ii) (hjx) ẳ (x) ỵ Sign(hjx)f jiẳSign j ị fmðhi xÞ À ! ðhj xÞ fmðhj xÞg, where j {j: À q j p & j 6¼ 0} ẳ [q^p] and !(hjx) ẳ 12[1 ỵ Sign(hjx)Sign(hphjx)( )] It can be seen that the mapping ’ is completely dened by (p ỵ q) free parameters: one parameter of the fuzziness measure of a primary term and (p ỵ 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}; H ẳ {Little} ẳ {h1}; q ẳ 1; Hỵ ¼ {Very} ¼ {h1}; p ¼ 1;  ¼ 0.5;  ẳ 0.5; ẳ 0.5 ( ỵ ẳ 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  0.5 ¼ 0.25; ’(Very small ) ẳ (small ) ỵ Sign(Very small ) (fm(Very small ) 0.5fm(Very small )) ẳ 0.25 ỵ (1)  0.5  0.5 Â0.5 ¼ 0.125; ’(Little small ) ẳ (small ) ỵ Sign(Little small ) (fm(Little small ) 0.5fm(Little small )) ẳ 0.25 ỵ (ỵ1) 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) ẳ (large) ỵ Sign(Very large) (fm(Very large) 0.5fm(Very large)) ẳ 0.75 ỵ (ỵ1) 0.5 0.5 0.5 ẳ 0.875; (Little large) ẳ (large) ỵ Sign(Little large)  (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 ) ẳ (Very small ) ỵ Sign(Very Very small )  (fm(Very Very small ) À 0.5fm(Very Very small )) ẳ 0.125 ỵ (1) 0.5 0.5  0.5  0.5 ¼ 0.0625; ’(Little Very small ) ẳ (Very small ) ỵ Sign(Little Very small )  (fm(Little Very small ) À 0.5fm(Little Very small )) ẳ 0.125 ỵ (ỵ1) 0.5 0.5 0.5  0.5 ¼ 0.1875; ’(Very Little small ) ¼ ’(Little small ) ỵ Sign(Very Little small ) (fm(Very Little small ) 0.5fm(Very Little small )) ẳ 0.375 ỵ (–1)  0.5  0.5  0.5  0.5 ¼ 0.3125; (Little Little small ) ẳ (Little small ) ỵ Sign(Little Little small )  (fm(Little Little small ) À 0.5fm (Little Little small )) ẳ 0.375 ỵ (ỵ1) 0.5  0.5  0.5  0.5 ¼ 0.4375; ’(Little Little large) ẳ (Little large) ỵ Sign(Little Little large) (fm(Little Little large) À 0.5fm (Little Little large)) ¼ 0.625 þ (–1)  0.5 Â0.5  0.5  0.5 ¼ 0.5625; (Very Little large) ẳ (Little large) ỵ Sign(Very Little large)  (fm(Very Little large) À 0.5fm(Very Little large)) ¼ 0.625 ỵ (ỵ1) 0.5 0.5 0.5 0.5 ẳ 0.6875; ](Little Very large) ẳ (Very large) ỵ Sign(Little Very large)  (fm(Little Very large) À 0.5fm(Little Very large)) ẳ 0.875 ỵ (1) 0.5 0.5 0.5  0.5 ¼ 0.8125; ’(Very Very large) ¼ ’(Very large) ỵ Sign(Very Very large) (fm(Very Very large) 0.5fm(Very Very large)) ẳ 0.875 ỵ (ỵ1) 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 Downloaded from jvc.sagepub.com at UNIV OF PITTSBURGH on February 12, 2015 2190 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 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 j (Very Very small ) = 0.0625 Figure Semantically quantifying mappings ’ 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_ 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_ x_ Ã2 (x_ Ã2 ¼ 0.6 m/s), and of the control force it is À6  106 u2  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 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 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 Three membership functions for x_ in its interval are established with values N, Z and P, as shown in Figure (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 (Guclu and Yazici, 2008) table (FAM table) is established as shown in Table 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 The linguistic labels in Table have to be transformed into the new ones, given in Tables 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_ are AX ¼ (X, G, C, H, ), where X ¼ x2 or x_ , G ¼ {small, large}, C ẳ {0, W, 1}, H ẳ {H, Hỵ} ẳ {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_ , however, their terms describe different quantitative semantics based on different real reference domains Downloaded from jvc.sagepub.com at UNIV OF PITTSBURGH on February 12, 2015 Duc et al 2191 Centroid Method Fuzzy Rule Base (FAM table) Fuzzifier State variables Defuzzifier Control voltage Fuzzy Inference Engine (Mamdani Method) Figure The configuration of the fuzzy controller FAM: fuzzy associative memory NB N Z PB P x_ x (m) −x *2 Table Fuzzy associative memory table for the actuator on the first floor x *2 Figure 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 Table Linguistic transformation for x2 and x_ x (m/s) −x *2 Figure Membership functions for x_ N: negative, Z: zero, P: positive NVB NB N Z 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 NB PB PVB Table 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 6×106 Figure 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 and (see Section 3) The configuration of the HAFC is shown in Figure 4.2.1 Semantization and desemantization Note that, for convenience in presenting the quantitative Table Parameters of semantically quantifying mapping s for x2 and x_ 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 Downloaded from jvc.sagepub.com at UNIV OF PITTSBURGH on February 12, 2015 2192 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_ 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:  ẳ ẵ0:4 0:6;  ẳ 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 for the actuator on the first floor Domain of x 0.0625 0.1875 0.3125 x *2 Domain of x 0.375 W Very Little large Little Very large Very Very large 0.5 0.6875 0.8125 0.9375 Semantization 0.75 0.625 Figure 11 Transformation: x_ to x_ 2s –6×106 –6×106 0.0625 0.9375 Domain of u2 Domain of u2s Figure 12 Transformation: u2 to u2s Linear Interpolation State variables −x *2 Table 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 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 The configuration of the hedge-algebras-based fuzzy controller HA: hedge algebra, SAM: semantic associative memory Downloaded from jvc.sagepub.com at UNIV OF PITTSBURGH on February 12, 2015 Duc et al 2193 Table 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 u2s 0.5 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: gẳ n X iẳ0 s x22 iị x_ 22 iị ỵ ẳ x2 ị2 x_ ị2 5ị 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 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 eẳ ỵ ỵ ỵ ỵ ỵ 6ị x2 ị2 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 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 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 Downloaded from jvc.sagepub.com at UNIV OF PITTSBURGH on February 12, 2015 2194 Journal of Vibration and Control 18(14) 0.3 Uncontrolled FC 0.2 0.1 The first floor displacements, m –0.1 –0.2 HAFC –0.3 0.3 OHAFC 10 15 20 21 22 23 24 25 30 35 40 45 50 26 27 28 29 30 0.2 0.1 –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 –0.1 –0.2 –0.3 OHAFC HAFC 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 –0.1 –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 Downloaded from jvc.sagepub.com at UNIV OF PITTSBURGH on February 12, 2015 Duc et al 2195 0.3 Uncontrolled FC 0.2 0.1 The third floor displacement, m –0.1 –0.2 OHAFC HAFC 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 –0.1 –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 –0.1 –0.2 –0.3 OHAFC HAFC 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 –0.1 –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 Downloaded from jvc.sagepub.com at UNIV OF PITTSBURGH on February 12, 2015 2196 Journal of Vibration and Control 18(14) Uncontrolled Floor FC HAFC OHAFC 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 Comparison of the effectiveness of the three controllers Controlled to uncontrolled displacement ratio (reduction ratio) Building floor Maximum uncontrolled displacement, m FC HAFC OHAFC 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 Uncontrolled FC 0.5 The first floor velocity, m/s –0.5 HAFC –1 OHAFC 10 15 20 –1 20 21 22 23 24 25 30 35 40 45 50 26 27 28 29 30 0.5 –0.5 25 Time, s Figure 19 Velocity x_ (m/s) versus time (s) FC: fuzzy controller, HAFC: hedge-algebras-based fuzzy controller, OHAFC: optimal hedge-algebras-based fuzzy controller Downloaded from jvc.sagepub.com at UNIV OF PITTSBURGH on February 12, 2015 Duc et al 2197 Uncontrolle FC 0.5 The second floor velocity, m/s –0.5 OHAFC HAFC –1 10 15 20 –1 20 21 22 23 24 25 30 35 40 45 50 26 27 28 29 30 0.5 –0.5 25 Time, s Figure 20 Velocity x_ (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 The third floor velocity, m/s –0.4 –0.8 HAFC –1.2 OHAFC 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 –0.4 –0.8 25 Time, s Figure 21 Velocity x_ (m/s) versus time (s) FC: fuzzy controller, HAFC: hedge-algebras-based fuzzy controller, OHAFC: optimal hedge-algebras-based fuzzy controller Downloaded from jvc.sagepub.com at UNIV OF PITTSBURGH on February 12, 2015 2198 Journal of Vibration and Control 18(14) 1.2 Uncontrolled FC 0.8 0.4 The fourth floor velocity, m/s –0.4 –0.8 HAFC –1.2 OHAFC 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 –0.4 –0.8 –1.2 20 Figure 22 Velocity x_ (m/s) versus time (s) FC: fuzzy controller, HAFC: hedge-algebras-based fuzzy controller, OHAFC: optimal hedge-algebras-based fuzzy controller 3.5 2.5 Uncontrolled HAFC FC OHAFC 1.5 Control error e 0.5 0 10 15 20 20 21 22 23 24 25 30 35 40 45 50 26 27 28 29 30 3.5 2.5 1.5 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 Downloaded from jvc.sagepub.com at UNIV OF PITTSBURGH on February 12, 2015 Duc et al 2199 x 106 FC Control force u2, N OHAFC HAFC –5 x 106 10 15 20 16 17 18 19 25 30 35 40 45 50 21 22 23 24 25 –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 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 Downloaded from jvc.sagepub.com at UNIV OF PITTSBURGH on February 12, 2015 2200 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 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(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. .. 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