Active control of a three-Story building using hedge-algebras-based fuzzy controller

Vietnam Journal of Mechanics, VAST, Vol. 33, No. 2 (2011), pp. 79 – 94 ACTIVE CONTROL OF A THREE - STORY BUILDING USING HEDGE - ALGEBRAS - BASED FUZZY CONTROLLER Bui Hai Le Hanoi University of Science and Technology Abstract. In this paper, conventional and hedge - algebras - based fuzzy controllers, respectively denoted by FC and HAFC, are designed to suppress vibrations of a three - story building against earthquake. The structural system is simulated against the ground accelerations of the El Centro earthquake in USA on May 18th, 1940; the Northridge earthquake in USA on January 17th, 1994 and the Kobe earthquake in Japan on January 16th, 1995. The control effects of FC and HAFC are compared via the time history of the story displacements of the structure. Keywords: Active control, fuzzy control, hedge algebras, earthquake. 1. INTRODUCTION Undesired vibrations result in structural fatigue, lowering the strength and safety of the structure, and reducing the accuracy and reliability of equipments. The problem of undesired vibration reduction is known for many years and it has become more attrac- tive nowadays to ensure the safety of structure, and increase the reliability and durability of equipment [1,2]. A critical aspect in the design of civil engineering structures is the reduction of response quantities such as velocities, deflections and forces induced by en- vironmental 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 [3-7]. Depending on the control methods, vibration control in the structure can be divided into two categories, namely, passive control and active control. The idea of passive struc- tural control is energy absorption, so as to reduce displacement in the structure. Recent development of control theory and technique has brought vibration control from passive to active and the active control method has become more effective in use. An active vibration controller is equipped with sensors, actuators, and it requires power [2,8]. Fuzzy set theory introduced by Zadeh in 1965 has provided a mathematical tool useful for modelling uncertain (imprecise) and vague data and been presented in many real situations. Recently, many researches on active fuzzy control of vibrating structures have been done [2,7,9-11]. 80 Bui Hai Le Hedge algebras (HAs) has been introduced and investigated since 1990 [12-19]. The authors of HAs discovered that: linguistic values can formulate an algebraic structure [12,13] and it is a Complete Hedge Algebras Structure [17,18] with a main property is that the semantic order of linguistic values is always guaranteed. It is even a rich enough algebraic structure [15] and, therefore, it can describe completely reasoning processes. In [19], HAs theory was begun applying to fuzzy control and it provided better results than FC, but studied objects in [19] are too simple to evaluate completely its control effect. That reason suggests us, in this paper, applying HAs in active fuzzy control of a three - story building against earthquake. 2. DYNAMIC MODEL OF THE STRUCTURAL SYSTEM In this paper, the simple structure model in [6] is used to study the control effect of HAFC in comparison with FC. The structure, which has three degrees of freedom all in a horizontal direction, is shown in Fig. 1. Fig. 1. The structural system The equations of motion of the system subjected to the ground acceleration x¨0 (see Fig. 2), with control force vector {F}, can be written as: [M ]{x¨}+ [C]{x˙}+ [K]{x} = {F} − [M ]{r}x¨0 (1) where, {x} = [x1 x2 x3] T , {F} = [−f 0 0]T , {r} = [1 1 1]T . f is the control force, the matrices [M ], [C] and [K], respectively representing the structural mass, damping and stiffness ones, are given as follow: [M ] =   m1 0 0 0 m2 0 0 0 m3   , [K] =   k1 + k2 −k2 0 −k2 k2 + k3 −k3 0 −k3 k3   , [C] = 0.1× [M ] + 0.003× [K]. (2) Active control of a three-story building using hedge-algebras-based fuzzy controller 81 Fig. 2. The ground acceleration x¨0, m/s 2 3. HEDGE ALGEBRAS In this section, the idea and basic formulas of HAs are summarized based on defi- nitions, theorems, propositions in [12-19]. By the term meaning 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 modelled by a poset (partially ordered set), a semantics - based order structure. Next, we explain how we can find out 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 ={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, 82 Bui Hai Le 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 observation: - Each primary term has a sign which expresses a semantic tendency. For instance, true has a tendency of "going up", called positive one, and it is denoted by c+, while false has a tendency of "going down", called negative one, denoted by c−. In general, we always have c+ ≥ c−, semantically. - Each hedge has also 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 a reverse effect and hence it is negative. Denote by H− the set of all negative hedges and by H+ the set of all positive ones of TRUTH. The term - set X can be considered as an abstract algebra AX = (X,G, C,H,≤), where G = {c−, c+}, C = {0 ,W , 1}, H = H+ ∪ H− and ≤ is a partially ordering relation on X . It is assumed that H− = {h−1, ..., h−q}, where h−1 < h−2 < ... < h−q, H+ = {h1, ..., hp}, where h1 < h2 < ... < hp. Fuzziness measure of vague terms and hedges of term-domains is defined as follow (Definition 2 - [19]): a fm : X → [0, 1] is said to be a fuzziness measure of terms in X if: fm(c−) + fm(c+) = 1 and ∑ h∈H fm(hu) = fm(u), for ∀u ∈ X. (3) For the constants 0 , W and 1 fm(0 ) = fm(W ) = fm(1 ) = 0. (4) For ∀x, y ∈ X, ∀h ∈ H , fm(hx) fm(x) = fm(hx) fm(y) (5) This proportion does not depend on specific elements, called fuzziness measure of the hedge h and denoted by µ(h). For each fuzziness measure fm on X , we have (Proposition 1 - [19]): fm(hx) = µ(h)fm(x), for every x ∈ X, (6) fm(c−) + fm(c+) = 1, (7) p∑ i=−q,i6=0 fm(hic) = fm(c), c ∈ {c −, c+}, (8) p∑ i=−q,i6=0 fm(hic) = fm(x), (9) −1∑ i=−q µ(hi) = α and p∑ i=1 µ(hi) = β where α, β > 0 and α + β = 1. (10) Active control of a three-story building using hedge-algebras-based fuzzy controller 83 A function Sign: X → {−1, 0, 1} is a mapping which is defined recursively as follows, for h, h′ ∈ H and c ∈ {c−, c+} (Definition 3 - [19]): Sign(c−) = −1, Sign (c+) = +1, (11) Sign (hc) = − Sign (c), if h is negative w.r.t. c, (12) Sign (hc) = + Sign (c), if h is positive w.r.t. c, (13) Sign (h′hx) = − Sign (hx), if h′hx 6= hx and h′ is negative w.r.t. h, (14) Sign (h′hx) = + Sign (hx), if h′hx 6= hx and h′is positive w.r.t. h, (15) Sign (h′hx) = 0 if h′hx = hx. (16) Let fm be a fuzziness measure on X . A semantically quantifying mapping (SQM) ϕ : X → [0, 1], which is induced by fm on X , is defined as follows (Definition 4 - [19]): ϕ(W ) = θ = fm(c−), ϕ(c−) = θ − αfm(c−) = βfm(c−), ϕ(c+) = θ + αfm(c+), (17) ϕ(hjx) = ϕ(x) + Sign(hjx){ j∑ i=Sign(j) fm(hix)− ω(hjx)fm(hjx)}, where j ∈ {j : −q ≤ j ≤ p and j 6= 0} = [−qp] and ω(hjx) = 1 2 [1 + Sign (hjx) Sign (hphjx)(β − α)] (18) It can be seen that the mapping ϕ is completely defined by (p+ q) free parameters: one parameter of the fuzziness measure of a primary term and (p+ q − 1) parameters of the fuzziness measure of hedges. To illustrate a close relationship between the meaning of terms and their fuzziness measure and the way to compute semantically quantifying mappings, we consider the following example. Example: Consider a hedge algebra AX =(X,G, C,H,≤), where G = {small, large}, C = {0 ,W , 1}, H− = {Little} = {h−1}, q = 1, H + = {Very} = {h1}, p = 1. We assume: θ = 0.5, α = 0.5. (19) It means that the semantically quantifying mapping of the neutral element and the sum of the fuzziness measure of the negative hedges are 0.5. Hence, - Using Equation (10) with q = 1, we have fuzziness measures of the hedges: µ(Little) = α = 0.5, µ(Very) = β = 1− α = 0.5. - Next, using Equations (17) and (7), we have fuzziness measures of the terms: fm(small) = θ = 0.5, fm(large) = 1− fm(small) = 0.5. 84 Bui Hai Le - Then, semantically quantifying mappings of the linguistic values are computed by using Equations (17) and (18) as follow: ϕ(W ) = θ = 0.5, ϕ(small) = θ − αfm(small) = 0.5− 0.5× 0.5 = 0.25, ϕ(Very small) = ϕ(small) + Sign (Very small)× (fm(Very small)− 0.5fm(Very small)) = 0.25 + (−1)× 0.5× 0.5× 0.5 = 0.125, ϕ(Little small) = ϕ(small) + Sign (Little small)× (fm(Little small)− 0.5fm(Little small)) = 0.25 + (+1)× 0.5× 0.5× 0.5 = 0.375 ϕ(large) = θ + αfm(large) = 0.5 + 0.5× 0.5 = 0.75, ϕ(Very large) = ϕ(large) + Sign (Very large)× (fm(Very large)− 0.5fm(Very large)) = 0.75 + (+1)× 0.5× 0.5× 0.5 = 0.875, ϕ(Little large) = ϕ(large) + Sign (Little large)× (fm(Little large)− 0.5fm(Little large)) = 0.75 + (−1)× 0.5× 0.5× 0.5 = 0.625, ϕ(Very Very small) = ϕ(Very small) + Sign (Very Very small)× (fm(Very Very small)− − 0.5fm(Very Very small)) = 0.125 + (−1)× 0.5× 0.5× 0.5× 0.5 = 0.0625, ϕ(Little Very small) = ϕ(Very small) + Sign (Little Very small)× (fm(Little Very small)− − 0.5fm(Little Very small)) = 0.125+ (+1)× 0.5× 0.5× 0.5× 0.5 = 0.1875, ϕ(Very Little small) = ϕ(Little small) + Sign (Very Little small)× (fm(Very Little small)− − 0.5fm(Very Little small)) = 0.375+ (−1)× 0.5× 0.5× 0.5× 0.5 = 0.3125, ϕ(Little Little small) = ϕ(Little small) + Sign (Little Little small)× (fm(Little Little small)− − 0.5fm(Little Little small)) = 0.375 + (+1)× 0.5× 0.5× 0.5× 0.5 = 0.4375, ϕ(Little Little large) = ϕ(Little large) + Sign (Little Little large)× (fm(Little Little large)− − 0.5fm(Little Little large)) = 0.625 + (−1)× 0.5× 0.5× 0.5× 0.5 = 0.5625, ϕ(Very Little large) = ϕ(Little large) + Sign (Very Little large)× (fm(Very Little large)− − 0.5fm(Very Little large)) = 0.625+ (+1)× 0.5× 0.5× 0.5× 0.5 = 0.6875, ϕ(Little Very large)ϕ(Very large) + Sign (Little Very large)× (fm(Little Very large)− − 0.5fm(Little Very large)) = 0.875+ (−1)× 0.5× 0.5× 0.5× 0.5 = 0.8125, ϕ(Very Very large) = ϕ(Very large) + Sign (Very Very large)× (fm(Very Very large)− − 0.5fm(Very Very large)) = 0.875 + (+1)× 0.5× 0.5× 0.5× 0.5 = 0.9375. The results obtained from above example will be used in the following section (see subsection 4.2). 4. FUZZY CONTROLLERS OF THE STRUCTURAL SYSTEM The fuzzy controllers are based on the closed-loop fuzzy system shown in Fig. 3. where, f is determined by above controllers, x1 and x˙1 are determined by Eqs. (1). The goal of controllers is to reduce displacement in the first storey, so as to reduce displacements in the structure. It is assumed that the universes of discourse of two state variables are Active control of a three-story building using hedge-algebras-based fuzzy controller 85 Fig. 3. Fuzzy controllers of the structural system −0.03 ≤ x1 ≤ 0.03(m),−0.3 ≤ x˙1 ≤ 0.3(m/s) and of the control force is −2 × 10 7 ≤ f ≤ 2× 107 (N). In the following parts of this section, establishing steps of the controllers will be presented. 4.1. Conventional fuzzy controller (FC) of the structure In this subsection, FC of the structure is established (establishing steps of a FC could refer in Mandal [20]) using Mamdani’s inference and centroid defuzzification method with fifteen control rules. The configuration of the FC is shown in Fig. 4. Fuzzy Rule Base (FAM table) Fuzzy Inference Engine (Mamdani Method) Fuzzifier Defuzzifier State variables Centroid Method Control force Fig. 4. The configuration of the FC 4.1.1. Fuzzifier Five membership functions for x1, three membership functions for x˙1 and seven membership functions for f in their intervals are established with values negative very big (NVB), negative big (NB), negative (N), zero (Z), positive (P), positive big (PB) and positive very big (PVB) as shown in Figs. 5 - 7, respectively [7]. Z PN 1 x 0.030 1 PBNB 0.03- Fig. 5. Membership functions for x1 Fig. 6. Membership functions for x˙1 86 Bui Hai Le Fig. 7. Membership functions for f 4.1.2. Fuzzy rule base The fuzzy associative memory table (FAM table) is established as shown in Table 1 [7]. Table 1. FAM table x1 x˙1 N Z P NB PVB PB P N PB P Z Z P Z N P Z N NB PB N NB NVB 4.2. Hedge-algebras-based fuzzy controller (HAFC) of the structure The linguistic labels in Table 1 have to be transformed into the new ones given in Ta- bles 2 and 3, that are suitable to describe linguistically reference domains of [0, 1] and can be modeled by suitable HAs. The HAs of x1 and x˙1 are AX = (X,G, C,H,≤), where X = x1 or x˙1, G = {small, large}, C = {0 ,W , 1}, H = {H −, H+} = {Little(L), Very(V )}, and the HAs of f is AF = (f, G, C,H,≤) with the same sets G,C and H as for x1 and x˙1, however, their terms describe different quantitative semantics based on different real reference domains. Table 2. Linguistic transformation for x1 and x˙1 NB N Z P PB small L small W L large large Table 3. Linguistic transformation for f NVB NB N Z P PB PVB V V small L V small V L small W V L Large L V large V V large The SQMs ϕ are determined and shown in Tables 4 and 5 (see section 3). The configuration of the HAFC is shown in Fig. 8. Active control of a three-story building using hedge-algebras-based fuzzy controller 87 Table 4. Parameters of SQMs for x1 and x˙1 small L small W L large large 0.25 0.375 0.5 0.625 0.75 Table 5. Parameters of SQMs for f V V small L V small V L small W V L Large L V large V V 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 force Desemantization Linear Interpolation Fig. 8. The configuration of the HAFC 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 reference do- main 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, e.g. the interval [a,b], of the linguistic variables and, therefore, we have to transform the in- terval [a,b] into [0,1] and, vice - versa. The transformation 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 [19]. Fig. 9. Transformation: x1 to x1s Fig. 10. Transformation: x˙1 to x˙1s The semantizations for each state variable are defined by the transformations given in Figs. 9 and 10. The semantization and desemantization for the control variable are defined by the transformation given in Fig. 11 (x1, x˙1 and f are replaced with x1s, x˙1s and fs when transforming from real domain to semantic one, respectively). 88 Bui Hai Le Fig. 11. Transformation: f to fs 4.2.2. HAs rule base We have the SAM (semantic associative memory) table based on FAM one (Table 1) with semantically quantifying mappings as shown in Table 6. Table 6. SAM table x1s x˙1s L small: 0.375 W : 0.5 L large: 0.625 small: 0.25 V V large: 0.9375 L V large: 0.8125 V L large: 0.6875 L small: 0.375 L V large: 0.8125 V L large: 0.6875 W : 0.5 W : 0.5 V L large: 0.6875 W : 0.5 V L small: 0.3125 L large: 0.625 W : 0.5 V L small: 0.3125 L V small: 0.1875 large: 0.75 V L small: 0.3125 L V small: 0.1875 V V small: 0.0625 4.2.3. HAs inference engine The Quantifying Semantic Curve describing the HAs inference method is established through the points that present the control rules occurring in Table 6 as shown in Fig. 12. Hence, fs is determined by linear interpolation through x1s and x˙1s. For example: if x1s = 0.7 and x˙1s = 0.5 (point X1s) then fs = 0.2375 (point Fs). Fig. 12. Quantifying semantic curve 5. RESULTS AND DISCUSSION The results include: time history of the storey displacements of the structure for both controlled and uncontrolled cases in order to compare control effect of FC and HAFC. Active control of a three-story building using hedge-algebras-based fuzzy controller 89 5.1. Results for the structure excited by the El Centro earthquake In this subsection, the following data will be used: m1 = m2 = m3 = m0 = 4 × 105(kg); k1 = k2 = k3 = k0 = 2× 10 8 (N/m). Fig. 13. Time responses of the first storey displacements - El Centro earthquake Figs 13 and 14 show the time responses of the first and top storey displacements, respectively. The maximum storey drifts are shown in Fig. 15. Comparison of the effec- tiveness of the two controllers is presented in Table 7. Table 7. Comparison of the effectiveness of the three controllers - El Centro earthquake Building Storey Maximum uncontrolled Controlled to uncontrolled displacement, m displacement ratio (reduction ratio) FC HAFC 1 0.048 0.302 0.204 2 0.084 0.558 0.502 3 0.107 0.607 0.586 5.2. Results for the structure excited by the Northridge earthquake In this subsection, the structural data will be changed as follow: m1 = m2 = m3 = m0+ 10%m0; k1 = k2 = k3 = k0 - 10%k0. Figs 16 and 17 show the time response of the top storey displacement and the max- imum storey drifts, respectively. Comparison of the effectiveness of the three controllers is presented in Table 8. 90 Bui Hai Le Fig. 14. Time responses of the top storey displacements - El Centro earthquake Fig. 15. The maximum storey drifts - El Centro earthquake Table 8. Comparison of the effectiveness of the three controllers - Northridge earthquake Building Storey Maximum uncontrolled Controlled to uncontrolled displacement, m displacement ratio (reduction ratio) FC HAFC 1 0.070 0.224 0.166 2 0.125 0.481 0.467 3 0.156 0.575 0.574 5.3. Results for the structure excited by the Kobe earthquake In this subsection, another structural parameter will be used as follow: m1 = m2 = m3 = m0 − 10%m0, k1 = k2 = k3 = k0 + 10%k0. Active control of a three-story building using hedge-algebras-based fuzzy controller 91 Fig. 16. Time responses of the top storey displacements - Northridge earthquake Fig. 17. The maximum storey drifts - Northridge earthquake Figs 18 and 19 show the time response of the top storey displacement and the max- imum storey drifts, respectively. Comparison of the effectiveness of the three controllers is presented in Table 9. Table 9. Comparison of the effectiveness of the three controllers - Kobe earthquake Building Storey Maximum uncontrolled Controlled to uncontrolled displacement, m displacement ratio (reduction ratio) FC HAFC 1 0.067 0.259 0.179 2 0.118 0.496 0.461 3 0.145 0.563 0.546 92 Bui Hai Le Fig. 18. Time responses of the top storey displacements - Kobe earthquake Fig. 19. The maximum storey drifts - Kobe earthquake 5.4. Discussion As shown in above - mentioned figures and tables, vibration amplitudes of the storeys are decreased successfully with FC and HAFC for the structure with the different structural data excited by three different earthquakes. It allows partially evaluating the stability and robustness capacities the proposed controller - HAFC. With the case of the El Centro earthquake, the reduction ratios (ratio of the con- trolled to uncontrolled response) for maximum displacement of the top floor of the struc- ture are about 61% and 59% for the FC and HAFC, respectively (Fig. 15 and Table 7). Therefore, it is seen that the HAFC is more effective than the FC in view of reducing the displacement response of the structure. Active control of a three-story building using hedge-algebras-based fuzzy controller 93 The effectiveness of two controllers in reducing the response of the structure due to other two earthquakes (Northridge and Kobe) is also shown for comparison in Figs. 17 and 19 and Tables 8 - 9. Almost the same behavior as for the El Centro earthquake can be observed for these earthquakes too. From Tables 2 - 5, it can be conceded that the semantic order of HAFC is always guaranteed. The semantization method (Figs. 9 - 11), the desemantization one (Fig. 11) and the inference one (Fig. 12) of HAFC are simpler than the fuzzification method (Figs. 5 - 7), the centroid defuzzification one and the inference one (Mamdani method) of FC, respectively. In order to describe three, five, seven,. . . , n linguistic labels by HAs, only two independent parameters (θ and α, 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. 6. CONCLUSIONS 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 allows lin- early establishing the Quantifying Semantic Curve through the points corresponding to the control rules. It is obtained that HAFC is simpler, effective and more understandable in comparison with FC. In fuzzy logic, many important concepts like fuzzy set, T - norm, S - norm, intersection, union, complement, composition. . . are used in approximate reasoning. This is an advantage for the process of flexible reasoning, but there are too many factors such as shape and number of membership functions, defuzzification method,. . . 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, approxi- mate reasoning based on hedge algebras, from the beginning, does not use fuzzy set concept and its precision is obviously not influenced by this concept. 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