General design method of hedge-algebras-based fuzzy controllers and an application for structural active control

Abstract

Hedge algebras (HAs) theory has been extensively developed with many effective applications in different fields of fuzzy theory such as data mining, fuzzy database system, fuzzy reasoning, and so on. We propose here a general design method of hedge-algebras-based fuzzy controllers (HACs) for a potential application in structural active control. The proposed method utilizes inherent order relationships between words in a word-domain to express the semantics of words instead of using fuzzy sets. This qualitative word semantics determines the so-called semantically quantifying mappings (SQMs), which are order-isomorphisms and map words of variable into numeric values, by only giving values of the fuzziness parameter of the variable. It is demonstrated that HAC and optimal HACs can be designed using this method for the active control of a structure, which is subjected to excitations of some different earthquakes. Simulation results show that the proposed method is simple, transparent and efficient in reducing the pick storey drift as well as the pick absolute acceleration of the structure. Moreover, computation time of HACs is significantly lower than that of ordinary fuzzy controllers.

Appendices

Appendix 1

In this section, the idea and basic formulas of HAs are summarized based on definitions, theorems, and propositions in [19-32, 34].

1.1 A1.1. Algebraic structure of term-domains

In the so-called analytic approach, the meaning of vague terms of linguistic variables is represented by fuzzy sets. In a certain aspect, this means that vague terms were understood as being not mathematical objects and, hence, fuzzy sets to represent their meaning have to be used. The motivation behind the algebraic approach to the semantics of terms comes from the observation that terms-domains of linguistic variables can be considered as partially ordered sets (posets), whose order relations are induced by the inherent meaning of vague terms. For instance, in virtue of vague terms of the linguistic variable VELOCITY in natural language, following relation could be found:

quick >medium >slow, Extremely_slow <Very_slow <slow, but that

Little_slow >Rather_slow >slow, and so on.

It leads to a viewpoint: term-domains can be modelled by a poset (partially ordered set), a semantics-based order structure. Next, the way to find out this structure could be explained as follows.

Consider the linguistic variable VELOCITY and let X be its term-set. Assume that its linguistic hedges used to express the VELOCITY are Extremely, Very, Approximately, Little, which for short are denoted by, respectively, E, V, A and L, and its primary terms are slow and quick. Then, X = {quick, V quick, E quick, EA quick, A quick, LA quick, Lquick, L slow, slow, A slow, V slow, E slow ...} ∪ {0, W, 1} is a term-domain of VELOCITY, where 0, W and 1 are specific constants called absolutely slow, neutral and absolutely quick, respectively. According to Zadeh, X can be produced by a context-free grammar and it may be finite or infinite. In general, its elements are of the form h _n ...h ₁ x, where h ₁,...,h _n ∈ H.

Now, an inherent semantics-based order structure of each term-domain is discovered. This structure can be axiomatized, based on semantic properties of term-domains that can be expressed in terms of the relation ≤ of VELOCITY, called semantics-based ordering relation. 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, quick has a tendency of “going up”, called positive one, and it is denoted by c ⁺, while slow has a tendency of “going down”, called negative one, denoted by c ⁻. In general, this always has c ⁺ ≥ 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 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 VELOCITY. If both hedges h and k belong together to H ⁺ or H ⁻, then it may happen that one of these hedges may modify the terms more strongly than the other. For example, it has L, A ∈ H ⁻ and L ≥ A, since L slow ≥ A slow ≥ slow. Note that I ≤ A and I ≤ V, where I is the identity on X. So, H ⁺ and H ⁻ become posets under the relation ≤, where the notation ≤ causes no confusion because H ⁺, H ⁻ and X are assumed to be disjoint.

• Further, each hedge may increase or decrease the semantic intensity of any other one. For instance, it has slow ≤ VA slow ≤ A slow, which shows that V decreases the intensity of A. If k decreases the semantic intensity of h, it is said to be negative w.r.t. h. Conversely, if k increases the semantic intensity of h, it is positive w.r.t. h.

• An important semantic property of hedges is the so-called heredity of hedges, which stems from the fact that for every h, the term hx inherits the meaning of x. This property may also be formulated in term of ≤: if the meaning of hx and kx can be expressed by hx ≤ kx, then it has h ^′ hx ≤ k ^′ kx, which says that h′ and k′ preserve the relative meaning of hx and kx and hence they cannot change the semantic-based order relationship between hx and kx. So, as a consequence, the inequality H(hx) ≤ H(kx) follows from hx ≤ kx, where H(y) = {h _n ...h ₁ y: ∀h ₁,...,h _n ∈ H, ∀n ∈ N}.

Based on these semantic properties of the primary terms and hedges, any term-set can be ordered and this is why an axiomatization of term-domains of linguistic variables could be introduced.

This observation allows assuming that there exists a semantics-based order relation ≤ defined on X. So, X can be considered as an abstract algebra AX = (X, G, C, H, ≤) is a set of generators, where G = {c ⁻, c ⁺}, C = {0, W, 1} is a set of constants, H = H ⁺∪ H ⁻ is a set of unary operations, X∖C = H(G) - the set generated from G - 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 ⁺ = {h ₁,...,h _p }, where h ₁ < h ₂ < ... < h _p

1.2 A1.2. Fuzziness measure of vague terms and hedges of term-domains

Fuzziness of vague terms is a concept which is very difficult to define because of its high level of abstraction. It is defined by taking full advantage of the shape of the fuzzy set which represents its meaning. Since fuzzy sets are very subjective and these definitions depend merely on the designed fuzzy sets, it seems to be not suitable to the concept of fuzziness which is a characteristic of the own linguistic terms more general than that defined by these definitions.

HAs theory provides an appropriate formal basis to define this concept. Since elements of H(x) still express a certain meaning stemming from x, the set H(x) as a model of the fuzziness of the term x is interpreted in [26, 27]. This observation allows using the “size” of the set H(x) to measure the fuzziness degree of x. Let the reference domain of the linguistic variable VELOCITY be a real interval [a, b]. Since, by a linear transformation, the domain [a, b] can be transformed into [0, 1], it could be assumed for normalization that the reference domains of all physical linguistic variables are the common unit interval [0, 1], called the semantic domain of linguistic variables.

An fm: X → [0, 1] is said to be a fuzziness measure of terms in X if:

$$\begin{array}{@{}rcl@{}} \textit{fm}(c^{-})+\textit{fm}(c^{+}) =1 \text{ and } {\sum}_{h\in H} \textit{ fm}(\textit{hu}) = \textit{fm}(u), \text{for}\ \forall u \in X; \end{array} $$

(9)

$$\begin{array}{@{}rcl@{}} \text{For the constants } \textit{\textbf{0}}, \textit{\textbf{W}}\ \text{and} \ \textit{\textbf{1}}, \textit{fm}(\textbf{{\textit{0}}}) =\textit{fm}(\textit{\textbf{W}}) =\textit{fm}(\textit{\textbf{1}}) =0; \end{array} $$

(10)

$$\begin{array}{@{}rcl@{}} \text{For}\ \forall x, y \in X, \forall h \in H, \frac{fm(hx)}{fm(x)}=\frac{fm(hy)}{fm(y)} \end{array} $$

(11)

This proportion (11) does not depend on specific elements, called the fuzziness measure of the hedge h and denoted by μ(h).

The condition (21) means that the primary terms and hedges under consideration are complete for modelling the semantics of the whole real interval of a physical variable. That is, except the primary terms and hedges under consideration, there are no more primary terms and hedges. The condition (10) is intuitively evident. The condition (11) seems also to be natural in the sense that applying a hedge h to different vague concepts, the relative modification effect of h is the same, i.e. this proportion does not depend on terms they apply to.

For each fuzziness measure fm on X, it has:

$$\begin{array}{@{}rcl@{}} \textit{fm}(\textit{hx}) = \mu (h)\textit{fm}(x), \text{for every x} \in X; \end{array} $$

(12)

$$\begin{array}{@{}rcl@{}} \textit{fm}(c^{\mathrm{-}}) + \textit{fm}(c^{\mathrm{+}}) = 1; \end{array} $$

(13)

$$\begin{array}{@{}rcl@{}} \sum\limits_{i=-q,i\ne 0}^{p} {fm(h_{i} c)=fm(c)} ,c\in \{ c^{\mathrm{-}},c^{\mathrm{+}}\}; \end{array} $$

(14)

$$ \sum\limits_{i=-q,i\ne 0}^{p} {fm(h_{i} x)=fm(x)} ; $$

(15)

$$\begin{array}{@{}rcl@{}} &&\!\!\!\!\!\sum\limits_{i=-q}^{-1} {\mu (h_{i} )=\alpha } \textit{ and }\sum\limits_{i=1}^{p} {\mu (h_{i} )=\beta } \textit{ where } \alpha , \beta > 0 \\ &&\!\!\!\!\!\textit{and } \alpha + \beta = 1. \end{array} $$

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1.3 A1.3. Quantification of term-domains of a linguistic variable

The relative sign between hedges is recalled first. Based on the action effect of hedges, one can discover that one hedge may have a relative sign with respect to another: h is positive (or, negative) with respect to k, denoted by sign (h, k) = +1 (or, sign (h, k) = −1), if it strengthens (or, weakens) the effect tendency of k. Then, every term x, x = h _m ...h ₁ c, which is not a fixed point, has also a sign defined by

$$ \textit{sign}\left( x \right)=\textit{sign}\left( {h_{m} ,h_{m-1} } \right)\mathellipsis \textit{sign}\left( {h_{2} ,h_{1} } \right)\textit{sign}\left( {h_{1} } \right)\textit{sign}\left( c \right) $$

(17)

The meaning of the sign of terms is the following statement:

$$\begin{array}{@{}rcl@{}} \textit{sign}\left( {hx} \right)=+1\Rightarrow hx\ge x \text{ and } \textit{sign}\left( {hx} \right)=-1\Rightarrow hx\le x \end{array} $$

(18)

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:

$$\begin{array}{@{}rcl@{}} \varphi (\textit{\textbf{W}})&=& \theta = \textit{fm}(c^{-}), \varphi (c^{-}) = \theta - \alpha \textit{fm}(c^{-})\\ &=& \beta \textit{fm}(c^{-}), \varphi (c^{+}) = \theta + \alpha \textit{fm}(c^{+}) \end{array} $$

(19)

$$\begin{array}{@{}rcl@{}} \varphi (h_{j}x) \!\! &=&\!\! \varphi (x) + \text{ Sign}(h_{j}x)\left\{\sum\limits_{i=\text{ Sign}(j)}^{j}\!\!\!\!\! fm(h_{i} x)-\omega (h_{j} x)fm(h_{j} x)\! \right\}\!, \end{array} $$

where

$$j \in \{j: -q\le j\le p\ {\&} j\ne \textit{0}\} = [-q^{\wedge}p]$$

and

$$ \omega (h_{j}x) = \frac{1}{2}\left[1 + \text{Sign}(h_{j}x)\text{Sign}(h_{p}h_{j}x)(\beta -\alpha )\right]. $$

(20)

It can be seen that the mapping φ is completely defined by (p + q) independent parameters: one parameter of the fuzziness measure of a primary term, and (p + q–1) parameters of the fuzziness measure of hedges.

Appendix 2

In this section, a simple FC described in [18] is presented.

Considering linearized differential equation describing dynamics of system given by following equation:

$$ \ddot{{\theta }}=\theta -u $$

(21)

Where, 𝜃 and u are state and control variables of the system, respectively. The system is subjected to initial condition: 𝜃(0) and $\dot {{\theta }}(0)$ are not simultaneously equal to zero. The main aim of the controller is to regulate the system into its stable state (𝜃 → 0 and $\dot {{\theta }} \quad \to$ 0).

It is assumed that the universe of discourse for state and control variables to be −a ₀ ≤ 𝜃 ≤ a ₀, $-b_{0} \le \dot {{\theta }}\le b_{0}$ and – c ₀ ≤ u ≤ c ₀, and they are fuzzified with linguistic values Negative Big (NB), Negative (N), Zero (Z), Positive (P), Positive Big (PB) as shown in Fig. 34.

Fig. 34

Fuzzification of variables

Rule base (FAM table) of the system is presented in Table 1. Mamdani and centre gravity are used as inference engine defuzzifier methods for the system, respectively.

Appendix 3

In this section, three typical earthquakes, which are the 1940 El Centro earthquake with Peak Ground Acceleration (PGA) 0.35 g, which can be found at http://www.vibrationdata.com/elcentro.htm, the 1994 Northridge one with PGA 0.57g and the 1979 Imperial Valley one with PGA 0.35g, see http://peer.berkeley.edu/smcat/search.html, are presented as shown in Figs. 35–37.

Fig. 35

El Centro earthquake excitation input to the structure

Fig. 36

Northridge earthquake excitation input to the structure

Fig. 37

Imperial Valley earthquake excitation input to the structure

Appendix 4

In this section, the construction of the fuzzy controller (FC) is realized by the design of the following factors which are the same as examined in [8], where, i = 2 or 15:

• (i) Fuzzifier: The vague terms of the both variables x ₂ and x ₁₅ are NB, NS, Z, PS and PB, of the both $\dot {{x}}_{2}$ and $\dot {{x}}_{15}$ are NS, Z and PS and of u ₂ and u ₁₅ are NB, NM, NS, Z, PS, PM and PB. The memberships of these terms are designed as depicted in Figs. 38–40, which are the same as examined in [8].

• (ii) Rule bases: The construction of the rule bases is also the same as examined in [8] and shown in Table 14.

• (iii) Inference engine: Mamdani method was used [8] as shown in Fig. 41.

• (iv) Defuzzifier is the centre gravity method, which was chosen also in [8].

Fig. 38

Membership functions for x _i

Fig. 39

Membership functions for $\dot {{x}}_{i}$

Fig. 40

Membership functions for u _i

Fig. 41

An example of Mamdani inference engine of u ₂ when [x ₂, $\dot {{x}}_{2}$] = [-0.04, 0.25]

Table 14 Rule base table for the actuator on the first and the fifteenth storeys [8]