# Hedge-Algebra-Based Phase-Locked Loop for Distorted Utility Conditions

Authors: Dinko Vukadinović; Tien Duy Nguyen; Cat Ho Nguyen; Nhu Lan Vu; Mateo Bašić; Ivan Grgić
Journal: Journal of Control Science and Engineering
 (2019)

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## Abstract

This paper presents the first application of the hedge-algebra theory in the field of grid synchronization. For this purpose, an optimized hedge-algebra controller (HAC) is developed and incorporated within the three-phase phase-locked loop (PLL) with moving average filters (MAFs) inside its feedback loop. Optimized fuzziness parameters and linguistic rule base of the HAC are obtained by a genetic algorithm using the integral of absolute error as the performance index during optimization. Calculated optimal parameter values of the HAC depend on the most frequently occurring disturbance in the electric grid. Two different PLL structures are proposed, depending on the types of disturbances occurring in the electric grid. The first structure is the conventional synchronous reference frame PLL with the nonadaptive MAF (i.e., MAF without order adjustment), but with the PI/PID controller in the phase loop replaced by the developed HAC. Such PLL structure is suitable for all analyzed disturbance types, expect for step-changes in the grid frequency. The second PLL structure introduces the adaptive MAF (i.e., MAF with order adjustment) and a new feedback signal in the output stage of the controller to achieve zero steady-state error in the case of step-changes in the grid frequency. The disturbance rejection capability of the two developed PLLs with the HAC (HAC-PLLs) is tested separately and compared experimentally with the PID- and fuzzy-controller-based PLLs.


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## Body

## 1. Introduction

The synchronization of power electronics converters with a distributive electric grid presents an intensive research area in the world. In three-phase systems, phase-locked loops (PLLs) present one of the most used synchronization methods. The conventional synchronous reference frame PLL (SRF-PLL) is shown in Figure1.Figure 1
Block diagram of the conventional SRF-PLL.The main parts of the conventional SRF-PLL are as follows: the phase detector, the low-pass filter, and the voltage controlled oscillator. The phase detector transforms the phase voltagesua,ub,uc to synchronous direct and quadrature voltage componentsud anduq, respectively. The direct voltage componentud is equal to the phase voltages amplitudeU for nondistorted phase voltages. Usually, the filter in Figure 1 is realized as the PI or PID controller, with the zero-voltage reference. Hence, the term controller is used below. The SRF-PLL outputs the estimated phase angle of the three-phase system ϑ^. This type of PLL shows its disadvantages in the case of an unbalanced phase voltage system, i.e., when there are negative- and zero-sequence components along with a positive-sequence component. In this case, a disturbance component whose frequency is equal to twice the fundamental grid frequency (e.g., 100 Hz when the nominal value is 50 Hz) appears in the direct and quadrature voltage componentsud anduq. Consequently, a power electronics inverter, which performs DC to AC inversion, is not properly synchronized with the electric grid.To deal with unbalanced phase voltage systems, many PLLs with added filters have been proposed [1]. The PLLs with filters in the phase loop can be classified as follows: moving average filter-based PLL (MAF-PLL) [2–4], cascaded delayed signal-cancellation (CDSC) operator [5], notch filter-based PLL [2, 6], variable sampling period filter-based PLL [7], multiple reference frame-based PLL [8], and in-loop complex-coefficient filter-based PLL [9].On the other hand, the PLLs with filters outside of the phase loop are often called PLLs with a prefiltering stage. This prefiltering stage significantly attenuates disturbance components before the phase loop input. Some of the PLLs with the prefiltering stage are the PLL with a multiple complex-coefficient filter [10], the PLL with the recursive discrete Fourier transform [11], the PLL with the sliding Goertzel discrete Fourier transform [12], the PLL with the moving average filter-based (MAF) prefiltering [13], the PLL with the dual second-order generalized integrators [14], and the delay signal cancelation-based PLL [5, 15].Scientific papers dealing with the PLLs in which artificial intelligence is applied are extremely rare. In review paper [1], there is no single example of artificial intelligence use in PLLs. In [16–18] the fuzzy logic theory is applied in the PLL. In [16], the particle swarm optimization algorithm is used to tune values of the fuzzy controller (FC) membership functions. In [17], the authors propose application of the same algorithm for both tuning of the FC membership functions and finding the optimal values of the low-pass filter. It is very important to point out that, in these papers, there is a nonzero phase error when unbalanced grid conditions appear. In [18], a FC in the three-phase PLL is applied, but only with the phase voltages containing harmonics or a DC component. The phase error is not shown in this paper.Generally, advantages that FCs have in nonlinear control systems over the classical PI/PID controller are numerous. For example, FCs do not require knowledge of a detailed mathematical model of the control system and also they are expressed in linguistic terms used in natural language. In addition, FCs have the capability of handling noisy signals and usually lead to better results compared to the classical PI/PID controller in terms of the control system performance [19].Hedge-algebra theory was first developed in 1990 [20] to model the order-based semantics of the terms in term-domains of linguistic variables. Hedge algebras form a formalism to immediately handle linguistic words and linguistic rule bases (LRBs) and their computational (compt.-) semantics instead of their fuzzy sets or fuzzy set expressions representing their inherent semantics. So, this theory is based on mapping of a few fuzziness parameters of each linguistic variable instead of using fuzzy sets. In [21], the authors point out that the structure of the hedge-algebra-based controller (HAC) is like the conventional FC, but it is simpler and more convenient. As a result, the HAC was applied in optimal fuzzy control of an inverted pendulum [22], in active vibration control of building structures subjected to seismic excitations [23–26], and in voltage control of a self-excited induction generator [27]. However, it was not yet applied in the field of grid synchronization.This paper introduces three-phase PLLs with the MAF inside the phase loop and the HAC used as a controller. The first proposed PLL is based on the MAF-PLL reported in [3], but with the PI/PID controller replaced by the HAC. For this purpose, the MAF with no order adjustment (nonadaptive MAF) is proposed. This PLL is suitable for all analyzed disturbance types, expect for step-changes in the grid frequency. The second PLL structure introduces the adaptive MAF (i.e., MAF with order adjustment) and a new feedback signal in the output stage of the controller to achieve zero steady-state error in the componentuq, in the case of step-changes in the grid frequency. For this PLL type, an application of the MAF is proposed with its order adjusted according to the linear interpolation method [4]. Both types of the PLLs are briefly named HAC-PLL. In addition, it is proposed that both the structure and parameters of the HAC-PLL are determined depending on the most frequently occurring disturbance in the electric grid. This approach requires that the power quality monitoring is carried out over one week [28]. Optimized fuzziness parameters and linguistic rule base of the HAC are obtained for each considered disturbance by means of a genetic algorithm. The integral of absolute error (IAE) is used as a performance index for finding these parameters and rules.In this paper, the following disturbances are considered: unbalanced voltage sag with/without harmonics distortion of the phase voltages, unbalanced low-frequency transients in the phase voltages, and step-changes in the grid frequency. It is important to point out that the HAC-PLL with parameters and linguistic rules optimized for a specific disturbance works stable with other analyzed disturbances as well. Finally, a comparison is shown between the HAC-PLL and two other MAF-PLLs: with the PID controller and with the FC.


## 2. Phase-Locked Loops with the Moving Average Filter

A MAF belongs to the group of finite impulse response filters which eliminate all harmonics whose frequency is multiplier of the reciprocal value of its window width. The MAFs are suitable for embedded computer systems because of their low computational requirements. On the other hand, their filtering capability decreases when the ratio of the window width and the sampling period is not an integer. In that case, the window width needs to be varied and adapted to the circumstances. Such MAFs are called hereinafterthe adaptive MAFs, whereasMAF only refers to the nonadaptive type, unless otherwise indicated.Figure2 shows the PLL with the nonadaptive MAF and a controller inside the phase loop [3]. The controller can be of a PID, fuzzy or HAC type, as described later.Figure 2
PLL with the nonadaptive MAF and a controller.The MAF with the input signalx and the output signal x- can be described by the following equations in the continuous-time and discrete-time domain, respectively: (1)x-t=1Tw∫t-TwTwxtdt(2)x-k=1N∑n=0N-1xk-nwhereTw= NTs is the MAF window width,Ts is the sampling time, andN is the number of samples within the window width.Using (1) and (2), the transfer functions in thes-domain andz-domain, respectively, are defined as(3)Gs=x-sxs=1-e-TwsTws(4)Gz=x-zxz=1N∑n=0N-1z-N=1N1-z-N1-z-1.By replacings with jω in (3), the magnitude and phase characteristics in thes-domain are given as(5)Gjω=sin⁡ωTw/2ωTw/2e-jωTw/2Similarly, by replacingz with ejωTs in (5), the magnitude and phase characteristics in thez-domain are given as(6)GejωTs=sin⁡ωNTs/2Nsin⁡ωTs/2e-jωN-1Ts/2Equation (5) shows that the MAF input signals whose angular frequency is a multiple of the reciprocal window width 1/Tw (f=n/Tw,n= 1, 2, 3,…) will be completely blocked. So, it is often said that the MAF acts as an ideal low-pass filter. In addition, (6) takes on the form of (5) if the sampling timeTs⟶ 0.As mentioned before, the analyzed disturbances in the phase voltage quadrature componentuq include a harmonic component whose frequency is 100 Hz when the grid frequency is 50 Hz. So, it is chosenTw = 0.01 s to completely remove the 100 Hz harmonic and all its multiples fromuq. This selection for the MAF window width is recommended when the grid harmonic content is unknown and the DC-offset may be present in the phase voltages. To determine the sampling timeTs, the fastest time-varying disturbance needs to be considered. In this paper, low-frequency transients in the phase voltages are considered, with an oscillation frequency less than 5 kHz, as the fastest time-varying disturbance [29]. The sampling time of 100 μs is, in this sense, sufficient, whereas the number of samples within the window widthN is equal to 100.Figure3 shows the Bode plots of the MAF obtained using (5) and withTw = 0.01 s andN= 100.Figure 3
MAF Bode plots obtained withTw=0.01 s andN =100.Equation (4) presents the nonadaptive MAF because its parameterN is constant. Such MAF is suitable when no significant frequency variation of the electric grid is expected. However, the problem arises when the grid frequency varies. In such cases, the nonadaptive MAF does not completely block the harmonic whose frequency is equal to twice the changed grid frequency. Therefore, some of the MAF variants with the variable parameterN should be applied [3]. The best filtering capabilities are achieved by using MAFs with the linear interpolation method [3, 4]. Using this approach, the output signal of the MAF in the kth sample instant x-(k) is defined as(7)x-k=TsTw∑i=0Nf-1xk-i+α1-αxk-Nf+1+αxk-NfwhereNf is the greatest integer less than or equal toTw/Ts and α =Tw/Ts- Nf.The parameterNf is calculated using thefloor function as follows:(8)Nf=floor1Ts·2f^gf=floorπTs·ω^gfwhere ω^gf is the estimated angular grid frequency, filtered by a low-pass filter described in Section 2.2.By using (7), the MAF becomes adaptive with regard to the grid frequency (i.e., the window width is variable). In this way, better filtering performance is achieved, especially for low-frequency components.


### 2.1. MAF-Based PLL with the PID Controller

The first considered PLL is the one with the nonadaptive MAF and PID controller inside the phase loop shown in Figure2. In this paper, this PLL is used for performance evaluation of the developed PLL, which is described in Section 3.The PID controller is written in the following form [3]:(9)Gs=kp1+τisτis1+τds1+βτdswherekp is the proportional gain, τi is the integral time constant, and τd is the derivative time constant.The parameterβ = 0.1 prevents the derivation action for the frequencies higher than 1/(βτd). The PID parameters which ensure the phase margin of 45°, the damping factor ζ = 0.707, and the natural frequency ωn = 2π20 rad/s can be obtained as(10)τi=2ζωn,τd=Tw2,kp=2ζωnU1+where U1+is the amplitude of the positive-sequence component of the phase voltage.More details about the described MAF-based PLL with the PID controller can be found in [3].


### 2.2. MAF-Based PLL for Rapid Changes in Grid Frequency

In the case of rapid changes in grid frequency, the FC within the PLL structure as in Figure2 is unable to achieve zero steady-state error in the componentuq. To deal with this problem, the new adaptive-MAF-based PLL structure with the FC or the HAC is proposed, as shown in Figure 4. In comparison with Figure 2, here it was necessary to add a new feedback signal in the output stage of the controller in order to obtain stable operation. Also, by using the adaptive MAF instead of the nonadaptive MAF it is possible to obtain lower IAE values. First, the FC is considered for the PLL in Figure 4, whereas the HAC is dealt with in more detail in Section 3.Figure 4
PLL for rapid changes in grid frequency with the fuzzy controller or the HAC and the adaptive MAF.In this study, the Mamdani fuzzy inference system is selected for the FC because it is more widely accepted and allows a more intuitive approach to design in comparison to the Sugeno type. Figure5 shows the membership functions of the Mamdani-type FC. These functions were obtained by the trial-and-error, whereas the IAE was used as a performance index. In Figure 5, N, P, and ZE stand fornegative,positive, andzero, respectively, whereas L, V, and VV stand forlittle,very, andvery very, respectively. Hence, for example, LN is interpreted aslittle negative. In addition,e is the error signal calculated as the difference between the voltageuq and its zero reference,ce is the change in error calculated as the difference between two consecutive values of the error signale, and cΔωg is the adjustment signal for the change in the estimated grid frequency. Variablese andce represent the inputs to the FC, whereas cΔωg represents the output variable in per unit notation. Such notation, however, requires scaling of the input and output signals. The scaling factors were selected according to the analyzed disturbance. For voltage sags (with and without harmonics) the scaling factors equal to 0.1, 3.5, and 85 were chosen fore,ce, and cΔωg, respectively, again using the IAE as the performance index. These factors are equal to 0.25, 2.8, and 0.1 for an unbalanced low-frequency transient, and, finally, they are equal to 0.093, 4, and 105 for rapid changes in grid frequency.Figure 5
Membership functions of the developed Mamdani-type FC.
(a) (b)The FC has 25 fuzzy rules, shown in Table1, whereas the centroid method was selected for defuzzification.Table 1
Rule base for FC.
ce VN LN ZE LP VP VN VVN VVN VN N ZE e LN VVN VN N ZE P ZE VN N ZE P VP LP N ZE P VP VVP VP ZE P VP VVP VVPThe low-pass filter in Figure4 outputs the feedback signal ω^gf, which presents the estimated grid frequency passed through the second-order filter. This filter has the damping factor ζ = 0.9 and the natural frequency ωn = 2π35 rad/s. The signal ω^gf, in comparison with the estimated grid frequency ω^g, has slower dynamics and it helps the FC to reach its steady-state setpoint. In addition, the parameterNf, which is the input parameter of the adaptive MAF, is calculated according to (8), whereby the estimated angular grid frequency is replaced by its filtered value.The rule base summarized in Table1 and the membership functions shown in Figure 5 also apply to the FC in Figure 4.


## 2.1. MAF-Based PLL with the PID Controller

The first considered PLL is the one with the nonadaptive MAF and PID controller inside the phase loop shown in Figure2. In this paper, this PLL is used for performance evaluation of the developed PLL, which is described in Section 3.The PID controller is written in the following form [3]:(9)Gs=kp1+τisτis1+τds1+βτdswherekp is the proportional gain, τi is the integral time constant, and τd is the derivative time constant.The parameterβ = 0.1 prevents the derivation action for the frequencies higher than 1/(βτd). The PID parameters which ensure the phase margin of 45°, the damping factor ζ = 0.707, and the natural frequency ωn = 2π20 rad/s can be obtained as(10)τi=2ζωn,τd=Tw2,kp=2ζωnU1+where U1+is the amplitude of the positive-sequence component of the phase voltage.More details about the described MAF-based PLL with the PID controller can be found in [3].


## 2.2. MAF-Based PLL for Rapid Changes in Grid Frequency

In the case of rapid changes in grid frequency, the FC within the PLL structure as in Figure2 is unable to achieve zero steady-state error in the componentuq. To deal with this problem, the new adaptive-MAF-based PLL structure with the FC or the HAC is proposed, as shown in Figure 4. In comparison with Figure 2, here it was necessary to add a new feedback signal in the output stage of the controller in order to obtain stable operation. Also, by using the adaptive MAF instead of the nonadaptive MAF it is possible to obtain lower IAE values. First, the FC is considered for the PLL in Figure 4, whereas the HAC is dealt with in more detail in Section 3.Figure 4
PLL for rapid changes in grid frequency with the fuzzy controller or the HAC and the adaptive MAF.In this study, the Mamdani fuzzy inference system is selected for the FC because it is more widely accepted and allows a more intuitive approach to design in comparison to the Sugeno type. Figure5 shows the membership functions of the Mamdani-type FC. These functions were obtained by the trial-and-error, whereas the IAE was used as a performance index. In Figure 5, N, P, and ZE stand fornegative,positive, andzero, respectively, whereas L, V, and VV stand forlittle,very, andvery very, respectively. Hence, for example, LN is interpreted aslittle negative. In addition,e is the error signal calculated as the difference between the voltageuq and its zero reference,ce is the change in error calculated as the difference between two consecutive values of the error signale, and cΔωg is the adjustment signal for the change in the estimated grid frequency. Variablese andce represent the inputs to the FC, whereas cΔωg represents the output variable in per unit notation. Such notation, however, requires scaling of the input and output signals. The scaling factors were selected according to the analyzed disturbance. For voltage sags (with and without harmonics) the scaling factors equal to 0.1, 3.5, and 85 were chosen fore,ce, and cΔωg, respectively, again using the IAE as the performance index. These factors are equal to 0.25, 2.8, and 0.1 for an unbalanced low-frequency transient, and, finally, they are equal to 0.093, 4, and 105 for rapid changes in grid frequency.Figure 5
Membership functions of the developed Mamdani-type FC.
(a) (b)The FC has 25 fuzzy rules, shown in Table1, whereas the centroid method was selected for defuzzification.Table 1
Rule base for FC.
ce VN LN ZE LP VP VN VVN VVN VN N ZE e LN VVN VN N ZE P ZE VN N ZE P VP LP N ZE P VP VVP VP ZE P VP VVP VVPThe low-pass filter in Figure4 outputs the feedback signal ω^gf, which presents the estimated grid frequency passed through the second-order filter. This filter has the damping factor ζ = 0.9 and the natural frequency ωn = 2π35 rad/s. The signal ω^gf, in comparison with the estimated grid frequency ω^g, has slower dynamics and it helps the FC to reach its steady-state setpoint. In addition, the parameterNf, which is the input parameter of the adaptive MAF, is calculated according to (8), whereby the estimated angular grid frequency is replaced by its filtered value.The rule base summarized in Table1 and the membership functions shown in Figure 5 also apply to the FC in Figure 4.


## 3. Hedge-Algebra-Based Phase-Locked Loop with Genetic Optimized LRBs

In this study, the LRBs are converted from the respective fuzzy (linguistic label) rule bases (FRBs) established by the experience of human domain experts. Though these domain-experts’ FRBs are very important, they are not always appropriate for specific application problems, particularly for complex problems. The construction of the FC component of the MAF-based PLL shown in Figures2 and 4 is considered as such a problem.A genetic method to design an optimized HAC is developed in order to replace the FC in the MAF-based PLL shown in Figures2 and 4. This one is called the optimized HAC (opHAC) developed based on the hedge-algebra-formalism (HA-formalism). It is important and novel that the main component of the HAC, its LRB, is also to be optimized. However, as an LRB is genetically established by computer instead of human experts, this problem must only be solved in a formalism that is able to immediately handle linguistic words with their own real-world-semantics. Another distinguished advantage of the HA-formalism is its ability to ensure the real-world-semantics (RWS-) interpretability of the designed HACs, which means that the expected behaviors of the designed HACs are actually the same as described by their designers in terms of their LRBs [29]. It is also required that its approximate reasoning method must be RWS-interpretable as well. So, it can be seen that the RWS-interpretability of a designed HAC is very essential and important to ensure the high performance of not only the designed HACs but also the designed fuzzy systems. Also, it is for the first time applied to design RWS-interpretable HACs to solve application problems. Finally, their LRBs, with their own semantics, are for the first time genetically constructed.


### 3.1. The Role of Human Language in Designing RWS-Interpretable Fuzzy Systems

In the fuzzy set framework, words assigned to the fuzzy sets are considered as linguistic labels and there is no formal basis to use words and linguistic sentences of human natural language with their owninherent semantics. So, there exists a gap between the fuzzy representations representing the inherent semantics of their respective words and of their respective LRBs. The main aim of the fuzzy control is to utilize human capability in handling words of the natural language. If there was no formal bridge established across the gap, one would not be able to indeed make use of human expert linguistic knowledge. The paper [30] introduces an RWS-approach to the interpretability of fuzzy systems in which the RWS-interpretability of the human language is utilized to design RWS-interpretable fuzzy systems. Assuming the RWS-interpretability of the natural language, it implies that the LRB equipped for a fuzzy system is able to properly describe the reality with which the system interfaces. Based on this feature, the ultimate goal of the RWS-approach is to establish a methodology to design RWS-interpretable fuzzy systems, whose RWS-interpretability means that their behavior is ensured to be compatible with the expected one described by the given LRB.The main specific features of the RWS-approach examined in [30] are the following:(i)
It is able to utilize the RWS-interpretability of the human language. In the fuzzy control, they usually describemonotonic dependences between two RW-variables.(ii)
Its formalism is able to immediately handle words and sentences of human expert languageswith their own RW-semantics as well as their compt.-semantics.These features are very crucial for any method to design a fuzzy (linguistic) controller to ensure its expected behavior when it interacts with its RW-counterpart.


### 3.2. Hedge Algebras of Variables, Their Quantification, and Their RWS-Interpretability

In order to ensure the RWS-interpretability of the designed HAC-PLLs, it is required to utilize the RWS-interpretability of human language to establish a mathematical formalization of word-domains for developing the HAC-PLLs. To do this, it is observed that the word-domain of every (linguistic) variable possesses the following two specific features:(i)
The word-domain can be generated from its atomic words, e.g., ‘small’ and ‘large’ of the current intensity variable, using linguistic hedges, e.g., ‘very’ or ‘extremely’. Then, every wordx is of the formhn…h1c, wherec∈small,large and hj’s are (linguistic) hedges and, hence, they can be considered as operations of an algebra.(ii)
There exists a linear or partial semantic order relation on this word-domain induced by theinherent semantics of its words.Thus, hedges with their own order-based semantics defined in their word-domain play an essential role in discovering order-based structures of the word-domains of variables. However, a large number of studies consider words only as linguistic labels of the constructed fuzzy sets, whereas there are relatively few studies thatimmediately deal with linguistic hedges with their own modification functionality, called hedge operators. For instance, such hedge operators are examined and applied in fuzzy logic and approximate reasoning [31, 32], in contrast intensification [33], and in fuzzy control [34]. In particular, it is actively examined in formal concept analysis of a dataset based on studying formal concept lattice with hedges, called truth-stressing hedges [35–38]. In these studies, hedges are utilized to modify concept-forming operators to control the number of conceptual clusters extracted from a given dataset represented by formal concepts of a concept lattice of the dataset. However, the inherent semantics of words, which may contain hedge occurrences, is still not taken into consideration and, therefore, ‘hedge operators’ were developed in these studies independently from the semantics of words and hedges, which in practice must be determined in the context of other words of their variable.Starting from the two aforementioned specific features of word-domains, HAs were proposed and developed in turn in [20, 30, 39, 40]. As argued in [30], all theories developed in an axiomatic way, whose axioms actually represent structural semantics of their respective RW-counterparts (e.g., math-theories or theoretical physics theories), are shown to be RWS-interpretable; in this section, we briefly explain that the theories of HAs and their quantification, which form the above formalism, are also RWS-interpretable.


#### 3.2.1. A Description of an Axiomatization of Word-Domains and Their Quantification

HAs introduced in [20, 30, 39, 40] establish an algebraic approach based on a formalization of their word-domains of variables in an axiomatic way to immediately handle the inherent order-based semantics of the words. It is natural that there are comparability elements in the human language, the presence of which is required by the human decision-making activities in which the comparability of decision criteria values is crucial. So, there exists a semantic order relation on every variable word-domain. For instance, considering the RW-variable IRW of the current intensity, this RW-variable may be examined on the standpoint either of a numeric formalism, in which its respective numeric variable is denoted by IN, or of a linguistic formalism, in which its respective linguistic variable is denoted by IL. For every hedgeh of IL, say ‘very’ (V), the pair of the words ‘very large’ and ‘large’ or the pair of the words ‘very small’ and ‘small’ of IL are always comparable. It is essential that the order of these words iscompatible with the order of the numeric values of the current intensity assigned to their respective words and, hence, the word-domain of IL must also be linearly ordered.For general case, let us consider an arbitrary linguistic variable denoted byX without its subscriptL, for simplicity. As analyzed above, the word-domain ofX becomes an order-based structure, denoted byAX = (Dom(X),G,C,H, ≤), whereG = c-,c+ – considered as the generators ofAX, wherec- <c+,c+ is called thepositive atomic word andc- – thenegative one;C is the set of constantsC = 0,W,1, which are, respectively, the least, neutral, and greatest words of Dom(X);H =H- ∪ H+ is the set of the hedges ofX, whereH- (orH+) is the set of the negative (or positive) hedges; e.g.,H- = R (Rather), L (Little), andH+ = M (More), V (Very); and ≤ is the semantic order relation. Syntactically, Dom(X) is the set of all the words ofX of the formx =hn… h1c,c∈G, whose length is denoted by x; i.e., x =n + 1. Many properties of the inherent semantics of the words ofX can be modeled in the order-based structure of Dom(X) and, more importantly, as the human language is RWS-interpretable, they are RWS-interpretable as well. To formalize the word-domains in an axiomatic way, it is necessary to discover few properties of the order-based semantics of words and hedges that can easily be verified based on their natural meaning so that they can be taken as axioms and their math-models and the remaining properties can be derived from them. Restricted to the purpose of the study, a very short description of this formalization is given, including the necessary knowledge for the study. More details can be found in [20, 30, 39, 40].(i) Words and hedges ofX have their own “algebraic” signs determined in this order-based structure as follows.(a)
For atom words, sign (c–) = –1 and sign(c+) = +1; for any hedgeh, sign (h) = +1 (or, sign (h) = –1) ifh∈H+ (or,h∈H–) which is identified byhc+ ≥c+ (or,hc+ ≤c+).(b)
Therelative sign ofh with respect tok, denoted by sign (h,k), is computed by sign (h,k) = –1 iff (∃c∈G)(kc ≤c ⇒kc ≤hkc) and sign (h,k) = +1 iff (∃c∈G)(kc ≤c ⇒hkc ≤kc). For illustration, the validity of Table 2 can be checked.(c)
Now, the sign ofx =hn …h1c is computed by sign (x) = sign (hn,hn-1) … sign (h1) sign (c) ∈–1, +1. There is an important fact that sign (hx) =–1 (or, +1) ⇔ hx ≤x (or,x ≤hx). For example, for L, R, M, and V stand forlittle,rather,more, andvery, respectively, it may be verified, for instance, that sign (VR_small) = sign (V,R)sign (R) × sign (small) = –1, which results in VR_small ≤ R_small.Table 2
The relative sign of a hedge in a row with respect to a hedge in a column.
V M R L V + + - +  M + + - +  R - - + -  L - - + -(ii) By the aforementioned functionality of the hedges ofX, for every hedgeh∈H and every wordx∈ Dom(X), the wordhx inherits the semantics of his parentx, calledhedge-inheritance, that can be formalized as follows:c– ≤c+ ⇒H(c–) ≤H(c+) andhx ≤kx ⇒H(hx) ≤H(kx), ∀h,k, whereH(z) = σz:σ =hn… h1∈H∗, ⌀∈H∗. It states that any strings of hedgesσ and σ′ applying tohx andkx cannot change their inherent semantics defined by the inequality direction ofhx ≤kx; i.e.,σhx ≤ σ′kx holds or, equivalently,H(hx) ≤H(kx). By this,H(x) can be considered in [39] as the fuzziness model ofx. The set of the fuzziness models, H(x):x∈ Dom(X), has many properties representing the RW-semantics of the words which can be proved from the axioms of the HA ofX [20]:(a)
The inclusionH(hx) ⊆H(x) represents the generality-specificity relation ofx andhx:x is more general thanhx and, conversely,hx is more specific thanx. Moreover, we have(11)0∪Hc-∪W∪Hc+∪1=DomXHLx∪HRx∪HMx∪HVx=Hx⋃h∈XkHhx=Hx,whereXk=x∈DomX;x=k(b)
The order of the fuzziness models of words is the same as the order of their words:(12)0≤Hc-≤W≤Hc+≤1HLx≤HRx≤x≤HMx≤HVx,forsign⁡Vx=+1HLx≥HRx≥x≥HMx≥HVx,forsign⁡Vx=–1


#### 3.2.2. Quantification of the Words of Variable-Domains [39]

In nature, the compt.-semantics, includingnumeric semantics,interval semantics, andfuzziness measure, of the words of Dom(X) must be determined from their inherent order-based semantics presented above. Hence, they are closely defined. The axiomatic way to quantify the words is described as follows:(  1) Numeric Semantics of the Words. A mappingf  : X→[0,1], where [0,1] is the normalized universe ofX, is said to be a numeric semantics interpretation if it is anorder-isomorphism, whose imagef(X) isdense in[0,1]. It is called aSemantically Quantifying Mapping (SQM) and its values can be interpreted as thenumeric word semantics.(  2) Interval Semantics and Fuzziness Measure of the Words. An SQMf maps eachH(x) to a subinterval of [0,1], which is the smallest subinterval including the image ofH(x),f(H(x)). It is called thefuzziness interval or aninterval semantics ofx and is denoted by I(x) = [f(H(x))] and its length is calledfuzziness measureofx, denoted byfm(x), i.e.,fm  : X → [0,1].(  3) Strict Relationships of the Compt.-Semantics of the Words. It can be seen that, given an SQMf, asf preserves the order of the words, by (11) and (12), it follows by induction that(i)
forx∈0,c–,W,c+,1, their numeric semantics, i.e., theirf-values, and the related fuzziness intervals are exactly located in [0,1] as represented in Figure 6(a);(ii)
forx with x =k > 1, assume that its fuzziness interval I(x) is already located in [0,1]. If sign (Vx) = -1 (sign (Vx) = +1), its numeric semanticsf(x) and its related fuzziness intervals are arranged as given in Figure 6(b) (are arranged so that a structure of the words generated from the atom c and the structure obtained from the former one by replacing c with the other atom word c' are mirror symmetrical).Figure 6
The structure of SQM-values and their related fuzziness intervals of the words ofC∪ c-,c+ that is given in (a) and of the wordx that is given in (b).
(a) (b)From these, it follows that(13)f0≤Ic-≤fW≤Ic+≤f1Ic-∪Ic+=0,1ILx≤IRx≤fx≤IMx≤IVx,ifsign⁡Vx=+1IVx≤IMx≤fx≤IRx≤ILx,ifsign⁡Vx=-1ILx∪IRx∪IMx∪IVx=Ix,forxwithx≥1From (13) it follows that the fuzziness measurefm ofX has the following properties:fm1
fmc-+fmc+=1;fm2
fmLx+fm(Rx)+fmMx+fmVx=fm(x);fm3
Fromfm2isinferredfmLx/fmx+fmRx/fmx+fmMx/fmx+fmVx/fmx=1.Assume that the fractionfm(hx)/fm(x) does not depend on particularx; it can be considered as the fuzziness measure ofh, denoted by μ(h),h∈H. Thus, it holdsfm4
fmhx=μhfmx and μL+μR+μM+μV=α+β=1,whereα = μ(L) + μ(R) and β = μ(M) + μ(V), which are the sums of the fuzziness measure of, respectively, the negative hedges and the positive ones.It is important that when the values offm(c-), μ(R), μ(M), and μ(V) are given,fm(x) can be calculated, for any wordx. These values are calledindependent fuzziness parameters.Now, interpreting (fm1) - (fm3) as axioms, a quantification theory is founded, in which the numeric semantics and the interval semantics of the words are easily computed [30, 39, 41], referring to Figure 6. This illustrates that the quantification of HAs can also developed in an axiomatic way and, hence, the quantification theory of HAs isRWS-interpretable.


### 3.3. Genetic Design of HAC-PLL

Two of the main components of HAC are its control LRB and approximate reasoning method (ARM) running on the graphical representation of its LRB. The HAC-PLL designed in this study has two new features: it is RWS-interpretable and its LRB is genetically constructed. Usually, the LRBs of FCs are formulated by human experts and, in general, they are very difficult to optimize. In this study, the LRBs of the designed HAC-PLLs are for the first time genetically optimally constructed, including the fuzziness parameters of their variables, while the RWS-interpretability of the designed HAC-PLLs is guaranteed. So, first, a method is examined to construct RWS-interpretable compt.-representations of LRBs and an RWS-interpretable ARM working on their compt.-representations constructed by this method. These concepts are first examined in [30] and improved in [40].


#### 3.3.1. A Method to Construct RWS-Interpretable Graphical Representations of LRBs

Consider an LRBB consisting ofn rules ofm input variables and one output variable(14)riIFX1,Lisxi,1&…&XmLisxi,m,THENXm+1,Lisxi,m+1;i=1tonwhere Xj,L’s denote the linguistic variables of the respective RW-variables Xj’s; similarly, Xj,N’s denote the numeric variables of the respective RW-variables Xj’s and their universes are denoted, respectively, byU(Xj,N)’s.As this LRB is generally assumed to be consistent, it describes alinguisticfunctional dependence ofXm+1,L on the remaining ones Xj,L’s defined in the Cartesian product Dom(X1L) × … × Dom(Xm+1,L). Methodologically, the method to construct compt.-representations of LRBs, written shortly as RMd, is crucial because of the technical requirement for studying the RWS-interpretability not only of the LRBs but also of the ARMs working on them. In addition, when methods to genetically design LRBs are examined, they must deal not only with the individual compt.-representations but also with the method itself. Thus, the functionality of an RMd is able to construct from any given LRBB a compt.-representation, denoted by RRMd(B), defined inU(X1,N) × … ×U(Xm+1,N). At the same time, as human language is RWS-interpretable, the linguistic function defined byB does describe an RW-functionfRW ofXm+1 on the remaining Xj’s. Denote byfN the numeric function modeling the samefRW; the both functions must becompatible with each other. This leads to the following constraint on RMd’s interpreted as a test condition whether an RMd is RWS-interpretable.Constraint on RMd: the output RRMd(B) of RMd must be a numeric function defined in the Euclid spaceU(X1,N) ×… ×U(Xm+1,N), denoted by fN,RMd(B), and if the linguistic functionB is increasingly or decreasingly monotonic function, then so is the fN,RMd(B).This compt.-representationfN,RMd(B) of the LRBB is calledgraphic representation ofB. It can be seen that it is an aggregation of the elementary linguistic predicates given inB. It should be emphasized that, in accordance with our knowledge, no aggregation operators aggregating these (fuzzy) linguistic predicates can preserve such an RW-semantics represented by the LRBB like this graphic representation. In relation to this, it is observed that practical applications of the math-theories and the theoretical physics theories show that the graphic representations of functions in Euclid space applied in these theories are RWS-interpretable.In this section, an RMd developed in the HA-formalism to construct graphic representation of LRBs, called graphical RMd and denoted byM, is shortly described as follows:(i)
Determine HAs of Dom(Xj,L),j = 1 tom+1, by selecting their negative and positive hedges and determining their relative signs and their fuzziness parameter values.(ii)
With these, the SQMsfXj,L’s are defined and map then linguistic points determined by the LRB ofB in Dom(X1,L) × … × Dom(Xm+1,L) inton points in the Euclid space 0,1m+1 that forms a grid of points, denoted byGM(B).To examine the RWS-interpretability ofGRMd(B) some notations are needed. Leta = (a1, …,am,am+1) be a point in 0,1m+1. The project ofa on the Cartesian space indicated byXi1,N × … ×Xik,N; i.e., aXi1×…×Xik= (ai1, …,aik) is denoted by aXi1×…×Xik. For any RMd, a point of GRMd(B) corresponding to a ruleri is denoted by GRMd(ri). By the constraint above, the RWS-interpretability of an RMd can be defined as follows, noting that the LRBs in many control applications are monotonic.Definition 1.
A given method RMd to produce a compt.-representation in a Cartesian productSm+1 from a given LRB is said to be RWS-interpretable provided that ifB is increasingly (decreasingly) monotonic, then(i)
GRMd(B) ⊆U(X1,N) × … ×U(Xm+1,N) must define a function. That is, for every ruleri ofB, GRMd(ri) is of the formai = (ai,1, …,ai,m,ai,+1), fori = 1, …,n, and for any such two vectorsaj = (aj,1, …,aj,m,aj,m+1) and aj′ = (aj′,1, …, aj′,m, aj′,m+1), the equality ajX1×…×Xm+1 = aj′X1×…×Xm implies that ajXm+1 = aj′Xm+1.(ii)
ForB being an increasingly (or decreasingly) monotonic function and itsi-th ruleri is denoted by xi = (xi,1, …,xi,m,xi,m+1), it holds xiX1×…×Xm ≤ xjX1×…×Xm⟹GRMdriXm+1 ≤ GRMdrjXm+1 (or, xiX1×…×Xm ≤ xjX1×…×Xm⟹GRMdriXm+1 ≥ GRMdrjXm+1).
For the RMdM, as the determined SQMs are isomorphisms, it can easily be verified that the monotonicity of a givenB is preserved byM; i.e., theGM(B) has the same monotonicity property asB. So, the following proposition holds:Proposition 2.
The RMdM proposed as above is RWS-interpretable.


#### 3.3.2. The RWS-Interpretability of Approximate Reasoning Method

Each ARM should be developed accompanied with a given RMd to solve an application problem and it also plays a key role to construct an RWS-interpretable fuzzy system. It strongly depends on a given LRBB as well as on a given RMd. Therefore, to define the RWS-interpretability of a given ARM, R, it is assumed that the given RMd accompanied with R is RWS-interpretable. LetB be an LRB of the form (14). For any input vectora = (a1, …,am) inX1,N × … ×Xm,N, denote by R(GRMd(B))(a) the output value ofa in Dom(Xm+1,N) produced by R working on GRMd(B). Then, the following definition may be introduced which is a modified version of the respective one examined in [30].Definition 3.
Let be given an RMd to produce compt.-representations of any given LRBs and let an ARMR be developed to work on the compt.-representations produced by the RMd. R is said to beRWS-interpretable provided that for any LRBB, which is increasingly (or decreasingly) monotonic, R should satisfy the following condition:(15)∀a,ba<b⇒RGRMdBa<RGRMdBbor,∀a,ba<b⇒RGRMdBa>RGRMdBb.The conditions in this definition are natural but very strong for the ARMs developed in the fuzzy set framework. In this study, the so-called semantically weighted interpolation-extrapolation reasoning method proposed in [40] is applied and described as follows.(i) Semantically Weighted Interpolation-Extrapolation Reasoning Method.(A) Interpolation: consider a graphic representation GRMd(B) ofB.(  1) Define semantic weight of the closeness of a given input vector falling in a grid-mesh of G R M d(B)to each grid-mesh point: a restriction is applied to the case of two input variables which is compatible to the HAC-PLL examined in this study. Assume that a mesh of the 2D-grid graphically representingB is exhibited in Figure 7, in which four points of the mesh are calculated from four respective linguistic points (x1i,x2j), (x1(i+1),x2j), (x1i,x2(j+1)), and (x1(i+1),x2(j+1)) and denoted by (a1i,a2j), (a1(i+1),a2j), (a1i,a2(j+1)), and (a1(i+1),a2(i+1)). That is,ak,l = fXk(xk,l),k = 1, 2 andl∈i,i+1,j,j+1. The fXm+1-values at these points are denoted byu11,u12,u21, andu22. To simplify the presentation, these values are used to name the respective points (a1i,a2j), (a1(i+1),a2j), (a1i,a2(j+1)), and (a1(i+1),a2(i+1)) and they indicate the rectangle represented in Figure 7. It is denoted by (u11,u12,u21,u22) and its area being denoted bySij. Consider an input vectora = (a1,a2), whose components fall, respectively, into [a1i,a1(i+1)] and [a2j,a2(j+1)]. The expected valueu of the input vector is computed by the weighted average ofu11,u12,u21, andu22 whose semantic weights are defined based on how close to each mesh point it is: the closer to a point, the larger its semantic weight. The point (a1,a2) decomposes the mesh (u11,u12,u21,u22) into four rectangles. The area of any of the rectangles determined by any pointc = (c1,c2) is denoted by the following rule: if it is located opposite to the position of the valueukl, it is denoted by Sklc. In Figure 7, S11a is opposite tou11, S21a is opposite tou21, S12a′ and S12b′ are opposite tou12, and so on.Figure 7
Semantically weighted inter-extrapolative reasoning method onG(B).The semantic weight of the closeness ofa to a mesh point (a1k,a2l), whose fXm+1-value isukl, denoted by wa(ukl)=wa(a1k,a2l), is defined by the following product:(16)waa1k,a2l=1-a1-a1ka1i+1-a1i1-a2-a2la2j+1-a2j=a1i+1-a1i-a1-a1ka2j+1-a2j-a2-a2lSijThe product in the numerator of (16) is just the area of the rectangle Skla opposite to the position ofukl, where no confusion occurs if the area of a rectangleSij is denoted also bySij.(  2) Compute the control value using the semantically weighted average: for the given input vectora,u is computed as the semantically weighted average as follows: (17)ua=S11au11+S12au12+S21au21+S22au22Sij(B) Extrapolation: assume that a′ falls outside of the 2D-grid GRMd(B). There are two cases. The first one is that a′ falls into a rectangle which have exactly two vertices belonging to GRMd(B), say the rectangle (u01,u02,u11,u12) in Figure 7. Similar as above,u is computed by (18)ua′=S11a′u11+S12a′u12S11a′+S12a′For a′ falling into a rectangle outside of GRMd(B) having exactly one common vertex with GRMd(B), e.g., the vertexu11 of the rectangle domain of the producte ×ce, similar as above,u is computed by(19)ua′=S11a′u11S11a′=u11Theorem 4.
For the graphical RMdM, the semantically weighted interpolation-extrapolation (SWIE) reasoning method working withMis RWS-interpretable; i.e., it preserves the monotonicity property of the graphic representation GRMd(B), for any LRBB.Proof.
LetB be increasing; thus, so is GRMd(B). In the case of interpolation, it is sufficient to consider the case where two input vectorsa = (a1,a2) andb = (b1,b2) in question satisfya ≤b and all fall into the same mesh of the grid GRMd(B). Based on these vectors, the rectangle (u11,u12,u21,u22) is divided into nine rectangles indicated by S22a, S22b∩S21a, S21b∩S21a, S22b∩S12a, S22b∩S11a, S21b∩S11a, S12b∩S12a, S12b∩S11a and S11a, as computed and shown in Figure 7. By the method SWIE, the outputsua andub are computed by(20)ua=S11au11+S12au12+S21au21+S22au22Sijandub=S11bu11+S12bu12+S21bu21+S22bu22SijExamine now the sign of the following quantity:(21)Sijub-ua=S11b-S11au11+S12b-S12au12+S21b-S21au21+S22b-S22au22Considering the above nine rectangles as elementary ones, it can be seen that(22)S11a=S22b∩S11a+S21b∩S11a+S12b∩S11a+S11bS12b=S12b∩S12a+S12b∩S11aS21a=S21b∩S21a+S21b∩S11aS22b=S22a+S22b∩S21a+S22b∩S12a+S22b∩S11aBy (22), the quantity (21) can be reduced to (23)Sijub-ua=S12b∩S11au12-u11+S21b∩S11au21-u11+S22b∩S11au22-u11+S22b∩S12au22-u12+S22b∩S21au22-u21By the increasing monotonicity of GRMd(B), it holdsu11 ≤u12 ≤u22 andu11 ≤u21 ≤u22 that leads to Sij(ub-ua)≥0.
In the case of extrapolation, assume for example that the input vectorsa′ = (a1, a2′) and b′ = (b1, b2′) fall into the rectangle (u01,u02,u11,u12). To simplify and to easily read the proof expressions, it is imagined that this rectangle is linearly transformed into the unit-rectangle whose vertexu01 locates at the center of the coordinate system. If it is proved that ua′ ≤ ub′ in the unit-rectangle, then, by the linear transformation, this respective inequality is also valid in the original rectangle. So, for simplicity, imagine thata andb are already linearly transformed in the unit-rectangle; i.e.,u01 = (0, 0),u02 = (0, 1), andu11 = (0, 1). By (18), it holds (24)ub′-ua′=S11b′u11+S12b′u12S11b′+S12b′-S11a′u11+S12a′u12S11a′+S12a′=1-b1b2′u11+b1b2′u121×b2′-1-a1a2′u11+a1a2′u121×a2′=a2′1-b1b2′u11+a2′b1b2′u12-1-a1a2′b2′u11+a1a2′b2′u12a2′×b2′=a2′b2′b1-a1u12-u11a2′×b2′≥0As the remaining cases are proved in a similar way, it can be concluded that the method SWIE preserves the monotonicity property of the given LRBB. By Definition 3, the method SWIE is RWS-interpretable.


#### 3.3.3. Genetic Algorithm to Design Optimized HAC for MAF-PLL

In this section, it is described how to develop a genetic algorithm (GA), denoted HAC-GA, to designopHAC for the MAF-PLL with HAC. The HAC-GA has three tasks: (i) to optimize LRBs, (ii) to optimize the variable fuzziness parameters, and (iii) to optimize the variable ranges. So, the HAC-GA has a functionality to simultaneously produce optimized LRB, optimized fuzziness parameters, and optimized variables ranges. To develop such a HAC-GA the following tasks should be performed.Task 1. Encode the Solution of the Optimization Problem of HACs.(a) Solutions of the optimization problem of HACs: each solution comprises three groups: LRB, fuzziness parameters, and ranges of the variablese,ce, andu. Assume that the HAs of these variables are syntactically the same; i.e., they have the same set of atomic wordsG = N (negative), P (positive) and the same set of hedgesH = L, V. Hence, Dom(e) = Dom(ce) = VN,LLN,ZE,LLP,VP and Dom(u) = VVN,VN,LVN,N,LLN,LN,VLN,ZE,VLP,LP,LLP,LVP,VP,VVP, which areordered from left to right. The limitation of Dom(e) and Dom(ce) of having only five words aims to be compatible with their five fuzzy sets of the FCs for comparative study in Section 4. A representation of an LRB of the HAC is given in the form of Table 3. It is assumed that this table isantonym in the sense defined as follows.Table 3
An Example of an LRBB.
ce VN LLN ZE LLP VP VN VVN VVN VN N ZE e LLN VVN VN N ZE P ZE VN N ZE P VP LLP N ZE P VP VVP VP ZE P VP VVP VVPFor everyX, two wordsx=hn… h1c andy= h′m…h′1c′ are said to be contradictory or antonyms ifhn… h1 = h′m…h′1 andc, c′ belonging to c–,c+ are antonym. For example,LLP andLLN are antonym. A cell of Table 3 can be written as (x,y,z),x∈ Dom(e),y∈ Dom(ce) andz∈ Dom(u). Two cells (x,y,z) and (x′, y′, z′) are said to be antonym if the pairs of the words (x, x′), (y, y′) and (z, z′) are antonym. Then, an LRB represented by Table 3 is said to be antonym if for any two its cells (x,y,z) and (x′, y′, z′) whose pairs (x, x′) and (y, y′) are antonym, then so is the pair (z, z′). As a consequence, the words of the cells lying on the secondary diagonal of the table are the same and equal to the neutral words (ZE) and, when the upper secondary diagonal of the table is known, the table of rules is completely determined. It can be seen that the LRB given in Table 3 is antonym.(b)Individual coding: Assume that thereal coding is applied. To genetically designopHAC, the following should be coded: its LRBB of the form given in Table 3, the independent fuzziness parameters, and the ranges of the variablese,ce, andu. For their fuzziness parameters and their range values, by the symmetricity of the variable ranges, it is assumed thatfme(c–) =fmce(c–) =fmu(c–) = 0.5 and, hence, the GA to be developed needs to seek for optimized values of μe(L), μce(L), and μu(L). So, the GA needs only three genes to represent the fuzziness parameters and only three genes to represent the variable ranges. For the words of the variableu in Table 3, it is required to know only ten words below the right diagonal of the table which are arranged fromleft to right and fromtop to bottom. For example, the LRBB given in Table 3 is represented by a chromosome of ten genes: (P P VP P VP VVP P VP VVP VVP). Each solution is encoded by an individual of 16 genes: the first three genes represent the values μe(L), μce(L), and μu(L), the next ten genes represent the ten above-mentioned words ofu uniquely determiningB, and the three remaining genes represent three positive variable range values. Assume that the words of Dom(u) = {VVN,VN,LVN,N,LLN,LN,VLN,ZE,VLP,LP,LLP,P,LVP,VP,VVP} are increasingly arranged and encoded by their positions, i.e., by the integers 0, 1, 2, …, 14. For example, three values μe(L), μce(L), and μu(L), three positive variable ranges, rgX’s, and theB given in Table 3 are represented by the solution individual: (μe(L), μce(L), μu(L), 11, 11, 13, 11, 13, 14, 11, 13, 14, 14,rge,rgce,rgu).Task 2. Construct a HAC, an Alternative Component for the FC (1)
Determine the inputs of a HAC They comprise the following:
(i)
the positive and negative hedges, their relative signs, fuzziness parameter values and the positive variable range values; (ii)
an LRBB given in the form of Table 3; (iii)
an ARM— a selected interpolative method working on the surface in0,13, which is computed using the SQMs fe, fce, and fu; (iv)
input vector of the control problem.These input values are needed in each iteration, noting that the values listed in (i) and (ii) are provided by an individual under consideration of a population generated by the HAC-GA.(2)
Construct components of a HAC of the HAC-PLLIts main components are shown in Figure8 which comprises the following:
(i)
the normalization to convert the universes ofe,ce, andu to their respective unit intervals [0,1]’s;(ii)
a given LRBB and its quantification to translateB into a graphic representation approximately modeling a surface Sreal3;(iii)
construct an RWS-interpretable interpolation-extrapolation reasoning method. In this study, the developed method SWIE is applied;(iv)
the denormalization to convert the numeric output ofu computed by the second component of HAC to its respective real value of the variableu.Figure 8
The components of the HAC.Task 3. Construct a GA, Denoted by GA(opHAC), to Produce the Optimized LRB B o p , the Optimized Fuzziness Parameters, and the Optimized Variable Ranges of the Desired opHAC.The HAC developed inTask 1 is a kernel-component of the desiredGA(opHAC).(1)
Inputs ofGA(opHAC):
(i)
The criterionJ1=∫0δe(t)dt (or =∑k=1ne(k), n=δ/Ts), called the performance index (Prf-index) in this study, and the time period of simulation δ for computing the Prf-indexJ1(ii)
GA running parameters: population size; generation number; disturbance type(2)
Outputs ofGA(opHAC):
(i)
The optimized values of the fuzziness parametersμe(L), μce(L), and μu(L) and the optimized variable ranges (-rge,rge), (-rgce,rgce), and (-rgu,rgu)(ii)
The optimized LRBBop(3)
General description ofGA(opHAC):
(i)
Population of solutions:GA(opHAC) will initialize an initial population. Every individual of a population consists of 16 genes, 3 genes for the fuzziness parameters, 3 genes for positive variable ranges values, and 10 genes for the words to determine LRB of the form in Table 3. Genetic operations like crossovers and mutation will be applied to generate the next population of the same size, in which only individuals with better fitness values are retained.(ii)
Method to compute the Prf-indexJ1 of an individual of a given population: the procedure ofGA(opHAC) to computeJ1 of each individual of the population in question, denoted by δ-LoopopHAC, is described by the scheme given in Figure 9.Figure 9
δ-LoopopHAC of the simulation model.(4)
Implementation ofGA(opHAC) in Matlab (see Box 1)Box 1
GA(opHAC) Input: (i) Simulation model ‘HACPLLfinal.slx’ suitable type for the expected disturbance; (ii)J1I=∑e(k)∈Setδ(e(k))e(k); // I is an individual of the population under consideration Output: The optimized HAC: ‘opHAC’; Method: Initialize parameters of the simulation model; Initialize parameters of the GA: num_var = number of variables; pop_size = population size; num_gen = number of generation; time_limit = time value; // to limit the running time of the algorithmGA(opHAC) Initialize an initial population; m = 1; t = 0; while ((m ≤ num_gen) and (t ≤ time_limit)) for each individualIm in the population call simulation model: ‘HACPLLfinal.slx’; computeJ1(Im)=∑e(k)∈Setδ(e(k))e(k); end_for m = m+1; Perform genetic operators; auto_update(t); end_while End_GA(opHAC)


## 3.1. The Role of Human Language in Designing RWS-Interpretable Fuzzy Systems

In the fuzzy set framework, words assigned to the fuzzy sets are considered as linguistic labels and there is no formal basis to use words and linguistic sentences of human natural language with their owninherent semantics. So, there exists a gap between the fuzzy representations representing the inherent semantics of their respective words and of their respective LRBs. The main aim of the fuzzy control is to utilize human capability in handling words of the natural language. If there was no formal bridge established across the gap, one would not be able to indeed make use of human expert linguistic knowledge. The paper [30] introduces an RWS-approach to the interpretability of fuzzy systems in which the RWS-interpretability of the human language is utilized to design RWS-interpretable fuzzy systems. Assuming the RWS-interpretability of the natural language, it implies that the LRB equipped for a fuzzy system is able to properly describe the reality with which the system interfaces. Based on this feature, the ultimate goal of the RWS-approach is to establish a methodology to design RWS-interpretable fuzzy systems, whose RWS-interpretability means that their behavior is ensured to be compatible with the expected one described by the given LRB.The main specific features of the RWS-approach examined in [30] are the following:(i)
It is able to utilize the RWS-interpretability of the human language. In the fuzzy control, they usually describemonotonic dependences between two RW-variables.(ii)
Its formalism is able to immediately handle words and sentences of human expert languageswith their own RW-semantics as well as their compt.-semantics.These features are very crucial for any method to design a fuzzy (linguistic) controller to ensure its expected behavior when it interacts with its RW-counterpart.


## 3.2. Hedge Algebras of Variables, Their Quantification, and Their RWS-Interpretability

In order to ensure the RWS-interpretability of the designed HAC-PLLs, it is required to utilize the RWS-interpretability of human language to establish a mathematical formalization of word-domains for developing the HAC-PLLs. To do this, it is observed that the word-domain of every (linguistic) variable possesses the following two specific features:(i)
The word-domain can be generated from its atomic words, e.g., ‘small’ and ‘large’ of the current intensity variable, using linguistic hedges, e.g., ‘very’ or ‘extremely’. Then, every wordx is of the formhn…h1c, wherec∈small,large and hj’s are (linguistic) hedges and, hence, they can be considered as operations of an algebra.(ii)
There exists a linear or partial semantic order relation on this word-domain induced by theinherent semantics of its words.Thus, hedges with their own order-based semantics defined in their word-domain play an essential role in discovering order-based structures of the word-domains of variables. However, a large number of studies consider words only as linguistic labels of the constructed fuzzy sets, whereas there are relatively few studies thatimmediately deal with linguistic hedges with their own modification functionality, called hedge operators. For instance, such hedge operators are examined and applied in fuzzy logic and approximate reasoning [31, 32], in contrast intensification [33], and in fuzzy control [34]. In particular, it is actively examined in formal concept analysis of a dataset based on studying formal concept lattice with hedges, called truth-stressing hedges [35–38]. In these studies, hedges are utilized to modify concept-forming operators to control the number of conceptual clusters extracted from a given dataset represented by formal concepts of a concept lattice of the dataset. However, the inherent semantics of words, which may contain hedge occurrences, is still not taken into consideration and, therefore, ‘hedge operators’ were developed in these studies independently from the semantics of words and hedges, which in practice must be determined in the context of other words of their variable.Starting from the two aforementioned specific features of word-domains, HAs were proposed and developed in turn in [20, 30, 39, 40]. As argued in [30], all theories developed in an axiomatic way, whose axioms actually represent structural semantics of their respective RW-counterparts (e.g., math-theories or theoretical physics theories), are shown to be RWS-interpretable; in this section, we briefly explain that the theories of HAs and their quantification, which form the above formalism, are also RWS-interpretable.


### 3.2.1. A Description of an Axiomatization of Word-Domains and Their Quantification

HAs introduced in [20, 30, 39, 40] establish an algebraic approach based on a formalization of their word-domains of variables in an axiomatic way to immediately handle the inherent order-based semantics of the words. It is natural that there are comparability elements in the human language, the presence of which is required by the human decision-making activities in which the comparability of decision criteria values is crucial. So, there exists a semantic order relation on every variable word-domain. For instance, considering the RW-variable IRW of the current intensity, this RW-variable may be examined on the standpoint either of a numeric formalism, in which its respective numeric variable is denoted by IN, or of a linguistic formalism, in which its respective linguistic variable is denoted by IL. For every hedgeh of IL, say ‘very’ (V), the pair of the words ‘very large’ and ‘large’ or the pair of the words ‘very small’ and ‘small’ of IL are always comparable. It is essential that the order of these words iscompatible with the order of the numeric values of the current intensity assigned to their respective words and, hence, the word-domain of IL must also be linearly ordered.For general case, let us consider an arbitrary linguistic variable denoted byX without its subscriptL, for simplicity. As analyzed above, the word-domain ofX becomes an order-based structure, denoted byAX = (Dom(X),G,C,H, ≤), whereG = c-,c+ – considered as the generators ofAX, wherec- <c+,c+ is called thepositive atomic word andc- – thenegative one;C is the set of constantsC = 0,W,1, which are, respectively, the least, neutral, and greatest words of Dom(X);H =H- ∪ H+ is the set of the hedges ofX, whereH- (orH+) is the set of the negative (or positive) hedges; e.g.,H- = R (Rather), L (Little), andH+ = M (More), V (Very); and ≤ is the semantic order relation. Syntactically, Dom(X) is the set of all the words ofX of the formx =hn… h1c,c∈G, whose length is denoted by x; i.e., x =n + 1. Many properties of the inherent semantics of the words ofX can be modeled in the order-based structure of Dom(X) and, more importantly, as the human language is RWS-interpretable, they are RWS-interpretable as well. To formalize the word-domains in an axiomatic way, it is necessary to discover few properties of the order-based semantics of words and hedges that can easily be verified based on their natural meaning so that they can be taken as axioms and their math-models and the remaining properties can be derived from them. Restricted to the purpose of the study, a very short description of this formalization is given, including the necessary knowledge for the study. More details can be found in [20, 30, 39, 40].(i) Words and hedges ofX have their own “algebraic” signs determined in this order-based structure as follows.(a)
For atom words, sign (c–) = –1 and sign(c+) = +1; for any hedgeh, sign (h) = +1 (or, sign (h) = –1) ifh∈H+ (or,h∈H–) which is identified byhc+ ≥c+ (or,hc+ ≤c+).(b)
Therelative sign ofh with respect tok, denoted by sign (h,k), is computed by sign (h,k) = –1 iff (∃c∈G)(kc ≤c ⇒kc ≤hkc) and sign (h,k) = +1 iff (∃c∈G)(kc ≤c ⇒hkc ≤kc). For illustration, the validity of Table 2 can be checked.(c)
Now, the sign ofx =hn …h1c is computed by sign (x) = sign (hn,hn-1) … sign (h1) sign (c) ∈–1, +1. There is an important fact that sign (hx) =–1 (or, +1) ⇔ hx ≤x (or,x ≤hx). For example, for L, R, M, and V stand forlittle,rather,more, andvery, respectively, it may be verified, for instance, that sign (VR_small) = sign (V,R)sign (R) × sign (small) = –1, which results in VR_small ≤ R_small.Table 2
The relative sign of a hedge in a row with respect to a hedge in a column.
V M R L V + + - +  M + + - +  R - - + -  L - - + -(ii) By the aforementioned functionality of the hedges ofX, for every hedgeh∈H and every wordx∈ Dom(X), the wordhx inherits the semantics of his parentx, calledhedge-inheritance, that can be formalized as follows:c– ≤c+ ⇒H(c–) ≤H(c+) andhx ≤kx ⇒H(hx) ≤H(kx), ∀h,k, whereH(z) = σz:σ =hn… h1∈H∗, ⌀∈H∗. It states that any strings of hedgesσ and σ′ applying tohx andkx cannot change their inherent semantics defined by the inequality direction ofhx ≤kx; i.e.,σhx ≤ σ′kx holds or, equivalently,H(hx) ≤H(kx). By this,H(x) can be considered in [39] as the fuzziness model ofx. The set of the fuzziness models, H(x):x∈ Dom(X), has many properties representing the RW-semantics of the words which can be proved from the axioms of the HA ofX [20]:(a)
The inclusionH(hx) ⊆H(x) represents the generality-specificity relation ofx andhx:x is more general thanhx and, conversely,hx is more specific thanx. Moreover, we have(11)0∪Hc-∪W∪Hc+∪1=DomXHLx∪HRx∪HMx∪HVx=Hx⋃h∈XkHhx=Hx,whereXk=x∈DomX;x=k(b)
The order of the fuzziness models of words is the same as the order of their words:(12)0≤Hc-≤W≤Hc+≤1HLx≤HRx≤x≤HMx≤HVx,forsign⁡Vx=+1HLx≥HRx≥x≥HMx≥HVx,forsign⁡Vx=–1


### 3.2.2. Quantification of the Words of Variable-Domains [39]

In nature, the compt.-semantics, includingnumeric semantics,interval semantics, andfuzziness measure, of the words of Dom(X) must be determined from their inherent order-based semantics presented above. Hence, they are closely defined. The axiomatic way to quantify the words is described as follows:(  1) Numeric Semantics of the Words. A mappingf  : X→[0,1], where [0,1] is the normalized universe ofX, is said to be a numeric semantics interpretation if it is anorder-isomorphism, whose imagef(X) isdense in[0,1]. It is called aSemantically Quantifying Mapping (SQM) and its values can be interpreted as thenumeric word semantics.(  2) Interval Semantics and Fuzziness Measure of the Words. An SQMf maps eachH(x) to a subinterval of [0,1], which is the smallest subinterval including the image ofH(x),f(H(x)). It is called thefuzziness interval or aninterval semantics ofx and is denoted by I(x) = [f(H(x))] and its length is calledfuzziness measureofx, denoted byfm(x), i.e.,fm  : X → [0,1].(  3) Strict Relationships of the Compt.-Semantics of the Words. It can be seen that, given an SQMf, asf preserves the order of the words, by (11) and (12), it follows by induction that(i)
forx∈0,c–,W,c+,1, their numeric semantics, i.e., theirf-values, and the related fuzziness intervals are exactly located in [0,1] as represented in Figure 6(a);(ii)
forx with x =k > 1, assume that its fuzziness interval I(x) is already located in [0,1]. If sign (Vx) = -1 (sign (Vx) = +1), its numeric semanticsf(x) and its related fuzziness intervals are arranged as given in Figure 6(b) (are arranged so that a structure of the words generated from the atom c and the structure obtained from the former one by replacing c with the other atom word c' are mirror symmetrical).Figure 6
The structure of SQM-values and their related fuzziness intervals of the words ofC∪ c-,c+ that is given in (a) and of the wordx that is given in (b).
(a) (b)From these, it follows that(13)f0≤Ic-≤fW≤Ic+≤f1Ic-∪Ic+=0,1ILx≤IRx≤fx≤IMx≤IVx,ifsign⁡Vx=+1IVx≤IMx≤fx≤IRx≤ILx,ifsign⁡Vx=-1ILx∪IRx∪IMx∪IVx=Ix,forxwithx≥1From (13) it follows that the fuzziness measurefm ofX has the following properties:fm1
fmc-+fmc+=1;fm2
fmLx+fm(Rx)+fmMx+fmVx=fm(x);fm3
Fromfm2isinferredfmLx/fmx+fmRx/fmx+fmMx/fmx+fmVx/fmx=1.Assume that the fractionfm(hx)/fm(x) does not depend on particularx; it can be considered as the fuzziness measure ofh, denoted by μ(h),h∈H. Thus, it holdsfm4
fmhx=μhfmx and μL+μR+μM+μV=α+β=1,whereα = μ(L) + μ(R) and β = μ(M) + μ(V), which are the sums of the fuzziness measure of, respectively, the negative hedges and the positive ones.It is important that when the values offm(c-), μ(R), μ(M), and μ(V) are given,fm(x) can be calculated, for any wordx. These values are calledindependent fuzziness parameters.Now, interpreting (fm1) - (fm3) as axioms, a quantification theory is founded, in which the numeric semantics and the interval semantics of the words are easily computed [30, 39, 41], referring to Figure 6. This illustrates that the quantification of HAs can also developed in an axiomatic way and, hence, the quantification theory of HAs isRWS-interpretable.


## 3.2.1. A Description of an Axiomatization of Word-Domains and Their Quantification

HAs introduced in [20, 30, 39, 40] establish an algebraic approach based on a formalization of their word-domains of variables in an axiomatic way to immediately handle the inherent order-based semantics of the words. It is natural that there are comparability elements in the human language, the presence of which is required by the human decision-making activities in which the comparability of decision criteria values is crucial. So, there exists a semantic order relation on every variable word-domain. For instance, considering the RW-variable IRW of the current intensity, this RW-variable may be examined on the standpoint either of a numeric formalism, in which its respective numeric variable is denoted by IN, or of a linguistic formalism, in which its respective linguistic variable is denoted by IL. For every hedgeh of IL, say ‘very’ (V), the pair of the words ‘very large’ and ‘large’ or the pair of the words ‘very small’ and ‘small’ of IL are always comparable. It is essential that the order of these words iscompatible with the order of the numeric values of the current intensity assigned to their respective words and, hence, the word-domain of IL must also be linearly ordered.For general case, let us consider an arbitrary linguistic variable denoted byX without its subscriptL, for simplicity. As analyzed above, the word-domain ofX becomes an order-based structure, denoted byAX = (Dom(X),G,C,H, ≤), whereG = c-,c+ – considered as the generators ofAX, wherec- <c+,c+ is called thepositive atomic word andc- – thenegative one;C is the set of constantsC = 0,W,1, which are, respectively, the least, neutral, and greatest words of Dom(X);H =H- ∪ H+ is the set of the hedges ofX, whereH- (orH+) is the set of the negative (or positive) hedges; e.g.,H- = R (Rather), L (Little), andH+ = M (More), V (Very); and ≤ is the semantic order relation. Syntactically, Dom(X) is the set of all the words ofX of the formx =hn… h1c,c∈G, whose length is denoted by x; i.e., x =n + 1. Many properties of the inherent semantics of the words ofX can be modeled in the order-based structure of Dom(X) and, more importantly, as the human language is RWS-interpretable, they are RWS-interpretable as well. To formalize the word-domains in an axiomatic way, it is necessary to discover few properties of the order-based semantics of words and hedges that can easily be verified based on their natural meaning so that they can be taken as axioms and their math-models and the remaining properties can be derived from them. Restricted to the purpose of the study, a very short description of this formalization is given, including the necessary knowledge for the study. More details can be found in [20, 30, 39, 40].(i) Words and hedges ofX have their own “algebraic” signs determined in this order-based structure as follows.(a)
For atom words, sign (c–) = –1 and sign(c+) = +1; for any hedgeh, sign (h) = +1 (or, sign (h) = –1) ifh∈H+ (or,h∈H–) which is identified byhc+ ≥c+ (or,hc+ ≤c+).(b)
Therelative sign ofh with respect tok, denoted by sign (h,k), is computed by sign (h,k) = –1 iff (∃c∈G)(kc ≤c ⇒kc ≤hkc) and sign (h,k) = +1 iff (∃c∈G)(kc ≤c ⇒hkc ≤kc). For illustration, the validity of Table 2 can be checked.(c)
Now, the sign ofx =hn …h1c is computed by sign (x) = sign (hn,hn-1) … sign (h1) sign (c) ∈–1, +1. There is an important fact that sign (hx) =–1 (or, +1) ⇔ hx ≤x (or,x ≤hx). For example, for L, R, M, and V stand forlittle,rather,more, andvery, respectively, it may be verified, for instance, that sign (VR_small) = sign (V,R)sign (R) × sign (small) = –1, which results in VR_small ≤ R_small.Table 2
The relative sign of a hedge in a row with respect to a hedge in a column.
V M R L V + + - +  M + + - +  R - - + -  L - - + -(ii) By the aforementioned functionality of the hedges ofX, for every hedgeh∈H and every wordx∈ Dom(X), the wordhx inherits the semantics of his parentx, calledhedge-inheritance, that can be formalized as follows:c– ≤c+ ⇒H(c–) ≤H(c+) andhx ≤kx ⇒H(hx) ≤H(kx), ∀h,k, whereH(z) = σz:σ =hn… h1∈H∗, ⌀∈H∗. It states that any strings of hedgesσ and σ′ applying tohx andkx cannot change their inherent semantics defined by the inequality direction ofhx ≤kx; i.e.,σhx ≤ σ′kx holds or, equivalently,H(hx) ≤H(kx). By this,H(x) can be considered in [39] as the fuzziness model ofx. The set of the fuzziness models, H(x):x∈ Dom(X), has many properties representing the RW-semantics of the words which can be proved from the axioms of the HA ofX [20]:(a)
The inclusionH(hx) ⊆H(x) represents the generality-specificity relation ofx andhx:x is more general thanhx and, conversely,hx is more specific thanx. Moreover, we have(11)0∪Hc-∪W∪Hc+∪1=DomXHLx∪HRx∪HMx∪HVx=Hx⋃h∈XkHhx=Hx,whereXk=x∈DomX;x=k(b)
The order of the fuzziness models of words is the same as the order of their words:(12)0≤Hc-≤W≤Hc+≤1HLx≤HRx≤x≤HMx≤HVx,forsign⁡Vx=+1HLx≥HRx≥x≥HMx≥HVx,forsign⁡Vx=–1


## 3.2.2. Quantification of the Words of Variable-Domains [39]

In nature, the compt.-semantics, includingnumeric semantics,interval semantics, andfuzziness measure, of the words of Dom(X) must be determined from their inherent order-based semantics presented above. Hence, they are closely defined. The axiomatic way to quantify the words is described as follows:(  1) Numeric Semantics of the Words. A mappingf  : X→[0,1], where [0,1] is the normalized universe ofX, is said to be a numeric semantics interpretation if it is anorder-isomorphism, whose imagef(X) isdense in[0,1]. It is called aSemantically Quantifying Mapping (SQM) and its values can be interpreted as thenumeric word semantics.(  2) Interval Semantics and Fuzziness Measure of the Words. An SQMf maps eachH(x) to a subinterval of [0,1], which is the smallest subinterval including the image ofH(x),f(H(x)). It is called thefuzziness interval or aninterval semantics ofx and is denoted by I(x) = [f(H(x))] and its length is calledfuzziness measureofx, denoted byfm(x), i.e.,fm  : X → [0,1].(  3) Strict Relationships of the Compt.-Semantics of the Words. It can be seen that, given an SQMf, asf preserves the order of the words, by (11) and (12), it follows by induction that(i)
forx∈0,c–,W,c+,1, their numeric semantics, i.e., theirf-values, and the related fuzziness intervals are exactly located in [0,1] as represented in Figure 6(a);(ii)
forx with x =k > 1, assume that its fuzziness interval I(x) is already located in [0,1]. If sign (Vx) = -1 (sign (Vx) = +1), its numeric semanticsf(x) and its related fuzziness intervals are arranged as given in Figure 6(b) (are arranged so that a structure of the words generated from the atom c and the structure obtained from the former one by replacing c with the other atom word c' are mirror symmetrical).Figure 6
The structure of SQM-values and their related fuzziness intervals of the words ofC∪ c-,c+ that is given in (a) and of the wordx that is given in (b).
(a) (b)From these, it follows that(13)f0≤Ic-≤fW≤Ic+≤f1Ic-∪Ic+=0,1ILx≤IRx≤fx≤IMx≤IVx,ifsign⁡Vx=+1IVx≤IMx≤fx≤IRx≤ILx,ifsign⁡Vx=-1ILx∪IRx∪IMx∪IVx=Ix,forxwithx≥1From (13) it follows that the fuzziness measurefm ofX has the following properties:fm1
fmc-+fmc+=1;fm2
fmLx+fm(Rx)+fmMx+fmVx=fm(x);fm3
Fromfm2isinferredfmLx/fmx+fmRx/fmx+fmMx/fmx+fmVx/fmx=1.Assume that the fractionfm(hx)/fm(x) does not depend on particularx; it can be considered as the fuzziness measure ofh, denoted by μ(h),h∈H. Thus, it holdsfm4
fmhx=μhfmx and μL+μR+μM+μV=α+β=1,whereα = μ(L) + μ(R) and β = μ(M) + μ(V), which are the sums of the fuzziness measure of, respectively, the negative hedges and the positive ones.It is important that when the values offm(c-), μ(R), μ(M), and μ(V) are given,fm(x) can be calculated, for any wordx. These values are calledindependent fuzziness parameters.Now, interpreting (fm1) - (fm3) as axioms, a quantification theory is founded, in which the numeric semantics and the interval semantics of the words are easily computed [30, 39, 41], referring to Figure 6. This illustrates that the quantification of HAs can also developed in an axiomatic way and, hence, the quantification theory of HAs isRWS-interpretable.


## 3.3. Genetic Design of HAC-PLL

Two of the main components of HAC are its control LRB and approximate reasoning method (ARM) running on the graphical representation of its LRB. The HAC-PLL designed in this study has two new features: it is RWS-interpretable and its LRB is genetically constructed. Usually, the LRBs of FCs are formulated by human experts and, in general, they are very difficult to optimize. In this study, the LRBs of the designed HAC-PLLs are for the first time genetically optimally constructed, including the fuzziness parameters of their variables, while the RWS-interpretability of the designed HAC-PLLs is guaranteed. So, first, a method is examined to construct RWS-interpretable compt.-representations of LRBs and an RWS-interpretable ARM working on their compt.-representations constructed by this method. These concepts are first examined in [30] and improved in [40].


### 3.3.1. A Method to Construct RWS-Interpretable Graphical Representations of LRBs

Consider an LRBB consisting ofn rules ofm input variables and one output variable(14)riIFX1,Lisxi,1&…&XmLisxi,m,THENXm+1,Lisxi,m+1;i=1tonwhere Xj,L’s denote the linguistic variables of the respective RW-variables Xj’s; similarly, Xj,N’s denote the numeric variables of the respective RW-variables Xj’s and their universes are denoted, respectively, byU(Xj,N)’s.As this LRB is generally assumed to be consistent, it describes alinguisticfunctional dependence ofXm+1,L on the remaining ones Xj,L’s defined in the Cartesian product Dom(X1L) × … × Dom(Xm+1,L). Methodologically, the method to construct compt.-representations of LRBs, written shortly as RMd, is crucial because of the technical requirement for studying the RWS-interpretability not only of the LRBs but also of the ARMs working on them. In addition, when methods to genetically design LRBs are examined, they must deal not only with the individual compt.-representations but also with the method itself. Thus, the functionality of an RMd is able to construct from any given LRBB a compt.-representation, denoted by RRMd(B), defined inU(X1,N) × … ×U(Xm+1,N). At the same time, as human language is RWS-interpretable, the linguistic function defined byB does describe an RW-functionfRW ofXm+1 on the remaining Xj’s. Denote byfN the numeric function modeling the samefRW; the both functions must becompatible with each other. This leads to the following constraint on RMd’s interpreted as a test condition whether an RMd is RWS-interpretable.Constraint on RMd: the output RRMd(B) of RMd must be a numeric function defined in the Euclid spaceU(X1,N) ×… ×U(Xm+1,N), denoted by fN,RMd(B), and if the linguistic functionB is increasingly or decreasingly monotonic function, then so is the fN,RMd(B).This compt.-representationfN,RMd(B) of the LRBB is calledgraphic representation ofB. It can be seen that it is an aggregation of the elementary linguistic predicates given inB. It should be emphasized that, in accordance with our knowledge, no aggregation operators aggregating these (fuzzy) linguistic predicates can preserve such an RW-semantics represented by the LRBB like this graphic representation. In relation to this, it is observed that practical applications of the math-theories and the theoretical physics theories show that the graphic representations of functions in Euclid space applied in these theories are RWS-interpretable.In this section, an RMd developed in the HA-formalism to construct graphic representation of LRBs, called graphical RMd and denoted byM, is shortly described as follows:(i)
Determine HAs of Dom(Xj,L),j = 1 tom+1, by selecting their negative and positive hedges and determining their relative signs and their fuzziness parameter values.(ii)
With these, the SQMsfXj,L’s are defined and map then linguistic points determined by the LRB ofB in Dom(X1,L) × … × Dom(Xm+1,L) inton points in the Euclid space 0,1m+1 that forms a grid of points, denoted byGM(B).To examine the RWS-interpretability ofGRMd(B) some notations are needed. Leta = (a1, …,am,am+1) be a point in 0,1m+1. The project ofa on the Cartesian space indicated byXi1,N × … ×Xik,N; i.e., aXi1×…×Xik= (ai1, …,aik) is denoted by aXi1×…×Xik. For any RMd, a point of GRMd(B) corresponding to a ruleri is denoted by GRMd(ri). By the constraint above, the RWS-interpretability of an RMd can be defined as follows, noting that the LRBs in many control applications are monotonic.Definition 1.
A given method RMd to produce a compt.-representation in a Cartesian productSm+1 from a given LRB is said to be RWS-interpretable provided that ifB is increasingly (decreasingly) monotonic, then(i)
GRMd(B) ⊆U(X1,N) × … ×U(Xm+1,N) must define a function. That is, for every ruleri ofB, GRMd(ri) is of the formai = (ai,1, …,ai,m,ai,+1), fori = 1, …,n, and for any such two vectorsaj = (aj,1, …,aj,m,aj,m+1) and aj′ = (aj′,1, …, aj′,m, aj′,m+1), the equality ajX1×…×Xm+1 = aj′X1×…×Xm implies that ajXm+1 = aj′Xm+1.(ii)
ForB being an increasingly (or decreasingly) monotonic function and itsi-th ruleri is denoted by xi = (xi,1, …,xi,m,xi,m+1), it holds xiX1×…×Xm ≤ xjX1×…×Xm⟹GRMdriXm+1 ≤ GRMdrjXm+1 (or, xiX1×…×Xm ≤ xjX1×…×Xm⟹GRMdriXm+1 ≥ GRMdrjXm+1).
For the RMdM, as the determined SQMs are isomorphisms, it can easily be verified that the monotonicity of a givenB is preserved byM; i.e., theGM(B) has the same monotonicity property asB. So, the following proposition holds:Proposition 2.
The RMdM proposed as above is RWS-interpretable.


### 3.3.2. The RWS-Interpretability of Approximate Reasoning Method

Each ARM should be developed accompanied with a given RMd to solve an application problem and it also plays a key role to construct an RWS-interpretable fuzzy system. It strongly depends on a given LRBB as well as on a given RMd. Therefore, to define the RWS-interpretability of a given ARM, R, it is assumed that the given RMd accompanied with R is RWS-interpretable. LetB be an LRB of the form (14). For any input vectora = (a1, …,am) inX1,N × … ×Xm,N, denote by R(GRMd(B))(a) the output value ofa in Dom(Xm+1,N) produced by R working on GRMd(B). Then, the following definition may be introduced which is a modified version of the respective one examined in [30].Definition 3.
Let be given an RMd to produce compt.-representations of any given LRBs and let an ARMR be developed to work on the compt.-representations produced by the RMd. R is said to beRWS-interpretable provided that for any LRBB, which is increasingly (or decreasingly) monotonic, R should satisfy the following condition:(15)∀a,ba<b⇒RGRMdBa<RGRMdBbor,∀a,ba<b⇒RGRMdBa>RGRMdBb.The conditions in this definition are natural but very strong for the ARMs developed in the fuzzy set framework. In this study, the so-called semantically weighted interpolation-extrapolation reasoning method proposed in [40] is applied and described as follows.(i) Semantically Weighted Interpolation-Extrapolation Reasoning Method.(A) Interpolation: consider a graphic representation GRMd(B) ofB.(  1) Define semantic weight of the closeness of a given input vector falling in a grid-mesh of G R M d(B)to each grid-mesh point: a restriction is applied to the case of two input variables which is compatible to the HAC-PLL examined in this study. Assume that a mesh of the 2D-grid graphically representingB is exhibited in Figure 7, in which four points of the mesh are calculated from four respective linguistic points (x1i,x2j), (x1(i+1),x2j), (x1i,x2(j+1)), and (x1(i+1),x2(j+1)) and denoted by (a1i,a2j), (a1(i+1),a2j), (a1i,a2(j+1)), and (a1(i+1),a2(i+1)). That is,ak,l = fXk(xk,l),k = 1, 2 andl∈i,i+1,j,j+1. The fXm+1-values at these points are denoted byu11,u12,u21, andu22. To simplify the presentation, these values are used to name the respective points (a1i,a2j), (a1(i+1),a2j), (a1i,a2(j+1)), and (a1(i+1),a2(i+1)) and they indicate the rectangle represented in Figure 7. It is denoted by (u11,u12,u21,u22) and its area being denoted bySij. Consider an input vectora = (a1,a2), whose components fall, respectively, into [a1i,a1(i+1)] and [a2j,a2(j+1)]. The expected valueu of the input vector is computed by the weighted average ofu11,u12,u21, andu22 whose semantic weights are defined based on how close to each mesh point it is: the closer to a point, the larger its semantic weight. The point (a1,a2) decomposes the mesh (u11,u12,u21,u22) into four rectangles. The area of any of the rectangles determined by any pointc = (c1,c2) is denoted by the following rule: if it is located opposite to the position of the valueukl, it is denoted by Sklc. In Figure 7, S11a is opposite tou11, S21a is opposite tou21, S12a′ and S12b′ are opposite tou12, and so on.Figure 7
Semantically weighted inter-extrapolative reasoning method onG(B).The semantic weight of the closeness ofa to a mesh point (a1k,a2l), whose fXm+1-value isukl, denoted by wa(ukl)=wa(a1k,a2l), is defined by the following product:(16)waa1k,a2l=1-a1-a1ka1i+1-a1i1-a2-a2la2j+1-a2j=a1i+1-a1i-a1-a1ka2j+1-a2j-a2-a2lSijThe product in the numerator of (16) is just the area of the rectangle Skla opposite to the position ofukl, where no confusion occurs if the area of a rectangleSij is denoted also bySij.(  2) Compute the control value using the semantically weighted average: for the given input vectora,u is computed as the semantically weighted average as follows: (17)ua=S11au11+S12au12+S21au21+S22au22Sij(B) Extrapolation: assume that a′ falls outside of the 2D-grid GRMd(B). There are two cases. The first one is that a′ falls into a rectangle which have exactly two vertices belonging to GRMd(B), say the rectangle (u01,u02,u11,u12) in Figure 7. Similar as above,u is computed by (18)ua′=S11a′u11+S12a′u12S11a′+S12a′For a′ falling into a rectangle outside of GRMd(B) having exactly one common vertex with GRMd(B), e.g., the vertexu11 of the rectangle domain of the producte ×ce, similar as above,u is computed by(19)ua′=S11a′u11S11a′=u11Theorem 4.
For the graphical RMdM, the semantically weighted interpolation-extrapolation (SWIE) reasoning method working withMis RWS-interpretable; i.e., it preserves the monotonicity property of the graphic representation GRMd(B), for any LRBB.Proof.
LetB be increasing; thus, so is GRMd(B). In the case of interpolation, it is sufficient to consider the case where two input vectorsa = (a1,a2) andb = (b1,b2) in question satisfya ≤b and all fall into the same mesh of the grid GRMd(B). Based on these vectors, the rectangle (u11,u12,u21,u22) is divided into nine rectangles indicated by S22a, S22b∩S21a, S21b∩S21a, S22b∩S12a, S22b∩S11a, S21b∩S11a, S12b∩S12a, S12b∩S11a and S11a, as computed and shown in Figure 7. By the method SWIE, the outputsua andub are computed by(20)ua=S11au11+S12au12+S21au21+S22au22Sijandub=S11bu11+S12bu12+S21bu21+S22bu22SijExamine now the sign of the following quantity:(21)Sijub-ua=S11b-S11au11+S12b-S12au12+S21b-S21au21+S22b-S22au22Considering the above nine rectangles as elementary ones, it can be seen that(22)S11a=S22b∩S11a+S21b∩S11a+S12b∩S11a+S11bS12b=S12b∩S12a+S12b∩S11aS21a=S21b∩S21a+S21b∩S11aS22b=S22a+S22b∩S21a+S22b∩S12a+S22b∩S11aBy (22), the quantity (21) can be reduced to (23)Sijub-ua=S12b∩S11au12-u11+S21b∩S11au21-u11+S22b∩S11au22-u11+S22b∩S12au22-u12+S22b∩S21au22-u21By the increasing monotonicity of GRMd(B), it holdsu11 ≤u12 ≤u22 andu11 ≤u21 ≤u22 that leads to Sij(ub-ua)≥0.
In the case of extrapolation, assume for example that the input vectorsa′ = (a1, a2′) and b′ = (b1, b2′) fall into the rectangle (u01,u02,u11,u12). To simplify and to easily read the proof expressions, it is imagined that this rectangle is linearly transformed into the unit-rectangle whose vertexu01 locates at the center of the coordinate system. If it is proved that ua′ ≤ ub′ in the unit-rectangle, then, by the linear transformation, this respective inequality is also valid in the original rectangle. So, for simplicity, imagine thata andb are already linearly transformed in the unit-rectangle; i.e.,u01 = (0, 0),u02 = (0, 1), andu11 = (0, 1). By (18), it holds (24)ub′-ua′=S11b′u11+S12b′u12S11b′+S12b′-S11a′u11+S12a′u12S11a′+S12a′=1-b1b2′u11+b1b2′u121×b2′-1-a1a2′u11+a1a2′u121×a2′=a2′1-b1b2′u11+a2′b1b2′u12-1-a1a2′b2′u11+a1a2′b2′u12a2′×b2′=a2′b2′b1-a1u12-u11a2′×b2′≥0As the remaining cases are proved in a similar way, it can be concluded that the method SWIE preserves the monotonicity property of the given LRBB. By Definition 3, the method SWIE is RWS-interpretable.


### 3.3.3. Genetic Algorithm to Design Optimized HAC for MAF-PLL

In this section, it is described how to develop a genetic algorithm (GA), denoted HAC-GA, to designopHAC for the MAF-PLL with HAC. The HAC-GA has three tasks: (i) to optimize LRBs, (ii) to optimize the variable fuzziness parameters, and (iii) to optimize the variable ranges. So, the HAC-GA has a functionality to simultaneously produce optimized LRB, optimized fuzziness parameters, and optimized variables ranges. To develop such a HAC-GA the following tasks should be performed.Task 1. Encode the Solution of the Optimization Problem of HACs.(a) Solutions of the optimization problem of HACs: each solution comprises three groups: LRB, fuzziness parameters, and ranges of the variablese,ce, andu. Assume that the HAs of these variables are syntactically the same; i.e., they have the same set of atomic wordsG = N (negative), P (positive) and the same set of hedgesH = L, V. Hence, Dom(e) = Dom(ce) = VN,LLN,ZE,LLP,VP and Dom(u) = VVN,VN,LVN,N,LLN,LN,VLN,ZE,VLP,LP,LLP,LVP,VP,VVP, which areordered from left to right. The limitation of Dom(e) and Dom(ce) of having only five words aims to be compatible with their five fuzzy sets of the FCs for comparative study in Section 4. A representation of an LRB of the HAC is given in the form of Table 3. It is assumed that this table isantonym in the sense defined as follows.Table 3
An Example of an LRBB.
ce VN LLN ZE LLP VP VN VVN VVN VN N ZE e LLN VVN VN N ZE P ZE VN N ZE P VP LLP N ZE P VP VVP VP ZE P VP VVP VVPFor everyX, two wordsx=hn… h1c andy= h′m…h′1c′ are said to be contradictory or antonyms ifhn… h1 = h′m…h′1 andc, c′ belonging to c–,c+ are antonym. For example,LLP andLLN are antonym. A cell of Table 3 can be written as (x,y,z),x∈ Dom(e),y∈ Dom(ce) andz∈ Dom(u). Two cells (x,y,z) and (x′, y′, z′) are said to be antonym if the pairs of the words (x, x′), (y, y′) and (z, z′) are antonym. Then, an LRB represented by Table 3 is said to be antonym if for any two its cells (x,y,z) and (x′, y′, z′) whose pairs (x, x′) and (y, y′) are antonym, then so is the pair (z, z′). As a consequence, the words of the cells lying on the secondary diagonal of the table are the same and equal to the neutral words (ZE) and, when the upper secondary diagonal of the table is known, the table of rules is completely determined. It can be seen that the LRB given in Table 3 is antonym.(b)Individual coding: Assume that thereal coding is applied. To genetically designopHAC, the following should be coded: its LRBB of the form given in Table 3, the independent fuzziness parameters, and the ranges of the variablese,ce, andu. For their fuzziness parameters and their range values, by the symmetricity of the variable ranges, it is assumed thatfme(c–) =fmce(c–) =fmu(c–) = 0.5 and, hence, the GA to be developed needs to seek for optimized values of μe(L), μce(L), and μu(L). So, the GA needs only three genes to represent the fuzziness parameters and only three genes to represent the variable ranges. For the words of the variableu in Table 3, it is required to know only ten words below the right diagonal of the table which are arranged fromleft to right and fromtop to bottom. For example, the LRBB given in Table 3 is represented by a chromosome of ten genes: (P P VP P VP VVP P VP VVP VVP). Each solution is encoded by an individual of 16 genes: the first three genes represent the values μe(L), μce(L), and μu(L), the next ten genes represent the ten above-mentioned words ofu uniquely determiningB, and the three remaining genes represent three positive variable range values. Assume that the words of Dom(u) = {VVN,VN,LVN,N,LLN,LN,VLN,ZE,VLP,LP,LLP,P,LVP,VP,VVP} are increasingly arranged and encoded by their positions, i.e., by the integers 0, 1, 2, …, 14. For example, three values μe(L), μce(L), and μu(L), three positive variable ranges, rgX’s, and theB given in Table 3 are represented by the solution individual: (μe(L), μce(L), μu(L), 11, 11, 13, 11, 13, 14, 11, 13, 14, 14,rge,rgce,rgu).Task 2. Construct a HAC, an Alternative Component for the FC (1)
Determine the inputs of a HAC They comprise the following:
(i)
the positive and negative hedges, their relative signs, fuzziness parameter values and the positive variable range values; (ii)
an LRBB given in the form of Table 3; (iii)
an ARM— a selected interpolative method working on the surface in0,13, which is computed using the SQMs fe, fce, and fu; (iv)
input vector of the control problem.These input values are needed in each iteration, noting that the values listed in (i) and (ii) are provided by an individual under consideration of a population generated by the HAC-GA.(2)
Construct components of a HAC of the HAC-PLLIts main components are shown in Figure8 which comprises the following:
(i)
the normalization to convert the universes ofe,ce, andu to their respective unit intervals [0,1]’s;(ii)
a given LRBB and its quantification to translateB into a graphic representation approximately modeling a surface Sreal3;(iii)
construct an RWS-interpretable interpolation-extrapolation reasoning method. In this study, the developed method SWIE is applied;(iv)
the denormalization to convert the numeric output ofu computed by the second component of HAC to its respective real value of the variableu.Figure 8
The components of the HAC.Task 3. Construct a GA, Denoted by GA(opHAC), to Produce the Optimized LRB B o p , the Optimized Fuzziness Parameters, and the Optimized Variable Ranges of the Desired opHAC.The HAC developed inTask 1 is a kernel-component of the desiredGA(opHAC).(1)
Inputs ofGA(opHAC):
(i)
The criterionJ1=∫0δe(t)dt (or =∑k=1ne(k), n=δ/Ts), called the performance index (Prf-index) in this study, and the time period of simulation δ for computing the Prf-indexJ1(ii)
GA running parameters: population size; generation number; disturbance type(2)
Outputs ofGA(opHAC):
(i)
The optimized values of the fuzziness parametersμe(L), μce(L), and μu(L) and the optimized variable ranges (-rge,rge), (-rgce,rgce), and (-rgu,rgu)(ii)
The optimized LRBBop(3)
General description ofGA(opHAC):
(i)
Population of solutions:GA(opHAC) will initialize an initial population. Every individual of a population consists of 16 genes, 3 genes for the fuzziness parameters, 3 genes for positive variable ranges values, and 10 genes for the words to determine LRB of the form in Table 3. Genetic operations like crossovers and mutation will be applied to generate the next population of the same size, in which only individuals with better fitness values are retained.(ii)
Method to compute the Prf-indexJ1 of an individual of a given population: the procedure ofGA(opHAC) to computeJ1 of each individual of the population in question, denoted by δ-LoopopHAC, is described by the scheme given in Figure 9.Figure 9
δ-LoopopHAC of the simulation model.(4)
Implementation ofGA(opHAC) in Matlab (see Box 1)Box 1
GA(opHAC) Input: (i) Simulation model ‘HACPLLfinal.slx’ suitable type for the expected disturbance; (ii)J1I=∑e(k)∈Setδ(e(k))e(k); // I is an individual of the population under consideration Output: The optimized HAC: ‘opHAC’; Method: Initialize parameters of the simulation model; Initialize parameters of the GA: num_var = number of variables; pop_size = population size; num_gen = number of generation; time_limit = time value; // to limit the running time of the algorithmGA(opHAC) Initialize an initial population; m = 1; t = 0; while ((m ≤ num_gen) and (t ≤ time_limit)) for each individualIm in the population call simulation model: ‘HACPLLfinal.slx’; computeJ1(Im)=∑e(k)∈Setδ(e(k))e(k); end_for m = m+1; Perform genetic operators; auto_update(t); end_while End_GA(opHAC)


## 3.3.1. A Method to Construct RWS-Interpretable Graphical Representations of LRBs

Consider an LRBB consisting ofn rules ofm input variables and one output variable(14)riIFX1,Lisxi,1&…&XmLisxi,m,THENXm+1,Lisxi,m+1;i=1tonwhere Xj,L’s denote the linguistic variables of the respective RW-variables Xj’s; similarly, Xj,N’s denote the numeric variables of the respective RW-variables Xj’s and their universes are denoted, respectively, byU(Xj,N)’s.As this LRB is generally assumed to be consistent, it describes alinguisticfunctional dependence ofXm+1,L on the remaining ones Xj,L’s defined in the Cartesian product Dom(X1L) × … × Dom(Xm+1,L). Methodologically, the method to construct compt.-representations of LRBs, written shortly as RMd, is crucial because of the technical requirement for studying the RWS-interpretability not only of the LRBs but also of the ARMs working on them. In addition, when methods to genetically design LRBs are examined, they must deal not only with the individual compt.-representations but also with the method itself. Thus, the functionality of an RMd is able to construct from any given LRBB a compt.-representation, denoted by RRMd(B), defined inU(X1,N) × … ×U(Xm+1,N). At the same time, as human language is RWS-interpretable, the linguistic function defined byB does describe an RW-functionfRW ofXm+1 on the remaining Xj’s. Denote byfN the numeric function modeling the samefRW; the both functions must becompatible with each other. This leads to the following constraint on RMd’s interpreted as a test condition whether an RMd is RWS-interpretable.Constraint on RMd: the output RRMd(B) of RMd must be a numeric function defined in the Euclid spaceU(X1,N) ×… ×U(Xm+1,N), denoted by fN,RMd(B), and if the linguistic functionB is increasingly or decreasingly monotonic function, then so is the fN,RMd(B).This compt.-representationfN,RMd(B) of the LRBB is calledgraphic representation ofB. It can be seen that it is an aggregation of the elementary linguistic predicates given inB. It should be emphasized that, in accordance with our knowledge, no aggregation operators aggregating these (fuzzy) linguistic predicates can preserve such an RW-semantics represented by the LRBB like this graphic representation. In relation to this, it is observed that practical applications of the math-theories and the theoretical physics theories show that the graphic representations of functions in Euclid space applied in these theories are RWS-interpretable.In this section, an RMd developed in the HA-formalism to construct graphic representation of LRBs, called graphical RMd and denoted byM, is shortly described as follows:(i)
Determine HAs of Dom(Xj,L),j = 1 tom+1, by selecting their negative and positive hedges and determining their relative signs and their fuzziness parameter values.(ii)
With these, the SQMsfXj,L’s are defined and map then linguistic points determined by the LRB ofB in Dom(X1,L) × … × Dom(Xm+1,L) inton points in the Euclid space 0,1m+1 that forms a grid of points, denoted byGM(B).To examine the RWS-interpretability ofGRMd(B) some notations are needed. Leta = (a1, …,am,am+1) be a point in 0,1m+1. The project ofa on the Cartesian space indicated byXi1,N × … ×Xik,N; i.e., aXi1×…×Xik= (ai1, …,aik) is denoted by aXi1×…×Xik. For any RMd, a point of GRMd(B) corresponding to a ruleri is denoted by GRMd(ri). By the constraint above, the RWS-interpretability of an RMd can be defined as follows, noting that the LRBs in many control applications are monotonic.Definition 1.
A given method RMd to produce a compt.-representation in a Cartesian productSm+1 from a given LRB is said to be RWS-interpretable provided that ifB is increasingly (decreasingly) monotonic, then(i)
GRMd(B) ⊆U(X1,N) × … ×U(Xm+1,N) must define a function. That is, for every ruleri ofB, GRMd(ri) is of the formai = (ai,1, …,ai,m,ai,+1), fori = 1, …,n, and for any such two vectorsaj = (aj,1, …,aj,m,aj,m+1) and aj′ = (aj′,1, …, aj′,m, aj′,m+1), the equality ajX1×…×Xm+1 = aj′X1×…×Xm implies that ajXm+1 = aj′Xm+1.(ii)
ForB being an increasingly (or decreasingly) monotonic function and itsi-th ruleri is denoted by xi = (xi,1, …,xi,m,xi,m+1), it holds xiX1×…×Xm ≤ xjX1×…×Xm⟹GRMdriXm+1 ≤ GRMdrjXm+1 (or, xiX1×…×Xm ≤ xjX1×…×Xm⟹GRMdriXm+1 ≥ GRMdrjXm+1).
For the RMdM, as the determined SQMs are isomorphisms, it can easily be verified that the monotonicity of a givenB is preserved byM; i.e., theGM(B) has the same monotonicity property asB. So, the following proposition holds:Proposition 2.
The RMdM proposed as above is RWS-interpretable.


## 3.3.2. The RWS-Interpretability of Approximate Reasoning Method

Each ARM should be developed accompanied with a given RMd to solve an application problem and it also plays a key role to construct an RWS-interpretable fuzzy system. It strongly depends on a given LRBB as well as on a given RMd. Therefore, to define the RWS-interpretability of a given ARM, R, it is assumed that the given RMd accompanied with R is RWS-interpretable. LetB be an LRB of the form (14). For any input vectora = (a1, …,am) inX1,N × … ×Xm,N, denote by R(GRMd(B))(a) the output value ofa in Dom(Xm+1,N) produced by R working on GRMd(B). Then, the following definition may be introduced which is a modified version of the respective one examined in [30].Definition 3.
Let be given an RMd to produce compt.-representations of any given LRBs and let an ARMR be developed to work on the compt.-representations produced by the RMd. R is said to beRWS-interpretable provided that for any LRBB, which is increasingly (or decreasingly) monotonic, R should satisfy the following condition:(15)∀a,ba<b⇒RGRMdBa<RGRMdBbor,∀a,ba<b⇒RGRMdBa>RGRMdBb.The conditions in this definition are natural but very strong for the ARMs developed in the fuzzy set framework. In this study, the so-called semantically weighted interpolation-extrapolation reasoning method proposed in [40] is applied and described as follows.(i) Semantically Weighted Interpolation-Extrapolation Reasoning Method.(A) Interpolation: consider a graphic representation GRMd(B) ofB.(  1) Define semantic weight of the closeness of a given input vector falling in a grid-mesh of G R M d(B)to each grid-mesh point: a restriction is applied to the case of two input variables which is compatible to the HAC-PLL examined in this study. Assume that a mesh of the 2D-grid graphically representingB is exhibited in Figure 7, in which four points of the mesh are calculated from four respective linguistic points (x1i,x2j), (x1(i+1),x2j), (x1i,x2(j+1)), and (x1(i+1),x2(j+1)) and denoted by (a1i,a2j), (a1(i+1),a2j), (a1i,a2(j+1)), and (a1(i+1),a2(i+1)). That is,ak,l = fXk(xk,l),k = 1, 2 andl∈i,i+1,j,j+1. The fXm+1-values at these points are denoted byu11,u12,u21, andu22. To simplify the presentation, these values are used to name the respective points (a1i,a2j), (a1(i+1),a2j), (a1i,a2(j+1)), and (a1(i+1),a2(i+1)) and they indicate the rectangle represented in Figure 7. It is denoted by (u11,u12,u21,u22) and its area being denoted bySij. Consider an input vectora = (a1,a2), whose components fall, respectively, into [a1i,a1(i+1)] and [a2j,a2(j+1)]. The expected valueu of the input vector is computed by the weighted average ofu11,u12,u21, andu22 whose semantic weights are defined based on how close to each mesh point it is: the closer to a point, the larger its semantic weight. The point (a1,a2) decomposes the mesh (u11,u12,u21,u22) into four rectangles. The area of any of the rectangles determined by any pointc = (c1,c2) is denoted by the following rule: if it is located opposite to the position of the valueukl, it is denoted by Sklc. In Figure 7, S11a is opposite tou11, S21a is opposite tou21, S12a′ and S12b′ are opposite tou12, and so on.Figure 7
Semantically weighted inter-extrapolative reasoning method onG(B).The semantic weight of the closeness ofa to a mesh point (a1k,a2l), whose fXm+1-value isukl, denoted by wa(ukl)=wa(a1k,a2l), is defined by the following product:(16)waa1k,a2l=1-a1-a1ka1i+1-a1i1-a2-a2la2j+1-a2j=a1i+1-a1i-a1-a1ka2j+1-a2j-a2-a2lSijThe product in the numerator of (16) is just the area of the rectangle Skla opposite to the position ofukl, where no confusion occurs if the area of a rectangleSij is denoted also bySij.(  2) Compute the control value using the semantically weighted average: for the given input vectora,u is computed as the semantically weighted average as follows: (17)ua=S11au11+S12au12+S21au21+S22au22Sij(B) Extrapolation: assume that a′ falls outside of the 2D-grid GRMd(B). There are two cases. The first one is that a′ falls into a rectangle which have exactly two vertices belonging to GRMd(B), say the rectangle (u01,u02,u11,u12) in Figure 7. Similar as above,u is computed by (18)ua′=S11a′u11+S12a′u12S11a′+S12a′For a′ falling into a rectangle outside of GRMd(B) having exactly one common vertex with GRMd(B), e.g., the vertexu11 of the rectangle domain of the producte ×ce, similar as above,u is computed by(19)ua′=S11a′u11S11a′=u11Theorem 4.
For the graphical RMdM, the semantically weighted interpolation-extrapolation (SWIE) reasoning method working withMis RWS-interpretable; i.e., it preserves the monotonicity property of the graphic representation GRMd(B), for any LRBB.Proof.
LetB be increasing; thus, so is GRMd(B). In the case of interpolation, it is sufficient to consider the case where two input vectorsa = (a1,a2) andb = (b1,b2) in question satisfya ≤b and all fall into the same mesh of the grid GRMd(B). Based on these vectors, the rectangle (u11,u12,u21,u22) is divided into nine rectangles indicated by S22a, S22b∩S21a, S21b∩S21a, S22b∩S12a, S22b∩S11a, S21b∩S11a, S12b∩S12a, S12b∩S11a and S11a, as computed and shown in Figure 7. By the method SWIE, the outputsua andub are computed by(20)ua=S11au11+S12au12+S21au21+S22au22Sijandub=S11bu11+S12bu12+S21bu21+S22bu22SijExamine now the sign of the following quantity:(21)Sijub-ua=S11b-S11au11+S12b-S12au12+S21b-S21au21+S22b-S22au22Considering the above nine rectangles as elementary ones, it can be seen that(22)S11a=S22b∩S11a+S21b∩S11a+S12b∩S11a+S11bS12b=S12b∩S12a+S12b∩S11aS21a=S21b∩S21a+S21b∩S11aS22b=S22a+S22b∩S21a+S22b∩S12a+S22b∩S11aBy (22), the quantity (21) can be reduced to (23)Sijub-ua=S12b∩S11au12-u11+S21b∩S11au21-u11+S22b∩S11au22-u11+S22b∩S12au22-u12+S22b∩S21au22-u21By the increasing monotonicity of GRMd(B), it holdsu11 ≤u12 ≤u22 andu11 ≤u21 ≤u22 that leads to Sij(ub-ua)≥0.
In the case of extrapolation, assume for example that the input vectorsa′ = (a1, a2′) and b′ = (b1, b2′) fall into the rectangle (u01,u02,u11,u12). To simplify and to easily read the proof expressions, it is imagined that this rectangle is linearly transformed into the unit-rectangle whose vertexu01 locates at the center of the coordinate system. If it is proved that ua′ ≤ ub′ in the unit-rectangle, then, by the linear transformation, this respective inequality is also valid in the original rectangle. So, for simplicity, imagine thata andb are already linearly transformed in the unit-rectangle; i.e.,u01 = (0, 0),u02 = (0, 1), andu11 = (0, 1). By (18), it holds (24)ub′-ua′=S11b′u11+S12b′u12S11b′+S12b′-S11a′u11+S12a′u12S11a′+S12a′=1-b1b2′u11+b1b2′u121×b2′-1-a1a2′u11+a1a2′u121×a2′=a2′1-b1b2′u11+a2′b1b2′u12-1-a1a2′b2′u11+a1a2′b2′u12a2′×b2′=a2′b2′b1-a1u12-u11a2′×b2′≥0As the remaining cases are proved in a similar way, it can be concluded that the method SWIE preserves the monotonicity property of the given LRBB. By Definition 3, the method SWIE is RWS-interpretable.


## 3.3.3. Genetic Algorithm to Design Optimized HAC for MAF-PLL

In this section, it is described how to develop a genetic algorithm (GA), denoted HAC-GA, to designopHAC for the MAF-PLL with HAC. The HAC-GA has three tasks: (i) to optimize LRBs, (ii) to optimize the variable fuzziness parameters, and (iii) to optimize the variable ranges. So, the HAC-GA has a functionality to simultaneously produce optimized LRB, optimized fuzziness parameters, and optimized variables ranges. To develop such a HAC-GA the following tasks should be performed.Task 1. Encode the Solution of the Optimization Problem of HACs.(a) Solutions of the optimization problem of HACs: each solution comprises three groups: LRB, fuzziness parameters, and ranges of the variablese,ce, andu. Assume that the HAs of these variables are syntactically the same; i.e., they have the same set of atomic wordsG = N (negative), P (positive) and the same set of hedgesH = L, V. Hence, Dom(e) = Dom(ce) = VN,LLN,ZE,LLP,VP and Dom(u) = VVN,VN,LVN,N,LLN,LN,VLN,ZE,VLP,LP,LLP,LVP,VP,VVP, which areordered from left to right. The limitation of Dom(e) and Dom(ce) of having only five words aims to be compatible with their five fuzzy sets of the FCs for comparative study in Section 4. A representation of an LRB of the HAC is given in the form of Table 3. It is assumed that this table isantonym in the sense defined as follows.Table 3
An Example of an LRBB.
ce VN LLN ZE LLP VP VN VVN VVN VN N ZE e LLN VVN VN N ZE P ZE VN N ZE P VP LLP N ZE P VP VVP VP ZE P VP VVP VVPFor everyX, two wordsx=hn… h1c andy= h′m…h′1c′ are said to be contradictory or antonyms ifhn… h1 = h′m…h′1 andc, c′ belonging to c–,c+ are antonym. For example,LLP andLLN are antonym. A cell of Table 3 can be written as (x,y,z),x∈ Dom(e),y∈ Dom(ce) andz∈ Dom(u). Two cells (x,y,z) and (x′, y′, z′) are said to be antonym if the pairs of the words (x, x′), (y, y′) and (z, z′) are antonym. Then, an LRB represented by Table 3 is said to be antonym if for any two its cells (x,y,z) and (x′, y′, z′) whose pairs (x, x′) and (y, y′) are antonym, then so is the pair (z, z′). As a consequence, the words of the cells lying on the secondary diagonal of the table are the same and equal to the neutral words (ZE) and, when the upper secondary diagonal of the table is known, the table of rules is completely determined. It can be seen that the LRB given in Table 3 is antonym.(b)Individual coding: Assume that thereal coding is applied. To genetically designopHAC, the following should be coded: its LRBB of the form given in Table 3, the independent fuzziness parameters, and the ranges of the variablese,ce, andu. For their fuzziness parameters and their range values, by the symmetricity of the variable ranges, it is assumed thatfme(c–) =fmce(c–) =fmu(c–) = 0.5 and, hence, the GA to be developed needs to seek for optimized values of μe(L), μce(L), and μu(L). So, the GA needs only three genes to represent the fuzziness parameters and only three genes to represent the variable ranges. For the words of the variableu in Table 3, it is required to know only ten words below the right diagonal of the table which are arranged fromleft to right and fromtop to bottom. For example, the LRBB given in Table 3 is represented by a chromosome of ten genes: (P P VP P VP VVP P VP VVP VVP). Each solution is encoded by an individual of 16 genes: the first three genes represent the values μe(L), μce(L), and μu(L), the next ten genes represent the ten above-mentioned words ofu uniquely determiningB, and the three remaining genes represent three positive variable range values. Assume that the words of Dom(u) = {VVN,VN,LVN,N,LLN,LN,VLN,ZE,VLP,LP,LLP,P,LVP,VP,VVP} are increasingly arranged and encoded by their positions, i.e., by the integers 0, 1, 2, …, 14. For example, three values μe(L), μce(L), and μu(L), three positive variable ranges, rgX’s, and theB given in Table 3 are represented by the solution individual: (μe(L), μce(L), μu(L), 11, 11, 13, 11, 13, 14, 11, 13, 14, 14,rge,rgce,rgu).Task 2. Construct a HAC, an Alternative Component for the FC (1)
Determine the inputs of a HAC They comprise the following:
(i)
the positive and negative hedges, their relative signs, fuzziness parameter values and the positive variable range values; (ii)
an LRBB given in the form of Table 3; (iii)
an ARM— a selected interpolative method working on the surface in0,13, which is computed using the SQMs fe, fce, and fu; (iv)
input vector of the control problem.These input values are needed in each iteration, noting that the values listed in (i) and (ii) are provided by an individual under consideration of a population generated by the HAC-GA.(2)
Construct components of a HAC of the HAC-PLLIts main components are shown in Figure8 which comprises the following:
(i)
the normalization to convert the universes ofe,ce, andu to their respective unit intervals [0,1]’s;(ii)
a given LRBB and its quantification to translateB into a graphic representation approximately modeling a surface Sreal3;(iii)
construct an RWS-interpretable interpolation-extrapolation reasoning method. In this study, the developed method SWIE is applied;(iv)
the denormalization to convert the numeric output ofu computed by the second component of HAC to its respective real value of the variableu.Figure 8
The components of the HAC.Task 3. Construct a GA, Denoted by GA(opHAC), to Produce the Optimized LRB B o p , the Optimized Fuzziness Parameters, and the Optimized Variable Ranges of the Desired opHAC.The HAC developed inTask 1 is a kernel-component of the desiredGA(opHAC).(1)
Inputs ofGA(opHAC):
(i)
The criterionJ1=∫0δe(t)dt (or =∑k=1ne(k), n=δ/Ts), called the performance index (Prf-index) in this study, and the time period of simulation δ for computing the Prf-indexJ1(ii)
GA running parameters: population size; generation number; disturbance type(2)
Outputs ofGA(opHAC):
(i)
The optimized values of the fuzziness parametersμe(L), μce(L), and μu(L) and the optimized variable ranges (-rge,rge), (-rgce,rgce), and (-rgu,rgu)(ii)
The optimized LRBBop(3)
General description ofGA(opHAC):
(i)
Population of solutions:GA(opHAC) will initialize an initial population. Every individual of a population consists of 16 genes, 3 genes for the fuzziness parameters, 3 genes for positive variable ranges values, and 10 genes for the words to determine LRB of the form in Table 3. Genetic operations like crossovers and mutation will be applied to generate the next population of the same size, in which only individuals with better fitness values are retained.(ii)
Method to compute the Prf-indexJ1 of an individual of a given population: the procedure ofGA(opHAC) to computeJ1 of each individual of the population in question, denoted by δ-LoopopHAC, is described by the scheme given in Figure 9.Figure 9
δ-LoopopHAC of the simulation model.(4)
Implementation ofGA(opHAC) in Matlab (see Box 1)Box 1
GA(opHAC) Input: (i) Simulation model ‘HACPLLfinal.slx’ suitable type for the expected disturbance; (ii)J1I=∑e(k)∈Setδ(e(k))e(k); // I is an individual of the population under consideration Output: The optimized HAC: ‘opHAC’; Method: Initialize parameters of the simulation model; Initialize parameters of the GA: num_var = number of variables; pop_size = population size; num_gen = number of generation; time_limit = time value; // to limit the running time of the algorithmGA(opHAC) Initialize an initial population; m = 1; t = 0; while ((m ≤ num_gen) and (t ≤ time_limit)) for each individualIm in the population call simulation model: ‘HACPLLfinal.slx’; computeJ1(Im)=∑e(k)∈Setδ(e(k))e(k); end_for m = m+1; Perform genetic operators; auto_update(t); end_while End_GA(opHAC)


## 4. Results and Discussion

In this section, the performance of the proposed HAC-PLL is validated through experimental studies with dSpace DS1103 R&D board. For this purpose, all the PLLs considered in this study were programmed in the MATLAB/Simulink environment. The comparison of the results obtained by the developed HAC-PLL with those obtained by both the PID-based MAF-PLL and the FC-based MAF-PLL is also shown.In addition to real-time execution of the analyzed PLL algorithms, the R&D board was used for grid emulation as well. In this regard, the D/A converters of the R&D board were utilized for generation of the phase voltages, but with reduced amplitude. The amplitude of the nondistorted phase voltages has been selected to cover ±10 V output range of the D/A converters when disturbances occur. The phase voltages were then sampled at 40 kHz sampling frequency by using the A/D converters of the same board. The sampling frequency of the analyzed PLLs was set to 10 kHz taking into consideration that the fastest time-varying disturbance analyzed in this paper has an oscillation frequency less than 5 kHz.Performance of the analyzed MAF-based PLL was tested under the following four conditions:( 1 ) Unbalanced voltage sag( 2 ) Unbalanced voltage sag with harmonics distortion (i.e., positive 5th and 7th order harmonics have been added with the amplitudes equal to 8% of the 1st phase voltage harmonic)( 3 ) Unbalanced transient in the phase voltages with the oscillation frequency of 500 Hz( 4 ) Step change of the grid frequency of +5 Hz and back againFor the disturbances(1), (2), and (4), the amplitude of the nondistorted phase voltages has been selected 8.6 V, whereas for the disturbance (3), it has been selected 1.7 V.Parameters of the FC and the HAC are obtained in simulations as mentioned in Sections2.2 and 3.3. Parameters of the analyzed disturbances are in accordance with the IEC classification of electromagnetic phenomena [42]. Distorted phase voltages and responses of the analyzed PLLs are shown in Figures 10–13. In this paper, it is selected that each disturbance lasts 200 ms exactly.Figure 10
Phase voltages (a), phase error (b), filtered quadrature voltage componentu-q (c), and integral of absolute error (d) for the unbalanced voltage sag in phasesa andc.
(a) (b) (c) (d)Figure 11
Phase voltages (a), phase error (b), filtered quadrature voltage componentu-q (c), and integral of absolute error (d) for the unbalanced voltage sag in phasesa andc with harmonics distortion.
(a) (b) (c) (d)Figure 12
Phase voltages (a), phase error (b), filtered quadrature voltage componentu-q (c), and integral of absolute error (d) for the unbalanced low-frequency transient in phase voltages.
(a) (b) (c) (d)Figure 13
Estimated grid frequency (a), phase error (b), filtered quadrature voltage componentu-q (c), and integral of absolute error (d) for the step-changes in frequency of ±5 Hz.
(a) (b) (c) (d)Figure10 shows the recorded responses under the unbalanced voltage sag in phasesa andc. The amplitudes of the faulted phasesa andc are 80% and 92% of the nondistorted phase voltages, respectively. Figures 10(b), 10(c), and 10(d) show traces of the phase error, the filtered quadrature voltage componentuq, and the IAE of the controllers, respectively. The HAC controller ensures the lowest IAE value - 4% lower compared to the FC and 44% lower compared to the PID.The phase voltages with harmonics under the unbalanced voltage sag in phasesa andc are considered in Figure 11.The traces of the phase error (Figure11(b)), the quadrature voltage componentuq (Figure 11(b)), and the IAE (Figure 11(c)) are very similar to the corresponding traces shown in Figure 10. This means that the influence of the harmonics is negligible. Again, the HAC controller ensures the lowest IAE value.The PLL response to the unbalanced transient in phase voltages with the oscillation frequency of 500 Hz is considered in Figure12. In this case, and in this case only, the HAC does not ensure the lowest IAE value—it is about 12% higher in comparison with the one obtained by the FC, but it is still lower compared to the PID controller.Figure13 shows the PLL response to step-changes in the grid frequency. Figure 13(a) shows the estimated grid frequency obtained by the nonadaptive MAF-PLL (Figure 2) with the PID controller and the estimated grid frequency obtained with the proposed PLL configuration (Figure 4) and with the HAC and FCs. Note that the nonadaptive MAF was here used in combination with the PID controller instead of the adaptive MAF because it was observed that the latter results in slightly higher IAE value.Choosing the settling time with a 2% criterion (i.e., the time after which the estimated frequency reaches and remains within the limits 49.9 Hz-50.1 Hz), 2.1 times shorter settling time is obtained in the PLL with the PID controller in comparison with the HAC-based PLL (Figure13(a)). However, the maximum phase error obtained by the HAC is 5.4 times lower than the one obtained by the PID controller (Figure 13(b)). On the other hand, when compared with the FC-based MAF-PLL, the HAC-based PLL has approximately the same settling time (Figure 13(a)) but 1.9 times lower maximum phase error (Figure 13(b)). Once more, the lowest IAE value is achieved by using the HAC-based PLL.For all analyzed PLLs, the possible occurrence of a DC-offset in phase voltages causes fundamental frequency oscillations in steady state in all estimated variables, causing incorrect operation of the proposed PLL. This effect will be the subject of a future research. On the other hand, a small DC component (up to 50 mV) caused by the A/D conversion process was present in the experiments, but its influence on the PLL performance was shown to be negligible.


## 5. Conclusion

In this paper, the new MAF-based PLLs with the HAC are successfully applied for the three-phase distorted signals. They can handle different types of disturbances and ensure zero steady-state error. Optimized fuzziness parameters and linguistic rule base of the HAC are successfully obtained by means of the genetic algorithm for each analyzed disturbance. The HAC with its parameters optimized for a certain disturbance can handle other analyzed disturbances as well, although with somewhat slower dynamic responses.The experimental comparison of the developed HAC-PLLs with both the PID-based MAF-PLL and the FC-based MAF-PLL has been carried out. It is shown that the HAC and the FC provide better dynamic performance in comparison to the PID controller, for all analyzed disturbance types. In addition, the HAC ensures the lowest IAE value for all of the analyzed disturbance types except for the unbalanced low-frequency transients, when the FC achieves the lowest IAE value (12% lower than the HAC). However, in this case, the maximum difference between the phase errors obtained by the HAC and FCs is less than 3°, which is considered negligible.The proposed PLLs do not work properly when there is a DC-offset present in phase voltages and this will be the subject of a future research. On the other hand, the influence of a small DC component (up to 50 mV) caused by the A/D conversion process is shown to be negligible.



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*Source: 3590527-2019-03-03.xml*