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Some studies tried to make transition from Type1 to interval Type2 membership functions, but they get the problems of choosing the footprint uncertainty size in the Interval Type2 Membership Functions. In this paper, our objective is to employ two optimization methods: Invasive Weed Optimization (IWO) and Particle Swarm Optimization (PSO) for tuning the Transitioning from Type1 To Interval Type2 Fuzzy Logic Controller for Boost DCDC Converters and compare their performances. Also, we will discuss the effects of the PID values in the operation of transition from Type1 to interval Type2 fuzzy logic Controller for Boost DCDC Converters. The simulation results show IWO optimization methods is helpful to Tuning the Transitioning from Type1 To Interval Type2 Fuzzy Logic Controller for Boost DCDC Converters. Moreover, when we tune both PID values and the FOU size in T2MFs of Interval Type2 Fuzzy Logic PIDcontroller, we will get the best performance for interval Type2 Fuzzy Logic PIDcontroller for Boost DCDC Converter. To sum up; the optimal footprint of uncertainty (FOU) size in interval Type2 membership functions, they have an essential role and good effect in the performance of Interval Type2 fuzzy logic Controller for Boost DCDC Converters.
IWO, Type2 fuzzy logic controller, Type1 membership functions, interval Type2 membership functions, footprint of uncertainty (FOU)
The membership functions (MFs) enable establishing a relationship between numerical values and linguistic labels.
Type1 fuzzy MFs (T1MF) are two dimensional and represent the membership degree μ for a variable x. Type2 fuzzy MFs (T2MF) are three dimensional:
They consider an uncertainty U of the membership degree. T1MFs are a particular case of T2MFs where the uncertainty value is 0. Membership functions are classified as (Figure 1 and Figure 2):
The concept of Type2 fuzzy set was initially proposed as an extension ofType1 fuzzy set by Prof. Zadeh [1].
Figure 1. Membership functions (a) singleton, (b) Interval Type1, (c) Type1
(a)
(b)
Figure 2. Membership functions (a) interval Type2, (b) general Type2
The Type2 fuzzy system is characterized by a fuzzy membership function, i.e., the membership grade for each element of this set is a fuzzy set in [0,1], contrary a Type1 fuzzy set where the membership grade is a crisp number in [0,1]. Such sets are very useful in circumstances where they are difficult to determine an exact membership function for a fuzzy set; hence, they are useful for incorporating uncertainties. Type2 fuzzy sets are appropriate for modeling uncertainty as Type2 fuzzy sets include FOU (Footprint of Uncertainty) and third dimension, offering extra degrees of freedom to Type2 fuzzy sets in comparison to Type1 fuzzy sets [2, 3].
The performance of Type2 Fuzzy Logic Controller (T1 FLC) is affected by the Footprint of uncertainty size in interval Type2 membership functions [48].
Some studies propose [7, 8] to transition from typel to interval Type2 fuzzy sets, through varying the size FOU (Footprint of uncertainty) in interval Type2 membership functions, but there are problems of tuning the footprint of uncertainty size parameter in The Interval Type2 Membership Functions.
In this work, we developed a new approach to transit from Type1 to interval Type2 fuzzy logic Controller using the optimization methods (IWO and PSO). Also, we used the optimization methods (IWO and PSO) for tuning the PID values and the footprint uncertainty size in the Interval Type2 Membership Functions of Interval Type2 Fuzzy Logic PID Controller for Boost DCDC Converters.
This paper will be organized as following:
Section 2: Converter modeling. section 3: The Type2 TSK Fuzzy Logic Controller. section 4: Invasive Weed Optimization Algorithm. section 5: Fuzzy PID Controller for Boost DCDC Converters. section 6: Simulation Phases and Results. section 7: Conclusion.
In this study we will use the DC–DC Boost converter (Figure 3 and Table 1).
Figure 3. The DC–DC boost converter
Table 1. The parameters of the DC–DC boost converter
the parameters of 
the DC–DC Boost converter 
Series Inductance 
$\mathrm{L}=20[\mathrm{mH}]$. 
Parallel Capacitance 
$\mathrm{C}=20[\mu \mathrm{f}]$ 
load resistance 
$\mathrm{R}=30[\Omega]$ 
Input Voltage 
$\mathrm{Vg}=15[\mathrm{V}]$ 
Switching frequency 
$\mathrm{f}_{\mathrm{SW}}=5[\mathrm{kHz}]$. 
V_{g}: Represents power supply voltage. i_{L}: The current through the inductance L. sw: An electronic switch. VD: Voltage of the diode. V_{c}: The voltage on the capacitor C. u_{0}(t) Voltage output across the resistive load R.
Continuous conduction mode (CCM) has two topologies depending on the position of switch swand. In this simulation, we use the Boost DCDC Converter operating in continuous conduction mode (CCM).
The following equations represent the first topology (Figure 4) of DC–DC Boost converter:
$L \frac{d}{d t} \mathrm{i}_{L}(t)=V_{\mathrm{g}}(t)$
$\frac{d}{d t} v_{\mathrm{c}}(t)=\frac{1}{C R} v_{c}(t)$ (1)
$u_{0}(t)=v_{c}(t)$
Figure 4. The first topology (sw closed, VD opened)
The following equations represent the second topology (Figure 5) of DC–DC Boost converter:
Figure 5. The second topology (sw opened, VD closed)
$L \frac{d}{d t} \mathrm{i}_{L}(t)=v_{\mathrm{g}}(t)v_{c}(t)$
$\frac{d}{d t} v_{c}(t)=\frac{1}{C R}\left(R \mathrm{i}_{L}(t)v_{\mathrm{c}}(t)\right)$ (2)
$u_{0}(t)=v_{c}(t)$
The state equation of the boost DCDC converter can be stated as [9]:
$\dot{x}=A_{i} x+B_{i} V_{\mathrm{g}}(t)$
$u_{0}=C_{i} x$ with $x=\left[i_{L} v_{\mathrm{c}}\right]^{T}$ (3)
where, subscript 1 stands for transistor ON, and subscript 2 stands for transistor OFF of the converter circuit.
$A_{i}, B_{i}$ and $C_{i}$ are system Matrices of the constituent linear circuits.
The system matrices can be obtained for different operating modes as:
$C_{1}=[0 \quad 1], C_{2}=[0 \quad 1] \cdot A_{1}=\left[\begin{array}{cc}0 & 0 \\ 0 & \frac{1}{R C}\end{array}\right],$ and $A_{2}=$$\left[\begin{array}{cc}0 & \frac{1}{L} \\ \frac{1}{c} & \frac{1}{R C}\end{array}\right] \cdot B_{1}=B_{2}=\left[\frac{1}{L} 0\right]^{T}$
The StateSpace Averaged model represented in the following equations [1013]:
$\left\{\begin{array}{c}\dot{x}=A_{a v g} x+B_{a v g} u \\ a n d \\ y=C_{a v g} x\end{array}\right.$ (4)
where, $\left\{\begin{array}{l}A_{a v g}=d A_{1}+(1d) A_{2} \\ B_{a v g}=d B_{1}+(1d) B_{2} \\ C_{a v g}=d C_{1}+(1d) C_{2}\end{array}\right.$
A Type2 TSK fuzzy logic controller was firstly introduced by Mendel and Liang. We have three models of T2 TSK fuzzy logics based on the type of the antecedent and consequent part of rules [1416] (Table 2 and Table 3).
Table 2. Classification of other t2 tsk fls models [15, 16]

Antecedent 
Consequent 
The Rule Base 
T2 TSK FLS Model I 
Type2 fuzzy sets 
Type1 fuzzy sets 
IF $\mathrm{x}_{1}$ is $\tilde{\mathrm{F}}_{1}^{\mathrm{i}}$ and... $\mathrm{x}_{\mathrm{p}}$ is $\tilde{\mathrm{F}}_{\mathrm{p}}^{\mathrm{i}}$ THEN $\mathrm{y}^{1}=\mathrm{C}_{0}^{1}+\mathrm{C}_{1}^{1} \mathrm{x}_{1} \ldots+\mathrm{C}_{\mathrm{p}}^{1} \mathrm{x}_{\mathrm{p}}$ 
T2 TSK FLS Model II 
Type2 fuzzy sets 
Crisp Numbers 
IF $\mathrm{x}_{1}$ is $\tilde{\mathrm{F}}_{1}^{\mathrm{i}}$ and... $\mathrm{x}_{\mathrm{p}}$ is $\tilde{\mathrm{F}}_{\mathrm{p}}^{\mathrm{i}}$ THEN $\mathrm{y}^{1}=\mathrm{C}_{0}^{1}+\mathrm{C}_{1}^{1} \mathrm{x}_{1} \ldots+\mathrm{C}_{\mathrm{p}}^{1} \mathrm{x}_{\mathrm{p}}$ 
T2 TSK FLS Model III 
Type2 fuzzy sets 
Type1 fuzzy sets 
IF $\mathrm{x}_{1}$ is $\tilde{\mathrm{F}}_{1}^{\mathrm{i}}$ and... $\mathrm{x}_{\mathrm{p}}$ is $\tilde{\mathrm{F}}_{\mathrm{p}}^{\mathrm{i}}$ THEN $\mathrm{y}^{1}=\mathrm{C}_{0}^{1}+\mathrm{C}_{1}^{1} \mathrm{x}_{1} \ldots+\mathrm{C}_{\mathrm{p}}^{1} \mathrm{x}_{\mathrm{p}}$ 
Table 3. The final output of other T2 TSK FLS models

The final output Of T2 TSK Models 
T2 TSK FLS Model I 
The final output is also an interval Type1 set and is calculated as follows [1517]: $Y\left(Y^{1}, \ldots Y^{M}, F^{1}, \ldots, F^{M}\right)=\left[y_{l}, y_{r}\right]=\int_{y^{1}} \ldots \int_{y^{M}} \int_{f^{1}} \ldots \int_{f^{M}} 1 / \frac{\sum_{i=1}^{M} f^{i} y^{i}}{\sum_{i=1}^{M} f^{i}}$ Where M is the number of rules fired, $y_{i} \in Y^{i},$ and $Y^{i}=\left[ y_{1}^{i}, y_{r}^{i}\right],(i=1 \ldots M)$ 
T2 TSK FLS Model II 
The final output Is a special case of (5), because now each $Y^{i}$ is a crisp value $y^{i}$. So $Y\left(f^{1}, \ldots, f^{M}\right)=\left[y_{l}, y_{r}\right]=\int_{f^{1}} \ldots \int_{f^{M}} 1 / \frac{\sum_{i=1}^{M} f^{i} y^{i}}{\sum_{i=1}^{M} f^{i}}$ 
T2 TSK FLS Model III 
The final output is special case of (5), because now each $F^{i}$ is a crisp value $f^{i}$ So $Y\left(Y^{1}, \ldots Y^{M}\right)=\left[y_{l}, y_{r}\right]=\int_{y^{1}} \ldots \int_{y^{M}} 1 / \frac{\sum_{i=1}^{M} f^{i} y^{i}}{\sum_{i=1}^{M} f^{i}}$ 
where, $i=1,2, \ldots \ldots \ldots, M, C_{k}^{i}(k=1,2, \ldots, p)$ are the consequent parameters whitch are Type1 fuzzy set, $c_{k}^{i}(k=1,2, \ldots, p)$ are the consequent parameters that are crisp numbers, $Y^{l}$ are the outputs of the $l^{\text {th }}$ rule, $\tilde{F}_{j}^{i}(j=1 \ldots . . p \text { ) are Type } 2$ fuzzy sets of input state $j$ in rule $M,$ given an inputs $x_{1}, x_{2} \ldots \ldots x_{p}, F_{j}^{i}(j=1 \ldots \ldots p)$ are Type 1 fuzzy sets.
The firing strength of the i^{th} rule $F^{i}(x)$ with meet operation under product or minimum tnorm is an interval Type1 set expressed as:
$F^{i}(x)=\left[\underline{f}^{i}(x), \bar{f}^{i}(x)\right]$ (5)
where,
$\underline{f}^{i}(x)=\underline{\mu}_{\tilde{F}_{1}^{i}}\left(x_{1}\right) * \ldots \underline{\mu}_{\tilde{F}_{p}^{i}}\left(x_{p}\right)$
$\bar{f}^{i}(x)=\bar{\mu}_{\tilde{F}_{1}^{i}}\left(x_{1}\right) * \ldots \bar{\mu}_{\tilde{F}_{p}^{i}}\left(x_{p}\right)$
To compute Y it is only necessary to compute its two endpoints y_{1} and y_{r} can also be computed more efficient by the KM Algorithm and the Defuzzified output is:
$y=\frac{y_{l}+y_{r}}{2}$ (6)
Invasive weed optimization (IWO) was developed by Mehrabian and Lucas in 2006 [18, 19], The invasive weed optimization technique is a populationbased evolutionary optimization method inspired by the behavior of weed colonies, Invasive weed optimization technique has been successfully used to a variety of optimization problems [2025]. The process is addressed in these steps (Figure 6):
Figure 6. Invasive weed optimization algorithm flowchart
The procedure starts off evolved with initializing a population. Its capacity that a populace of preliminary options is randomly generated over the problem space. Then contributors of the population produce seeds relying on their relative fitness in the population. In other words, the range of seeds for every member is starting with the value of $\mathrm{S}_{\min }$ for the worst member and increases linearly to $\mathrm{S}_{\max }$ for the firstrate member. For next step, these seeds are randomly scattered over the search area by way of generally distributed random numbers with mean equal to zero and an adaptive standard deviation [18, 19]. the standard deviation (SD) [18, 19] for every generation is in:
$\sigma_{\text {iter }}=\frac{\left(\text { iter }_{\text {max }}\text { iter }\right)^{\mathrm{n}}}{\left(\text { iter }_{\text {max }}\right)^{\mathrm{n}}}\left(\sigma_{\text {init }}\sigma_{\text {final }}\right)+\sigma_{\text {final }}$ (7)
n: is the nonlinear modulation index, $\sigma_{\text {iter }}$ is the standard deviation at the current iteration and iter $_{\max }$: is the maximum number of iterations. The produced seeds, accompanied through their parents are viewed as the practicable solutions for the subsequent generation. Finally step, a competitive exclusion is conducted in the algorithm, i.e., after a number of iterations the population reaches its maximum, and an elimination mechanism have to be employed. To this end, the seeds and their parents are ranked together and those with better fitness live on and end up reproductive [18, 19].
Fuzzy PID Controller systems (Figure 7 and Figure 8) have double inputs and single output. The error (e) and the change of error (de) are used as the inputs and the change of the control signal $\widehat{d_{1}}$ is used as the output of the FLC
Figure 7. The FLCPID controller for boost dcdc converters
Figure 8. Structure of a fuzzy logic PID Controller (FLCPID)
$K_{\mathrm{pi}}, K_{\mathrm{pd}}$ are values of PID.
$G_{1}, G_{2}$ values of the gains normalization of the fuzzy system inputs.
$\text { where, } G_{1}=0.5 \text { and } G_{2}=9$
The reference voltage: $\operatorname{Vref}=37.5 \mathrm{V}$. sensor of gain: $K_{\operatorname{sen}}$=0.04
And $V_{\text {refim }}=K_{\text {sen }} * V_{\text {ref }}, e=V_{\text {refim }}K_{\text {sen }} * u_{0}$
This system is constructed from the human experience formulated in a collection of fuzzy rules in the following form:
$\mathrm{j}^{\mathrm{th}}: \mathrm{IF} \mathrm{e}$ is $\mathrm{E}_{0}^{\mathrm{j}}$ and $\operatorname{de}$ is $\mathrm{E}_{1}^{\mathrm{j}} \mathrm{THEN} $ $\hat{\mathrm{d}}_{1}=\mathrm{C}_{\mathrm{j}}(\mathrm{e}, \mathrm{de})$
With $\mathrm{E}_{0}^{\mathrm{j}} \mathrm{E}_{1}^{\mathrm{j}}$ are respectively the fuzzy sets of the error voltage e and its time derivative de $C_{\mathrm{j}}$ is the $\mathrm{j}^{\text {th }}$ output singleton.
The strategy of fuzzy control is derived using the following knowledge on the system:
The change of duty cycle $\hat{\mathrm{d}}{1}$ must be large, when $u_{0}$ is far from the reference.
Vreffor provides a small response time.
The small change of duty cycle $\hat{\mathrm{d}}{1}$ is sufficient to reach the reference providing that $u_{0}$ approaches the reference.
The duty cycle must be unchanged as long as $u_{0}$ is in the vicinity of the reference with a sufficient approaching speed, for preventing the output overshoot.
When $u_{0}$ reaches the reference and continue growing up: first, we decrease the duty cycle change, then if $u_{0}$ remains closer to the reference, the duty cycle changes must be zero otherwise, it must be negative.
Thus, the final control action $\hat{\mathrm{d}}$ applied to the converter is given by:
$\hat{\mathrm{d}}=\mathrm{G}_{1} \hat{\mathrm{d}}_{1}+\mathrm{G}_{2} \int \widehat{\mathrm{d}}_{1} \mathrm{d}$ (8)
We obtain 25 fuzzy rules with 17 output singletons issued from the human expertise (Table 4).
Table 4. Rules and output membership functions [8, 26]
e\de 
PH 
PL 
Z 
NL 
NH 
PH 
1 
0.81 
0.49 
0.36 
0.25 
PL 
0.64 
0.36 
0.16 
0.04 
0 
Z 
0.16 
0.04 
0 
0.04 
0.16 
NL 
0 
0.04 
0.16 
0.36 
0.64 
NH 
0.25 
0.36 
0.49 
0.81 
1 
In this work, we propose using the optimizations methods (IWO and PSO) to tuning the footprint of uncertainty size (FOU) in interval Type2 fuzzy sets. Besides; We discuss the effects of the PID values in the operation of transition from Type1 to interval Type2 fuzzy logic Controller for Boost DCDC Converters (Figure 9).
Figure 9. The FLCPID controller with optimizations methods
We use the two inputs fuzzy sets (e and de) are composed of five membership functions [8, 26] are defined in (Figure 10).
Figure 10. The two inputs fuzzy sets (e and de)
The transition from Type1 membership functions to interval Type2 membership functions shown in (Figure 11). [7, 8, 27]. The uncertainty size parameter (U) in the interval Type2 membership functions is to be determined and optimized, using optimization algorithms (IWO and PSO).
Figure 11. Interval Type2 membership functions [7, 8, 26]
After make transition from Type1membership functions (Figure 12) to interval Type2 membership functions
Figure 12. Type1 membership functions
where,
$\Delta \mathbf{x}_{1}=\mathbf{x}_{2}\mathbf{x}_{1} ; \Delta \mathbf{x}_{2}=\mathbf{x}_{3}\mathbf{x}_{2}$
The uncertainty (U) in the membership functions is the third considered parameter. we need to avoid overlapping between: $\mathrm{x}_{1}^{\text {lower }}$ and $\mathrm{x}_{3}^{\text {lower }},$ so, we propose: $\mathrm{U} \in[\mathbf{0}, \mathbf{1}] \Rightarrow \Delta \mathbf{x}_{1} * \mathbf{U} \leq$$\Delta \mathbf{x}_{1}$ and $\Delta \mathbf{x}_{2} * \mathbf{U} \leq \Delta \mathbf{x}_{2}$
$x_{1}^{\text {Upper }}=\max \left(1, x_{1}\Delta x_{1} * \mathbf{U} / 2\right)$.
$x_{1}^{l o w e r}=\max \left(1+\Delta x_{1} * \mathbf{U}, x_{1}+\Delta x_{1} * \mathbf{U} / 2\right)$.
$x_{3}^{\text {lower }}=\min \left(1\Delta x_{2} * \mathbf{U}, x_{3}\Delta x_{2} * \mathbf{U} / 2\right)$.
$x_{3}^{\text {Upper }}=\min \left(x_{3}+\Delta x_{2} * \mathbf{U} / 2,1\right)$.
The uncertainty size Parameter (U) is to be determined and optimized, using optimization algorithms (IWO and PSO).
The objective function as following:
$\operatorname{cost}$ function $=\frac{\left(10^{3}\right)}{n} \sum_{t=0}^{n}[e(t)]^{2}$ (9)
In this work we have two types of simulation.
Firstly: tuning the FOU size in T2MFsof Interval Type2 Fuzzy Logic PID controller.
where,
Secondly: tuning both the PID values and the FOU size in T2MFsof Interval Type2 Fuzzy Logic PID controller.
where,
where, $\mathrm{K}_{\mathrm{pd}} \in[0,0.59]$ And $\mathrm{K}_{\mathrm{pi}} \in[0,450][26]$
In this paper, we developed a new approach to transit from Type1 to interval Type2 fuzzy logic Controller using the optimization methods (IWO and PSO). We have two types of simulation.
Table 5. The optimal controllers’ parameters

Optimizing the FOU size parameter (U) 
IT2FPID controller with WO 
0.8534 
IT2FPID controller with PSO 
0.8534 
Fuzzy Logic PID Controller (T1FPID) [26] 
0 
Table 6. Compare results obtained by IWO and PSO

ISE 
IAE 
T_{R} 
Bestcost function 
IT2FPID controller with IWO 
0.1087 
0.1503 
0.0206 
108.7 
IT2FPID controller with PSO 
0.1087 
0.1503 
0.0206 
108.7 
Fuzzy Logic PID Controller (T1FPID) [26] 
0.1559 
0.2250 
0.0306 
\ 
Table 7. The optimal controllers parameters

K_{pd} 
K_{pi} 
the FOU size parameter (U) 
IT2FPID controller with IWO 
0.590 
392.6647 
0.8219 
IT2FPID controller with PSO 
0.590 
382.7533 
0.8182 
Fuzzy Logic PID Controller (T1FPID) [26] 
0.25 
255 
0 
Table 8. Compare results obtained by IWO and PSO

ISE 
IAE 
T_{R} 
Bestcost function 
IT2FPID controller with IWO 
0.0958 
0.1202 
0.0148 
95.7732 
IT2FPID controller with PSO 
0.0963 
0.1234 
0.0160 
96.3237 
Fuzzy Logic PID Controller (T1FPID) [26] 
0.1559 
0.2250 
0.0306 
\ 
Figure 13. Iterative convergence curve
Figure 14. The Type2 membership function of fuzzy sets of inputs interval Type2 fuzzy logic controller after tuning by the IWO (the FOU size parameter U=0.8534)
Simulation results show:
The Invasive weed optimization (IWO) algorithm converges quicker than the Particle Swarm Optimization algorithm (PSO) (Figure 13 and Figure 16). The superiority of the IT2FPID controller with IWO comparing with both the IT2FPID controller with PSO and fuzzy Logic PID controllers(Figure 15, Figure 18, Table 6 and Table 8), where: the IT2FPID controller with IWO has minimum the Rise time (Tr), achieve lower the integral of square of errors (ISE) and the integral of the absolute errors (IAE) comparing with the other controllers. Finally; Invasive Weed Optimization Algorithm is helpful to tuning the PID values and the footprint of uncertainty size (FOU) in Interval Type2 fuzzy logic PID controller for Boost DCDC Converter.
Figure 15. The output voltage of different simulation cases
Figure 16. Iterative convergence curve
Figure 17. The Type2 membership function of fuzzy sets of inputs Interval Type2 fuzzy logic controller after tuning by the IWO (the FOU size parameter U=0.8219)
Figure 18. The output voltage of different simulation cases
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