Type-1 and Interval Type-2 Fuzzy Systems

equation1 A T1 fuzzy set $X$ is comprised of a domain $D_X$ of real numbers (also called the universe of discourse of $X$) together with a membership function (MF) $u_{_X}(x):\ D_X \to [0,1]$, i.e.,

$X=\int_{D_X}u_{_X}(x)/x .$ $(1)$

A trapezoidal T1 fuzzy set is shown in Figure $1$.

Figure 1:
A trapezoidal T1 fuzzy set.

The membership grade of $x$ can be computed as:

A triangular T1 fuzzy set is a special case of a trapezoidal T1 fuzzy set, when $b=c$.

Figure $2$ shows a specific example of a T1 fuzzy set "Moderate". When only integer numbers are considered in the domain, the T1 fuzzy set can be represented as $\{0/1, 0.5/2, 1/3, 1/4, 0.5/5, 0/6\}$, where $0/1$ means that $x=1$ has a membership grade of 0 in Moderate, and $0.5/2$ means $x=2$ has a membership grade of $1$ in Moderate. Figure $2$ shows that there is a complete agreement (certainty) for $x\in[0,1]\cup[3,4]\cup[6,+\infty)$, and a partial agreement for $x\in(1,3)\cup(4,6)$. In contrast, for a crisp set, the membership grade of each element in it is either 0 or 1; there is no value (e.g., 0.3) in between, i.e., a crisp set cannot model a partial agreement.

Try to drag the blue dot and change the membership grade of $x$.

Figure 2: An example of a T1 fuzzy set.

Figure $3$ shows the diagram of a typical T1 fuzzy system, which consists of four components: fuzzifier, rulebase, inference engine, and defuzzifier. The fuzzifier transforms crisp inputs into T1 fuzzy sets. The inference engine performs inference on the T1 fuzzy sets, combining the outputs of the rules in the rulebase into a new T1 fuzzy set. The defuzzifier converts the mapped T1 fuzzy set into a crisp output.

There are two kinds of rules for fuzzy systems, Zadeh (Mamdani), in which rule consequents are words modeled by T1 fuzzy sets , and Takagi-Sugeno-Kang (TSK), in which rule consequents are mathematical formulas . The latter is popular due to their simplicity and flexibility.

T1 TSK rules are:

$R_k: \mbox{ IF } X_1 \mbox{ is } X_{k,1} \mbox{ AND } \cdots \mbox{ AND } X_d \mbox{ is } X_{k,d}, \mbox{ THEN } y_k(\mathbf{x}) = \sum_{i=1}^d a_{k,i}x_i + b_k, \quad k=1,...,K$

where $d$ is the input dimensionality, $K$ is the number of rules, $X_{k,i}$ is a linguistic term (modeled by a T1 fuzzy set) for linguistic variable $X_i$ ($i=1,...,d$) in the $k^{\mathrm{th}}$ rule, $x_i$ is the crisp input for $X_i$, and $a_{k,i}$ and $b_k$ ($k=1,...,K$) are adjustable regression coefficients. Many times all $a_{k,i}$ are set to 0, i.e., each rule consequent is a constant.

For the car following example, $d=2$, $X_1$ denotes Distance, $X_2$ denotes Relative Speed, $K=9$ if both Distance and Relative Speed have three linguistic terms (T1 fuzzy sets), and $x_1$ and $x_2$ are the current input values of Distance and Relative Speed, respectively.

For a particular input $\mathbf{x}=(x_1,...,x_d)$, the operations in a TSK fuzzy system are:

Units

1. Fuzzification:

   Compute $u_{X_{k,i}}(x_i)$, the membership grade of $x_i$ on $X_{k,i}$ ($k=1,...,K$; $i=1,...,d$).

2. Inference:

   Compute the firing level of each rule:
   

   $f_k(\mathbf{x})=u_{X_{k,1}}(x_1)\times\cdots\times u_{X_{k,d}}(x_d),$

   
   Note that the minimum operation could also be used to replace the multiplication above. The inference engine output is:

   Units $y(\mathbf{x})=\frac{\sum_{k=1}^K f_k(\mathbf{x})\cdot y_k(\mathbf{x})}{\sum_{k=1}^K f_k(\mathbf{x})}.$ $(3)$

3. Defuzzification:

   Since the inference output is already a crisp number, no defuzzification is needed in a T1 TSK fuzzy system. The inference output $y(\mathbf{x})$ is the final output.

   However, for Mamdani T1 fuzzy systems, since the rule consequents are words modeled by T1 fuzzy sets, the inference output is also a T1 fuzzy set, which must be defuzzified to obtain a crisp output. One popular defuzzification approach is to compute the centroid of that T1 fuzzy set.

$(3)$ fuses the consequents of multiple rules to give an output. If we view each rule as an expert, then a TSK fuzzy system is functionally equivalent to a mixture of expert model . Alternatively, if we view each rule as a base model, then a TSK fuzzy system may also be understood as an adaptive ensemble model .

This subsection presents a simple example to illustrate the computations in a T1 TSK fuzzy system. Consider the car following example introduced at the beginning of this paper.

Assume the distance has three linguistic terms (modeled by T1 fuzzy sets), Close, Okay and Far, as depicted in Figure $4(a)$, and the relative speed also has three linguistic terms (modeled by T1 fuzzy sets), Slow, Okay and Fast, as depicted in Figure $4(b)$. Note that the same word "Okay" for the distance and the relative speed could have different T1 FS definitions.

(a)

(b)

Figure 4:
T1 fuzzy sets for the distance and the relative speed, in the car following example.

There are nine rules, whose consequents are crisp numbers:

• $R_1: \mbox{ IF distance is Close AND relative speed is Slow, THEN acceleration is } y_1=0$

• $R_2: \mbox{ IF distance is Close AND relative speed is Okay, THEN acceleration is } y_2=-5$

• $R_3: \mbox{ IF distance is Close AND relative speed is Fast, THEN acceleration is } y_3=-10$

• $R_4: \mbox{ IF distance is Okay AND relative speed is Slow, THEN acceleration is } y_4=3$

• $R_5: \mbox{ IF distance is Okay AND relative speed is Okay, THEN acceleration is } y_5=0$

• $R_6: \mbox{ IF distance is Okay AND relative speed is Fast, THEN acceleration is } y_6=-3$

• $R_7: \mbox{ IF distance is Far AND relative speed is Slow, THEN acceleration is } y_7=10$

• $R_8: \mbox{ IF distance is Far AND relative speed is Okay, THEN acceleration is } y_8=5$

• $R_9: \mbox{ IF distance is Far AND relative speed is Fast, THEN acceleration is } y_9=0$

Assume the current distance to the front car is 47 meters, and the relative speed is 4 KM/h. The membership grades of the corresponding T1 fuzzy sets are shown in Figure $5$:

$\quad u_{_{\mathrm{Close}}}($47$)=$0.60 $\quad u_{_{\mathrm{Okay}}}($47$)=$0.40 $\quad u_{_{\mathrm{Far}}}($47$)=$0.00

$\quad u_{_{\mathrm{Slow}}}($4$)=$0.00 $\quad u_{_{\mathrm{Okay}}}($4$)=$0.20 $\quad u_{_{\mathrm{Fast}}}($4$)=$0.80

Try to drag the dots to change the distance and relative speed values. The membership grades and all subsequent calculations will be updated accordingly.

(a)

(b)

Figure 5:
Membership grades on the T1 fuzzy sets. (a) Distance is 47 meters, the membership interval on Close is 0.60, on Okay is 0.40, and on Far is 0.00; (b) Relative speed is 4 KM/h, the membership interval on Slow is 0.00, on Okay is 0.20, and on Fast is 0.80.

Try to change the rule consequent values in Table $1$ by direct typing or clicking on the sidebar arrows. After adjusting the rulebase, click the "Update" button to update all calculations.

Figure 6:
Input-output mapping of the T1 fuzzy system. The figure can be zoomed and rotated by using the mouse or the top right buttons.

An IT2 fuzzy set $\tilde{X}$ is defined as

where $x$, called the primary variable, is in domain $D_{\tilde{X}}$; $u\in [0,1]$ is the secondary variable; and, the secondary grade of $\tilde{X}$ equals 1 for $\forall x\in D_{\tilde{X}}$ and $\forall u\in [0,1]$.

An example of an IT2 fuzzy set $\tilde{X}$ is shown in Figure $7$. Observe that unlike a T1 fuzzy set, whose membership for each $x$ is a number, the membership of an IT2 fuzzy set is an interval. For example, the membership of $x=3$ is $[0.25, 1]$, and the membership of $x=5$ is $[0.75, 1]$. A membership interval $[0.25, 1]$ means that some people think the membership of $x=3$ is $0.25$, some think the membership is $1$, and others may think the membership is any value between $0.25$ and $1$, i.e., an IT2 fuzzy set can model inter-personal uncertainties, whereas a T1 fuzzy set cannot.

Figure $7$ shows that an IT2 fuzzy set is bounded from the above and below by two T1 fuzzy sets, which are called upper MF (UMF) and lower MF (LMF). The area between the LMF and the UMF is the footprint of uncertainty (FOU).

Figure $8$ shows the schematic diagram of an IT2 fuzzy system. It is similar to its T1 counterpart, with the major difference being that at least one of the fuzzy sets in the rulebase is an IT2 fuzzy set. Hence, the outputs of the inference engine are IT2 fuzzy sets, and a type-reducer is usually needed (it can be bypassed , too) to convert them into a T1 fuzzy set before defuzzification can be carried out.

Assume the rulebase of an IT2 TSK fuzzy system consisting of $K$ rules in the following form:

$\tilde{R}_k: \mbox{ IF } x_1 \mbox{ is } \widetilde{X}_{k,1} \mbox{ AND } \cdots \mbox{ AND } x_d \mbox{ is } \widetilde{X}_{k,d}, \mbox{ THEN } Y_k(\mathbf{x}) = [\underline{y}_k,\ \overline{y}_k], \quad k=1,...,K$

For a particular input $\mathbf{x}=(x_1,...,x_d)$, the output of the IT2 TSK fuzzy system is computed as:

1. Fuzzification:

   Compute the membership interval $[u_{\underline{X}_{k,i}}(x_i),u_{\overline{X}_{k,i}}(x_i)]$ of $x_i$ on each $\widetilde{X}_{k,i}$ ($k=1,...,K$; $i=1,...,d$).

2. Inference:

   Compute the firing interval of the $k^{\mathrm{th}}$ rule, $F_k(\mathbf{x})$:

   equation1

   $F_k (\mathbf{x}) = [u_{\underline{X}_{k,1}}(x_1)\times\cdots\times u_{\underline{X}_{k,d}}(x_d), u_{\overline{X}_{k,1}}(x_1)\times\cdots\times u_{\overline{X}_{k,d}}(x_d)] \equiv [\underline{f}_k,\,\overline{f}_k], \quad k=1,...,K$ $(6)$

   Note that the minimum operation could also be used to replace the multiplication above. Inference is performed by:

   $Y(\mathbf{x}) = \bigcup\limits_{f_k \in F_k(\mathbf{x}) \atop y_k \in Y_k(\mathbf{x})} \frac{\sum\limits_{k=1}^K f_k y_k} {\sum\limits_{k=1}^K f_k}.$ $(7)$

   $(7)$is a direct extension of $(3)$, when all $f_k(\mathbf{x})$ and $y_k(\mathbf{x})$ in $(3)$ become intervals.

3. Type-reduction:

   Many different approaches can be used to compute $(7)$. When the popular center-of-sets type-reducer is used, the type-reduced output is:

   $Y(\mathbf{x}) =[y_l,\ y_r],$ $(8)$

   where:

   $y_l=\min\limits_{l\in[1,K-1]}\frac{\sum_{k=1}^l\overline{f}_k\underline{y}_k+\sum_{k=l+1}^K \underline{f}_k\underline{y}_k} {\sum_{k=1}^l\overline{f}_k+\sum_{k=l+1}^K\underline{f}_k}\equiv \frac{\sum_{k=1}^L\overline{f}_k\underline{y}_k+\sum_{k=L+1}^K\underline{f}_k\underline{y}_k} {\sum_{k=1}^L\overline{f}_k+\sum_{k=L+1}^K\underline{f}_k},$ $(9)$

   $y_r=\max\limits_{r\in[1,K-1]}\frac{\sum_{k=1}^r\underline{f}_k\overline{y}_k+\sum_{k=r+1}^K \overline{f}_k\overline{y}_k} {\sum_{k=1}^r\underline{f}_k+\sum_{k=r+1}^K\overline{f}_k}\equiv \frac{\sum_{k=1}^R\underline{f}_k\overline{y}_k+\sum_{k=R+1}^K\overline{f}_k\overline{y}_k} {\sum_{k=1}^R\underline{f}_k+\sum_{k=R+1}^K\overline{f}_k},$ $(10)$

   in which the switch points $L$ and $R$ are determined by

   $\underline{y}_L \le y_l\le \underline{y}_{L+1},$ $(11)$

   $\overline{y}_R \le y_r\le \overline{y}_{R+1},$ $(12)$

   and $\{\underline{y}_k\}_{k=1}^K$ and $\{\overline{y}_k\}_{k=1}^K$ have been sorted in ascending order, separately.

   $y_l$ and $y_r$ can be computed using the original Karnik-Mendel algorithms , the enhanced Karnik-Mendel algorithms , the iterative algorithm with stop condition (IASC), or the enhanced IASC (EIASC) , all of which have been included in the official Matlab Fuzzy Logic Toolboxhttps://www.mathworks.com/help/fuzzy/type-2-fuzzy-inference-systems.html. See, also for other algorithms.

4. Defuzzification:

   The defuzzified crisp output is:

   $y = \frac{y_l+y_r} {2}.$ $(13)$

Essentially, $y_l$ is the minimum of $\frac{\sum_{k=1}^K f_k y_k}{\sum_{k=1}^K f_k}$, and $y_r$ is the maximum, when each $f_k$ and $y_k$ can assume value in an interval.

Clearly, the minimum of $y_k$, $\underline{y}_k$, should be used in computing $y_l$; and, the maximum of $y_k$, $\overline{y}_k$, should be used in computing $y_r$, as $y_k$ only appears in the numerator of $(6)$.

Intuitively, to compute $y_l$ (the minimum), a large $f_k$ ($\overline{f}_k$) should be used for a small $\underline{y}_k$, and a small $f_k$ ($\underline{f}_k$) should be used for a large $\underline{y}_k$. Assume $\{\underline{y}_k\}_{k=1}^K$ has been sorted in ascending order. Then, the switch point, $l$, determines when we should switch from $\overline{f}_k$ to $\underline{f}_k$. $y_r$ (the maximum) is computed in a similar way, but the switch point $r$ determines when we should switch from $\underline{f}_k$ to $\overline{f}_k$.

The switch points $l$ and $r$, and the corresponding $y_l$ and $y_r$, do not have a closed-form solution. They are computed by the iterative type-reduction algorithms. One of the most efficient type-reduction algorithm, EIASC , is shown in Table $2$.

The main idea of EIASC is to find the switch points for $y_l$ and $y_r$. Take $y_l$ for examples, $y_l$ is the minimum of $(7)$. Since $\underline{y}_k$ increases from the left to the right along the horizontal axis of Figure $9(a)$, we should choose a large weight (upper membership grade) for $\underline{y}_k$ on the left and a small weight (lower membership grade) for $\underline{y}_k$ on the right. EIASC finds the switch point $L$. For $k\leq L$, the upper membership grades are used to calculate $y_l$; for $k>L$, the lower membership grades are used. This ensures $y_l$ is the minimum.

(a)

(b)

Figure 9:
Illustration of the switch points $L$ and $R$ in computing $y_l$ and $y_r$, respectively. (a) Computing $y_l$: the used firing level in $(9)$ switches from the upper firing level to the lower firing level, shown as the blue curve; (b) computing $y_r$: the used firing level in $(10)$ switches from the lower firing level to the upper firing level, shown as the blue curve.

This subsection uses the car following example again to illustrate the computations in an IT2 TSK fuzzy system.

Assume the distance has three linguistic terms (modeled by IT2 fuzzy sets), Close, Okay and Far, as depicted in Figure $10(a)$, and the relative speed also has three IT2 linguistic terms (modeled by IT2 fuzzy sets), Slow, Okay and Fast, as depicted in Figure $10(b)$. Note again that the word "Okay" for the distance and the relative speed has different IT2 fuzzy set representations.

(a)

(b)

Figure 10:
IT2 fuzzy sets for the distance and the relative speed, in the car following example.

There are again nine rules, whose consequents are intervals:

• $\widetilde{R}_1: \mbox{ IF distance is Close AND relative speed is Slow, THEN acceleration is } Y_1=[-1,1]$
• $\widetilde{R}_2: \mbox{ IF distance is Close AND relative speed is Okay, THEN acceleration is } Y_2=[-6,-4]$
• $\widetilde{R}_3: \mbox{ IF distance is Close AND relative speed is Fast, THEN acceleration is } Y_3=[-11,-9]$
• $\widetilde{R}_4: \mbox{ IF distance is Okay AND relative speed is Slow, THEN acceleration is } Y_4=[2,4]$
• $\widetilde{R}_5: \mbox{ IF distance is Okay AND relative speed is Okay, THEN acceleration is } Y_5=[-1,1]$
• $\widetilde{R}_6: \mbox{ IF distance is Okay AND relative speed is Fast, THEN acceleration is } Y_6=[-4,-2]$
• $\widetilde{R}_7: \mbox{ IF distance is Far AND relative speed is Slow, THEN acceleration is } Y_7=[9,11]$
• $\widetilde{R}_8: \mbox{ IF distance is Far AND relative speed is Okay, THEN acceleration is } Y_8=[4,6]$
• $\widetilde{R}_9: \mbox{ IF distance is Far AND relative speed is Fast, THEN acceleration is } Y_9=[-1,1]$

Assume the current distance to the front car is 47 meters, and the relative speed is 4 KM/h. Let $\underline{\mathrm{Close}}$ and $\overline{\mathrm{Close}}$ be the LMF and UMF of $\mathrm{Close}$, respectively. LMFs and UMFs of other words are denoted in a similar way. The membership grades on the corresponding LMFs and UMFs are shown in Figure $11$:

$\quad [u_{_{\underline{\mathrm{Close}}}}($ 47 $),u_{_{\overline{\mathrm{Close}}}}($ 47 $)]=[$ 0.20 $,$ 0.60 $]$ $\quad [u_{_{\underline{\mathrm{Okay}}}}($ 47 $),u_{_{\overline{\mathrm{Okay}}}}($ 47 $)]=[$ 0.00 $,$ 0.67 $]$ $\quad [u_{_{\underline{\mathrm{Far}}}}($ 47 $),u_{_{\overline{\mathrm{Far}}}}($ 47 $)]=[$ 0.00 $,$ 0.00 $]$

$\quad [u_{_{\underline{\mathrm{Slow}}}}($ 4 $),u_{_{\overline{\mathrm{Slow}}}}($ 4 $)]=[$ 0.00 $,$ 0.00 $]$ $\quad [u_{_{\underline{\mathrm{Okay}}}}($ 4 $),u_{_{\overline{\mathrm{Okay}}}}($ 4 $)]=[$ 0.00 $,$ 0.43 $]$ $\quad [u_{_{\underline{\mathrm{Fast}}}}($ 4 $),u_{_{\overline{\mathrm{Fast}}}}($ 4 $)]=[$ 0.40 $,$ 0.80 $]$

Figure11

Try to drag the dots to change the distance and relative speed values. The membership intervals and all subsequent calculations will be updated accordingly.

(a)

(b)

Figure 11:
Membership intervals on the IT2 fuzzy sets. For each fuzzy set, the UMF is less saturated, and the LMF is more saturated. (a) Distance is 47 meters, the membership interval on Close is [0.20, 0.60], on Okay is [0.00, 0.67], and on Far is [0.00, 0.00]; (b) Relative speed is 4 KM/h, the membership interval on Slow is [0.00, 0.00], on Okay is [0.00, 0.43], and on Fast is [0.40, 0.80].

The firing intervals of the nine rules are:

Try to change the rule consequent values in Table $1$ and Table $3$ by direct typing or clicking the sidebar arrows. After adjusting the rulebases, click the "Update" button to automatically update all calculations.

(a)

(b)

Figure 12:
(a) Input-output mapping of the IT2 fuzzy system; (b) difference between the input-output mappings of the IT2 fuzzy system and the T1 fuzzy system. The figure can be zoomed and rotated by using the mouse or the top right buttons.