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Fuzzy Logic with Engineering Applications

Timothy Ross

Chapter 14

Miscellaneous Topics - all with Video Answers

Educators

Chapter Questions

Problem 1

The feedforward transfer function for a unit-feedback control system is $1 /(s-1)$, as shown in Fig. P14.1. A unit step signal is input to the system. Determine the minimum error of the system response by using a fuzzy optimization method for the time period, $0<t<10 \mathrm{~s}$, and a fuzzy constraint given by

$$
u_c(t)= \begin{cases}1, & 0 \leq t \leq 1 \\ e^{1-t}, & t>1\end{cases}
$$

FIGURE P14.1

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09:30

Problem 2

A beam structure is forced by an axial load $P$ (Fig. P14.2). When $P$ is increased to its critical value, the beam will buckle. Prove that the critical force $P$ to cause buckling can be expressed by a function

$$
P=\frac{n^2 \pi^2 E I}{L}
$$

where $E I=$ stiffness of the beam
$L=$ span length of the beam
$n=$ number of sine waves the beam shape takes when it buckles (assume it to be continuous)

If $0 \leq n \leq 2$, assume that $n$ is constrained by the fuzzy member function

$$
u_c(n)= \begin{cases}1-n, & 1 \leq n \leq 2 \\ 0, & n<1\end{cases}
$$

Ahmad Reda
Ahmad Reda
Numerade Educator

Problem 3

Suppose that the beam structure in Problem 14.2 also has a transverse load $P$ applied at the middle of the beam. Then the maximum bending stress can be calculated by the equation $\sigma_b=P l / 4 w_z$, where $w_z$, with units $\mathrm{m}^3$, is a coefficient based on the shape and size of the cross section of the beam, and $l$ is in meters. If $0 \leq \sigma_b \leq 60 \mathrm{MPa}$, and the fuzzy constraint function for $\sigma_b$ is

$$
\mu_c\left(\sigma_b\right)= \begin{cases}\frac{1}{(x+1)^2}, & 0 \leq x \leq 1 \\ 0, & x>1\end{cases}
$$

where $\quad x=\frac{\sigma}{60 \mathrm{MPa}}$
combine the conditions given in Problem 14.2 to find the optimum load $P$, in newtons.
Hint: This problem involves multiple constraints.

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02:04

Problem 4

In the metallurgical industry, the working principle for a cold rolling mill is to extrude a steel strip through two rows of working rollers, as shown in Fig. P14.4. The size of the roller is very important. The stress between the roller and the strip can be expressed by the following function:

$$
\sigma_H=0.564 \sqrt{\frac{P E}{L R}}
$$

where $E=$ Young's modulus $\left(\mathrm{kN} / \mathrm{cm}^2\right)$
$P=$ loading force (N)
$L=$ contact length between roll and strip (cm)
$R=$ radius of a roller (cm)
If $\sigma_H=2.5 \mathrm{kN} / \mathrm{cm}^2$ and $10<R<20$, find the minimum $R$ in which $\sigma_H$ has a maximum value. The radius $R$ has a fuzzy constraint of

$$
u_c(R)= \begin{cases}1, & 10 \leq R \leq 15 \\ \frac{20-R}{5}, & 15<R \leq 20\end{cases}
$$

Surendra Kumar
Surendra Kumar
Numerade Educator

Problem 5

In a fuzzy relation equation, $\mathrm{A} \circ \mathrm{R}=\mathrm{B}, r_i$ is known as $(0.5,0.7,0.9)$ and $b_i$ is 0.6 . Use the Tsukamoto method for an inverse fuzzy equation to find $a_i(i=1,2,3)$.

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Problem 6

A fuzzy relation has an expression given as

$$
\left\{a_1, a_2\right\} \circ\left[\begin{array}{ll}
0.4 & 0.6 \\
0.8 & 0.1
\end{array}\right]=\left[\begin{array}{ll}
0.3 & 0.1
\end{array}\right]
$$

Find $a_i(i=1,2)$ by using the Tsukamoto method for inverse fuzzy relations.

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02:07

Problem 7

A system having a single degree of freedom has the following ordinary differential governing equation:

$$
a_1 \ddot{x}+a_2 \dot{x}+a_3 x=b
$$

If $b$ is the input signal, in which $b_1=0.5$ and $b_2=0.6$, and two sets of response data of $x$ are $(0.4,0.6,0.8)$ and $(0.5,0.7,0.9)$, respectively, find the system coefficients $a_i(i=1,2,3)$.

James Kiss
James Kiss
Numerade Educator

Problem 8

Show that as the parameter $h$ increases, the furziness of the output increases in the five-point regression problem (Example 14.7) [Kikuchi and Nanda, 1991]. Use the simplex method for values of $h=0.2$ and 0.8 .

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Problem 9

Risk assessment is fast becoming the basis of many EPA guidelines that determine whether a site contaminated with hazardous substances needs to be remediated or not. Risk is defined as the likelihood of an adverse health impact to the public due to exposure to environmental hazards. Risk assessment consists of four parts: (1) hazard identification; (2) dose-response assessment - assessing the health response to a certain dose (concentration) of the chemical; (3) exposure assessment - assessing the duration and concentration of exposure; and (4) risk characterization - quantification and presentation of risks. Part (2) of the risk assessment process consists of exposing a controlled population of animals to various doses of the chemical and fitting a dose-response curve to the experimental data. The following table comprises the data derived from the tests:
Assuming that the dose-response relationship can be expressed by a fuzzy linear regression model, $Y=A_0+A_1 x$, where $x$ represents the dose in $\mathrm{mg} / \mathrm{L}$, determine the fuzzy coefficients $\mathrm{A}_0$ and $\mathrm{A}_1$. Use an $h$ value of 0.5 .

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Problem 10

In a survey on costs for the construction of new houses, the number of rooms (including bedrooms, kitchen, bathroom, and living rooms) in a house was compared with the material costs of the house. The following table gives the results of the survey:
Assuming an $h$ of 0.5 , determine the coefficients of the fuzzy one-dimensional linear regression model, $Y=\underset{\sim}{\mathrm{A}}+\underset{\sim}{\mathrm{A}_1} x$.

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Problem 11

In fuzzy regression, the output $y$ is a triangular fuzzy number with the spread $e_j$ representing the error in measurement (Fig. P14.11).
The accompanying table shows the fuzzy output and the corresponding crisp input:
Using Eqs. (14.41)-(14.43) for fuzzy output, and $h=0.4$, determine the fuzzy coefficients for a simple fuzzy linear regression model, $\underset{\sim}{\mathrm{Y}}=\underset{\sim}{\mathrm{A}_0}+\underset{\sim}{\mathrm{A}_1} x$.

Shu Naito
Shu Naito
Numerade Educator

Problem 12

For the information in Example 14.3 find the stabilized state vector corresponding to an initial state vector of $[0,1,0,0,0]$

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Problem 13

For the information in Example 14.3 find the stabilized state vector corresponding to a fuzzy linguistic effect on the path from $C_1$ to $C_2$ that is "Much."

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Problem 14

For the information in Example 14.3 find the stabilized state vector corresponding to an initial state vector of $[0,0,1,0,0]$ and a fuzzy linguistic effect on the path from $C_2$ to $C_3$ that is "Some."

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