Chapter 9, Rule-base Reduction Methods Video Solutions, Fuzzy Logic with Engineering Applications

Problem 1

This problem will demonstrate the progression of the singular value approximation using the MATLAB command "svd" in finding a singular value decomposition. A singular value decomposition diagonalizes and decomposes a matrix $A$ into three matrices, one diagonal and two orthogonal.
(a) Given

$$
A=\left[\begin{array}{llll}
4 & 0 & 0 & 0 \\
0 & 4 & 0 & 0 \\
0 & 0 & 4 & 0 \\
0 & 0 & 0 & 4
\end{array}\right]
$$

find its singular value decomposition using the MATLAB command *" $[\mathrm{U}, \mathrm{S}, \mathrm{V}]=$ $\operatorname{svd}(\mathrm{A}) . "$
(b) A matrix approximation is a summation of the products of all the singular values with their corresponding column vectors from matrices U and V as

$$
\mathrm{A}=u_1 s_1 v_1^{\mathrm{T}}+\cdots+u_n s_n v_n^{\mathrm{T}}
$$

Perform a rank $1,2,3$, and 4 approximation of the matrix in part (a).

Jack Chen

Numerade Educator

00:10

Problem 2

Repeat Problem $9.1 a$ and $b$ on the following matrix:

$$
A=\left[\begin{array}{llll}
9 & 3 & 2 & 7 \\
8 & 5 & 7 & 3 \\
9 & 2 & 1 & 3 \\
3 & 2 & 2 & 7
\end{array}\right]
$$

What are the two largest singular values?

Amy Jiang

Numerade Educator

01:31

Problem 3

Plot the set A and its convex hull

$$
A=\left[\begin{array}{cc}
1 & 2 \\
1 & 3 \\
4 & 3 \\
4 & 5 \\
7 & 2 \\
5 & 2.5
\end{array}\right]
$$

Lucía Guerrero

Numerade Educator

01:51

Problem 4

Using the matrix

$$
\mathrm{A}=\left[\begin{array}{cccc}
0.3 & 0.2 & 0.4 & 0.1 \\
-0.4 & 0.5 & 0.1 & 0.8 \\
1.2 & -0.4 & -0.3 & 0.5 \\
0.15 & 0.7 & 0.35 & -0.2
\end{array}\right]
$$

show its corresponding doubly stochastic matrix.

Mahnoor Amin

Numerade Educator

04:51

Problem 5

This problem demonstrates the use of the "orth" command in MATLAB and how with an orthogonal matrix one can prove the "sum-normal" condition.
(a) Using the "orth" command in MATLAB verify that the orthonormal basis to matrix A is U .

$$
A=\left[\begin{array}{lll}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 0
\end{array}\right], \quad U=\left[\begin{array}{ccc}
-0.230357 \ldots & -0.396071 \ldots & -0.888855 \ldots \\
-0.607283 \ldots & -0.655207 \ldots & 0.449343 \ldots \\
-0.760356 \ldots & 0.643296 \ldots & -0.089595 \ldots
\end{array}\right]
$$

(b) Prove the sum-normal condition using matrix U .

Chris Trentman

Numerade Educator

Problem 6

For a given type of building with natural frequency $\omega$, a vulnerability surface on damage has been developed based on analyst opinion and past measurements where damage ranges from 1 (slight) to 6 (collapse). The following table results:
(a) Calculate U, S, and V using the MATLAB command, "[U S V] $=\operatorname{svd}(\mathrm{A}) . "$
(b) Condition U and V according to the three steps outlined in this chapter.

Note: Peak ground acceleration corresponds to U and maximum displacement corresponds to V .
- Keep two singular values
convex hull, $\mathrm{U}=\left[\begin{array}{l}\mathrm{U}_2(1,:) \\ \mathrm{U}_2(3,:) \\ \mathrm{U}_2(4,:)\end{array}\right]$
convex hull, $\mathrm{V}=\left[\begin{array}{l}\mathrm{V}_2(1,:) \\ \mathrm{V}_2(2,:) \\ \mathrm{V}_2(5,:)\end{array}\right]$
(c) Form the matrix of rule consequent values, $\mathrm{Z}_r$.

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Problem 7

Using a truth table prove the set-theoretic equivalence of the IRC with the URC.

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Problem 8

Input measurements are taken for fuzzy sets $\mathrm{A}, \mathrm{B}$, and $\underset{\sim}{\mathrm{C}}$. They yield the following membership values:

$$
\begin{array}{lll}
\mathrm{A}(x)_{\text {Low }}=0.25 & \mathrm{~B}(y)_{\text {Low }}=0.63 & \mathrm{C}(z)_{\text {Low }}=0.18 \\
\mathrm{~A}(x)_{\text {Mellium }}=0.4 & \mathrm{~B}(y)_{\text {Median }}=0.37 & \mathrm{C}(z)_{\text {Medium }}=0.04 \\
\mathrm{~A}(x)_{\text {High }}=0.9 & \mathrm{~B}(y)_{\text {High }}=0.1 & \mathrm{C}(z)_{\text {High }}=0.0 \\
& \text { Output Low }=2 & \\
& \text { Output Medium }=5 \\
\text { Output High }=10
\end{array}
$$
Fill in the following union rule matrix and calculate the final output, ZURC.

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03:45

Problem 9

Take the outer sum of the following vectors:

$$
\mathrm{A}=\left[\begin{array}{l}
1.2 \\
7.6 \\
9.4 \\
3.3
\end{array}\right] \quad \mathrm{B}=\left[\begin{array}{l}
5.5 \\
3.9 \\
2.0 \\
6.8
\end{array}\right]
$$

Fan Yang

Numerade Educator

Problem 10

Repeat Problem 9.8 , except now $A, B$, and $C$ are mapped to the following output sets:

$$
\begin{array}{lll}
\text { Output } \mathrm{A}_{\text {Low }}=2 & \text { Output } \mathrm{B}_{\text {Low }}=3 & \text { Output } \mathrm{C}_{\text {Low }}=1 \\
\text { Output } \mathrm{A}_{\text {Medium }}=5 & \text { Output } \mathrm{B}_{\text {Medium }}=7 & \text { Output } \mathrm{C}_{\text {Medum }}=2 \\
\text { Output } \mathrm{A}_{\text {High }}=10 & \text { Output } \mathrm{B}_{\text {High }}=8 & \text { Output } \mathrm{C}_{\text {High }}=3
\end{array}
$$

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