📄 18.htm
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(4.48)
<P>其中：x<SUB>1</SUB>，x<SUB>2</SUB>，……，x<SUB>m</SUB>为输入精确量
<P>A<SUB>1i</SUB>(x<SUB>1</SUB>)为x<SUB>1</SUB>对A<SUB>1i</SUB>的隶属度，其余A<SUB>1i</SUB>(x<SUB>2</SUB>)，……，A<SUB>mi</SUB>(x<SUB>m</SUB>)同理。
<P>所有的控制规则产生的总输出为y：
<TABLE height=148 cellSpacing=0 cellPadding=0 width="80%" align=center
border=0>
<TBODY>
<TR>
<TD width="73%" height=41><IMG height=39 src="18.files/7.2.ht1.gif"
width=128 border=0></TD>
<TD width="27%" height=41>(4.49)</TD></TR>
<TR>
<TD width="73%" height=22><FONT
size=2>模糊量A<SUB>ki</SUB>(x<SUB>k</SUB>)用钟形函数或高斯函数表示，故而有：</FONT></TD>
<TD width="27%" height=22></TD></TR>
<TR>
<TD width="73%" height=49><IMG height=47 src="18.files/7.2.ht2.gif"
width=307 border=0></TD>
<TD width="27%" height=49>(4.50)</TD></TR></TBODY></TABLE></P></TD></TR>
<TR>
<TD width="100%" height=18>
<P>或者 </P>
<P>A<SUB>ki</SUB>(x<SUB>k</SUB>)=exp(-{[(x<SUB>k</SUB>-a<SUB>ki</SUB>)/b<SUB>ki</SUB>]<SUP>2</SUP>}<SUP>cki</SUP>)
(4.51)</P>
<P>其中：a<SUB>ki</SUB>，b<SUB>ki</SUB>，c<SUB>ki</SUB>是函数的参数；改变这些参数则函效会产生相应的变化。故而模糊量的形状也产生变化。<BR>当系统是一个双输入单输出系统时，则模糊推理网络的结构如图4—11所示。</P>
<P>在这个网络中有两个输入，每个输入被划分成3个模糊量。对于输入x<SUB>1</SUB>，有L<SUB>1</SUB>(大)，M<SUB>1</SUB>(中)，s<SUB>1</SUB>(小)这3个隶属函数；对于输入x<SUB>2</SUB>，有I<SUB>2</SUB>(大)，M<SUB>2</SUB>(中)，S<SUB>2</SUB>(小)这3个隶属函数。故而有9条控制规则：</P>
<P>if x<SUB>1</SUB> is L<SUB>1</SUB> and x<SUB>2</SUB> is L<SUB>2</SUB>
then y<SUB>1</SUB>=f<SUB>1</SUB>(x<SUB>1</SUB>,x<SUB>2</SUB>)</P>
<P>if x<SUB>1</SUB> is L<SUB>1</SUB> and x<SUB>2</SUB> is M<SUB>2</SUB>
then y<SUB>2</SUB>=f<SUB>2</SUB>(x<SUB>1</SUB>,x<SUB>2</SUB>)</P>
<P>if x<SUB>1</SUB> is L<SUB>1</SUB> and x<SUB>2</SUB> is S<SUB>2</SUB>
then y<SUB>3</SUB>=f<SUB>3</SUB>(x<SUB>1</SUB>,x<SUB>2</SUB>)</P>
<P>if x<SUB>1</SUB> is M<SUB>1</SUB> and x<SUB>2</SUB> is L<SUB>2</SUB>
then y<SUB>4</SUB>=f<SUB>4</SUB>(x<SUB>1</SUB>,x<SUB>2</SUB>)</P>
<P>if x<SUB>1</SUB> is M<SUB>1</SUB> and x<SUB>2</SUB> is M<SUB>2</SUB>
then y<SUB>5</SUB>=f<SUB>5</SUB>(x<SUB>1</SUB>,x<SUB>2</SUB>)</P>
<P>if x<SUB>1</SUB> is M<SUB>1</SUB> and x<SUB>2</SUB> is S<SUB>2</SUB>
then y<SUB>6</SUB>=f<SUB>6</SUB>(x<SUB>1</SUB>,x<SUB>2</SUB>)</P>
<P>if x<SUB>1</SUB> is S<SUB>1</SUB> and x<SUB>2</SUB> is L<SUB>2</SUB>
then y<SUB>7</SUB>=f<SUB>7</SUB>(x<SUB>1</SUB>,x<SUB>2</SUB>)</P>
<P>if x<SUB>1</SUB> is S<SUB>1</SUB> and x<SUB>2</SUB> is M<SUB>2</SUB>
then y<SUB>8</SUB>=f<SUB>8</SUB>(x<SUB>1</SUB>,x<SUB>2</SUB>)</P>
<P>if x<SUB>1</SUB> is S<SUB>1</SUB> and x<SUB>2</SUB> is S<SUB>2</SUB>
then y<SUB>9</SUB>=f<SUB>9</SUB>(x<SUB>1</SUB>,x<SUB>2</SUB>)</P>
<P>对应于图4—11中的模糊划分，则输入空间被划分成9个模糊子空间，如图4—12中所示。</P>
<P align=center><IMG height=362 src="18.files/7.2.ht3.gif" width=627
border=0></P>
<P align=center>图4-11 简化的模糊推理网络SFIN</P></TD></TR>
<TR>
<TD width="100%" height=534>
<P>图4—11所示的模糊推理网络表示的控制规则中，由于只有两个输入，故而有 </P>
<P>y<SUB>i</SUB>=f<SUB>i</SUB>(X)=f<SUB>i</SUB>(x<SUB>1</SUB>,x<SUB>2</SUB>),
i=1,2,......,9</P>
<P>即是：y<SUB>i</SUB>=p<SUB>0i</SUB>+p<SUB>1i</SUB>x<SUB>1</SUB>+p<SUB>2i</SUB>x<SUB>2</SUB>
(4.52)</P>
<P>如果对图4—11所示的网络进行优化，则相应对于控制规则的前件和后件进行优化。</P>
<P>当对控制规则进行前件优化时，也即对网络的第二层神经元进行优化，这时也即是对径向基函数的参数a<SUB>ki</SUB>，b<SUB>ki</SUB>，c<SUB>ki</SUB>进行优化；其中k是输入量下标号，即k＝1，2，i是控制规则条数号，即i=1，2，…，9。从控制规则集中可知，有模物量：</P>
<P>A<SUB>11</SUB>=A<SUB>12</SUB>=A<SUB>13</SUB>=L<SUB>1</SUB>
A<SUB>21</SUB>=A<SUB>22</SUB>=A<SUB>23</SUB>=L<SUB>2</SUB></P>
<P>A<SUB>14</SUB>=A<SUB>15</SUB>=A<SUB>16</SUB>=M<SUB>1</SUB>
A<SUB>24</SUB>=A<SUB>25</SUB>=A<SUB>26</SUB>=M<SUB>2</SUB></P>
<P>A<SUB>17</SUB>=A<SUB>18</SUB>=A<SUB>19</SUB>=S<SUB>1</SUB>
A<SUB>27</SUB>=A<SUB>28</SUB>=A<SUB>29</SUB>=S<SUB>2</SUB></P>
<P>当对控制规则的后件进行优化时，也即是对f<SUB>i</SUB>(x<SUB>1</SUB>，x<SUB>2</SUB>)中的系数p<SUB>0i</SUB>，p<SUB>1i</SUB>，p<SUB>2i</SUB>进行优化。</P>
<P align=center><IMG height=406 src="18.files/7.2.ht4.gif" width=554
border=0></P>
<P align=center>图4-12 模糊子空间</P></TD></TR>
<TR>
<TD width="100%" height=147>
<P>2．神经模糊控制器的优化<BR>当在简化的模糊推理网络SFIN中的X<SUB>1</SUB>端输入偏差信号e(k)＝y<SUB>d</SUB>(k)-y(k)；其中y<SUB>d</SUB>(k)为给定信号，y(k)为对象实际输出信号。在x<SUB>2</SUB>端输入偏差变化率<SPAN
style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">Δ</SPAN>e(k)＝e(k)-e(k-1)。而在SFIN输出端输出的是增量控制信号<SPAN
style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">Δ</SPAN>u(k)＝u(k)-u(k-1)。则SFIN可以看作一个参数化的模糊控制器。
</P>
<P>对SFIN考虑二个优化过程。一个是SFIN初始化的监视学习优化；另一个SFIN的在线自适应学习优化。</P>
<P>(1)SFIN的初始化监视学习</P>
<P>给出用于SFIN初始化的教师信号数据。</P>
<P>T{[e(i),<SPAN
style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">Δ</SPAN>e(i)]——<SPAN
style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">Δ</SPAN>u(i)|i=1:N}
(4.53)</P>
<P>其中：e(i)，<SPAN
style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">Δ</SPAN>e(i)是输入</P>
<P><SPAN
style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">Δ</SPAN>u(i)是输出。</P>
<P>初始化教师数据来自系统控制人员的控制数据，优化时采用的目标函数为I<SUB>0</SUB>：</P>
<TABLE cellSpacing=0 cellPadding=0 width="80%" align=center border=0>
<TBODY>
<TR>
<TD width="80%"><IMG height=42 src="18.files/7.2.ht5.gif" width=366
border=0></TD>
<TD width="20%">(4.54)</TD></TR>
<TR>
<TD width="80%"><FONT
size=2>其中：SFIN[e(i)，Ae(i)]表示数据对e(i)，Ae(I)输入SFIN之后得到的输出。<BR>用W表示SFIN中的普通参数，它可以用下式进行校正：</FONT></TD>
<TD width="20%"></TD></TR>
<TR>
<TD width="80%"><IMG height=42 src="18.files/7.2.ht6.gif" width=324
border=0></TD>
<TD width="20%">(4.55)</TD></TR></TBODY></TABLE></TD></TR>
<TR>
<TD width="100%" height=325>
<P>其中：<SPAN
style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">η</SPAN>(k)是学习速率，1><SPAN
style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">η</SPAN>(k)>0。
</P>
<P>(i)前件参数估计</P>
<P>偏差e所取的模糊量个数表示为N<SUB>e</SUB>。i偏差变化率<SPAN
style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">Δ</SPAN>e所取的模糊量个数表示为N<SUB><SPAN
style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">Δ</SPAN>e</SUB>。</P>
<P>根据先验知识，偏差e和偏差变化率<SPAN
style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">Δ</SPAN>e的边界分别是：</P>
<P>S<SUB>e</SUB><SPAN
style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">≤</SPAN>e<SPAN
style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">≤</SPAN>B<SUB>e</SUB></P>
<P>S<SUB><SPAN
style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">Δ</SPAN>e</SUB><SPAN
style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">≤Δ</SPAN>e<SPAN
style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">≤</SPAN>B<SUB><SPAN
style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">Δ</SPAN>e</SUB></P>
<P>则前件的初始参数可以选择如下</P>
<P><IMG height=23 src="18.files/7.2.ht7.gif" width=394 border=0></P>
<P><IMG height=22 src="18.files/7.2.ht8.gif" width=336 border=0></P>
<P><IMG height=20 src="18.files/7.2.ht9.gif" width=178 border=0></P>
<P>(ii)后件参数估计</P>
<P>从上可知，在SFIN中有控制规则为N<SUB>r</SUB>条：</P>
<P>N<SUB>r</SUB>=N<SUB>e</SUB>*N<SUB><SPAN
style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">Δ</SPAN>e</SUB>
(4.56)</P>
<P>由于控制规则的后件为y<SUB>i</SUB>＝f<SUB>i</SUB>(x)＝p<SUB>0i</SUB>+p<SUB>1i</SUB>x<SUB>1</SUB>+p<SUB>2i</SUB>x<SUB>2</SUB>，也即有参数p<SUB>0i</SUB>，p<SUB>1i</SUB>，p<SUB>2i</SUB>。所以，总的后件参数为Ng；</P>
<P>Ng=3Nr
(4.57)</P>
<P>要求取Ng个参数，则要求有Ng个输入输出教师数据对：</P>
<P>{[e(q),<SPAN
style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">Δ</SPAN>e(q)]——<SPAN
style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">Δ</SPAN>u(q)},q=1:Ng
(4.58)</P>
<P>并通过这Ng个数据对才能形成Ng个方程的方程组．以求取后件的Ng个参数。</P>
<P>当在SFIN输入Ng对[e(q)，e(q)]，q＝1:Ng数据时，则从式(4．49)有：</P>
<TABLE cellSpacing=0 cellPadding=0 width="80%" align=center border=0>
<TBODY>
<TR>
<TD width="82%"><IMG height=48 src="18.files/7.2.ht10.gif" width=512
border=0></TD>
<TD width="18%">(4.59)</TD></TR>
<TR>
<TD width="82%"><FONT
size=2>在上面式(4.59)中，需要求f<SUB>i</SUB>中的参数p<SUB>0i</SUB>,p<SUB>1i</SUB>,p<SUB>2i</SUB>，则可写成</FONT></TD>
<TD width="18%"></TD></TR>
<TR>
<TD width="82%"><IMG height=50 src="18.files/7.2.ht11.gif" width=477
border=0></TD>
<TD width="18%">(4.60)</TD></TR></TBODY></TABLE></TD></TR>
<TR>
<TD width="100%" height=62>
<P>用P表示后件f<SUB>i</SUB>的参数向量．M为Di的Ng*Ng短阵．U＝[<SPAN
style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">Δ</SPAN>u(1)，<SPAN
style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">Δ</SPAN>u(2)，……，<SPAN
style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">Δ</SPAN>u(Ng)]T，则式(4.60)可以写成如下矩阵形式：
</P>
<P>MP=U (4.61)</P>
<P>显然，解矩阵方程式(4．61)，则有</P>
<P>P=M<SUP>-1</SUP>U</P>
<P>从而可得后件参数向量P，也即得出后件参数，p<SUB>0i</SUB>，p<SUB>1i</SUB>，p<SUB>2i</SUB>，i＝1:9。</P>
<P>(2)SFIN的在线自适应学习优化</P>
<P>模糊推理网络SFIN是作为模糊控制器的，由它组成的模糊控制系统的框图如图4-13所示。</P>
<P align=center><IMG height=79 src="18.files/7.2.ht12.gif" width=488
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