📄 11.htm
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<P><IMG height=42 src="11.files/5.3.ht18.gif" width=448 border=0></P>
<P>从而可知</P>
<P><IMG height=45 src="11.files/5.3.ht19.gif" width=323 border=0></P>
<P>最后有</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>W(k)]<SUP>2</SUP>-[<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>W(k-1)]<SUP>2</SUP><0
(2.68)</P>
<P>式(2．68)说明引理的性质(1)成立。</P>
<P>根据性质(1)，则当k——<SPAN
style="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>，则有w(k)＝W<SUB>0</SUB>．故而在式(2．66)中两边都为0。这也就是必定有</P>
<DIV align=center>
<CENTER>
<TABLE cellSpacing=0 cellPadding=0 width="80%" border=0>
<TBODY>
<TR>
<TD width="50%"><IMG height=51 src="11.files/5.3.ht20.gif" width=288
border=0></TD>
<TD width="50%">(2.69)</TD></TR></TBODY></TABLE></CENTER></DIV></TD></TR>
<TR>
<TD width="100%" height=414>
<P>可见，引理的性质(2)成立。 </P>
<P>证毕。</P>
<P>有了上面的引理，就可以给出由式(2．53)—(2．59)组成的控制结构对对象式(2．52)执行适应控制的闭环性质定理。</P>
<P>定理：在对象由式(2．52)描述的控制中，式(2．53)—(2．59)构成的适应控制有如下的闭环性质：</P>
<P>(1)输入信号u(t)，输出信号y(t)都是有界的。</P>
<P><IMG height=30 src="11.files/5.3.ht21.gif" width=230 border=0></P>
<P>证明：</P>
<P>设系统的跟踪误差用e'(k)表示</P>
<P>e'(k)=y(k)-r(k)
(2.70)</P>
<P>y(k)由式(2．61)给出。</P>
<P>r(k)可由式(2．57)，(2．58)，(2．59)求出，先用W<SUB>n+1</SUB>(k)乘〔2．59)两边，则有</P>
<P>W<SUB>n+1</SUB>(k-1)u(k-1)=r(k)+W<SUB>1</SUB>(k-1)(-X<SUB>1</SUB>(k-1))+.....,+W<SUB>n</SUB>(k-1)(-X<SUB>n</SUB>(k-1))+W<SUB>n+2</SUB>(k-1)(-X<SUB>n+2</SUB>(k-1))+......,Wn+m(k-1)(-Xn+m(k-1))</P>
<P>整理后有</P>
<P>r(k)=W<SUB>1</SUB>(k-1)X<SUB>1</SUB>(k-1)+......,+W<SUB>n</SUB>(k-1)X<SUB>n</SUB>(k-1)+W<SUB>n+1</SUB>(k-1)u(k-1)+W<SUB>n+2</SUB>(k-1)X<SUB>n+2</SUB>(k-1)+......,+W<SUB>n+m</SUB>(k-1)X<SUB>n+m</SUB>(k-1)</P>
<P align=right>(2.71) </P>
<P>由于 u(k-1)=Xn+1(k-1)</P>
<P>故而有</P>
<TABLE cellSpacing=0 cellPadding=0 width="80%" align=center border=0>
<TBODY>
<TR>
<TD width="71%"><IMG height=43 src="11.files/5.3.ht22.gif" width=288
border=0></TD>
<TD width="29%">(2.72)</TD></TR>
<TR>
<TD width="71%"><FONT size=2>从式(2.61)和式(2.72)，则有</FONT></TD>
<TD width="29%"></TD></TR>
<TR>
<TD width="71%"><IMG height=111 src="11.files/5.3.ht23.gif"
width=448 border=0></TD>
<TD width="29%">(2.73)</TD></TR>
<TR>
<TD width="71%"><FONT size=2>从引理的性质(2)有</FONT></TD>
<TD width="29%"></TD></TR>
<TR>
<TD width="71%"><IMG height=52 src="11.files/5.3.ht25.gif" width=274
border=0></TD>
<TD width="29%">(2.74)</TD></TR></TBODY></TABLE></TD></TR>
<TR>
<TD width="100%" height=2>
<P>只要证明x<SUP>T</SUP>(k-1)x(k-1)是有界的，就可以证明e(<SPAN
style="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>)＝0，也就可以证明定理中的性质(2)。
</P>
<P>下面证明<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>x(k-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">‖</SPAN>有限。</P>
<P>从对象式(2．52)有关条件，对象的输入输出信号满足</P>
<TABLE cellSpacing=0 cellPadding=0 width="80%" align=center border=0>
<TBODY>
<TR>
<TD width="71%"><IMG height=33 src="11.files/5.3.ht24.gif" width=288
border=0></TD>
<TD width="29%">(2.75)</TD></TR></TBODY></TABLE></TD></TR>
<TR>
<TD width="100%" height=39>
<P>其中：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">≤</SPAN>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">≤</SPAN>k；m<SUB>1</SUB><<SPAN
style="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>；m<SUB>2</SUB><<SPAN
style="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>
</P>
<P>根据式(2．52)对象的满足条件，从式(2．53)则有</P>
<TABLE cellSpacing=0 cellPadding=0 width="80%" align=center border=0>
<TBODY>
<TR>
<TD width="71%"><IMG height=36 src="11.files/5.3.ht26.gif" width=512
border=0></TD>
<TD width="29%">(2.76)</TD></TR>
<TR>
<TD width="71%"><FONT size=2>既然，给定信号r是有界的，所以跟踪误差有</FONT></TD>
<TD width="29%"></TD></TR>
<TR>
<TD width="71%"><IMG height=31 src="11.files/5.3.ht27.gif" width=364
border=0></TD>
<TD width="29%">(2.77)</TD></TR></TBODY></TABLE></TD></TR>
<TR>
<TD width="100%" height=6>
<P>从而有|e'(k)|+m3<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>|y(k)|
</P>
<P>由此，式(2．76)可以写为：</P>
<TABLE cellSpacing=0 cellPadding=0 width="80%" align=center border=0>
<TBODY>
<TR>
<TD width="86%"><IMG height=42 src="11.files/5.3.ht28.gif" width=701
border=0></TD>
<TD width="14%">(2.78)</TD></TR></TBODY></TABLE></TD></TR>
<TR>
<TD width="100%" height=70>
<P>其中：0<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>C<SUB>1</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><SPAN
style="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>；0<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>C<SUB>2</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><SPAN
style="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>。
</P>
<P>假设跟踪误差e'(k)有界，则从式(2．78)可知：<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>x(k)<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>同样有界；这样从式(2．74)可知</P>
<TABLE cellSpacing=0 cellPadding=0 width="80%" align=center border=0>
<TBODY>
<TR>
<TD width="71%"><IMG height=35 src="11.files/5.3.ht30.gif" width=128
border=0></TD>
<TD width="29%">(2.79)</TD></TR>
<TR>
<TD width="71%"><FONT size=2>显然，定理的性质(2)成立。
<BR>假设跟踪误差e'(k)无界，则存在时刻序列|kn|，令</FONT></TD>
<TD width="29%"></TD></TR>
<TR>
<TD width="71%"><IMG height=42 src="11.files/5.3.ht29.gif" width=160
border=0></TD>
<TD width="29%">(2.80)</TD></TR></TBODY></TABLE></TD></TR>
<TR>
<TD width="100%">
<P>取m4＝max(1，<SPAN
style="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">ε</SPAN>)
</P>
<P>考虑</P>
<P><IMG height=57 src="11.files/5.3.ht31.gif" width=284 border=0></P>
<TABLE cellSpacing=0 cellPadding=0 width="80%" align=center border=0>
<TBODY>
<TR>
<TD width="70%"><IMG height=247 src="11.files/5.3.ht32.gif"
width=320 border=0></TD>
<TD width="30%">(2.81)</TD></TR>
<TR>
<TD width="70%"><FONT size=2>对式(2.81)取极限有</FONT></TD>
<TD width="30%"></TD></TR>
<TR>
<TD width="70%"><IMG height=159 src="11.files/5.3.ht33.gif"
width=464 border=0></TD>
<TD width="30%">(2.82)</TD></TR></TBODY></TABLE></TD></TR>
<TR>
<TD width="100%" height=1174>
<P>这个极限存在说明e'(K)有界，假设其无界不成立。 </P>
<P>由于e'(k)有界，故式(2．79)是必定成立的。由于e'(k)＝y(k)-r(k),而r(k)有界,所以，y(k)有界。从式(2．75)可知u(k)也有界。则定理的两个性质成立。</P>
<P>证毕。</P>
<P>四、系统实际运行情况</P>
<P>当对象的结构不同时，可以用于检验图2-16所示的神经适应控制系统的运行结果。对象仿真器PE，神经控制器NC分别由式(2.52)-(2.55)和式(2.57)-(2.59)所描述；学习时采用式(2．56)和式(2．58)。</P>
<P>1．对有噪声的稳定对象的控制</P>
<P>对象由下式表示</P>
<P><IMG height=55 src="11.files/5.3.ht34.gif" width=366 border=0></P>
<P>设对象仿真器PE和神经控制器NC输入的向量为6个元素，有n＝m＝3。在训练学习时PE的权系数向量更新取<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">α</SPAN>和<SPAN
style="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">ε</SPAN>的值如下：
<P><IMG height=52 src="11.files/5.3.ht35.gif" width=93 border=0>
<P>权系数向量的初始值取
<P>W(0)=[0,0,0,1,0,0]<SUP>T</SUP>
<P align=center><IMG height=342 src="11.files/5.3.ht36.gif" width=467
border=0>
<P align=center>图2-19 给定值r和对象输出y
<P align=center><IMG height=253 src="11.files/5.3.ht37.gif" width=461
border=0>
<P align=center>图2-20 NC产生的控制信号u
<P align=center><IMG height=237 src="11.files/5.3.ht38.gif" width=481
border=0>
<P align=center>图2-21 PE的学习过程W(k)的变化 </P></TD></TR>
<TR>
<TD width="100%" height=1319>
<P>噪声是平均值为零的高斯白噪声。 </P>
<P>给定输入r是幅值为1的方波；每方波周期采样80次。</P>
<P>控制结果和情况如图2—19和图2—20所示。其中图2—19是对象输出和给定值的情况；图2—20是NC产生的控制信号u(k)。<BR>很明显，对象仿真器能正确地预测对象的动态过程。</P>
<P>图2—21给出了对象仿真器PE的学习过程。</P>
<P>2．对不稳定对象的控制</P>
<P>不稳定对象由下式表示</P>
<P><IMG height=54 src="11.files/5.3.ht39.gif" width=390 border=0></P>
<P>在系统中，PE和NC的输入都采用6个元素的向量，故n＝m＝3。在训练学习时．PE权系数向量更新取<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">α</SPAN>和<SPAN
style="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">ε</SPAN>的值为</P>
<P><IMG height=51 src="11.files/5.3.ht40.gif" width=94 border=0></P>
<P>权系数向量初始化取值为</P>
<P>W(0)=[0,0,0,1,0,0]<SUP>T</SUP></P>
<P>给定输入r为幅度为1的方波，方波每周期采样80次。</P>
<P>控制情况和结果以及邢学习时的w(k)变化情况分别如图2—22，图2—23，图2—24所示。对于不稳定对象，显然在过渡过程中有较大的超调；但在PE学习之后，对象输出能跟踪给定r。</P>
<P align=center><IMG height=355 src="11.files/5.3.ht41.gif" width=480
border=0></P>
<P align=center>图2-22 给定r和对象输出y的波形</P>
<P align=center><IMG height=238 src="11.files/5.3.ht42.gif" width=457
border=0></P>
<P align=center>图2-23 NC产生的控制信号U的波形</P>
<P align=center><IMG height=324 src="11.files/5.3.ht43.gif" width=472
border=0></P>
<P align=center>图2-24 PE学习时W(k)的变化情况 </P></TD></TR>
<TR>
<TD width="100%" height=17>
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