Fuzzy模糊推导(Matlab实现)

问题呈述

在模糊控制这门课程中，学到了与模糊数学及模糊推理相关的内容，但是并不太清楚我们在选择模糊规则时应该如何处理，是所有的规则都需要由人手工选择，还是仅需要选择其中的一部分就可以了。因此，在课程示例的基础上做了如下的探究。

设计一个以E、EC作为输入，U作为输出的模糊推理系统，令E、EC、U的隶属度函数为如下：

1	0.6	0.2	0	0	0	0	0	0
0.2	0.6	1	0.6	0.2	0	0	0	0
0	0	0.2	0.6	1	0.6	0.2	0	0
0	0	0	0	0.2	0.6	1	0.6	0.2
0	0	0	0	0	0	0.2	0.6	1

分别给定“中心十字规则”以及“最强对角线规则”作为初始规则，观察由此推导出的结果，以验证初始模糊规则库应该如何选择。

结果

中心十字规则

其中，列索引代表E，行索引代表EC，中间的数据区域代表U。1代表负大(NB)，2代表负中(NM)，3代表零(Z)，4代表正中(PB)，5代表正大(PB)。

最强对角线

结果分析

从上面的结果可以分析得出：

1. 当提供部分规则时，其它规则可由这些规则导出；
2. 强对角线规则作为初始规则时，推导效果较好；
3. 在强对角线中，左下角和右上角的隶属度为零，这与人的主观判断相同，即“误差正大，但是误差速度为负大，即误差减小（趋于零）的速度最大”，此时不应有主观判断，即维持原态即可。

Additional

tight_subplot.m

function ha = tight_subplot(Nh, Nw, gap, marg_h, marg_w)

% tight_subplot creates "subplot" axes with adjustable gaps and margins
%
% ha = tight_subplot(Nh, Nw, gap, marg_h, marg_w)
%
%   in:  Nh      number of axes in hight (vertical direction)
%        Nw      number of axes in width (horizontaldirection)
%        gap     gaps between the axes in normalized units (0...1)
%                   or [gap_h gap_w] for different gaps in height and width 
%        marg_h  margins in height in normalized units (0...1)
%                   or [lower upper] for different lower and upper margins 
%        marg_w  margins in width in normalized units (0...1)
%                   or [left right] for different left and right margins 
%
%  out:  ha     array of handles of the axes objects
%                   starting from upper left corner, going row-wise as in
%                   going row-wise as in
%
%  Example: ha = tight_subplot(3,2,[.01 .03],[.1 .01],[.01 .01])
%           for ii = 1:6; axes(ha(ii)); plot(randn(10,ii)); end
%           set(ha(1:4),'XTickLabel',''); set(ha,'YTickLabel','')

% Pekka Kumpulainen 20.6.2010   @tut.fi
% Tampere University of Technology / Automation Science and Engineering


if nargin<3; gap = .02; end
if nargin<4 || isempty(marg_h); marg_h = .05; end
if nargin<5; marg_w = .05; end

if numel(gap)==1; 
    gap = [gap gap];
end
if numel(marg_w)==1; 
    marg_w = [marg_w marg_w];
end
if numel(marg_h)==1; 
    marg_h = [marg_h marg_h];
end

axh = (1-sum(marg_h)-(Nh-1)*gap(1))/Nh; 
axw = (1-sum(marg_w)-(Nw-1)*gap(2))/Nw;

py = 1-marg_h(2)-axh; 

ha = zeros(Nh*Nw,1);
ii = 0;
for ih = 1:Nh
    px = marg_w(1);

    for ix = 1:Nw
        ii = ii+1;
        ha(ii) = axes('Units','normalized', ...
            'Position',[px py axw axh], ...
            'XTickLabel','', ...
            'YTickLabel','');
        px = px+axw+gap(2);
    end
    py = py-axh-gap(1);
end

中心十字规则

clc;
E = [1,0.6,0,0,0,0,0,0,0;0.2,0.6,1,0.6,0.2,0,0,0,0;0,0,0.2,0.6,1,0.6,0.2,0,0;0,0,0,0,0.2,0.6,1,0.6,0.2;0,0,0,0,0,0,0.2,0.6,1];
EC = E;
U = E;


% ----------------------------------------------------------------------------------
% Calculate R
% Deduct relationship
% ----------------------------------------------------------------------------------
R = zeros(81,9);
for i = 1:5
	A = E(i,:)';
	B = EC(3,:);
	C = U(i,:);
	AB = min(repmat(A,1,9), repmat(B,9,1));
	AB = reshape(AB, [81,1]);
	RC = min(repmat(AB,1,9), repmat(C, 81,1));
	R = max(R,RC);
end

for i = [1,2,4,5]
	A = E(3,:)';
	B = EC(i,:);
	C = U(i,:);
	AB = min(repmat(A,1,9), repmat(B,9,1));
	AB = reshape(AB, [81,1]);
	RC = min(repmat(AB,1,9), repmat(C, 81,1));
	R = max(R,RC);
end

% ----------------------------------------------------------------------------------
% Calculate C
% Relationship induction
% ----------------------------------------------------------------------------------

C = zeros(9,5,5);
for i = 1:5
	for j = 1:5
		A = E(i,:)';
		B = EC(j,:);
		AB = min(repmat(A,1,9), repmat(B,9,1));
		AB = reshape(AB, [81,1]);
		C(:,i,j) = max(min(repmat(AB, 1, 9), R));
	end
end

% ----------------------------------------------------------------------------------
% Plot
% ----------------------------------------------------------------------------------
figure(2);clf;
x = (1:9)/9;
ha = tight_subplot(5,5,[.0 .0],[.0 .0],[.0 .0]);
for i = 1:5
	for j = 1:5
		axes(ha(i*5-5+j));
		h = plot(x, C(:,i,j));
		ylim([0,1.2]);
		xlim([min(x), max(x)]);
		set(gca,'XTick',[])
		set(gca,'YTick',[])
	end
end

最强对角线规则

clc;
E = [1,0.6,0,0,0,0,0,0,0;0.2,0.6,1,0.6,0.2,0,0,0,0;0,0,0.2,0.6,1,0.6,0.2,0,0;0,0,0,0,0.2,0.6,1,0.6,0.2;0,0,0,0,0,0,0.2,0.6,1];
EC = E;
U = E;


% ----------------------------------------------------------------------------------
% Calculate R
% Deduct relationship
% ----------------------------------------------------------------------------------
R = zeros(81,9);
for i = 1:5
	A = E(i,:)';
	B = EC(i,:);
	C = U(i,:);
	AB = min(repmat(A,1,9), repmat(B,9,1));
	AB = reshape(AB, [81,1]);
	RC = min(repmat(AB,1,9), repmat(C, 81,1));
	R = max(R,RC);
end


% ----------------------------------------------------------------------------------
% Calculate C
% Relationship induction
% ----------------------------------------------------------------------------------

C = zeros(9,5,5);
for i = 1:5
	for j = 1:5
		A = E(i,:)';
		B = EC(j,:);
		AB = min(repmat(A,1,9), repmat(B,9,1));
		AB = reshape(AB, [81,1]);
		C(:,i,j) = max(min(repmat(AB, 1, 9), R));
	end
end

% ----------------------------------------------------------------------------------
% Plot
% ----------------------------------------------------------------------------------
figure(2);clf;
x = (1:9)/9;
ha = tight_subplot(5,5,[.0 .0],[.0 .0],[.0 .0]);
for i = 1:5
	for j = 1:5
		axes(ha(i*5-5+j));
		h = plot(x, C(:,i,j));
		ylim([0,1.2]);
		xlim([min(x), max(x)]);
		set(gca,'XTick',[])
		set(gca,'YTick',[])
	end
end

模糊合成的定义

设(P)是(U imes V) 上的模糊关系，(Q)是(V imes W)上的模糊关系，则(R)是(U imes W)上的模糊关系，它是(Pcirc Q)的合成，其隶属函数被定义为

[mu_{R}left(u,w ight)Leftrightarrowmu_{P,Q}left(u,w ight)=vee_{vin V}left{ mu_{P}left(u,v ight)wedgemu_{Q}left(v,w ight) ight} ]

若式中牌子(wedge)代表“取小–(min)”，(vee)代表“取大–(max)”，这种合成关系即为最大值(cdot)最小值合成，合成关系(R=Pcirc Q)。

示例：

[A=egin{bmatrix}{0.4} & {0.5} & {0.6}\ {0.1} & {0.2} & {0.3} end{bmatrix},B=egin{bmatrix}0.1 & 0.2\ 0.3 & 0.4\ 0.5 & 0.6 end{bmatrix}. ]

则(Acirc B=egin{bmatrix}0.5 & 0.6\ 0.3 & 0.3 end{bmatrix}), (Bcirc A=egin{bmatrix}{0.1} & {0.2} & {0.2}\ {0.3} & {0.3} & {0.3}\ {0.4} & {0.5} & {0.5} end{bmatrix})。

有定义为

[A imes B = A^mathrm{T}circ B. ]

模糊推导示例

已知一个双输入单输出的模糊系统，其输入量为(x)和(y),输出量为(z)，其输入输出的关系可用如下两条模糊规则描述：

• (R_{1}):如果(x)是(A_{1}) and (y)是(B_{1})，则(z)是(C_{1})

• (R_{2}):如果(x)是(A_{2}) and (y)是(B_{2})，则(z)是(C_{2})

[egin{array}{ccc} {A_{1}}=frac{1}{{a_{1}}}+frac{{0.5}}{{a_{2}}}+frac{0}{{a_{3}}} & {B_{1}}=frac{1}{{b_{1}}}+frac{{0.6}}{{b_{2}}}+frac{{0.2}}{{b_{3}}} & {C_{1}}=frac{1}{{c_{1}}}+frac{{0.4}}{{c_{2}}}+frac{0}{{c_{3}}}\ {A_{2}}=frac{0}{{a_{1}}}+frac{{0.5}}{{a_{2}}}+frac{1}{{a_{3}}} & {B_{2}}=frac{{0.2}}{{b_{1}}}+frac{{0.6}}{{b_{2}}}+frac{1}{{b_{3}}} & {C_{2}}=frac{0}{{c_{1}}}+frac{{0.4}}{{c_{2}}}+frac{1}{{c_{3}}} end{array} ]

(感觉被恶心到了，不知道为什么这儿的array环境始终出不来)

现已知输入(x)为(A'), (y)为(B’),试求输出量。

[egin{array}{cc} A'=frac{{0.5}}{{a_{1}}}+frac{1}{{a_{2}}}+frac{{0.5}}{{a_{3}}} & B'=frac{{0.6}}{{b_{1}}}+frac{1}{{b_{2}}}+frac{{0.6}}{{b_{3}}}\ end{array} ]

---

[egin{aligned} {A_{1}} imes{B_{1}} & =A_{1}^{T}circ{B_{1}}={left[{egin{array}{ccc} 1 & {0.5} & 0end{array}} ight]^{T}}left[{egin{array}{ccc} 1 & {0.6} & {0.2}end{array}} ight]\ & =left[{egin{array}{ccc} 1 & {0.6} & {0.2}\ {0.5} & {0.5} & {0.2}\ 0 & 0 & 0 end{array}} ight] end{aligned} ]

将其按行展开得（把矩阵压扁为一行向量）

[{R_{1}}=ar{R}_{{A_{1}} imes{B_{1}}}^{T}wedge{C_{1}}=left[{egin{array}{c} 1\ {0.6}\ {0.2}\ {0.5}\ {0.5}\ {0.2}\ 0\ 0\ 0 end{array}} ight]wedgeleft[{egin{array}{ccc} 1 & {0.4} & 0end{array}} ight]=left[{egin{array}{ccc} 1 & {0.4} & 0\ 1 & {0.4} & 0\ {0.2} & {0.2} & 0\ {0.5} & {0.4} & 0\ {0.5} & {0.4} & 0\ {0.2} & {0.2} & 0\ 0 & 0 & 0\ 0 & 0 & 0\ 0 & 0 & 0 end{array}} ight] ]

同理：

[{R_{2}}=ar{R}_{{A_{2}} imes{B_{2}}}^{T}wedge{C_{2}}=left[{egin{array}{ccc} 0 & 0 & 0\ 0 & 0 & 0\ 0 & 0 & 0\ 0 & {0.2} & {0.2}\ 0 & {0.4} & {0.5}\ 0 & {0.4} & {0.5}\ 0 & {0.2} & {0.2}\ 0 & {0.4} & {0.6}\ 0 & {0.4} & 1 end{array}} ight] ]

总的蕴含关系为

[R={R_{1}}cup{R_{2}}=left[{egin{array}{ccc} 1 & {0.4} & 0\ {0.6} & {0.4} & 0\ {0.2} & {0.2} & 0\ {0.5} & {0.4} & {0.2}\ {0.5} & {0.4} & {0.5}\ {0.2} & {0.4} & {0.5}\ 0 & {0.2} & {0.2}\ 0 & {0.4} & {0.6}\ 0 & {0.4} & 1 end{array}} ight] ]

计算输入量的模糊集合

[A' ext{ and }B'=A' imes B'=left[{egin{array}{c} {0.5}\ 1\ {0.5} end{array}} ight]wedgeleft[{egin{array}{ccc} {0.6} & 1 & {0.6}end{array}} ight]=left[{egin{array}{ccc} {0.5} & {0.5} & {0.5}\ {0.6} & 1 & {0.6}\ {0.5} & {0.5} & {0.5} end{array}} ight] ]

[ar{R}_{A' imes B'}^{T}=left[{egin{array}{ccccccccc} {0.5} & {0.5} & {0.5} & {0.6} & 1 & {0.6} & {0.5} & {0.5} & {0.5}end{array}} ight] ]

[C'=ar{R}_{A' imes B'}circ R=left[{egin{array}{ccc} {0.5} & {0.4} & {0.5}end{array}} ight] ]

[C'=frac{{0.5}}{{c_{1}}}+frac{{0.4}}{{c_{2}}}+frac{{0.5}}{{c_{3}}} ]