一、直线型一阶倒立摆模型建立
一级倒立摆系统是一个不稳定的系统，需要对其进行机理建模。 在研究过程中，应忽略空气摩擦、等，而后可将倒立摆系统进行抽象化，认为其由小车和匀质刚性杆两部分组成并对这两部分进行如图所示的受力分析:
其中为小车的质量和摆杆的质量；b、F 和 x分别为小车的摩擦系数、施加在小车上的作用力和小车的位置[8]；I和 分别为摆杆的惯量和摆杆转动轴心到质心的长度； 和 分别为摆杆与竖直向上方向和竖直向下方向的夹角；N 和P 分别为摆杆作用力的水平与竖直分量。
小车水平方向的合力：
摆杆水平方向的合力:
摆杆水平方向运动方程：
摆杆力矩平衡方程：
可得到摆杆在竖直方向的运动方程：
摆杆竖直方向运动方程：
将作用力 F 用 u代替,同时进行线性化,即得到:
二、部分源代码
function varargout = dlb(varargin) % GUI主程序
gui_Singleton = 1;
gui_State = struct('gui_Name', mfilename, ...
'gui_Singleton', gui_Singleton, ...
'gui_OpeningFcn', @dlb_OpeningFcn, ...
'gui_OutputFcn', @dlb_OutputFcn, ...
'gui_LayoutFcn', [] , ...
'gui_Callback', []);
if nargin & isstr(varargin{1})
gui_State.gui_Callback = str2func(varargin{1});
end
if nargout
[varargout{1:nargout}] = gui_mainfcn(gui_State, varargin{:});
else
gui_mainfcn(gui_State, varargin{:});
end
%-----------------------打开界面------------------------%
function dlb_OpeningFcn(hObject, eventdata, handles, varargin)
handles.output = hObject;
guidata(hObject, handles);
global h1 h2 h3;
axes(handles.sys1);
axis equal;
axis([0 5 -20 20 0 20]);
grid on;
view(80,30);
reall=15;
ll=4+reall;
v=[0 0 0;0 5 0;3 5 0;3 0 0;0 0 3;0 5 3;3 5 3;3 0 3];
f=[1 2 3 4;2 6 7 3;4 3 7 8;1 5 8 4;1 2 6 5;5 6 7 8];
g_v=[1 2 3;1 3 3;2 3 3;2 2 3;1 2 ll;1 3 ll;2 3 ll;2 2 ll];
g_f=[1 2 3 4;2 6 7 3;4 3 7 8;1 5 8 4;1 2 6 5;5 6 7 8];
h1=patch('Faces',f,'Vertices',v,'FaceColor','b');
h2=patch('Faces',g_f,'Vertices',g_v,'FaceColor','r');
hold on;
[ballx,bally,ballz]=sphere(8);
ballo=[1.5 2.5 ll];
h3=surf(ballx*1.1+ballo(1),bally*1.1+ballo(2),ballz*1.1+ballo(3));
colormap(autumn);
axes(handles.sys3);
sys3data=imread('model.jpg'); %打开倒立摆示意图
image(sys3data);
set(gca,'Xtick',[],'Ytick',[],'box','on');
function varargout = dlb_OutputFcn(hObject, eventdata, handles) %输出函数
varargout{1} = handles.output;
function normal(hObject, eventdata, handles) %公共调用函数
if ispc
set(hObject,'BackgroundColor','white');
else
set(hObject,'BackgroundColor',get(0,'defaultUicontrolBackgroundColor'));
end
%----------------------------倒立摆模型参数创建---------------------------%
function mc_CreateFcn(hObject, eventdata, handles) %M,小车质量
normal(hObject, eventdata, handles);
function mc_Callback(hObject, eventdata, handles)
function l_CreateFcn(hObject, eventdata, handles) %L，摆杆长度
normal(hObject, eventdata, handles);
function l_Callback(hObject, eventdata, handles)
function mq_CreateFcn(hObject, eventdata, handles) %m，摆杆质量
normal(hObject, eventdata, handles);
function mq_Callback(hObject, eventdata, handles)
%-------------------------LQR参数创建-------------------------%
function q1_CreateFcn(hObject, eventdata, handles) %q1
normal(hObject, eventdata, handles);
function q1_Callback(hObject, eventdata, handles)
function q2_CreateFcn(hObject, eventdata, handles) %q2
normal(hObject, eventdata, handles);
function q2_Callback(hObject, eventdata, handles)
function q3_CreateFcn(hObject, eventdata, handles) %q3
normal(hObject, eventdata, handles);
function q3_Callback(hObject, eventdata, handles)
function q4_CreateFcn(hObject, eventdata, handles) %q4
normal(hObject, eventdata, handles);
function q4_Callback(hObject, eventdata, handles)
function r_CreateFcn(hObject, eventdata, handles) %R
normal(hObject, eventdata, handles);
function r_Callback(hObject, eventdata, handles)
%-----------------采用LQR,计算增益K------------------%
function lqrok_Callback(hObject, eventdata, handles)
global A B C D K;
M_str=get(handles.mc,'string');
M=str2double(M_str); %获取小车质量M
m_str=get(handles.mq,'string');
m=str2double(m_str); %获取摆杆质量m
L_str=get(handles.l,'string');
L=str2double(L_str); %获取摆杆长度L
q1_str=get(handles.q1,'string');
q1=str2double(q1_str);
q2_str=get(handles.q2,'string');
q2=str2double(q2_str);
q3_str=get(handles.q3,'string');
q3=str2double(q3_str);
q4_str=get(handles.q4,'string');
q4=str2double(q4_str);
R_str=get(handles.r,'string');
R=str2double(R_str);
Q=[q1,0,0,0;0,q2,0,0;0,0,q3,0;0,0,0,q4];
%Inverted Pendulum Model
g=9.8;
I=1/12*m*L^2;
l=1/2*L;
t1=m*(M+m)*g*l/[(M+m)*I+M*m*l^2];
t2=-m^2*g*l^2/[(m+M)*I+M*m*l^2];
t3=-m*l/[(M+m)*I+M*m*l^2];
t4=(I+m*l^2)/[(m+M)*I+M*m*l^2];
A=[0,1,0,0;t1,0,0,0;0,0,0,1;t2,0,0,0];
B=[0;t3;0;t4];
C=[0,0,1,0];
D=[0];
[K,sz,ez]=lqr(A,B,Q,R); %利用LQR方法计算K值
set(handles.k1,'string',K(1));
set(handles.k2,'string',K(2));
set(handles.k3,'string',K(3));
set(handles.k4,'string',K(4));
%--------------------K的创建-----------------------%
function k1_CreateFcn(hObject, eventdata, handles) %K1
normal(hObject, eventdata, handles);
function k1_Callback(hObject, eventdata, handles)
function k2_CreateFcn(hObject, eventdata, handles) %K2
normal(hObject, eventdata, handles);
function k2_Callback(hObject, eventdata, handles)
function k3_CreateFcn(hObject, eventdata, handles) %K3
normal(hObject, eventdata, handles);
function k3_Callback(hObject, eventdata, handles)
function k4_CreateFcn(hObject, eventdata, handles) %K4
normal(hObject, eventdata, handles);
function k4_Callback(hObject, eventdata, handles)
%------------------------摆角度和小车位置初始值设定-----------------------%
function in_po_CreateFcn(hObject, eventdata, handles) %小车水平位置创建
normal(hObject, eventdata, handles);
function in_po_Callback(hObject, eventdata, handles) %小车水平位置回调
Current_Val=str2num(get(hObject,'string'));
set(handles.slider_po,'Value',Current_Val);
function slider_po_CreateFcn(hObject, eventdata, handles) %小车水平拖动条创建
usewhitebg = 1;
if usewhitebg
set(hObject,'BackgroundColor',[.9 .9 .9]);
else
set(hObject,'BackgroundColor',get(0,'defaultUicontrolBackgroundColor'));
end
function slider_po_Callback(hObject, eventdata, handles) %小车水平拖动条回调
global h1 h2 h3;
Value=get(hObject,'Value');
set(handles.in_po,'String',Value);
axes(handles.sys1);
axis equal;
axis([0 5 -20 20 0 20]);
reall=15;
ll=4+reall;
v=[0 0 0;0 5 0;3 5 0;3 0 0;0 0 3;0 5 3;3 5 3;3 0 3];
f=[1 2 3 4;2 6 7 3;4 3 7 8;1 5 8 4;1 2 6 5;5 6 7 8];
g_v=[1 2 3;1 3 3;2 3 3;2 2 3;1 2 ll;1 3 ll;2 3 ll;2 2 ll];
g_f=[1 2 3 4;2 6 7 3;4 3 7 8;1 5 8 4;1 2 6 5;5 6 7 8];
delete(h1);delete(h2);delete(h3);
i=get(handles.slider_po,'Value')*10;
bbb=[0 i 0;0 i 0;0 i 0;0 i 0;0 i 0;0 i 0;0 i 0;0 i 0;];
v1=v+bbb;
af=get(handles.slider_ag,'Value');
gt1=0.5*cos(-af)-0.5;
gt2=0.5*sin(-af)-0;
gt3=0.5/cos(af)+(ll-0.5*tan(af))*sin(af)-cos(af)+0.5;
gt4=(ll-0.5*tan(af))*cos(af)+sin(af)-ll;
gt5=0.5/cos(af)+(ll-0.5*tan(af))*sin(af)-0.5;
gt6=(ll-0.5*tan(af))*cos(af)-ll;
jiao=[0 -gt1 -gt2;0 gt1 gt2;0 gt1 gt2;0 -gt1 -gt2;0 gt3 gt4;0 gt5 gt6;0 gt5 gt6;0 gt3 gt4];
g_v1=g_v+jiao+bbb;
h1=patch('Faces',f,'Vertices',v1,'FaceColor','b');
h2=patch('Faces',g_f,'Vertices',g_v1,'FaceColor','r');
hold on;
[ballx,bally,ballz]=sphere(8);
ballo=[1.5 2.5 ll];
bally_1=bally*1.1+i+ballo(2)+ll*sin(af);
ballz_1=ballz*1.1+ll*cos(af);
h3=surf(ballx*1.1+ballo(1),bally_1,ballz_1);
colormap(autumn);
function in_ag_CreateFcn(hObject, eventdata, handles) %摆杆角度创建
normal(hObject, eventdata, handles);
function in_ag_Callback(hObject, eventdata, handles) %摆杆角度回调
Current_Val=str2num(get(hObject,'string'));
set(handles.slider_ag,'Value',Current_Val);
function slider_ag_CreateFcn(hObject, eventdata, handles) %摆杆角度拖动条创建
usewhitebg = 1;
if usewhitebg
set(hObject,'BackgroundColor',[.9 .9 .9]);
else
set(hObject,'BackgroundColor',get(0,'defaultUicontrolBackgroundColor'));
end
function slider_ag_Callback(hObject, eventdata, handles) %摆杆角度拖动条回调
global h1 h2 h3;
Value=get(hObject,'Value');
set(handles.in_ag,'String',Value);
axes(handles.sys1);
axis equal;
axis([0 5 -20 20 0 20]);
v=[0 0 0;0 5 0;3 5 0;3 0 0;0 0 3;0 5 3;3 5 3;3 0 3];
f=[1 2 3 4;2 6 7 3;4 3 7 8;1 5 8 4;1 2 6 5;5 6 7 8];
reall=15;
ll=4+reall;
g_v=[1 2 3;1 3 3;2 3 3;2 2 3;1 2 ll;1 3 ll;2 3 ll;2 2 ll];
g_f=[1 2 3 4;2 6 7 3;4 3 7 8;1 5 8 4;1 2 6 5;5 6 7 8];
delete(h1);delete(h2);delete(h3);
i=get(handles.slider_po,'Value')*10;
bbb=[0 i 0;0 i 0;0 i 0;0 i 0;0 i 0;0 i 0;0 i 0;0 i 0;];
v1=v+bbb;
af=get(handles.slider_ag,'Value');
gt1=0.5*cos(-af)-0.5;
gt2=0.5*sin(-af)-0;
gt3=0.5/cos(af)+(ll-0.5*tan(af))*sin(af)-cos(af)+0.5;
gt4=(ll-0.5*tan(af))*cos(af)+sin(af)-ll;
gt5=0.5/cos(af)+(ll-0.5*tan(af))*sin(af)-0.5;
gt6=(ll-0.5*tan(af))*cos(af)-ll;
jiao=[0 -gt1 -gt2;0 gt1 gt2;0 gt1 gt2;0 -gt1 -gt2;0 gt3 gt4;0 gt5 gt6;0 gt5 gt6;0 gt3 gt4];
g_v1=g_v+jiao+bbb;
h1=patch('Faces',f,'Vertices',v1,'FaceColor','b');
h2=patch('Faces',g_f,'Vertices',g_v1,'FaceColor','r');
hold on;
三、运行结果
四、matlab版本及参考文献
1 matlab版本 2014a
2 参考文献 [1] 包子阳,余继周,杨杉.智能优化算法及其MATLAB实例（第2版）[M].电子工业出版社，2016. [2]张岩,吴水根.MATLAB优化算法源代码[M].清华大学出版社，2017. [3]周品.MATLAB 神经网络设计与应用[M].清华大学出版社，2013. [4]陈明.MATLAB神经网络原理与实例精解[M].清华大学出版社，2013. [5]方清城.MATLAB R2016a神经网络设计与应用28个案例分析[M].清华大学出版社，2018.
