非線性大作業(yè)-二階機(jī)械臂

1、精選優(yōu)質(zhì)文檔-傾情為你奉上南京航空航天大學(xué)非線性系統(tǒng)控制大作業(yè)二階機(jī)械臂的非線性反饋設(shè)計(jì)姓 名： XXX學(xué) 號(hào)： XXX專(zhuān) 業(yè)： XXX2011年6月10日專(zhuān)心-專(zhuān)注-專(zhuān)業(yè)實(shí)驗(yàn)對(duì)象：二階機(jī)械臂的非線性反饋設(shè)計(jì)一 二階機(jī)械臂的動(dòng)力學(xué)控制模型圖1 模型在垂直平面內(nèi)的投影機(jī)械臂操作端的位置方程為（1）（2）上述方程的Jacobian矩陣為機(jī)械臂的動(dòng)力學(xué)方程為（3）其中，為總的驅(qū)動(dòng)力矩矢量，為轉(zhuǎn)動(dòng)角度矢量；為轉(zhuǎn)動(dòng)慣量矩陣，為Coriolis離心轉(zhuǎn)矩，為重力矢量。、的數(shù)學(xué)表達(dá)式如下：（4）（5）（6）其中定義狀態(tài)變量（7）設(shè)（8）則機(jī)械臂模型的狀態(tài)方程可以寫(xiě)成（9）系統(tǒng)輸出方程為（10）二 二階機(jī)械臂模

2、型線性化1）計(jì)算式（4）、（5）、（6）中各參數(shù)的值：，、。2）計(jì)算系統(tǒng)相對(duì)階：有上面的表達(dá)式可知，的后兩列全為0，的前兩行全為0（），因此同理，對(duì)輸出，有矩陣非奇異，故系統(tǒng)在某個(gè)的鄰域內(nèi)，系統(tǒng)的相對(duì)階為。3）選擇變換坐標(biāo)為（11）4）將系統(tǒng)轉(zhuǎn)換為標(biāo)準(zhǔn)型（12）輸出為（13）5）利用狀態(tài)反饋陣使系統(tǒng)成為一個(gè)完全可控的線性系統(tǒng)，令其中通過(guò)反饋和坐標(biāo)映射，系統(tǒng)可變?yōu)锽runovsky標(biāo)準(zhǔn)型即（14）記三 二階機(jī)械臂的最優(yōu)控制是Brunovsky標(biāo)準(zhǔn)型線性系統(tǒng)中的控制輸入，擬運(yùn)用LQR方法進(jìn)行求解，確定最優(yōu)控制律，使系統(tǒng)實(shí)際輸出盡可能逼近理想輸出。該問(wèn)題屬于無(wú)限時(shí)間、定常輸出跟蹤問(wèn)題。首先定義性能指

3、標(biāo)（15）式中，為、為的正定對(duì)稱(chēng)常數(shù)矩陣，為誤差輸出向量，是期望輸出。根據(jù)二次型最優(yōu)控制理論，可以知道當(dāng)性能指標(biāo)最小時(shí)，輸入應(yīng)為其中，為常值伴隨向量，為正定矩陣，滿足如下Riccati方程（16）最優(yōu)控制信號(hào)依賴(lài)于狀態(tài)變量和Riccati方程的解，在靜態(tài)狀態(tài)變量下假設(shè)時(shí)，可使系統(tǒng)的狀態(tài)趨近于0，是=0時(shí)的常值矩陣，在此種情況下，Riccati方程被簡(jiǎn)化為上式被稱(chēng)作Riccati Algebraic方程。取利用Matlab工具求解Riccati方程得則正定，滿足Riccati方程。利用Matlab進(jìn)行LQR求解，得而與求解Riccati方程所得結(jié)果一致。四 控制對(duì)象的MATLAB仿真在Simul

4、ink仿真環(huán)境下搭建完整控制系統(tǒng)，如Error! Reference source not found.所示：圖2 Simulink仿真框圖選擇期望輸入信號(hào)為輸入信號(hào)給定的理想輸出軌跡線設(shè)定為長(zhǎng)半軸為0.7，短半軸為0.3，焦點(diǎn)在軸上的橢圓形軌跡。利用上面搭建的Simulink仿真圖運(yùn)行程序，可得到給定軌跡曲線、跟蹤軌跡曲線、誤差軌跡曲線。圖3 軌跡跟蹤曲線圖4 跟蹤誤差曲線圖3中青色粗線為期望軌跡，紅色帶方塊虛線為機(jī)械臂實(shí)際運(yùn)動(dòng)軌跡；圖4中紅色實(shí)線為方向跟蹤誤差，藍(lán)色點(diǎn)劃線為方向跟蹤誤差。由圖3、圖4的軌跡跟蹤曲線和跟蹤誤差曲線可以看出，實(shí)際輸出很接近理想輸出，而且在響應(yīng)速度上也很快。跟蹤誤

5、差曲線的幅值很小（平穩(wěn)運(yùn)行階段），能夠滿足性能指標(biāo)要求。因此，應(yīng)用上述控制方法可以達(dá)到很好的控制效果。五 實(shí)驗(yàn)感想實(shí)際生活生產(chǎn)中，所面臨的系統(tǒng)極少會(huì)是純粹的線性系統(tǒng)，可以肯定地說(shuō)，任何投入實(shí)際生產(chǎn)運(yùn)行的系統(tǒng)都會(huì)或多或少地具有非線性因素。所謂的線性系統(tǒng)只是為了研究對(duì)象的控制問(wèn)題所作的合理簡(jiǎn)化。因此，對(duì)非線性系統(tǒng)理論的了解和學(xué)習(xí)，有助于培養(yǎng)正確的認(rèn)識(shí)觀，同時(shí)掌握一些實(shí)際有效的處理非線性問(wèn)題的方法。在本次大作業(yè)中，控制對(duì)象是具有很強(qiáng)非線性的二階機(jī)械臂。實(shí)驗(yàn)利用坐標(biāo)變換陣和非線性反饋陣，結(jié)合LQR最優(yōu)控制理論，實(shí)現(xiàn)了多輸入-多輸入系統(tǒng)的精確線性化，完成了對(duì)系統(tǒng)很好的控制效果，達(dá)到了最初的設(shè)定目標(biāo)。通過(guò)

6、本課程的學(xué)習(xí)，對(duì)非線性系統(tǒng)有了進(jìn)一步深入的認(rèn)識(shí)和理解，掌握了反饋線性化、自適應(yīng)觀測(cè)器設(shè)計(jì)等內(nèi)容，能夠?qū)?jiǎn)單的非線性系統(tǒng)進(jìn)行實(shí)際分析和控制律設(shè)計(jì)。這對(duì)我們以后課題的研究具有很大的促進(jìn)作用。另外，本課程為雙語(yǔ)教學(xué)，在學(xué)習(xí)過(guò)程中，通過(guò)上課和查閱了相關(guān)英文文獻(xiàn)，使自己的專(zhuān)業(yè)英語(yǔ)水平和專(zhuān)業(yè)知識(shí)都得到了提高。在課程教學(xué)中，姜老師除了講授課件知識(shí)，還為我們拓展了很多很實(shí)用的知識(shí)，比如對(duì)控制領(lǐng)域內(nèi)的重要學(xué)術(shù)會(huì)議、國(guó)內(nèi)外核心期刊和國(guó)內(nèi)外知名院士的介紹，使學(xué)生的學(xué)術(shù)科研和文獻(xiàn)查閱有了更強(qiáng)的目標(biāo)性，獲益良多?！痉抡娉绦蛞?jiàn)附件】附件/*alphax.m*/function a_out = a(u)x1=u(1); x

7、2=u(2); x3=u(3); x4=u(4);a_x1 = -125/*(74217*cos(x1)+*cos(x1)*cos(x2)-*cos(x1+x2)/(-sin(x1+x2)*cos(x1)+sin(x1)*cos(x1+x2)*(-54/125*cos(x1)-54/125*cos(x1+x2)*x3-54/125*cos(x1+x2)*x4)*x3+(-54/125*cos(x1+x2)*x3-54/125*cos(x1+x2)*x4)*x4+(-54/125*sin(x1)-54/125*sin(x1+x2)*(-/(-+*cos(x2)+*cos(x2)2)*(55663

8、/85248*sin(x2)*(x3*x4+1/2*x42)-77017/416*cos(x1)-88305/664*cos(x1+x2)+40992*(74217+*cos(x2)/(-+*cos(x2)+*cos(x2)2)*(-55663/70496*sin(x2)*x12-88305/664*cos(x1+x2)-54/125*sin(x1+x2)*(40992*(74217+*cos(x2)/(-+*cos(x2)+*cos(x2)2)*(55663/85248*sin(x2)*(x3*x4+1/2*x42)-77017/416*cos(x1)-88305/664*cos(x1+x2

9、)-*(54044+55663*cos(x2)/(-+*cos(x2)+*cos(x2)2)*(-55663/70496*sin(x2)*x12-88305/664*cos(x1+x2)-125/*(74217*sin(x1)+*sin(x1)*cos(x2)-*sin(x1+x2)/(-sin(x1+x2)*cos(x1)+sin(x1)*cos(x1+x2)*(-54/125*sin(x1)-54/125*sin(x1+x2)*x3-54/125*sin(x1+x2)*x4)*x3+(-54/125*sin(x1+x2)*x3-54/125*sin(x1+x2)*x4)*x4+(54/12

10、5*cos(x1)+54/125*cos(x1+x2)*(-/(-+*cos(x2)+*cos(x2)2)*(55663/85248*sin(x2)*(x3*x4+1/2*x42)-77017/416*cos(x1)-88305/664*cos(x1+x2)+40992*(74217+*cos(x2)/(-+*cos(x2)+*cos(x2)2)*(-55663/70496*sin(x2)*x12-88305/664*cos(x1+x2)+54/125*cos(x1+x2)*(40992*(74217+*cos(x2)/(-+*cos(x2)+*cos(x2)2)*(55663/85248*s

11、in(x2)*(x3*x4+1/2*x42)-77017/416*cos(x1)-88305/664*cos(x1+x2)-*(54044+55663*cos(x2)/(-+*cos(x2)+*cos(x2)2)*(-55663/70496*sin(x2)*x12-88305/664*cos(x1+x2);a_x2 = 125/*(-74217*cos(x1)+*cos(x1+x2)*cos(x2)/(-sin(x1+x2)*cos(x1)+sin(x1)*cos(x1+x2)*(-54/125*cos(x1)-54/125*cos(x1+x2)*x3-54/125*cos(x1+x2)*x4

12、)*x3+(-54/125*cos(x1+x2)*x3-54/125*cos(x1+x2)*x4)*x4+(-54/125*sin(x1)-54/125*sin(x1+x2)*(-/(-+*cos(x2)+*cos(x2)2)*(55663/85248*sin(x2)*(x3*x4+1/2*x42)-77017/416*cos(x1)-88305/664*cos(x1+x2)+40992*(74217+*cos(x2)/(-+*cos(x2)+*cos(x2)2)*(-55663/70496*sin(x2)*x12-88305/664*cos(x1+x2)-54/125*sin(x1+x2)*

13、(40992*(74217+*cos(x2)/(-+*cos(x2)+*cos(x2)2)*(55663/85248*sin(x2)*(x3*x4+1/2*x42)-77017/416*cos(x1)-88305/664*cos(x1+x2)-*(54044+55663*cos(x2)/(-+*cos(x2)+*cos(x2)2)*(-55663/70496*sin(x2)*x12-88305/664*cos(x1+x2)+125/*(-74217*sin(x1)+*sin(x1+x2)*cos(x2)/(-sin(x1+x2)*cos(x1)+sin(x1)*cos(x1+x2)*(-54/

14、125*sin(x1)-54/125*sin(x1+x2)*x3-54/125*sin(x1+x2)*x4)*x3+(-54/125*sin(x1+x2)*x3-54/125*sin(x1+x2)*x4)*x4+(54/125*cos(x1)+54/125*cos(x1+x2)*(-/(-+*cos(x2)+*cos(x2)2)*(55663/85248*sin(x2)*(x3*x4+1/2*x42)-77017/416*cos(x1)-88305/664*cos(x1+x2)+40992*(74217+*cos(x2)/(-+*cos(x2)+*cos(x2)2)*(-55663/70496

15、*sin(x2)*x12-88305/664*cos(x1+x2)+54/125*cos(x1+x2)*(40992*(74217+*cos(x2)/(-+*cos(x2)+*cos(x2)2)*(55663/85248*sin(x2)*(x3*x4+1/2*x42)-77017/416*cos(x1)-88305/664*cos(x1+x2)-*(54044+55663*cos(x2)/(-+*cos(x2)+*cos(x2)2)*(-55663/70496*sin(x2)*x12-88305/664*cos(x1+x2);a_out = a_x1;a_x2;/*betax.m*/funct

16、ion b_out = b(u)x1=u(1); x2=u(2);bx1=125/*(74217*cos(x1)+*cos(x1)*cos(x2)-*cos(x1+x2)/(-sin(x1+x2)*cos(x1)+sin(x1)*cos(x1+x2) ;bx2=125/*(74217*sin(x1)+*sin(x1)*cos(x2)-*sin(x1+x2)/(-sin(x1+x2)*cos(x1)+sin(x1)*cos(x1+x2);bx3=-125/*(-74217*cos(x1)+*cos(x1+x2)*cos(x2)/(-sin(x1+x2)*cos(x1)+sin(x1)*cos(x

17、1+x2);bx4=-125/*(-74217*sin(x1)+*sin(x1+x2)*cos(x2)/(-sin(x1+x2)*cos(x1)+sin(x1)*cos(x1+x2);b_out = bx1 ,bx2;bx3, bx4;/*hx.m*/function h_out = h(u)x1=u(1); x2=u(2);h_x = 54/125*cos(x1)+54/125*cos(x1+x2); 54/125*sin(x1)+54/125*sin(x1+x2);h_out = h_x;/*Gx.m*/function g_out = g(u)x2=u(2);g_x31= -/(-+*c

18、os(x2)+*cos(x2)2) ;g_x32=40992*(74217+*cos(x2)/(-+*cos(x2)+*cos(x2)2);g_x41=40992*(74217+*cos(x2)/(-+*cos(x2)+*cos(x2)2);g_x42= -*(54044+55663*cos(x2)/(-+*cos(x2)+*cos(x2)2);g_out = 0 0; 0 0; g_x31 g_x32; g_x41 g_x42 ;/*Fx.m*/function f_out = f(u)x1=u(1); x2=u(2); x3=u(3); x4=u(4);f_x = x3; x4; -/(-

19、+*cos(x2)+*cos(x2)2)*(55663/85248*sin(x2)*(x3*x4+1/2*x42)-77017/416*cos(x1)-88305/664*cos(x1+x2)+40992*(74217+*cos(x2)/(-+*cos(x2)+*cos(x2)2)*(-55663/70496*sin(x2)*x12-88305/664*cos(x1+x2); 40992*(74217+*cos(x2)/(-+*cos(x2)+*cos(x2)2)*(55663/85248*sin(x2)*(x3*x4+1/2*x42)-77017/416*cos(x1)-88305/664*

20、cos(x1+x2)-*(54044+55663*cos(x2)/(-+*cos(x2)+*cos(x2)2)*(-55663/70496*sin(x2)*x12-88305/664*cos(x1+x2);f_out = f_x;/*fai_out.m*/function fai_out = fai(u)x1=u(1); x2=u(2); x3=u(3); x4=u(4);fai_x = 54/125*cos(x1)+54/125*cos(x1+x2); (-54/125*sin(x1)-54/125*sin(x1+x2)*x3-54/125*sin(x1+x2)*x4; 54/125*sin

21、(x1)+54/125*sin(x1+x2); (54/125*cos(x1)+54/125*cos(x1+x2)*x3+54/125*cos(x1+x2)*x4 ;fai_out = fai_x;%系統(tǒng)線性化/*linearization.m*/M1=15.19; M2=11.36; L1=0.432; L2=0.432; g=9.81;syms x1 ; syms x2 ; syms x3 ; syms x4;a1=(1/3)*M1*L12+(1/3)*M2*L22+M2*L12;a2=M2*L1*L2;a3=(1/3)*M2*L22;a4=0.5*M1*g*L1+M2*g*L2;a5=0

22、.5*M2*g*L2;M_x= a1+a2*cos(x2) a3+a2*cos(x2); a3+a2*cos(x2) a3 ;N_x= -(a2*sin(x2)*(x3*x4+x42/2); (a2*sin(x2)*x12/2 ;G_x=a4*cos(x1)+a5*cos(x1+x2); a5*cos(x1+x2) ;D_x = M_x;E_x = -( N_x +G_x );D_xni = inv(D_x);f_x = x3; x4; D_xni*E_x;g_x = 0 0; 0 0; D_xni;h1_x = L1*cos(x1)+L2*cos(x1+x2);h2_x = L1*sin(x1)+L2*sin(x1+x2);h_x =h1_x; h2_x;Lgh1_x = diff(h1_x,x1) diff(h1_x,x2) d