控制系统计算及辅助设计实验.docx
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控制系统计算及辅助设计实验.docx
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控制系统计算及辅助设计实验
控制系统计算及辅助设计实验
1.
解:
函数:
functiondx=rossler1(t,x,a,b,c)
dx=[-x
(2)-x(3);x
(1)+a*x
(2);b+(x
(1)-c)*x(3)];
>>a=0.2;b=0.2;c=5.7;
>>[t,y]=ode45(@rossler1,[0,100],x0,[],a,b,c);
>>plot3(y(:
1),y(:
2),y(:
3)),grid
三维相轨迹:
>>view(0,90)
在x-y平面投影
2.
求解:
functiony=c3exmobj(x)
y=x
(1)^2-2*x
(1)+x
(2);
function[c,ce]=c3exmcon(x)
ce=[];c=[4*x
(1)^2+x
(2)^2-4]
>>A=[];B=[];Aeq=[];Beq=[];xm=[0;0];xM=[];x0=[5,5];
>>ff=optimset;ff.TolX=1e-10;ff.TolFun=1e-20;
>>while
(1)
[x,a,b]=fmincon(@c3exmobj,x,A,B,Aeq,Beq,xm,xM,@c3exmcon,ff);
ifb>0,break;end
i=i+1;
end
3.
求解:
(a)>>symss,G=(s^3+4*s+2)/(s^3*(s^2+2)*((s^2+1)^3+2*s+5))
结果:
G=
(s^3+4*s+2)/s^3/(s^2+2)/((s^2+1)^3+2*s+5)
(b)>>z=tf('z',0.1);
>>H=(z^2+0.568)/((z-1)*(z^2-0.2*z+0.99))
Transferfunction:
z^2+0.568
-----------------------------------
z^3-1.2z^2+1.19z-0.99
Samplingtime:
0.1
4.
解:
>>A=[0,1,0;0,0,1;-5,-4,-13];B=[0;0;2];C=[1,0,0];D=0;
>>G=ss(A,B,C,D)
a=
x1x2x3
x1010
x2001
x3-5-4-13
b=
u1
x10
x20
x32
c=
x1x2x3
y1100
d=
u1
y10
Continuous-timemodel.
>>G1=tf(G)%系统传递函数
Transferfunction:
2
----------------------
s^3+13s^2+4s+5
>>GG=zpk(G1)%零极点模型
Zero/pole/gain:
2
-----------------------------------
(s+12.72)(s^2+0.2836s+0.3932)
5.
6.
解:
functionH=feedback(G1,G2,key)
ifnargin==2;key=-1;end,H=G1/(sym
(1)-key*G1*G2);H=simple(H)
%使feedback函数能处理符号运算
>>G1=(s+1)/(j*s^2+2*s+5)
G1=
(s+1)/(j*s^2+2*s+5)
>>Gc=(kp*s+ki)/s
Gc=
(kp*s+ki)/s
>>G=feedback(G1*Gc,1)
G=
(s+1)*(kp*s+ki)/(j*s^3+2*s^2+5*s+kp*s^2+s*ki+kp*s+ki)
7.
解:
(a)>>s=tf('s');
>>G1=(211.87*s+317.64)/((s+20)*(s+94.34)*(s+0.1684))
>>Gc1=(169.6*s+400)/(s*(s+4));
>>H1=1/(0.01*s+1);
>>GG1=feedback(G1*Gc1,H1)
Transferfunction:
359.3s^3+3.732e004s^2+1.399e005s+127056
-----------------------------------------------------------------------------
0.01s^6+2.185s^5+142.1s^4+2444s^3+4.389e004s^2+1.399e005s
+127056
>>GG10=zpk(GG1)
Zero/pole/gain:
35933.152(s+100)(s+2.358)(s+1.499)
-----------------------------------------------------------------------
(s^2+3.667s+3.501)(s^2+11.73s+339.1)(s^2+203.1s+1.07e004)
(b)>>z=tf('z');
>>G2=(35786.7*z^-1+108444)/((z^-1+4)*(z^-1+20)*(z^-1+74.04));
>>Gc2=1/(z^-1-1);
>>H2=1/(0.5*z^-1-1);
>>GG2=feedback(G2*Gc2,H2)
Transferfunction:
-108444z^6+1.844e004z^5+1.789e004z^4
-------------------------------------------------------------------------
1.144e005z^6+2.876e004z^5+274.2z^4+782.4z^3+47.52z^2+0.5z
Samplingtime:
unspecified
>>G20=zpk(GG2)
Zero/pole/gain:
-0.94821z^4(z-0.5)(z+0.33)
------------------------------------------------------------
z(z+0.3035)(z+0.04438)(z+0.01355)(z^2-0.11z+0.02396)
Samplingtime:
unspecified
8.
解:
>>s=tf('s');
>>G1=1/(s+1);
>>G2=s/(s^2+2);G3=1/s^2;
>>H1=50;H2=(4*s+2)/(s+1)^2;H3=(s^2+2)/(s^3+14);
>>c1=feedback(G3,H1);
>>c2=feedback(G1*G2,H2);
>>c3=feedback(c1*c2,H3);
>>G3=3*c3
Transferfunction:
3s^6+6s^5+3s^4+42s^3+84s^2+42s
-----------------------------------------------------------------------------------------
s^10+3s^9+55s^8+175s^7+300s^6+1323s^5+2656s^4+3715s^3+7732s^2+5602s+1400
9.
解:
(1)连续:
>>s=tf('s');
>>G=((s+1)^2*(s^2+2*s+400))/((s+5)^2*(s^2+3*s+100)*(s^2+3*s+2500));
>>step(G,7)
(2)T=1
>>G1=c2d(G,1)
Transferfunction:
8.625e-005z^5-4.48e-005z^4+6.545e-006z^3+1.211e-005z^2-3.299e-006z+1.011e-007
------------------------------------------------------------------------------------
z^6-0.0419z^5-0.07092z^4-0.0004549z^3+0.002495z^2-3.347e-005z+1.125e-007
Samplingtime:
1
>>step(G1,10)
(3)T=0.1
>>G2=c2d(G,0.1)
Transferfunction:
0.0003982z^5-0.0003919z^4-0.000336z^3+0.0007842z^2-0.000766z+0.0003214
-----------------------------------------------------------------------------------
z^6-2.644z^5+4.044z^4-3.94z^3+2.549z^2-1.056z+0.2019
Samplingtime:
0.1
>>step(G2,7)
(4)T=0.01
>>G3=c2d(G,0.01)
Transferfunction:
4.716e-005z^5-0.0001396z^4+9.596e-005z^3+8.18e-005z^2-0.0001289z+4.355e-005
------------------------------------------------------------------------------------
z^6-5.592z^5+13.26z^4-17.06z^3+12.58z^2-5.032z+0.8521
Samplingtime:
0.01
>>step(G3,7)
10.
(1)>>s=tf('s');
>>G1=1/(s^3+2*s^2+s+2)
Transferfunction:
1
-------------------
s^3+2s^2+s+2
>>pzmap(G1)
>>eig(G1),isstable(G1)
ans=
-2.0000
-0.0000+1.0000i
-0.0000-1.0000i
ans=
1
系统临界稳定。
(2)>>G2=1/(6*s^4+3*s^3+2*s^2+1);
>>eig(G2),isstable(G2)
ans=
-0.5099+0.5585i
-0.5099-0.5585i
0.2599+0.4732i
0.2599-0.4732i
ans=
0
系统不稳定。
(3)>>G3=1/(s^4+s^3-3*s^2-s+2);
>>eig(G3),isstable(G3)
ans=
-2.0000
-1.0000
1.0000
1.0000
ans=
0
系统不稳定。
11.
解:
(1)
>>num=[-32];den=[10.2-0.250.05];
>>H1=tf(num,den,'Ts',0.1)
Transferfunction:
-3z+2
-----------------------------
z^3+0.2z^2-0.25z+0.05
Samplingtime:
0.1
>>v=abs(eig(H1)),isstable(H1)
v=
0.6777
0.2716
0.2716
ans=
1
系统是稳定的。
(2)
>>num=[3-0.39-0.09];den=[11.71.040.2680.024];
>>H2=tf(num,den,'Ts',0.1)
Transferfunction:
3z^2-0.39z-0.09
------------------------------------------
z^4+1.7z^3+1.04z^2+0.268z+0.024
Samplingtime:
0.1
>>v=abs(eig(H2)),isstable(H2)
v=
0.6000
0.5000
0.4000
0.2000
ans=
1
系统是稳定的。
12.
解:
>>A=[-0.2,0.5,0,0,0;0,-0.5,1.6,0,0;0,0,-14.3,85.8,0;0,0,0,-33.3,100;0,0,0,0,-10];
>>B=[0;0;0;0;30];
>>C=zeros(5,1);
>>D=0;
>>G=ss(A,B,C,D);
>>eig(G),pzmap(G),isstable(G)
ans=
-0.2000
-0.5000
-14.3000
-33.3000
-10.0000
ans=
1
系统是稳定的。
13.
解:
(1)数值解:
>>f=@(t,x)[-5*x
(1)+2*x
(2);-4*x
(2);-3*x
(1)+2*x
(2)-4*x(3)-x(4);-3*x
(1)+2*x
(2)-4*x(4)];
>>t_final=10;
>>x0=[1200];
>>[t,x]=ode45(f,[0,t_final],x0);
>>plot(t,x)
(2)解析解:
function[Ga,Xa]=ss_augment(G,cc,dd,X)
G=ss(G);Aa=G.a;Ca=G.c;Xa=X;Ba=G.b;D=G.d;
if(length(dd)>0&sum(abs(dd)>1e-5)),
if(abs(dd(4))>1e-5),
Aa=[Aadd
(2)*Ba,dd(3)*Ba;...
zeros(2,length(Aa)),[dd
(1),-dd(4);dd(4),dd
(1)]];
Ca=[Cadd
(2)*Ddd(3)*D];Xa=[Xa;1;0];Ba=[Ba;0;0];
else,
Aa=[Aadd
(2)*B;zeros(1,length(Aa))dd
(1)];
Ca=[Cadd
(2)*D];Xa=[Xa;1];Ba=[B;0];
end
end
if(length(cc)>0&sum(abs(cc))>1e-5),M=length(cc);
Aa=[AaBazeros(length(Aa),M-1);zeros(M-1,length(Aa)+1)...
eye(M-1);zeros(1,length(Aa)+M)];
Ca=[CaDzeros(1,M-1)];Xa=[Xa;cc
(1)];ii=1;
fori=2:
M,ii=ii*i;Xa(length(Aa)+i)=cc(i)*ii;
end,end
Ga=ss(Aa,zeros(size(Ca')),Ca,D);
>>c=0;d=[0,0,0,0];
>>x0=[1;2;0;1];
>>A=[-5,2,0,0;0,-4,0,0;-3,2,-4,-1;-3,2,0,-4];B=zeros(4,1);C=zeros(1,4);D=0;
>>G=ss(A,B,C,D);
>>Ga.a,xx0'
>>symst;y=Ga.c*expm(Ga.a*t)*xx0
y=
0
14.
解:
(a)>>s=tf('s');
>>G1=((s+6)*(s-6))/(s*(s+3)*(s+4+4i)*(s+4-4i))
Transferfunction:
s^2-36
----------------------------
s^4+11s^3+56s^2+96s
>>rlocus(G1)
增益K>0。
(b)>>num=[1,2,2];den=[1,1,14,8,0];
>>G2=tf(num,den)
Transferfunction:
s^2+2s+2
------------------------
s^4+s^3+14s^2+8s
>>rlocus(G2)
根据图像知,增益K>0。
15.
>>s=tf('s');
>>G=(s-1)/(s+1)^5;G.ioDelay=2;
>>rlocus(pade(G,2))
所以K值范围是0-1.15。
16.
解:
(a)
>>s=tf('s');
>>G=(8*(s+1))/(s^2*(s+15)*(s^2+6*s+10));
>>bode(G)
>>nyquist(G),grid
>>set(gca,'Xlim',[-61])
>>nichols(G),grid
>>[gm,pm,wg,wp]=margin(G)
gm=%幅值裕量
30.4686
pm=
4.2340
wg=%相位裕量
1.5811
wp=
0.2336
>>G1=feedback(G,1)
Transferfunction:
8s+8
------------------------------------------
s^5+21s^4+100s^3+150s^2+8s+8
>>pzmap(G1)
所以系统是稳定的。
>>step(G1,1000)
(b)
>>z=tf('z','Ts',0.1);
>>H=0.45*((z+1.31)*(z+0.054)*(z-0.957))/(z*(z-1)*(z-0.368)*(z-0.99))
Transferfunction:
0.45z^3+0.1832z^2-0.5556z-0.03046
------------------------------------------
z^4-2.358z^3+1.722z^2-0.3643z
Samplingtime:
0.1
>>bode(H)
>>nyquist(H),grid
>>nichols(H),grid
>>H1=feedback(H,1)
Transferfunction:
0.45z^3+0.1832z^2-0.5556z-0.03046
------------------------------------------------
z^4-1.908z^3+1.905z^2-0.9199z-0.03046
Samplingtime:
0.1
>>pzmap(H1)
系统是不稳定的。
>>step(H1,1000)
17.
>>z=[-2.5];p=[0;-0.5;-50];
>>G=zpk(z,p,100/2.5*0.5*50);
>>z=[-1;-2.5];p=[-0.5;-50];
>>Gc=zpk(z,p,1000);
>>G1=feedback(G*Gc,1)
Zero/pole/gain:
1000000(s+1)(s+2.5)^2
------------------------------------------------------
(s+1)(s^2+4.99s+6.239)(s^2+95.01s+1.002e006)
>>bode(G1)
系统是稳定的。
>>step(G1)
第二部分:
解:
y的仿真结果曲线:
3.
4.
解:
5.
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