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1、关于作业,sys=tf(24,18,3,1 2 10 0 0)sys1=feedback(sys,1);key=0;t=0:0.1:50stre(key,sys1,t)stre(1,sys1,0:0.1:50)stre(2,sys1,0:0.1:50),stre.m,Design constraint,第7章 频域分析,参见112页5.4节和262页8.3节,内容,绘制Nyquist图绘制Bode图绘制Nichols图分析频率特性性能，如稳定性、稳定裕度计算频域性能指标应用频域法设计校正装置,7.1 控制系统的频率特性,设s=j,系统的频域响应可由下式直接求出：,建模已知传递函数模型num=0
2、.5;den=1 2 1 0.5;w=0.01:0.01:1000;Gwnum=polyval(num,i*w);Gwden=polyval(den,i*w);Gw=Gwnum./Gwden;如果是状态空间模型则可以Gw=C*inv(i*w*eye(n)-A)*B+D;,计算系统的频率特性的方法,计算系统的频率特性的方法,real(g)计算实频响应;imag(g)计算虚频响应a1=real(Gw);b1=imag(Gw);plot(a1,b1)abs(g)计算幅频响应;angle(g)计算相频响应h1=abs(Gw)f1=angle(Gw);subplot(2,1,1),semilogx(w,
3、20*log(h1)subplot(2,1,2),plot(w,f1)注意频率w=0.01:0.01:1000;w=logspace(-2,-1,1000);,7.2 绘制Nyquist图,nyquist(),re,im,w=nyquist(num,den,w),re=实部向量im=虚部向量,G(s)=num/den,用户选择的频率（可选）,nyquist(sys)nyquist(sys,wmin,wmax)nyquist(sys,w)nyquist(sys1,sys2,.,w)nyquist(sys1,r,sys2,y-,sys3,gx).re,im=nyquist(sys,w)re,im,
4、w=nyquist(sys),其他调用格式有,若想使得闭环系统稳定，则开环系统G(s)H(s)的nyquist图逆时针地绕(-1,j0)点的圈数必须等于G(s)H(s)位于s右半平面开环极点数。,nyquist分析闭环系统的稳定性：,Ex1:,num=0.5;den=1 2 1 0.5；roots(den)ans=-1.5652-0.2174+0.5217i-0.2174-0.5217ire,im=nyquist(num,den);plot(re,im),grid,(-1,j0)点,num=0.5 5;d1=0.5 1 0;d2=1/2500 0.6/50 1;den=conv(d1,d2);
5、nyquist(num,den),axis(-0.1,0,-0.15,0.15),7.3 绘制Bode图,mag,phase,w=bode(num,den,w),mag=幅值向量phase=相角向量,G(s)=num/den,用户选择的频率（可选）,bode(num,den)bode(num,den,w)mag,phase=bode(num,den),注意要得到幅频需用分贝表示dB MAGDB=20*log10(mag),其他调用格式有,bode(sys)bode(sys,wmin,wmax)bode(sys,w)bode(sys1,sys2,.,w)bode(sys1,r,sys2,y-,s
6、ys3,gx).mag,phase=bode(sys,w)mag,phase,w=bode(sys),logspace():指定频率点w,w=logspace(a,b,n),w=用对数尺度表示的频率点数据向量,介于10a与10b之间的n点,Ex 2：,num=0.5 5;d1=0.5 1 0;d2=1/2500 0.012 1;den=conv(d1,d2);w=logspace(-1,3,200);%指定频率范围mag,phase,w=bode(num,den,w)semilogx(w,20*log10(mag),gridxlabel(Frequencyrad/sec),ylabel(20l
7、og(mag)dB),实际幅频特性与近似幅频特性比较,相频特性,semilogx(w,phase),gridxlabel(frequencyrad/sec);ylabel(phase);,完整Bode图,bode(num,den),grid,7.4 求增益裕度和相角裕度,margin(),Gm,Pm,Wcg,Wcp=margin(num,den),Gm=增益裕度（dB）Pm=相角裕度（deg）,G(s)=num/den,Gm,Pm,Wcg,Wcp=margin(sys)Gm_dB=20*log10(Gm)Gm,Pm,Wcg,Wcp=margin(mag,phase,w)margin(num,d
8、en)直接在Bode图上标明margin(sys),Wcg=相角-180对应频率Wcp=增益0dB对应频率,bode(num,den)margin(num,den),Gm=26.9405Pm=48.5831Wcg=47.5445Wcp=2.939,bode(num,den)Gm,Pm,Wcg,Wcp=margin(num,den),title(Gm=,num2str(Gm),Pm=,num2str(Pm),7.5 绘制Nichols图,mag,phase,w=nichols(num,den,w),mag=幅值向量phase=相角向量,G(s)=num/den,用户选择的频率（可选）,nicho
9、ls(num,den)nichols(num,den,w)mag,phase=nichols(num,den),单位负反馈系统的幅相频率特性,其他调用格式有,nichols(sys)nichols(sys,wmin,wmax)nichols(sys,w)nichols(sys1,sys2,.,w)nichols(sys1,r,sys2,y-,sys3,gx).mag,phase=nichols(sys,w)mag,phase,w=nichols(sys),Nichols图框架下的等M()和等()线,ngrid在已有的nichols图中绘制等M()和等()线。ngrid(new)产生nichol
10、s图框架下的等M()和等()线，以便绘制nichols图，命令自动保持图形，相当于hold on。,num=0.0001 0.0218 1.0436 9.3599;den=0.0006 0.0268 0.6365 6.2711;sys=tf(num,den);ngrid(new);nichols(sys),7.6 BODE串联校正设计工具,1.超前校正设计 校正原理：超前校正的两个转折频率应分设在未校正系统的剪切频率的两侧。相频特性具有正相移，幅频特性具有正斜率。校正后，低频段不变，剪切频率比原系统大，说明快速性提高。,设计方法,ess K 画出校正前原系统的 Bode 计算出稳定裕度及剪切频
11、率PMk c由m=PMd-PMk+5,求 值.L()=10lg dB.确定校正后的系统的剪切频率m.根据m计算校正器的零极点的转折频率.,6.画出校正后的系统Bode图,校验系统性能指标7.使系统构成闭环,验证闭环系统的响应,已知单位负反馈系统被控对象的传函,设计校正器使之满足:,斜坡信号作用下,稳态误差小于0.001;相角裕度在4348 之间,figure(2);sysc=feedback(sys,1);step(sysc),Gm0=1.0100Pm0=0.0584wcg0=100.0000wcp0=99.4863,k0=1000;n1=1;d1=conv(conv(1 0,0.1 1),0
12、.001 1);figure(1);sys=tf(k0*n1,d1);bode(sys)Gm0,Pm0,wcg0,wcp0=margin(k0*n1,d1)hold on,期望的稳定裕度d=45,gamad=45;phai=gamad-Pm0+5;pha=phai*pi/180;alpha=(1+sin(pha)/(1-sin(pha);mag,phase,w=bode(sys);adb=20*log10(mag);am=-10*log10(alpha);wgc=spline(adb,w,am);%插值T=1/(wgc*sqrt(alpha);alphat=alpha*T;Gc=tf(alph
13、at 1,T 1),Transfer function:0.01669 s+1-0.002218 s+1,根据校正后系统的结构与参数,校验其是否满足要求,sopen=sys*Gc;figure(1);bode(sopen)margin(sopen),sclose=feedback(sopen,1);figure(2);step(sclose),2.滞后校正设计 校正原理：滞后校正的两个转折频率应比未校正系统的剪切频率小很多。相频特性具有负相移，幅频特性具有负斜率。校正后，低频段不变，剪切频率比原系统小，说明快速性变差，意味着牺牲系统的快速性换取稳定性。,设计方法,ess K 画出校正前原系统的
14、 Bode 由(gc)=-180+PMd+5,确定校正后的系统的剪切频率 gcL(gc)=20lg,get=10(L/20).根据gc计算校正器的零极点的转折频率.,5.画出校正后的系统Bode图,校验系统性能指标6.使系统构成闭环,验证闭环系统的响应,已知单位负反馈系统被控对象的传函,设计校正器使之满足:,斜坡信号作用下,Kv30s-1;相角裕度45;(剪切频率c 2.3s-1),k0=30;n1=1;d1=conv(conv(1 0,0.1 1),0.2 1);figure(1);sys=tf(k0*n1,d1);mag,phase,w=bode(sys)figure(1);margin(
15、k0*n1,d1)hold onfigure(2);sysc=feedback(sys,1);step(sysc),gama=45pha=gama+5-180;wgc=spline(pu,w,pha),gama=45wgc=2.4588,na=polyval(k0*n1,j*wgc);da=polyval(d1,j*wgc);g=na/dag1=abs(g);h=20*log10(g1);beta=10(h/20);T=10/wgc;betat=beta*T;Gc=tf(T 1,betat 1),Transfer function:4.183 s+1-46.06 s+1,sopen=sys*G
16、c;figure(1);bode(sopen)margin(sopen)sclose=feedback(sopen,1);figure(2);step(sclose),3.滞后-超前校正设计 校正原理：实现滞后校正与超前校正的综合。滞后校正把剪切频率左移。从而减小了系统在剪切频率处的相位滞后，超前校正的作用是新剪切频率提供一个相位超前量，用以增大一个相位超前量，用以增大系统相位稳定裕度，使其满足动态性能要求,设计方法,ess K 画出校正前原系统的 Bode，获取系统的剪切频率 g1确定滞后校正的参数，取=810，1/T1=0.1 g1确定校正后的系统的剪切频率g2，使这一点超前校正器能满足，
17、并在该点综合后幅频衰减为0db.L()=10lg dB.根据g2计算校正器的零极点的转折频率.,6.画出校正后的系统Bode图,校验系统性能指标7.使系统构成闭环,验证闭环系统的响应,已知单位负反馈系统被控对象的传函,设计校正器使之满足:,斜坡信号作用下,Kv=10s-1;相角裕度45;剪切频率c 1.5s-1时域性能指标:%25%,Tp 2s,Ts 6s,k0=20;n1=1;d1=conv(conv(1 0,1 1),1 2);figure(1);sys=tf(k0*n1,d1);mag,phase,w=bode(sys);figure(1);margin(k0*n1,d1)hold on
18、figure(2);sysc=feedback(sys,1);step(sysc),wg1=1.5;beta=9.5;T1=1/(0.1*wg1);betat=beta*T1;Gc1=tf(T1 1,betat 1),求滞后校正器，取 g1=1.5s-1,=9.5,1/T1=0.1 g1,Transfer function:6.667 s+1-63.33 s+1,Transfer function:133.3 s+20-63.33 s4+191 s3+129.7 s2+2 s,求超前校正器，已知 g2,na=polyval(num,j*wg2);da=polyval(den,j*wg2);g=
19、na/da;g1=abs(g);h=20*log10(g1);alpha=10(-h/10);T2=1/(wg2*(alpha)(1/2);alphat=alpha*T2;Gc2=tf(alphat 1,T2 1),Transfer function:2.13 s+1-0.2086 s+1,sopen=sys*Gc1*Gc2;figure(1);bode(sopen)margin(sopen)sclose=feedback(sopen,1);figure(2);step(sclose),校正的解析法,频域作业,编写串联校正的函数。输入参数：开环传函，校正器的种类，输出参数：校正器传函,频域作业
20、,已知单位负反馈系统被控对象的传函,设计校正器使之满足:,斜坡信号作用下,Kv=10s-1;相角裕度45;剪切频率c 1.5s-1时域性能指标:%25%,Tp 2s,Ts 6s,附:系统串联校正的程序设计,Leadcmpst.m:超前校正,function modelcompensator,Wcnew,Wcold,alpha=leadcmpst(sysopen,Pmd)%To get a model of a lead compensator%sysopen-a open system%Pmd-desired phase margin%modelcompensator-Model of the
21、 new system%Wcnew-Crossover frequency of the new system%Wcold-Crossover frequency of the open system%alpha-coefficient of the compensator,Gmo,Pmo,Wcgo,Wcpo=margin(sysopen);phacmp=Pmd-Pmo+5;phc=phacmp*pi/180;alpha=(1-sin(phc)/(1+sin(phc);Gaincmp=10*log10(alpha);mag,phase,W=bode(sysopen);l,n,c=size(ma
22、g);mag1=zeros(c,1);for i=1:c mag1(i)=20*log10(mag(1,1,i);endWcold=interp1(mag1,W,0,spline);%插值Wcnew=interp1(mag1,W,Gaincmp,spline);%插值Zc=Wcnew*sqrt(alpha);Pc=Zc/alpha;modelcompensator=zpk(-Zc,-Pc,1/alpha);,Pmd=50;%degreeKv=20;disp(Model of the original system:);sys=zpk(,0-2,4)disp(open loop system m
23、odel which meets steady target:)sysopen,kc=modtraget(sys,Kv);sysopendisp(Model of the compensator:);Cmp,Wcnew,Wcold,alpha=leadcmpst(sysopen,Pmd);Cmpdisp(Model of new open-loop system:);sysnew=Cmp*sysopen,Amp407.m,disp(Margin of the open loop system:);Gmold,Pmold,Wcgold,Wcpold=margin(sysopen);GmoldPm
24、oldWcgoldWcpolddisp(Margin of the new system:);Gmnew,Pmnew,Wcgnew,Wcpnew=margin(sysnew);GmnewPmnewWcgnewWcpnew,subplot(1,2,1);bode(sysnew,r,sysopen,b),grid;subplot(1,2,2);sysclose1=feedback(sysnew,1);sysclose2=feedback(sysopen,1);step(sysclose1,r-,sysclose2,b:),modtraget.m 子程序，求满足稳态性能要求的开环传函,functio
25、n sysopen,kc=modtraget(sysold,kd)%This function is to get a system model which meets%steady traget kd%systemold-the original system%kd-the gived steady performence traget%sysopen-a system model which meets steady traget kd%kc-compensative gain,sys=zpk(sysold);z,p,k=zpkdata(sys,v);ind=find(p=0);p1=ze
26、ros(1,length(ind)for i=1:length(ind)p1(i)=p(ind(i);endsys1=zpk(z,p1,k);num,den=tfdata(sys1,v);k1=polyval(num,0)/polyval(den,0);kc=kd/k1;sysopen=kc*sys;,Lagcmpst.m:滞后校正,lagcmpst.mfunction modelcompensator,Wcnew,Wcold,alpha=lagcmpst(sysopen,Pmd)%To get a model of a lag compensator%sysopen-a open syste
27、m%Pmd-desired phase margin%modelcompensator-Model of the new system%Wcnew-Crossover frequency of the new system%Wcold-Crossover frequency of the open system%alpha-coefficient of the compensator,mag,phase,W=bode(sysopen);l,n,c=size(mag);mag1=zeros(c,1);for i=1:c mag1(i)=20*log10(mag(1,1,i);endWcold=i
28、nterp1(mag1,W,0,spline);l,n,c=size(phase);pha1=zeros(c,1);for i=1:c pha1(i)=phase(1,1,i);endphawc=-180+Pmd+15;Wcnew=interp1(pha1,W,phawc,spline);Magwcn=interp1(W,mag1,Wcnew,spline);beta=10(Magwcn/20);Zc=Wcnew/5;Pc=Zc/beta;modelcompensator=zpk(-Zc,-Pc,1/beta);,Pmd=40;%degreeKv=5;disp(Model of the ori
29、ginal system:);sys=zpk(,0-1-2,2)disp(open loop system model which meets steady traget:)sysopen,kc=modtraget(sys,Kv);sysopendisp(Model of the compensator:);Cmp,Wcnew,Wcold,alpha=lagcmpst(sysopen,Pmd);Cmpdisp(Model of new open-loop system:);sysnew=Cmp*sysopen,disp(Margin of the open loop system:);Gmol
30、d,Pmold,Wcgold,Wcpold=margin(sysopen);GmoldPmoldWcgoldWcpolddisp(Margin of the new system:);Gmnew,Pmnew,Wcgnew,Wcpnew=margin(sysnew);GmnewPmnewWcgnewWcpnew,subplot(1,2,1);bode(sysnew,r,sysopen,b),grid;subplot(1,2,2);sysclose1=feedback(sysnew,1);sysclose2=feedback(sysopen,1);step(sysclose1,r-,sysclose2,b:),
