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22 Problem A9.10 OGATA 4ed(PID compensator)



    Let's do the programming by Scilab of compensator that the system verifies the following values: static velocity constant error $K_{v}=4$, phase margin 50 and gain margin $\geq 10dB$. The open-loop system is:

    \begin{displaymath}G(s)=\frac{1}{(s^{2}+1)}\end{displaymath}


    Let's draw the bode plot by Scilab of:

    \begin{displaymath}G_{1}(s)=\frac{4}{(s^{2}+1)}\end{displaymath}


    Scilab program:
    s=%s/(2*%pi);
    g=4/(s*(s^2+1));
    gs=syslin('c',g);
    clf();
    bode(gs,0.1,10);
    
    Image ProblemaA9_10
    Let's draw the bode plot by Scilab of:

    \begin{displaymath}G_{1}(s)\cdot G_{ad}=\frac{4}{(s^{2}+1)}\cdot (5\cdot s+1)\end{displaymath}


    Scilab program:
    s=%s/(2*%pi);
    g=4/(s*(s^2+1));
    gcad=(5*s+1);
    gs=syslin('c',g);
    gs2=syslin('c',g*gcad);
    clf();
    bode(gs2,'compensated');
    
    Image ProblemaA9_10a
    Let's draw the bode plot by Scilab of:

    \begin{displaymath}G_{1}(s)\cdot G_{ad}=\frac{4}{(s^{2}+1)}\cdot (5\cdot s+1)\cdot (0.25\cdot s+1)\end{displaymath}


    Scilab program:
    s=%s/(2*%pi);
    g=4/(s*(s^2+1));
    gcad=(5*s+1);
    gcad2=(0.25*s+1);
    gs=syslin('c',g);
    gs2=syslin('c',g*gcad*gcad2);
    clf();
    bode(gs2,0.01,100,'compensated');
    
    Image ProblemaA9_10b
    Let's draw the unit-step response by Scilab.



    Scilab program:
    s=%s;
    g=4/(s*(s^2+1));
    gcad=(5*s+1);
    gcad2=(0.25*s+1);
    gt1=g*gcad*gcad2;
    glc=g /. 1;
    glc2=gt1 /. 1;
    gs=syslin('c',glc);
    gs2=syslin('c',glc2);
    t=0:0.1:14;
    y=csim('step',t,gs);
    y2=csim('step',t,gs2);
    clf();
    subplot(2,1,1);
    plot(t,y,'g');
    legends('no compensated',3,opt=1)
    xgrid;
    xtitle('Unit-step response','','y(t)')
    subplot(2,1,2)
    plot(t,y2,'b');
    legends('compensated',2,opt=4);
    xtitle('','t(seg)','y(t)')
    xgrid;
    
    Image ProblemaA9_10c
    Let's draw the ramp response by Scilab.



    Scilab program:
    s=%s;
    g=4/(s*(s^2+1));
    gcad=(5*s+1);
    gcad2=(0.25*s+1);
    gt1=g*gcad*gcad2;
    glc=g /. 1;
    glc2=gt1 /. 1;
    gs=syslin('c',glc);
    gs2=syslin('c',glc2);
    t=0:0.1:20;
    y=csim(t,t,gs);
    y2=csim(t,t,gs2);
    clf();
    subplot(2,1,1);
    plot(t,y,'g');
    legends('no compensated',3,opt=1)
    xgrid;
    xtitle('Ramp response','','y(t)')
    subplot(2,1,2)
    plot(t,y2,'b');
    plot(t,t,'r');
    legends(['compensated';'ramp'],[2;5],opt=4);
    xtitle('','t(seg)','y(t)')
    xgrid;
    
    Image ProblemaA9_10d

cajael