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Automatic Control: Specifications for 3rd-order systems and higher ?

  1. Jun 28, 2008 #1
    I'm designing a lead-lag controller to control a robotics-arm (part of an excercise).
    I have found diagrams showing the correlation between overshoot and phasemargin for a second order system but in my excercise I have a third-order system. Can I apply the the relations for a 2nd-order system or can I somehow divide the problem into parts and apply the knowledge of 2nd and 1st order systems? I have the following system:
    1.9/(9*s^3+96.5s^2+40s)
    with the following specifications:
    Risetime < 1.72s
    Overshoot < 5%
    Control signal < 100
    Stationary error <5% when unit ramp is reference signal.

    I tried an approach with 2 lead-compensators in series to reduce N in
    Flead=N(s+b)/(s+bN)
    Flag=(s+a)/(s+a/M)

    If you want to see it, my matlab code follows (A Proportional controller was tried first and used as a reference). Appearantly, the expected speed-increase isn't there at all either is any of the other improvements.

    %%
    %Problem 1 %(FrCoef=b, GearFact=n in the instructions)
    J=4.5;Lm=2;Rm=21;FrCoef=1;Ktau=38;Km=0.5;GearFact=1/20;
    s=tf('s');
    G=GearFact*Ktau/(s*(s*Lm+Rm)*(J*s+FrCoef)+s*Km*Ktau);
    G

    %%
    %Problem 2, risetime, loop-gain
    Gc=feedback(G,1);
    %Identification gives:
    Q=1.9;P=9*s^3+96.5*s^2+40*s;
    rlocus(Q/P);
    %Kp=4.41 has relative damping of 1/sqrt(2) (Increasing Kp implies shorter rise-time). We end up at Kp=4.57.
    figure
    step(feedback(4.57*G,1),100);

    %%
    %Problem 3, Cross-over frequency, phasemargin och bandwidth.
    [aaa Phasemargin ccc CrossoverFrequency]=margin(4.57*G);
    Bandwidth=bandwidth(feedback(4.57*G,1),-3);

    Phasemargin %64.34°
    CrossoverFrequency %=0.1974
    Bandwidth %=0.3129

    %%
    %Problem 4
    bode(G)
    %We calculate the phase at frequency=0.7896 (4 times faster system gives cross-overfrequency at 0.7896). Phase(0.7896)= -156°.
    %5% Overshoot implies 64° phasemargin (from the infamous diagram for 2nd order systems) (från figur 5.16 sid 94 Glad &
    %Ljung). Required increase in phasemargin is 64°-(180°-156°-11.3° (We preemtively consider the cominig lag-compensator))=51.3°. Uppdelat
    %2 lead-compensators in series give 25.65°/compensator. Solving for N in
    %arctan(0.5(sqrt(N)-1/sqrt(N))
    %gives:
    y=25.65; %y=required phase-increase for simplicity
    N=1+2*tan(y)+sqrt((1+2*tan(y))^2-1);
    N
    %N=6.4812
    %b is calculated in accordance with b="desired cross-over frequency"/sqrt(N)
    b=0.7896/sqrt(N);
    b
    %We calculate K=1/(sqrt(N)*sqrt(N)*abs(G(i*"desired cross-over frequency"))
    K=1/(sqrt(N)*abs(evalfr(G,0.7896*i)));
    K
    a=0.1.*0.7896
    Flead=K*(N*((s+a)/(s+b*N)))^2;
    M=2;
    Flag=((s+a)/(s+(a/M)));
    figure
    bode(feedback(G*Flead*Flag,1));
    figure
    step((1/s)*feedback(G*Flead*Flag,1)) %We multiply the feedback function with 1/s so that matlab-function step gives the ramp-response (a ramp is an integrated step).
     
  2. jcsd
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