I'm designing a lead-lag controller to control a robotics-arm (part of an excercise).(adsbygoogle = window.adsbygoogle || []).push({});

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).

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

Can you offer guidance or do you also need help?

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