Matlab create transfer function for given settling time and overshooting

Question

I want to find a second order transfer function with a non minimum phase zero z=36.6 which has 2% overshooting and a 2% settling time of 0.2s. I created a tunable transfer function but I don’t know how to find the values for the tunable parameters w and xi that allows the performances I want.

Is there an automatic way to find them ? I tried to sample the variables and then choose the best function, but I think that this is not the best way to do.

w = realp('w',25);
xi = realp('xi',0.8);
z = 36.6; 
G = tf(w^2*[-1/z 1],[1 2*xi*w w^2]);

if you know w and xi and the equation is second order, you can calculate it by the formoula of Ts and OS. — Pranav Totala, Feb 20 '18 at 10:33
The problem is that I don't know xi and w. They are the unknowns. — ardayigit, Mar 08 '18 at 13:45

Pranav Totala · Answer 1 · 2018-02-20T10:41:07.280

0

w = realp('w',25);
xi = realp('xi',0.8);
z = 36.6; 
G = tf(w^2*[-1/z 1],[1 2*xi*w w^2]);

step(G);
val = stepinfo(G);
ts=val.SettlingTime;
Os=val.Overshoot;

Hopefully this works... the step response was a bit weird though.

I am feeling this is a analogous second order system, so you can directly calculate from w and zi.

edited Feb 20 '18 at 10:41

answered Feb 19 '18 at 19:35

Pranav Totala

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I'm trying to find w and xi and I know ts and Os. – ardayigit Mar 08 '18 at 13:46

score 0 · Answer 2 · answered Nov 11 '19 at 10:09

The overshoot is given by sigma = exp(-pi*zeta/sqrt(1-zeta^2)), so, for a given overshoot sigma in [0,1], the damping factor becomes zeta = -log(sigma)/sqrt(pi^2+(log(sigma))^2).

The settling time is approximated as ts = 4/(zeta*wn).

An example:

ts = 0.5;
sigma = 0.2;

zeta = -log(sigma)/sqrt(pi^2+(log(sigma))^2);
wn = 4/zeta/ts;

K = 1;

H = tf(K*wn^2,[1,2*zeta*wn,wn^2])

val = stepinfo(H)
ts_meas = val.SettlingTime
Os = val.Overshoot

Matlab create transfer function for given settling time and overshooting

2 Answers2