Settling time of sixth order denominator transfer function

Question

I have a system whose transfer function is as follows

$$ \frac{V_o(s)}{V_i(s)}=\frac {\text{numerator}(s)}{a_1 s^6 + a_2 s^5 + a_3 s^4 + a_4 s^3 + a_5 s^2 +a_6 s + a_7}. $$

I am interested in the settling time exhibited by the system, hence the numerator is not in focus (hope I am correct there). I do know that the sixth order polynomial has no analytical solution. Hence I used Matlab to find the numerical solution and settling time. I would like to know which coefficients of the denominator has the most influence on settling time. Any ideas on how to analyze that?

On the other hand, for a given set of values for coefficients, if I am able to figure it out the dependence of settling time on coefficients, for example \$ \text{Settling time} = f(\frac{a_1}{a_2})\$ and then if I change the values of \$a_1\$ and/or \$a_2\$, to lower the settling time then will that work? I reckon that the moment I change the coefficents then the settling time will now be dependent on other coefficients, such as, for instance \$\text{Settling time} = f(\frac{a_3}{a_4})\$, am I correct? The aim is to reduce settling time by choosing right values of two variables which appear only in the denominator of the transfer function

Answer 1

You cannot find the rule you hope. Assuming the system is stable a long settling time is caused by a pole or conjugate pole pair which has near zero negative real part. You may separate from the denominator factor \$s^2 + 2As + A^2 + B^2\$ if you have complex pole pair \$A+Bj\$ and \$A-Bj\$. Let A be radically closer to zero than the real parts of other poles. When this is multiplied with the other (4th order) part of the denominator, the critical real part \$A\$ affects all but the sixth order term.

ADD after more info was inserted to the question:

So, you are able to use advanced software like Matlab and you already have made simulation runs. Go on and find a good combination of the 2 variables which you can set. I guess Matlab can be programmed to make the search for you. Unfortunately I'm not a programmer nor have such advanced software. Sorry.

BTW. Often people must take into the account also other things than the settling time. The general method is to build a cost function which is tried to be minimized.

Answer 2

I'll add a few bits besides user287001's answer (who has the answer you should be selecting) and jonk's comments. The settling time is generally accepted to be the time it takes for the response to settle within 1% of the final value. As per the answer above, the closer the real parts of the roots are to the \$j\omega\$ axis, the longer the settling time, and the lower frequency these poles have, the more they influence the output.

Consider three sets of roots, each with three complex conjugate pairs: r1 has a pole closer to the origin by a factor of around 10, compared to the others, r2 has twice that value of the real part for the first pair, and r3 has the realpart of the 3rd pair close to the \$j\omega\$ axis. The three resulting polynomials will be their denominators. For convenience, the transfer functions are for an all-pole lowpass.

r1=[-1-i, -1+i, -10-5i, -10+5i, -20-30i, -20+30i];
r2=[-1-r1(1:2) r1(3:end)];
r3=[r1(1:4) 19+r1(5:6)];
p1=poly(r1),
p2=poly(r2),
p3=poly(r3),
p1 =

        1       62     2347    35570   228950   387000   325000

p2 =

        1       64     2470    40200   297625   805000   812500

p3 =

        1       24     1112    20446   151297   261790   225250

Their values are all quite similar, with the minor exceptions that p2 has 8.125e5 instead of 3.25e5, and p3 has lower \$s^5\$ term (and onward), but similar for \$s^0\$. If you were to look at any of these, without any other comparison, would you be able to tell which one had the fastest settling time? Looking at all three, you might guess that the \$s^0\$ term for all will influence the most their corner frequency and, while true, it wouldn't be quite so true numerically:

Despite the large difference between p1 and p2, the difference is less than 20%, and that's because the last term has, in fact, the power of 6. OTOH, that doesn't tell you anything about the damping factor(s) of the forming 2nd order sections, those would influence the other terms except \$s^6\$, and it's these ones that affect the settling time:

Out of the three, p2 influences the most the settling time because the magnitude of the pole is 1.58 times greater and still closer to the origin than the others, while p2 and p3, despite an almost twice as large difference in magnitude for the last pole, their responses are similar because they are also 20 times farther from the unit circle.

So, the conclusion to take is that there is no analytical solution, and even "informed guessing" can't help much. Also, what you said in the comment is not applicable, because if you can simply discard the greater term, what's stopping you from discarding the next largest, and then on and on? And a 5th order has no analytical roots, a 4th does, but have you seen the horrors? This is one root for the generic \$x^4+ax^3+bx^2+cx+d\$:

I dare you to find a numerically-friendly way to calculate this (i mean in terms of numerical accuracy). You, yourself, are talking about root finding, which are complex operations -- doesn't this make you ponder about the possibility of finding simplicity in such complex operations?

Still, if the settling time is not your concern but the step response is (the time for the response to reach the 50% value), then you could use free term and the larger the value, the faster the response. As for the rising time (10% to 90%), that's also dependent on the damping(s), so that's also off.

All in all, I'd say that your quest for finding that one magical formula to determine the settling time for higher order systems is not worth it, because higher order systems are inherently complex and, thus, impossible to describe in a few words. Except, maybe, "they're complex".

Answer 3

But firstly, you have to determine if the system is stable or not or else it will never settle. Use RH criteria to determine the system stability. This will give you a lot of constraints on the value of coefficients in denominator. That would be the first step of analyzing.