# Electronic – Settling time of sixth order denominator transfer function

damping-factorlaplace transformtransfer function

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

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.