Analyzing effects of sixth order transfer function

Question

This question is in relation to this question asked before (Settling time of sixth order denominator transfer function), but formulated in greater details.

So I have a transfer function which looks as follows: $$ \frac{V_o(s)}{V_i(s)}=\frac {\text{numerator}(s)}{a_1 s^6 + a_2 s^5 + a_3 s^4 + a_4 s^3 + a_5 s^2 +a_6 s + a_7}. $$ Each coefficient contains number of variables, but there are only two variables i have the freedom to play with, the rest are fixed. For example: if \$ a6= C1\cdot Rx+C1\cdot R1+C2\cdot R2+C3\cdot Ry + C3\cdot (R1+R2)\$, then I have the freedom to play with \$ R1 \$ and \$R2 \$ and other variables are constant. The rest of the co-efficients i.e \$a1,a2,a3,a4,a5\$ are an algebraic addition and subtraction of \$ R1 \$ and \$R2 \$ and more variable i.e \$a1,a2...a6=f(R1,R2,Rx,Cx,Lx...) \$. Also \$ R1 \$ and \$R2 \$ do not appear in the numerator and hence numerator is ignored. Unfortunately it is not possible to post the circuit due to legal reasons.

The goal is to find how these two variables affect settling time.

Now I need to elaborate my question, which I found lacking in the previous post as pointed out by others. What I would like to know is which direction should i proceed from the present values of \$ R1 \$ and \$R2 \$ such that settling time is minimized. I know there is no analytical solution to the sixth order denominator, but I am not trying to find a solution, but a relation between these two variables and settling time. Numerical analysis will give me two values of \$ R1 \$ and \$R2 \$ for which minimum settling time can be obtained. But is it possible by numerical analysis to graph the approximate relation between \$ R1 \$, \$R2 \$ and settling time?

What is the purpose or use case to find the relation?
This is a complicated circuit, and based on the environment this circuit is employed in, the values of other variables, apart from \$ R1 \$ and \$R2 \$ can change slightly, and hence \$ R1 \$ and \$R2 \$ need to be tweaked in order to either minimize the settling time or to bring it back to original value, if the settling time has increased. I will not always have access to a computer, let alone numerical analysis tools. Also in controlled environment there is a need to understand the effects of the two variables on settling time.

From the post earlier, I have come to following inferences.

1. The denominator is of the sixth order hence there are six poles. If the denominator can be factorized, assuming there are complex conjugate poles \$A+Bj\$ and \$A-Bj\$ to seperate out \$s^2 + 2As + A^2 + B^2\$ from the denominator, then there are four more poles remaining, which when expanded lead to 4th order polynomial, i.e \$den=(s+A+Bj)\cdot (s+A-Bj) \cdot(s+p3) \cdot(s+p4) \cdot(s+p5) \cdot(s+p6)\$. Now if A is very close to zero then it wont affect \$ s+p6\$ when multiplied, hence the effect of sixth order term can be isolated. The problem however is the fact that the complex poles only exist for a specific set of values of \$ R1 \$ and \$R2 \$. The moment i change the values of \$ R1 \$ and \$R2 \$, the complex roots might not exist close to zero and hence this analysis wont be valid. Although a very nice idea, courtesy @user287001. It could be helpful to to others, and I hope i understood user287001's answer correctly.

2. Another way is to take the derivative of coefficients to settling time. This will tell me how much does the settling time change by change of one coefficient, keeping other coefficients unchanged. That is to take \$ \frac{d settling time}{d a1},\frac{d settling time}{d a2}...\$. individually. In this way one can find which co-efficient affects the most on settling time. For example if \$ a5 \$ affects the most then one can concentrate only on \$ a5 \$ leading to an approximate graphical solution which can be converted to an analytical solution by curve fitting. I can change the absolute values of each coefficient after substituting the numerical value of \$ R1 \$ and \$ R2 \$. If i find out that \$ a3 \$ affects the settling time the most, then changing values of \$ R1 \$ and \$ R2 \$ in \$ a3\$ will also change the other coefficients (\$a1,a2,a4,a5,a6\$, since they are a function of \$ R1 \$ and \$ R2 \$) which could make the system unstable. This is chicken and egg problem. Had \$ R1 \$ and \$ R2 \$ appeared in only three or less coefficients i think this solution could have been feasible.

3. At all times system stability needs to be guaranteed. Hence finding the bounds of each coefficient keeping other coefficients unchanged will atleast put some constrains on \$ R1 \$ and \$ R2 \$. But I think it will lead to the same problem as in the second point.

I have run out of ideas, but I am certain that this is not a new problem, and there must be people out there dealing with higher order systems to find approximate analysis. Like I mentioned before, I am not after the solution of the denominator polynomial, but to use numerical analysis to get to an approximate relation between \$ R1 \$ and \$ R2 \$ and settling time. Any thoughts comments will be more than welcome.

Answer 1

I'll reply given your information that you added in the comments:

it is a matching network with only resistors caps and inductors, hence it is difficult to isolate them into second order systems. No opamps in the circuit, hence no high impedance nodes

What you want is impossible without a numerical solver, or a symbolic one that can handle that polynomial. This is because a completely passive network cannot be isolated in sections due to the loading effect. Therefore the transfer function for such a circuit must be determined a priori and, once placed in the circuit, it should not be altered. Any modifications will affect the entire transfer function, because almost all the coefficients will be affected and, thus, all the roots will be displaced.

Simple 3rd order case:

The transfer function is seen in the Laplace source, above (ignore the opening '+' at the beginning of the lines). The two plots show the differences between the original transfer function and the one with only a modified Rx (0.4, instead of 0.3). The response is shown with the group delay, rather than the phase, and the impulse response, to show the time domain differences. And here are the roots, shown in full precision to show that even the single, real root is affected (a bit), despite the plot below:

-0.4312319661454532
 1.037404105974689i-0.28855068359394
-1.037404105974689i-0.28855068359394

-0.431294440041699
 1.02877420167959i-0.3176861133124839
-1.02877420167959i-0.3176861133124839

Small differences but you can see effects overall. Therefore the conclusion remains: an exact solution is not possible without numerical solvers, and changing any component will affect the entire tranfer function.

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If you can live with these differences then I have to wonder why bother asking so many questions about 6th order polynomials when you can simply "ogle" and change values as needed. This is why your whole question seems strange: you say you need to perform changes "on the field" and without helpers, but you are looking for deep analysis of 6th order polynomials; you say it's an impedance matching network, yet you use the control-system, analysis, stability tags. What sort of field intervention requires one changing a passive network for impedance adaptation, used in a control system with stability concerns, and without proper tools but which needs a polynomial evaluation in order to reduce the step response?