This probably is a bandaid of what you really want to accomplish.
What he likely needs is a transfer function S to t domain differential equation driven control system with a washout depending on S domain to adapt to situation at hand. I'm in a process dynamics and controls lecture and then I find yaw damper code for 747 by Boeing published by Matlab as tutorial and I see it gets even "shittier" than my simplistic textbook models without answers in back of book. Basically you have an input into a system, you have outputs say voltage versus time or pitch angle versus time and these oscillate, you have another plot trying to converge on it in real time. They have open loop and closed loop systems. You can ultimately fine tune servo movements by velocity or time by a given oscillating system in real time and have outputs progressively decrease.
The problem is with Boeing they give you inputs in matrices for a 747 aileron bank angle etc and a bunch of proprietary numbers. The numbers they found somehow Anyone's guess knows how for that object. The rocket wouldn't behave like the 747. It will have a lot more rapid inputs with progressively less pitch angles on fins as the rocket velocity increases. It's just a complex thing to model where y=mx+b linear algebra won't solve it. You can size capacitors and inductors in labs to a resistor for a desired voltage response through differential equations or guessing if you hate diff. And a secondary response follows a similar amplitude and frequency but less amplitude until it's neutral again and stabilizes on an oscilloscope. But I'm just a dumb undergrad mechanical student trying to graduate. Only time I used complex numbers were phasers in circuits long ago. This goes from real to imaginary or vice versa and it's infuriating with differentials. I didn't take a mechanical vibrations course but I'm certain some of the frequency aspects could get pulled off the airframe somehow. That other guy Dave was mentioning how the inputs were specific to airframe and the scary part is I haven't even learned how one would get those numbers. They were experimentally found somehow. Heck NASA had like 400 pages of the X-31 test program where they developed some dial a gain system a control system. And it crashed anyways.
I think he wants his gain to vary. Boeing multiplied their gain by H(s)=s/(s+a) but Matlab was doing the operations from proprietary data. How they found what rad/s went where beats me!
Perhaps they had a slight idea then iterated by tests. Currently doing beam deflection labs with vibrations math and there's this LVDT method with strain gauges by omega to get natural frequency but that's child's play compared to a hobby rocket airframe in flight. Supposedly you can similulink an arduino from Matlab. Dumb mechanical so they didn't teach that bit. Guess that's electrical engineer or on the job. Maybe someone smarter could get an actual actuator to response dampened to an input using the method. Tweak boeing's tutorial code is my opinion.
Not trying to snark. I feel f---ing stupid myself looking at it. The easy stuff starts with partial fraction decomposition and Laplace transforms if Laplace even applies! And the prof is check marking answers we know we missed. There's not answers in book back but I'm trying.
So with airplanes like Cessnas the control input by cable/pulley wasn't linear with velocity. You had regions near stall velocity where full control authority with maximized control deflection described as mushy feeling was required while near velocity never exceed you might need only a very slight touch rather than tens of pounds of effort to deflect control surfaces maybe a few degrees. The response was increasing with velocity despite decreasing forces applied. And that was a manual cable trainer no fly by wire gizmos or yaw dampers at a puny fifty to hundred knots. The controls system sorta changes the input from manual into a automated precise varying output reaction of what is sensed through coding and electronics/sensors. And you can adapt linkages, servos, motors, and all kinds of mechanical systems to it. I don't fault anyone for trying bandaids. It would be nice to average a linear model equation but if the control input changes proportional to output needed in flight you need a system like a control system using differentials that can adapt. Basically your control surface on an airplane has force components of lift when it's angled that generate the rolling or pitching. The drag on that surface increases as it deflects and it varies in force. And it causes oscillations on airframe elsewhere. And it's not linear. Putting flaps to 20 degrees on prolonged approaches caused the tail to bobble I remember. A damper via a control system could counter that by having elevator oscillate via math model but Cessna was pure mechanical linkages. And if on rocket the fin vibrates then god help you because it could chase the vibration with the control system picking up false inputs.
And the the theoretical is just as scary as how the heck did manufacturer X or Y actually develop that control system for Z problem. Nobody says hey start with math function this one. Is it suppose to be that complicated or is it just hard? I feel stupid. There's math tutors running away from control textbooks. Also seen control systems on VW based car factory robots using PLC server logic making cars every eight seconds. Door gates and part placement was inputs for when robot arms would move with relation to other robots. They had teams of programmers and electrical engineers. Teams. Beyond my understanding and pay grade...
