# How can I find characteristic equation of complex systems?

I am working on an Overhead crane, where I am supposed to design a controller that controls the position of the crane while minimizing the load's sway angle. I have derived the transfer functions for the system and am using a PID-PD Controller in the system, as shown in the figure below.

I am using Particle Swarm Optimization to tune the controller, and I would like to use Routh-Hurwitz Criterion to evaluate the stability of the system of the particles in the beginning. However, it is not clear how to minimize the system and find its characteristic equation. I have tried minimizing based on the following derivation:

$$CLTF1 = \frac{1}{1-TF2*PD}$$

$$CLTF2 = \frac{PID*CLTF1*TF1}{1+PID*CLTF1*TF1}$$

I have tried using CLTF2 and comparing its results with the original system to verify the result, but I got very different results, as shown in the figure below Is my minimization technique, or approach, incorrect? If so, can you guide me on how to use Routh Hurwitz Criterion for such a system?

Thank you

• The two equations appear to be correct. What do you mean by "original system" when you say "I have tried using CLTF2 and comparing its results with the original system ..." ? Please provide more details about your attempts to compare CLTF2 and original_system.
– AJN
Commented Apr 26, 2022 at 13:07
• @AJN I mean the original Simulink model shown in the first figure. I have appended the figure of the comparison I made between "CLTF2" and "Position" Transfer functions in the main post. This is where I got different responses. Commented Apr 26, 2022 at 14:35
• Did you check if the saturation block was saturating ? Have you checked after removing the saturation block ? Are the original system and the CLTF2 stable systems ?
– AJN
Commented Apr 26, 2022 at 15:16
• @AJN Yes I have. Indeed it is saturating. However, even after removing the saturation, the response is still very different. The original system is naturally unstable. However, with the two controllers implemented, it is stable. CLTF2, on the other hand, is unstable. Commented Apr 26, 2022 at 15:46
• Are the "structure" (equations) of PD and PID same as in Simulink ? e.g. some write PD as $K (e_p + K_D e_d)$ instead of $K_P e_p + K_D e_d$. There are many such variations. Are you using the same "kind" of PID and PD "structure" as Simulink in your derivations ? Also, does the PID block in simulink have internal saturation logic ?
– AJN
Commented Apr 26, 2022 at 16:12