Optimisation of PID parameters implemented in Python, using scipy.optimize.minimize.
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Looking for a little-known way to tune a PID?
Python can help. Curious? 👇
If you know the model of the plant (linear, non-linear, acausal, doesn't really matter) you can use this approach.
Let's see how!
The equation of a PID controller with filtered derivative and back-calculation anti-windup is:
U(s) = K_p E(s) + (K_i E(s) + K_aw (U_sat(s) - U(s)))*1/s + K_d E(s) * s/(T_c s + 1)
where U_sat(s) is the command U(s) saturated (always necessary to match the actuator's capability) and E(s) = R(s) - Y(s).
The idea is to find the optimal combination of the PID parameters ((K_p, K_i, K_d, K_{aw}, T_c)) that minimises a cost function:
J = sum[W_e*(r_i - y_i)^2 + W_u*u_sat_i^2]
where:
✅ r: setpoint ✅ y: controlled variable ✅ u_sat: saturated command ✅ W_e: weighting coefficient related to the control error ✅ W_u: weighting coefficient related to u_sat
The choice of W_e and W_u determines the compromise between minimising the error and minimising the control effort:
✅ W_e > W_u: minimising the error is more important than minimising the control effort
✅ W_e < W_u: minimising the control effort is more important than minimising the error
The example in the carousel uses a plant that I have shown before:
An object of mass m = 10 kg, sliding on a surface where it's subject to viscous friction of coefficient k = 0.5 N*s/m, pushed by a force F.
The dynamic equation is:
d^2z/dt^2 m = F - k * dz/dt
The Python code (link to GitHub in the comments) consists of:
✅ A "PID" class that implements a PID controller with filtered derivative and back-calculation anti-windup, with a method "Step" that executes a step of the controller.
✅ An "Object" class that represents the plant described before, with a method "Step" that updates the position of the object based on the applied force F.
✅ A function "Simulation" that runs a simulation of 80 seconds where the PID controls the object's position following a step setpoint of 500 m.
✅ A function "Cost" that runs the function "Simulation" with a set of PID parameters and computes the cost function J
✅ A main code that runs the optimisation 3 times with different combinations of W_e and W_u and plots the different performances
Here are the 3 different configurations and their results:
Configuration 1
W_e = 1, W_u = 1
Result: Kp = 1.65, Ki = 0.228, Kd = 4.87, Kaw = 0.301, T_c = 0.506
Configuration 2
W_e = 30, W_u = 1
Result: Kp = 1.07, Ki = 0.0119, Kd = 4.99, Kaw = 0.0123, T_c = 1.01
Configuration 3
W_e = 1, W_u = 30
Result: Kp = 1.04, Ki = 0.0941, Kd = 4.99, Kaw = 0.176, T_c = 0.981
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#controlsystems #embeddedsystems #softwareengineering #embeddedsoftware #coding #controltheory #python #PID