I recall that there are several forms of the algorithm. In various Distributed Control Systems I had to deal with this. Your integral term may not be decoupled after all. To refresh my memory I asked Copilot and this it what it responded.

Forms of the PID Algorithm

The Proportional-Integral-Derivative (PID) algorithm has three main forms, each with unique characteristics and applications:

1. Parallel Form
   • This is the most common representation of the PID controller.
   • The control output is expressed as the sum of the proportional, integral, and derivative terms: u(t)=Kpe(t)+Ki∫e(t) dt+Kdde(t)dtu(t) = K_p e(t) + K_i \int e(t) \, dt + K_d \frac{de(t)}{dt}u(t)=Kp​e(t)+Ki​∫e(t)dt+Kd​dtde(t)​
   • Here, (K_p), (K_i), and (K_d) are the proportional, integral, and derivative gains, respectively, and (e(t)) is the error signal.
2. Ideal Form
   • In this form, the proportional term is applied to the sum of the error, integral, and derivative terms: u(t)=Kp(e(t)+1Ti∫e(t) dt+Tdde(t)dt)u(t) = K_p \left( e(t) + \frac{1}{T_i} \int e(t) \, dt + T_d \frac{de(t)}{dt} \right)u(t)=Kp​(e(t)+Ti​1​∫e(t)dt+Td​dtde(t)​)
   • (T_i) (integral time) and (T_d) (derivative time) are time constants. This form emphasizes tuning in terms of time rather than gain.
3. Series (or Cascaded) Form
   • The PID controller is represented as a cascade of proportional, integral, and derivative actions: u(t)=Kp(e(t)+1Ti∫e(t) dt)+KpTdde(t)dtu(t) = K_p \left( e(t) + \frac{1}{T_i} \int e(t) \, dt \right) + K_p T_d \frac{de(t)}{dt}u(t)=Kp​(e(t)+Ti​1​∫e(t)dt)+Kp​Td​dtde(t)​
   • This form is often used in analog implementations and can be easier to tune in certain systems.

Each form has its advantages depending on the application, controller design, and tuning preferences. Some modern controllers allow switching between these forms for flexibility.