PID feedback control

A proportional–integral–derivative controller (PID controller) is a control loop feedback mechanism control technique widely used in control systems.

A PID controller continuously calculates an error value ${\displaystyle e(t)}$ as the difference between a desired setpoint and a measured process variable and applies a correction based on proportional, integral, and derivative terms. PID is an initialism for Proportional-Integral-Derivative, referring to the three terms operating on the error signal to produce a control signal.

The theoretical understanding and application dates from the 1920s, and they are implemented in nearly all analogue control systems; originally in mechanical controllers, and then using discrete electronics and latterly in industrial process computers. The PID controller is probably the most-used feedback control design.

Referring to the equation below;

If u(t) is the control signal sent to the system, y(t) is the measured output and r(t) is the desired output, and tracking error ${\displaystyle e(t)=r(t)-y(t)}$, a PID controller has the general form

${\displaystyle u(t)=K_{P}e(t)+K_{I}\int e(\tau ){\text{d}}\tau +K_{D}{\frac {\text{de(t)}}{{\text{d}}t}}.}$

The desired closed loop dynamics is obtained by adjusting the three parameters ${\displaystyle K_{P}}$, ${\displaystyle K_{I}}$ and ${\displaystyle K_{D}}$, often iteratively by "tuning" and without specific knowledge of a plant model. Stability can often be ensured using only the proportional term. The integral term permits the rejection of a step disturbance (often a striking specification in process control). The derivative term is used to provide damping or shaping of the response. PID controllers are the most well established class of control systems: however, they cannot be used in several more complicated cases, especially if MIMO systems are considered.

Applying Laplace transformation results in the transformed PID controller equation

${\displaystyle u(s)=K_{P}e(s)+K_{I}{\frac {1}{s}}e(s)+K_{D}se(s)}$

${\displaystyle u(s)=\left(K_{P}+K_{I}{\frac {1}{s}}+K_{D}s\right)e(s)}$

with the PID controller transfer function

${\displaystyle C(s)=\left(K_{P}+K_{I}{\frac {1}{s}}+K_{D}s\right).}$

There exists a nice example of the closed-loop system discussed above. If we take

PID controller transfer function in series form

${\displaystyle C(s)=K\left(1+{\frac {1}{sT_{i}}}\right)(1+sT_{d})}$

1st order filter in feedback loop

${\displaystyle F(s)={\frac {1}{1+sT_{f}}}}$

linear actuator with filtered input

${\displaystyle P(s)={\frac {A}{1+sT_{p}}}}$, A = const

and insert all this into expression for closed-loop transfer function H(s), then tuning is very easy: simply put

${\displaystyle K={\frac {1}{A}},T_{i}=T_{f},T_{d}=T_{p}}$

and get H(s) = 1 identically.

For practical PID controllers, a pure differentiator is neither physically realisable nor desirable due to amplification of noise and resonant modes in the system. Therefore, a phase-lead compensator type approach is used instead, or a differentiator with low-pass roll-off.