pid_pas

PID controller model written in Pascal.

Derivation of discrete form (1)

Parallel form of PID

The conventional parallel form of a proportional - integral - derivate controller is:

$$OP(t) = Kp.\epsilon (t) + Ki.\int_0^t \epsilon (t) \partial t + Kd.\frac{\partial \epsilon (t)}{\partial t} \tag{1} \label{eq:1}$$

Where $OP(t)$ is the controller output at time $t$, $\epsilon$ is the error between the controller setpoint $SP$ and the process value $PV$: $\epsilon = SP - PV$.
This can be solved using a simple discrete formulation (to reduce computational power):

$$OP(k) = Kp.\epsilon (k) + \underbrace { Ki. \sum_{j=0}^{k} \epsilon (j) \Delta t}_{\text{I(k)}} + \underbrace { Kd. \left( \frac{\epsilon (k) - \epsilon (k-1)}{\Delta t} \right)}_{D(k)} \tag{2} \label{eq:2}$$

The integral term $I(k)$ is further limited in magnitude to prevent unbounded growth of this term:

$$\big| I(k) \big| \leq I_{max} \label{eq:3} \tag{3}$$

where

$$I_{max} = \frac{OP_{max}}{Ki} \tag{4} \label{eq:4}$$

The calculated derivative is sensitive to changes in $SP$, even thought the $PV$ may not be changing, leading to a sudden change in the output of the controller. To prevent setpoint changes from affecting the derivative action, and noting that $\left( \frac{\partial \epsilon(t)}{\partial t} \right)_{SP=const} = -\frac{\partial PV(t)}{\partial t}$ the derivative term can be calculated as follows:

$$D(k) = -Kd. \left( \frac{PV(k) - PV(k-1)}{\Delta t} \right) \tag{5} \label{eq:5}$$

The derivative term $D(k)$ is sensitive to measurement noise of the $PV$. To limit the impact of noise on the derivate term a low pass filter is used:

$$\begin{gather} D_{filtered}(k) = \alpha D(k-1) + (1-\alpha)D(k) \tag{6} \label{eq:6} \\\ 0 \leq \alpha < 1 \end{gather}$$

The filtered derivate term $D_{filtered}(k)$ is then used when calculating the output of the PID controller. Note that when large values for $Kd$ are used, the output of the PID controller can oscillate even if little or no noise is present in the $PV$. This type of oscillations can to some extend be eliminated by specifying an $\alpha$ value close to 1.

Standard form of PID

The series (or standard) form of a proportional - integral - derivate controller is:

$$OP(t) = Kg \left( \epsilon (t) + \frac{1}{Ti}\int_0^t \epsilon (t) \partial t + Td.\frac{\partial \epsilon (t)}{\partial t}\right) \tag{7} \label{eq:7}$$

This is mathematically identical to the first equation, with the gain calculated as $Kg = Kp$, the integral reset time calculated as $Ti = \frac{Kp}{Ki}$ and the integration time is calculated as $Td = Kp.Kd$. This form is often used in industry, hence the option is available to specify the PID settings in this form. Internally the parameters are converted to the equivalent parallel form and solved via equations above.

Derivation of discrete form (2)

An alternative formulation for solving the PID equation is to use equation $\ref{eq:2}$ and calculate the difference between timestep $k$ and $k-1$:

$$OP(k) = OP(k-1) + Kp.(\epsilon (k) - \epsilon (k-1)) + Ki.\epsilon (k) \Delta t + Kd. \left( \frac{D(k) - D(k-1)}{\Delta t}\right) \tag{8} \label{eq:8}$$

This is sometimes called the velocity form.