$1$ PID Controller and Its Practical Implementation

A continuous$-$time PID controller is given by

u$($t$)=$K$\_($P$)$e$($t$)+$K$\_($I$)\backslash$int$\_0^$t e$(\backslash$tau $)$d$\backslash$tau $-$K$\_($D$)$y$^(\text{'})($t$)$

where e$($t$)=$r$($t$)-$y$($t$)$ is the tracking error, r$($t$)$ is the reference value $($or set$-$point$),$ and y$($t$)$ is the measured output.

Note that, to avoid the "derivative kick" issue, the derivative component is calculated with the plant's output y rather

than the tracking error e$,$ and the negative sign before K$\_($D$)$ is because we estimate e$^(\text{'})($t$)=-$y$^(\text{'})($t$).$

A PID controller implemented in a computer program will be executed with discrete sampling periods $\backslash$Delta $\_($t$)\backslash$Delta $\_($t$)=1$ second$\backslash$Delta $\_($t$)$ as the time between sampling instances. It also replaces the derivative with either a filtered

version of the derivative or another method to approximate the instantaneous slope of the measured plant's output.

u$($t$)=$K$\_($P$)$e$($t$)+$K$\_($I$)\backslash$sum$\_($i$=0)^$t e$($i$)\backslash$Delta $\_($t$)-$K$\_($D$)($y$($t$)-$y$($t$-1))/(\backslash$Delta $\_($t$))$

In an iterative implementation, assume that u$\_($I$)($t$)$ is the accumulated integral value until time t:u$\_($I$)($t$)=\backslash$sum$\_($i$=0)^$t e$($i$)\backslash$Delta $\_($t$)=$

u$\_($I$)($t$-1)+$e$($t$)\backslash$Delta $\_($t$),$ with u$\_($I$)(-1)=0.$ Then the above equation can be calculated as

u$\_($I$)($t$)=$u$\_($I$)($t$-1)+$e$($t$)\backslash$Delta $\_($t$)$

u$($t$)=$K$\_($P$)$e$($t$)+$K$\_($I$)$u$\_($I$)($t$)-$K$\_($D$)($y$($t$)-$y$($t$-1))/(\backslash$Delta $\_($t$))$

Integral Windup Issue: An important feature of a controller with an integral term is to consider the case when the

controller output u$($t$)$ saturates at an upper or lower bound for an extended period of time. For example, in the case of

the TCL system, the heater power command is limited to between $0\%$ and $100\%,$ so if the controller output is below $0\%$

or above $100\%,$ it must be saturated to be within the limits$.$ This causes the integral term u$\_($I$)($t$)$ to accumulate to a large

summation that causes the controller to stay at the saturation limit until the integral summation is reduced; this issue

is known as integral windup. Anti$-$reset windup ensures that the integral term does not accumulate if the controller

output is saturated at an upper or lower limit$.$ In this project, we will use a practical anti$-$reset windup method where the

integral component is switched off or reset $($i$.$e$.,$ it does not accumulate$)$ when the controller output is saturated. Below

is the description of this method, where l and h are the lower and higher limits of the controller output u :

Save the previous value of the integral term u$\_($I$)($t$-1)$ of the PID controller

Update the integral term given the current error value as above: u$\_($I$)($t$)=$u$\_($I$)($t$-1)+$e$($t$)\backslash$Delta $\_($t$)$

Calculate u$($t$)$ as above

If u$($t$)>$hu$\_($I$)$u$\_($I$)($t$)=$u$\_($I$)($t$-1)$u$($t$)$ or u$($t$)>$h then:

a

bu$($t$)$ to the limit range l$,$h :

u$($t$)=\{($l if u$($t$)$h$)$:$\}$

Return u$($t$)$ and u$\_($I$)($t$)($which will be used in the next sampling time step$)$

$2$ Tuning PID Controller for FOPDT Models

For tuning the PID parameters for an FOPDT model with the transfer function $($K$\_($p$))/(\backslash$tau $\_($p$)$s$+1)$e$^(-\backslash$theta $\_($ps$)),$ we will use the following

methods:

PI control:

PI$-$IMC method: $\backslash$tau $\_($c$)=$max$(0\mathrm{.}1\backslash$tau $\_($p$),0\mathrm{.}8\backslash$theta $\_($p$))$;$,$K$\_($c$)=(1)/($K$\_($p$))(\backslash$tau $\_($p$))/((\backslash$theta $\_($p$)+\backslash$tau $\_($c$)))$;$,\backslash$tau $\_($I$)=\backslash$tau $\_($p$)$

PI$-$ITAE method: K$\_($c$)=(0\mathrm{.}586)/($K$\_($p$))((\backslash$theta $\_($p$))/(\backslash$tau $\_($p$)))^(-0\mathrm{.}916)$;$,\backslash$tau $\_($I$)=(\backslash$tau $\_($p$))/(1\mathrm{.}03-0\mathrm{.}165((\backslash$theta $\_($p$))/(\backslash$tau $\_($p$))))$

PID control: PID$-$IMC method:

$\backslash$tau $\_($c$)=$max$(0\mathrm{.}1\backslash$tau $\_($p$),0\mathrm{.}8\backslash$theta $\_($p$))$;$,$K$\_($c$)=(1)/($K$\_($p$))(\backslash$tau $\_($p$)+0\mathrm{.}5\backslash$theta $\_($p$))/((\backslash$tau $\_($c$)+0\mathrm{.}5\backslash$theta $\_($p$)))$;$,\backslash$tau $\_($I$)=\backslash$tau $\_($p$)+0\mathrm{.}5\backslash$theta $\_($p$)$;$,\backslash$tau $\_($D$)=(\backslash$tau $\_($p$)\backslash$theta $\_($p$))/(2\backslash$tau $\_($p$)+\backslash$theta $\_($p$))$

Then, the PID parameters are given by:

K$\_($P$)=$K$\_($c$)$;$,$K$\_($I$)=($K$\_($c$))/(\backslash$tau $\_($I$))$;$,$K$\_($D$)=$K$\_($c$)\backslash$tau $\_($D$)$