Qualitative Effect of P, PI, PID parameters

So far in this course, we have done the following:

• Developed dynamic models for chemical processes

• Studied the effect of changing inputs on the outputs and states

• Developed a procedure for interconnection of systems In this lecture, we will address the issue of what happens when we connect a controller to a process. In particular, we will study how the parameters of the controller (which we can tune), affect the outputs.

P Controller

Consider the following process:

\begin{aligned}\frac{d}{dt} [ \begin{matrix}X_{1} \\ X_{2} \\ X_{3} \end{matrix} ] & = & [ \begin{matrix}−1 & 0 & 0 \\ 1 & −1 & 0 \\ 0 & 10 & −10 \end{matrix} ] [ \begin{matrix}X_{1} \\ X_{2} \\ X_{3} \end{matrix} ] + [ \begin{matrix}1 \\ 0 \\ 0 \end{matrix} ] U \\ Y & = & [ 0 0 1 ] [ \begin{matrix}X_{1} \\ X_{2} \\ X_{3} \end{matrix} ] \end{aligned}

and the P controller

\begin{aligned}U & = & k_{c}E \\ & = & k_{c}(Y_{sp} − Y) \end{aligned}  (2)

The closed-loop system is given by

\begin{aligned}\frac{d}{dt} [ \begin{matrix}X_{1} \\ X_{2} \\ X_{3} \end{matrix} ] & = & [ \begin{matrix}−1 & 0 & −k_{c} \\ 1 & −1 & 0 \\ 0 & 10 & −10 \end{matrix} ] [ \begin{matrix}X_{1} \\ X_{2} \\ X_{3} \end{matrix} ] + [ \begin{matrix}k_{c} \\ 0 \\ 0 \end{matrix} ] Y_{sp} \\ Y & = & [ 0 0 1 ] [ \begin{matrix}X_{1} \\ X_{2} \\ X_{3} \end{matrix} ] \end{aligned}

(3)

Suppose Y_{sp} undergoes a step change from 0 to 1. Clearly, we would like Y to reach the new value of the setpoint as soon as possible. What is the effect of the value of k_{c} on the output Y?

• For small k_{c}, response is smooth and stable, but there is a large off-set.

• For medium k_{c}, response is oscillatory and stable, but the off-set is small.

• For large k_{c}, the response is oscillatory and unstable.

PI Controller

Consider the following process:

\begin{aligned}\frac{d}{dt} [ \begin{matrix}X_{1} \\ X_{2} \\ X_{3} \end{matrix} ] & = & [ \begin{matrix}−1 & 0 & 0 \\ 1 & −1 & 0 \\ 0 & 10 & −10 \end{matrix} ] [ \begin{matrix}X_{1} \\ X_{2} \\ X_{3} \end{matrix} ] + [ \begin{matrix}1 \\ 0 \\ 0 \end{matrix} ] U \\ Y & = & [ 0 0 1 ] [ \begin{matrix}X_{1} \\ X_{2} \\ X_{3} \end{matrix} ] \end{aligned}

and the PI controller

\begin{aligned}\frac{dX_{4}}{dt} & = & Y_{sp} − Y \\ U & = & k_{c}E + \frac{k_{c}}{τ_{I}} X_{4} \end{aligned}  (5)

The closed-loop system is given by

\begin{aligned}\frac{d}{dt} [ \begin{matrix}X_{1} \\ X_{2} \\ X_{3} \\ X_{4} \end{matrix} ] & = & [ \begin{matrix}−1 & 0 & −k_{c} & \frac{k_{c}}{τ_{I}} \\ 1 & −1 & 0 & 0 \\ 0 & 10 & −10 & 0 \\ 0 & 0 & −1 & 0 \end{matrix} ] [ \begin{matrix}X_{1} \\ X_{2} \\ X_{3} \\ X_{4} \end{matrix} ] + [ \begin{matrix}k_{c} \\ 0 \\ 0 \\ 1 \end{matrix} ] Y_{sp} \\ Y & = & [ 0 0 1 0 ] [ \begin{matrix}X_{1} \\ X_{2} \\ X_{3} \\ X_{4} \end{matrix} ] \end{aligned}

Suppose Y_{sp} undergoes a step change from 0 to 1. Clearly, we would like Y to reach the new value of the setpoint as soon as possible. What is the effect of the value of k_{c} and τ_{I} on the output Y?

• Integral action eliminates off-set.

• As τ_{I} is decreased (=> \frac{1}{τ_{I}} is increased), the range of values for k_{c} which lead to stable output response, decreases.

PID Controller

Consider the following process:

\begin{aligned}\frac{d}{dt} [ \begin{matrix}X_{1} \\ X_{2} \\ X_{3} \end{matrix} ] & = & [ \begin{matrix}−1 & 0 & 0 \\ 1 & −1 & 0 \\ 0 & 10 & −10 \end{matrix} ] [ \begin{matrix}X_{1} \\ X_{2} \\ X_{3} \end{matrix} ] + [ \begin{matrix}1 \\ 0 \\ 0 \end{matrix} ] U \\ Y & = & [ 0 0 1 ] [ \begin{matrix}X_{1} \\ X_{2} \\ X_{3} \end{matrix} ] \end{aligned}

and the PID controller

\begin{aligned}\frac{dX_{4}}{dt} & = & E \\ \frac{dX_{5}}{dt} & = & −\frac{1}{ατ_{D}} X_{5} + \frac{1}{ατ_{D}} E \\ U & = & k_{c}(1 + \frac{1}{α})E + \frac{k_{c}}{τ_{I}} X_{4} − \frac{k_{c}}{ατ_{D}} X_{5} \end{aligned}  (8)

Suppose Y_{sp} undergoes a step change from 0 to 1. Clearly, we would like Y to reach the new value of the setpoint as soon as possible. What is the effect of the value of k_{c}, τ_{I}, and τ_{D} on the output Y?

The value of α is set to 0.1 for most commercial controllers. Thus there are only three parameters to vary (k_{c}, τ_{I}, and τ_{D})

• Integral action eliminates off-set.

• Derivative action has a stabilizing effect on the process. Thus, the presence of the τ_{D} term allows the use of higher values of k_{c} as compared to the PI controller.

Summary

• Proportional action alone is not sufficient to tightly control the output to the desired set-point. Increasing k_{c} reduces offset but destabilizes the process.

• Integral action eliminates offset. However as the integral term becomes larger (τ_{I} becomes small), the response is destabilized.

• Derivative action has a stabilizing effect on the closed-loop system which allows for higher values of proportional and integral terms to be used for faster response.