Dynamics of Commercial Controllers

A controller is an electronic device that helps to:

1. regulate a process at its desired set-point despite disturbances to the process.

2. take the system to a new set-point safely, with desired dynamic characteristics.

The vast majority of controllers used in the process industry (over 80%) are P, PI or PID controllers. In this lecture, we develop the dynamic models for these commercial controllers.

P = Proportional PI = Proportional-Integral PID = Proportional-Integral-Derivative

Y_{sp}: Desired set-point Y: Signal from sensor E: Error(input to controller) E = Y_{sp} − Y U: Signal to actuator (output of controller)

1. P Controller

U = k_{c}E  (1)

The variable k_{c} is called the controller gain and is a tunable parameter, which is set by the user.

2. PI Controller

\begin{aligned}\frac{de_{I}}{dt} & = & E \\ U & = & k_{c}E + \frac{k_{c}}{τ_{I}} e_{I} \end{aligned}  (2)

The variable τ_{I} is called the integral time constant. The PI controller has two tunable parameters: k_{c} and τ_{I}.

3. PID Controller

\begin{aligned}\frac{de_{I}}{dt} & = & E \\ \frac{de_{F}}{dt} & = & −\frac{1}{λ}e_{F} + \frac{1}{λ}E \\ U & = & k_{c}E + \frac{k_{c}}{τ_{I}} e_{I} + k_{c}τ_{D} (−\frac{1}{λ}e_{F} + \frac{1}{λ}E) \end{aligned}

(3)

The variable τ_{D} is called the derivative time constant. The variable e_{F} is the filtered derivative of the error with filter constant λ. The value of λ is not arbitrary but is set as the following function of τ_{D}.

λ = ατ_{D}  (4)

In most commercial controllers, α is set at a value of 0.1. Thus, the PID controller has three tunable parameters: k_{c}, τ_{I} and τ_{D}.

Commercial Controllers in Standard Form

1. P Controller

U = 0.X + k_{c}.E  (5)

Note that there is no differential equation involved and thus the ’A’ and ’B’ matrices are both zero.

2. PI Controller

\begin{aligned}\frac{de_{I}}{dt} & = & 0.e_{I} + 1.E \\ U & = & \frac{k_{c}}{τ_{I}} .e_{I} + k_{c}.E \end{aligned}  (6)

\begin{aligned}\frac{d}{dt} [ \begin{matrix}e_{I} \\ e_{F} \end{matrix} ] & = & [ \begin{matrix}0 & 0 \\ 0 & −\frac{1}{λ} \end{matrix} ] [ \begin{matrix}e_{I} \\ e_{F} \end{matrix} ] + [ 1 ; \frac{1}{λ} ] E \\ U & = & [\frac{k_{c}}{τ_{I}} − \frac{k_{c}τ_{D}}{λ} ] [ \begin{matrix}e_{I} \\ e_{F} \end{matrix} ] + [k_{c} + \frac{k_{c}τ_{D}}{λ} ] E \end{aligned}