AS 3/2017

CONTROLLO

tecnica

Aprile 2017

Automazione e Strumentazione

where the weights, and , of the reference signal are introduced

in order to have the possibility to do an independent processing

of the reference and output signals. In such case we are facing a

two degree of freedom (2-DoF) PID controller. Whereas the set-

point weight usually takes values between 0 and 1, the value is

usually set to 0 in order to avoid derivatives of the reference signal

that can generate the well known derivative

kick

uSort: an unified PID tuning

As PID controller tuning has been in place during almost 80 years,

there are out there numerous proposals on how to select the con-

troller gains.. The book

[1]

provides a quite panoramic and cata-

logued view on existing methods. Tuning rules and suggestions

for controller gains are usually provided depending on the con-

troller structure (PI, PID, 1-DoF, 2DoF ecc.), controlled process

model (first order, second order, integrative, unstable, etc) and, of

course, specifications for the closed-loop behaviour. The elevate

number of variations and possibilities constitutes a nightmare for

the PID operator that has to change the setup depending on the

concrete scenario.

Whereas it may be understandable that a change in the closed-loop

specifications should impose a change in the way the controller

gains are parameterised, it may in turn be desirable that the fact of

having a PI or a PID should not change the controller gains. This

is one of the aspects this note is referring to when stating about an

unified PID tuning approach.

Tuning is usually a compromise between performance and robust-

ness. In fact, information about the process to be controlled takes

the form of a model but this is always incomplete. Therefore,

robustness is needed in order to preserve the basic properties that

the model-based tuning provides. Among them, stability of the

controlled system is a first need. Also, to minimize the degrada-

tion of the performance is desirable. As a basic tradeoff, as more

robustness is imposed, the model-based tuning tends to provide

lower performance. This is why some tunings focuses exclusively

on loop performance, whereas others are aimed to ensure robust

stability, or a compromised mix of both, etc. Among the consid-

ered loop performance measures, integrated indexes are among

the most popular, being the Integrated Absolute Error (IAE)

the one that has gained most acceptance as being correlated with

economic considerations

[3]

. Minimum-IAE tuning, although

maximizing performance, tends to leave a low margin of stability.

Moreover, as higher-performance controllers are used, the stabil-

ity margin decreases. It is therefore needed to achieve a reasonable

trade-off among both considerations.

[2]

the optimal IAE PI/PID tuning is addressed by also consid-

ering the optimal way of degrading the performance in such a way

certain degree of robustness is achieved. Very simple tuning rules

are provided for PI/PID controller based on controlled process

models of first as well as second order plus dead-time. For such

purpose, a process model of the following generic form is assumed

with 0 ≤

≤ 1. Notice that for the particular case a=0 the usual first

order plus time delay process model is recovered. Simple tuning

rules for the PI/PID controller parameters are provided for optimal

servo and regulation operation under the parameterizations dis-

played in

Table 1

The values for the

and

are provided in

[2]

depending

on the process model constants as well as desired robustness for

the closed loop system. For the regulatory tuning, it may happen

that very poor set-point-following performance is observed. Then,

to improve the servo-control performance, the set-point weight is

also provided. It is important to highlight that this table provides

parameter values for PI and PID controllers in an unified way.

Example

To illustrate the use of the uSort tuning, we consider a process

whose dynamics is represented by the following transfer function:

As approximations of the dynamics described by

P(s)

, the follow-

ing first order plus time delay (FOPDT) and second order plus

time delay (SOPDT) models were obtained:

Whereas

(s)

is the usual process model used for PID design,

here we will also make use of

(s)

as a better model for

P(s)

. In

Table 2

, the corresponding PID controller gains are provided

by using

(s)

and

(s)

as models for controller design. For each

case, two desired robustness are considered. All the designs are

optimized for a regulatory operation and including the set-point

weight for set-point tracking improvement.

Table 1 - uSort PI/PID Tuning relations