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7.PID Controllers

7.0Overview

7.1PID controller variants

7.2 Choice of controller type

7.3 Specifications and performance criteria

7.4 Controller tuning based on frequency response

7.5Controller tuning based on step response

7.6Model-based controller tuning

7.7Controller design by direct synthesis

7.8Internal model control

7.9Model simplification

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7.0Overview

PID controller (pee-i-dee) is a generic name for a controller containing a linear combination of

proportional (P)

integral (I)

derivative (D)

terms acting on a control error (or sometimes the process output).

All parts need not be present. Frequently I and/or D action is missing, giving a controller like

P, PI, or PD controller

It has been estimated that of all controllers in the world

95 % are PID controllers

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7. PID Controllers

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7.1PID controller variants

An ideal PID controller is described by the control law

(7.1)

is the controller output

is the control error, which is the difference between the setpoint and the measured process output

is the proportional gain

is the integral time

is the derivative time

is the normal value of the controller output

The transfer function of the PID controller is

(7.2)

is the Laplace transform of

is the Laplace transform of the control error

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7.1.1Ideal PID controller

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Depending on the values of and , the transfer function of the PID controller can have

real or complex valued zeros

Complex zeros might be useful for control of underdamped systems with complex poles.

A PI controller is obtained from a PID controller by letting . Its transfer function is

(7.3)

A PD controller is obtained from a PID controller by letting . Its transfer function is

(7.4)

The ideal PID controller is sometimes referred to as

the parallel form of a PID controller

the (ISA) standard form

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7.1.1 Ideal PID controller

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7.1.2The series form of a PID controller

In the pre-digital era it was convenient to implement an analog PID controller as a PI controller and a PD controller in series. This form of a PID controller is called the series form. Occasionally, the terms interactive form or classical form are used. The controller has the transfer function

(7.5)

where is used to distinguish the parameters from the parameters of the parallel form.

The series form of a PID controller can only have real valued zeros. This means that the series form is less general than the parallel form.

It is easy to find the controller parameters of the series form by frequency analytic methods by so-called lead-lag design.

Exercise 7.1

Which is the control law in the time domain for a series form PID controller?

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7.1 PID controller variants

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7.1.3A PID controller with derivative filter

A drawback with the ideal PID controller (7.1) is that the derivative part cannot be realized exactly in a real controller. For example, if the control error changes as a step, the derivate in (7.1) becomes infinitely large. This problem can be remedied by

filtering the signal to be differentiated.

This also has the practical advantage that (high-frequency) noise is filtered before differentiation.

The transfer function of a parallel form PID controller with a derivative filter is

(7.6)

The transfer function of a series form PID controller with a derivative filter is

(7.7)

and are filter constants, usually 10-30 % of corresp. derivative time.

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7.1 PID controller variants

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Relationships between parallel and series form

If the parameters of the series form are known, the corresponding parameters of the parallel form can be calculated according to

, , , (7.8)

For calculation of the parameters of the series form from the parameters of the parallel form, we define the parameter

(7.9)

If , the zeros of the parallel PID are real. Then, there exists a series form PID controller which is equivalent to the parallel form according to

, , , (7.10)

The condition for in terms of the controller parameters is

(7.11)

i.e., the derivative time has to be small enough.

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7.1.3 A PID controller with derivative filter

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7.1.4Differentiation of the measured output

Even if we have a derivative filter, a step change in the setpoint tends to affect the derivative part much more strongly than a disturbance in the output . A remedy to this is to

differentiate the (filtered) output instead of the control error .

The ideal control law (7.1) then becomes

(7.12a)

(7.12b)

In the Laplace domain we get

(7.13)

which is a combination of a PI controller and a PID controller

(7.14)

This kind of 2-degrees-of-freedom (2DOF) controller can be tuned separately for setpoint tracking and disturbance rejection.

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7.1 PID controller variants

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Exercise 7.2

Which is the control law, both in the time domain and the Laplace domain, for the series form of a PID controller with differentiation of the filtered output measurement?

A simple way of obtaining 2DOF PID controller is to use setpoint weighting. With the definitions

, , (7.15)

where and are setpoint weights, the control law becomes

(7.16a)

(7.16b)

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7.1.4 Differentiation of the measured output

7.1.5Setpoint weighting

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In the Laplace domain the control law with setpoint weighting is

(7.17)

where

(7.18)

and is as in (7.6).

With suitable choices of and , all previously treated PID controllers on parallel form can be obtained.

and do not affect the controllers ability to reject disturbances in the output, only the ability to track setpoint changes.

can be tuned for setpoint tracking and for disturbance rejection (i.e., , and need not have the same values in and ).

Exercise 7.3

Include setpoint weighting in the series form of a PID controller.

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7.1.5 Setpoint weighting

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7.1.6Non-interactive form of a PID controller

In the control laws treated so far, the proportional part alone cannot be disconnected by letting because that would disconnect all parts; it would put the controller on manual with .

Tuning the proportional part by adjusting will affect all controller parts (however, this is often a desired feature); hence, it is an interactive controller form.

The non-interactive form

(7.19)

is a more flexible control law. In the Laplace domain it can be written

(7.20)

where

(7.21a)

(7.21b)

Note: It is essential to know which form is used when tuning a controller!

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7.1 PID controller variants

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7.2Choice of controller type

The choice between controller types such as P, PI, PD, PID is considered. In principle, the simplest controller that can do the job should be chosen.

An on-off controller is the simplest type of controller, where the control signal has only two levels. If the variables are defined such that a positive control error should be corrected by an increase o