7. PID Controllers - Åbo · PDF fileControl Laboratory 7. PID Controllers 7.0 Overview 7.1 PID controller variants 7.2 Choice of controller type ... 7.8 Internal model control 7.9

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Laboratory7. PID Controllers7.0 Overview

7.1 PID controller variants

7.2 Choice of controller type

7.3 Specifications and performance criteria

7.4 Controller tuning based on frequency response

7.5 Controller tuning based on step response

7.6 Model-based controller tuning

7.7 Controller design by direct synthesis

7.8 Internal model control

7.9 Model simplification

KEH Process Dynamics and Control 7–1

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7. PID Controllers

7.0 OverviewPID 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


An ideal PID controller is described by the control law

𝑢 𝑡 = 𝐾c 𝑒 𝑡 +1

𝑇i 0𝑡𝑒 𝜏 d𝜏 + 𝑇d

d𝑒(𝑡)

d𝑡+ 𝑢0 (7.1)

𝑢(𝑡) is the controller output 𝑒 𝑡 = 𝑟 𝑡 − 𝑦(𝑡) is the control error, which is the difference

between the setpoint 𝑟(𝑡) and the measured process output 𝑦(𝑡) 𝐾c is the proportional gain 𝑇i is the integral time 𝑇d is the derivative time 𝑢0 is the “normal” value of the controller output

The transfer function of the PID controller is

𝐺PID =𝑈(𝑠)

𝐸(𝑠)= 𝐾c 1 +

1

𝑇i𝑠+ 𝑇d𝑠 =

𝐾c

𝑇i𝑠1 + 𝑇i𝑠 + 𝑇i𝑇d𝑠

2 (7.2)

𝑈(𝑠) is the Laplace transform of 𝑢 𝑡 − 𝑢0 𝐸(𝑠) is the Laplace transform of the control error


7.1.1 Ideal PID controller

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Depending on the values of 𝑇i and 𝑇d, 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 𝑇d = 0. Its transfer function is

𝐺PI = 𝐾c 1 +1

𝑇i𝑠=

𝐾c

𝑇i𝑠1 + 𝑇i𝑠 (7.3)

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

𝐺PD = 𝐾c 1 + 𝑇d𝑠 (7.4)

The ideal PID controller is sometimes referred to as

the parallel form of a PID controller

the (ISA) standard form


7.1.1 Ideal PID controller

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7. PID Controllers

7.1.2 The 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

𝐺PIPD = 𝐾c′ 1 +

1

𝑇i′𝑠

1 + 𝑇d′𝑠 =

𝐾c′

𝑇i′𝑠

1 + 𝑇i′𝑠 1 + 𝑇d

′𝑠 (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. PID Controllers

7.1.3 A 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 𝑒(𝑡) 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

𝐺PIDf = 𝐾c 1 +1

𝑇i𝑠+

𝑇d𝑠

𝑇f𝑠+1(7.6)

The transfer function of a series form PID controller with a derivative filter is usually stated in the form

𝐺PIPDf = 𝐾c′ 1 +

1

𝑇i′𝑠

𝑇d′𝑠+1

𝑇f′𝑠+1

(7.7)

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



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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

𝑇i = 𝑇i′ + 𝑇d

′ − 𝑇f′ , 𝑇d = 𝑇d

′ 𝑇i′

𝑇i− 𝑇f

′ , 𝑇f = 𝑇f′ , 𝐾c = 𝐾c

′ 𝑇i′

𝑇i(7.8)

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

𝛿 = 1 −4𝑇i(𝑇d+𝑇f)

(𝑇i+𝑇f)2 (7.9)

If 𝛿 ≥ 0, 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

𝑇i′ =

(𝑇i+𝑇f)

21 + 𝛿 , 𝑇d

′ = 𝑇i + 𝑇f − 𝑇i′ , 𝑇f

′ = 𝑇f , 𝐾c′ = 𝐾c

𝑇i′

𝑇i(7.10)

The condition for 𝛿 ≥ 0 in terms of the controller parameters is

𝑇d ≤(𝑇i−𝑇f)

2

4𝑇i(7.11)

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


7.1.3 A PID controller with derivative filter

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7. PID Controllers

7.1.4 Differentiation of the measured output

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

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

The ideal control law (7.1) then becomes


𝑇i 0𝑡𝑒 𝜏 d𝜏 − 𝑇d

d𝑦f(𝑡)

d𝑡+ 𝑢0 (7.12a)

𝑇fd𝑦f(𝑡)

d𝑡+ 𝑦f 𝑡 = 𝑦(𝑡) (7.12b)

In the Laplace domain we get

𝑈 𝑠 = 𝐾c 1 +1

𝑇i𝑠𝑅 𝑠 − 𝐾𝑐 1 +

1

𝑇i𝑠+

𝑇d𝑠

𝑇f𝑠+1𝑌(𝑠) (7.13)

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

𝑈 𝑠 = 𝐺PI𝑅 𝑠 − 𝐺PIDf𝑌(𝑠) (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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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

𝑒p = 𝑏𝑟 − 𝑦 , 𝑒 = 𝑟 − 𝑦 , 𝑒d = 𝑐𝑟 − 𝑦f (7.15)

where 𝑏 and 𝑐 are setpoint weights, the control law becomes

𝑢 𝑡 = 𝐾c 𝑒p 𝑡 +1

𝑇i 0𝑡𝑒 𝜏 d𝜏 + 𝑇d

d𝑒d(𝑡)

d𝑡+ 𝑢0 (7.16a)

𝑇fd𝑦f(𝑡)

d𝑡+ 𝑦f 𝑡 = 𝑦(𝑡) (7.16b)


7.1.4 Differentiation of the measured output

7.1.5 Setpoint weighting

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In the Laplace domain the control law with setpoint weighting is

𝑈 𝑠 = 𝐺vPID𝑅 𝑠 − 𝐺PIDf𝑌(𝑠) (7.17)where

𝐺vPID = 𝐾c 𝑏 +1

𝑇i𝑠+ 𝑐𝑇d𝑠 (7.18)

and 𝐺PIDf is as in (7.6).

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

𝑏 and 𝑐 do not affect the controller’s ability to reject disturbances in the output, only the ability to track setpoint changes.

𝐺vPID can be tuned for setpoint tracking and 𝐺PIDf for disturbance rejection (i.e., 𝐾c, 𝑇i and 𝑇d need not have the same values in 𝐺vPIDand 𝐺PIDf).

Exercise 7.3

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


7.1.5 Setpoint weighting

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7. PID Controllers

7.1.6 Non-interactive form of a PID controller

In the control laws treated so far, the proportional part alone cannot be disconnected by letting 𝐾c = 0 because that would disconnect all parts; it would put the controller on “manual” with 𝑢 𝑡 = 𝑢0.

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

The non-interactive form

𝑢 𝑡 = 𝐾c𝑒𝑝 𝑡 + 𝐾i 0𝑡𝑒 𝜏 d𝜏 + 𝐾d

d𝑒d(𝑡)

d𝑡+ 𝑢0 (7.19)

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

𝑈 𝑠 = 𝐺vP+I+D𝑅 𝑠 − 𝐺P+I+Df𝑌(𝑠) (7.20)where

𝐺vP+I+D = 𝐾c𝑏 + 𝐾i𝑠−1 + 𝑐𝐾d𝑠 (7.21a)

𝐺P+I+Df = 𝐾c + 𝐾i𝑠−1 + 𝐾d𝑠(𝑇f𝑠 + 1)−1 (7.21b)

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



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7. PID Controllers

7.2 Choice of controller typeThe 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 𝑒(𝑡) should be corrected by an increase of the control signal 𝑢(𝑡), the control law is

𝑢 𝑡 =

𝑢max if 𝑒 𝑡 > 𝑒hi𝑢0 or unchanged if 𝑒lo ≤ 𝑒 𝑡 ≤ 𝑒hi𝑢min if 𝑒 𝑡 < 𝑒lo

(7.23)

where 𝑢max, 𝑢0, 𝑢min is the high, normal, low value of the control signal. The interval (𝑒lo, 𝑒hi) is a dead zone. In the simplest case, 𝑒lo = 𝑒hi = 0.

The on-off controller is inexpensive, but it causes oscillations in the pro-cess. It is often used for temperature control in simple appliances such as ovens, irons, refrigerators and freezers, where oscillations are tolerated.


7.2.1 On-off controller

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7. PID Controllers

7.2.2 P controller

A P controller implements the simple control law

𝑢 𝑡 = 𝐾c𝑒 𝑡 + 𝑢0 (7.24)

where 𝐾c is the adjustable controller gain and 𝑢0 is the normal value of the control signal, which is also be adjustable. In principle, 𝑢0 is selected to make the control error 𝑒 𝑡 = 0 at the nominal operating point.

If the output is changed by a disturbance or a setpoint change, the P controller is unable to bring the control error to zero, i.e., there will be a remaining control error.

The higher the controller gain, the smaller the control error. Thus, P control is used when a (small) control error is allowed and a high controller gain can be used without risk of instability.

A typical application for P control is level control in liquid tank. Another situation when P control is often sufficient is as an inner loop (a secon-dary loop) in so-called cascade control.



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7. PID Controllers

7.2.3 PI controller

A PI controller is by far the most common type of controller. The ideal PI controller implements the control law


𝑇i 0𝑡𝑒 𝜏 d𝜏 + 𝑢0 (7.25)

where the gain 𝐾c and the integral time 𝑇i are adjustable parameters; 𝑢0 is less important due to the integral.

The main advantage of the PI controller is that there will be no remaining control error after a setpoint change or a process disturbance. A dis-advantage is that there is a tendency for oscillations.

PI control is used when no steady-state error is desired and there is no reason to use derivative action. Measurement noise is often a reason for not using derivative action.

PI control is suitable for noisy processes, integrating processes and processes resembling first-order systems. The most typical application is flow control. PI control might also be preferable for processes with large time delays.



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7. PID Controllers

7.2.4 PD controller

The ideal form of a PD controller implements the control law

𝑢 𝑡 = 𝐾c 𝑒 𝑡 + 𝑇dd𝑒(𝑡)

d𝑡+ 𝑢0 (7.26)

where the gain 𝐾c and the derivative time 𝑇d are adjustable para-meters; 𝑢0 is chosen as for a P controller.

A PD controller is preferred when integral action is not needed, but the dynamics of the process are so slow that the predictive nature of derivative action is useful.

Many thermal processes, where energy is stored with small heat losses (e.g., ovens), usually have slow dynamics, almost as integrating systems. A PD controller might then be suitable for temperature control.

Another typical application for PD control is in servo mechanisms such as electrical motors, which usually behave as second-order integrating systems.



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7. PID Controllers

7.2.5 PID controller

As has been shown in Section 7.1, there are many variants of PID controllers.

The ideal form and the classical series form have 3 adjustable parameters in addition to 𝑢0 : the proportional gain, the integral time, and the derivative time.

If a derivative filter is included, there are 4 adjustable parameters, but the filter time constant is usually selected as a given fraction (e.g., 10 %) of the derivative time.

In addition, the setpoint can be weighted in the proportional part and the derivative part.

If there is no reason to exclude integral action or derivative action, a PID controller is the natural choice. Typically PID control is used for under-damped processes, processes with slow dynamics and not very large time delays, and systems of second and higher order.

Typical applications are control of temperature and chemical compositionwhen the process is not close to an integrating system.



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7. PID Controllers


The task of a controller is to control a system to behave in a desired waydespite unknown disturbances and an inaccurately known system.

The controlled system should satisfy performance criteria such as:

The controlled system must be stable; this is absolutely necessary.

The effect of disturbances on the controlled output is minimized; this is especially important for regulatory control.

The controlled output should follow setpoint changes fast and smoothly; this is especially important for setpoint tracking.

The control error is minimized or kept within certain limits,

The control signal variations should be moderate or at least not be excessively large; more variations wear out control equipment faster.

The control system should be robust (insensitive) against moderate changes in system properties, which introduce model uncertainty.

The importance of these criteria varies from case to case. Since many cri-teria are conflicting, compromises have to be made in the control design.


7.3.1 General performance criteria

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7. PID Controllers

7.3.2 Fundamental limitations

One reason to the fact that there are usually good solutions to the conflicting control criteria is that feedback control is used.

However, feedback also introduces limitations because a control error is required for the controller to take action.

The fact that the available resources for control are always limited, also limit the achievable performance.

In addition to the general limitations above, there are also limitations that depend on the process to be controlled, e.g.,

the dynamics of the process

nonlinearities

model and process uncertainty

disturbances

control signal limitations



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The process dynamics is often the performance limiting factor. Such factors are

time delays as well as RHP (right-half plane) poles and zeros high-order dynamics

In practice, all processes are nonlinear. Such a process

cannot be described accurately at different operating points by a linear model with constant parameters; thus there is model/process uncertainty.

Disturbances such as load disturbances and measurement noise limit how well a variable can be controlled.

Efficient control of load disturbances often require derivative action, but measurement noise is bad for the derivative.

Large load disturbances can cause the control variable to reach its (physical) maximum or minimum value. This is especially troublesome if the controller contains an integrator. Proportional band and integrator windup are two concepts that deal with this limitation.


7.3.2 Fundamental limitations

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7. PID Controllers

7.3.3 Proportional band and integrator windup

Proportional band

A controller’s proportional band (PB) denotes the maximum control error the controller can handle with the available control signal. The PB is defined for a P controller, but it can be extended to a full PID controller.

If the control signal is limited by 𝑢min ≤ 𝑢(𝑡) ≤ 𝑢max , a P controller can according to (7.24) handle a control error that satisfies

𝑢min−𝑢0

𝐾c≡ 𝑒min ≤ 𝑒(𝑡) ≤ 𝑒max ≡

𝑢max−𝑢0

𝐾c(7.27)

The PB is equal to 𝑒max − 𝑒min = 𝑦hi − 𝑦lo, where 𝑦hi is the highest output (𝑒min = 𝑟 − 𝑦hi) and 𝑦lo is the lowest output (𝑒max = 𝑟 − 𝑦lo) the controller can handle. Usually, the PB is defined in percent of the total measurable output interval 𝑦min, 𝑦max . Then, the PB is

𝑃b =𝑦hi−𝑦lo

𝑦max−𝑦min100% =

𝑢max−𝑢min

𝑦max−𝑦min⋅100%

𝐾c(7.28)



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If the proportional band is known, the controller gain is given by

𝐾c =𝑦hi−𝑦lo

𝑦max−𝑦min100% =

𝑢max−𝑢min

𝑦max−𝑦min⋅100%

𝑃b(7.29)

In (old) automation systems, the signals are often expressed as a fraction or percentage of the total signal interval (0-1 or 0-100%). The PB is then

𝑃b = 100%/𝐾c (7.30)

Note that the controller gain here has to be expressed in terms of the normalized signals, which means that the controller gain is dimensionless.

The practical usefulness of the PB is that it tells something about the size of control errors that can be handled without reaching an input signal constraint. If 𝑢0 is in the middle of the interval 𝑢min, 𝑢max , a P controller with 𝑃b = 50 % can handle an instantaneous control error equal to ±25 % (i.e., 50 % in total) of the total output signal range.

Note that the PB is an adjustable controller parameter — if it is to small, it can be increased (corresponding to a decrease of 𝐾c).


Proportional band

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Integrator windup

Usually controllers are tuned for stability and performance, not for signal limits. Therefore, it is not uncommon that a control signal reaches a constraint. If the controller contains integral action, this can be very damaging to the control performance unless the situation is handled properly.

Consider the figure, where the PI control law (7.25) has been used. A strong disturbance causes the process output to fall well below the set-point. The controller is not able to elimi-nate the control error (A) because thecontrol signal has reached a constraint.During this time, the positive control errorwill increase the integral in the controller.If the disturbance later disappears, thecontroller will still keep the control signalat the constraint due to the large value ofthe integral, even if the control error goesbelow zero. This will cause the output (B),which is entirely due to the controller. Illustration of integral windup.


7.3.3 PB and integrator windup

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The described phenomenon is called integral windup (also reset windup).

There are sophisticated as well as simple methods for handling the problem. The term anti-windup is used for such arrangements.

A simple solution is to stop integrating when a control signal reaches a constraint. This requires that

it is known when the control signal reaches a constraint (e.g., through measurement)

there is some built-in logic to interrupt the integration

In the case of digital control, which nowadays is customary, automatic anti-windup can be built into the control law.


Integral windup

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7.3.4 Design specifications

Above, some general performance criteria and fundamental limitations to achievable control performance have been considered.

Here, some ways of making more specific design specifications will introduced.

If a process model is available, the specifications make it possible to calculate controller parameters.

Step-response specifications

It is of often desired that the closed-loop response to a step change in the setpoint resembles an underdamped second-order system. Therefore, parameters familiar from the step-response of such a system can be used to specify the desired behaviour. Such parameters are

the maximum relative overshoot 𝑀

the rise time 𝑡r the settling time 𝑡𝛿 the relative damping 𝜁

the ratio between successive relative overshoots (or undershoots) 𝑀R



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According to the relationships in Section 5.3.3:

The two parameters 𝑀 and 𝑡r are sufficient to determine the transfer function of an underdamped second-order system with a given gain.

The settling time 𝑡𝛿 can be used instead of 𝑀 or 𝑡r , but the relationships are then only approximate.

The relative damping 𝜁 or the overshoot ratio 𝑀R can be specified instead of 𝑀.

Some classical tuning recommendations are based on the specification 𝑀R = 1/4.

This may be acceptable for regulatory control, but not for setpoint tracking. 𝑀R = 1/4 corresponds to 𝑀 = 0.5 (i.e., a 50 % overshoot) and 𝜁 = 0.22 .

For setpoint tracking, 𝑀 ≈ 0.1 (𝜁 ≈ 0.6) is usually more appropriate.

If an overdamped closed-loop response is desired, this cannot be achieved with a specification 𝜁 > 1 , because the other parameters require an underdamped system. Instead, the closed-loop transfer function can be directly specified and controller parameters calculated by direct synthesis (Section 7.7), for example.


Step-response specifications

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Error integrals

In principle, a small overshoot, rise time and settling time are desired. In practice, the overshoot and settling time will increase with decreasing rise time, and vice versa. Therefore, compromises have to be made.

One way of solving this problem in an optimal way is to specify some error integral to be minimized. Examples of such error integrals are

𝐽IAE = 0𝑡s 𝑒(𝑡) d𝑡 , 𝐽ISE = 0

𝑡s 𝑒(𝑡)2 d𝑡

𝐽ITAE = 0𝑡s 𝑡 𝑒(𝑡) d𝑡 , 𝐽ITSE = 0

𝑡s 𝑡𝑒(𝑡)2 d𝑡(7.31)

where the acronyms are

– IAE = “integrated absolute error”

– ISE = “integrated square error”

– ITAE = “integrated time-weighted absolute error”

– ITSE = “integrated time-weighted square error”

The weighting with time forces the control error towards zero as time in-creases. In principle, the integration time should be infinite, but because the minimization has to be done numerically, a finite 𝑡s has to be used.



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It is of interest to consider how the error integrals relate to step-response specifications when the controlled system is of second order, i.e.,

𝐺 𝑠 =𝜔n2

𝑠2+2𝜁𝜔n𝑠+𝜔n2 (7.32)

In the figure, IAE and ISE are normalized with 𝜔n , ITAE and ITSE with 𝜔n2.

As can be seen, every normalized error integral has a minimum for a given relative damping 𝜁 .This damping as well as thecorresponding relative over-shoot 𝑀 are shown below.

Table 7.1 Optimal relativedamping for 2nd order system.

Error integrals as function of 𝜁.


Error integrals

Error integral M (%)

ISE 0.50 16.3

ITSE 0.59 10.1

IAE 0.66 6.3

ITAE 0.75 2.8

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7.4 Tuning based on frequency response

An ideal PID controller of interactiveform can be tuned experimentallyby making closed-loop control experi-ments with the real process. Thestandard feedback structure is used.

1. A P controller (𝐺c = 𝐾c) is used for the first experiment. A low value is chosen for the gain 𝐾c . Note that 𝐾c must have the same sign as 𝐾p .

2. A change in the setpoint 𝑅 is introduced. (Some other disturbance could also be used.) The controller gain 𝐾c is increased until the output 𝑌 starts to oscillate with a constant amplitude (see next slide).

3. The value of the controller gain yielding constant oscillations is denoted 𝐾c,max . The period of the oscillations is denoted 𝑃c .

4. The controller gain is changed to 𝐾c = 0.5𝐾c,max . If the intention was to tune a P controller, this is the final tuning.


7.4.1 Experimental tuning

G

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5. To tune a controller with integralaction (PI or PID), an experimentis done with a PI controller using𝐾c = 0.5𝐾c,max . A large value isinitially used for the integral time 𝑇i .

6. A change in the setpoint 𝑅 (or someother disturbance) is introduced. Theintegral time 𝑇i is reduced until 𝑌starts to oscillate with a constantamplitude. This occurs at 𝑇i = 𝑇i,min .

7. The integral time for a PI or PID controller is chosen as 𝑇i = 3𝑇i,min .

8. To tune the derivative part of a PID (or PD) controller, an experiment is done with such a controller using 𝐾c = 0.5𝐾c,max , 𝑇i = 3𝑇i,min (if a PID controller). The derivative time is initially set at 𝑇d = 0 .

9. A change in the setpoint 𝑅 (or some other disturbance) is introduced.The derivative time 𝑇d is increased until the output 𝑌 starts to oscillate with a constant amplitude. This occurs when 𝑇d = 𝑇d,max.

10. The derivative time for a PD or PID controller is set at 𝑇d =1

3𝑇d,max .



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If the control performance obtained by the above tunings turns out to be unsatisfactory, the controller parameters can be adjusted by “trial and error”.

The next figure shows how changes of the controller gain 𝐾c and the integral time 𝑇i typically affect the control performance. The optimal performance is in this case obtained by 𝐾c = 3 and 𝑇i = 11 .

𝑇i = 5 𝑇i = 11 𝑇i = 20

𝐾c = 5

𝐾c = 3

𝐾c = 1

ProcessControl

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7. PID Controllers

7.4.2 Ziegler-Nichols’s recommendations

In 1942, Ziegler and Nichols suggested tunings for P, PI and PID controllers based on 𝐾c,max and 𝑃c only. To obtain this information, it is sufficient to do steps 1–3 in the experimental procedure.

The tunings are primarily intended for regulatory control (i.e., disturbance rejection). For setpoint tracking, setpoint weighting is suggested, e.g. 𝑏 = 0.5.

The controller tuning should Table 7.2. Ziegler-Nichols’s controllerpreferably not be used out- tuning recommendations based onside the range 0.1 < 𝜅 < 0.5, frequency response (0.1 < 𝜅 < 0.5).where

𝜅 = 𝐾𝐾c,max−1

.

𝐾 is the process gain.

The critical frequency 𝜔c isoften used instead of 𝑃c:

𝜔c = 2𝜋/𝑃c.



Controller

c c,max/K K i c/T P d c/T P

P 0.5 – –

PI 0.45 0.8 –

PID 0.6 0.5 0.125

ProcessControl

Laboratory

7. PID Controllers

7.4.3 Åström’s and Hägglund’s correlations

In 2006, Åström and Hägglund showed that, in general, 𝐾c,max and 𝑃calone do not provide sufficient information for good controller tuning.

In addition to 𝐾c,max and 𝑃c , Åström and Hägglund also use the

parameter 𝜅 = 𝐾𝐾c,max−1

in their controller tuning correlations.

The tuning correlations are primarily intended for regulatory control; for setpoint tracking, setpoint weighting is suggested.

The correlations should Table 7.3. Åström-Hägglund’s controllernot be used below the tuning correlations based on frequencyrange 𝜅 > 0.1 . Response (𝜅 > 0.1).Large time delays areallowed, but clearlyunderdamped systemsare less suitable.



Controller c c,max/K K i c/T P d c/T P

PI 0.16 1(1 4.5 ) –

PID 40.3 0.1 0.6

1 2

0.15(1 )

1 0.95

ProcessControl

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7. PID Controllers

7.5 Tuning based on step responseA drawback with generating the frequency response is that it is quite cumbersome and time-consuming to generate oscillations with constant amplitude by adjusting a controller parameter.

An alternative is to use a step response for the process.

The figure illustrates how theneeded parameters are obtainedfrom a unit-step response, i.e., astep with size 𝑢step = 1 expressedin the units used for the controlvariable.

The method is based on the(modified) tangent method, buthere it is not necessary to waitfor the new steady state; onlythe parameters 𝑎 and 𝐿 need Characteristic parameters from ato be determined. monotonous unit-step response.


L

iy

it

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7. PID Controllers


7.5 Tuning based on step response

Instead of taking the 𝑎 parameter from the point, where the tangent through the inflexion point (i.e., the point where the slope is highest) of the step response crosses the vertical axis, it can be calculated when the coordinates (𝑡i, 𝑦i) of the inflexion point are known. The calculation is valid for any size of 𝑢step . The formula for 𝑎 is

𝑎 =𝐿𝑦i

𝑢step(𝑡i−𝐿)(7.34)

Another useful parameter is

𝜃 = 𝐿/𝑇eq , 𝑇eq = 𝑡63 − 𝐿 (7.35)

where 𝑇eq is the equivalent time constant of the system and 𝑡63 is the time it takes to reach 63% of the total output change.

The step response of a purely integrating system is a ramp that changes linearly with time, i.e., it has a constant slope. Any point on the ramp can then be used as a pair of coordinates (𝑡i, 𝑦i) for calculation of 𝑎according to (7.34).

caKi /T Td /T LcaKi /T Td /T L

ProcessControl

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7. PID Controllers

7.5.1 Ziegler-Nichols’s recommendations

In 1942, Ziegler and Nichols also suggested tunings for P, PI and PID controllers based on the information that can be obtained from a step test. Their recommendations for an ideal controller are given in Table 7.4.

The method requires 𝐿 > 0 and preferably 0.1 ≤ 𝜃 ≤ 1.

Table 7.4. Ziegler-Nichols’s controller tuningrecommendations based on step response.

Note that Ziegler-Nichols’s recommendations based on frequency response and step response do not necessarily give the same controller tuning for the same process.KEH Process Dynamics and Control 7–35


Controller caK i /T L d /T L

P 1.0 – –

PI 0.9 3 –

PID 1.2 2 0.5

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7. PID Controllers

7.5.2 The CHR method

In 1952, Chien, Hrones and Reswick suggested improvements to Ziegler’s and Nichols’s recommendations based on a step response. The CHR-method gives

different tunings for regulatory control and setpoint tracking

tunings for aggressive control (with ~20 % overshoot) and cautious control (no overshoot)

The method requires 𝐿 > 0 and preferably 0.1 ≤ 𝜃 ≤ 1.

The CHR tunings (even the aggressive one) are less aggressive than the ZN tuning.

Note that the different tunings for regulatory control and setpoint tracking can directly be used in a 2DOF controller.



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Table 7.5. Controller tuning for regulatory control by the CHR method.

Table 7.6. Controller tuning for setpoint tracking by the CHR method.


7.5.2 The CHR method

Controller No overshoot 20 % overshoot

caK i /T L d /T L caK i /T L d /T L

P 0.3 – – 0.7 – –

PI 0.6 4.0 – 0.7 2.3 –

PID 0.95 2.4 0.42 1.2 2.0 0.42

Controller No overshoot 20 % overshoot

caK i /T T d /T L caK i /T T d /T L

P 0,3 – – 0,7 – –

PI 0,35 1,2 – 0,6 1,0 –

PID 0,6 1,0 0,5 0,95 1,4 0,47

i eq/T T i eq/T T

ProcessControl

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7. PID Controllers

7.5.3 Åström’s and Hägglund’s correlations

In 2006, Åström and Hägglund presented improved controller tunings based on a step response. In addition to 𝑎 and 𝐿 , they use 𝜃 in their correlations, which can be used for all 𝜃 ≥ 0. However, for 𝜃 < 0.4 , they tend to be conservative. For an integrating process, 𝜃 = 0 is used.

The tunings are primarily intended for regulatory control. For setpoint tracking, setpoint weighting can be used as follows:

PI control: 𝑏 = 1 if 𝜃 > 0.4 , 𝑏 < 1 if 𝜃 ≤ 0.4 (optimal 𝑏 is unclear)

PID control: 𝑏 = 1 if 𝜃 > 1 , 𝑏 = 0 if 𝜃 ≤ 1

Table 7.7. Åström’s and Hägglund’s controller tuning correlations.



Controller caK i /T L d /T L

PI 20.35 0.15

(1 )

2

130.35

1 12 7

–

PID 0.45 0.2 8 4

1 10

0.5

1 0.3

ProcessControl

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7. PID Controllers

7.6 Model-based controller tuningThe controller tuning methods in Sections 7.4 and 7.5 employ parameters that can be determined from an experiment with an existing process.

If a process model is known, the same parameters can be determined

through a simulation experiment

possibly by direct calculation from the process model

For example, a first-order system with a time delay has the transfer function

𝐺 𝑠 =𝐾

𝑇𝑠+1e−𝐿𝑠 (7.36)

from which the parameters 𝑎 and 𝜃 can be calculated according to

𝑎 =𝐾𝐿

𝑇, 𝜃 =

𝐿

𝑇(7.37)

The same tuning methods as in Sections 7.4 and 7.5 can then be used.

However, the methods in Sections 7.4 and 7.5 are “general purpose”methods that are not optimized for any specific model type.

For a given model, better controller tunings probably exist.


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7. PID Controllers

7.6.1 First-order system with a time delay

The transfer function is defined in (7.36) and the parameter 𝜃 in (7.37).

Minimization of error integrals

Controller tunings that minimize IAE and ITAE when 0.1 ≤ 𝜃 ≤ 1.

Table 7.8. IAE and ITAE minimizing controller tunings for regulatory control.

Table 7.9. IAE and ITAE minimizing controller tunings for setpoint tracking.



Error

integral

P controller PI controller PID controller

cKK cKK i /T T

cKK i /T T d /T T

IAE 0.9850.902 0.9860.984 0.7071.645 0.9211.435 0.7491.139 1.1370.482

ITAE 1.0840.490 0.9770.859 0.6801.484 0.9471.357 0.7381.188 0.9950.381

Error

integral

PI controller PID controller

cKK i /T T

cKK i /T T d /T T

IAE 0.8610.758 1(1.020 0.323 ) 0.8691.086 1(0.740 0.130 ) 0.9140.348

ITAE 0.9160.586 1(1.030 0.165 ) 0.8550.965 1(0.796 0.147 ) 0.9290.308

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Other optimality criteria

The controller tunings for minimizing the error integrals IAE and ITAE in Tables 7.8 and 7.9 do not give any robustness guarantees. Thus, the control performance can be bad if the model contains errors.

Cvejn (2009) has derived controller tunings that have a certain robustnesseven for systems with large time delays, i.e., for large 𝜃 values.

Table 7.10. Cvejn’s tunings for regulatory control and setpoint tracking.

The PI controller tunings tend to give better robustness than the PID controller tunings, which tend to give better performance.


7.6.1 First-order system with time delay

Control

PI controller PID controller

cKK i /T T

cKK i /T T d /T T

Regulatory 1

2

5.92

1 5.92

3.26

4

3.91

1 3.91 3

3.26

Tracking 1

2 1

3

4

13

3

ProcessControl

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7. PID Controllers

7.6.2 Second-order no-zero system with a time delay

We shall consider second-order systems with a time delay but no zeros. Such a system has the transfer function

𝐺 𝑠 =𝐾𝜔n

2

𝑠2+2𝜁𝜔n𝑠+𝜔n2 e

−𝐿𝑠 (7.40)

In Cvejn’s method for tracking control, the controller 𝐺c(𝑠) is tuned to give the loop transfer 𝐺k(𝑠) = 𝐺(𝑠)𝐺c(𝑠) such that

𝐺k 𝑠 =1

2𝐿𝑠e−𝐿𝑠 (7.38)

or

𝐺k 𝑠 =1

41 +

3

𝐿𝑠e−𝐿𝑠 (7.39)

Tuning by (7.38) gives better stability, (7.39) gives better performance.

Exercise 7.3

Use Cvejn’s method for tracking control to tune a PID controller for the system (7.40).



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Overdamped system without zeros

For an overdamped (or critically damped) second-order system, 𝜁 ≥ 1. In this case, (7.40) is more conveniently written as

𝐺 𝑠 =𝐾

(𝑇1𝑠+1)(𝑇2𝑠+1)e−𝐿𝑠, 𝑇1 ≥ 𝑇2 (7.41)

Cvejn’s method can be used also in this case, but Åström and Hägglund(2006) suggest the following tuning when the system is overdamped:

𝐾𝐾c = 0.19 + 0.37𝜃1−1 + 0.18𝜃2

−1 + 0.02𝜃1−1𝜃2

−1

𝐾𝐾c𝐿/𝑇i = 0.48 + 0.03𝜃1−1 − 0.0007𝜃2

−1 + 0.0012𝜃1−1𝜃2

−1 (7.42)

𝐾𝐾c𝑇d/𝐿 = 0.29 + 0.16𝜃1−1 − 0.2𝜃2

−1 + 0.28𝜃1−1𝜃2

−1 𝜃1+𝜃2

𝜃1+𝜃2+𝜃1𝜃2

where𝜃1 = 𝐿/𝑇1 , 𝜃2 = 𝐿/𝑇2 (7.43)


7.6.2 Second-order system with delay

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Second-order system including integration

A second-order no-zero system including an integrator has the transfer function

𝐺 𝑠 =𝐾

𝑠 (𝑇2𝑠+1)e−𝐿𝑠 (7.44)

For this kind of system, Åström and Hägglund (2006) suggest the tuning:

𝐾𝐾c𝐿 = 0.37 + 0.02𝜃2−1

𝐾𝐾c𝐿2/𝑇i = 0.03 + 0.0012𝜃2

−1 (7.45)

𝐾𝐾c𝑇d = 0.16 + 0.28𝜃2−1

If the system is a double integrator with the transfer function

𝐺 𝑠 =𝐾

𝑠2e−𝐿𝑠 (7.46)

the suggested tuning is

𝐾𝐾c𝐿2 = 0.02

𝐾𝐾c𝐿3/𝑇i = 0.0012

𝐾𝐾c𝑇d𝐿 = 0.28


Overdamped system

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Second-order system with a zero

An overdamped 2nd order system with a zero has the transfer function

𝐺 𝑠 =𝐾(𝑇3𝑠+1)

(𝑇1𝑠+1)(𝑇2𝑠+1)e−𝐿𝑠 (7.47)

Such a system can often be approximated by a first-order system or a second-order system without a zero (see Section 7.9).

Integrating second-order system with a zero

An IPZ system (1 integrator, 1 pole, 1 zero) has a transfer function

𝐺 𝑠 =𝐾(𝑇3𝑠+1)

𝑠 (𝑇2𝑠+1)e−𝐿𝑠 , 𝑇3 > 𝑇2 > 0 (7.48)

An IPZ system is difficult to approximate by a simpler one, esp. if 𝑇3 ≫ 𝑇2.

In Table 7.11, Table 7.11 Slätteke’s regulatory tuning for an IPZ process.𝜃2 = 𝐿/𝑇2. ForPID control, aderivative filter𝑇f = 0.1𝑇d isused. For set-point tracking,𝑏 < 1 is used.



Controller 3 cT KK i /T L d 2/T T

PI 120.0767(3 1)

2

2

100 17

11 94

–

PID 120.115(3 1)

22 2

22 2

835 842 277

3(55 386 241 )

22 2

22 2

3 176 736

500(1 2 )

ProcessControl

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7. PID Controllers

7.7 Controller design by direct synthesisIn the previous sections, equations for controller tuning have been given for first- and second-order no-zero systems.

The equations are usually the result of optimization of some criterion that is considered to imply “good control”.

However, what is “good control” varies from case to case depending on the compromise between stability and performance.

A drawback of the tuning equations is that the user cannot influence the tuning according to his/her opinion of “good control”.

In this section, a method is introduced whereby

the user can influence the controller tuning in a systematic way according to his/her opinion of “good control”

more model types than in previous sections can be handled, e.g., systems with a zero


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7. PID Controllers

7.7.1 Closed-loop transfer functions

Consider the closed-loopsystem in the figure with thefollowing transfer functions:– 𝐺 𝑠 process being controlled– 𝐺c 𝑠 controller– 𝐺d 𝑠 disturbance system Block diagram of closed-loop system

Standard block diagram algebra gives

𝑌 =𝐺𝐺c

1+𝐺𝐺c𝑅 +

𝐺d

1+𝐺𝐺c𝑉 (7.49)

where

𝐺r =𝐺𝐺c

1+𝐺𝐺c, 𝐺v =

𝐺d

1+𝐺𝐺c(7.50,51)

are the closed-loop transfer functions from the setpoint 𝑅 and the disturbance 𝑉 to the output 𝑌.

The user can specify the desired 𝐺r for setpoint tracking or 𝐺v for regu-latory control. For setpoint tracking, the required controller is given by

𝐺c =1

𝐺

𝐺r

(1−𝐺r)(7.52)


7.7 Controller tuning by direct synthesis

( )Y s

( )V s

c ( )G s( )R s

( )G s

d ( )G s

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7. PID Controllers

7.7.2 Low-order minimum-phase systems

First-order system

A strictly proper first-order system without a time delay has the transfer function

𝐺 =𝐾

𝑇𝑠+1(7.53)

Assume that we want the controlled system to behave as a first-order system with the time constant 𝑇r . Then,

𝐺r =1

𝑇r𝑠+1, which gives

𝐺r

1−𝐺r=

1

𝑇r𝑠(7.54)

Substitution of (7.53) and (7.54) into (7.52) gives

𝐺c =𝑇𝑠+1

𝐾

1

𝑇r𝑠=

𝑇

𝐾𝑇r1 +

1

𝑇𝑠(7.55)

which is a PI controller with the parameters

𝐾c =𝑇

𝐾𝑇r, 𝑇i = 𝑇 (7.56)

Here, 𝑇r is a design parameter, by which the performance of the control system can be affected.



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Second-order system with no zero

A second-order system with no zero and no time delay has the transfer function

𝐺 𝑠 =𝐾𝜔n

2

𝑠2+2𝜁𝜔n𝑠+𝜔n2 (7.57)

Even if the uncontrolled system is of second order, we can specify the controlled system to be of first order. Substitution of (7.54) and (7.57) into (7.52) then gives

𝐺c =𝑠2+2𝜁𝜔n𝑠+𝜔n

2

𝐾𝜔n2

1

𝑇r𝑠=

2𝜁

𝐾𝜔n𝑇r1 +

𝜔n

2𝜁𝑠+

𝑠

2𝜁𝜔n(7.58)

which is an ideal PID controller with the parameters

𝐾c =2𝜁

𝐾𝜔n𝑇r, 𝑇i =

2𝜁

𝜔n, 𝑇d =

1

2𝜁𝜔n(7.59)

Also here, 𝑇r is a design parameter which only affects the controller gain.


7.7.2 Low-order min-phase systems

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Overdamped second-order system with a LHP zero

An overdamped second-order system with a zero in the left half of the complex plane (LHP) has the transfer function

𝐺 𝑠 =𝐾(𝑇3𝑠+1)

(𝑇1𝑠+1)(𝑇2𝑠+1), 𝑇𝑖 ≥ 0 (7.60)

We can specify the controlled system to be of first order. Substitution of (7.54) and (7.60) into (7.52) gives

𝐺c =(𝑇1𝑠+1)(𝑇2𝑠+1)

𝐾(𝑇3𝑠+1)

1

𝑇r𝑠=

1

𝐾𝑇r𝑠

𝑇1𝑇2𝑠2+ 𝑇1+𝑇2 𝑠+1

𝑇3𝑠+1

=1

𝐾𝑇r𝑠1 + 𝑇1 + 𝑇2 − 𝑇3 𝑠 +

𝑇1𝑇2− 𝑇1+𝑇2−𝑇3 𝑇3

𝑇3𝑠+1𝑠2

or

𝐺c = 𝐾c 1 +1

𝑇i𝑠+

𝑇d𝑠

𝑇f𝑠+1(7.61)

where

𝐾c =𝑇1+𝑇2−𝑇3

𝐾𝑇r, 𝑇i = 𝑇1 + 𝑇2 − 𝑇3 , 𝑇d =

𝑇1𝑇2

𝑇1+𝑇2−𝑇3− 𝑇3 , 𝑇f = 𝑇3 (7.62)

This is a PID controller with a derivative filter.


7.7.2 Low-order min-phase systems

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7. PID Controllers

7.7.3 High-order minimum-phase systems

A high-order minimum-phase system with real poles and zeros, but with no time delay, has the transfer function

𝐺 = 𝐾 𝑗=𝑛+1𝑛+𝑚 (𝑇𝑗𝑠+1)

𝑖=1𝑛 (𝑇𝑖𝑠+1)

, 𝑇𝑖 > 0 , 𝑇𝑗 > 0 , 𝑛 > 2 (7.63)

If 𝑛 = 3 and 𝑚 = 0 or 1 , a closed-loop system of second order can be obtained by a full PID controller.

If 𝑛 > 3, it is not possible to obtain a closed-loop system of lower order than 3 by a PID controller and an exact design by specifying 𝐺ris thus not practical.

In the case of 𝑛 > 3 , two possibilities are to specify a closed-loop system of first or second order and then to

first calculate a 𝐺c according to (7.52), then to approximate 𝐺c by a PID controller;

first approximate 𝐺 by a model of at most third order, then to calculate the PID controller according to (7.52).

In Section 7.9, the latter approach will be described.



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7. PID Controllers

7.7.4 Second-order system with RHP zero

A second-order system with real poles and a right half plane (RHP) zero has the transfer function

𝐺 𝑠 =𝐾(−𝑇3𝑠+1)

(𝑇1𝑠+1)(𝑇2𝑠+1), 𝑇𝑖 ≥ 0 (7.71)

Now division by 𝐺 in (7.52) will result in an unstable controller with a RHP pole if 𝐺r is chosen as in the previous sections.

One possible solution is to approximate the unstable controller by a stable controller. This tends to result in too aggressive control because the controller is then designed as if there were no RHP zero in 𝐺 .

Another solution is to include the same RHP zero in 𝐺r as in 𝐺 ; it will then be cancelled out in (7.52) and the controller will automatically be stable. This means that the choice of 𝐺r is restricted, but otherwise the control performance tends to be as expected.

In this section, the latter approach is used.



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Closed-loop system of first order

The closed-loop transfer function is chosen as

𝐺r =−𝑇3𝑠+1

𝑇r𝑠+1, which gives

𝐺r

1−𝐺r=

−𝑇3𝑠+1

(𝑇r+𝑇3)𝑠(7.72)


𝐺c =(𝑇1𝑠+1)(𝑇2𝑠+1)

𝐾

1

(𝑇r+𝑇3)𝑠=

𝑇1+𝑇2

𝐾(𝑇r+𝑇3)1 +

1

𝑇1+𝑇2 𝑠+

𝑇1𝑇2𝑠

𝑇1+𝑇2(7.73)

which is a PID controller with the parameters

𝐾c =𝑇1+𝑇2

𝐾(𝑇r+𝑇3), 𝑇i = 𝑇1 + 𝑇2 , 𝑇d =

𝑇1𝑇2

𝑇1+𝑇2(7.74)


7.7.4 System with RHP zero

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Laboratory


Closed-loop system of second order

A first-order system with a zero is proper, but not strictly proper. If a zero is present, a strictly proper system has to be at least second order. Hence, a more natural choice for 𝐺r is

𝐺r =(−𝑇3𝑠+1)𝜔r

2

𝑠2+2𝜁r𝜔r𝑠+𝜔r2 , which gives

𝐺r

1−𝐺r=

(−𝑇3𝑠+1)𝜔r2

𝑠(𝑠+2𝜁r𝜔r+𝑇3𝜔r2)

(7.75)

To simplify the derivation of controller parameters, we define

𝑇f = 1/(2𝜁r𝜔r + 𝑇3𝜔r2) (7.76)

Substitution of (7.71) and (7.75) into (7.52), gives with (7.76)

𝐺c =(𝑇1𝑠+1)(𝑇2𝑠+1)𝑇f𝜔r

2

𝐾 𝑇f𝑠+1 𝑠=

𝑇f𝜔r2

𝐾𝑠

𝑇1𝑇2𝑠2+ 𝑇1+𝑇2 𝑠+1

𝑇f𝑠+1(7.77)

Analogously with the derivation of (7.62), this gives the PID controller parameters (7.76) and

𝐾c =𝑇f𝜔r

2

𝐾(𝑇1 + 𝑇2 − 𝑇f), 𝑇i = 𝑇1 + 𝑇2 − 𝑇f , 𝑇d =

𝑇1𝑇2

𝑇1+𝑇2−𝑇f− 𝑇f (7.78)

where 𝑇f is the derivative filter time constant in a PID controller (7.61).


7.7.4 System with RHP zero

ProcessControl

Laboratory

7.7.4 Second-order system with RHP zero

Choice of closed-loop system parameters

In (7.75), there are two design parameters, the relative damping 𝜁r, and the undamped natural frequency 𝜔r. The meanings of these parameters are discussed in Section 5.3, especially Subsection 5.3.3.

The choice of design parameters can be simplified in the following two ways.

Let 𝐺r have two equally large real poles at −1/𝑇r . This corresponds to 𝜁r = 1 and 𝜔r = 1/𝑇r , which for (7.76) gives

𝑇f =𝑇r2

2𝑇r+𝑇3

Let 𝐺r have real poles at −1/𝑇r and −1/𝑇3 . This corresponds to

𝜁r = 0.5(𝑇r + 𝑇3)𝜔r and 𝜔r = 1/ 𝑇r𝑇3 , which for (7.76) gives

𝑇f =𝑇r𝑇3

𝑇r+2𝑇3


Closed-loop system of 2nd order

ProcessControl

Laboratory

7. PID Controllers

7.7.5 First-order system with a time delay

To illustrate how systems with a time delay can be handled by direct synthesis, a first-order system with a time delay will be studied. Such a system has the transfer function

𝐺 𝑠 =𝐾

𝑇𝑠+1e−𝐿𝑠 (7.79)

Calculation of a controller by (7.52) will then result in a controller containing a time delay — there is no practical way to avoid this by the choice of 𝐺r.

There are methods to implement a controller resulting from (7.52) (see Section 7.8), but not by a regular PID controller.

If a PID controller is desired, the time delay has to be approximated in some way.



ProcessControl

Laboratory


Time-delay approximation in the model

A standard way of approximating a time delay is to use a Padé approxi-mation. I first-order Padé approximation

e−𝐿𝑠 ≈1−0.5𝐿𝑠

1+0.5𝐿𝑠(7.80)

gives the model

𝐺 𝑠 =𝐾(−0.5𝐿𝑠+1)

(𝑇𝑠+1)(0.5𝐿𝑠+1)(7.81)

A natural choice for 𝐺r is then

𝐺r =−0.5𝐿𝑠+1

(𝑇r𝑠+1)(0.5𝐿𝑠+1), which gives

𝐺r

1−𝐺r=

−0.5𝐿𝑠+1

𝑠(0.5𝑇r𝐿𝑠+𝑇r+𝐿)(7.82)

Substitution of (7.81) and (7.82) into (7.52) gives a PID controller with the parameters

𝐾c =𝑇+0.5𝐿−𝑇f

𝐾(𝑇r+𝐿), 𝑇i = 𝑇 + 0.5𝐿 − 𝑇f , 𝑇d =

0.5𝐿𝑇

𝑇+0.5𝐿−𝑇f, 𝑇f =

0.5𝐿𝑇r

𝑇r+𝐿(7.83)

Here, 𝑇f is the time constant of a derivative filter in the PID controller (7.61).


7.7.5 1st order system with a delay

ProcessControl

Laboratory


Time-delay approximation in the controller

If e−𝐿𝑠 is retained in the model, it also has to be part of 𝐺r , because it is impossible for the closed-loop system to have a shorter time-delay than the uncontrolled system.

If 𝐺r is chosen to be first order with a time delay

𝐺r =1

𝑇r𝑠+1e−𝐿𝑠 , which gives

𝐺r

1−𝐺r=

e−𝐿𝑠

𝑇r𝑠+1−e−𝐿𝑠 (7.84)


𝐺c =𝑇𝑠+1

𝐾(𝑇r𝑠+1−e−𝐿𝑠)

(7.85)

Unfortunately, this controller cannot be implemented by a PID controller in a regular feedback loop. In order to do that, the time delay in (7.85) has to be approximated by a rational expression.

If the approximation (7.80) is used, the controller parameters will be as in (7.83).

The simpler approximation e−𝐿𝑠 ≈ 1 − 𝐿𝑠 gives a PI controller with

𝐾c =𝑇

𝐾(𝑇r+𝐿), 𝑇i = 𝑇 (7.86)


7.7.5 1st order system with a delay

ProcessControl

Laboratory

7. PID Controllers

7.8 Internal model control“Internal model control” (IMC) is closely related to “direct synthesis” (DS). As in DS, a model of the system to be controlled is explicitly built into the controller, but in a different way.

An advantage with IMC is that it is easier to implement more complex control laws than regular PID controllers. For example, the controller transfer function (7.85) can easily be implemented exactly with IMC.

Even if the controller design is based on IMC, it is often desirable to implement the controller as a regular PID controller. In such cases, the IMC approach offers better possibilities to deal with robustness issuesthan DS.


ProcessControl

Laboratory

7. PID Controllers

7.8.1 The IMC structure

Consider the closed-loopsystem in the figure with thefollowing transfer functions:– 𝐺 𝑠 true process– 𝐺 𝑠 process model– 𝐺IMC 𝑠 a controller– 𝐺d 𝑠 disturbance system

Standard block diagram algebra The IMC structure.gives 𝑈 = 𝐺IMC(𝐸 + 𝐺𝑈) from which𝑈

𝐸= 𝐺c = 𝐼 − 𝐺IMC

𝐺−1𝐺IMC = 𝐺IMC 𝐼 − 𝐺𝐺IMC

−1=

𝐺IMC

1− 𝐺𝐺IMC(7.87)

Assume that𝐺IMC = 𝐺−1𝐺f (7.88)

where 𝐺f is a “filter”. Substitution of (7.88) into (7.87) gives

𝐺c = 𝐺−1𝐺f 𝐼 − 𝐺f−1 =

1

𝐺

𝐺f

(1−𝐺f)(7.89)

If the filter is chosen as 𝐺f = 𝐺r (and 𝐺 = 𝐺), this is the same as (7.52) !



( )G s

ˆ ( )G s

( )E s

ProcessControl

Laboratory

7. PID Controllers

7.8.2 Handling of time delays without approximation

Consider a system modelled as a first-order system with a time delay, i.e., 𝐺 = 𝐾e−𝐿𝑠/(𝑇𝑠 + 1). Choose the IMC filter as 𝐺f = e−𝐿𝑠/(𝑇r𝑠 + 1) . Substitution into (7.88) now gives

𝐺IMC =1

𝐾

𝑇𝑠+1

𝑇r𝑠+1=

1

𝐾1 +

𝑇−𝑇r

𝑇r𝑠+1𝑠 (7.90)

which is a PD controller with a derivative filter having the parameters 𝐾𝑐 = 1/𝐾 , 𝑇d = 𝑇 − 𝑇r , 𝑇f = 𝑇r . Substitution of (7.90) and the model 𝐺 into (7.87) gives

𝐺c =𝑇𝑠+1

𝐾(𝑇r𝑠+1−e−𝐿𝑠)

(7.91)

which identical with (7.85). The difference is that (7.91) can be implemen-ted exactly with the IMC structure without time-delay approximation.

Note that there is no integration in 𝐺IMC , but the feedback of 𝐺 in the IMC structure introduces integration if 𝐺IMC has been calculated using the same 𝐺 in (7.88); integration is achieved even if 𝐺 ≠ 𝐺 .

Exercise. Calculate the closed-loop transfer function 𝐺r when 𝐺 ≠ 𝐺 . Show that there will be no steady-state error, i.e., that 𝐺r 0 = 1 .



ProcessControl

Laboratory

7. PID Controllers

7.8.3 The predictive character of the IMC structure

The previous block diagram of the IMC structure is drawn to empha-size how 𝐺IMC combined with the feedback of 𝐺 is equivalent to 𝐺c.

The block diagram can also be drawn to emphasize the predictive character of the IMC structure, as shown below. (Note that the two diagrams are completely equivalent.)

– The control signal is an input to the real system 𝐺 and the model 𝐺.

– 𝐺 predicts the output 𝑌, which is compared with the true output 𝑌.

– Only the prediction error 𝐸 = 𝑌 − 𝑌 is fed back, not the entire 𝑌.

The latter property is a clearadvantage in controller design.If 𝐺 = 𝐺 (i.e., 𝐸 = 0)

𝐺r = 𝐺𝐺IMC (7.93)

which means that the closed-loop transfer function dependslinearly on 𝐺IMC making designof 𝐺IMC easier than design of 𝐺c. Predictive nature of IMC structure.



ˆ ( )G s

( )G s

ProcessControl

Laboratory

7. PID Controllers

7.8.4 Controller design

The following conclusions can be drawn from (7.93):

A stable closed-loop system 𝐺r requires a stable IMC controller 𝐺IMC ; in particular, the IMC controller may not contain integral action.

Non-minimum phase properties (i.e., RHP zeros and time delays) in 𝐺will also be present in 𝐺r because they cannot be cancelled out by a stable and realizable 𝐺IMC.

From (7.88) it follows that

the filter 𝐺f has to be chosen to cancel out non-minimum phase prop-erties of 𝐺 — this is equivalent to the choice of 𝐺r in direct synthesis.

In practice, the IMC design is done differently. Instead of guaranteeing the stability and realizability of 𝐺IMC by the choice of 𝐺f , it is handled by the choice of 𝐺 to be inverted: non-minimum phase parts of 𝐺 are not inverted.



ProcessControl

Laboratory


The process model 𝐺 can always be factorized as

𝐺 = 𝐺⊕ 𝐺⊖ (7.94)

where 𝐺⊕ contains all non-minimum-phase elements of 𝐺, but no minimum-phase elements, and normalized so that 𝐺⊕ 0 = 1 (i.e., it has the static gain 1). This means that 𝐺⊕ contains all RHP zeros and time delays of 𝐺 ; if there are no such elements, 𝐺⊕ = 1.

When 𝐺IMC is calculated according to (7.88), only 𝐺⊖ is inverted. Thus,

𝐺IMC = 𝐺⊖ −1𝐺f (7.95)

Note that the full 𝐺 should be used as internal model as illustrated by the IMC block diagrams — the use of 𝐺⊖ is only a technical aid for the calculation of 𝐺IMC .

The IMC filter 𝐺f could be chosen as the desired closed-loop transfer function without any non-minimum phase elements (not even a time delay), but in practice a low-pass filter

𝐺f =1

(𝑇r𝑠+1)𝑛 (7.96)

is chosen. Here, 𝑛 is an integer, usually 𝑛 = 1, sometimes 𝑛 > 1.


7.8.4 Controller design

ProcessControl

Laboratory

7. PID Controllers

7.8.5 Implementation with a regular PID controller

An advantage of the IMC structure is that time delays can be handled exactly, but often a regular PID controller is preferred, because it is standard software in all automation systems.

If an IMC controller 𝐺IMC has been designed, the corresponding “regular” controller 𝐺c can be calculated according to (7.87). If 𝐺 contains a time delay, it will also be present in 𝐺c. In such cases, the time delay has to be approximated in a suitable way.

Table 7.12 shows IMC-based tunings of regular PID controllers for some typical model structures.

The tunings can also be used for models of lower degree or no time delay as long as

𝑇1 > 0 , 𝑇2 ≥ 0 , 𝑇3 ≥ 0 , 𝐿 ≥ 0 (7.101)

The tunings can be used for (underdamped) models expressed by the relative damping and the natural frequency by the substitutions

𝑇1 + 𝑇2 = 2𝜁/𝜔n , 𝑇1𝑇2 = 1/𝜔n2 (7.103)

Usually 𝑇r is chosen such that 𝐿 ≤ 𝑇r < 𝑇 (but no clear consensus).



ProcessControl

Laboratory


Table 7.12. IMC-based tuning of ideal PID controller.

The desired time constant of the close-loop system is 𝑇r . 𝜆 , which is used in the calculations, is closely related to 𝑇r . Note that the calculated integral time 𝑇i is used in several expressions.


7.8.5 Implementation with a PID controller

( )G s cK K iT dT

1

e

1

LsK

T s

i /T 1

1 2T L 1

1 i2/LT T 1

r 2T L

3

1 2

( 1)e

( 1)( 1)

LsK T s

T s T s

i /T 1 2 3T T T 1 2 i 3( / )TT T T rT L

3

1 2

( 1)e

( 1)( 1)

LsK T s

T s T s

i /T 1 2 3( / )T T T L 1 2 i 3( / ) ( / )TT T T L r 3T T L

e LsK

s

2

i /T 2 1 1i2 2

(1 / )L L T 1r 2

T L

2

e

( 1)

LsK

s T s

2

i /T 22 T L 2 2 i(1 / )T T T rT L

ProcessControl

Laboratory

7. PID Controllers

7.9 Model simplificationMany controller tuning methods have been presented in the previous sections.

Section 7.4: Controller tuning based on frequency-response para-meters 𝐾c,max , 𝑃c (or 𝜔c) and 𝜅. These methods are “general-purpose methods” not optimized for any specific model type.

Section 7.5: Controller tuning based on step-response parameters 𝑎 (or 𝑡i, 𝑦i), 𝐿 and 𝜃. These methods are also general-purpose methods not optimized for any specific model type.

Section 7.6: Model-based tuning optimized for given model structures and control criteria with no user interaction.

Section 7.7: Direct synthesis for low-order models according to desired closed-loop response.

Section 7.8: Internal model control mainly for low-order models according to desired closed-loop response.

In this section, methods to reduce high-order models to first- or second-order models are presented. Any controller tuning method can be used.


ProcessControl

Laboratory

7. PID Controllers

7.9.1 Skogestad’s method

Skogestad and Grimholt (2012) have presented a method to simplify a high-order model with real poles and zeros to a first- or second-order model with a time delay but with no zeros.

The transfer function to be simplified is factorized into a minimum-phase part 𝐺⊖ and a non-minimum-phase part 𝐺⊕ , i.e.,

𝐺 𝑠 = 𝐺⊕(𝑠)𝐺⊖(s) (7.106)

Any left-half plane (LHP) zeros of 𝐺⊖(s) and RHP zeros of 𝐺⊕(𝑠) are eliminated by suitable approximations.

Elimination of LHP zeros

If the poles and zeros are real, the minimum-phase part has the form

𝐺⊖ 𝑠 =𝐾 𝑇𝑛+1𝑠+1 𝑇𝑛+2𝑠+1 …(𝑇𝑛+𝑚𝑠+1)

𝑇1𝑠+1 𝑇2𝑠+1 …(𝑇𝑛𝑠+1)(7.107)

where 𝑇1 ≥ 𝑇2 ≥ ⋯ ≥ 𝑇𝑛 > 0, 𝑇𝑛+1 ≥ 𝑇𝑛+2 ≥ ⋯ ≥ 𝑇𝑛+𝑚 > 0 , 𝑛 > 𝑚. The simplification procedure now goes as follows.

The numerator time constants 𝑇𝑛+1, 𝑇𝑛+2, …, 𝑇𝑛+𝑚 are considered in that order. Assume that 𝑇𝑛+𝑗 is the one currently being considered.



ProcessControl

Laboratory


Next, the smallest remaining denominator time constant 𝑇𝑖 such that 𝑇𝑖 ≥ 𝑇𝑛+𝑗 is selected. If there is no such time constant, or if 𝑇𝑖 ≫𝑇𝑛+𝑗, the smaller 𝑇𝑖 closest to 𝑇𝑛+𝑗 is chosen. It is considered that

𝑇𝑖 ≫ 𝑇𝑛+𝑗 if 𝑇𝑖 > 𝑇𝑛+𝑗2 /𝑇𝑖+1 and 𝑇𝑛+𝑗/𝑇𝑖+1 < 1.6 .

The ratio (𝑇𝑛+𝑗𝑠 + 1)/(𝑇𝑖𝑠 + 1) is now approximated as

𝑇𝑛+𝑗𝑠+1

𝑇𝑖𝑠+1≈

𝑇𝑛+𝑗/𝑇𝑖 if 𝑇𝑖 ≥ 𝑇𝑛+𝑗 ≥ 5𝑇r a5𝑇r/𝑇𝑖

5𝑇r−𝑇𝑛+𝑗 𝑠+1if 𝑇𝑖 ≥ 5𝑇r ≥ 𝑇𝑛+𝑗 b

1

𝑇𝑖−𝑇𝑛+𝑗 𝑠+1if 5𝑇r ≥ 𝑇𝑖 ≥ 𝑇𝑛+𝑗 c

𝑇𝑛+𝑗/𝑇𝑖 if 𝑇𝑛+𝑗≥ 𝑇𝑖 ≥ 𝑇r (d)

𝑇𝑛+𝑗/𝑇r if 𝑇𝑛+𝑗≥ 𝑇r ≥ 𝑇𝑖 (e)

1 if 𝑇r ≥ 𝑇𝑛+𝑗 ≥ 𝑇𝑖 (f)

(7.108)

Here, 𝑇r is the desired closed-loop time constant. If this is not known, the suggested value is 𝑇r = 𝐿 , which is the time delay in the simplified model. Since this is not initially known, one may have to iterate (i.e., first guessing 𝐿, then possibly correcting with the new 𝐿).



ProcessControl

Laboratory


The above procedure gives an approximate minimum-phase part 𝐺⊖ of the form

𝐺⊖ 𝑠 = 𝐾

𝑇1𝑠+1 𝑇2𝑠+1 …( 𝑇 𝑛𝑠+1)(7.109)

Note that the gain as well as the values and number of denominator time constants may have changed from the original 𝐺⊖.

Elimination of RHP zeros and the half rule

The transfer function 𝐺 𝑠 = 𝐺⊕(𝑠) 𝐺⊖(s) now has the form

𝐺 𝑠 = 𝐾 −𝑇𝑛+𝑚+1𝑠+1 −𝑇𝑛+𝑚+2𝑠+1 …(−𝑇𝑛+𝑚+𝑝𝑠+1)

𝑇1𝑠+1 𝑇2𝑠+1 …( 𝑇 𝑛𝑠+1)e−𝐿𝑠 (7.110)

where 𝑇1 ≥ 𝑇2 ≥ ⋯ ≥ 𝑇 𝑛 > 0, 𝑇𝑛+𝑚+1 ≥ 𝑇𝑛+𝑚+2 ≥ ⋯ ≥ 𝑇𝑛+𝑚+𝑝 > 0 .

Skogestad’s half rule

If an approximate model of order 𝑛 is desired, the 𝑛 largest denomi-nator time constants are retained in the model with the modification that half of 𝑇 𝑛+1 is added to 𝑇 𝑛. Half of 𝑇 𝑛+1 is also added to the time delay as well as all remaining smaller denominator time constants. In addition, all negative numerator time constants are subtracted from the time delay.



ProcessControl

Laboratory


Approximation by first-order system

If a first-order model is desired, the half rule gives

𝐺 𝑠 = 𝐾

𝑇𝑠+1e− 𝐿𝑠 (7.111a)

𝑇 = 𝑇1 +1

2 𝑇2 , 𝐿 = 𝐿 + 1

2 𝑇2 + 𝑖=3

𝑛 𝑇𝑖 + 𝑗=1𝑝

𝑇𝑛+𝑚+𝑗 (7.111b)

Approximation by second-order system

If a second-order model is desired, the half rule gives

𝐺 𝑠 = 𝐾

( 𝑇1𝑠+1)( 𝑇2𝑠+1)e− 𝐿𝑠 (7.112a)

𝑇2 = 𝑇2 +1

2 𝑇3 , 𝐿 = 𝐿 + 1

2 𝑇3 + 𝑖=4

𝑛 𝑇𝑖 + 𝑗=1𝑝

𝑇𝑛+𝑚+𝑗 (7.112b)


Elimination of RHP zeros and the half rule

ProcessControl

Laboratory


Example 7.2. IMC via model reduction by Skogestad’s method.

Simplify the model

𝐺 𝑠 =(16𝑠+1)(4𝑠+1)(−8𝑠+1)e−2𝑠

(50𝑠+1)(20𝑠+1)(12𝑠+1)(6𝑠+1)(3𝑠+1)(𝑠+1)

to a second-order model by Skogestad’s method and determine the parameters of a PID controller by IMC-based tuning for this model. Use a first-order filter time constant 𝑇r = 10.

Here

𝐺⊖ 𝑠 =(16𝑠+1)(4𝑠+1)

(50𝑠+1)(20𝑠+1)(12𝑠+1)(6𝑠+1)(3𝑠+1)(𝑠+1).

According to (7.108c), 16𝑠+1

20𝑠+1≈

1

4𝑠+1. The numerator factor (4𝑠 + 1) can

now be cancelled out against the new denominator factor, which gives

𝐺⊖ 𝑠 =1

(50𝑠+1)(12𝑠+1)(6𝑠+1)(3𝑠+1)(𝑠+1)

and

𝐺 𝑠 =(−8𝑠+1)e−2𝑠

(50𝑠+1)(12𝑠+1)(6𝑠+1)(3𝑠+1)(𝑠+1).



ProcessControl

Laboratory


The resulting second-order model is

𝐺 𝑠 =1

( 𝑇1𝑠+1)( 𝑇2𝑠+1)e− 𝐿𝑠

with 𝑇1 = 50 , 𝑇2 = 12 + 1

2⋅ 6 = 15 , 𝐿 = 2 + 1

2⋅ 6 + 3 + 1 + 8 = 17.

Thus 𝐺 𝑠 =

1

(50𝑠+1)(15𝑠+1)e−17𝑠 .

According to Table 7.12 for IMC-based tuning of second-order model:

– 𝜆 = 𝑇r + 𝐿 = 10 + 17 = 27

– 𝑇i = 𝑇1 + 𝑇2 = 50 + 15 = 65

– 𝐾c = 𝑇i/( 𝐾𝜆) = 65/(1 ⋅ 27) = 2.4

– 𝑇d = 𝑇1 𝑇2/𝑇i = 50 ⋅ 15/65 = 11.5


Example 7.2

ProcessControl

Laboratory

7. PID Controllers

7.9.2 Isaksson’s and Graebe’s method

Isaksson and Graebe (1999) have presented a method to simplify a high-order model, where the fast and slow dynamics are combined to yield a model with a desired number of poles and zeros. If the original model contains a time delay, it is either left intact or substituted by a Padéapproximation.

To describe the method, both factorized and polynomial forms of the original transfer function are employed. If the numerator order is 𝑚 and the denominator order is 𝑛 , the transfer function is

𝐺 𝑠 = 𝐾𝑇𝑛+1𝑠+1 𝑇𝑛+2𝑠+1 …(𝑇𝑛+𝑚𝑠+1)

𝑇1𝑠+1 𝑇2𝑠+1 …(𝑇𝑛𝑠+1)(7.113a)

= 𝐾𝑏0𝑠

𝑚+⋯+𝑏𝑚−2𝑠2+𝑏𝑚−1𝑠+1

𝑎0𝑠𝑛+⋯+𝑎𝑛−2𝑠

2+𝑎𝑛−1𝑠+1(7.113b)

where 𝑇1 ≥ 𝑇2 ≥ ⋯ ≥ 𝑇𝑛 > 0 (i.e., a stable system) and |𝑇𝑛+1| ≥|𝑇𝑛+2| ≥ ⋯ ≥ |𝑇𝑛+𝑚| . The numerator time constants can be positive or negative.



ProcessControl

Laboratory


If a model with the numerator order 𝑚 and the denominator order 𝑛 is desired, the simplified model is

𝐺 𝑠 = 𝐾𝑇𝑛+1𝑠+1 …(𝑇𝑛+ 𝑚𝑠+1) + 𝑏𝑚− 𝑚𝑠 𝑚+⋯+𝑏𝑚−1𝑠+1

𝑇1𝑠+1 …(𝑇 𝑛𝑠+1) + 𝑎𝑛− 𝑛𝑠 𝑛+⋯+𝑎𝑛−1𝑠+1

(7.114)

Complex-conjugated poles or zeros is no problem, except if they occur as poles number 𝑛 and 𝑛 + 1 or zeros number 𝑛 + 𝑚 and 𝑛 + 𝑚 + 1. One solution is then to use the real part of the complex conjugate as 𝑇 𝑛or 𝑇𝑛+ 𝑚 .

If the model is to be used for controller tuning, a strictly proper first- or second-order model, possibly with a time delay, is usually desired. Then

𝐺 𝑠 =𝐾

1

2𝑇1+𝑎𝑛−1 𝑠+1

(1st order) (7.115)

𝐺 𝑠 =𝐾

1

2𝑇𝑛+1+𝑏𝑚−1 𝑠+1

1

2𝑇1𝑇2+𝑎𝑛−2 𝑠2+

1

2𝑇1+𝑇2+𝑎𝑛−1 𝑠+1

(2nd order) (7.116)

where

𝑏𝑚−1 = 𝑗=1𝑚 𝑇𝑛+𝑗 , 𝑎𝑛−1 = 𝑖=1

𝑛 𝑇𝑖 , 𝑎𝑛−2 =1

2 𝑖=1𝑛 𝑇𝑖

2− 𝑖=1

𝑛 𝑇𝑖2 (7.117)


7.9.2 Isakssons’s and Graebe’s method

ProcessControl

Laboratory


Example 7.3. IMC via model reduction by Isaksson–Graebe’s method.

Solve the same problem as in Example 7.2 by Isaksson’s and Graebe’smodel reduction method.

The model gives

𝑏𝑚−1 = 16 + 4 − 8 = 12 , 𝑎𝑛−1 = 50 + 20 + 12 + 6 + 3 + 1 = 92

𝑎𝑛−2 =1

2922−(502+202+122+62+32+12) = 2687

from which

𝐺 𝑠 =1

216+12 𝑠+1

1

21000+2687 𝑠2+

1

270+92 𝑠+1

e−2𝑠 =(14𝑠+1)e−2𝑠

1843.5𝑠2+81𝑠+1

This model has complex-conjugated poles, but according to (7.103), 𝑇1 +𝑇2 = 81 and 𝑇1𝑇2 = 1843.5 can be used in the controller calculations. Table 7.12 for IMC-based tuning of second-order model then gives

– 𝜆 = 𝑇r + 𝐿 = 10 + 2 = 12

– 𝑇i = 𝑇1 + 𝑇2 − 𝑇3 = 81 − 14 = 67

– 𝐾c = 𝑇i/(𝐾𝜆) = 67/(1 ⋅ 12) = 5.6 (much bigger than in Ex. 7.2!)

– 𝑇d = 𝑇1 𝑇2/𝑇i − 𝑇3 = 1843.5/67 −14 = 13.5


7.9.2 Isakssons’s and Graebe’s method

7. PID Controllers - Åbo · PDF fileControl Laboratory 7. PID Controllers 7.0 Overview 7.1 PID controller variants 7.2 Choice of controller type ... 7.8 Internal model control 7.9

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