Process Laboratory Control 7. PID Controllers Control Laboratory 7. PID Controllers 7.0 Overview 7.1...
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Process Control Laboratory 7. PID Controllers 7.0 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 7β1
Transcript of Process Laboratory Control 7. PID Controllers Control Laboratory 7. PID Controllers 7.0 Overview 7.1...
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Laboratory 7. PID Controllers7.0 Overview7.1 PID controller variants7.2 Choice of controller type7.3 Specifications and performance criteria7.4 Controller tuning based on frequency response7.5 Controller tuning based on step response7.6 Model-based controller tuning7.7 Controller design by direct synthesis7.8 Internal model control7.9 Model simplification
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7. PID Controllers
7.0 OverviewPID controller (βpee-i-deeβ) 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
7.1 PID controller variants
An ideal PID controller is described by the control law
π’π’ π‘π‘ = πΎπΎc ππ π‘π‘ + 1ππiβ«0π‘π‘ ππ ππ dππ + ππd
dππ(π‘π‘)dπ‘π‘
+ π’π’0 (7.1)
π’π’(π‘π‘) is the controller output ππ π‘π‘ = ππ π‘π‘ β π¦π¦(π‘π‘) is the control error, which is the difference between the
setpoint ππ(π‘π‘) and the measured process output π¦π¦(π‘π‘) πΎπΎc is the proportional gain ππi is the integral time ππd is the derivative time π’π’0 is the βnormalβ value of the controller output
The transfer function of the PID controller is
πΊπΊPID = ππ(π π )πΈπΈ(π π )
= πΎπΎc 1 + 1ππiπ π
+ ππdπ π = πΎπΎcππiπ π
1 + ππiπ π + ππiππdπ π 2 (7.2)
ππ(π π ) is the Laplace transform of π’π’ π‘π‘ β π’π’0 πΈπΈ(π π ) is the Laplace transform of the control error
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7.1.1 Ideal PID controller
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7.1 PID controller variants
Depending on the values of ππi and ππd, 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 ππd = 0. Its transfer function is
πΊπΊPI = πΎπΎc 1 + 1ππiπ π
= πΎπΎcππiπ π
1 + ππiπ π (7.3)
A PD controller is obtained from a PID controller by letting ππi = β. Its transfer function is
πΊπΊPD = πΎπΎc 1 + ππdπ π (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. PID Controllers
7.1.2 The series form of a PID controllerIn 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
πΊπΊPIPD = πΎπΎcβ² 1 + 1ππiβ²π π
1 + ππdβ²π π = πΎπΎcβ²
ππiβ²π π
1 + ππiβ²π π 1 + ππdβ²π π (7.5)
where β² 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 relatively easy to find the controller parameters of the series form by
frequency analytic methods by so-called lead-lag design.
Exercise 7.1Which 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. PID Controllers
7.1.3 A PID controller with derivative filterA 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 ππ(π‘π‘)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
πΊπΊPIDf = πΎπΎc 1 + 1ππiπ π
+ ππdπ π ππfπ π +1
(7.6)
The transfer function of a series form PID controller with a derivative filter is usually stated in the form
πΊπΊPIPDf = πΎπΎcβ² 1 + 1ππiβ²π π
ππdβ²π π +1
ππfβ²π π +1
(7.7)
ππf and ππfβ² are filter constants, usually 10-30 % of corresponding derivative time.
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7.1 PID controller variants
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7.1 PID controller variants
Relationships between parallel and series formIf the parameters of the series form are known, the corresponding parameters of the parallel form can be calculated according to
ππi = ππiβ² + ππdβ² β ππfβ² , ππd = ππdβ²
ππiβ²
ππiβ ππf
β² , ππf = ππfβ² , πΎπΎc = πΎπΎcβ²
ππiβ²
ππi(7.8)
For calculation of the parameters of the series form from the parameters of the parallel form, we define the parameter
πΏπΏ = 1 β 4ππi(ππd+ππf)(ππi+ππf)2
(7.9)
If πΏπΏ β₯ 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
ππiβ² = (ππi+ππf)2
1 + πΏπΏ , ππdβ² = ππi + ππf β ππiβ² , ππfβ² = ππf , πΎπΎcβ² = πΎπΎc
ππiβ²
ππi(7.10)
The condition for πΏπΏ β₯ 0 in terms of the controller parameters is
ππd β€(ππiβππf)2
4ππi(7.11)
i.e., the derivative time has to be βsmall enoughβ.
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7.1.3 A PID controller with derivative filter
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7. PID Controllers
7.1.4 Differentiation of the measured outputEven if we have a derivative filter, a step change in the setpoint ππ(π‘π‘) tends to affect the derivative part much more strongly than a disturbance in the output π¦π¦(π‘π‘). A remedy to this is to differentiate the (filtered) output instead of the control error ππ(π‘π‘).
The ideal control law (7.1) then becomes
π’π’ π‘π‘ = πΎπΎc ππ π‘π‘ + 1ππiβ«0π‘π‘ ππ ππ dππ β ππd
dπ¦π¦f(π‘π‘)dπ‘π‘
+ π’π’0 (7.12a)
ππfdπ¦π¦f(π‘π‘)dπ‘π‘
+ π¦π¦f π‘π‘ = π¦π¦(π‘π‘) (7.12b)
In the Laplace domain we get
ππ π π = πΎπΎc 1 + 1ππiπ π
π π π π β πΎπΎc 1 + 1ππiπ π
+ ππdπ π ππfπ π +1
ππ(π π ) (7.13)
which is a combination of a PI controller and a PID controller
ππ π π = πΊπΊPIπ π π π β πΊπΊPIDfππ(π π ) (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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7.1 PID controller variants
Exercise 7.2Which 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 a 2DOF PID controller is to use setpoint weighting. With the definitions
ππp = ππππ β π¦π¦ , ππ = ππ β π¦π¦ , ππd = ππππ β π¦π¦f (7.15)
where ππ and ππ are setpoint weights, the control law becomes
π’π’ π‘π‘ = πΎπΎc ππp π‘π‘ + 1ππiβ«0π‘π‘ ππ ππ dππ + ππd
dππd(π‘π‘)dπ‘π‘
+ π’π’0 (7.16a)
ππfdπ¦π¦f(π‘π‘)dπ‘π‘
+ π¦π¦f π‘π‘ = π¦π¦(π‘π‘) (7.16b)
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7.1.4 Differentiation of the measured output
7.1.5 Setpoint weighting
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7.1 PID controller variants
In the Laplace domain the control law with setpoint weighting is
ππ π π = πΊπΊvPIDπ π π π β πΊπΊPIDfππ(π π ) (7.17)where
πΊπΊvPID = πΎπΎc ππ + 1ππiπ π
+ ππππdπ π (7.18)
and πΊπΊPIDf is as in (7.6).
With suitable choices of ππ and ππ, all previously treated PID controllers on parallel form can be obtained.
ππ and ππ do not affect the controllerβs ability to reject disturbances in the output, only the ability to track setpoint changes.
πΊπΊvPID can be tuned for setpoint tracking and πΊπΊPIDf for disturbance rejection(i.e., πΎπΎc, ππi and ππd need not have the same values in πΊπΊvPID and πΊπΊPIDf).
Exercise 7.3Include setpoint weighting in the series form of a PID controller.
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7.1.5 Setpoint weighting
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7.1.6 Non-interactive form of a PID controller In the control laws treated so far, the proportional part alone cannot be
disconnected by letting πΎπΎc = 0 because that would disconnect all parts; it would put the controller on βmanualβ with π’π’ π‘π‘ = π’π’0.
Tuning the proportional part by adjusting πΎπΎc will affect all controller parts(however, this is often a desired feature); hence, it is an interactive controller form.
The non-interactive form
π’π’ π‘π‘ = πΎπΎcππππ π‘π‘ + πΎπΎi β«0π‘π‘ ππ ππ dππ + πΎπΎd
dππd(π‘π‘)dπ‘π‘
+ π’π’0 (7.19)
is a more flexible control law. In the Laplace domain it can be written
ππ π π = πΊπΊvP+I+Dπ π π π β πΊπΊP+I+Dfππ(π π ) (7.20)where
πΊπΊvP+I+D = πΎπΎcππ + πΎπΎiπ π β1 + πππΎπΎdπ π (7.21a)
πΊπΊP+I+Df = πΎπΎc + πΎπΎiπ π β1 + πΎπΎdπ π (ππfπ π + 1)β1 (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. 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 ππ(π‘π‘) should be corrected by an increase of the control signal π’π’(π‘π‘), the control law is
π’π’ π‘π‘ = οΏ½π’π’max if ππ π‘π‘ > ππhiπ’π’0 or unchanged if ππlo β€ ππ π‘π‘ β€ ππhiπ’π’min if ππ π‘π‘ < ππlo
(7.22)
where π’π’max, π’π’0, π’π’min are the high, normal, low value of the control signal. The interval [ππlo, ππhi] is a dead zone. In the simplest case, ππlo = ππhi = 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.
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7.2.1 On-off controller
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7. PID Controllers
7.2.2 P controllerA P controller implements the simple control law
π’π’ π‘π‘ = πΎπΎcππ π‘π‘ + π’π’0 (7.23)
where πΎπΎc is the adjustable controller gain and π’π’0 is the normal value of the control signal, which is also be adjustable. In principle, π’π’0 is selected to make the control error ππ π‘π‘ = 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 a liquid tank. Another situation when P control is often sufficient is as an inner loop (a secondary loop) in so-called cascade control.
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7.2 Choice of controller type
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7. PID Controllers
7.2.3 PI controllerA PI controller is by far the most common type of controller. The ideal PI controller implements the control law
π’π’ π‘π‘ = πΎπΎc ππ π‘π‘ + 1ππiβ«0π‘π‘ ππ ππ dππ + π’π’0 (7.24)
where the gain πΎπΎc and the integral time ππi are adjustable parameters; π’π’0 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 disadvantage 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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7.2 Choice of controller type
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7. PID Controllers
7.2.4 PD controllerThe ideal form of a PD controller implements the control law
π’π’ π‘π‘ = πΎπΎc ππ π‘π‘ + ππddππ(π‘π‘)dπ‘π‘
+ π’π’0 (7.25)
where the gain πΎπΎc and the derivative time ππd are adjustable parameters; π’π’0 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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7.2 Choice of controller type
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7. PID Controllers
7.2.5 PID controllerAs 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 π’π’0 : 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 underdampedprocesses, 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 composition when the process is not close to an integrating system.
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7.2 Choice of controller type
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7. PID Controllers
7.3 Specifications and performance criteria
The task of a controller is to control a system to behave in a desired way despite 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) to moderate changes in
system properties, which introduce model uncertainty.The importance of these criteria varies from case to case. Since many criteria are conflicting, compromises have to be made in the control design.
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7.3.1 General performance criteria
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7.3.2 Fundamental limitationsOne 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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7.3 Specifications and performance criteria
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7.3 Specifications and performance criteria
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.
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7.3.2 Fundamental limitations
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7.3.3 Proportional band and integrator windupProportional bandA controllerβs 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 π’π’min β€ π’π’(π‘π‘) β€ π’π’max , a P controller can according to (7.23) handle a control error that satisfies
π’π’minβπ’π’0πΎπΎc
β‘ ππmin β€ ππ(π‘π‘) β€ ππmax β‘π’π’maxβπ’π’0
πΎπΎc(7.26)
The PB is equal to ππmax β ππmin = π¦π¦hi β π¦π¦lo, where π¦π¦hi is the highest output (ππmin = ππ β π¦π¦hi) and π¦π¦lo is the lowest output (ππmax = ππ β π¦π¦lo) the controller can handle. Usually, the PB is defined in percent of the total measurable output interval π¦π¦min,π¦π¦max . Then, the PB is
ππb = π¦π¦hiβπ¦π¦loπ¦π¦maxβπ¦π¦min
100% = π’π’maxβπ’π’minπ¦π¦maxβπ¦π¦min
β 100%πΎπΎc
(7.27)
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7.3 Specifications and performance criteria
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7.3.3 Proportional band and integrator windup
If the proportional band is known, the controller gain is given by
πΎπΎc = π¦π¦hiβπ¦π¦loπ¦π¦maxβπ¦π¦min
100% = π’π’maxβπ’π’minπ¦π¦maxβπ¦π¦min
β 100%ππb
(7.28)
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
ππb = 100%/πΎπΎc (7.29)
Note that the controller gain here has to be expressed in terms of 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 π’π’0 is in the middle of the interval π’π’min,π’π’max , a P controller with ππb = 50 %can handle an instantaneous control error equal to Β±25 % (i.e., 50 % in total) of the total output signal range.
Note that the PB is an adjustable controller parameter β if it is to small, it can be increased (corresponding to a decrease of πΎπΎc).
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Proportional band
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7.3 Specifications and performance criteria
Integrator windupUsually 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 (7.24) is used. A strong disturbance causes the process output to fall well below the set-point. The controller is not able to eliminate the control error (A)because the control signal has reached aconstraint. During this time, the positivecontrol error will increase the integral inthe controller. If the disturbance laterdisappears, the controller will still keepthe control signal at the constraint dueto the large value of the integral, evenIf the control error goes below zero.This will cause the output (B), which isentirely due to the controller.
Illustration of integral windup.
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7.3.3 PB and integrator windup
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7.3.3 Proportional band and integrator windup
The described phenomenon is called integrator windup, integral windup, or 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.
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Integrator windup
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7. PID Controllers
7.3.4 Design specificationsAbove, some general performance criteria and fundamental limitations to achievable control performance have been considered. Here, some ways of making more specific design specifications will be
introduced. If a process model is available, the specifications make it possible to calculate
controller parameters.
Step-response specificationsIt 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 ππ the rise time π‘π‘r the settling time π‘π‘πΏπΏ the relative damping ππ the ratio between successive relative overshoots (or undershoots) ππR
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7.3 Specifications and performance criteria
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7.3.4 Design specifications
According to the relationships in Section 5.3.3: The two parameters ππ and π‘π‘r are sufficient to determine the transfer
function of an underdamped second-order system with a given gain. The settling time π‘π‘πΏπΏ can be used instead of ππ or π‘π‘r , but the relationships
are then only approximate. The relative damping ππ or the overshoot ratio ππR can be specified instead
of ππ.
Some classical tuning recommendations are based on the specification ππR =1/4. This may be acceptable for regulatory control, but not for setpoint tracking. ππR = 1/4 corresponds to ππ = 0.5 (i.e., a 50 % overshoot) and ππ = 0.22 .
For setpoint tracking, ππ β 0.1 (ππ β 0.6) is usually more appropriate.
If an overdamped closed-loop response is desired, this cannot be achieved with a specification ππ > 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.
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Step-response specifications
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7.3 Specifications and performance criteria
Error integralsIn 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
π½π½IAE = β«0π‘π‘s ππ(π‘π‘) dπ‘π‘ , π½π½ISE = β«0
π‘π‘s ππ(π‘π‘)2 dπ‘π‘
π½π½ITAE = β«0π‘π‘s π‘π‘ ππ(π‘π‘) dπ‘π‘ , π½π½ITSE = β«0
π‘π‘s π‘π‘ππ(π‘π‘)2 dπ‘π‘(7.30)
where the acronyms areβ IAE = βintegrated absolute errorββ ISE = βintegrated square errorββ ITAE = βintegrated time-weighted absolute errorββ ITSE = βintegrated time-weighted square errorβThe weighting with time forces the control error towards zero as time increases. In principle, the integration time should be infinite, but because the minimization has to be done numerically, a finite π‘π‘s has to be used.
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7.3.4 Design specifications
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7.3.4 Design specifications
It is of interest to consider how the error integrals relate to step-response specifications when the closed-loop system is of second order, i.e.,
πΊπΊ π π = ππn2
π π 2+2ππππnπ π +ππn2 (7.31)
In the figure, IAE and ISE are normalized with ππn , ITAE and ITSE with ππn2. As can be seen, every normalized error integral has a minimum for a given relative damping ππ . This damping as wellas the corresponding relativeovershoot ππ are shown below.
Table 7.1 Optimal relativedamping for 2nd order system.
Error integrals as function of ππ.
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Error integrals
Error integral ΞΆ 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 (πΊπΊc = πΎπΎc) is used for the first experiment. A low value is chosen for the gain πΎπΎc . Note that πΎπΎc must have the same sign as πΎπΎp .
2. A change in the setpoint π π is introduced. (Some other disturbance could also be used.) The controller gain πΎπΎc is increased until the output ππ starts to oscillate with a constant amplitude (see next slide).
3. The value of the controller gain yielding constant oscillations is denoted πΎπΎc,max . The period of the oscillations is denoted ππc .
4. The controller gain is changed to πΎπΎc = 0.5πΎπΎc,max . If the intention was to tune a P controller, this is the final tuning.
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7.4.1 Experimental tuning
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7.4 Tuning based on frequency response
5. To tune a controller with integralaction (PI or PID), an experimentis done with a PI controller usingπΎπΎc = 0.5πΎπΎc,max . A large value isinitially used for the integral time ππi.
6. A change in the setpoint π π (or someother disturbance) is introduced. Theintegral time ππi is reduced until ππstarts to oscillate with a constantamplitude. This occurs at ππi = ππi,min .
7. The integral time for a PI or PIDcontroller is chosen as ππi = 3ππi,min .
7. To tune the derivative part of a PID (or PD) controller, an experiment is done with such a controller using πΎπΎc = 0.5πΎπΎc,max , ππi = 3ππi,min (if a PID controller). The derivative time is initially set at ππd = 0 .
9. A change in the setpoint π π (or some other disturbance) is introduced.The derivative time ππd is increased until the output ππ starts to oscillate with a constant amplitude. This occurs when ππd = ππd,max.
10. The derivative time for a PD or PID controller is set at ππd = 13ππd,max .
KEH Process Dynamics and Control 7β29
7.4.1 Experimental tuning
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7.4 Tuning based on frequency response
If the control performance obtained by the above tunings turns out to be unsatisfactory, the controller parameters can be adjusted by βtrial and errorβ.
The next figure shows how changes of the controller gain πΎπΎc and the integral time ππi typically affect the control performance. The optimal performance is in this case obtained by πΎπΎc = 3 and ππi = 11 .
ππi = 5 ππi = 11 ππi = 20
πΎπΎc = 5
πΎπΎc = 3
πΎπΎc = 1
KEH Process Dynamics and Control 7β30
7.4.1 Experimental tuning
ProcessControl
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7. PID Controllers
7.4.2 Ziegler-Nicholsβs recommendationsIn 1942, Ziegler and Nichols suggested tunings for P, PI and PID controllers based on πΎπΎc,max and ππc only. To obtain this information, it is sufficient to do steps 1β3 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. ππ = 0.5.
The controller tuning should Table 7.2. Ziegler-Nicholsβs controllerpreferably not be used out- tuning recommendations based onside the range 0.1 < π π < 0.5, frequency response (0.1 < π π < 0.5).where
π π = πΎπΎπΎπΎc,maxβ1
.
πΎπΎ is the process gain.
The critical frequency ππc isoften used instead of ππc:
ππc = 2ππ/ππc.
KEH Process Dynamics and Control 7β31
7.4 Tuning based on frequency response
Controller c c,max/K K i c/T P d c/T P
P 0.5 β β PI 0.45 0.8 β
PID 0.6 0.5 0.125
ProcessControl
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7. PID Controllers
7.4.3 Γ strΓΆmβs and HΓ€gglundβs correlationsIn 2006, Γ strΓΆm and HΓ€gglund showed that, in general, πΎπΎc,max and ππc alone do not provide sufficient information for good controller tuning.
In addition to πΎπΎc,max and ππc , Γ strΓΆm and HΓ€gglund also use the parameter π π = πΎπΎπΎπΎc,max
β1in 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. Γ strΓΆm-HΓ€gglundβs controllernot be used below the tuning correlations based on frequencyrange π π > 0.1 . response (π π > 0.1).
Large time delays areallowed, but clearlyunderdamped systemsare less suitable.
KEH Process Dynamics and Control 7β32
7.4 Tuning based on frequency response
Controller c c,max/K K i c/T P d c/T P
PI 0.16 1(1 4.5 )ΞΊ β+ β
PID 40.3 0.1ΞΊβ 0.6
1 2ΞΊ+ 0.15(1 )
1 0.95ΞΊΞΊ
ββ
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 π’π’step = 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 ππ and πΏπΏ need Characteristic parameters from ato be determined. monotonous unit-step response.
KEH Process Dynamics and Control 7β33
L
iy
it
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7. PID Controllers
Instead of taking the ππ 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 (π‘π‘i,π¦π¦i) of the inflexion point are known. The calculation is valid for any size of π’π’step . The formula for ππ is
ππ = πΏπΏπ¦π¦iπ’π’step(π‘π‘iβπΏπΏ)
(7.32)
Another useful parameter is
ππ = πΏπΏ/ππeq , ππeq = π‘π‘63 β πΏπΏ (7.33)
where ππeq is the equivalent time constant of the system and π‘π‘63 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 (π‘π‘i,π¦π¦i) for calculation of ππ according to (7.32).
KEH Process Dynamics and Control 4β34
7.5 Tuning based on step responsecaKi /T Td /T LcaKi /T Td /T L
ProcessControl
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7. PID Controllers
7.5.1 Ziegler-Nicholsβs recommendationsIn 1942, Ziegler and Nichols suggested tunings for P, PI and PID controllers based also 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 πΏπΏ > 0 and preferably 0.1 β€ ππ β€ 1.
Table 7.4. Ziegler-Nicholsβs controller tuningrecommendations based on step response.
Note that Ziegler-Nicholsβs 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 7β35
7.5 Tuning based on step response
Controller caK i /T L d /T L
P 1.0 β β PI 0.9 3 β
PID 1.2 2 0.5
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7. PID Controllers
7.5.2 The CHR methodIn 1952, Chien, Hrones and Reswick suggested improvements to Zieglerβs and Nicholsβs 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 πΏπΏ > 0 and preferably 0.1 β€ ππ β€ 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.
KEH Process Dynamics and Control 7β36
7.5 Tuning based on step response
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7.5 Tuning based on step response
Controller No overshoot 20 % overshoot
caK i /T T d /T L caK i /T T d /T L
P 0,3 β β 0,7 β β PI 0,35 1,2 β 0,6 1,0 β
PID 0,6 1,0 0,5 0,95 1,4 0,47
Table 7.5. Controller tuning for regulatory control by the CHR method.
Table 7.6. Controller tuning for setpoint tracking by the CHR method.
KEH Process Dynamics and Control 7β37
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 β β 0.7 β β PI 0.6 4.0 β 0.7 2.3 β
PID 0.95 2.4 0.42 1.2 2.0 0.42
i eq/T T i eq/T T
ProcessControl
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7. PID Controllers
7.5.3 Γ strΓΆmβs and HΓ€gglundβs correlationsIn 2006, Γ strΓΆm and HΓ€gglund presented improved controller tunings based on a step response. In addition to ππ and πΏπΏ , they use ππ in their correlations, which can be used for all ππ β₯ 0. However, for ππ < 0.4 , the tunings tend to be conservative. For an integrating process, ππ = 0 is used.
The tunings are primarily intended for regulatory control. For setpoint tracking, setpoint weighting can be used as follows: PI control: ππ = 1 if ππ > 0.4 , ππ < 1 if ππ β€ 0.4 (optimal ππ is unclear) PID control: ππ = 1 if ππ > 1 , ππ = 0 if ππ β€ 1
Table 7.7. Γ strΓΆmβs and HΓ€gglundβs controller tuning correlations.
KEH Process Dynamics and Control 7β38
7.5 Tuning based on step response
Controller caK i /T L d /T L
PI 20.35 0.15(1 )
ΞΈΞΈΞΈ
+ β+
2130.35
1 12 7ΞΈ ΞΈ+
+ + β
PID 0.45 0.2ΞΈ+ 8 41 10
ΞΈΞΈ
++
0.51 0.3ΞΈ+
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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
πΊπΊ π π = πΎπΎπππ π +1
eβπΏπΏπ π (7.34)
from which the parameters ππ and ππ can be calculated according to
ππ = πΎπΎπΏπΏππ
, ππ = πΏπΏππ
(7.35)
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 βgeneral purposeβ methods that are not optimized for any specific model type.
For a given model, better controller tunings probably exist.
KEH Process Dynamics and Control 7β39
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7. PID Controllers
7.6.1 First-order system with a time delayThe transfer function is defined in (7.34) and the parameter ππ in (7.35).
Minimization of error integralsController tunings that minimize IAE and ITAE when 0.1 β€ ππ β€ 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.
KEH Process Dynamics and Control 7β40
7.6 Model-based controller tuning
Error integral
P controller PI controller PID controller
cKK cKK i /T T cKK i /T T d /T T
IAE 0.9850.902 ΞΈ β 0.9860.984ΞΈ β 0.7071.645ΞΈ 0.9211.435ΞΈ β 0.7491.139ΞΈ 1.1370.482ΞΈ
ITAE 1.0840.490 ΞΈ β 0.9770.859ΞΈ β 0.6801.484ΞΈ 0.9471.357 ΞΈ β 0.7381.188ΞΈ 0.9950.381ΞΈ
Error integral
PI controller PID controller
cKK i /T T cKK i /T T d /T T
IAE 0.8610.758ΞΈ β 1(1.020 0.323 )ΞΈ ββ 0.8691.086ΞΈ β 1(0.740 0.130 )ΞΈ ββ 0.9140.348ΞΈ
ITAE 0.9160.586 ΞΈ β 1(1.030 0.165 )ΞΈ ββ 0.8550.965ΞΈ β 1(0.796 0.147 )ΞΈ ββ 0.9290.308ΞΈ
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7.6 Model-based controller tuning
Other optimality criteriaThe 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 robustness even for systems with large time delays, i.e., for large ππ values.
Table 7.10. Cvejnβs tunings for regulatory control and setpoint tracking.
The PI controller tunings tend to give better robustness than the PID controllertunings, which tend to give better performance.
KEH Process Dynamics and Control 7β41
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
2ΞΈ 5.92
1 5.92ΞΈ
ΞΈ+ 3.26
4ΞΈ
ΞΈ+ 3.91
1 3.91 3ΞΈ ΞΈ
ΞΈ+
+
3.26ΞΈ
ΞΈ+
Tracking 1
2ΞΈ 1 3
4ΞΈ
ΞΈ+ 1
3ΞΈ+
3ΞΈ
ΞΈ+
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7. PID Controllers
7.6.2 Second-order no-zero system with a time delayWe shall consider second-order systems with a time delay but no zeros. Such a system has the transfer function
πΊπΊ π π = πΎπΎππn2
π π 2+2ππππnπ π +ππn2 eβπΏπΏπ π (7.36)
In Cvejnβs method for tracking control, the controller πΊπΊc(π π ) is tuned to give the loop transfer πΊπΊβ(π π ) = πΊπΊ(π π )πΊπΊc(π π ) such that
πΊπΊβ π π = 12πΏπΏπ π
eβπΏπΏπ π (7.37)or
πΊπΊβ π π = 14
1 + 3πΏπΏπ π
eβπΏπΏπ π (7.38)
Tuning by (7.37) gives better stability, (7.38) gives better performance.
Exercise 7.3Use Cvejnβs method for tracking control to tune a PID controller for the system (7.36).
KEH Process Dynamics and Control 7β42
7.6 Model-based controller tuning
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7.6 Model-based controller tuning
Overdamped system without zerosFor an overdamped (or critically damped) second-order system, ππ β₯ 1. In this case, (7.36) is more conveniently written as
πΊπΊ π π = πΎπΎ(ππ1π π +1)(ππ2π π +1)
eβπΏπΏπ π , ππ1 β₯ ππ2 (7.39)
Cvejnβs method can be used also in this case, but Γ strΓΆm and HΓ€gglund (2006) suggest the following tuning when the system is overdamped:
πΎπΎπΎπΎc = 0.19 + 0.37ππ1β1 + 0.18ππ2β1 + 0.02ππ1β1ππ2β1
πΎπΎπΎπΎcπΏπΏ/ππi = 0.48 + 0.03ππ1β1 β 0.0007ππ2β1 + 0.0012ππ1β1ππ2β1 (7.40)
πΎπΎπΎπΎcππd/πΏπΏ = 0.29 + 0.16ππ1β1 β 0.2ππ2β1 + 0.28ππ1β1ππ2β1ππ1+ππ2
ππ1+ππ2+ππ1ππ2where
ππ1 = πΏπΏ/ππ1 , ππ2 = πΏπΏ/ππ2 (7.41)
KEH Process Dynamics and Control 7β43
7.6.2 Second-order system with delay
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7.6.2 Second-order system with delay
Second-order system including integrationA second-order no-zero system including an integrator has the transfer function
πΊπΊ π π = πΎπΎπ π (ππ2π π +1)
eβπΏπΏπ π (7.42)
For this kind of system, Γ strΓΆm and HΓ€gglund (2006) suggest the tuning:
πΎπΎπΎπΎcπΏπΏ = 0.37 + 0.02ππ2β1
πΎπΎπΎπΎcπΏπΏ2/ππi = 0.03 + 0.0012ππ2β1 (7.43)πΎπΎπΎπΎcππd = 0.16 + 0.28ππ2β1
If the system is a double integrator with the transfer function
πΊπΊ π π = πΎπΎπ π 2
eβπΏπΏπ π (7.44)the suggested tuning is
πΎπΎπΎπΎcπΏπΏ2 = 0.02πΎπΎπΎπΎcπΏπΏ3/ππi = 0.0012 (7.45)πΎπΎπΎπΎcππdπΏπΏ = 0.28
KEH Process Dynamics and Control 7β44
Overdamped system
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7.6 Model-based controller tuning
Second-order system with a zeroAn overdamped 2nd order system with a zero has the transfer function
πΊπΊ π π = πΎπΎ(ππ3π π +1)(ππ1π π +1)(ππ2π π +1)
eβπΏπΏπ π (7.46)
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 zeroAn IPZ system (1 integrator, 1 pole, 1 zero) has a transfer function
πΊπΊ π π = πΎπΎ(ππ3π π +1)π π (ππ2π π +1)
eβπΏπΏπ π , ππ3 > ππ2 > 0 (7.47)
An IPZ system is difficult to approximate by a simpler one, esp. if ππ3 β« ππ2.
In Table 7.11, Table 7.11. SlΓ€ttekeβs regulatory tuning for an IPZ process.ππ2 = πΏπΏ/ππ2. ForPID control, aderivative filterππf = 0.1ππd isused. For set-point tracking,ππ < 1 is used.
KEH Process Dynamics and Control 7β45
7.6.2 Second-order system with delay
Controller 3 cT KK i /T L d 2/T T
PI 120.0767(3 1)ΞΈ β + 2
2
100 1711 94
ΞΈΞΈ
++
β
PID 120.115(3 1)ΞΈ β +
22 2
22 2
835 842 2773(55 386 241 )
ΞΈ ΞΈΞΈ ΞΈ
+ ++ +
2
2 22
2 2
3 176 736500(1 2 )
ΞΈ ΞΈΞΈ ΞΈ
+ ++ +
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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 βgood controlβ.
However, what is βgood controlβ 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 βgood controlβ.
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 βgood controlβ
more model types than in previous sections can be handled, e.g., systems with a zero
KEH Process Dynamics and Control 7β46
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7. PID Controllers
7.7.1 Closed-loop transfer functionsConsider the closed-loopsystem in the figure with thefollowing transfer functions:β πΊπΊ π π process being controlledβ πΊπΊc π π controllerβ πΊπΊd π π disturbance system Block diagram of closed-loop systemStandard block-diagram algebra gives
ππ = πΊπΊπΊπΊc1+πΊπΊπΊπΊc
π π + πΊπΊd1+πΊπΊπΊπΊc
ππ (7.48)where
πΊπΊr = πΊπΊπΊπΊc1+πΊπΊπΊπΊc
, πΊπΊv = πΊπΊd1+πΊπΊπΊπΊc
(7.49,50)
are the closed-loop transfer functions from the setpoint π π and the disturbance ππ to the output ππ.
The user can specify the desired πΊπΊr for setpoint tracking or πΊπΊv for regulatory control. For setpoint tracking, the required controller is given by
πΊπΊc = 1πΊπΊ
πΊπΊr(1βπΊπΊr)
(7.51)
KEH Process Dynamics and Control 7β47
7.7 Controller tuning by direct synthesis
( )Y s
( )V s
c ( )G s( )R s
+β
++
( )G s
d ( )G s
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7. PID Controllers
7.7.2 Low-order minimum-phase systemsFirst-order systemA strictly proper first-order system without a time delay has the transfer function
πΊπΊ = πΎπΎπππ π +1
(7.52)
Assume that we want the controlled system to behave as a first-order systemwith the time constant ππr . Then,
πΊπΊr = 1ππrπ π +1
, which gives πΊπΊr1βπΊπΊr
= 1ππrπ π
(7.53)
Substitution of (7.52) and (7.53) into (7.51) gives
πΊπΊc = πππ π +1πΎπΎ
1ππrπ π
= πππΎπΎππr
1 + 1πππ π
(7.54)
which is a PI controller with the parameters
πΎπΎc = πππΎπΎππr
, ππi = ππ (7.55)
Here, ππr is a design parameter, by which the performance of the control system can be affected.
KEH Process Dynamics and Control 7β48
7.7 Controller tuning by direct synthesis
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7.7 Controller tuning by direct synthesis
Second-order system with no zeroA second-order system with no zero and no time delay has the transfer function
πΊπΊ π π = πΎπΎππn2
π π 2+2ππππnπ π +ππn2 (7.56)
Even if the uncontrolled system is of second order, we can specify the controlled system to be of first order. Substitution of (7.53) and (7.56) into (7.51) then gives
πΊπΊc = π π 2+2ππππnπ π +ππn2
πΎπΎππn2
1ππrπ π
= 2πππΎπΎππnππr
1 + ππn2πππ π
+ π π 2ππππn
(7.57)
which is an ideal PID controller with the parameters
πΎπΎc = 2πππΎπΎππnππr
, ππi = 2ππππn
, ππd = 12ππππn
(7.58)
Also here, ππr is a design parameter which only affects the controller gain.
KEH Process Dynamics and Control 7β49
7.7.2 Low-order minimum-phase systems
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7.7 Controller tuning by direct synthesis
Overdamped second-order system with a LHP zeroAn overdamped second-order system with a zero in the left half of the complex plane (LHP) has the transfer function
πΊπΊ π π = πΎπΎ(ππ3π π +1)(ππ1π π +1)(ππ2π π +1)
, ππππ β₯ 0 (7.59)
We can specify the controlled system to be of first order. Substitution of (7.53) and (7.59) into (7.51) gives
πΊπΊc = (ππ1π π +1)(ππ2π π +1)πΎπΎ(ππ3π π +1)
1ππrπ π
= 1πΎπΎππrπ π
ππ1ππ2π π 2+ ππ1+ππ2 π π +1ππ3π π +1
= 1πΎπΎππrπ π
1 + ππ1 + ππ2 β ππ3 π π + ππ1ππ2β ππ1+ππ2βππ3 ππ3ππ3π π +1
π π 2
orπΊπΊc = πΎπΎc 1 + 1
ππiπ π + ππdπ π
ππfπ π +1(7.60)
where
πΎπΎc = ππ1+ππ2βππ3πΎπΎππr
, ππi = ππ1 + ππ2 β ππ3 , ππd = ππ1ππ2ππ1+ππ2βππ3
β ππ3 , ππf = ππ3 (7.61)
This is a PID controller with a derivative filter.
KEH Process Dynamics and Control 7β50
7.7.2 Low-order minimum-phase systems
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7. PID Controllers
7.7.3 High-order minimum-phase systemsA high-order minimum-phase system with real poles and zeros, but with no time delay, has the transfer function
πΊπΊ = πΎπΎβππ=ππ+1ππ+ππ (πππππ π +1)βππ=1ππ (πππππ π +1)
, ππππ > 0 , ππππ > 0 , ππ > 2 (7.62)
If ππ = 3 and ππ = 0 or 1 , a closed-loop system of second order can be obtained by a full PID controller.
If ππ > 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 πΊπΊr is thus not practical.
In the case of ππ > 3 , two possibilities are to specify a closed-loop system of first or second order and then to
first calculate a πΊπΊc according to (7.51), then to approximate πΊπΊc by a PID controller;
first approximate πΊπΊ by a model of at most third order, then to calculate the PID controller according to (7.51).
In Section 7.9, the latter approach will be described.
KEH Process Dynamics and Control 7β51
7.7 Controller tuning by direct synthesis
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7. PID Controllers
7.7.4 Second-order system with RHP zeroA second-order system with real poles and a right half-plane (RHP) zero has the transfer function
πΊπΊ π π = πΎπΎ(βππ3π π +1)(ππ1π π +1)(ππ2π π +1)
, ππππ β₯ 0 (7.63)
Now division by πΊπΊ in (7.51) will result in an unstable controller with a RHP pole if πΊπΊr 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 πΊπΊ .
Another solution is to include the same RHP zero in πΊπΊr as in πΊπΊ ; it will then be cancelled out in (7.51) and the controller will automatically be stable. This means that the choice of πΊπΊr is restricted, but otherwise the control performance tends to be as expected.
In this section, the latter approach is used.
KEH Process Dynamics and Control 7β52
7.7 Controller tuning by direct synthesis
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7.7 Controller tuning by direct synthesis
Closed-loop system of first orderThe closed-loop transfer function is chosen as
πΊπΊr = βππ3π π +1ππrπ π +1
, which gives πΊπΊr1βπΊπΊr
= βππ3π π +1(ππr+ππ3)π π
(7.64)
Substitution of (7.63) and (7.64) into (7.51) gives
πΊπΊc = (ππ1π π +1)(ππ2π π +1)πΎπΎ
1(ππr+ππ3)π π
= ππ1+ππ2πΎπΎ(ππr+ππ3)
1 + 1ππ1+ππ2 π π
+ ππ1ππ2π π ππ1+ππ2
(7.65)
which is a PID controller with the parameters
πΎπΎc = ππ1+ππ2πΎπΎ(ππr+ππ3)
, ππi = ππ1 + ππ2 , ππd = ππ1ππ2ππ1+ππ2
(7.66)
KEH Process Dynamics and Control 7β53
7.7.4 Second-order system with RHP zero
ProcessControl
Laboratory
7.7 Controller tuning by direct synthesis
Closed-loop system of second orderA 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 πΊπΊr is
πΊπΊr = (βππ3π π +1)ππr2
π π 2+2ππrππrπ π +ππr2 , which gives πΊπΊr
1βπΊπΊr= (βππ3π π +1)ππr
2
π π (π π +2ππrππr+ππ3ππr2)
(7.67)
To simplify the derivation of controller parameters, we define
ππf = 1/(2ππrππr + ππ3ππr2) (7.68)
Substitution of (7.63) and (7.67) into (7.51), gives, with (7.68),
πΊπΊc = (ππ1π π +1)(ππ2π π +1)ππfππr2
πΎπΎ ππfπ π +1 π π = ππfππr
2
πΎπΎπ π ππ1ππ2π π 2+ ππ1+ππ2 π π +1
ππfπ π +1(7.69)
Analogously with the derivation of (7.61), this gives the PID controller parameters
πΎπΎc = ππfππr2
πΎπΎ(ππ1 + ππ2 β ππf), ππi = ππ1 + ππ2 β ππf , ππd = ππ1ππ2
ππ1+ππ2βππfβ ππf (7.70)
where ππf , given by (7.68), is the derivative filter time constant in a PID controller (7.60).
KEH Process Dynamics and Control 7β54
7.7.4 Second-order system with RHP zero
ProcessControl
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7.7.4 Second-order system with RHP zero
Choice of closed-loop system parametersIn (7.67), there are two design parameters, the relative damping ππr, and the undamped natural frequency ππr. 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 πΊπΊr have two equally large real poles at β1/ππr . This corresponds to ππr = 1 and ππr = 1/ππr , which for (7.68) gives
ππf = ππr2
2ππr+ππ3(7.71)
Let πΊπΊr have real poles at β1/ππr and β1/ππ3 . This corresponds to ππr = 0.5(ππr + ππ3)ππr and ππr = 1/ ππrππ3 , which for (7.68) gives
ππf = ππrππ3ππr+2ππ3
(7.72)
KEH Process Dynamics and Control 7β55
Closed-loop system of 2nd order
ProcessControl
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7. PID Controllers
7.7.5 First-order system with a time delayTo 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
πΊπΊ π π = πΎπΎπππ π +1
eβπΏπΏπ π (7.73)
Calculation of a controller by (7.51) will then result in a controller containing a time delay β there is no practical way to avoid this by the choice of πΊπΊr.
There are methods to implement a controller resulting from (7.51) (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.
KEH Process Dynamics and Control 7β56
7.7 Controller tuning by direct synthesis
ProcessControl
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7.7 Controller tuning by direct synthesis
Time-delay approximation in the modelA standard way of approximating a time delay is to use a PadΓ© approximation. A first-order PadΓ© approximation
eβπΏπΏπ π β 1β0.5πΏπΏπ π 1+0.5πΏπΏπ π
(7.74)gives the model
πΊπΊ π π = πΎπΎ(β0.5πΏπΏπ π +1)(πππ π +1)(0.5πΏπΏπ π +1)
(7.75)
A natural choice for πΊπΊr is then
πΊπΊr = β0.5πΏπΏπ π +1(ππrπ π +1)(0.5πΏπΏπ π +1)
, which gives πΊπΊr1βπΊπΊr
= β0.5πΏπΏπ π +1π π (0.5ππrπΏπΏπ π +ππr+πΏπΏ)
(7.76)
Substitution of (7.75) and (7.76) into (7.51) gives a PID controller with the parameters
πΎπΎc = ππ+0.5πΏπΏβππfπΎπΎ(ππr+πΏπΏ)
, ππi = ππ + 0.5πΏπΏ β ππf , ππd = 0.5πΏπΏππππ+0.5πΏπΏβππf
, ππf = 0.5πΏπΏππrππr+πΏπΏ
(7.77)
Here, ππf is the time constant of a derivative filter in the PID controller (7.60).
KEH Process Dynamics and Control 7β57
7.7.5 First-order system with a delay
ProcessControl
Laboratory
7.7 Controller tuning by direct synthesis
Time-delay approximation in the controllerIf eβπΏπΏπ π is retained in the model, it also has to be part of πΊπΊr , because it is impossible for the closed-loop system to have a shorter time-delay than the uncontrolled system.
If πΊπΊr is chosen to be first order with a time delay
πΊπΊr = 1ππrπ π +1
eβπΏπΏπ π , which gives πΊπΊr1βπΊπΊr
= eβπΏπΏπΏπΏ
ππrπ π +1βeβπΏπΏπΏπΏ(7.78)
Substitution of (7.73) and (7.78) into (7.51) gives
πΊπΊc = πππ π +1πΎπΎ(ππrπ π +1βeβπΏπΏπΏπΏ)
(7.79)
Unfortunately, this controller cannot be implemented by a PID controller in a regular feedback loop. In order to do that, the time delay in (7.79) has to be approximated by a rational expression. If the approximation (7.74) is used, the controller parameters will be as in
(7.77). The simpler approximation eβπΏπΏπ π β 1 β πΏπΏπ π gives a PI controller with
πΎπΎc = πππΎπΎ(ππr+πΏπΏ)
, ππi = ππ (7.80)
KEH Process Dynamics and Control 7β58
7.7.5 First-order system with a delay
ProcessControl
Laboratory
7. PID Controllers
7.8 Internal model controlβInternal model controlβ (IMC) is closely related to βdirect synthesisβ (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 (7.79) 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 issues than DS.
KEH Process Dynamics and Control 7β59
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7. PID Controllers
7.8.1 The IMC structureConsider the closed-loopsystem in the figure with thefollowing transfer functions:β πΊπΊ π π true processβ οΏ½πΊπΊ π π process modelβ πΊπΊIMC π π a controllerβ πΊπΊd π π disturbance systemStandard block diagram algebra The IMC structure.gives ππ = πΊπΊIMC(πΈπΈ + οΏ½πΊπΊππ) from which
πππΈπΈ
= πΊπΊc = πΌπΌ β πΊπΊIMC οΏ½πΊπΊβ1πΊπΊIMC = πΊπΊIMC πΌπΌ β οΏ½πΊπΊπΊπΊIMC
β1 = πΊπΊIMC1β οΏ½πΊπΊπΊπΊIMC
(7.81)
Assume thatπΊπΊIMC = οΏ½πΊπΊβ1πΊπΊf (7.82)
where πΊπΊf is a βfilterβ. Substitution of (7.82) into (7.81) gives
πΊπΊc = οΏ½πΊπΊβ1πΊπΊf πΌπΌ β πΊπΊf β1 = 1οΏ½πΊπΊ
πΊπΊf(1βπΊπΊf)
(7.83)
If the filter is chosen as πΊπΊf = πΊπΊr (and οΏ½πΊπΊ = πΊπΊ), this is the same as (7.51) !
KEH Process Dynamics and Control 7β60
7.8 Internal model control
( )G s
Λ ( )G s
( )E s
ProcessControl
Laboratory
7. PID Controllers
7.8.2 Handling of time delays without approximationConsider a system modelled as a first-order system with a time delay, i.e., οΏ½πΊπΊ = πΎπΎeβπΏπΏπ π /(πππ π + 1). Choose the IMC filter as πΊπΊf = eβπΏπΏπ π /(ππrπ π + 1) . Substitution into (7.82) now gives
πΊπΊIMC = 1πΎπΎπππ π +1ππrπ π +1
= 1πΎπΎ
1 + ππβππrππrπ π +1
π π (7.84)
which is a PD controller with a derivative filter having the parameters πΎπΎππ = 1/πΎπΎ , ππd = ππ β ππr , ππf = ππr . Substitution of (7.84) and the model οΏ½πΊπΊ into (7.81) gives
πΊπΊc = πππ π +1πΎπΎ(ππrπ π +1βeβπΏπΏπΏπΏ)
(7.85)
which is identical with (7.79). The difference is that (7.85) can be implemented exactly by the IMC structure without time-delay approximation.
Note that there is no integration in πΊπΊIMC , but the feedback of οΏ½πΊπΊ in the IMC structure introduces integration if πΊπΊIMC is calculated using the same οΏ½πΊπΊ in (7.82); integration is achieved even if οΏ½πΊπΊ β πΊπΊ .
Exercise. Calculate the closed-loop transfer function πΊπΊr when οΏ½πΊπΊ β πΊπΊ . Show that there will be no steady-state error, i.e., that πΊπΊr 0 = 1 .
KEH Process Dynamics and Control 7β61
7.8 Internal model control
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 emphasize how πΊπΊIMC combined with the feedback of οΏ½πΊπΊ is equivalent to πΊπΊc.
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.)β The control signal is an input to the real system πΊπΊ and the model οΏ½πΊπΊ.β οΏ½πΊπΊ predicts the output οΏ½ππ, which is compared with the true output ππ.β Only the prediction error πΈπΈ = ππ β οΏ½ππ is fed back, not the entire ππ.
The latter property is a clearadvantage in controller design.If οΏ½πΊπΊ = πΊπΊ (i.e., πΈπΈ = 0)
πΊπΊr = πΊπΊπΊπΊIMC (7.86)
which means that the closed-loop transfer function dependslinearly on πΊπΊIMC making designof πΊπΊIMC easier than design of πΊπΊc. Predictive nature of IMC structure.
KEH Process Dynamics and Control 7β62
7.8 Internal model control
Λ ( )G s
( )G s
ProcessControl
Laboratory
7. PID Controllers
7.8.4 Controller designThe following conclusions can be drawn from (7.86): A stable closed-loop system πΊπΊr requires a stable IMC controller πΊπΊIMC ;
in particular, the IMC controller may not contain integral action. Non-minimum phase properties (i.e., RHP zeros and time delays) in πΊπΊ will
also be present in πΊπΊr because they cannot be cancelled out by a stable and realizable πΊπΊIMC.
From (7.82) it follows that the filter πΊπΊf has to be chosen to cancel out non-minimum phase properties of πΊπΊ β this is equivalent to the choice of πΊπΊr in direct synthesis.
In practice, the IMC design is done differently. Instead of guaranteeing the stability and realizability of πΊπΊIMC by the choice of πΊπΊf , it is handled by the choice of οΏ½πΊπΊ to be inverted; non-minimum phase parts of οΏ½πΊπΊ are not inverted.
KEH Process Dynamics and Control 7β63
7.8 Internal model control
ProcessControl
Laboratory
7.8 Internal model control
The process model οΏ½πΊπΊ can always be factorized asοΏ½πΊπΊ = οΏ½πΊπΊβ οΏ½πΊπΊβ (7.87)
where οΏ½πΊπΊβ contains all non-minimum-phase elements of οΏ½πΊπΊ, but no minimum-phase elements, and normalized so that οΏ½πΊπΊβ 0 = 1 (i.e., it has the static gain 1). This means that οΏ½πΊπΊβ contains all RHP zeros and time delays of οΏ½πΊπΊ ; if there are no such elements, οΏ½πΊπΊβ = 1.
When πΊπΊIMC is calculated according to (7.82), only οΏ½πΊπΊβ is inverted. Thus,
πΊπΊIMC = οΏ½πΊπΊβ β1πΊπΊf (7.88)
Note that the full οΏ½πΊπΊ is to be used as internal model as illustrated by the IMC block diagrams β the use of οΏ½πΊπΊβ is only a technical aid for the calculation of πΊπΊIMC .
The IMC filter πΊπΊf 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
πΊπΊf = 1(ππrπ π +1)ππ
(7.89)
is chosen. Here, ππ is an integer, usually ππ = 1, sometimes ππ > 1.
KEH Process Dynamics and Control 7β64
7.8.4 Controller design
ProcessControl
Laboratory
7. PID Controllers
7.8.5 Implementation with a regular PID controllerAn 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 πΊπΊIMC has been designed, the corresponding βregularβ controller πΊπΊc can be calculated according to (7.81). If οΏ½πΊπΊ contains a time delay, it will also be present in πΊπΊc. 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 asππ1 > 0 , ππ2 β₯ 0 , ππ3 β₯ 0 , πΏπΏ β₯ 0 (7.90)
The tunings can be used for (underdamped) models expressed by the relative damping and the natural frequency by the substitutions
ππ1 + ππ2 = 2ππ/ππn , ππ1ππ2 = 1/ππn2 (7.91)
Usually ππr is chosen such that πΏπΏ β€ ππr < ππ (but no clear consensus).
KEH Process Dynamics and Control 7β65
7.8 Internal model control
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7.8 Internal model control
Table 7.12. IMC-based tuning of ideal PID controller.
The desired time constant of the closed-loop system is ππr . ππ , which is used in the calculations, is closely related to ππr . Note that the calculated integral time ππi is used in several expressions.
KEH Process Dynamics and Control 7β66
7.8.5 Implementation with a PID controller
( )G s cK K iT dT Ξ»
1
e1
LsKT s
β
+ i /T Ξ» 1
1 2T L+ 11 i2 /LT T 1
r 2T L+
3
1 2
( 1)e( 1)( 1)
LsK T sT s T s
β++ +
i /T Ξ» 1 2 3T T T+ β 1 2 i 3( / )T T T Tβ rT L+
3
1 2
( 1)e( 1)( 1)
LsK T sT s T s
ββ ++ +
i /T Ξ» 1 2 3( / )T T T L Ξ»+ + 1 2 i 3( / ) ( / )T T T T L Ξ»β r 3T T L+ +
e LsKs
β 2
i /T Ξ» 2Ξ» 1 1i2 2(1 / )L L Tβ 1
r 2T L+
2
e( 1)
LsKs T s
β
+ 2
i /T Ξ» 22 T LΞ» + β 2 2 i(1 / )T T Tβ 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 parameters πΎπΎc,max , ππc (or ππc) and π π . These methods are βgeneral-purpose methodsβ not optimized for any specific model type.
Section 7.5: Controller tuning based on step-response parameters ππ (or π‘π‘i, π¦π¦i), πΏπΏ and ππ. 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.
KEH Process Dynamics and Control 7β67
ProcessControl
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7. PID Controllers
7.9.1 Skogestadβs methodSkogestad 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 πΊπΊβ and a non-minimum-phase part πΊπΊβ , i.e.,
πΊπΊ π π = πΊπΊβ(π π )πΊπΊβ(s) (7.92)
Any left-half plane (LHP) zeros of πΊπΊβ(s) and RHP zeros of πΊπΊβ(π π ) are eliminated by suitable approximations.Elimination of LHP zerosIf the poles and zeros are real, the minimum-phase part has the form
πΊπΊβ π π = πΎπΎ ππππ+1π π +1 ππππ+2π π +1 β¦(ππππ+πππ π +1)ππ1π π +1 ππ2π π +1 β¦(πππππ π +1)
(7.93)
where ππ1 β₯ ππ2 β₯ β― β₯ ππππ > 0, ππππ+1 β₯ ππππ+2 β₯ β― β₯ ππππ+ππ > 0 , ππ > ππ. The simplification procedure now goes as follows. The numerator time constants ππππ+1, ππππ+2, β¦, ππππ+ππ are considered in that
order. Assume that ππππ+ππ is the one currently being considered.
KEH Process Dynamics and Control 7β68
7.9 Model simplification
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7.9.1 Skogestadβs method
Next, the smallest remaining denominator time constant ππππ such that ππππ β₯ ππππ+ππ is selected. If there is no such time constant, or if ππππ β« ππππ+ππ, the smaller ππππ closest to ππππ+ππ is chosen. It is considered that ππππ β« ππππ+ππif ππππ > ππππ+ππ2 /ππππ+1 and ππππ+ππ/ππππ+1 < 1.6 .
The ratio (ππππ+πππ π + 1)/(πππππ π + 1) is now approximated as
ππππ+πππ π +1πππππ π +1
β
ππππ+ππ/ππππ if ππππ β₯ ππππ+ππ β₯ 5ππr a5ππr/ππππ
5ππrβππππ+ππ π π +1if ππππ β₯ 5ππr β₯ ππππ+ππ b
1ππππβππππ+ππ π π +1
if 5ππr β₯ ππππ β₯ ππππ+ππ c
ππππ+ππ/ππππ if ππππ+ππβ₯ ππππ β₯ ππr (d)ππππ+ππ/ππr if ππππ+ππβ₯ ππr β₯ ππππ (e)1 if ππr β₯ ππππ+ππ β₯ ππππ (f)
(7.94)
Here, ππr is the desired closed-loop time constant. If this is not known, the suggested value is ππr = οΏ½πΏπΏ , which is the time delay in the simplified model. Since this is not initially known, one may have to iterate (i.e., first guessing οΏ½πΏπΏ, then possibly correcting with the new οΏ½πΏπΏ).
KEH Process Dynamics and Control 7β69
Elimination of LHP zeros
ProcessControl
Laboratory
7.9.1 Skogestadβs method
The above procedure gives an approximate minimum-phase part οΏ½πΊπΊβ of the form
οΏ½πΊπΊβ π π =οΏ½πΎπΎ
οΏ½ππ1π π +1 οΏ½ππ2π π +1 β¦( οΏ½πποΏ½πππ π +1)(7.95)
Note that the gain as well as the values and number of denominator time constants may have changed from the original πΊπΊβ.
Elimination of RHP zeros and the half ruleThe transfer function οΏ½πΊπΊ π π = πΊπΊβ(π π ) οΏ½πΊπΊβ(s) now has the form
οΏ½πΊπΊ π π =οΏ½πΎπΎ βππππ+ππ+1π π +1 βππππ+ππ+2π π +1 β¦(βππππ+ππ+πππ π +1)
οΏ½ππ1π π +1 οΏ½ππ2π π +1 β¦( οΏ½πποΏ½πππ π +1)eβπΏπΏπ π (7.96)
where οΏ½ππ1 β₯ οΏ½ππ2 β₯ β― β₯ οΏ½πποΏ½ππ > 0, ππππ+ππ+1 β₯ ππππ+ππ+2 β₯ β― β₯ ππππ+ππ+ππ > 0 .
Skogestadβs half ruleIf an approximate model of order οΏ½ππ is desired, the οΏ½ππ largest denominator time constants are retained in the model with the modification that half of οΏ½πποΏ½ππ+1 is added to οΏ½πποΏ½ππ. Half of οΏ½πποΏ½ππ+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.
KEH Process Dynamics and Control 7β70
Elimination of LHP zeros
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7.9.1 Skogestadβs method
Approximation by first-order systemIf a first-order model is desired, the half rule gives
οΏ½πΊπΊ π π =οΏ½πΎπΎ
οΏ½πππ π +1eβοΏ½πΏπΏπ π (7.97a)
οΏ½ππ = οΏ½ππ1 + 12οΏ½ππ2 , οΏ½πΏπΏ = πΏπΏ + 1
2οΏ½ππ2 + βππ=3
οΏ½ππ οΏ½ππππ + βππ=1ππ ππππ+ππ+ππ (7.97b)
Approximation by second-order systemIf a second-order model is desired, the half rule gives
οΏ½πΊπΊ π π =οΏ½πΎπΎ
( οΏ½ππ1π π +1)( οΏ½ππ2π π +1)eβοΏ½πΏπΏπ π (7.98a)
οΏ½ππ2 = οΏ½ππ2 + 12οΏ½ππ3 , οΏ½πΏπΏ = πΏπΏ + 1
2οΏ½ππ3 + βππ=4
οΏ½ππ οΏ½ππππ + βππ=1ππ ππππ+ππ+ππ (7.98b)
KEH Process Dynamics and Control 7β71
Elimination of RHP zeros and the half rule
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Laboratory
7.9 Model simplification
Example 7.2. IMC via model reduction by Skogestadβs method.Simplify the model
πΊπΊ π π = (16π π +1)(4π π +1)(β8π π +1)eβ2πΏπΏ
(50π π +1)(20π π +1)(12π π +1)(6π π +1)(3π π +1)(π π +1)
to a second-order model by Skogestadβs method and determine the parameters of a PID controller by IMC-based tuning for this model. Use a first-order filter time constant ππr = 10.
HereπΊπΊβ π π = (16π π +1)(4π π +1)
(50π π +1)(20π π +1)(12π π +1)(6π π +1)(3π π +1)(π π +1).
According to (7.94c), 16π π +120π π +1
β 14π π +1
. The numerator factor (4π π + 1) can now be cancelled out against the new denominator factor, which gives
οΏ½πΊπΊβ π π = 1(50π π +1)(12π π +1)(6π π +1)(3π π +1)(π π +1)
andοΏ½πΊπΊ π π = (β8π π +1)eβ2πΏπΏ
(50π π +1)(12π π +1)(6π π +1)(3π π +1)(π π +1).
KEH Process Dynamics and Control 7β72
7.9.1 Skogestadβs method
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7.9.1 Skogestadβs method
The resulting second-order model is
οΏ½πΊπΊ π π = 1( οΏ½ππ1π π +1)( οΏ½ππ2π π +1)
eβοΏ½πΏπΏπ π
with οΏ½ππ1 = 50 , οΏ½ππ2 = 12 + 12 β 6 = 15 , οΏ½πΏπΏ = 2 + 1
2 β 6 + 3 + 1 + 8 = 17. Thus,
οΏ½πΊπΊ π π = 1(50π π +1)(15π π +1)
eβ17π π .
According to Table 7.12 for IMC-based tuning of second-order model:
β ππ = ππr + οΏ½πΏπΏ = 10 + 17 = 27β ππi = οΏ½ππ1 + οΏ½ππ2 = 50 + 15 = 65β πΎπΎc = ππi/(οΏ½πΎπΎππ) = 65/(1 β 27) = 2.4β ππd = οΏ½ππ1 οΏ½ππ2/ππi = 50 β 15/65 = 11.5
KEH Process Dynamics and Control 7β73
Example 7.2
ProcessControl
Laboratory
7. PID Controllers
7.9.2 Isakssonβs and Graebeβs methodIsaksson 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 PadΓ© approximation.
To describe the method, both factorized and polynomial forms of the original transfer function are employed. If the numerator order is ππ and the denominator order is ππ , the transfer function is
πΊπΊ π π = πΎπΎ ππππ+1π π +1 ππππ+2π π +1 β¦(ππππ+πππ π +1)ππ1π π +1 ππ2π π +1 β¦(πππππ π +1)
(7.99a)
= πΎπΎ ππ0π π ππ+β―+ππππβ2π π 2+ππππβ1π π +1ππ0π π ππ+β―+ππππβ2π π 2+ππππβ1π π +1
(7.99b)
where ππ1 β₯ ππ2 β₯ β― β₯ ππππ > 0 (i.e., a stable system) and |ππππ+1| β₯ |ππππ+2| β₯β― β₯ |ππππ+ππ| . The numerator time constants can be positive or negative.
KEH Process Dynamics and Control 7β74
7.9 Model simplification
ProcessControl
Laboratory
7.9 Model simplification
If a model with the numerator order οΏ½ππ and the denominator order οΏ½ππ is desired, the simplified model is
οΏ½πΊπΊ π π = πΎπΎ ππππ+1π π +1 β¦(ππππ+οΏ½πππ π +1) + ππππβοΏ½πππ π οΏ½ππ+β―+ππππβ1π π +1ππ1π π +1 β¦(πποΏ½πππ π +1) + ππππβοΏ½πππ π οΏ½ππ+β―+ππππβ1π π +1
(7.100)
Complex-conjugated poles or zeros is no problem, except if they occur as poles number οΏ½ππ and οΏ½ππ + 1 or zeros number ππ + οΏ½ππ and ππ + οΏ½ππ + 1. One solution is then to use the real part of the complex conjugate as πποΏ½ππ or ππππ+ οΏ½ππ .
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
οΏ½πΊπΊ π π = πΎπΎ12 ππ1+ππππβ1 π π +1
(1st order) (7.101)
οΏ½πΊπΊ π π =πΎπΎ 1
2 ππππ+1+ππππβ1 π π +112 ππ1ππ2+ππππβ2 π π 2+12 ππ1+ππ2+ππππβ1 π π +1
(2nd order) (7.102)
whereππππβ1 = βππ=1ππ ππππ+ππ , ππππβ1 = βππ=1ππ ππππ , ππππβ2 = 1
2 βππ=1ππ ππππ
2ββππ=1ππ ππππ
2 (7.103)
KEH Process Dynamics and Control 7β75
7.9.2 Isakssonsβs and Graebeβs method
ProcessControl
Laboratory
7.9 Model simplification
Example 7.3. IMC via model reduction by IsakssonβGraebeβs method.Solve the same problem as in Example 7.2 by Isakssonβs and Graebeβs model reduction method.The model gives
ππππβ1 = 16 + 4 β 8 = 12 , ππππβ1 = 50 + 20 + 12 + 6 + 3 + 1 = 92ππππβ2 = 1
2 922β(502+202+122+62+32+12) = 2687from which
οΏ½πΊπΊ π π =12 16+12 π π +1
12 1000+2687 π π 2+12 70+92 π π +1
eβ2π π = (14π π +1)eβ2πΏπΏ
1843.5π π 2+81π π +1
This model has complex-conjugated poles, but according to (7.91), ππ1 + ππ2 = 81and ππ1ππ2 = 1843.5 can be used in the controller calculations. Table 7.12 for IMC-based tuning of second-order model then givesβ ππ = ππr + πΏπΏ = 10 + 2 = 12β ππi = οΏ½ππ1 + οΏ½ππ2 β οΏ½ππ3 = 81 β 14 = 67β πΎπΎc = ππi/(πΎπΎππ) = 67/(1 β 12) = 5.6 (much bigger than in Ex. 7.2!)β ππd = οΏ½ππ1 οΏ½ππ2/ππi β οΏ½ππ3 = 1843.5/67 β14 = 13.5
KEH Process Dynamics and Control 7β76
7.9.2 Isakssonsβs and Graebeβs method
