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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
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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‘ = πΎc π π‘ +1
πi 0π‘π π dπ + πd
dπ(π‘)
dπ‘+ π’0 (7.1)
π’(π‘) is the controller output π π‘ = π π‘ β π¦(π‘) is the control error, which is the difference
between the setpoint π(π‘) and the measured process output π¦(π‘) πΎc is the proportional gain πi is the integral time πd is the derivative time π’0 is the βnormalβ value of the controller output
The transfer function of the PID controller is
πΊPID =π(π )
πΈ(π )= πΎc 1 +
1
πiπ + πdπ =
πΎc
πiπ 1 + πiπ + πiπdπ
2 (7.2)
π(π ) is the Laplace transform of π’ π‘ β π’0 πΈ(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 πi and πd, 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 πd = 0. Its transfer function is
πΊPI = πΎc 1 +1
πiπ =
πΎc
πiπ 1 + πiπ (7.3)
A PD controller is obtained from a PID controller by letting πi = β. Its transfer function is
πΊPD = πΎc 1 + πdπ (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 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
πΊPIPD = πΎcβ² 1 +
1
πiβ²π
1 + πdβ²π =
πΎcβ²
πiβ²π
1 + πiβ²π 1 + πd
β²π (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 easy to find the controller parameters of the series form by frequency analytic methods by so-called lead-lag design.
Exercise 7.1
Which is the control law in the time domain for a series form PID controller?
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7.1 PID controller variants
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7. 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 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
πΊPIDf = πΎc 1 +1
πiπ +
πdπ
πfπ +1(7.6)
The transfer function of a series form PID controller with a derivative filter is usually stated in the form
πΊPIPDf = πΎcβ² 1 +
1
πiβ²π
πdβ²π +1
πfβ²π +1
(7.7)
πf and πfβ² are filter constants, usually 10-30 % of corresp. 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 form
If the parameters of the series form are known, the corresponding parameters of the parallel form can be calculated according to
πi = πiβ² + πd
β² β πfβ² , πd = πd
β² πiβ²
πiβ πf
β² , πf = πfβ² , πΎc = πΎc
β² πiβ²
πi(7.8)
For calculation of the parameters of the series form from the parameters of the parallel form, we define the parameter
πΏ = 1 β4πi(πd+πf)
(πi+πf)2 (7.9)
If πΏ 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
πiβ² =
(πi+πf)
21 + πΏ , πd
β² = πi + πf β πiβ² , πf
β² = πf , πΎcβ² = πΎc
πiβ²
πi(7.10)
The condition for πΏ β₯ 0 in terms of the controller parameters is
πd β€(πiβπf)
2
4πi(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 output
Even if we have a derivative filter, a step change in the setpoint π(π‘)tends to affect the derivative part much more strongly than a disturbance in the output π¦(π‘). A remedy to this is to
differentiate the (filtered) output instead of the control error π(π‘).
The ideal control law (7.1) then becomes
π’ π‘ = πΎc π π‘ +1
πi 0π‘π π dπ β πd
dπ¦f(π‘)
dπ‘+ π’0 (7.12a)
πfdπ¦f(π‘)
dπ‘+ π¦f π‘ = π¦(π‘) (7.12b)
In the Laplace domain we get
π π = πΎc 1 +1
πiπ π π β πΎπ 1 +
1
πiπ +
πdπ
πfπ +1π(π ) (7.13)
which is a combination of a PI controller and a PID controller
π π = πΊPIπ π β πΊPIDfπ(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 7.2
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
πp = ππ β π¦ , π = π β π¦ , πd = ππ β π¦f (7.15)
where π and π are setpoint weights, the control law becomes
π’ π‘ = πΎc πp π‘ +1
πi 0π‘π π dπ + πd
dπd(π‘)
dπ‘+ π’0 (7.16a)
πfdπ¦f(π‘)
dπ‘+ π¦f 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
π π = πΊvPIDπ π β πΊPIDfπ(π ) (7.17)where
πΊvPID = πΎc π +1
πiπ + ππdπ (7.18)
and πΊPIDf is as in (7.6).
With suitable choices of π and π, all previously treated PID controllers on parallel form can be obtained.
π and π do not affect the controllerβs ability to reject disturbances in the output, only the ability to track setpoint changes.
πΊvPID can be tuned for setpoint tracking and πΊPIDf for disturbance rejection (i.e., πΎc, πi and πd need not have the same values in πΊvPIDand πΊPIDf).
Exercise 7.3
Include setpoint weighting in the series form of a PID controller.
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7.1.5 Setpoint weighting
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7. 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 πΎc = 0 because that would disconnect all parts; it would put the controller on βmanualβ with π’ π‘ = π’0.
Tuning the proportional part by adjusting πΎc 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‘ = πΎcππ π‘ + πΎi 0π‘π π dπ + πΎd
dπd(π‘)
dπ‘+ π’0 (7.19)
is a more flexible control law. In the Laplace domain it can be written
π π = πΊvP+I+Dπ π β πΊP+I+Dfπ(π ) (7.20)where
πΊvP+I+D = πΎcπ + πΎiπ β1 + ππΎdπ (7.21a)
πΊP+I+Df = πΎc + πΎiπ β1 + πΎdπ (πfπ + 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 π(π‘) should be corrected by an increase of the control signal π’(π‘), the control law is
π’ π‘ =
π’max if π π‘ > πhiπ’0 or unchanged if πlo β€ π π‘ β€ πhiπ’min if π π‘ < πlo
(7.23)
where π’max, π’0, π’min is the high, normal, low value of the control signal. The interval (πlo, πhi) is a dead zone. In the simplest case, πlo = πhi = 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 controller
A P controller implements the simple control law
π’ π‘ = πΎcπ π‘ + π’0 (7.24)
where πΎc is the adjustable controller gain and π’0 is the normal value of the control signal, which is also be adjustable. In principle, π’0 is selected to make the control error 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 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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7.2 Choice of controller type
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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
π’ π‘ = πΎc π π‘ +1
πi 0π‘π π dπ + π’0 (7.25)
where the gain πΎc and the integral time πi are adjustable parameters; 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 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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7.2 Choice of controller type
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7. PID Controllers
7.2.4 PD controller
The ideal form of a PD controller implements the control law
π’ π‘ = πΎc π π‘ + πddπ(π‘)
dπ‘+ π’0 (7.26)
where the gain πΎc and the derivative time πd are adjustable para-meters; 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 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 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 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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7.2 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 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.
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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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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. PID Controllers
7.3.3 Proportional band and integrator windup
Proportional band
A 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’min β€ π’(π‘) β€ π’max , a P controller can according to (7.24) handle a control error that satisfies
π’minβπ’0
πΎcβ‘ πmin β€ π(π‘) β€ πmax β‘
π’maxβπ’0
πΎc(7.27)
The PB is equal to πmax β πmin = π¦hi β π¦lo, where π¦hi is the highest output (πmin = π β π¦hi) and π¦lo is the lowest output (πmax = π β π¦lo) the controller can handle. Usually, the PB is defined in percent of the total measurable output interval π¦min, π¦max . Then, the PB is
πb =π¦hiβπ¦lo
π¦maxβπ¦min100% =
π’maxβπ’min
π¦maxβπ¦minβ 100%
πΎc(7.28)
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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
πΎc =π¦hiβπ¦lo
π¦maxβπ¦min100% =
π’maxβπ’min
π¦maxβπ¦minβ 100%
πb(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
πb = 100%/πΎc (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 π’0 is in the middle of the interval π’min, π’max , a P controller with πb = 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 πΎc).
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Proportional band
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7.3 Specifications and performance criteria
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.
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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 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.
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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 π
the rise time π‘r the settling time π‘πΏ the relative damping π
the ratio between successive relative overshoots (or undershoots) πR
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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 π and π‘r are sufficient to determine the transfer function of an underdamped second-order system with a given gain.
The settling time π‘πΏ can be used instead of π or π‘r , but the relationships are then only approximate.
The relative damping π or the overshoot ratio πR can be specified instead of π.
Some classical tuning recommendations are based on the specification πR = 1/4.
This may be acceptable for regulatory control, but not for setpoint tracking. πR = 1/4 corresponds to π = 0.5 (i.e., a 50 % overshoot) and π = 0.22 .
For setpoint tracking, π β 0.1 (π β 0.6) is usually more appropriate.
If an overdamped closed-loop response is desired, this cannot be achieved with a specification π > 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 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
π½IAE = 0π‘s π(π‘) dπ‘ , π½ISE = 0
π‘s π(π‘)2 dπ‘
π½ITAE = 0π‘s π‘ π(π‘) dπ‘ , π½ITSE = 0
π‘s π‘π(π‘)2 dπ‘(7.31)
where the acronyms are
β IAE = βintegrated absolute errorβ
β ISE = βintegrated square errorβ
β ITAE = βintegrated time-weighted absolute errorβ
β ITSE = βintegrated time-weighted square errorβ
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 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 controlled system is of second order, i.e.,
πΊ π =πn2
π 2+2ππnπ +πn2 (7.32)
In the figure, IAE and ISE are normalized with πn , ITAE and ITSE with πn2.
As can be seen, every normalized error integral has a minimum for a given relative damping π .This damping as well as thecorresponding relative over-shoot π are shown below.
Table 7.1 Optimal relativedamping for 2nd order system.
Error integrals as function of π.
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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. PID Controllers
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 (πΊc = πΎc) is used for the first experiment. A low value is chosen for the gain πΎc . Note that πΎc must have the same sign as πΎp .
2. A change in the setpoint π is introduced. (Some other disturbance could also be used.) The controller gain πΎc is increased until the output π starts to oscillate with a constant amplitude (see next slide).
3. The value of the controller gain yielding constant oscillations is denoted πΎc,max . The period of the oscillations is denoted πc .
4. The controller gain is changed to πΎc = 0.5πΎc,max . If the intention was to tune a P controller, this is the final tuning.
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7.4.1 Experimental tuning
G
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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 usingπΎc = 0.5πΎc,max . A large value isinitially used for the integral time πi .
6. A change in the setpoint π (or someother disturbance) is introduced. Theintegral time πi is reduced until πstarts to oscillate with a constantamplitude. This occurs at πi = πi,min .
7. The integral time for a PI or PID controller is chosen as πi = 3πi,min .
8. To tune the derivative part of a PID (or PD) controller, an experiment is done with such a controller using πΎc = 0.5πΎc,max , πi = 3πi,min (if a PID controller). The derivative time is initially set at πd = 0 .
9. A change in the setpoint π (or some other disturbance) is introduced.The derivative time πd is increased until the output π starts to oscillate with a constant amplitude. This occurs when πd = πd,max.
10. The derivative time for a PD or PID controller is set at πd =1
3πd,max .
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7.4.1 Experimental tuning
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7.4 Tuning based on frequency response
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7.4.1 Experimental tuning
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 πΎc and the integral time πi typically affect the control performance. The optimal performance is in this case obtained by πΎc = 3 and πi = 11 .
πi = 5 πi = 11 πi = 20
πΎc = 5
πΎc = 3
πΎc = 1
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7. PID Controllers
7.4.2 Ziegler-Nicholsβs recommendations
In 1942, Ziegler and Nichols suggested tunings for P, PI and PID controllers based on πΎc,max and πc 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. π = 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 < 0.5, frequency response (0.1 < π < 0.5).where
π = πΎπΎc,maxβ1
.
πΎ is the process gain.
The critical frequency πc isoften used instead of πc:
πc = 2π/πc.
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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
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7. PID Controllers
7.4.3 Γ strΓΆmβs and HΓ€gglundβs correlations
In 2006, Γ strΓΆm and HΓ€gglund showed that, in general, πΎc,max and πcalone do not provide sufficient information for good controller tuning.
In addition to πΎc,max and πc , Γ strΓΆm and HΓ€gglund also use the
parameter π = πΎπΎc,maxβ1
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. Γ strΓΆm-HΓ€gglundβs controllernot be used below the tuning correlations based on frequencyrange π > 0.1 . Response (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 ) β
PID 40.3 0.1 0.6
1 2
0.15(1 )
1 0.95
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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’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 π and πΏ 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
KEH Process Dynamics and Control 4β34
7.5 Tuning based on step response
Instead of taking the π 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‘i, π¦i) of the inflexion point are known. The calculation is valid for any size of π’step . The formula for π is
π =πΏπ¦i
π’step(π‘iβπΏ)(7.34)
Another useful parameter is
π = πΏ/πeq , πeq = π‘63 β πΏ (7.35)
where πeq is the equivalent time constant of the system and 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‘i, π¦i) for calculation of πaccording to (7.34).
caKi /T Td /T LcaKi /T Td /T L
ProcessControl
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7. PID Controllers
7.5.1 Ziegler-Nicholsβs 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 πΏ > 0 and preferably 0.1 β€ π 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 method
In 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 πΏ > 0 and preferably 0.1 β€ π 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
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
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
i eq/T T i eq/T T
ProcessControl
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7. PID Controllers
7.5.3 Γ strΓΆmβs and HΓ€gglundβs correlations
In 2006, Γ strΓΆm and HΓ€gglund presented improved controller tunings based on a step response. In addition to π and πΏ , they use π in their correlations, which can be used for all π β₯ 0. However, for π < 0.4 , they tend to be conservative. For an integrating process, π = 0 is used.
The tunings are primarily intended for regulatory control. For setpoint tracking, setpoint weighting can be used as follows:
PI control: π = 1 if π > 0.4 , π < 1 if π β€ 0.4 (optimal π is unclear)
PID control: π = 1 if π > 1 , π = 0 if π 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 )
2
130.35
1 12 7
β
PID 0.45 0.2 8 4
1 10
0.5
1 0.3
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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
πΊ π =πΎ
ππ +1eβπΏπ (7.36)
from which the parameters π and π can be calculated according to
π =πΎπΏ
π, π =
πΏ
π(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 βgeneral purposeβmethods 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 delay
The transfer function is defined in (7.36) and the parameter π in (7.37).
Minimization of error integrals
Controller tunings that minimize IAE and ITAE when 0.1 β€ π 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 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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7.6 Model-based controller tuning
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 π values.
Table 7.10. Cvejnβs 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.
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
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
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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
πΊ π =πΎπn
2
π 2+2ππnπ +πn2 e
βπΏπ (7.40)
In Cvejnβs method for tracking control, the controller πΊc(π ) is tuned to give the loop transfer πΊk(π ) = πΊ(π )πΊc(π ) such that
πΊk π =1
2πΏπ eβπΏπ (7.38)
or
πΊk π =1
41 +
3
πΏπ eβπΏπ (7.39)
Tuning by (7.38) gives better stability, (7.39) gives better performance.
Exercise 7.3
Use Cvejnβs method for tracking control to tune a PID controller for the system (7.40).
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 zeros
For an overdamped (or critically damped) second-order system, π β₯ 1. In this case, (7.40) is more conveniently written as
πΊ π =πΎ
(π1π +1)(π2π +1)eβπΏπ , π1 β₯ π2 (7.41)
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:
πΎπΎc = 0.19 + 0.37π1β1 + 0.18π2
β1 + 0.02π1β1π2
β1
πΎπΎcπΏ/πi = 0.48 + 0.03π1β1 β 0.0007π2
β1 + 0.0012π1β1π2
β1 (7.42)
πΎπΎcπd/πΏ = 0.29 + 0.16π1β1 β 0.2π2
β1 + 0.28π1β1π2
β1 π1+π2
π1+π2+π1π2
whereπ1 = πΏ/π1 , π2 = πΏ/π2 (7.43)
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 integration
A second-order no-zero system including an integrator has the transfer function
πΊ π =πΎ
π (π2π +1)eβπΏπ (7.44)
For this kind of system, Γ strΓΆm and HΓ€gglund (2006) suggest the tuning:
πΎπΎcπΏ = 0.37 + 0.02π2β1
πΎπΎcπΏ2/πi = 0.03 + 0.0012π2
β1 (7.45)
πΎπΎcπd = 0.16 + 0.28π2β1
If the system is a double integrator with the transfer function
πΊ π =πΎ
π 2eβπΏπ (7.46)
the suggested tuning is
πΎπΎcπΏ2 = 0.02
πΎπΎcπΏ3/πi = 0.0012
πΎπΎcπdπΏ = 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 zero
An overdamped 2nd order system with a zero has the transfer function
πΊ π =πΎ(π3π +1)
(π1π +1)(π2π +1)eβπΏπ (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
πΊ π =πΎ(π3π +1)
π (π2π +1)eβπΏπ , π3 > π2 > 0 (7.48)
An IPZ system is difficult to approximate by a simpler one, esp. if π3 β« π2.
In Table 7.11, Table 7.11 SlΓ€ttekeβs regulatory tuning for an IPZ process.π2 = πΏ/π2. ForPID control, aderivative filterπf = 0.1πd isused. For set-point tracking,π < 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)
2
2
100 17
11 94
β
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 )
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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 functions
Consider the closed-loopsystem in the figure with thefollowing transfer functions:β πΊ π process being controlledβ πΊc π controllerβ πΊd π disturbance system Block diagram of closed-loop system
Standard block diagram algebra gives
π =πΊπΊc
1+πΊπΊcπ +
πΊd
1+πΊπΊcπ (7.49)
where
πΊr =πΊπΊc
1+πΊπΊc, πΊv =
πΊd
1+πΊπΊc(7.50,51)
are the closed-loop transfer functions from the setpoint π and the disturbance π to the output π.
The user can specify the desired πΊr for setpoint tracking or πΊv for regu-latory control. For setpoint tracking, the required controller is given by
πΊc =1
πΊ
πΊr
(1βπΊr)(7.52)
KEH Process Dynamics and Control 7β47
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
πΊ =πΎ
ππ +1(7.53)
Assume that we want the controlled system to behave as a first-order system with the time constant πr . Then,
πΊr =1
πrπ +1, which gives
πΊr
1βπΊr=
1
πrπ (7.54)
Substitution of (7.53) and (7.54) into (7.52) gives
πΊc =ππ +1
πΎ
1
πrπ =
π
πΎπr1 +
1
ππ (7.55)
which is a PI controller with the parameters
πΎc =π
πΎπr, πi = π (7.56)
Here, πr 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 zero
A second-order system with no zero and no time delay has the transfer function
πΊ π =πΎπn
2
π 2+2ππnπ +πn2 (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
πΊc =π 2+2ππnπ +πn
2
πΎπn2
1
πrπ =
2π
πΎπnπr1 +
πn
2ππ +
π
2ππn(7.58)
which is an ideal PID controller with the parameters
πΎc =2π
πΎπnπr, πi =
2π
πn, πd =
1
2ππn(7.59)
Also here, πr is a design parameter which only affects the controller gain.
KEH Process Dynamics and Control 7β49
7.7.2 Low-order min-phase systems
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7.7 Controller tuning by direct synthesis
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
πΊ π =πΎ(π3π +1)
(π1π +1)(π2π +1), ππ β₯ 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
πΊc =(π1π +1)(π2π +1)
πΎ(π3π +1)
1
πrπ =
1
πΎπrπ
π1π2π 2+ π1+π2 π +1
π3π +1
=1
πΎπrπ 1 + π1 + π2 β π3 π +
π1π2β π1+π2βπ3 π3
π3π +1π 2
or
πΊc = πΎc 1 +1
πiπ +
πdπ
πfπ +1(7.61)
where
πΎc =π1+π2βπ3
πΎπr, πi = π1 + π2 β π3 , πd =
π1π2
π1+π2βπ3β π3 , πf = π3 (7.62)
This is a PID controller with a derivative filter.
KEH Process Dynamics and Control 7β50
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
πΊ = πΎ π=π+1π+π (πππ +1)
π=1π (πππ +1)
, ππ > 0 , ππ > 0 , π > 2 (7.63)
If π = 3 and π = 0 or 1 , a closed-loop system of second order can be obtained by a full PID controller.
If π > 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 πΊris thus not practical.
In the case of π > 3 , two possibilities are to specify a closed-loop system of first or second order and then to
first calculate a πΊc according to (7.52), then to approximate πΊc by a PID controller;
first approximate πΊ 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.
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 zero
A second-order system with real poles and a right half plane (RHP) zero has the transfer function
πΊ π =πΎ(βπ3π +1)
(π1π +1)(π2π +1), ππ β₯ 0 (7.71)
Now division by πΊ in (7.52) will result in an unstable controller with a RHP pole if πΊr 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 πΊ .
Another solution is to include the same RHP zero in πΊr as in πΊ ; it will then be cancelled out in (7.52) and the controller will automatically be stable. This means that the choice of πΊr 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 order
The closed-loop transfer function is chosen as
πΊr =βπ3π +1
πrπ +1, which gives
πΊr
1βπΊr=
βπ3π +1
(πr+π3)π (7.72)
Substitution of (7.71) and (7.72) into (7.52) gives
πΊc =(π1π +1)(π2π +1)
πΎ
1
(πr+π3)π =
π1+π2
πΎ(πr+π3)1 +
1
π1+π2 π +
π1π2π
π1+π2(7.73)
which is a PID controller with the parameters
πΎc =π1+π2
πΎ(πr+π3), πi = π1 + π2 , πd =
π1π2
π1+π2(7.74)
KEH Process Dynamics and Control 7β53
7.7.4 System with RHP zero
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7.7 Controller tuning by direct synthesis
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 πΊr is
πΊr =(βπ3π +1)πr
2
π 2+2πrπrπ +πr2 , which gives
πΊr
1βπΊr=
(βπ3π +1)πr2
π (π +2πrπr+π3πr2)
(7.75)
To simplify the derivation of controller parameters, we define
πf = 1/(2πrπr + π3πr2) (7.76)
Substitution of (7.71) and (7.75) into (7.52), gives with (7.76)
πΊc =(π1π +1)(π2π +1)πfπr
2
πΎ πfπ +1 π =
πfπr2
πΎπ
π1π2π 2+ π1+π2 π +1
πfπ +1(7.77)
Analogously with the derivation of (7.62), this gives the PID controller parameters (7.76) and
πΎc =πfπr
2
πΎ(π1 + π2 β πf), πi = π1 + π2 β πf , πd =
π1π2
π1+π2βπfβ πf (7.78)
where πf is the derivative filter time constant in a PID controller (7.61).
KEH Process Dynamics and Control 7β54
7.7.4 System with RHP zero
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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 πr, and the undamped natural frequency πr. 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 πΊr have two equally large real poles at β1/πr . This corresponds to πr = 1 and πr = 1/πr , which for (7.76) gives
πf =πr2
2πr+π3
Let πΊr have real poles at β1/πr and β1/π3 . This corresponds to
πr = 0.5(πr + π3)πr and πr = 1/ πrπ3 , which for (7.76) gives
πf =πrπ3
πr+2π3
KEH Process Dynamics and Control 7β55
Closed-loop system of 2nd order
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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
πΊ π =πΎ
ππ +1eβπΏπ (7.79)
Calculation of a controller by (7.52) will then result in a controller containing a time delay β there is no practical way to avoid this by the choice of πΊr.
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.
KEH Process Dynamics and Control 7β56
7.7 Controller tuning by direct synthesis
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7.7 Controller tuning by direct synthesis
Time-delay approximation in the model
A standard way of approximating a time delay is to use a PadΓ© approxi-mation. I first-order PadΓ© approximation
eβπΏπ β1β0.5πΏπ
1+0.5πΏπ (7.80)
gives the model
πΊ π =πΎ(β0.5πΏπ +1)
(ππ +1)(0.5πΏπ +1)(7.81)
A natural choice for πΊr is then
πΊr =β0.5πΏπ +1
(πrπ +1)(0.5πΏπ +1), which gives
πΊr
1βπΊr=
β0.5πΏπ +1
π (0.5πrπΏπ +πr+πΏ)(7.82)
Substitution of (7.81) and (7.82) into (7.52) gives a PID controller with the parameters
πΎc =π+0.5πΏβπf
πΎ(πr+πΏ), πi = π + 0.5πΏ β πf , πd =
0.5πΏπ
π+0.5πΏβπf, πf =
0.5πΏπr
πr+πΏ(7.83)
Here, πf is the time constant of a derivative filter in the PID controller (7.61).
KEH Process Dynamics and Control 7β57
7.7.5 1st order system with a delay
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7.7 Controller tuning by direct synthesis
Time-delay approximation in the controller
If eβπΏπ is retained in the model, it also has to be part of πΊr , because it is impossible for the closed-loop system to have a shorter time-delay than the uncontrolled system.
If πΊr is chosen to be first order with a time delay
πΊr =1
πrπ +1eβπΏπ , which gives
πΊr
1βπΊr=
eβπΏπ
πrπ +1βeβπΏπ (7.84)
Substitution of (7.79) and (7.84) into (7.52) gives
πΊc =ππ +1
πΎ(πrπ +1βeβπΏπ )
(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 eβπΏπ β 1 β πΏπ gives a PI controller with
πΎc =π
πΎ(πr+πΏ), πi = π (7.86)
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7.7.5 1st order system with a delay
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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 (7.85) 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.
KEH Process Dynamics and Control 7β59
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7. PID Controllers
7.8.1 The IMC structure
Consider the closed-loopsystem in the figure with thefollowing transfer functions:β πΊ π true processβ πΊ π process modelβ πΊIMC π a controllerβ πΊd π disturbance system
Standard block diagram algebra The IMC structure.gives π = πΊIMC(πΈ + πΊπ) from whichπ
πΈ= πΊc = πΌ β πΊIMC
πΊβ1πΊIMC = πΊIMC πΌ β πΊπΊIMC
β1=
πΊIMC
1β πΊπΊIMC(7.87)
Assume thatπΊIMC = πΊβ1πΊf (7.88)
where πΊf is a βfilterβ. Substitution of (7.88) into (7.87) gives
πΊc = πΊβ1πΊf πΌ β πΊfβ1 =
1
πΊ
πΊf
(1βπΊf)(7.89)
If the filter is chosen as πΊf = πΊr (and πΊ = πΊ), this is the same as (7.52) !
KEH Process Dynamics and Control 7β60
7.8 Internal model control
( )G s
Λ ( )G s
( )E s
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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., πΊ = πΎeβπΏπ /(ππ + 1). Choose the IMC filter as πΊf = eβπΏπ /(πrπ + 1) . Substitution into (7.88) now gives
πΊIMC =1
πΎ
ππ +1
πrπ +1=
1
πΎ1 +
πβπr
πrπ +1π (7.90)
which is a PD controller with a derivative filter having the parameters πΎπ = 1/πΎ , πd = π β πr , πf = πr . Substitution of (7.90) and the model πΊ into (7.87) gives
πΊc =ππ +1
πΎ(πrπ +1βeβπΏπ )
(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 πΊIMC , but the feedback of πΊ in the IMC structure introduces integration if πΊIMC has been calculated using the same πΊ in (7.88); integration is achieved even if πΊ β πΊ .
Exercise. Calculate the closed-loop transfer function πΊr when πΊ β πΊ . Show that there will be no steady-state error, i.e., that πΊr 0 = 1 .
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7.8 Internal model control
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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 πΊIMC combined with the feedback of πΊ is equivalent to πΊc.
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 πΊ and the model πΊ.
β πΊ predicts the output π, which is compared with the true output π.
β Only the prediction error πΈ = π β π is fed back, not the entire π.
The latter property is a clearadvantage in controller design.If πΊ = πΊ (i.e., πΈ = 0)
πΊr = πΊπΊIMC (7.93)
which means that the closed-loop transfer function dependslinearly on πΊIMC making designof πΊIMC easier than design of πΊc. Predictive nature of IMC structure.
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7.8 Internal model control
Λ ( )G s
( )G s
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7. PID Controllers
7.8.4 Controller design
The following conclusions can be drawn from (7.93):
A stable closed-loop system πΊr requires a stable IMC controller πΊIMC ; in particular, the IMC controller may not contain integral action.
Non-minimum phase properties (i.e., RHP zeros and time delays) in πΊwill also be present in πΊr because they cannot be cancelled out by a stable and realizable πΊIMC.
From (7.88) it follows that
the filter πΊf has to be chosen to cancel out non-minimum phase prop-erties of πΊ β this is equivalent to the choice of πΊr in direct synthesis.
In practice, the IMC design is done differently. Instead of guaranteeing the stability and realizability of πΊIMC by the choice of πΊf , it is handled by the choice of πΊ to be inverted: non-minimum phase parts of πΊ are not inverted.
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7.8 Internal model control
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7.8 Internal model control
The process model πΊ can always be factorized as
πΊ = πΊβ πΊβ (7.94)
where πΊβ contains all non-minimum-phase elements of πΊ, but no minimum-phase elements, and normalized so that πΊβ 0 = 1 (i.e., it has the static gain 1). This means that πΊβ contains all RHP zeros and time delays of πΊ ; if there are no such elements, πΊβ = 1.
When πΊIMC is calculated according to (7.88), only πΊβ is inverted. Thus,
πΊIMC = πΊβ β1πΊf (7.95)
Note that the full πΊ should be used as internal model as illustrated by the IMC block diagrams β the use of πΊβ is only a technical aid for the calculation of πΊIMC .
The IMC filter πΊf 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
πΊf =1
(πrπ +1)π (7.96)
is chosen. Here, π is an integer, usually π = 1, sometimes π > 1.
KEH Process Dynamics and Control 7β64
7.8.4 Controller design
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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 πΊIMC has been designed, the corresponding βregularβ controller πΊc can be calculated according to (7.87). If πΊ contains a time delay, it will also be present in πΊc. 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
π1 > 0 , π2 β₯ 0 , π3 β₯ 0 , πΏ β₯ 0 (7.101)
The tunings can be used for (underdamped) models expressed by the relative damping and the natural frequency by the substitutions
π1 + π2 = 2π/πn , π1π2 = 1/πn2 (7.103)
Usually πr is chosen such that πΏ β€ πr < π (but no clear consensus).
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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 close-loop system is πr . π , which is used in the calculations, is closely related to πr . Note that the calculated integral time πi is used in several expressions.
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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
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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 πΎc,max , πc (or πc) and π . These methods are βgeneral-purpose methodsβ not optimized for any specific model type.
Section 7.5: Controller tuning based on step-response parameters π (or π‘i, π¦i), πΏ and π. 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.
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7. PID Controllers
7.9.1 Skogestadβs 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 πΊβ and a non-minimum-phase part πΊβ , i.e.,
πΊ π = πΊβ(π )πΊβ(s) (7.106)
Any left-half plane (LHP) zeros of πΊβ(s) and RHP zeros of πΊβ(π ) are eliminated by suitable approximations.
Elimination of LHP zeros
If the poles and zeros are real, the minimum-phase part has the form
πΊβ π =πΎ ππ+1π +1 ππ+2π +1 β¦(ππ+ππ +1)
π1π +1 π2π +1 β¦(πππ +1)(7.107)
where π1 β₯ π2 β₯ β― β₯ ππ > 0, ππ+1 β₯ ππ+2 β₯ β― β₯ ππ+π > 0 , π > π. The simplification procedure now goes as follows.
The numerator time constants ππ+1, ππ+2, β¦, ππ+π are considered in that order. Assume that ππ+π is the one currently being considered.
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7.9 Model simplification
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7.9.1 Skogestadβs method
Next, the smallest remaining denominator time constant ππ such that ππ β₯ ππ+π is selected. If there is no such time constant, or if ππ β«ππ+π, the smaller ππ closest to ππ+π is chosen. It is considered that
ππ β« ππ+π if ππ > ππ+π2 /ππ+1 and ππ+π/ππ+1 < 1.6 .
The ratio (ππ+ππ + 1)/(πππ + 1) is now approximated as
ππ+ππ +1
πππ +1β
ππ+π/ππ if ππ β₯ ππ+π β₯ 5πr a5πr/ππ
5πrβππ+π π +1if ππ β₯ 5πr β₯ ππ+π b
1
ππβππ+π π +1if 5πr β₯ ππ β₯ ππ+π c
ππ+π/ππ if ππ+πβ₯ ππ β₯ πr (d)
ππ+π/πr if ππ+πβ₯ πr β₯ ππ (e)
1 if πr β₯ ππ+π β₯ ππ (f)
(7.108)
Here, πr is the desired closed-loop time constant. If this is not known, the suggested value is πr = πΏ , which is the time delay in the simplified model. Since this is not initially known, one may have to iterate (i.e., first guessing πΏ, then possibly correcting with the new πΏ).
KEH Process Dynamics and Control 7β69
Elimination of LHP zeros
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7.9.1 Skogestadβs method
The above procedure gives an approximate minimum-phase part πΊβ of the form
πΊβ π = πΎ
π1π +1 π2π +1 β¦( π ππ +1)(7.109)
Note that the gain as well as the values and number of denominator time constants may have changed from the original πΊβ.
Elimination of RHP zeros and the half rule
The transfer function πΊ π = πΊβ(π ) πΊβ(s) now has the form
πΊ π = πΎ βππ+π+1π +1 βππ+π+2π +1 β¦(βππ+π+ππ +1)
π1π +1 π2π +1 β¦( π ππ +1)eβπΏπ (7.110)
where π1 β₯ π2 β₯ β― β₯ π π > 0, ππ+π+1 β₯ ππ+π+2 β₯ β― β₯ ππ+π+π > 0 .
Skogestadβs half rule
If an approximate model of order π is desired, the π largest denomi-nator time constants are retained in the model with the modification that half of π π+1 is added to π π. Half of π π+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.
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Elimination of LHP zeros
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7.9.1 Skogestadβs method
Approximation by first-order system
If a first-order model is desired, the half rule gives
πΊ π = πΎ
ππ +1eβ πΏπ (7.111a)
π = π1 +1
2 π2 , πΏ = πΏ + 1
2 π2 + π=3
π ππ + π=1π
ππ+π+π (7.111b)
Approximation by second-order system
If a second-order model is desired, the half rule gives
πΊ π = πΎ
( π1π +1)( π2π +1)eβ πΏπ (7.112a)
π2 = π2 +1
2 π3 , πΏ = πΏ + 1
2 π3 + π=4
π ππ + π=1π
ππ+π+π (7.112b)
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Elimination of RHP zeros and the half rule
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7.9 Model simplification
Example 7.2. IMC via model reduction by Skogestadβs method.
Simplify the model
πΊ π =(16π +1)(4π +1)(β8π +1)eβ2π
(50π +1)(20π +1)(12π +1)(6π +1)(3π +1)(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 πr = 10.
Here
πΊβ π =(16π +1)(4π +1)
(50π +1)(20π +1)(12π +1)(6π +1)(3π +1)(π +1).
According to (7.108c), 16π +1
20π +1β
1
4π +1. The numerator factor (4π + 1) can
now be cancelled out against the new denominator factor, which gives
πΊβ π =1
(50π +1)(12π +1)(6π +1)(3π +1)(π +1)
and
πΊ π =(β8π +1)eβ2π
(50π +1)(12π +1)(6π +1)(3π +1)(π +1).
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7.9.1 Skogestadβs method
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7.9.1 Skogestadβs method
The resulting second-order model is
πΊ π =1
( π1π +1)( π2π +1)eβ πΏπ
with π1 = 50 , π2 = 12 + 1
2β 6 = 15 , πΏ = 2 + 1
2β 6 + 3 + 1 + 8 = 17.
Thus πΊ π =
1
(50π +1)(15π +1)eβ17π .
According to Table 7.12 for IMC-based tuning of second-order model:
β π = πr + πΏ = 10 + 17 = 27
β πi = π1 + π2 = 50 + 15 = 65
β πΎc = πi/( πΎπ) = 65/(1 β 27) = 2.4
β πd = π1 π2/πi = 50 β 15/65 = 11.5
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Example 7.2
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7. PID Controllers
7.9.2 Isakssonβs and Graebeβs 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 PadΓ©approximation.
To describe the method, both factorized and polynomial forms of the original transfer function are employed. If the numerator order is π and the denominator order is π , the transfer function is
πΊ π = πΎππ+1π +1 ππ+2π +1 β¦(ππ+ππ +1)
π1π +1 π2π +1 β¦(πππ +1)(7.113a)
= πΎπ0π
π+β―+ππβ2π 2+ππβ1π +1
π0π π+β―+ππβ2π
2+ππβ1π +1(7.113b)
where π1 β₯ π2 β₯ β― β₯ ππ > 0 (i.e., a stable system) and |ππ+1| β₯|ππ+2| β₯ β― β₯ |ππ+π| . The numerator time constants can be positive or negative.
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7.9 Model simplification
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7.9 Model simplification
If a model with the numerator order π and the denominator order π is desired, the simplified model is
πΊ π = πΎππ+1π +1 β¦(ππ+ ππ +1) + ππβ ππ π+β―+ππβ1π +1
π1π +1 β¦(π ππ +1) + ππβ ππ π+β―+ππβ1π +1
(7.114)
Complex-conjugated poles or zeros is no problem, except if they occur as poles number π and π + 1 or zeros number π + π and π + π + 1. One solution is then to use the real part of the complex conjugate as π πor ππ+ π .
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
πΊ π =πΎ
1
2π1+ππβ1 π +1
(1st order) (7.115)
πΊ π =πΎ
1
2ππ+1+ππβ1 π +1
1
2π1π2+ππβ2 π 2+
1
2π1+π2+ππβ1 π +1
(2nd order) (7.116)
where
ππβ1 = π=1π ππ+π , ππβ1 = π=1
π ππ , ππβ2 =1
2 π=1π ππ
2β π=1
π ππ2 (7.117)
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7.9.2 Isakssonsβs and Graebeβs method
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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 Graebeβsmodel reduction method.
The model gives
ππβ1 = 16 + 4 β 8 = 12 , ππβ1 = 50 + 20 + 12 + 6 + 3 + 1 = 92
ππβ2 =1
2922β(502+202+122+62+32+12) = 2687
from which
πΊ π =1
216+12 π +1
1
21000+2687 π 2+
1
270+92 π +1
eβ2π =(14π +1)eβ2π
1843.5π 2+81π +1
This model has complex-conjugated poles, but according to (7.103), π1 +π2 = 81 and π1π2 = 1843.5 can be used in the controller calculations. Table 7.12 for IMC-based tuning of second-order model then gives
β π = πr + πΏ = 10 + 2 = 12
β πi = π1 + π2 β π3 = 81 β 14 = 67
β πΎc = πi/(πΎπ) = 67/(1 β 12) = 5.6 (much bigger than in Ex. 7.2!)
β πd = π1 π2/πi β π3 = 1843.5/67 β14 = 13.5
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7.9.2 Isakssonsβs and Graebeβs method