Lecture9 Controller tuning

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ChE 170
Lecture 9
Controller tuning
Criteria for good control
• Most operators of processes know what they want in the form of a
response to a change in set point or load.
• For example, a response that gives minimum overshoot and ¼ decay ratio is
often considered as a satisfactory response.
• In many cases, tuning is done by trial and error until such a response is
obtained.
Performance Criteria for Closed-Loop Systems
1. The closed-loop system must be stable
2. The effects of disturbances are minimized, providing good disturbance
rejection
3. Rapid, smooth responses to set-point changes are obtained, that is, good
set-point tracking
4. Steady-state error (offset) is eliminated
5. Excessive control action is avoided.
6. The control system is robust, that is, insensitive to changes in process
conditions and to inaccuracies in the process model
Tradeoffs between Performance and Robustness
Rapid
Smooth
Little
oscillation
Balance
Performance
Robustness
Satisfactory
performance to wide
range of condition.
Conservative
controller settings.
Tradeoffs between set point tracking and disturbance
rejection
Balance
Excellent set point
tracking will provide
sluggish response to
disturbance
Set point
Tracking
Disturbance
Rejection
Excellent disturbance
rejection will produce
large overshoot in
set-point changes.
Performance Criteria
Quarter-Amplitude Decay
If a loop is oscillating, each peak deviation should be only onefourth of the previous peak deviation on the same side of set
point.
Also called “Quarter-wave decay”, “Quarter-wave damping”
Quarter-Amplitude Decay Response
Quarter-Amplitude Decay Response
Quarter-wave decay ratio response
•A control loop tuned to have a Quarter-wave
Amplitude decay response in a set point change
will show a sluggish response in a load
disturbance.
•A control loop tuned to have a Quarter-wave
Amplitude decay response in a load upset will
show a too oscillatory response in a set point
change.
Quarter-wave decay ratio response
• For many applications, the quarter-amplitude-decay criterion
provides acceptable damping of the oscillations that follow a set
point change. For other applications, quarter-amplitude decay
may be too oscillatory. A plant’s operations and engineering staff
may be willing to accept a more sluggish response to a load upset
in order to minimize the overshoot that follows a set point
change.
• Some operators prefer to see loops that are critically damped;
that is, they want the measurement to rise rapidly to the new set
point value and yet avoid overshoot.
Quarter-wave decay ratio response
• For many control loops, the set point is rarely changed. The
purpose of these loops is to minimize the effect of disturbances.
Even so, because set point changes are usually made more easily
than load changes, in actual practice many loops are tuned for a
suitable response to a set point change. Then, the resulting
response for a load upset is accepted even though it may not be
the best.
• A preferable tuning strategy would be to tune the controller for
best response to a load change, then use one of the set point
“softening” techniques to ameliorate the effect of occasional set
point changes.
Performance Criteria for Closed-Loop Systems
Decay ratio alone does not provide a complete performance
specification.
Other criteria that are sometimes used to specify, or measure,
control loop performance include:
 Rise time—the time between a set point change and the first
crossing of the set point;
 Overshoot ratio—the ratio of the magnitude of the first overshoot
above set point to the magnitude of the set point change itself;
 Settling time—the time, following a set point change or
disturbance, that it takes for the oscillation to become so small
that the deviation does not exceed some specified amount.
Integral Error Criteria
Integral Error Criteria
 based on minimizing the integral of
some function of the error.
Note:
Graphical interpretation of
IAE. The shaded area is the
IAE value.
Integral Error Criteria
The optimum settings minimize an integral error criterion.
Note:
Techniques for determine controller settings
1.
2.
3.
4.
5.
6.
Direct Synthesis (DS) method
Internal Model Control (IMC)
Controller tuning relations
Frequency response techniques
Computer Simulation
On-line tuning after the control system is installed
Note:
Techniques 1 – 5 are based on process models, however, initial controller
setting are typically adjusted after the control system is installed. Techniques 1
– 5 provide a good initial controller setting for the subsequent on-line tuning.
On-line tuning after the control system is installed
On-site tuning
Field tuning
Controller tuning
If preliminary settings are not satisfactory, alternative settings
are obtained from simple experimental tests. And if necessary,
the settings are fine-tuned by a modest trial-and-error
i.e. Continuous Cycling Method
Continuous Cycling Method:
• Refers to sustained oscillation with a constant amplitude
• Closed-loop tuning method (automatic mode)
• Determined empirically to provide closed-loop response that
have a ¼ decay ratio
•
(ultimate gain) and (ultimate period) be obtained from a
closed-loop test of the actual process.
• Example:
• Ziegler-Nichols (Z-N) controller settings
• Tyreus-Luyben controller settings
Experimental determination of the ultimate gain
Controller setting based on continuous cycling method
The ultimate period Pu is defined as the period of the sustained cycling that would occur
if a proportional controller with gain Kcu were used.

=  ,   
Continuous Cycling Method Steps
Step 1. After the process has reached steady state (at least approximately),
eliminate the integral and derivative control action by setting to zero
and
to the largest possible value.
Step 2. Set Kc equal to a small value (e.g., 0.5) and place the controller in
the automatic mode.
Step 3. Introduce a small, momentary set-point change so that the
controlled variable moves away from the set point. Gradually increase Kc
in small increments until continuous cycling occurs. The numerical value
of Kc that produces continuous cycling (for proportional-only control) is
called the ultimate gain, Kcu. The period of the corresponding sustained
oscillation is referred to as the ultimate period, Pu.
Continuous Cycling Method
Step 4. Calculate the PID controller settings using the Ziegler-Nichols (Z-N)
tuning relations
Step 5. Evaluate the Z-N controller settings by introducing a small set-point
change and observing the closed-loop response. Fine-tune the settings, if
necessary.
Disadvantages of Continuous Cycling Method
• Time-consuming if several trials are required and the process dynamics are
slow. The long experimental tests may result in reduced production or poor
product quality.
• In many applications, continuous cycling is objectionable because the process
is pushed to the stability limits.
• Not applicable to integrating or open-loop unstable processes because their
control loops typically are unstable at both high and low values of Kc, while
being stable for intermediate values.
• For first-order and second-order models without time delays, the ultimate
gain does not exist because the closed-loop system is stable for all values of
Kc, providing that its sign is correct. However, in practice, it is unusual for a
control loop not to have an ultimate gain.
Direct synthesis method (Controller synthesis)
Consider the block diagram for a
standard feedback control system.
The closed-loop transfer function
for set-point changes
K mGcGvG p
Y

Ysp 1  GcGvG p Gm
For simplicity, let
and assume
Gc G
Y

Ysp 1  Gc G
(12-1)
(12-2)
Rearranging and solving for Gc
1  Y / Ysp
Gc  
G  1  Y / Ysp



(12-3a)
•Let (Y/Ysp)d be the desired closed-loop transfer function
What do we choose for
?
 The specification of (Y/Ysp)d is the key design decision
 Ideally we want (Y/Ysp)d = 1, a perfect control, but this can not
be achieve for feedback control
 For process w/o time delay, 1st order model is a reasonable
choice:
 
where

Desired Closed-Loop Transfer Function
For processes without time delays, the first-order model is
 Y

 Ysp

1


d  c s  1
(12-4)
• The model has a settling time of ~ 4
• Because the steady-state gain is one, no offset occurs for set-point changes.
• By substituting (12-4) into (12-3b) and solving for Gc, the controller design
equation becomes:
Recall PID controller:
If

 =
Desired Closed-Loop Transfer Function
If the process transfer function contains a known time
delay,
 θs


Y
e



Y
 sp d τc s  1
(12-6)
• The time-delay term in (12-6) is essential because it is physically impossible for the controlled variable to
respond to a set-point change at t = 0, before t = .
• If the time delay is unknown,  must be replaced by an estimate.
• Combining Eqs. 12-6 and 12-3b gives:
28
Approximating the time-delay term in the denominator of (12-7) with a truncated
Taylor series expansion:
e  θ s  1  θs
(12-8)
Substituting (12-8) into the denominator of Eq. 12-7 and rearranging gives
If
in a FOPTD Model
Substituting Eq. 12-10 into Eq. 12-9 and rearranging gives a PI controller, with
the following controller settings:
Gc  K c 1  1/ τ I s 
Where
1 τ
Kc 
,
K θ  τc
τI  τ
(12-11)
If
in an SOPTD Model
Substitution into Eq. 12-9 and rearrangement gives a PID controller in
parallel form,


1
Gc  K c 1 
 τD s 
(12-13)
 τI s

where:
1 τ1  τ 2
Kc 
,
K τc  
τ I  τ1  τ 2 ,
τ1τ 2
τD 
τ1  τ 2
(12-14)
Internal Model Control (IMC)
• A more comprehensive model-based design method, Internal Model Control
(IMC), was developed by Morari and coworkers (Garcia and Morari, 1982;
Rivera et al., 1986).
• The IMC method, like the DS method, is based on an assumed process
model and leads to analytical expressions for the controller settings.
• These two design methods are closely related and produce identical
controllers if the design parameters are specified in a consistent manner.
• Current modern technique in design controller
•The IMC method is based on the simplified
block diagram shown in Fig. 12.6b
is the TF of the ideal process model
is the TF of the actual process
is the ideal process model response
is the actual response
∗
is the IMC controller
is the TF of the ideal process model
is the TF of the actual process
is the ideal process model response
is the actual response
∗
is the IMC controller
• The model response is subtracted from the actual response Y, and the difference, Y  Y
*
is used as the input signal to the IMC controller, Gc
• In general, Y  Y due to modeling errors
that are not accounted for in the model.
 G  G 
and unknown disturbances  D  0 
• It can be shown that the two block diagrams are identical if controllers
Gc and Gc* satisfy the relation
Gc 
Gc*
(12-16)
1  Gc*G
• Thus, any IMC controller Gc* is equivalent to a standard feedback
controller Gc, and vice versa.
• The following closed-loop relation for IMC can be derived from Fig.
12.6b using the block diagram algebra of Chapter 11:
Y
Gc*G
1  Gc*

G  G

Ysp 
1  Gc*G
1  Gc*

G  G

D
(12-17)
For the special case of a perfect model, G  G, (12-17) reduces to


Y  G c* G Y sp  1  G c* G D
(12-18)
The IMC controller is designed in two steps:
Step 1. The process model is factored as
G  G  G 
(2-19)
where G  contains any time delays and right-half plane zeros.
• In addition, G  is required to have a steady-state gain equal to one in order
to ensure that the two factors in Eq. 12-19 are unique.
Step 2. The controller is specified as
1
*
Gc 
f

G
(12-20)
where f is a low-pass filter with a steady-state gain of one. It typically has
the form:
f 
1
 τc s  1
r
(12-21)
In analogy with the DS method, τc is the desired closed-loop time
constant. Parameter r is a positive integer. The usual choice is r = 1.


For the ideal situation where the process model is perfect G  G , substituting Eq.
12-20 into (12-18) gives the closed-loop expression


Y  G  fYsp  1  fG  D
(12-22)
Thus, the closed-loop transfer function for set-point changes is
Y
 G  f
Ysp
(12-23)
Selection of
• The choice of design parameter
and IMC design methods.
is a key decision in both the DS
• In general, increasing produces a more conservative controller
because Kc decreases while increases.
• Several IMC guidelines for
12-10:

and
have been published for the model in Eq.
Rivera et al., 1986
Chien and Fruehauf, 1990
(Skogestad, 2003)
Solution: a.
Table 12.1 IMC-Based PID Controller Settings for Gc(s) (Chien and
Fruehauf, 1990). See the text for the rest of this table.
Table 12.1 IMC-Based PID Controller Settings for Gc(s) (Chien and
Fruehauf, 1990). See the text for the rest of this table.
Tuning Relations Based on Integral Error Criteria
• Developed for FOPTD model
• Parallel form of the PID Controller (noninteractive)
• Disturbance and Process Transfer function are identical (Gd=G).
• Optimal set-point changes and step disturbances are different.
Controller Design relations Based on the ITAE Performance Index and
FOPTD Model
Controller Design relations Based on the ITAE Performance Index
and FOPTD Model (For disturbance and set-point change)
DESIGN RELATION:
P only:
PI:
PID:
For set-point change, DESIGN
RELATION FOR INTEGRAL MODE:
Other tuning relation
Other tuning relation
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