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PID Options and Solutions ISA Mentor Program Presentation by:
Gregory McMillan
Presenter
• Gregory K McMillan is a retired Senior Fellow from Solutia/Monsanto and an ISA Fellow. Greg was an adjunct professor in the Washington University Saint Louis Chemical Engineering Department 2001-2004. Greg received the ISA “Kermit Fischer Environmental” Award for pH control in 1991, the Control Magazine “Engineer of the Year” Award for the Process Industry in 1994, was inducted into the Control “Process Automation Hall of Fame” in 2001, was honored by InTech Magazine in 2003 as one of the most influential innovators in automation, and received the ISA Life Achievement Award in 2010. Greg is the author of numerous ISA books on process control, his most recent being Advances in Reactor Measurement and Control and Good Tuning: A Pocket Guide - 4th Edition. Greg has been the monthly “Control Talk” columnist for Control magazine since 2002. Greg is the founder and co-leader with Hunter Vegas of the ISA Mentor Program for users. Greg’s expertise is available on the web sites: http://www.controlglobal.com/blogs/controltalkblog/ http://automation.isa.org/author/gregmcmillan/
2
Self-Regulating Process Response
3
Near-Integrating: looks like ramp in time frame
of PID response (4 dead times)
If 4>o
o
θτ
, process is classified as Near-Integrating !
Time (seconds)
∆% CO
∆% PV
θo
Ko = ∆% PV / ∆% CO
0.63∗∆% PV
% CO
% PV
Self-regulating process primary negative feedback time constant
Open loop self-regulating process gain (%/%)
Response to change in controller output with controller in manual
observed total loop dead time
ideally τp τo
Maximum speed in 4 dead times
is critical speed
Noise Band
% P
roce
ss V
aria
ble
(%PV
) or
%
Con
trol
ler O
utpu
t (%
CO
)
o
oi
KK
τ=
Equivalent open loop integrating
process gain:
Figure 1 – Self-Regulating Process Open Loop Response
True Integrating Process Response
4
Time (seconds) θo
Ki = { [ % PV2 / ∆t2 ] − [ % PV1 / ∆t1 ] } / ∆% CO
∆% CO
ramp rate is ∆% PV1 / ∆t1
ramp rate is ∆% PV2 / ∆t2
% CO
% PV
Open loop integrating process gain (%/sec/%)
Response to change in controller output with controller in manual
observed total loop dead time
Maximum ramp rate in 4 dead times is used to estimate integrating
process gain
% P
roce
ss V
aria
ble
(%PV
) or
%
Con
trol
ler O
utpu
t (%
CO
)
Figure 2 – True Integrating Process Open Loop Response
Continual ramp: no deceleration
and no steady state
Runaway Process Response
5
Response to change in controller output with controller in manual
Noise Band
Acceleration !
∆% PV
∆% CO
1.72∗∆% PV
K’o = ∆% PV / ∆% CO Open loop
runaway process gain (%/%) derived from differential equations
% P
roce
ss V
aria
ble
(%PV
) or
%
Con
trol
ler O
utpu
t (%
CO
)
Time (seconds) observed total loop dead time
runaway process primary positive feedback time constant
derived from differential equations
For safety reasons, tests are terminated within 4 dead times before noticeable acceleration
τ’ o must be τ’ p θo
% PV
% CO
Figure 3 – Runaway Process Open Loop Response
Examples
• Near Integrating Processes – Continuous column composition, pH and temperature control – Continuous vessel composition, pH and temperature control
– crystallizers, evaporators, fermenters, neutralizers, reactors • True Integrating Processes
– Level and gas pressure (level or pressure doesn’t appreciably affect flow) – Batch column composition, pH and temperature control – Batch vessel composition, pH and temperature control
– crystallizers, evaporators, fermenters, neutralizers, reactors • Runaway Processes
– Polymerization reactor temperature control (highly exothermic) – Biological reactor cell concentration control (exponential growth phase) – Axial compressor speed control (precipitous drop in flow at start of surge) – Strong acid and base pH control (approach to neutral point)
6
Many Loop Objectives
• Minimum PV peak error in load response to prevent: – Compressor surge, SIS activation, relief, undesirable reactions, poor cell health
• Minimum PV integrated error in load or setpoint response to minimize: – total amount of off-spec product to enable closer operation to optimum setpoint
• Minimum PV overshoot of SP in setpoint response to prevent: – Compressor surge, SIS activation, relief, undesirable reactions, poor cell health
• Minimum Out overshoot of FRV in setpoint response to prevent: – Interaction with heat integration and recycle loops in hydrocarbon gas unit ops
• Minimum PV time to reach SP in setpoint response to minimize: – Batch cycle time, startup time, transition time to new products and operating rates
• Minimum split range point crossings to prevent: – Wasted energy-reactants-reagents, poor cell health (high osmotic pressure)
• Maximum absorption of variability in level control to prevent: – Passing of changes in input flows to output flows upsetting downstream unit ops
• Optimum transfer of variability from controlled to manipulated variable
7
FRV is Final Resting Value of PID output. Overshoot of FRV is necessary for setpoint and load response for integrating and runaway processes
Different Worlds
8
• Hydrocarbon processes and other gas unit operations with plug flow, heat integration & recycle streams (e.g. crackers, furnaces, reformers)
– Fast self-regulating responses, interactions and complex secondary responses with sensitivity to SP and FRV overshoot, split range crossings and utility interactions.
• Chemical batch and continuous processes with vessels and columns – Important loops tend to have slow near or true integrating and runaway responses
with minimizing peak and integrated errors and rise time as key objectives.
• Utility systems (e.g., boilers, steam headers, chillers, compressors) – Important loops tend to have fast near or true integrating responses with
minimizing peak and integrated errors and interactions as key objectives.
• Pulp, paper, food and polymer inline, extrusion and sheet processes – Fast self-regulating responses and interactions with propagation of variability into
product (little to no attenuation of oscillations by back mixed volumes) with extreme sensitive to variability and resonance. Loops (particularly for sheets) can be dead time dominant due to transportation delays unless there are heat transfer lags.
• Biological vessels (e.g., fermenters and bioreactors) – Most important loops tend have slow near or true integrating responses with
extreme sensitivity to SP and FRV overshoot, split range crossings and utility interactions. Load disturbances originating from cells are incredibly slow.
Automation System and Process Dynamics in a Control Loop
9
Y fraction of small lag that is equivalent dead time is a logarithmic function of the ratio of the small to largest lag (Y = 0.28, 0.88 for ratios = 1.0 and 0.01, respectively)
10
Kick from filtered
derivative mode (e.g., filter = 1/8 x rate time)
∆% CO2 = ∆% CO1
∆% SP
∆% CO1
Time (seconds)
Signal (%)
Step from proportional mode Repeat from
Integral mode
No setpoint filter or lead-lag
Structure of PID on error (β=1 and γ=1)
seconds/repeat
Contribution of Each PID Mode
Why Reset Time is Set Too Small for Near-True Integrating & Runaway Processes
11
Looking at digital values humans think steam valve should be open, which is what will happen if there is more reset than gain action
Should steam or water valve be open for this liquid reactor ?
SP PV CO
52 48 ?
TC-100 Reactor Temperature
steam valve opens
water valve opens
50%
SP
Temperature
time
PV
Overshoot if steam valve is open
If integral action dominates, water valve won’t open
until error changes sign !
Peak and Integrated Error Metrics for Load Response
12
In gamesmanship, proponents of tuning methods and software show how close they get to the
optimum in simulation tests when in reality plants
are too nonlinear and non-ideal to make this realistic. The real value of figures and associated
equations is in realization of how dynamics and tuning relatively
affect loop performance.
“So Many Tuning Rules, So Little Time”, Control Global white paper: http://www.controlglobal.com/whitepapers/2014/so-many-tuning-rules-so-little-time/
The time to reversal of direction of the PV and start of the approach back to setpoint is the arrest time,
which is lambda for integrating process tuning rules.
Ratio of peak error to open loop error becomes 1 (no reduction in peak error) for dead time dominant processes and becomes dead time to time constant ratio for near integrating processes: Peak error is maximum PV excursion
Integrated error is area between PV and SP
SP
PV
Rise Time, Overshoot & Undershoot Metrics for Setpoint Response
13
offci
r SPKKCOKSPT θ+
∆∗+∆∗∆
=)%)(|,%|(min
%
max
Controller gain (dimensionless)
Integrating process gain
(%/sec)/% => 1/sec
Rise Time (sec) Open loop dead time
(sec)
Maximum allowable change in controller output (%)
PD on PV, I on E PID Structure
and slow reset time
The time for the PV to reach 63% of setpoint change is the closed loop time constant,
which is lambda for self-regulating process tuning rules.
Setpoint change (%)
Setpoint feedforward gain (dimensionless)
Ultimate Limit to Performance (Process Input Load Disturbance)
14
ooo
ox EE ∗
+=
)( τθθ
ooo
oi EE ∗
+=
)(
2
τθθ
Peak error is proportional to the ratio of loop deadtime to 63% response time (Important to prevent SIS trips, relief device activation, surge prevention, and RCRA pH violations)
Integrated error is proportional to the ratio of loop deadtime squared to 63% response time (Important to minimize quantity of product off-spec and total energy and raw material use)
For a sensor lag (e.g. electrode or thermowell lag) or signal filter that is much larger than the process time constant, the unfiltered actual process variable error can be
found from the equation for attenuation
Total loop deadtime that is often set by automation design
Largest lag in loop that is ideally set by large process volume
Open loop error for fastest and largest load disturbance
Attenuation by Process and by Automation System Time Constants
15
f
oof
tAAτπ ∗
=2
*
The attenuation of oscillations can be estimated from the expression of the Bode plot equation for the attenuation of oscillations slower than the break frequency where (τf ) is
the filter time constant, electrode or thermowell lag, or a mixed volume residence time
Equation is also useful for estimating original process oscillation amplitude from filtered oscillation amplitude to better know actual process variability (measurement lags and filters provide an attenuated view of real world)
Filtered Amplitude
Original Amplitude
Oscillation Period
Time Constant
Practical Limit to Performance (Process Input Load Disturbance)
16
oco
x EKK
E ∗∗+
=)1(
1
oco
fxii E
KKtT
E ∗∗
+∆+=
τ
Peak error decreases as the controller gain increases but is essentially the open loop error for systems when total deadtime >> process time constant
Integrated error decreases as the controller gain increases and reset time decreases but is essentially the open loop error multiplied by the reset time plus signal delays and lags for systems when total deadtime >> process time constant
Peak and integrated errors cannot be better than ultimate limit - The errors predicted by these equations for enhanced PID and deadtime compensators cannot be better
than the ultimate limit set by the loop deadtime and process time constant
Open loop error for fastest and largest load disturbance
Dead Time is the Key
• Without Dead Time, I would be Out of a Job – Controller would immediately see and correct for load upsets and setpoint change – No high limit to maximum controller gain and no low limit as to minimum reset time – Control could be perfect if there is no noise or interaction
• PID tuning settings for max loop performance reduce to a function of dead time (gain inversely proportional to dead time and reset time and rate time directly proportional to dead time for near and true integrating and runway processes)
– Lambda = 0.6 x dead time for max performance for fixed and well known dynamics – Lambda = 3.0 x dead time for adverse changes in loop dynamics less than 5 – Lambda = 6.0 x dead time for adverse changes in loop dynamics less than 10 – Adverse changes are multiplicative (decreases in open loop time constant for self-
regulating and increases in open loop gain and loop dead time for all processes) • Ultimate Period is 2x to 4x dead time for self-regulating processes and is 4x
dead time for other processes using first order plus dead time approximation • Resonance can occur for oscillation periods between 2x and 10x dead time • Scan time and PID execution rates should not exceed 0.1x dead time • Oscillation periods greater than 10x dead time are indicative of too low of a PID
gain and valve problems (e.g., backlash, stiction and poor positioner designs) 17
Minimizing Measurement Dead Time
18
mxmmmmawm YY θθθττθθθ +++∗+∗++= 2121
wireless dead time
(sec)
analyzer dead time
(sec)
sensor lag
(sec)
transmitter damping
(sec)
transport delay (sec)
sensor delay (sec)
resolution or sensitivity delay
(sec)
wxw t∆∗= 5.0θ
wireless update rate
(sec)
azaxa tt ∆∗+∆∗= 5.05.1θ
analyzer cycle time
(sec)
tPVSm
mx ∆∆=
/%θ
sensor resolution or threshold sensitivity
(%)
PV rate of change (%/sec)
88.028.0 =>≅YMeasurement
dead time (sec)
analyzer multiplex time
(sec)
Effect of PID Gain on Fast Disturbance Oscillations
19
Resonance results in PV amplitude in automatic greater than in manual
Filtering by process lag
Effect of PID Gain on Slow Disturbance Oscillations
20
Hi PID gain can effectively eliminate slow oscillation
Effect of Hi PID Gain on Near-Integrating Processes
21
Effect of Hi PID Gain on True Integrating Processes
22
Effect of Hi PID Gain on Runaway Processes
23
The Question
• If a loop is oscillating, should we decrease the PID gain?
24
Effect of Lo PID Gain on Near-Integrating Processes
25
Same scales as for Hi PID gain tests to show increases in oscillation period and amplitude
Effect of Lo PID Gain on True Integrating Processes
26
Same scales as for Hi PID gain tests to show increases in oscillation period and amplitude
Effect of Lo PID Gain on Runaway Processes
27
Same scales as for Hi PID gain tests to show increases in oscillation period and amplitude
Effect of Backlash Deadband in Integrating Processes
28
Same scales as for Hi PID gain tests to show increases in oscillation period and amplitude
Window of Allowable PID Controller Gains
29
iic TK
K∗
>2
controller gain (dimensionless)
reset time (sec)
iic TK
K∗
>5.0
oic K
Kθ∗
=8.0
integrating process gain
(%/sec)/% => 1/sec
dead time (sec)
oic K
Kθ∗
<6.1
To prevent the start of significant fast oscillations:
To prevent severe fast oscillations:
oic K
Kθ∗
<2.1
Maximum gain for smooth response:
To prevent the start of significant slow oscillations:
To prevent severe slow oscillations:
Simple Solution – Lambda Tuning for Integrating Processes
30
oiT θλ +∗= 2
2)( oi
ic K
TKθλ +∗
=
maximum dead time
(sec)
control objective arrest time
(sec)
controller gain (dimensionless)
reset time (sec)
Use Lambda Integrating Process Tuning Rules for Near-Integrating, True Integrating, and Runaway Processes
(All tuning rules shown are for ISA Standard Form PID)
[ ]),5.0(,25.0 soid MaxTMinT τθ∗∗=rate time (sec)
Setting reset time first and then controller gain per above tuning rules prevents violation of
low gain limit & prevents violation of high gain limit if arrest time is greater than maximum dead time !
Tuning for Level Control
31
max
max
FLAt fsr ∆
∆∗∗=
ρ level span (m)
flow span (kg/sec)
full scale residence time
(sec)
arrest time (sec)
Maximum allowable level change (%)
∆∆
∗=max
max
%%,
COPVtMax fsroθλ
Maximum allowable PID output change (%)
For level control by manipulation of liquid inlet or outlet flow the theoretical process dead time would be small. The actual total loop deadtime can be much larger. Sources of large level loop dead time is the time till correct response in boiling liquids due to inverse response, time it takes level rate of change to get through sensor noise or resolution or threshold sensitivity limit, heat transfer surface lags from manipulation of heat input, and time it takes PID output rate of change to get through VFD or valve deadband and resolution.
Modified Lambda Tuning Rules Self-Regulating Processes
32
[ ]ooi MaxT τθ ,4.0 ∗=
)( oo
ic K
TKθλ +∗
=
oo
oc K
Kθ
τ∗
∗= 5.0
[ ]),5.0(,25.0 soid MaxTMinT τθ∗∗=
A low limit is added to traditional equations to prevent a reset time smaller than ¼ the dead time for loops where time constant is very small compared to dead time to provide some gain action:
For maximum unmeasured disturbance rejection, a lambda equal to the dead time is used yielding equation seen in most other tuning methods (gain about ½ Ziegler Nichols Reaction Curve gain):
If the primary time constant is greater than ½ the dead time, derivative action may be beneficial
Closed loop time constant
(sec)
reset time (sec)
controller gain (dimensionless)
rate time (sec)
PID Options
• Integral Deadband: stops integral action when error is within deadband • External Reset Feedback: Prevents integral mode output from changing
faster than the response of the secondary loop PV or valve position – is the key feature for suppressing oscillations from backlash, stiction, and cascade rule violation, enabling move suppression and enhanced PID
• Setpoint rate limits: Provides directional move suppression to reduce split range crossings and to provide smooth optimization with fast recovery for abnormal conditions (e.g., surge and valve position control)
• Enhanced PID: Prevents integral mode from changing output if PV has not updated (critical for wireless update rates and analyzer cycle times)
• PID Structure: Enables user to specify what modes are active and if active, whether they act only changes in PV or also on changes in SP
• Anti Reset Windup Limits: Normally set equal to output limits but can be set inside output limits to provide faster recovery from heat transfer limit and to get valve open when there is solids buildup or high seal stiction
• Feedforward control: Flow and speed ratio control corrected by PID 33
Advantages of PID Proportional Mode Gained through PID Options
• PID gain: – Provides a more immediate response whose magnitude decreases as the PV
approaches setpoint and stops changing if process variable has settled – Essential for minimizing peak error and rise time – Essential for missing self-regulation and overdriving output past final resting value
(FRV) for integrating and runaway processes – Essential in valve positioner to give immediate response – Reduces dead time and oscillations in PID controllers from valve backlash and
stiction and from measurement sensitivity and resolution limit – Judicious signal filter (e.g., 0.2 deadtime) prevents valve movement from gain action
on noise (PID output fluctuations kept less than valve deadband and resolution)
• External Reset Feedback turned on with BKCAL that is fast readback of manipulated variable PV (e.g., flow or valve position) enables:
– Suppression of oscillations from violation of cascade rule – Directional move suppression by setpoint rate limits on valve or secondary loop
– Prevention of unnecessary crossings of split range point – Gradual optimization and fast get away for valve position control and surge control – Suppression oscillations and optimize the transfer of variability from the controlled to the
manipulated variable achieving tight PV control with minimal upset to other loops – PID gain to be inverse of dead time dominant open loop self-regulating gain 34
Procedure
• Determine loop objective(s) with operators and process engineers. • With PID in manual track down and mitigate oscillations. For
remaining oscillation amplitude and period determine if limit cycle, low PID gain limit violation, noise or if resonance can occur.
• For noise, use small PV filter time less than 0.2x dead time. • Choose PID options accordingly (e.g., use PID structure with no gain
or rate action on SP to minimize overshoot). Use ISA Standard Form. • Use signal characterization in DCS to help linearize system. • Identify dynamics for different production rates, weather, products,
fouling, catalyst activity etc. (identify dead time, open loop process gain, open loop (primary) time constant for self-regulating processes, secondary time constant, and lead time). Find the worst case and determine if process should be treated as integrating or runaway.
• Tune for loop objectives(s) using adaptive control as needed making sure the PID window of allowable gains is open enough for changes and unknowns. First test load response by momentarily putting PID in manual making output change and finally test setpoint response. 35
PID Structure
36
1. PID action on error (β = 1 and γ = 1)
2. PI action on error, D action on PV (β = 1 and γ = 0)
3. I action on error, PD action on PV (β = 0 and γ = 0) {equivalent to setpoint filter equal to reset time }
4. PD action on error (β = 1 and γ = 1) (no I action)
5. P action on error, D action on PV (β = 1 and γ = 0) (no I action)
6. ID action on error (γ = 1) (no P action)
7. I action on error, D action on PV (γ = 0) (no P action)
8. Two degrees of freedom controller (2DOF) (β and γ adjustable 0 to 1)
The β and γ factors do not affect the load response!
Effect of Beta on PID %PV Response in 2DOF Structure for Setpoint Change
37
Beta = 1 Beta = 0.5 Beta = 0.25
Beta = 0
Effect of Beta on PID %CO Response in 2DOF Structure for Setpoint Change
38
Beta = 1
Beta = 0.5
Beta = 0.25
Beta = 0
Series (Real) Form
39
Σ
∗
%SP
β
∆ proportional
integral
derivative
∗
Gain
∗
∗ ∗
∗
Inverse Reset Time
∗
Rate Time
∆
∆
γ
%CO
filter
filter
%PV filter
Filter Time = α ∗ Rate Time
Σ
Switch position for no derivative action
All signals are % of scale in PID algorithm but inputs and outputs are in engineering units
Most books show a more concise generic Laplace Transform Block Diagram - simpler
than this Time Domain Block Diagram
Series Form in analog controllers and early DCS available as a choice in most modern DCS
Settings are interactive in time domain but non-interactive in frequency domain
Parallel (Independent) Form
40
Parallel Form in a few early DCS and many PLC and in many control theory textbooks
Σ
∗
%SP
β
∆ proportional
integral
derivative
∗
Proportional Gain Setting
∗
∗
∗
∆
∆
γ
%CO filter
%PV filter
Integral Gain Setting
Derivative Gain Setting
All signals are % of scale in PID algorithm but inputs and outputs are in engineering units
ISA Standard Form
41
Σ
∗
%SP
β
∆ proportional
integral
derivative
∗
Gain
∗
∗ ∗
∗
∗
∆
∆
γ
%CO
filter
filter
%PV filter
Filter Time = α ∗ Rate Time
All signals are % of scale in PID algorithm but inputs and outputs are in engineering units
Rate Time
Default ISA Standard Form in most modern DCS
Inverse Reset Time
ISA Standard Form with External Reset Feedback
42
Gain
E-R is external reset (e.g. secondary % PVs)
Σ
% SP
β
derivative
∗
∗
∗ ∗ ∗
∆
∆
γ
% CO
filter
filter
% PV filter Filter Time = α ∗ Rate Time
Σ
filter
Filter Time = Reset Time
E-R
Positive Feedback
All signals are % of scale in PID algorithm but inputs and outputs are in engineering units
Out1
Out2 D
*P
∗
−
+
−
+
∆ +
−
*P = (β −1) ∗ Gain ∗ % SP
Rate Time
Switch position for external
reset feedback feedback
∆
For zero error Out1 = 0
For reverse action, Error = % SP − % PV
I
PI
Bias
Bias is used as input to Reset time filter block when there is “no integral action”. Bias is the PID output when the error is zero and is filtered by the reset time whose best setting is reduced to be about the dead time.
∆
Σ
Smart Preload
For structures with “No P action”: this Gain is zero for proportional mode, this Gain is one for integral mode and this Gain is equal to PID block gain setting for the derivative mode
This is the only known Time Domain Block Diagram showing positive feedback implementation of integral mode with external reset feedback for
ISA Standard Form
General and Specific PID Solutions
43
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