special nonlinear pid controllers

SPECIAL NONLINEAR PID CONTROLLERS

Special Nonlinear PID ControllersWhere to Apply Them

The Loop Analysis Report screen lists a table of P, PI, and PID tuningparameters. From the table, you select P, PI, or PID tuning parametersfor slow, medium, or fast response to remote setpoint or load changes inaccordance with the recommendations outlined in Table 5.2 of thismanual. Along with the optimum PID tuning parameters and thecontroller filter time constant selection, you must also configure yourcontroller to work as one of the various nonlinear PID controllersdiscussed in this section. Many digital controllers have listed theavailable controller types in their configuration options. In other cases,you may need to program the controller to emulate the controlleralgorithms presented in this section.

PID, PI-D, and I-PDError Reference on Setpoint Options

In general, a controller has many different requirements. It should havegood transient response to setpoint changes and reject load disturbances.In the textbook PID controller, an attempt is made to satisfy all thedemands with a single mechanism. (Such controllers are called onedegree of freedom controllers). The introduction of the derivative on PVand the proportional on PV control options allow for different paths forthe setpoint and load disturbance responses (two degrees of freedomcontrollers), which add more flexibility to satisfy the control objectives.Many controller brands allow the user to select if the controller P and Dtuning parameters operate on (SP-PV) or (-∆PV). Thus, all the errorreference configurations work exactly the same on load upsets, butproduce dramatically different output changes when a setpoint change ismade.

PID Type

CHAPTER

6


Output = + +

∫Kp e

Tie dt Td

de

dt

1

PI-D Type

Output = + +−

∫Kp e

Tie dt Td

d PV

dt

1 ( )∆

I-PD Type

Output = − + +−

∫Kp PV

Tie dt Td

d PV

dt∆

∆1 ( )

Where:Kp = Controller GainTi = Integral in Time per RepeatTd = Derivative in Timee = Error (SP-PV)∆∆PV = Change in PV

Equation 6.1 - PID, PI-D, and I-PD Controller Types

The PID controller equations illustrated in Equation 6.1 are a simplifieddifferential equation of how error reference values are implemented.

PID Type

As you can see from the formula for PID implementation, all three PIDterms act on the error. The controller proportional action acts on themagnitude of the error, and the derivative action acts on the rate ofchange of the error. Since setpoint changes are typically abrupt, it isundesirable in most applications to have derivative action act on setpointchanges. In normal applications, it is undesirable to kick the valve openon a small setpoint change when derivative action is used.

PI-D Type

The PI-D type in some controller manuals is referred to as derivative onPV. In the PI-D implementation, the derivative acts on -∆PV and noterror. Thus, a step change in the setpoint does not result in anundesirable kick in the controller output. This implementation ispreferred anytime derivative action is used.


I-PD Type

In the I-PD implementation, both the controller proportional andderivative act on -∆PV and not on error. Thus, a step change in thesetpoint does not result in either a proportional or derivative kick in thecontroller output. In self-regulating processes, when the process variablemeasurement signal contains measurement lag the I-PD controllerprevents setpoint overshoot.

Many self-regulating processes contain lag in the measurement. Intemperature control loops the thermowell adds lag to the temperaturemeasurement. In flow loops, filtering in either the controller ortransmitter, add lag to the measurement. In both these examples, theactual measurement of the controlled variable is not seen by the controlsystem due to the lag in the measurement. The controller tuning willcalculate a large controller gain because the measurement lag will bemuch larger than the process deadtime. A step change in setpoint usinga standard PID controller will appear to provide very good control, butin fact, the real process variable follows the valve position and willovershoot the setpoint.

In level control loops, the proportional gain setting is often very large.Even a small setpoint change with a standard PID controller will resultin a very large change in the valve position. If the controlled variable,which is either feeding the tank or controlling the flow from the tankcannot stand the large kick, the I-PD controller should be used. Therefore,you will need to implement the I-PD controller to eliminate theunmeasured overshoot.

The I-PD controller implementation protects the system from abruptoperator entered setpoint changes and should be used in all applications,unless; there is a reason that a fast response to a setpoint change isrequired, or no concern with overshoot in the real measurement, orconcern about large and fast changes in the valve position that mayupset other processes in the system.


Figure 6.1 - Normal PID Response to a Setpoint Change With Filtering in theController

Figure 6.1 illustrates the overshoot in the real unfiltered PV signal andhow the filtered PV variable appearing on the screen appears to providestable response with no overshoot when PV filtering is present and anormal linear PID controller is used.


Figure 6.2 - I-PD Response to a Setpoint Change With Filtering in theController

Figure 6.2 clearly illustrates the benefits of the I-PD controller whencontroller filtering is used. The response of the unfiltered PV to asetpoint change is fast and responsive with no overshoot of the setpoint.The filtered PV signal (as seen by the operator) appears to be slow.


Figure 6.3 - PID Response to a Setpoint Change on a Level Loop

Figure 6.3 illustrates the response to a setpoint change on a level looputilizing a standard PID controller. Level loops typically require a largecontroller proportional gain and a slow integral. With a normal PIDcontroller, the large proportional gain setting results in a very largechange in the controller output in response to a step change in thesetpoint. In many applications, this is unacceptable.


Figure 6.4 - I-PD Response to a Setpoint Change on a Level Loop

Figure 6.4 illustrates the response of a typical level loop to a setpointchange utilizing the I-PD controller algorithm configuration. Asillustrated, the I-PD algorithm makes the response to a step change insetpoint very slow, preventing the large change in the controller output.In many applications, this response minimizes upsets in other loopsbeing controlled and is the preferred response.

Error Gap ControllerTo Solve Rangeability ProblemsIn a gap controller, a deadband is placed around the error. If the error isin the gap range, the controller error is set to zero. The error calculationsin a gap action controller are:


Output = + +

∫Kp e

Tie dt Td

de

dt

1

e (SP PV)if e e Then e 0if e e Then e ee deadband%

gap

gap

gap

= −≤ => == ±

Equation 6.2 - Error Calculation

In some applications, where both wide rangeability and precise controlare required, a small valve and a large valve are used. The small valve isused to provide the precision and the large valve is used to provide therangeability.

FIC

GAP

Protuner1600PC

SP = 50%

Test 1Step FIC with GAP

in manual

FT1

Test 1Response in PV

Test 2Step GAP with

FIC in Auto

Test 2Response inPV (FIC PD)

Figure 6.5 - Error Gap Action Controller Application for Large Valve, SmallValve Rangeability Problem

As illustrated in Figure 6.5, a small valve is used to control the flow. Thegap action controller is used to control the large valve position to keep


the small valve operating near mid-range. The correct tuning procedureis:

1. Connect the Protunertm to record the three control signals asillustrated.

2. Record a Loop Analysis test on the FIC loop in manual, calculate thePID tuning parameters, enter the parameters in the controller, andplace FIC back in automatic control.

3. Place the gap controller in manual and record a Loop Analysis teston the GAP loop, calculate the PID tuning parameters, enter theparameters in the controller.

4. Along with the PID tuning parameters, enter an error gap of 10%into the GAP controller algorithm configuration and set its setpointto 50%.

In closed loop operation, the flow controller will control the flow and thesmall valve will operate from 40% to 60% travel. If the system demand issuch that the small valve position is required to go below 40% or above60%, the GAP controller will cause the big valve to make a positionchange to bring the small valve back into the desired operating range.Therefore, the tuning procedure and the tuning parameters aredetermined in the same manner as with normal controllers. The PIDcontroller with the gap function is required to keep the system stable andto prevent cycling due to interaction.

Error Squared PI ControllerControl of Surge and Averaging Level Loops

It is possible to create a controller with a continuous nonlinear functionwhose control action increases with error. This type of controller is calledan error squared controller. The error squared controller configurationcontrols averaging level and surge level control systems.

The Error Squared controller is only used for PI control modes and notPID control. There are two ways to correctly implement the errorsquared controller:


Series or Ideal Algorithms Output = + ∫

K p

e e e

Tie dt

100

2

10000

Parallel Algorithm Output = + ∫K pe e e

Tie dt

100

2

10000

Where:|e| = Absolute Value of e to maintain directionKp = Controller GainTi = Controller Integral Timee = Error (SP-PV)

Equation 6.3 - Correct Implementation of the Error Squared PI ControlAlgorithm for Integrating Processes

The equations in Equation 6.3 illustrate the correct implementation ofthe Error Squared controller algorithm for a PI controller used for thecontrol of surge and averaging tank level control. Notice, the effectivecontroller gain becomes smaller as the error approaches zero, theeffective integral time becomes longer. Thus, the effective controller gain,times the effective controller integral time, remains constant as a functionof error and the tuning parameters remain stable for the integratingprocess being controlled.

NOTEThere are a number of control system vendors that offer an Error Squared PIcontroller where the implementation is not correct for the control of integratingprocesses. In some controller implementations, (e*e)/100 is simply substitutedfor the error term in the standard PI controller. This implementation is of courseincorrect, and will result in unstable control because the effective gain, times theeffective integral time, does not remain constant as a function of error. Anumber of DCS manufacturers implement an error squared on gain controller.When using this controller implementation to control surge vessels, be sure touse P only tuning because any I tuning parameter will result in unstablecontrol and cyclic control. For an integrating process to be stable, the controllergain must be able to balance the process faster than the integral can ramp thevalve open. Thus, the integral time is inversely proportional to the controllergain. It is essential that you check that the PI controller implementationconfigured by the manufacturer is correct before using.


Averaging Level Control

In an averaging level control system, you want to both minimize themovement of the level control valve and have the level control loop settleout at setpoint after a load disturbance.

LT

Protuner

Protuner1600PC

PV

PD

Serial Cable

LIC

Flow In

Flow Out

MinimizeValve Movement

Return to Setpointafter loadchange

PI Controllerwith Error Squared

Algorithm

Figure 6.6 - Averaging Level Control Loop

Figure 6.6 illustrates an averaging level control loop on a mixing tank. Inthis application, a number of input flows are blended in the tank. Thecontrol objective is to minimize movement of the valve, thus minimizeflow disturbances to the process downstream of the vessel. The othercontrol objective is not allow the tank to go empty, or overflow duringlarge disturbances, but to return the level to setpoint after a change in theload to insure proper mixing in the tank.

The Loop Analysis test procedure for calculating the optimum tuningparameters of the level controller LIC, uses the same test and analysisprocedures as an ordinary integrating level control loop. Using theProtunertm calculated tuning parameters displayed on the Loop Analysis


Report screen, and the Error Squared Controller implementation for thePI controller, will result in the desired control.

Surge Level Control

Surge tanks are intended to absorb process upsets as well as average outload disturbances in the downstream process. For surge control systems,there is no need to return the level to setpoint, doing so would actuallyreduce the effective capacity of the tank.

LT

Protuner

Protuner1600PC

PV

PD

Serial Cable

LIC

Flow In

Flow Out

MinimizeValve Movement

Ponly orPI Controllerwith Error Squared

Algorithm

Level atHigh In Flow Rate

Level atLow In Flow Rate

Setpoint

Figure 6.7 - Surge Level Control

Consider for example, a surge tank as illustrated in Figure 6.7 feeding adownstream process. If the production is high, the level in the surge tankshould also be high, the most likely event is a decrease in the inflow tothe tank, then the entire volume of the tank is available to feed thedownstream process. Conversely, if the production is low, the tank levelshould be low, so that the maximum volume is available to absorb aninput change which is certain to be positive.

The Loop Analysis test procedure for calculating the optimum tuningparameters, uses the same test and analysis procedures as an ordinary


integrating level control loop. For surge level control loops, implement Ponly control utilizing the Error Squared PI controller.

NOTEIn commissioning any P only controller, you must use the manual resetfunction (output bias) to insure that the level is at setpoint under normal loadconditions.

While the Error Squared controller is useful in the control of averagingand surge vessel control, it is not recommended on boilers, reboilers, andother vessels where thermal and hydraulic effects are prominent, andtight control is required under all operating conditions.

Error Squared on I ControllerSolving Hysteresis Cycling Problems in Level LoopsThe Error Squared on Integral implementation is another nonlinear PIDcontroller configuration developed to prevent integral or hysteresis plusdeadband cycling in level control loops. The Error Squared on Integral isimplemented as follows:

Series or Ideal Algorithms

Output = + ∫K eTi

e edtp( )

1

100Parallel Algorithm

Output = + ∫K eTi

e ep

1

100

Where:|e| = Absolute Value of e to maintain directionKp = Controller GainTi = Controller Integral Time

Equation 6.4 - Error Squared on Integral Controller Algorithms

Error Squared on Integral controllers is typically used on level controlloops where fast controller proportional action is required to arrest loadupsets, making the flow out of the tank equal to the flow into the tank.Integral action in the controller is required to bring the level back to


setpoint. If you are tuning such a process, and the control valve containshysteresis or deadband, the integral action in the PI controller will causecontinuous cycling at steady state. Figure 6.8 illustrates the closed loopresponse of a level control loop where the control valve has a 2%deadband subjected to load upset. The controller is a standard PIcontroller tuned with the optimum Protunertm calculated PI tuningparameters.

Figure 6.8 - Linear PI Control of Level with 2% Deadband in the Valve

Figure 6.8 illustrates how integral action in a controller used to controlan integrating process will result in continuous steady state cycling ifthere is deadband in the control valve. Making the integral settingslower will change the frequency of the cycle, but will not eliminate itcompletely. Turning off the controller integral action will eliminate thecycling, but with no integral action in the controller the process variablewill never be at setpoint.


Figure 6.9 - Error Squared on Integral to Eliminate Cycling

Figure 6.9 illustrates the closed loop response to a 10% load disturbanceutilizing the same PI tuning parameters as shown in the example inFigure 6.8, implementing the Error Squared on Integral algorithm.

Comparing the results in the two graphs, you will notice that the ErrorSquared on Integral controller both returns the process variable tosetpoint and eliminates the steady state cycling. Also, the integral actionin the Error Squared on Integral controller takes a much longer time toeliminate the error, in that the integral time becomes infinitely slow asthe process variable approaches setpoint. Think of the Error Squared onIntegral controller as a P only controller with automatic manual reset.


Deadband Reset Scheduling ControllerSolving Stick-Slip Cycling

As many as one in five control loops demonstrate a continuous cycling atsteady state when tuned with the optimum PI or PID tuning parameterscalculated using the Protunertm. (The same cycling occurs if tuned usingLamda, Ziegler Nichols Ultimate Sensitivity, and other tuningcalculation methods). In most cases, the cycling can be directly traced tothe nonlinear behavior of pneumatically actuated control valves. The twomost common types of motion nonlinear control valve responses arehysteresis plus deadband and stick-slip.

As presented in Error Squared on I in this Chapter, hysteresis plusdeadband will cause steady state cycling in properly tuned integratingloops, unless the valve is fixed or the Error Squared on Integralcontroller configuration is used. Stick-slip action in the control valve willresult in steady state cycling in self-regulating loops unless the valve isfixed or the Deadband Reset Scheduling controller configuration is used.

Stick-slip response is commonly observed in pneumatically actuatedcontrol valves with positioners. To move the valve, the air pressure inthe actuator must be increased to overcome the friction in the actuator,linkages, and the valve itself. The air pressure, from the positioner to theactuator dome, will continue to increase without change in the valveposition during the stick phase. The stored energy in the actuator resultsin the valve popping to a new position. This is the slip phase.

The new valve position is beyond the desired setpoint. Pneumaticpositioners are also nonlinear devices and thus contribute to the stick-slip problem. At a constant ramp input signal to a positoner, thepositioner gain starts out small and thus loads the actuator slowly. Whenthe magnitude of the ramp input exceeds some predetermined value thepositioner gain increases and loads the actuator dome at a higher rate.


Figure 6.10 - Typical Closed Loop Stick-Slip Cycling

Stick-slip cycling as illustrated in Figure 6.10 occurs when the controllerintegral action continuously increases the controller output without acorresponding change in the actual valve position. When the valvefinally does move, it pops and the process variable overshoots thesetpoint. At that point, the error becomes negative and the controllerintegral action drives the output in the other direction. This results in thedistinctive continuous limit cycle known as a stick-slip cycle. The processvariable appears as a square wave oscillating around the setpoint. Thecontroller output appears as a triangular wave with a frequencydependent on the tuning parameters, the valve, and the process gain.

There are three traditional solutions to the stick-slip cycling problem.The first is to repair or replace the valve. A suspect valve, when removedfrom service, will often pass the typical bench tests. Despite the verifiedintegrity of the valve, maintenance personnel will sometimes install anew valve and discard the old one. This can be a very expensiveresponse to the stick-slip problem, and one that does not guaranteesuccess. The second is to place the controller in manual. This is an


effective approach to eliminate the cycling but is clearly unacceptable.The third traditional solution is to de-tune the controller integral settingsuch that the ramp rate as a function of error is so slow the stick-slipcycle is eliminated. Unfortunately, substantial de-tuning of control loopscauses problems which are often more serious than the effects of thecycling itself.

Deadband Reset Scheduling (DRS) is the name given to an algorithm inwhich the controller integral setting is adjusted between a fast settingand a slow setting depending on the size of the control error (SP-PV).

Output = + ∫K eTi K

e dtpr

(*

)1

Where: if |e| ≤ errorgap then Kr = 10 to 20

else Kr = 1Where:

Kp = Controller GainTi = Controller Integral TimeKr = Reset Ratio Factor (user adjustable typically between 10 and 20)e = Error (SP - PV)errorgap = Width of Error Deadband around setpoint

Equation 6.5 - PI Time Domain Deadband Reset Scheduling (DRS) Equation

The DRS implementation, essentially increases the controller integraltime setting (Ti) when the actual error (e) is smaller than the errorgap,

thus, de-tuning the controller integral setting when the error is small.With the controller de-tuned near setpoint, the rate at which thecontroller integral action drives the controller output is too slow for astick-slip cycle to be maintained. When the controller is forced torespond to an upset or setpoint change, the optimum tuning is used toinsure fast and stable response.

Determination of the errorgap and Kr constant is done as part of thestandard loop tuning procedure. After entering the Protunertm calculatedtuning parameters, the closed loop response of the process is recorded. Ifa stick-slip cycle is detected, use the Statistical Analysis function todocument the range of the cycle on the PV signal. Set the errorgap equal to


the range of oscillation in the PV signal. Since the absolute value of theerror is compared to the errorgap this represents a safety factor of 2X.Selection of Kr is made by simply observing the cycling at steady stateand increasing Kr until the cycle disappears. Typically, a Kr settingbetween 10 to 20 is sufficient to stop the cycling.

Though not intended to replace good valve selection and maintenanceprocedures, the Deadband Reset Scheduling DRS algorithm provides agood compromise between the competing requirements of steady statestability and speed of response.

Reset Gap ControllerSolving Cycling Problems when Using Motor OperatedValves on Self-Regulating Processes

Standard integral action in a controller continuously changes thecontroller output in an attempt to bring the measured error to zero. Allelectric motor actuated valves have a deadband gap and a minimum onand off time setting. Therefore, when the size of the error (valve positionsetpoint - actual valve position) exceeds the deadband gap, the motor isturned on and changes the valve position at least proportional to thesmallest on time increment.


Figure 6.11 - Typical Closed Loop Cycle on Motor Operated Valve UsingStandard PI Controller

Figure 6.11 illustrates the closed loop control of water flow in a watertreatment plant. The final control element is a large butterfly valve withan electric operator. The actuator was set up by the manufacturer with aminimum deadband and cycle time. As you can see, the cycle looks verysimilar to the stick-slip cycle discussed in Deadband Reset Schedulingin this Chapter. The controller output, as a function of the Protunertm

calculated Medium PI tuning parameters, ramps the controller output toeliminate the error and the motor turns on and makes the smallestchange possible and overshoots the setpoint.

The new error, causes the controller output to ramp in the oppositedirection. When the motor deadband is again overcome, the actuatorturns on and makes the smallest change. This cycle will continue forever,or until the motor over heats and shuts off, or the actuator just wears out.For this reason a continuous integral action in the controller is notappropriate for this application.


To control this process use the Deaband Reset Scheduling (DRS)algorithm shown in Equation 6.5 with the reset ratio factor Kr set to999999 to turn the controller integral action off in the errorgap. Again, theerror gap is found by determining the size of the limit cycle in the PV inthe closed loop test with the Protunertm calculated tuning parameters inthe controller.

Conditional IntegrationSolving Overshoot Problems During Setpoint Changes

Batch processes are typically integrating type processes with a large lagtime. The optimum control is a P+D action controller without I. Any Iaction tuning, in a controller controlling an integrating process, willresult in an overshoot following a setpoint change. In real processes,integral action is required following load disturbances to integrate anysmall error to maintain the process variable at setpoint. The preferredcontroller algorithm is a PID controller with conditional integration. Theconditional integration feature turns off the integral action when theerror is large, and turns on the integral action when the error issufficiently small.

Output = + +

≤ =

=

∫K eTi

e dt Tde

dt

If eK

then Ti Ti

else Ti

p d

p

( )1

100

9999999

Equation 6.6 - Conditional Integration Algorithm

Figure 6.12 illustrates the difference between the setpoint response in abatch reactor with and without the use of the conditional integrationalgorithm.


Figure 6.12 - Setpoint Response Comparing standard PID and a PIDcontroller with the Conditional Integral Algorithm Response

Split Range ControlScaling the Protunertm

It is often necessary to configure control loops to control two valves witha single controller. This technique is called split range control. Splitranging can be accomplished in a number of ways. The most common isto split range the signal using the valve positioner calibration. In manydigital control systems, split ranging can be done in software. Thepurpose of this section is to cover the two most common split rangeimplementations and the correct methodology to scale the controlleroutput signals.


Split Range Implementation In Valve Positioners

Figure 6.13 illustrates a typical chemical reactor temperature control loopconfiguration controlling both heating and cooling valves with a singlePID controller. The heating and cooling valves are split ranged in thefield.

M

Return

HeatExchanger

FC9 -15 PSIG

F03 - 9 PSIG

HeatingMedium

Coolant

I/P

PV

PDTIC

Protuner1600PC

TT1

Figure 6.13 - Split Ranged in Field

In this example, the temperature controller TIC sends a single outputfrom 0% to 100% to either heat or cool the reactor. The cooling valve is afail open with its positioner calibrated for 3 to 9 psi (0.2 to 0.5 bar). Theheating valve is a fail closed design with its positioner calibrated for 9 to15 psi (0.5 to 1 bar). In this example, the Protunertm is connected to boththe controller PV and PD signals and scaled in the normal manner, thistemperature control loop is an integrating process. The open loop testingis conducted by first allowing the process to come to a steady statecondition, placing the loop in manual and conducting a standard openloop integrating step test both above and below 50% controller output toinsure that the loop dynamics are the same for both heating and cooling.


Split Range Implementation in a Digital Controller

Figure 6.14 illustrates a typical chemical reactor temperature control loopconfiguration controlling both heating and cooling valves with a singlePID controller. In this example, the controller output is sent in softwareto a pair of split range blocks and separate outputs are then sent to theindividual valves.

M

Return

HeatExchanger

FC3 -15 PSIG

F03 - 15 PSIG

HeatingMedium

Coolant

PV PDTIC

Protuner1600PC

TT1

Input = 0% to 50%Output = 0% to 100%

Input = 50% to 100%Output = 0% to 100%

I/P

I/P

PD2

PD1

Split Range In SoftwareRequires Special PD Scaling

Figure 6.14 - Split Ranged in the Controller

The Protunertm signal cables are connected as shown to record theoutputs to both valves. Scaling of the output signals is critical to insurethe correct process gain is recorded. To correctly scale the controlleroutput signals (PD1 and PD2) the following scaling procedure isrecommended:

1. Connect the Protunertm leads as shown.2. Place the controller output at 50%.3. Click the Channel corresponding to the cooling valve PD signal.


4. Click Calculate Voltage (Two Point Scaling) enter 50 in theController Output 1.

5. Change the controller output to 45% and enter 45 in ControllerOutput 2.

6. Click the Channel corresponding to the heating valve PD signal.7. Place the controller output at 50%.8. Click Calculate Voltage (Two Point Scaling) enter 50 in the

Controller Output 1.9. Change the controller output to 55% and enter 55 in Controller

Output 2.

Scaling the PD signals in this manner will correctly record the percentchange in the controller output vs. the changes in the controller PVsignal. For changes below 50% controller output, the change in PD1 vs.PV are analyzed, above 50% controller output, PD2 vs. PV are analyzed.

In many batch processes, heating and cooling result in substantiallydifferent process gains. To test the loop use the following steps:

1. Place the controller in manual at 50% output.2. Step the controller output to 60% and record the rate of temperature

increase.3. Once the temperature increases, step the controller output back to

50% to stop the test. Use this test data to determine tuningparameters for heating.

4. Step the output from 50% to 40% and record the rate at which theprocess cools.

5. Step the contoller output back to 50% and use this data to calculatethe tuning parameters for the cooling mode.

In many batch applications it is necessary to use different PID tuning forheating and cooling. Therefore, configure your controller to change PIDsettings as a function of controller output.

special nonlinear pid controllers

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