Fully Automated PID and Lead/Lag Compensator Design tool forIndustrial Use

P. D. Rohitha S. Senadheera and Jeff K. Pieper

Abstract— This paper presents the development procedureof a fully automated software based One-Touch-and-Go PIDcontroller and Lead/Lag compensator design tool. The key goalis to help industrial process control engineers by providingunderlying theory and implementation techniques behind somePID controller design tools already available in the marketand to present how these techniques could be developed toautomatically tune other controllers. The design tool is designedto communicate with the plant and handle the controllertuning and control tasks automatically. Results obtained byimplementing the developed design tool to an experimentalliquid level control and temperature control systems showpromise.

I. INTRODUCTION

Auto-tuning techniques tune controller parameters auto-matically, at the demand of an operator [1] or a supervisorycontrol system when retuning of the controllers is required.Development of design tools for tuning PID controllershave been reported widely due to the extensive use of PIDcontrollers in industry [2], [3]. Some currently availablesystems use step response method [4], [5]. Some others aspresented in [13] use relay feedback technique [6], [7]. Thesetechniques are applicable for multivariable systems as wellif the controllers are decentralized [8]. However accordingto [9], most of the controllers in many cases operate verypoorly.

An experimental technique that uses a variable relayhysteresis to design Lead and Lag compensators is presentedin [10], [11]. Estimated phase margin in this method becomesless accurate if the plant output oscillations are not sinu-soidal. An improvement to this method is suggested in [7]and [12]that uses pure time delay instead of relay hysteresis.In [12], differentiation of the plant output sensor signal isused to add extra 90 degrees of phase up-shift in the feedbackpath. This differentiator concept is effective if obtained by anadditional sensor attached to the plant output for example usea velocity sensor instead of a position sensor but becomesless practicable if a separate differentiator has to be addedexternally. Because differentiation of high frequency noisenormally present in sensor signals distorts the sensor signalgiving very poor signal to noise ratio. Low pass filters maynot be a good solution to damp noise because filters missinterprets plant dynamics to the auto-tuner.

P. D. R. S. Senadheera is a PhD student at the Department of Mechan-ical and Manufacturing Engineering, University of Calgary, Calgary, AB,T2N1N4 Canada. pdrssena@ucalgary.ca

J. K. Pieper is with the Department of Mechanical and Manufactur-ing Engineering, University of Calgary, Calgary, AB, T2N1N4 Canada.pieper@ucalgary.ca

This paper highlights the importance of developing adesign tool to automate design procedure of Lead and Lag [4]compensators ahead of the widely used design tools that de-sign PID controllers. Variable pure time delay together witha relay in an efficient algorithm is used to estimate phase andgain margins of the plant. Based on these margins requiredLead and Lag compensator parameters are estimated. Sec. IIpresents the basic relay feedback algorithm normally used inmost of the advanced PID controller auto-tuners available inthe market. Basic structure and behavior of Lead, Lag andLead/Lag compensators is presented in Sec. III highlightingadvantages of using them instead of PID. Sec. IV and Vsummarizes functionality of the auto-tuning procedure ofLead and Lag compensators. Implementation of the designtool is presented in Sec. VI. Behavior plots together withsome step responses of PID and Lead compensators designedby the design tool are evaluated in Sec. VII and

II. REVIEW OF PID CONTROLLER AUTO-TUNER

A. Experimental Estimation of the Gain Margin

Fig. 1 shows the basic relay feed back arrangementnormally used in auto-tuning PID controllers. Plant inputand output when the switch is set to the relay position isshown in Fig. 2. The purpose of this arrangement is to forcethe plant output to oscillate at phase cross over frequency[1], [13]. In other words to maintain a phase difference of-180 degrees between the plant input and the output. Thisallows experimental calculation of gain margin (1/Ku) of theplant using Eq. 1 [15]. Here ∈ is the relay hysteresis andother variables as shown in Fig. 2. Based on the period ofoscillations (Tu) and the amplitude of oscillation (1/Ku) atthis point and by referring to Table I, the PID controllerparameters of Fig. 3 will be decided [14], [4].

Ku =4d

aπ

(√1 −

(∈a

)2

− i∈a

)(1)

B. Implementation Procedure of the PID Auto-Tuner

The steps need to implement the One-Touch-and-Go de-sign tool is summarized in the following steps:

Fig. 1. The control loop with relay and PID control configurations.

Proceedings of the2005 IEEE Conference on Control ApplicationsToronto, Canada, August 28-31, 2005

TC2.4

0-7803-9354-6/05/$20.00 ©2005 IEEE 1009

Fig. 2. Plant input(dotted) and output(solid).

Fig. 3. The PID controller of Fig. 1.

1. Select a value for ∈ just enough to stop chattering ofthe relay output due to signal noise. [6]

2. Put the switch in Fig. 1 to the relay position and increasethe relay output, d gradually until the plant output startsto oscillate.

3. Once the plant output has achieved stable oscillations,measure the period of oscillation Tu and amplitude ofoscillation a.

4. Use Eq. 1 to calculate Ku.5. Use the estimated Tu and Ku values appropriately in

Table I to obtain required controller parameters, KP ,TI and TD.

TABLE I

PID CONTROLLER PARAMETERS.

CONTROLLER KP TI TDP 0.5*KU 99999 0PI 0.4*KU 0.8*TU 0

PID 0.6*KU 0.6*TU 0.12*TU

6. Now the parameters of the PID controller in Fig. 3 areknown, thus turning the switch back to the PID positionallows the controller to handle the plant.

III. LEAD AND LAG COMPENSATORS

The Lead compensator given in Eq. 2 is a better versionof the PD (Proportion Derivative) controller given in Eq. 3.According to Eq. 2, the Lead compensator is a PD controllercombined with a low pass filter. One advantage of this lowpass filter is the damping of unnecessary high frequencynoise picked up by the plant and the sensors that causecontrol problems and reduced lifetime of plant actuators.The name “Lead compensator” is given because it improves(leads) the phase margin of the plant thereby improvingstability and transient response of the closed loop system.

DLEAD(s) = KTs + 1αTs + 1

; α < 1 (2)

DPD(s) = K(TDs + 1) (3)

DLAG(s) = KTs + 1αTs + 1

; α > 1 (4)

DPI(s) = KTIs + 1

TIs(5)

Here, K, α, T , TD and TI are parameters need to be designedand s is the Laplace variable.

The Lag compensator given in Eq. 4 is a better version ofthe PI (Proportional Integral) controller given in Eq. 5. ThePI and Lag compensators help to increase steady state gainof the plant to reduce steady state error. The drawbacks ofPI and Lag compensators are the reduction of phase marginthat degrades transient responses of the closed loop system.The PI controller reduces the phase for all the frequenciesless than 1/TI rad/sec whereas the Lag compensator reducesphase only for a small range of frequencies around 1/TI

radians per second. Therefore, it is advantages to use a Leadcompensator instead of PI controller.

Accordingly, the Lead/Lag compensator that combinesboth Lead and Lag compensators is clearly, a better versionof the PID controller.

IV. AUTO-TUNING LEAD COMPENSATOR

A. Experimental Estimation of the Phase Margin

The main task in implementing Lead and Lag compen-sators is to estimate the phase margin of the uncompensatedplant. The following experimental technique is used. Inserta delay block between the relay and the switch of Fig. 1 andreplace the PID controller by the Lead compensator. Here, atime delay of Td seconds introduces a phase lag of Td ∗ ωradians where ω is the frequency of relay feedback oscillationof the plant output in radians per second.

The increase in time delay Td reduces the frequency ofoscillations and increase the plant output amplitude a of Fig.2, which then reduces the value of Ku of Eq. 1. The valueof Td when Ku becomes unity when multiplied by ω givesTd ∗ ω, the phase margin of the system in rad/sec.

B. Steps to implement the Lead Compensator Auto-tuner

Following steps need to be automated to achieve One-Touch-and-Go design of the Lead compensator.

1. Consider the closed loop system with a proportionalgain (Kss) and increase its value until the steady stateerror requirement (ESS) is satisfied. This value ofproportional gain is equal to the K of Eq. 2.

2. While performing relay feedback oscillations increaseTd until unity gain mentioned in Sec. IV-A is achieved.At this point measure the frequency of relay feedback oscillations (ω rad/sec). The uncompensated phasemargin of the plant (PM) is (Td*ω). Now determine

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the extra phase margin needed to be added in order toachieve the design phase margin (DPM) which is equalto Φ = DPM(Td ∗ ω) + β). Here, β is some extramargin, may be a value between 5 and 10 degrees.

3. If Φ > 0 proceeds to Step 3 otherwise a lead compen-sator is not required because DPM already exists.

4. Choose α of Eq. 2 as (1 − sin(Φ))/(1 + sin(Φ)) [4].5. Choose T of Eq. 2 as 1/(ω

√α) [4].

6. Test performance of the designed Lead compensator.

V. AUTO-TUNING THE LAG COMPENSATOR

The auto-tuning steps of the PID and Lead compensatorscan be rearranged to auto-tune the Lag compensator.

A. Auto-Tuning Steps of Lag Compensator

1. Follow Step 1 of Sec. IV-B and use the obtained K inEq. 4.

2. While performing relay feedback oscillations increasethe time delay Td until DPM= (Td ∗ ω)-β. Here, β isthe same extra margin defined in Step 2 of Sec. IV-B.

3. Continue performing relay feedback oscillations andadjust the open loop proportional gain (KssHold) ofthe closed loop system until unity gain is achieved.Measure the frequency of relay feedback oscillationsat this point, which is the gain cross over frequency ωc

rad/sec of the plant. This KssHold is equal to the open-loop gain that will meet the phase-margin requirementwithout compensation.

4. Determine α of Eq. 4, which is K obtained in Step 1divided by KssHold (K/KssHold) [4].

5. Choose T of Eq. 4 as 10/ωc. This ensures that the zeroof the lag compensator in Eq. 4 to be selected as onedecade below the ωc obtained in Step 3.

6. Test performance of the designed Lag compensator.

VI. IMPLEMENTATION ISSUES

A. The Overall Control Structure

Complete Software and Hardware implementation struc-ture is shown on the left side of Fig. 4. This paper presentsonly the top layer (the GUI), top part of the second layer(Signal processing and design tool implementation) and theconnection of external hardware shown on the very bottomlayer. Right side of Fig. 4 shows the implemented structureof the “signal processing and design tool implementation”part shown on the left side of Fig. 4.

Fig. 4. Software and Hardware state transition diagram.

The total structure is organized into 6 modes where onlyone mode is active at a time. Each mode is constructed byimplementing the steps presented in Sec. II-B, IV-B and V-A,as explained in the following.

Mode 1: Estimates the gain required achieving steady stateerror requirement (Step 1 of Sec. IV-B).

Mode 2: Perform relay oscillations and estimate PID con-troller parameters (Steps 1 to 4 of Sec. II-B).

Mode 3: If Lead Control switch of the GUI is activatedestimate the Phase margin of the system (Step 2of Sec. IV-B) otherwise go to Mode 6.

Mode 4: If the phase margin of the system is less thanthe design phase margin plus an extra 5 degreesdesign a Lead controller and control the plantappropriately (Steps 3 to 6 of Sec. IV-B).

Mode 5: If condition for starting Mode 4 is not satisfieddesign a Lag compensator and control the plantappropriately. (Steps 2 to 6 of Sec. V-A).

Mode 6: If PID control switch of the GUI is activated,calculate the PID controller parameters and controlthe plant appropriately using the PID controller(Steps 5 and 6 of Sec. II-B).

B. The Graphical User Interface (GUI)

Left and right side of Fig. 5 shows the two states ofthe GUI where data entry and switch panels are installed.First the sensor calibration constant that converts electricalsignal obtained from the sensors into physical parameterslike liquid level, temperature etc. has to be entered. Then theminimum allowable steady state error (Ess), expected phasemargin of the compensated plant (DPM) and the operatingreference value (Ref) has to be entered. Now when the startswitch is pressed, it turns to a stop switch and the plant willstart to run. The ‘Reset’/‘Retune and Hold’ switch can beused to reset the controller parameters and enable retuningof the controllers. The ‘OneWay Control’/‘TwoWay Control’switch is used to manually select between one-way and two-way controlled plants.

Fig. 5. The graphical user interface (GUI).

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Fig. 6. Liquid level and temperature control systems.

For example leftmost plant of Fig. 6 is a two-way plantbecause the pump can both pump in and out liquid whereasthe other two are one-way plants where only pump in andpositive temperatures above room temperature is possible.The ‘PID control/Lead control’ switch allow the operator toselect between PID and Lead controller. Pressing the plotswitch generates a popup menu from which necessary plantparameters (in form of digital meters and graphs) can beselected to visualize online behavior of the plant.

Only the fully automatic operation of PID and Leadcontrollers are implemented so far. Implementation of Lagand Lead/Lag compensators and manual operation possibilitytogether with safety alarm systems is recommended forfuture work.

VII. RESULTS

Fig. 7 and Fig. 8 show first three modes of the designtool where PID and Lead controller tuning steps are handledfor the two liquid level control and temperature controlexperimental plants shown in Fig. 6. The leftmost and theright most plants are first order plus dead time (FOPDT)systems and that on the middle is a second order plus deadtime (SOPDT) system. The SOPDT plant is a relativelyslow system (open loop 90% rise time for a step input is 145seconds) with long transport delay (3 seconds). The leftmostplant has a open loop 90% rise time of 57 seconds and

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a transport delay of 0.04 seconds. The temperature controlplant when sensor Y1 is used in the feedback loop has a openloop 90% rise time of 50 seconds and a transport delay of0.172 seconds and that with sensor Y2 has a rise time of 55seconds and a transport delay of 0.6 seconds.

As can be seen from Fig. 7, Mode 1 of the second orderplant has taken about 200 seconds and that of the first orderplant is about 48 seconds shows slow response of the SOPDTplant. Note also the behavior of the first order and secondorder plant outputs during Mode 1 where the second orderplant output is somewhat oscillatory and difficult to handle.

During mode 2 of all plants the relay output gets auto-matically adjusted until stable and symmetrical oscillationsare observed. As mentioned earlier a pure time delay willbe added to the closed loop during Mode 3 and as a result,frequency of plant output oscillations decrease and amplitudeof oscillations increase in all plants. It was also mentionedearlier that the mode 3 ends when the plant input and outputamplitudes become equal (unity gain) concluding PID andLead controller design procedure.

User specified parameters via the GUI and the controllerparameters estimated by the design tool during Mode 1, 2and 3 are summarized in Table II.

TABLE II

COMPARISON OF PID AND LEAD TUNING PARAMETERS.

Category Parameter

PlantTank Tank Temp. Temp.2nd 1st control controlorder order Y1 Y2

Input ESS (cm) 0.2 0.2 0.005 0.01given DPM (deg) 45.0 45.0 45.000 45.00

to GUI Ref (cm) 10.0 10.0 2.000 2.00Kss 2.74 2.84 130.30 67.00

Tu (Sec.) 29.24 1.39 4.16 5.51Output Ku 2.19 6.14 96.76 133.80from Kp=0.6*Ku 1.30 3.70 58.06 80.30the Kd=0.125*Tu 3.60 0.17 0.52 0.69

design Ki=1/(0.5*Tu) 0.07 1.40 0.48 0.36tool PM (deg) 9.00 72.80 6.92 6.55

α (Eq. 2) 0.20 NA 0.18 0.18T (Eq. 2) 13.25 NA 2.88 2.90

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As shown in Table II, a lead controller was not necessaryfor the first order plant because original phase margin ofthe plant (72.8 deg) satisfies the phase margin requirement(DPM). Therefore the plant will switch to the PID controlleror may be replaced by a lag compensator once implemented.The increase in derivative gain Kd of with the increaseof plant transport delay (Table II) shows the first glanceaffectivity of the designed PID controllers.

In order to compare performance of the designed Leadcontrollers with that implemented using existing PID tech-niques, step responses of the plant outputs are plotted in Fig.9 to Fig. 12 and the performance parameters measured fromthese figures are summarized in Table III.

In Sec. III we encouraged the use of Lead compensatorahead of the PID controller. However only 90% rise timeof the second order plant and the settling times of the tem-perature control plant showed improvements achieved withthe Lead compensator and all other performance parametersshowed poorer results. But still careful investigation of stepresponse figures show that actuator saturation of the lead

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Fig. 9. Step response of the two-tank sys. for PID and Lead controllers.

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compensator is one of the reasons for poorer performancesof the lead compensator. So far acturator saturation controlis implemented only with the PID controller. For example,according to Fig. 9, PID controller shows less overshootthan that of the Lag compensator because actuator saturationcompensation is used with the PID controller but not withthe Lag compensator. As a result Lead control signal, whichis zero during the overshoot, could not reduce the overshoot.

Another reason is that PID controller contain both prop-erties of PD and PI where as Lead controller only containsPD properties.

PID control signal of Fig. 9 is quite noisy and the filteringeffect of the Lead controller has reduced the noise level. As

TABLE III

COMPARISION OF STEP RESPONSE DATA.

ParameterTwo One Temp. Sys. Temp. Sys.

Tank sys. Tank sensor Y1 sensor Y2

PID Lead PID PID Lead PID LeadRise time 18.6 14.9 1.5 3.1 8.8 5.9 10.0

Settling time 41.1 60.6 2.5 27.0 10.7 25.5 13.6% Overshoot 3.8 21.0 4.8 32.0 0.0 30.0 0.0

Saturation 4.0 32.6 1.1 3.0 3.1 2.6 High

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can be seen from PID control signal, the derivative action hasto be switched off around 1440 seconds as the plant outputreaches the reference level.

All step responses show that plant output with the PIDcontroller oscillates about the reference level whereas Leadcontroller gradually drives it to the reference level as itrecovers from the actuator saturation.

VIII. CONCLUSIONS AND FUTURE WORKS

A. Conclusion

Presented is the development procedure and functionalityof a fully automated software based One-Touch-and-Go Pro-portional Integral and Derivative (PID) and Lead controllerdesign tool. The main advantage of such a design toolsis that detailed knowledge of the plant to be controlled isnot required to design controllers. Such design tools oncesuccessfully connected to a plant through a PC or a PLCsystem will only require the operator to enter performancerequirements via the graphical user interface and activate thesystem by pressing a button. The design tool then handlesthe controller tuning and control tasks automatically. Resultsobtained by implementing the developed design tool toan experimental water level control system through a PCand data acquisition system showed the affectivity. It canbe concluded that once both lead and lag controllers areimplemented and actuator saturation control is included, farbetter performances are expected from controllers designedby the one-touch-and-go design tool.

B. Future Works

According to the presented performance investigation, itis required to include the Lag compensator tuning stepsdeveloped in Sec. V into the design tool to form a completeLead/Lag compensator auto-tuner. Requirement of actuatorsaturation control in this Lead/Lag controller is the nextimportant development required.

REFERENCES

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[12] D.E. Schinstock and M.D. Trumble, “Relay tuning of controllers inmachine axes for high-speed tracking”, American Society for PrecisionEngineering, vol. 32, Spring topical meeting, 2004, pp. 22-27.

[13] K.J. Astrom and T. Hagglund, PID Controllers: Theory, Design, andTuning, Ed. 2, Instrument Society of America, Research Triangle Park,NC; 1995.

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