Examining the Benefits of Internal Model Controlin a Motor-Tachometer System

Alexander Harding, Thomas Ignaczak, Logan Williamson, Sammir LeSage

University of Rochester, Rochester NY 14627, March 6, 2017

As technology advances, and manufacturing becomes both cheaper andmore efficient through automation, an understanding of control pro-cesses has become increasingly valuable. The following project was com-pleted by the University of Rochester Senior Class of 2017 as part of acourse on Process Dynamics and Control taught by Professor Eldred H.Chimowitz of the Department of Chemical Engineering. It consists ofcontrolling a first order motor-tachometer system using three differentcontrol structures. The first control scheme is the PID (Proportional,Integral, and Derivative) controller, which is commonly used in indus-try. Another type of control structure is called Internal Model Control(IMC), which is a model-based scheme that requires insight into thephysical processes. The closer the model is to the actual process, thebetter the system behaves. Traditional PID and IMC can be integratedand shown to be equivalent in many cases, resulting in an IMC-PIDcontroller. For this project the system’s response for each of the threemethods are compared, then some of the advantages and drawbacks areinvestigated. “Pure” IMC performs better than PID-type controllerswhen operating at or near system limitations, due to PID controllersexhibiting reset windup, resulting in poor process control. IMC cal-culations also allow for the elimination of unnecessary guesswork whendetermining characteristic process parameters.

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1. Internal Model Control Theory

Standard feedback controllers are in-credibly robust and operate with no in-formation about the process. IMC at-tempts to incorporate information knownabout the process into a model in order toestablish a precise control method with-out incorporating PID control parame-ters.

Figure 1: IMC Block Diagram

By using the Laplace transform, alge-braic operations can be carried out to es-tablish equations as a function of the fre-quency domain (s). Once the equationshave been established an inverse Laplacetransform is used to find the controlleroutput and system response as functionsof time. See Appendix A for nomencla-ture.

u(θ) = M(θ)−MMin =

τpλKp

(r(θ)− y(θ) + y(θ))

+ 1λKp

∫ θ

0(r(t)− y(t) + y(t))dt

(1)

y(θ) = 1τp

∫ θ

0{Kpu(t)− y(t)}dt (2)

The controller transfer function of aPID, c(s), and the controller of an IMC,

q(s), are related through the followingequations:

c(s) = q(s)1− g(s)q(s) (3)

q(s) = c(s)1 + c(s)g(s) (4)

As stated before, IMC and standardfeedback controllers are directly related.This relationship is illustrated in Figure2.

Figure 2: Evolution of an IMC Struc-ture

Figure 3 will show the relation be-tween the PID, c(s), and the IMC Con-troller, q(s), are related as shown byequations 3-4.

Figure 3: Controller Comparison

For the motor-tachometer system,the first-order process model is:

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g(s) = Kp

τps+ 1 (5)

And the control algorithm for a first-order process is:

q(s) = τps+ 1Kp(λs+ 1) (6)

Combining equations 3, 5, and 6 re-sults in:

c(s) = τps+ 1λKp

+ 1λKps

(7)

Notice that equation 7 is simply equa-tion 1 in the frequency domain, and bothare proportional and integral (PI) con-trollers. The proportional control param-eter is τps+1

λKpand the integration parame-

ter is 1λKps

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2. The Motor-TachometerSystem

A. Characteristics andParameters

It has been shown[2] that a perma-nent magnet DC motor can be dynam-ically modeled using the Laplace trans-form to establish an equation for thetransfer function between an input volt-age and an output voltage signal. Themotor-tachometer system used in thisproject consists of two gears which areinterlocked. One is driven by a signalfrom a LabJack between 0 and 5 Volts;the other is driven by the first, whichis registered by an analog tachometeras an output voltage signal. The set-point and the system response are com-pared in real time using an IMC con-troller and an IMC-PID controller, im-plemented through LabVIEW programsfound in Appendix B.

Before the programs can be effec-tively run, there are several characteristicparameters that must be determined forthe motor-tachometer system: the timeconstant, τp, governs the speed of the re-sponse to changes in the setpoint; theprocess gain, Kp, dictates the magnitudeof the response to changes in input volt-age; and the minimum driving voltage,MMin, establishes the signal required tomaintain a dynamic state. Additionally,the motor requires a minimum voltage inorder to overcome static friction and be-gin dynamic operation. For the given ex-perimental system, the minimum voltageis 2.6 Volts.

B. Identifying SystemParameters

τp,Kp, and MMin are found by first,probing the system in an open loop witha known harmonic function, second, de-veloping a model for how the motor re-sponds, and, third, carrying out a simpleregression to find the best fitting param-eters. To ensure dynamic operation, themotor was set to follow a sinusoidal set-point established by Equation 8.

M(θ) = Mo +MAsin(ωθ) (8)

Equation 9 establishes a functionalcontrol algorithm for the system to fol-low the set point.

c(θ) = Kp[(Mo −MMin)[1− e−θτp ]+

KpMA

τp(ω2 + 1τ2p

)[ωe

−θτp + sin(ωθ + φ)

τpcos(φ) ](9)

When the system reaches steady-state, the exponential decay terms ap-

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proach zero and the response simplifiesto Equation 10.

c(θ) = Kp[(Mo −MMin)+

MA

τ2pω

2 + 1sin(ωθ + φ)cos(φ) ]

(10)

The characteristic parameters τp,Kp, and MMin were regressed by mini-mizing the sum of the squared residualsbetween the actual system response andthe calculated response using Equation 9.Table 1 displays the IMC control param-eters found using this method, and Table2 displays the IMC parameters found towork over the widest range of input fre-quencies and the IMC-PID proportionaland integral control parameters fromthem. Also displayed in Table 2 are PIDparameters found empirically.

Table 1: Parameters (top) are fixedinputs, system characteristics (mid-dle) are altered to minimize the sumof the squared residuals (bottom)between the system response and themodel.

C. Tuning λOnce the characteristic parameters

of the first-order system are determined,these values can be used in the IMC al-gorithm and the program can be exe-cuted to determine how well the system

responds to a setpoint. The only otherparameter that needs to be tuned is λ,a tunable time constant for the outputresponse. As λ is decreased, the systemwill respond more quickly to error. How-ever, for processes where disturbances arefrequent, a small λ may cause the con-troller to over correct and lead to poorprocess control because of this noise am-plification effect. In these cases, a morerobust model may be needed which willin effect be able to “ignore” more processnoise. To institute a more robust model,all that is needed is a larger λ. The opti-mal value of λ for the motor system wasfound to be 0.1, and this value was usedto calculate the IMC-PID parameters inTable 2.

PID IMC IMC-PIDkp = 1.44 τp = 0.85 kp = 2.17ki = 2.5 Kp = 3.89 ki = 2.57kd = 0.30 MMin = 2.43V kd = 0Table 2. Parameters that yielded best

control

3. Results

A. Comparison of PID, IMC,and IMC-PID Controllers

These three control strategies werecompared at the same operating condi-tions using the parameters in Table 2to see which gave the most accurate re-sponse. The conditions were as follows:after reaching a steady-state the con-trollers set the motor to follow a sinewave of frequency 1.2 Hz, amplitude 1.25Volts, and an offset of 2.75 Volts. At leasttwo periods at a time were captured inthe following figures.

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Figure 4: PID Standard Operation

Figure 5: IMC Standard Operation

In all charts included in this paperthe white line represents the system re-sponse, while the red line represents thesystem set point. All three controllers ef-fectively follow the sine wave. However,there are some differences. The pure IMCcontroller in Figure 5 is by far the noisi-est of the three control systems. The in-creased noise is a result of the controllerresponding too quickly to the differencebetween the system response, the model’scalculated response, and the system’s set-point. An increase in λ would mitigatethis noise but it would also causes thesystem’s response to lag behind the set-point.

Figure 6: IMC-PID Standard Opera-tion

B. Reset WindupThe term reset or integral windup

refers to a flaw in feedback controllerswhere a manipulated variable hits a phys-ical constraint but an integral term in thecontroller continues to accumulate error.Thus, the integral term will request an in-creasingly greater response from the sys-tem, even though the variable is physi-cally constrained. This is a well docu-mented drawback of all PID controllers.However, an IMC controller is able tocancel out the effects of reset windup byincluding the model response in the errorcalculation, which follows the set pointeven when the system can’t. As long asthe model is well-behaved, the IMC willnot accumulate unnecessary error.

Reset windup was purposefully in-duced in the motor-tachometer system byasking for a greater response than thesystem is capable of giving. The sinewave parameters for this experiment wereas follows: an amplitude of 1.25 Volts,a frequency of 0.6 Hz, and an offset of3.75 Volts. The motor-tachometer sys-tem is unable to output a signal greaterthan 4.2 Volts, therefore whenever thesetpoint rose above this value, the sys-tem would continually output its maxi-mum response.

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Figure 7: IMC Avoiding ResetWindup

Figure 8: IMC-PID with ResetWindup

As seen in Figures 7 and 8, the IMCand IMC-PID controllers behave simi-larly before reaching their maximum op-erating condition of 4.2 Volts, but theybehave differently once the setpoint dropsto an operable range. The IMC con-troller in Figure 7 begins to respond tothe operable range within one half sec-ond, and it fully recovers after about 2.5seconds. On the other hand, Figure 8does not respond well because it suffersfrom reset windup. The integration termrecords an increasingly large error whichit adds to the system’s response, but itfails to account for the system’s physicallimitations. Therefore, when the systemreturns to an operable range it continuesto send a positive voltage signal to themotor until a large enough negative error

has accumulated to induce the integra-tor to lower the voltage signal. The sys-tem takes nearly 2 full seconds before re-sponding and sending an appropriate sig-nal. It does not reach the local minimum,suffers from a phase shift, and barely re-turns to the setpoint before once morereturning to the inoperable range.

C. Time DelaysMany slow processes have a period

of time – known as a time delay or“dead time” – before the system respondsto a variation in the setpoint. Thesecannot be modeled using standard IMCand are also particularly problematic forPID controllers which will suffer from re-set windup. However, since the motor-tachometer system is a fast process, con-trolled by electrical signals, dead timemay be ignored in this instance.

D. Response to DisturbancesThe IMC and IMC-PID systems

were run and allowed to reach a “dy-namic steady-state” before the motor-tachometer system was brought to a fullstop in order to simulate an extreme ex-ternal disturbance. The system was al-lowed to return to dynamic operation,and the response was recorded for eachcontrol method. The disturbances wererepeated twice for each type of con-trol with one at the maximum operatingspeed and the other at the minimum.

Figures 9 and 10 and show a compar-ison of the two control methods respond-ing to a sudden disturbance at their max-imum speed. As seen in Figure 10, theIMC responds with a jump that under-shoots the setpoint, followed by a slowrise before returning to steady-state op-eration a full 3 seconds after the distur-

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bance. On the other hand, the IMC-PID, as seen in Figure 10 responds to themaximum disturbance with a jump thatovershoots the setpoint, requiring nearly4 seconds before recovering steady-stateoperation.

The difference in disturbance re-sponse arises largely because of the in-tegrating term present in both controlschemes and the way each calculates thenext controller input. The controller in-put for feedback control – which can alsobe thought of as the error signal (ε) –is ε = r − y for IMC-PID, whearas anIMC has an error signal of ε = r− y + y.When the IMC-PID records an error, itstores that error in the system memoryas part of an integral and then sends alarger signal to the control output, as oc-curs with reset windup. This causes anovercompensation for the disturbance, re-sulting in the observed overshoot in boththe maximum and minimum cases. Sincethe error signal is larger in the maximumcase, the system overshoots the setpointmore and, in turn, takes longer to returnto steady-state. On the other hand, theIMC does not experience an overshoot ineither case.

Figure 9: IMC Disturbance at Maxi-mum

Figure 10: IMC-PID Disturbance atMaximum

Figure 11: IMC Disturbance at Mini-mum

Figure 12: IMC-PID Disturbance atMinimum

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E. Physical LimitationsThe motor-tachometer system is only

able to operate within a certain rangeof amplitudes and frequencies. Since theLabJack is limited to voltages between 0and 5 V, the motor is likewise limited.It can neither exceed an output voltageof 4.2 V, nor output a signal below 0 V.This limits the system’s acceleration anddeceleration. In the case of Figure 13 andFigure 14, IMC and IMC-PID control re-spectively, the motor-tachometer systemis unable to accelerate quickly enough tofollow the set point when operating at afrequency of 5 Hz and an amplitude of1 V. A combination of a high frequencyand a large amplitude cause the systemto physically break down even thoughthe model can still follow the set pointat these operating conditions. The sys-tem can however follow a sine wave at 5Hz with an amplitude of 0.5 V, althoughnew parameters must be derived in or-der to establish good control. Therefore,increases in frequency must be offset bydecreases in amplitude to account for themotor’s acceleration/deceleration limita-tions.

Figure 13: IMC at 5 Hz and 1 V

Figure 14: IMC-PID at 5 Hz and 1 V,(kp = 14.58, ki = 0.69)

4. Conclusions

IMC is a useful control scheme thatimproves upon some of the shortcomingsof traditional feedback control such as re-set windup and unnecessary guessworkwhen determining characteristic processparameters. No update of equipmentor algorithms is necessary to implementcertain IMC schemes because, as shown,IMC calculations can be used to designIMC-PID feedback controllers. For pro-cesses with no time delay, as long as in-puts do not run up against a constraintof the process, IMC-PID controllers willresult in nearly the same process controlas IMC. However, there is a clear dif-ference in these two controller’s distur-bance response and due to the PID struc-ture’s storage of past error. As long asthe model accurately reflects the physi-cal process, a “pure” IMC should respondto disturbances just as well if not betterthan a PID-type controller. Though IMChas many benefits, a final word of warn-ing, model accuracy is not often accom-plished as easily as it was in this instanceand a “pure” IMC can never handle un-stable processes. Such a process must al-ways be controlled with a standard feed-back or IMC-PID controller.

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References1) Internal Model Control, Rivera D.

and Flores M.2) ALTAS, Ismail H. Dynamic Model

of a Permanent Magnet DC Motor. Ka-radeniz Technical University Faculty ofEngineering Electrical and Electronics En-gineering Web. 20 Jan. 2017.

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Appendix A - Nomenclature

M(θ): Driving FunctionMo: Offset voltageMA: wave amplitudeω: harmonic frequencyθ : Timec(s): system responseq(s): controller transfer functionu(s) system outputgp(s): process transfer functiongp(s): model transfer functionz(s): Setpointy(s): model outputy(s) system responseφ :phase shiftλ: Filter function constants: independent variable of frequency domainKp: process gainkp: proportional constantki: integration constantkd: derivative constantSSR: Sum of the Squared Residuals AdjSSR: Adjusted Sum of the SquaredResiduals

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Appendix B - Labview Programs

Figure 15: LabVIEW Block diagram of PID/IMC-PID Controller

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Figure 16: LabVIEW Block diagram of IMC Controller

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Figure 17: LabVIEW block diagram of Open Loop Harmonic Probe

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