Characterization of temperature sensors using Peltier cells · 1 Characterization of temperature...

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1 Characterization of temperature sensors using Peltier cells Jo˜ ao F. M. Ventura Mestrado em Engenharia Electrot´ ecnica e de Computadores, Instituto Superior T´ ecnico Email: [email protected] Abstract—In the present work the behaviour of a Peltier Cell is studied with the purpose of precision temperature control in testing and characterization of thermal sensors. An overview of the state of the art is made and a theoretical model is derived for the real system and two controllers are developed. The first controller, based on the classic PID and a second one employing the linear quadratic regulator applied to the cell. Both controllers are able to reach a precision of 100m C, a feat unattainable by current thermal chambers. The controllers are firstly simulated then implemented and their performance is compared based on the following figures of merit: time to reach a desired temperature within a threshold, overshoot and above all, steady state error. With the second controller, an error of 5.5m C in 320 seconds can be achieved in the worst case scenario, an improvement of 3 orders of magnitude over the conventional methods of thermal sensor characterization. I. I NTRODUCTION Nowadays, device reliability is of utmost importance. So much in fact that current equipment must have solutions which render them fault tolerant. Temperature has a severe impact in the performance of electronic devices since resistances, capacities, inductances and dielectric constants are dependent on that variable. CMOS characteristics are also affected by temperature namely the threshold voltage and mobilities [1], [2]. Temperature also reduces device lifetime, especially when subject to high temperatures. In a report by [3], 40% of every failure registered were caused by abnormal temperature, pointing in the direction of stricter temperature control and regulation. This is done via temperature sensors embedded on the System-on-Chip which must be tested and characterized in order to guarantee its correct performance. Usually this is done using ovens or thermostreams, both of which are long processes (about 30 minutes) and capable of low precision (±1 C). The proposed work aims to study an alternative to this testing methodology by using Peltier cells the actuator in the characterization process due to their fast response, high precision and low maintenance. The project requirements for the present work are a precision of 100m C or better for regular tests, a precision of 10m C or better for high precision tests, under 5 minutes, ability to regulate the temperature in a set point for an indefinite period of time with minimal overshoot. Peltier cells are non linear thermoelectric devices based on the Peltier effect, discovered by Jean Peltier, in which heat is produced in a junction by applying a current [4]. The effect is formally characterized by: ˙ Q = (Π A - Π B )I (1) Where ˙ Q is the Peltier heat per second [W], Π A and Π B are the Peltier coefficients of the junction [V] and I is the current applied to the junction [A]. The cells can act either as an actuator or as a sensor, the former being the interest of this work. As a sensor, Peltier cells receive a temperature and output a current thusly being able to measure temperature differences between its plates. As an actuator, the cell receives a current and outputs a temperature difference between its plates. If one of the plates has its temperature fixed by an heat sink, one can control the temperature of the opposite plate by applying the desired current. The temperature difference can even be negative if the polarity o f the applied current is reversed, allowing a full range of control. Naming the plates of the cell as ”hot” for the highest temperature plate and ”cold” for the plate with the lowest temperature, the equations that describe the behavior of the cell are For the hot plate: P h = R m I 2 2 + IαT h (2) For the cold plate: P c = R m I 2 2 - IαT c (3) In which R m is an internal resistance of the cell, α is the Seebeck coefficient, T h is the temperature of the hot plate in K and T c is the temperature of the cold plate in K. In the literature there have been some authors who wrote about Peltier cells and have developed models and controllers for it, namely [5], [6], [7], [8], [9], [10]. The models developed are very important because they allow the simulation of the actuator and the prediction of its behavior as well as the de- velopment of robust controllers given the accurate information provided by the model. The solutions proposed as state of the art are most often used in cooling applications and in a fairly low spectrum of temperatures, usually linearizing the problem around ambient temperature and reducing the model to either a one pole system, a two pole system or a two pole one zero system. In the current work, a broader range is required to fully characterize and test the temperature sensors and as such, a model of the cell will have to be obtained in order to find a solution that meets the desired requirements.

Transcript of Characterization of temperature sensors using Peltier cells · 1 Characterization of temperature...

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Characterization of temperature sensorsusing Peltier cells

Joao F. M. VenturaMestrado em Engenharia Electrotecnica e de Computadores, Instituto Superior Tecnico

Email: [email protected]

Abstract—In the present work the behaviour of a Peltier Cellis studied with the purpose of precision temperature control intesting and characterization of thermal sensors. An overview ofthe state of the art is made and a theoretical model is derivedfor the real system and two controllers are developed. The firstcontroller, based on the classic PID and a second one employingthe linear quadratic regulator applied to the cell. Both controllersare able to reach a precision of 100m◦C, a feat unattainable bycurrent thermal chambers.

The controllers are firstly simulated then implemented andtheir performance is compared based on the following figures ofmerit: time to reach a desired temperature within a threshold,overshoot and above all, steady state error.

With the second controller, an error of 5.5m◦C in 320 secondscan be achieved in the worst case scenario, an improvement of 3orders of magnitude over the conventional methods of thermalsensor characterization.

I. INTRODUCTION

Nowadays, device reliability is of utmost importance. Somuch in fact that current equipment must have solutions whichrender them fault tolerant. Temperature has a severe impactin the performance of electronic devices since resistances,capacities, inductances and dielectric constants are dependenton that variable. CMOS characteristics are also affected bytemperature namely the threshold voltage and mobilities [1],[2]. Temperature also reduces device lifetime, especially whensubject to high temperatures.

In a report by [3], 40% of every failure registered werecaused by abnormal temperature, pointing in the directionof stricter temperature control and regulation. This is donevia temperature sensors embedded on the System-on-Chipwhich must be tested and characterized in order to guaranteeits correct performance. Usually this is done using ovens orthermostreams, both of which are long processes (about 30minutes) and capable of low precision (±1◦C). The proposedwork aims to study an alternative to this testing methodologyby using Peltier cells the actuator in the characterizationprocess due to their fast response, high precision and lowmaintenance. The project requirements for the present workare a precision of 100m◦C or better for regular tests, aprecision of 10m◦C or better for high precision tests, under 5minutes, ability to regulate the temperature in a set point foran indefinite period of time with minimal overshoot.

Peltier cells are non linear thermoelectric devices based onthe Peltier effect, discovered by Jean Peltier, in which heat is

produced in a junction by applying a current [4]. The effectis formally characterized by:

Q = (ΠA −ΠB)I (1)

Where Q is the Peltier heat per second [W], ΠA and ΠB arethe Peltier coefficients of the junction [V] and I is the currentapplied to the junction [A].

The cells can act either as an actuator or as a sensor, theformer being the interest of this work. As a sensor, Peltiercells receive a temperature and output a current thusly beingable to measure temperature differences between its plates.As an actuator, the cell receives a current and outputs atemperature difference between its plates. If one of the plateshas its temperature fixed by an heat sink, one can controlthe temperature of the opposite plate by applying the desiredcurrent. The temperature difference can even be negative ifthe polarity o f the applied current is reversed, allowing a fullrange of control.

Naming the plates of the cell as ”hot” for the highesttemperature plate and ”cold” for the plate with the lowesttemperature, the equations that describe the behavior of thecell are

For the hot plate: Ph =RmI

2

2+ IαTh (2)

For the cold plate: Pc =RmI

2

2− IαTc (3)

In which Rm is an internal resistance of the cell, α is theSeebeck coefficient, Th is the temperature of the hot plate inK and Tc is the temperature of the cold plate in K.

In the literature there have been some authors who wroteabout Peltier cells and have developed models and controllersfor it, namely [5], [6], [7], [8], [9], [10]. The models developedare very important because they allow the simulation of theactuator and the prediction of its behavior as well as the de-velopment of robust controllers given the accurate informationprovided by the model. The solutions proposed as state of theart are most often used in cooling applications and in a fairlylow spectrum of temperatures, usually linearizing the problemaround ambient temperature and reducing the model to eithera one pole system, a two pole system or a two pole one zerosystem. In the current work, a broader range is required to fullycharacterize and test the temperature sensors and as such, amodel of the cell will have to be obtained in order to find asolution that meets the desired requirements.

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II. CELL MODEL

In [11] a SPICE model is proposed in order to accuratelysimulate the Peltier cell. This model, represented in Figure 1takes into account not only the cell itself but also its envi-ronment, featuring an heat sink, an object in contact with theopposite plate of the cell to be heated or cooled and interfacethermal resistances. In the model, the cell is comprised bythe capacitors Cc and Ch, the resistances Rm and Km, thedependent current sources Pe and Px, the dependent voltagesource Vα and the voltage source V . The heat sink and itsinterface connections are represented by the capacitor Cs andthe resistors Rka and Rs. Similarly, the object to be measuredis represented by the capacitor Co and the resistors Rco andRs. The influence of the ambient temperature is also taken intoaccount by having the voltage source Ta. The sources Px, Peand Vα are defined as

Px = αThI −1

2I2Rm (4)

Pe = α(Th − Tc)I + I2Rm (5)Vα = α(Th − Tc) (6)

Fig. 1: Cell model proposed by [11]

Following the approach proposed by [9] and defining thecurrents and voltages of the circuit as in Figure 2, de circuit

equations are:

ia =1

Rka(Ta − Ts) (7)

is =1

Rs(Ts − Th) (8)

im =1

Km(Tc − Th) (9)

is3 =1

Rs(To − Tc) (10)

ia2 =1

Rco(Ta − To) (11)

is2 = ia − is (12)ih = is + Pe + Px + im (13)ic = is3 − im + Px − Px (14)io = ia2 − is3 (15)

CsTs = ia + is (16)

CoTo = is2 + ia2 (17)

ChTh = Px + Pe − is − im (18)

CcTc = −Px + im − is2 (19)

Fig. 2: Circuit definitions

Solving the system formed by equations 7 to 19 we obtain:

Ts = is21

Cs

Th = ih1

Ch

Tc = ic1

Cc

To = io1

Co

(20)

Replacing

x1 = To

x2 = Tc

x3 = Th

x4 = Ts

u = i

(21)

We get the non linear model for the cell with an heat sink

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given by:

x1(t) =− (1

CoRs+

1

CoRco)x1(t) +

1

CoRsx2(t) +

TaCoRco

x2(t) =1

CcRsx1(t)− (

1

CcKm+

1

CcRs)x2(t)+

(1

CcKm− uα

Cc)x3(t) +

Rmu(t)2

2Cc

x3(t) =(1

ChKm− α

Ch)x2(t)+

Ch+uα

Ch− 1

ChKm− 1

ChRs)x3(t) +

1

ChRsx4(t)+

Rmu(t)2

2Ch

x4(t) =1

CsRsx3(t)− (

1

CsRka+

1

CsRs)x4(t) +

TaCsRka

y(t) =x1(t)(22)

This model contains powers and products in its state vari-ables and inputs, rendering it non linear. This is an undesirabletrait because control theory has more robust techniques todeal with linear control problems than with non linear controlproblems. The solution to this problem is to linearize thesystem in order to obtain an approximation surrounding anoperating point. This will be done in a later chapter.

III. CONTROLLERS

As mentioned before, there have been a number of proposedcontrollers to deal with the Peltier cell [6], [7], [8], [9], [10],[12], [13], [14]. [8] proposes an hardware only solution to thecontrol of Peltier cells. In the work by [6] a pseudo derivativefeedback controller, a take on the classic PID controller is usedeffectively to drive the Peltier element. [9] takes a differentapproach by using a state controller with a state observer inorder to control the thermoelectric element.

Based on the works studied as state of the art, two controlarchitectures were chosen and developed. The first controller isa classic PID controller with no information about the detailsof the cell; The second controller is a Kalman filter basedlinear quadratic regulator (LQR) with state space observerwhich uses the internal model of the cell derived in section II.

A. PID controller

Due to its simplicity, fast implementation and widespreadusage, the PID control architecture was chosen as a firstapproach to control the Peltier cell. This controller achieves aset point by having three contributions of the feedback error.A proportional contribution which is simply the feedback errormultiplied by a gain Kp, an integral contribution which is theintegral of the feedback error multiplied by a gain Ki anda differential contribution representing the derivative of thefeedback error multiplied by a gain Kd. The equations of thePID controller are as follows:

u(t) = Kpe(t) +KiI(t) +KdD(t) (23)

I(t) =

∫e(t) (24)

D(t) =d

dte(t) (25)

Here, u(t) is the controller output and e(t) is the feedbackerror.

Figure 3 illustrates the classic PID controller block diagram.

Fig. 3: PID controller block diagram

In order to be implemented in a computer, the PID controllermust be discretized. As such, subject to a sampling time Ts,kth iteration k, the finite differences method for differentiation[15] and trapezoidal approximation for the integral [16] we getfor the differential contribution

ud(k) =e(k)− e(k − 1)

Ts(26)

and for the integral contribution

ui(k) = ui(k − 1) +e(k)− e(k − 1)

2Ts(27)

Taking the z transform of Equation 26 and of equation 27we get respectively:

Ud(z) =E(z)− z−1E(z)

Ts(28)

⇔ Ud(z)

E(z)=z − 1

zTs(29)

Ui(z) = z−1Ui(z) +E(z)− z−1E(z)

2Ts(30)

⇔ Ui(z)

E(z)=

z + 1

(z − 1)2Ts(31)

The transfer function of the discrete PID controller is givenby:

U(z)

E(z)=Up(z)

E(z)+Ui(z)

E(z)+Ud(z)

E(z)(32)

= Kp +z + 1

(z − 1)2Ts+z − 1

zTs(33)

Switching to the discrete time domain, the difference equa-tion of the controller is given by:

u(k) =u(k − 1) + (Kp +KiTs

2+Kd

Ts)e(k)+

(−Kp +KiTs

2− Kd

Ts)e(k − 1) + (

Kd

Ts)e(k − 2)

(34)

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In order to prevent integrator wind-up, the controller must benotified of the actuation limits. To do so a runtime check isadded such that:{

u(k) = umax, if u(t) > umax

u(k) = umin, if u(t) < umin(35)

By this point, a PID controller has been developed and iseasily implemented in software but its gains still need to bedetermined.

Using Ziegler-Nichols tuning method [17] all gains but Kp

were set to zero and Kp was incrementally raised until anoscillation behavior was registered. The gain at which thisoscillation was verified ,Kpu, was registered as well as theoscillation frequency fu. According to the tuning method, theoptimal gains for better disturbance rejection are given by:

Kp = 0.2Kpu,Ki =0.5

fu,Kd =

0.33

fu(36)

Figure 4 depicts the oscillating behaviour observed whenKp = 12.

Fig. 4: Experimental data with Kp = Kpu = 12

The gains obtained by the Ziegler-Nichols tuning methodare:

Kp = 2.4,Ki = 0.1,Kd = 0.066 (37)

These gains were found not to give the best response possible,mainly because of the theoretical nature of the algorithm andits ignorance of the internal workings of the cell. Despite this,the effort didn’t go to waste because the gains were used asa starting point for manually tuning the controller. After thissecond tuning the gains obtained are:

Kp = 3,Ki = 0.2,Kd = 1 (38)

when heating the cell and

Kp = 4,Ki = 0.2,Kd = 0.2 (39)

when cooling the cell.With these gains, the PID controller is able to achieve a

precision of 100m◦C in all its operating range. Despite itsease of implementation, simplicity and lack of information

about the cell, the PID controller can achieve a precision ofone order of magnitude more than the conventional methodsos thermal testing. However, the way its gains are obtainedbegs the question if those gains are actually the best one couldachieve. With this in mind, the second controller of this workwas developed.

B. Linear Quadratic Regulator

Due to the limitations of the PID controller, namely itstuning and lack of information about the plant, another con-troller was developed and implemented. Based on the Kalmanfilter proposed by [18], the linear quadratic regulator is astandard in state space regulation theory that results in aclosed form solution directly implementable. This controllerhas been studied extensively in the literature, namely in [19],[20], [21]. The LQR controller has several advantages overthe PID controller. The first advantage is that it calculatesexactly the required input to drive the system from one statewith no need of guesswork as in the PID controller. Secondly,the LQR controller generates a current that doesn’t oscillate,which promotes the lifetime of the cell. Thirdly, the controllerstill works in a feedback loop so any mismatches between thecalculated input and the real necessary input can be promptlyfixed in real time with no interference from the user. Thecontroller has one disadvantage over the PID controller whichis the need of an internal model that in some cases may behard to obtain.

1) Model linearization and discretization: The first step inthe design of this new controller is to collect real data of theresponse of the system to pseudo-random stimuli. Those dataare then used to fit the model derived in section II in order toachieve a good approximation of the real system. Such stimuliand data fits are depicted in Figures 5 and 6, where fits of96.84% were obtained, indicating a good matching betweenmodel and reality.

Fig. 5: Model fit to training set

The parameters obtained through this process are those ofTable I

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Fig. 6: Model fit to validation set

TABLE I: Parameters obtained through data fit

Parameter Value Parameter ValueRka 0.2066 Cc 6.5845Rs 2.5308 Ro 2.9436 × 104

Cs 4.2397 × 103 Co 9.8416Ch 1.0281 α 0.0584Km 1.4200 Rm 0.9837

To convert the model to state space a linearisation is inorder. As such, following a first order Taylor series expansionis given by:

δx

δt+δ∆x

δt≈ f(x, u) +

δf(x, u)

δx

∣∣∣∣x,u

∆x+δf(x, u)

δu

∣∣∣∣x,u

∆u

(40)The state space representation of the system is given by:

x(t) =Ax(t) +Bu(t) (41)y(t) =Cx(t) +Du(t) (42)

Where x(t) is the system state, u(t) is the input, y(t) is theoutput and A, B, C and D are given by:

A =δf(x, u)

δx

∣∣∣∣x,u

(43)

B =δf(x, u)

δu

∣∣∣∣x,u

(44)

C =[1 0 0 0

](45)

D =[0]

(46)

Applying the linearisation the A and B matrices are given by:

A =

−( 1

CoRs+ 1

CoRco) 1

CoRs0 0

1CcRs

−( 1CcKm

+ 1CcRs

) 1CcKm

− αCcu 0

0 ( 1ChKm

− αCh

) ( αCh− 1

ChKm− 1

ChRs) + α

Chu 1

ChRs

0 0 1CsRs

−( 1CsRka

+ 1CsRs

)

∣∣∣∣∣∣∣∣x,u

(47)

B =

0

− αCcx3 + Rm

Ccu

αChx3 + +Rm

Chu

0

∣∣∣∣∣∣∣∣x,u

(48)

The linearisation is made around the point x, u, fixing x1 asthe desired temperature. The remaining variables are obtainedby solving Equation 22.

The resulting system is then discretized using the result [22]:

e

A B0 0

Ts

=

[Ad Bd0 I

](49)

Where Ad and Bd are the discrete counterparts of matrices Aand B.

The final model has now been obtained and can be used inthe LQR controller.

2) Controller design: The LQR controller minimizes theperformance index

J =

∫ ∞t0

xTQx+ uTRudt (50)

And in the discrete time domain:N−1∑τ=0

(xTτ Qxτ + uTτ Ruτ ) (51)

In [21] it’s shown that the solution to the discrete problemhas a closed form given by the discrete Riccati equation:

X = ATXA−(ATXB)(R+BTXB)−1(BTXA)+Q (52)

The Riccati equation is widely known in control theory,having been discussed in the works of [23], [24], [25],[26], [27], [28] and [29] who propose numerical solution tothe problem. The optimal control signal that minimizes thediscrete time performance index is given by:

u = −Kx (53)

Where K is a gain matrix [30] defined as:

K = (BTXB +R)−1(BTXA) (54)

Since only one state variable is available, a state observerarchitecture is necessary in order to correctly estimate theremaining state variables. As such, a current sample observer[21] is chosen as it accurately estimates the state variables us-ing the most recent state observation. This controller euationsare given by:

x(k) = Ax(k−1)+Bu(k−1)+L[y(k)−C(Ax(k−1)+Bu(k−1))](55)

where L(k) is the observer gain matrix calculated as:

L = (CPCT +R)−1(CPAT ) (56)

Where P is the solution to the discrete time Riccati equation:

P = APAT − (APCT )(RE + CPCT )−1(CPAT ) +QE(57)

The block diagram of the observer is depicted in Figure 7Introducing the reference signal, the block diagram of the

controller becomes that of Figure 8.In order to use the feedback error e(k), we choose N = 0

and M = −L, resulting in the following equation for thecontroller:

x(k) = [A−BK − LCA+ LCBK]x(k − 1)− Le(k) (58)

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Fig. 7: Current observer block diagram

Fig. 8: Controller block diagram with reference tracking

Simplifying the notation:

Φ = A−BK − LCA+ LCBK (59)Γ = −L (60)

The controller in state space form is described by:

z(k + 1) = Φz(k) + Γe(k) (61)n(k) = −Kz(k) (62)

Where z(k) is the new state and n(k) is the controller outputto the plant.

Figure 9 shows the block diagram of the controller.

Fig. 9: Simplified block diagram of the controller with refer-ence

This controller has tunable parameters (Q, QE , R and RE)which dictate its aggressiveness. This means that the approachto the target temperature can be customized in order to meetdesign criteria such as overshoot or setting time.

3) State variable saturation: Similarly to the PID con-troller, the LQ regulator has no knowledge of actuation limitswhich can lead to state variable saturation, an undesirabletrait since it severely compromises response time. In order toovercome this advantage, the state feedback equation is alteredsuch that:

nsat(k) = −Kz(k)f (63)

Wheref =

nsatn(k)

(64)

In Figure 10, a simulation with and without this parameterillustrates the importance of such mechanism.

(a) Simulation for To = −10 withf = 1

(b) Simulation for To = −10 withf =

nsat

n(k)

Fig. 10: Simulations with and without state variable saturationcorrection

4) Delay prediction: In real systems there are delays in thecommunication between instruments, controller and actuator toprevent such situation from deteriorating the performance ofthe controller, a limited window prediction was added to it sothat when a delay is present, the controller can still recover andact accordingly. To do so, the controller matrices are alteredsuch that:

AD =

A B 0 0 . . . 00 0 1 0 . . . 00 0 0 1 . . . 0

0 0 0 0. . . 0

0 0 0 0 . . . 10 0 0 0 . . . 0

(65)

BD =

0...01

(66)

andCD =

[C 0 . . . 0

](67)

In Figure 11 a simulation is shown where there is acommunication delay of 10 samples and the response of thesystem with and without delay prediction.

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(a) Delay of 10 samples withoutcorrection

(b) Delay of 10 samples with cor-rection

Fig. 11: Simulation of the effect of delays in controllerbehaviour

5) Error minimization: To further improve the controllererror, a small integral component is added in order to the-oretically eliminate steady state error an integral saturatedcontribution of the error is added in parallel with the errorsignal that feeds into the controller such that:

ei(k) = eimax ,if ei(k) > eimax

ei(k) = e(k)−e(k−1)Ts

ei(k) = eimin ,if ei(k) < eimin

(68)

The block diagram of the controller with the addition of theintegral effect is depicted in Figure 12

Fig. 12: Block diagram with error correction

This addition to the controller structure proves to be bene-ficial as the error drops several orders of magnitude as seenin Figure 13 where a simulation is presented with and withoutintegral action.

(a) System error without integralaction

(b) System error with integral ac-tion

Fig. 13: Simulation of system performance with and withoutintegral action

IV. RESULTS

A. Experimental set-up

The experimental set-up used in this work consists of aBOP 20-10M current source, a Keithley 2000 multimeter,

TABLE II: Cell characteristics

Qmax(W ) Imax(A) Vmax(V ) ∆Tmax(◦C)

29 10.3 7.87 87

a Keithley 2302 battery simulator, a vacuum pump and aLaird MS2,102,22,22,17,17,11,W8 Peltier Cell with the char-acteristics presented in Table II. The set-up block diagramis shown in Figure 14. The controllers were developed in

Fig. 14: Experimental Set-up block diagram

Matlab and implemented in Python. All communication withthe instruments was made through GPIB.

B. Result comparison

The two controller were implemented in the real system andtested. In Figure 15 a cooling behaviour and current evolutionis shown for both the PID and LQR controllers. Figure 16displays a heating behaviour for both controllers as well asthe current applied to the cell.

(a) PID (b) LQ Regulator

Fig. 15: Results with To = −10◦C

(a) PID (b) LQ Regulator

Fig. 16: Results with To = 80◦C

As expected, both controllers are able to regulate thetemperature in a commanded set point. Analysing the resultsthe current calculated by the PID controller has an oscilla-tory behaviour, which is undesirable since it degrades device

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performance and lifetime while the LQ regulator exhibits nosuch behaviour despite having a slight overshoot in heatingmode. The two controllers were submitted to further testingand the results obtained are summarized in Tables III and IVwhere Ti is the initial temperature, To is the objective setpoint, Min Error was the maximum error attained in the last10% of the 10 minute tests, tMin error is the time neededto reach this error, tError=0.1 is the time needed to reach anerror of ±0.1◦C and tError=0.01 is the time needed to reachan error of ±0.01◦C. An error of less than 0.01◦C couldonly be attained by the LQ Regulator due to the restrictionsof the PID controller. On average, the LQ regulator is slowerthan the PID controller in reaching an error of 0.1◦C but itcan achieve errors one order of magnitude smaller, makingit crucial for high precision characterization. A comparisonof both controllers with the state of the art is made in Tabletab:statecomparisson where the results stated are obtained foran error of 0.1◦C.

TABLE III: Data PID

To Ti Overshoot Min Error tMin error tError=0.1 tError=0.01

-10 25 0.0952 0.0466 527.65 91.74 -0 25 0.0661 0.0633 211.71 60.86 -50 25 0.1083 0.1003 58.732 58.73 -80 25 0.1082 0.0939 509.955 459.61 -

TABLE IV: Data LQR

To Ti Overshoot Min Error tMin error tError=0.1 tError=0.01

-10 25 0.2106 0.0055 321.34 139.08 264.360 25 0.3737 0.0040 353.97 96.706 206.850 25 4.2735 0.0041 534.23 89.029 235.5880 25 1.5576 0.0036 456.91 97.09 256.049

Another test is proposed, a pre-programmed sequence ofobjective temperatures in which the controllers must achievea temperature and proceed to the next one only when theprevious temperature was stabilized. The results of this testare in Figure fig:sequences where it can be seen that despitebeing theoretically slower, the LQ regulator managed to finishthe sequence in less time than the PID controller. As seenbefore, the current profile is better for device performance andlifetime when calculate by the LQ regulator, indicating it asthe better choice for regular and prolonged usage.

TABLE V: State of the art comparison

[8]Heating

[8]Cooling [10] [6] [9]

Ti/To 15/-40 25/120 25/21 8/5 26.25/22t[s] 360 240 200 30 10

tPID[s] - - 40.52 26.78 39.87tLQR[s] - - 90.21 82.16 91.74

(a) PID(b) LQ Regulator

Fig. 17: Results with To = 80◦C

V. CONCLUSIONS

In the current work the problem of fine temperature sen-sor testing and calibration was tackled by proposing twocontrollers able to regulate the temperature of a Peltier cellin a set point for an indefinite amount of time. The firstcontroller proposed is able to reach an error of 100m◦C inunder a minute and a half with minimal overshoot. The secondcontroller developed, an LQ regulator is able to attain an errorof 5.5m◦C in less than 10 minutes, being able to reach theerror of the PID controller in about a minute and a half. Thiscontroller exhibits a slight overshoot of 5◦C in the worst casescenario. A comparison was made between the controllers andthe state of the art, indicating a clear superiority of the LQregulator with regard to the error attainable. Regarding settingtime, the controllers perform about the same, the PID havinga slight advantage over the regulator.

Both controllers were implemented in a personal computerand all communication with the instruments was done re-motely through GPIB interfaces. The code is modular andeasily implementable in already developed frameworks.

One of the main achievements of this work is the applicationof state space theory to Peltier cells, resulting in a fine controlsystem able to achieve sufficiently small errors in order toaccurately test and characterize thermal sensors.

As future work it is proposed the application of the method-ology described to other thermal actuators such as ovens andthermostreams so that big scale testing can be done in anaccurate and timely fashion.

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