International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:3 No:06 22 Linear Dynamic Modelling and Identification of Hybrid hermoelectric Refrigerator System M. Saifizi, Sazali Yaacob, M. S. Mohamad, M. Z. Ab Muin, Abstract his paper reports on modeling carried out for Hybrid hermoelectric Refrigerator (H-ER) system. he system is highly non-linear and exhibits varying model parameters and dead-time, hence the objective of the study is to investigate the mathematical model that are not based on detailed advance plant knowledge for changing system dynamics. 2nd orders of stochastic (ARMAX) and deterministic (ARX) mathematical models have been identified for the H-ER system using real-time operating system. he H-ER system has been identified using both the Recursive Extended Least Squares (RELS) and Recursive Least Squares (RLS) methods. Since RELS has been shown to give biased estimates, the values found from the RLS have been chosen for the model. A 2 nd order ARX model is found to adequately represent the system as it gives the best fit with better properties than the ARMAX model. Validation procedures show that the derived model is indeed a good enough representation of the H-ER. Index erm black box, adaptive, control, self-tuning, thermoelectric, refrigerator I. INRODUCION he design of control system is frequently divided into two steps: determination of a mathematical model and design of a control strategy. It is a requirement in most control algorithms that a model of the system to be controlled in available a- priori. Mathematical modeling based on input-output data is commonly known as identification. Hence identification is first performed using input-output data to extract the system dynamic model before implementation of the appropriate control algorithm. he plant that wants to model is Hybrid hermoelectric Refrigerator (H-ER). H-ER is developed by configuration of two type thermoelectric device. hermoelectric device is a kind of technology with solid state and no moving parts. he combination of cooling/heating ability makes them attractive for heating and cooling applications that use a small electrical energy. hermoelectric couples are connected in parallel thermally and in series electrically, are integrated into modules. ypical thermoelectric modules (EMs) contain from 3 to 27 thermocouples []. hermoelectric devices are not the option for all cooling problem. Alternatively, it should considered in case the design requirement of the system include such as factor high consistency, small size or capacity, inexpensive, less height, intrinsic safety for risky electrical conditions, and precise temperature control [2-4]. hese systems are energized by a DC power input. In order to study the performance of a system, it is often necessary to establish its mathematical models. he choice of system identification method is based on understanding the method and the system characteristics. Numerous research studies are available on the identification of engineering systems as, summarized by [5-9]. Due to the nonlinear, time varying deterministic and stochastic nature of H-ER system, the application of adaptive control techniques to H-ER system has an attention in this research. II. HYBRID HERMOELECRIC REFRIGERAOR SYSEM he main components of the proposed system, which were studied, consisted of a aluminium box (2 cm width x 2 cm length x 0 cm height with a thickness of 3.5 mm), two thermoelectric devices was applied: direct thermoelectric heat pump (operating 2 VDC, 2.6 Amp) and air to air thermoelectric heat pump (operating 2 VDC, 4 Amp), finned heat sink and fans (2 cm width x 2 cm length operating 2 VDC, 0.8A) as shown in Fig.. M. Saifizi, Advanced Intelligent Computing and Sustainability Research Group, School of Mechatronic, Universiti Malaysia Perlis KampusPauh Putra, 02600 Arau, Perlis, MALAYSIA (e-mail: saifizi@unimap.edu.my). M. Z. Ab Muin, School of Manufacturing, University Malaysia Perlis Kampus Pauh Putra, 02600 Arau, Perlis, Malaysia (email:zakimuin@unimap.edu.my). Sazali Yaacob, School of Mechatronic, Universiti Malaysia Perlis KampusPauh Putra, 02600 Arau, Perlis, MALAYSIA (e-mail: s.sazali@unimap.edu.my). M. S. Mohamad, School of Mechatronic, Universiti Malaysia Perlis Kampus Pauh Putra, 02600 Arau, Perlis, MALAYSIA. (e-mail: sofwanmohamad@unimap.edu.my). Fig.. hermoelectric refrigerator Control system he air-air thermoelectric heat pump device was cool the
International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:3 No:06 23 aluminium chamber by convection the cool air to the chamber. he direct thermoelectric heat pump was cool the chamber by transfer the heat from the aluminium chamber with touch to the thermoelectric device. A digital-analog IO DAQ provide digital-analog signal channel allow for closed loop control through connection to MOSFE to control the input actuator (H-ER). he thermoelectric module operates directly from DC power, being controlled by a computer program to maintains the temperature inside the chamber by controlling the current. A resistance temperature sensor, called a RD, was used to determine the temperature inside the refrigerator. he energy balance in conservation equation for thermoelectric materials, there are three effect must be considered: the heat transferred by conduction (Fourier s heat conduction law), the thermoelectric effect (Peltier effect), and the heat generated (Joule effect). Conservative form: C k J J () t Non-conservative form: C 2 k J J (2) t where:, temperature; J, current density (electrical current per unit area); t, hompson coefficient;, electrical resistivity; C, heat capacity;, density; k, thermal conductivity. Assuming uniform hot and cold side temperature the temperature distribute through it can be consider as one dimensional, and the current density through it as constant. hen, equation (2) is reduced to: C I I k t x x A x A (3) where: I, current through the EM; A, area of the EM. he solution of equation (3) gives the temperature distribution though the EM, from its cold surface temperature to its hot surface temperature. he cold side temperature of the EM has to be lower than the required cold temperature in order to pump heat from the cooling load, while the hot side temperature has to be higher than the ambient temperature in order to reject the heat pumped from the load plus the heat generated by the Joule effect. III. MODELING he selection of an adequate thermoelectric device is a difficult process. he performance of thermoelectric devices relies on numerous variables including operating temperature, voltage and current supplied, performance of the heat sink on the hot side, and the power pumped on the cold side. ypically a thermoelectric system does not function at a specific circumstance. he changing temperature environment and various power degenerated make the thermoelectric system become dynamic. herefore, a lumped model is adequate for 2 the analysis of this complex thermoelectric system. A. Recursive method Parameter estimation is the determination of values of parameters that govern the dynamic behaviour of a process with the structure of the process model assumed to be known. A recursive algorithm has the advantage of providing significant savings in computation as non-recursive version would need the recalculate the whole estimate, requiring the storage of all pervious data. Fig. 2 provides a useful way to visualize the estimation process where new input/output data become available at each sample interval. he model based on past information and stored in matrix θ(k-) is used to obtain and estimate of current output y(k). his will then compared with the observed output y(k) to generate an error e(k). Subsequently, this used to generate an update to the model which makes a correction to θ(k-) to approach the new θ(k). Fig. 2. Recursive estimation representation he aim is to select a value of θ(t) so that the error is minimized according to the sum of squares of errors: N 2 J e () t e e (4) t he algorithm implemented for parameter estimation of model under study is summarized as follows: For RLS algorithm, at time step (t+) i. From x(t+) using the new data ii. iii. From ε(t+) using ( t ) y( t ) x ( t ) ( t) (5) From P(t+) using x( t ) x ( t ) P( t) P( t ) P( t) I m (6) x ( t ) P( t) x( t ) iv. Update θ(t) ( t ) ( t) P( t ) x( t ) ( t ) (7) v. Wait for the next time to elapse and loop back to step i. For the stochastic modelling, where the system output is corrupted by noise, the polynomial C in Equation (6) cannot be assumed to be unity. For the RELS algorithm, at time step (t+):
International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:3 No:06 24 i. From φ(t+) using the new data u(t+), y(t+) and ε(t+) ii. From P(t+) using ( t ) ( t ) P( t) P( t ) P( t) I m (8) ( t ) P( t) ( t ) iii. Update θ(t) ( t ) ( t) P( t ) x( t ) ( t ) (9) his algorithm is used to estimate the system parameter a, a2, b0, b, c and c2 in Equation 6. B. Autoregressive with exogenous input(arx) model he input-output relationship of an ARX model is given by a linear differential equation as follows: y( t) a y( t )... a y( t n ) na a (0) b u( t d)... b u( t d n ) e( t) 0 nb b It is also kwon as an equation error model. Since the noise e(t) enters as a direct error in equation (4). he adjustable parameters of this structure are expressed as below: a a... a b b... b () 2 na 0 2 nb Given that na A( q ) a q... a q na (2) and B( q ) b n b q... b q b (3) 0 n b where q is the backward shift operator, the following inputoutput model structure is informed. t A( q ) y( t) B( q ) u( t) e( t) (4) Bq ( ) y( t) u( t) e( t) (5) A( q ) A( q ) C. Autoregressive with exogenous input (ARMAX) model he different of the ARX model is the absence of the properties of the error term. Accordingly, with the consideration of the error term, the ARMAX model is developed and given by the following equation. y( t) a y( t )... a y( t n ) b u( t d)... na a 0 b u( t d n ) e( t) c e( t )... c e( t n ) nb b nc c (6) he ARMAX model is suitable in areas of control, processes and econometrics for both system modeling and control scheme design. he adjustable parameters in this structure are expressed as follows: a a... a b b... b c c... c (7) Given that 2 na 0 2 nb 2 nc A( q ) y( t) B( q ) u( t) C( q ) e( t) (2) or B( q ) C( q ) y( t) u( t) e( t) (3) A( q ) A( q ) IV. EXPERIMENS AND RESULS A. Step response H-ER system A simple method to characterize the dynamic properties of the estimated model is by examining the transient response of the system. his is step response in the time domain. he open-loop temperature response curve is the dynamic response of the H-ER system toward the step input signal of 5% PWM cycle; correspond to a increase of about 2.6 C, from the current temperature. Fig. 3. Step response of H-ER system. For the temperature considered, the computer program has been set up so that initially, the H-ER system is run at a steady state temperature value before a step input signal is applied to the H-ER at time t=60 sec of the data acquisition period. he curve of figure exhibits a first-order system response. Furthermore, looking at each of the step response curve of the H-ER, it can be seen that there is a delay between the input signal and the H-ER temperature response about 20 sec. his is due to the transport delay and the temperature response of the system. his is time delay is approximated from the step response curve and considered as the system delay of the H-ER. It will be taken into the account in the designing of the PRBS signal. able show the approximate values of the H-ER system time constant, settling time, and the system delay for the respective temperature range. his is shown in Fig. 3 where the response of the real time H-ER model has been presented. It can be seen that the model provide good transient with without overshoot and slow settling time B. Real time identification of H-ER system In a real-time, online environment of parameter estimation, the magnitude and sequence length of the PRBS signal should
International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:3 No:06 25 be selected properly as too small a magnitude of the input signal will not be simulating sufficient and the system response will be improperly represented which will consequently cause parameter estimates is to be inaccurate. he choice of input signal has a very substantial influence on the observed data [0]. On the other hand, too large a value will drive the system into non-linearity and even instability. he signal is varied between two levels and it can switch from one level to the other at time t=0, Δt, 2Δt, 3Δt, where Δt is the time interval or clock period. In this experiment, the clock period is calculated according to equation (8): t (0.2 0.5) (8) c However, since there is a delay in the system response, this must taken into account in the calculation of the clock period must consider it. he clock period chosen in designing the PRBS is then become: t (0.2 0.5) + time delay (9) c Referring to step response of H-ER system, using equation (28), the clock period Δt for the temperature range is calculated as follows: t (0.2 0.5) 620 20 (44 330) s (20) From this, the clock period chosen for temperature range in this study is 44s. he determination of sequence length of PRBS input should consider the total duration time of the signal to exhibit a complete transient behavior of the system. Choosing the settling time of the H-ER system to be approximately equal to the sequence period of the PRBS signal, the minimum PRBS sequence length can be determined by: t N (2) Using settling time 500 s from the step response, N 500 44 (22) However, the sequence length from this calculation is too short. By using trial and error procedure, a PRBS signal with maximum length sequence of 3 with time period of s used during data collection. he input magnitude of the PRBS signal is chosen to be between 90 and 95 is chosen for the PRBS signal. he estimated model is then inserted into the model board and its various properties can be examined. It is then convenient to choose the most appropriate model representing the system as comparison of the properties of the different models can be observed either from the output response plots or the fit criterion. Besides that, information about the estimated model can be obtained and the estimated parameters with their standard deviations can be observed from Excel command window. Fig. 4. Response of measured and simulated 4 th order ARX Model Fig. 5. Response of measured and simulated 4 th order ARX Model In the first attempt, a fourth order model of ARX and ARMAX models are tested. he measured and simulated output responses of ARX and ARMAX model are shown in Figure 4 and Figure 5 where the best fit is given as 0.00000563 and 0.000054 respectively. he criterion of fit is computed as the root of the mean square value of the difference measured and stimulated output. Obviously then the smaller value, the better is the estimates. Fig. 6. Response of measured and simulated 2 nd order ARX Model
International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:3 No:06 26 he second order model might give best description of the system. Even it is does not give an excellent fit, in many cases such model is adequate for our control purpose. In any case it is best to go through the process validation to ensure that we have indeed made the right choice. Fig. 7. Response of measured and simulated 2 nd order ARMAX Model An eight-order model was found to give worse result with the best fit increasing to 0.00003529 for ARX model and 0.000047 for ARMAX model, While a second order model gives a best fit of ARX model and ARMAX model are 0.00000353 and 0.0000554 respectively. he response of the measured and simulated output of this model is shown in Figure 6, 7, 8 and 9. herefore, a second order system of ARX and ARMAX model are selected to represent H-ER model. Such a low order model will also ease the design of an adaptive controller. Fig. 0. Convergence of parameters using RLS Fig. 8. Response of measured and simulated 8 th order ARX Model Fig. 9. Response of measured and simulated 8 th order ARMAX Model Fig.. Convergence of parameters using RELS
International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:3 No:06 27 he use of RLS estimation with a constant forgetting factor 0.98 in the presence is unbiased estimates. he noise has been assumed to be white. his can be seen in Figure 0. Figure shows the system parameter estimates using the RELS algorithm. It can be seen that the convergence is slow and more significantly, the parameter values are biased. he results therefore confirm theoretical expectation that RLS is capable of estimating the H-ER system without bias. he parameter of the system and the standard deviations are given in able I. able I Parameter using RELS Parameter a a2 b b2 c c2 2nd Order -0.007-0.922 0.0000002 0.0000004 0.5 0.057 Std Deviation 0.23 0.23 0.0005 0.002 0.35 0.26 he value obtained from the RLS technique is shown in able II, where in this case c and c2 are not given as they are assumed to be unity. able II Parameters using RLS Parameter a a2 b b2 2nd Order -0.996-0.003 0.0000003-0.00000006 Std Deviation 0.007 0.007 0.000056 0.000058 C. Validation of H-ER system he use of residual analysis is another aid for validation purposes. Residual analysis consists two parts: esting cross-correlation function between input and the residuals, that is the difference between the real output data and the output reproduced by the model. If the model obtained is good, the residual should be independent of the input and the cross-correlation function should lie within the specified confidence interval. Auto-correlation test for the residual to ensure that the residuals are mutually independent. Again, for an ideal model this function should be entirely inside the confidence interval. Fig. 3. Auto-correlation and cross-correlation functions of residues showing 99% confidence limits Fig. 3 shows the cross and auto-correlation function of the residuals for the estimated second order of ARX model. he two functions lie well inside the 99% confidence region for most for the time, indicating a reasonably good model. he peaks which are outside the confidence interval shown in the cross correlation function indicate that there are elements in the output which has not been properly described by the model. his can be through of as non-linearities which have been approximated during the modeling. CONCLUSIONS Identification of thermoelectric refrigerator control system has been carried out successfully. An input-output model is appropriately chosen since this suitable for the implementation of adaptive controllers. RLS estimation technique is thus found suitable to estimate the system parameters. he H-ER system has been identified using both the RLS and RELS. Since RELS has been shown to give biased estimates, the value found from RLS has been chosen for the model. A second order model is found to adequately represent the system as it gives best fit with better properties than the fourth order model one. he reduced order of system is selected since it appears to derive optimal approximation of the original system. It is expected that the performance of the controller will not be significantly degraded by the model order reduction while the advantage is that the controller design will be greatly simplified. Furthermore, validation procedures show that the derived model is indeed a good
International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:3 No:06 28 enough representation of the H-ER system. ACKNOWLEDGMEN Special thanks to all members of UniMAP Advanced Intelligent Computing and Sustainability Research Group and School Of Mechatronics Engineering, Universiti Malaysia Perlis (UniMAP) for providing the research equipment s and internal foundations. Especially for my supervisor Prof Mohd Zaki Ab. Muin and Prof Sazali Yaacob for his advice. REFERENCES [] Melcor. 2003. hermoelectric Design/Selection Guide, http://www.melcor.com/download.htm [2] Andersen JR. hermoelectric air conditioner for sub-marines. Adv Energy Conv 962; 2: 24-8 [3] Marlow R, Buist RJ, Nelson JL. System aspects of thermoelectric cooler for hand held thermal viewer. Fourth International Conference on hermoelectric Energy Power Conversion (IEEE Calalog No. 82CH763-2), 982. p. 25-9. [4] Mei VC, Chen FC, Mathiprkasam B. Comparison of hermoelectric and Vapor Cycle echnologies for Ground Water Heat Pump Application. SME J Solar Energy Eng 989; : 353-7. [5] Bishop, C.M, Neural Network for Pattern Recognition, Clarendon Press, Oxford, 995. [6] Schalkoff, R.J., Pattern Recognition: Statistical, Structural, and Neural Approaches, John Wiley & Sons, New York, 992. [7] Wasserman, P.D., Neural Computing: heory and Practice, Van Nostrand Reinhold, New York, 989. [8] Venkatasubramanian, V., Rengaswamy et al., A Review of Process Fault Detection and Diagnosis- Part I: Quantitative Model-based Methods. Computers in Chemical Engineering, 27:293-3, 2003. [9] Wellstead, P.E and M.B. Zarrop, Self-uning System, Control and Signal Processing, John Willey and Sons, 99. [0] Ljung, L., System Identification: heory for the User, Englewood Cliffs, NJ., Prentice-Hall, Inc. M. Saifizi Saidon received his Bachelor Engineering degree in Electrical & Electronic Engineering from Fukui University, Japan in 2008 and completed his MSc from University Malaysia Perlis in 202. He is members of Board of Engineer Malaysia. His research interest is in modeling and design control system, heat transfer and thermodynamic system. Sazali Yaccob He is a Professor at School of Mechatronic, University Malaysia Perlis. His research interest is in Modeling and Control, and Satellite System. M. S. Mohamad received his Bachelor Engineering degree in Mechanical Engineering from Kyoto Institute of echnology, Japan in 2007 and completed his MSc from University Sains Malaysia in 203. His research interest is in hermo-fluid, Heat ransfer, Electronic Cooling and thermodynamic system. M. Z Ab. Muin received his Bachelor Engineering degree in Mechanical Engineering from eesside Polytechnic, United Kingdom in 975, MSc in hermo-fluids from Stratchclyde University, and PhD. in Modeling and Control from University Malaya, Malaysia. He is a Professor at School of Manufacturing, University Malaysia Perlis. His research interest is in Modeling and Control, and hermodynamic System.