Nonlinear FOPDT Model Identification for the Superheat Dynamic in a Refrigeration System

Nonlinear FOPDT Model Identification for the Superheat Dynamic in a Refrigeration System Zhenyu Yang Department of Energy Technology Aalborg University, Esbjerg Campus Niels Bohrs Vej 8 Esbjerg 67, Denmark Email: yang@et.aau.dk Zhen Sun Department of Energy Technology Aalborg University, Esbjerg Campus Niels Bohrs Vej 8 Esbjerg 67, Denmark Email: zhen@et.aau.dk Casper Andersen Intelligent Reliable Systems Program Aalborg University, Esbjerg Campus Niels Bohrs Vej 8 Esbjerg 67, Denmark Email: caspera@gmail.com Abstract An on-line nonlinear FOPDT system identification method is proposed and applied to model the superheat dynamic in a supermarket refrigeration system. The considered nonlinear FOPDT model is an extension of the standard FOPDT model by means that its parameters are time dependent. After the considered system is discretized, the nonlinear FOPDT identification problem is formulated as a Mixed Integer Non-Linear Programming problem, and then an identification algorithm is proposed by combining the Branch-and-Bound method and Least Square technique, in order to on-line identify these timedependent parameters. The proposed method is firstly tested through a number of numerical examples, and then applied to model the superheat dynamic in a supermarket refrigeration system based on experimental data. As shown in these studies, the proposed method is quite promising in terms of reasonable accuracy, large flexibility and low computation load. The study on the superheat also clearly showed time varying properties of superheat dynamic, which indicate some necessity of adaptive mechanism of efficient superheat control. I. INTRODUCTION A typical refrigeration system follows the vaporcompression principle by using some type of refrigerant as the heat transfer medium. In general, a refrigeration system consists of four basic components, namely expansion valve, evaporator, compressor and condenser [5]. One important variable that greatly affects the efficiency of this type of system is the refringent filling of the evaporator [1]. The critical factor to evaluate this filling is the superheat, which is defined as the difference between the outlet temperature of the gas and evaporation temperature in the evaporator [5]. The superheat can be controlled by adjusting the opening degree of the expansion valve. In order to maximally utilize the potential of the evaporator, the superheat needs to be kept as low as possible [7], [1], [12]. Most of the existing commercial refrigeration systems use either a thermostatic expansion valve or a kind of on-off control of the expansion valve [5], [1]. These types of control are easy and simple for design and implementation, however they often do not lead to (smooth) comfort and energy-efficient performance. Some advanced feedback control methods are expected for this type of system [5], [7], [1]. Nevertheless, no matter what kind of control methods, in general, a mathematical model of the considered superheat dynamic is often required in order to have a systematical control design and tuning procedure. The superheat dynamic of a refrigeration system can be very complicated, including high nonlinearities and time varying features. The detailed model of the evaoprator/superheat can be set up based on the conservation of mass, momentum and energy on the refrigerant, air and tube wall etc [3]. However, this type of detailed model often causes some difficulties in the control design stage due to the complexity. From a simple modeling point of view, Li et al proposed an empirical model for decoupling the superheat control and capacity control in [7], where the superheat dynamic is modeled by a First-Order Plus Dead- (FOPDT) model. However, it is well known that a FOPDT model can only make sense for some local operating region [5], [12]. Thereby a kind of nonlinear First- Order (FO) model is proposed in [1] according to the first modeling principle. The considered nonlinear FO model is an extension of the standard FO model by means that system s gain and time constant both are functions of the inputs and disturbances, and thereby an adaptive control of superheat was developed using back-stepping method [1]. Even though, the acquisition of this nonlinear FO model need a lot of assumptions due to the physical modeling principle, while many of these assumptions are either impossible or hard to be checked in the reality. Moreover, the time- feature of the superheat dynamic is not explicitly expressed in this model either. Bearing all above observations in the mind, in the following, we propose a type of nonlinear FOPDT, named -Varying FOPDT (TV-FOPDT) model, to model the superheat dynamic in a supermarket refrigeration system. The TV-FOPDT model is an extension of the standard FOPDT by allowing the system parameters (system gain, time constant and time ) to be time dependent variables. The benefit of using a FOPDT type of model for control purpose is quite obvious, i.e., if a FOPDT type model can be obtained/identified, the design of autotuning regulator or adaptive control can be straightforward by using some standard methods [1], [2], [6], [9], [11]. We observed that the nonlinear FOPDT models have been used in a number of nonlinear control applications [2], [6], [9]. The model used in [2] is called adaptive (FOPDT) transfer 978-1-61284-971-3/11/$26. 211 IEEE 582

function and this model can be obtained by linearizing the nonlinear dynamic model at each sampling time, so that the system parameters of this adaptive FOPDT model are time dependent. This method requires the nonlinear system model as a prior knowledge. Lee et al proposed to obtain a nonlinear FOPDT model by linearizing the nonlinear system at a number of different operating points, so that the system parameters of the obtained FOPDT are operating-point dependent [6]. However, this proposed method is not recommended for online identification if a precise nonlinear model can not be obtained beforehand. An on-line nonlinear FOPDT identification method is proposed in [9] by using the so-called longrange predictive identification method. However, the formulated system identification is a nonlinear optimization problem due to the unknown time-dependent time. Therefore, four different potential time scenarios are assumed, before converting the nonlinear optimization problem into a Least-Square (LS) problem using the spectral factorization technique. The assumption of time s limits the proposed method in [9] to be applied for any other systems except these two specific patient cases they have studied. In the following, an on-line TV-FOPDT model identification method is proposed based on a formulated Mixed Integer Non-Linear Programming (MINLP) problem. Compared with the existing nonlinear FOPDT model acquisition methods, the proposed algorithm has the following distinguishable features: (i) The proposed method is a kind of black-box identification method, i.e., only a number of sampled input and output data are required by the proposed algorithm; (ii) The proposed method is at an on-line manner in terms that the system parameters can be updated at every sampling step by using the moving window technique; (iii) The proposed algorithm is directly oriented to efficient numerical computation by combining the Branch-and-Bound(BB) method and LS technique to cope with the formulated Mixed Integer Non-Linear Programming (MINLP) problem. Of course, the proposed method could have some potential drawback in the identification accuracy. For instance, the largest steady-state estimation error of the system time is one sampling period, which could further cause some steady-state estimation errors for system gain and time constant. However, these estimation errors can be reduced by increasing the sampling rate if feasible. The proposed method is firstly tested through a number of numerical examples, and then applied to model the superheat dynamic in a supermarket refrigeration system based on experimental data. For the comparing purpose, the standard pem method provided in Matlab system identification toolbox is also applied to examples. The rest of the paper is organized in the following: Section II formulates the considered identification problem; Section III introduces the proposed method for TV-FOPDT model identification; Section IV discusses testing results for numerical examples; Section V explores the superheat model identification; and finally, we conclude the paper in Section VI. II. PROBLEM FORMULATION A -Varying FOPDT (TV-FOPDT) model is defined in the following 1 Y (s) =G t (s)u(s), (1) with a time-varying transfer function Kt G t (s) = Tps t +1 exp T d s, (2) where Y (s)/u(s) is the Laplace-transform of the system output/input (y(t)/u(t)). K t, Tp t and Td t are the system gain, time constant, and time (dead-time) of the TV-FOPDT model, respectively. Different with the standard FOPDT model [1], [8], all these system parameters of TV-FOPDT model can be time-dependent, e.g., they can be some timed functions with proper smooth properties. This time-dependent feature is represented by the subscript t. The TV-FOPDT system identification can be described as: how to correctly estimate system parameters K t, Tp t and Td t for a system modeled by (1) with (2) based on measured system input and output data, in an on-line manner? III. TV-FOPDT MODEL IDENTIFICATION The proposed TV-FOPDT model identification consists of two steps, namely discretized problem and iterative LS Prediction algorithm. A. Discretized Problem Following the similar procedure as we did in [12], the model (1) with (2) can be approximated by its discrete-time equivalence, i.e., Y (z) =G k (z)u(z), with G k (z) = Kk (1 α k ) z lk (z α k ), (3) where α k e Ts Tp k, and Ts is the sample interval. It should be noticed that K k in (3) is not same as K t in (2). K k is a piecewise-constant (constant during every sampling period) timed function, while K t in (2) is a real-timed function. The relationship of them is that K k is equal to K t at each sampling point, i.e., K k = K t when t = kt s for any k. Thereby, we call K k the kth sampled (time-varying) system gain. The similar situation is considered for parameter Tp t and Td t in (2). T p k /Td k is called the kth sampled (time-varying) time constant/time for discrete-time system (3), i.e., Tp k = Tp t and Td k = T d t when t = kt s for any k. l k in (3) is the discrete approximation of the kth sampled system Td k, and it is defined as an integer with the property: l k T s T k d < (l k +1)T s. (4) Define β k K k (1 α k ), then TV-FOPDT model (3) can be transferred into a difference equation described as y(k) =α k y(k 1) + β k u(k l k 1), (5) 1 We noticed that this expression (1) is not strictly mathematically correct. However, we would use it as an illustration of concept, instead of a math equation. t 583

for k = l k +1,l k +2,. The original continuous-time model identification problem of (2) with parameters K t, Tp t and Td t is converted to estimate parameter sequences of α k, β k and l k for a discrete-time system (5) based on a number of sampled input and output data. This discrete-time system identification problem is called the discreteized approximation of the original continuous-time identification problem [12]. B. Iterative LSP Method The method, named Least-Square Prediction (LSP) in [12] for a standard FOPDT identification, is extended in the following in order to handle the TV-FOPDT identification problem. Assume the considered system (1) currently is at kth sampling step and take N number of latest pair samples of the output and input into consideration, where N is the length of a moving window for selecting sampled data for each estimation step. Define Υ k [α k β k ] T, then the parameter identification of system (5) at the kth sampling step can be formulated as a MINLP problem, which is defined as l k : min positive integer Υ k Ω k B k N A k N (l k )Υ k 2 2, (6) where BN k is a stack of N latest measured outputs at the current kth sampling step, i.e., B k N [y(k) y(k 1) y(k N +1)] T. (7) A k N (lk ) is a stack of N measured inputs and outputs at the current kth sampling step, depending on the parameter l k, i.e., A k N (l k ) y(k 1) u(k l k 1) y(k 2) u(k l k 2).. y(k N) u(k l k N). (8) Ω k represents the possible range of Υ k, which is determined by the limits of the original system gain K t and time constant T t p in (2) at the current sampling point (kt s ). If the concerned system has no any time, or the time is a piece of prior knowledge, the optimization problem (6) is simplified to a standard LS problem. In general, to solve this MINLP problem (6) may face to some non-convex issue due to the unknown time l k [4]. However, in case that some pre-knowledge about the system time can be obtained, such as the potential upper and lower limits of the time (s) for the entire system or each sampling step, an iterative numerical algorithm can be proposed by combining the BB method, which is one typical method for MINLP problem [4], and the LS technique for efficiently solving the MINLP problem (6). We call this proposed algorithm an Iterative LSP algorithm, which is summarized in the following: Pre-knowledge: The upper and lower limits for system time (s) in terms of some integer number multiplying with sampling period, without losing generality, we assume that lmin k lk lmax k and lmin k, lk max are known beforehand. Initialization: selection of the sampling rate and the sliding window. Window selection includes the window type, length (N) and potential weighting etc. Data collection period: Since the start, the algorithm only collect the sampled data until the process reaches a specific sampling step first time, denoted this step as k ini, where there is N + lmax kini = k ini. It is to guarantee that there are enough data for constructing matrix (8) 2. Iteration period: The iterative identification starts from the k ini step. At each sampling step, denoted as k and there is k k ini, a while-loop needs to be constructed w.r.t. l k starting from lmin k and ending at lk max by taking the unit increment. For each iteration (t) ofl t (lmin k lt lmax), k solve the LS problem (6) and record the corresponding prediction error. The analytical LS solution has the format as Υ k (l t )=((A k N (l t )) T A k N (l t )) 1 (A k N (l t )) T BN k. (9) The pair of (Υ k (l t ), l t ) which leads to the minimal prediction error among all iterations moving from lmin k to lk max, denoted as (Υ k, l k ), is the optimal solution for (6) at the current step. The optimal estimation of sampled system parameters of (3), Tp k and K k, can be obtained from Υ k =[α k β k ] T by Tp k = T s ln α k, and Kk = βk, (1) 1 αk and the sampled time Td k is estimated as lk T s. Repeat the above steps when a (couple) new data of input and output is obtained. C. PEM Method in Matlab In order to evaluate the proposed method, the system identification toolbox in Matlab is also used for the following study. An on-line system identification is constructed by iteratively call the built-in function pem in SI toolbox. We refer to [8] for more details about this function and method. IV. TEST OF NUMERICAL EXAMPLES A. A Standard FOPDT Model A standard FOPDT model with parameters Td t = 2.5, Tp t =2and K t =4is used to test the proposed method. The tests are carried out w.r.t. different sampling rates and (rectangular) window lengths, and compared with PEM method. The symmetric relay cycle input signal and the corresponding output are shown in Fig. 1. When the sampling period is 2 We assume that there always is enough data for (8) since the k ini th step. Otherwise, the following identification steps will temporarily stop until there is enough data for (8) again. During the waiting-data period, the estimated system parameters obtained in the previous step will be used as the current estimation result. 584

1 2 3 4 5 6 4 input output output input 2 2 15 2 1 4 5 Fig. 1. Input and output data for the standard FOPDT 1 2 3 4 5 6 3 Fig. 4. Delay using PEM with N =5, T s =.1, standard FOPDT 2.5 2 1.5 1 1 2 3 4 5 6 Fig. 2. Delay using LSP with N =5, T s =.1, standard FOPDT 3.86 Parameter K 3.84 3.82 3.8 1 2 3 4 5 6 1.94 1.93 Parameter Tp Fig. 5. Parameters using LSP with N = 1, T s =.25, standard FOPDT 1.92 1.91 1.9 1 2 3 4 5 6 TABLE I THE COMPUTATION TIMES FOR THE STANDARD FOPDT Fig. 3. K k, T k p using LSP with N =5, Ts =.1, standard FOPDT T s =.1sec, lmin k =and lk max =5are given for any k, the system estimated using a rectangular window with 5 samples is shown in Fig. 2. The identification procedure start from 1 sec (the data collection period is (5+5).1 =1 sec). The estimation stayed at value of 2 all the way, the observed small deviation (2.5%) is due to the fact that this identification resolution is determined by T s (see (4)). The estimated system gain and time constant are illustrated in Fig. 3. It can be observed that the identified system parameters quickly converge to some steady-sate values after about 8 iterations. However, some small steady-state estimation errors (about 4.5%) can be observed. This is mainly due to the effect of the unavoidable time estimation error. There is no doubt that these estimation offsets can be reduced by increasing the sampling rate. The estimated using the PEM method is shown in Fig. 4. It is obvious that this PEM method couldn t succeed under this condition. The estimations of the other parameters are also heavily fluctuated. We suspect that this is mainly due to a too short window, so that there is not enough data to excite pem function to work properly. This suspicion is confirmed by a later test, where a window with 1 samples is used. Both LSP method and PEM method worked perfectly. The results using LSP are at the same precision level as the case using 5 samples. While PEM method leaded to a much more precise steady-state results: T d is estimated as 1.954, while K is 4. and T p is 2.. In order to illustrate the influence of the sampling rate, one LSP test result with a 1-length window and T s =.25sec is shown in Fig. 5. The slow sampling rate caused a poor estimation of time, while the estimation of the other parameters are still within ±5% steady-sate errors. The study using PEM method obtained the steady-state values as T d = 1.7727sec, K =4. and T p =2.. LSP Method PEM Method Condition CUP T23, RAM 1GB, software matlab 7.6. 5 Samples.531928 seconds 16.54329 seconds 1 Samples.548145 seconds 18.222629 seconds Fig. 6. Delay using LSP with N = 1, T s =.1, switching FOPDT The computation times of the LSP and PEM methods are listed in Table I. It is the running time of the whole procedure under the same simulation and computation conditions. There is no doubt that the proposed LSP method is much faster than the PEM method. B. A Switching FOPDT Model The second numerical example is a switching FOPDT model. i.e., during the period of [, 3] sec, system parameters are K =3, T p =1, T d =3.5. After 3 sec., the system parameters are switched immediately to K =4,T p =2,T d = 2.5. One result under the testing condition: T s =.1 sec, N = 1 and lmax k =4, lmin k =, is shown in Fig.6 and Fig.7, respectively. It is clear that the LSP method leaded to a more precise estimation of the time, while the PEM method leaded to a better results for the other two parameters. The computation time used by LSP method is.66577 seconds, while the time used by PEM method is 1267.936578 seconds. It has been observed that if the system time had the property of over 4 samples, the PEM method often returned a 585

Fig. 9. One set of input and output data from the testing system 25 2 Fig. 7. K k, Tp k using LSP with N = 1,Ts =.1, switching FOPDT 15 1 5 7 75 8 85 9 Fig. 1. Delay estimation using LSP for considered system 1 Parameter K 12 14 16 18 7 75 8 85 9 Fig. 8. Front and side faces of the testing refrigeration system 18 16 Parameter Tp 14 warning and the estimation result often went to worse. But the LSP method didn t have this kind of problem. Furthermore, the length of the window need to be carefully selected in order to balance the estimation accuracy and the detectability of time varying characteristics for both methods. In general, it can be observed that the proposed LSP method is quite promising for FOPDT model identification in terms of reasonable accuracy, large flexibility and low computation load. In the following, this method is further applied to model the superheat dynamic in a refrigeration system. V. SUPERHEAT DYNAMIC IDENTIFICATION The considered refrigeration system is a supermarket display case cooler as shown in Fig.8. Compared with a freezer, the display case cooler has a less efficient (adjustable) air curtain. Two sensors are installed to gain the superheat measurement. One pressure sensor is placed close to the inlet tube of the evaporator. Then the evaporation temperature is estimated based on this pressure measurement and the knowledge of refrigerant type. A thermostat transducer is placed at the evaporator outlet to measure the gaseous refrigerant temperature. The experimental data is collected from this testing system. The sampling period T s is selected as 2 sec. Moreover, it has been noticed that the system time is no more than 3 sec., i.e., we can set up the upper limit of the time as 15 samples. The input data is the measurement of the openness percentage of the expansion valve, and the output data is 12 1 7 75 8 85 9 Fig. 11. K k and T k p estimations using LSP for considered system the calculated superheat (temperature) based on two sensor measurements. In order to significantly excite the considered system, the designed input signal consists of a number of asymmetrical relay cycles. One set of input and output data is illustrated in Fig.9. A rectangular window with a length of 2 samples is used. Thereby the first estimation result comes at 7 sec, i.e., (2 (window length) + 15 (maximal )) T s =7 sec.. The estimated system time is indicated in Fig. 1. It can be noticed that during the period from the beginning to the 744th sampling step, the estimation stayed at a value of 32 sec. From the 746th sampling step, the estimated value stabilized around a value of 234 sec. The estimated system only significantly changed twice regarding to this tested experiment. The identification results of (sampled) system gain and time constant can be seen in Fig. 11. The time varying feature of these two parameters is quite obvious. In general, the estimated system gain has a trend to slightly increase until reaching some steady-state while the estimated system time constant has a trend to slightly decrease until reaching some steady-state. This test also showed that the superheat gradually converge to its expected working point (1 degree for this case). It has been found in [1] that the system parameters of a 586

.2 Fig. 12. 25 2 15 1 5 5 25 2 15 1 5 5 Fig. 13. output 5 1 estimation.4 input 3-D plot of w.r.t. input and output signals (stable test) output 5 estimation 1.4.6.5 input 3-D plot of w.r.t. input and output (unstable test) nonlinear FO model of the superheat dynamic are relevant to system input, output and disturbance as well. The coupling between the superheat dynamic and the compressor behavior is also studied in [7]. Thereby, a 3-D plot of the estimated time w.r.t. the input and output signals is shown in Fig. 12. From this observation, it seems that the system time mainly depends on the output value. Another test exhibited quite different results. The input signal is different from the above discussed test. This input signal didn t lead the superheat to gradually converge to a steady-state working point, instead, the superheat oscillated quite lot for the entire testing period. The 3-D plot of the estimated time w.r.t. the input and output signals, is shown in Fig. 12. It indicated a much more complicated situation. The investigation of these relationships need to take the physical modeling [3], [1] into consideration. For instance, the model (2) may need to be extended as G t (s) = K(t,y(t),u(t),d(t)) T (t,y(t),u(t),d(t)) p.6.8.7 (t,y(t),u(t),d(t)) 1.8 s +1 exp T d s, (11) where X (t,y(t),u(t),d(t)) indicates that the value of time-varying parameter X depends on the time t, output y(t), input u(t) and disturbance d(t). The disturbance can include the heat exchange of the evaporator with ambient environment [1], and compressor influence as well [7]. Maybe this general model (11) can be simplified by only taking the critical variable(s) into consideration, such as the time may be dominantly determined by the superheat. The investigation of these issues will be part of our future work. VI. CONCLUSION An on-line nonlinear FOPDT system identification method is proposed and applied to model the superheat dynamic for a supermarket refrigeration system. The considered nonlinear FOPDT model is an extension of standard FOPDT model by means that system parameters, such as system gain, time constant, and time, are real-time variables. After the considered system is discretized, the nonlinear FOPDT identification problem is formulated as a MINLP problem. By combining the BB method and LS technique, an iterative algorithm is proposed to on-line identify these time-dependent parameters. The proposed method is firstly tested through a number of numerical examples, and then applied to model the superheat dynamic in a supermarket refrigeration system. The performance of the proposed algorithm is also compared with that of an algorithm constructed using the built-in function pem in Matlab SI toolbox. These studies discovered that the proposed method is quite promising for on-line nonlinear FOPDT model identification in terms of reasonable accuracy, large flexibility and low computation load. The study on the superheat dynamic indicated some necessity of adaptive mechanism of efficient superheat control. Acknowledgement: The authors would like to thank Dr. R. Izdi-Zamanabadi and Dr. Lars F. S. Larsen from Danfoss A/S for valuable discussions and their support for testings. REFERENCES [1] K. J. Åström and T. Hägglund, PID Controllers: Theory, Design and Tuning, Instrument Society of America, 1995. [2] K. Brazauskas and D. Levisauskas, Adaptive transfer function-based control of nonlinear process. Case study: Control of temperature in industrial methane tank, Int. J. Adapt. Control Signal Process., Vol.21,pp.871-884, 27. [3] E.W. Gralda and J.W. MacArthurb, A moving-boundary formulation for modeling time-dependent two-phase flows, International Journal of Heat and Fluid Flow, Vol.13, No.3, pp.266-272, 1992. [4] I. E. Grossman and N. V. Sahinidis (eds), Special issue on mixed integer programming and its application to engineering, Part I, Optim. 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