Time-Varying FOPDT System Identification with ... - IEEE Xplore

2012 IEEE International Conference on Control Applications (CCA) Part of 2012 IEEE Multi-Conference on Systems and Control October 3-5, 2012. Dubrovnik, Croatia

Time-Varying FOPDT System Identification with Unknown Disturbance Input Zhen Sun and Zhenyu Yang Abstract— The Time-Varying First Order Plus Dead Time (TV-FOPDT) model is an extension of the conventional FOPDT by allowing the system parameters, which are primarily defined on the transfer function description, i.e., the DC-gain, time constant and time delay, to be time dependent. The TV-FOPDT identification problem turns to estimate these time-varying parameters based on measured control input and system output. This work considers a TV-FOPDT identification problem in the presence of an unknown disturbance input. By regarding the unknown input as one extra system parameter, the considered identification problem is formulated as a Stochastic Mixed Integral Programming (SMIP) problem after discretizing the original problem. The sliding window technique with forgetting factor is employed to cope with time resolution issue, and the Least Mean Square (LMS) method is used to obtain the optimal solution of each individual optimization problem based on different time delay assumptions. The proposed method is firstly tested through a number of numerical examples, and then it is applied to estimate a TV-FOPDT model of the superheat dynamic of a supermarket refrigeration system.

I. INTRODUCTION To develop mathematical models for real systems or processes is normally the first task in order to employ the model-based design method for control development. The mathematical model and the modeling methods can be very sophisticated for complicated systems, for instance, to model the superheat dynamic in the refrigeration systems [4], [7], [10]. Some tradeoff between the model/modeling complexity and the modeling precision needs to be taken. In our previous work [8], [10], a Time-Varying FOPDT (TV-FOPDT) model is proposed to model the superheat dynamic at a large operating range for a class of supermarket refrigeration system. The TV-FOPDT model is an extension of the standard FOPDT by allowing the system parameters (i.e., the system gain, time constant and time delay) to be possibly time dependent [5]. Of course, this time dependency could also be extended due to the fact that some of these parameters are directly dependent on some system variables, such as the input dependent time delay studied in [8]. However, this proposed model and the corresponding identification method is only suitable for SISO system situation. In this work, we propose a MISO TV-FOPDT model and its corresponding identification method. The proposed model and method are also applied to describe the superheat dynamic in a supermarket refrigeration system in a more realistic manner, i.e., the system model will consist of two Zhen Sun is with the Department of Energy Technology, Aalborg University, 6700 Esbjerg, Denmark [email protected] Zhenyu Yang is with the Department of Energy Technology, Aalborg University, 6700 Esbjerg, Denmark [email protected]

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classes of inputs: the openness degree of the expansion valve as the control input, and all other influences to superheat dynamic as the disturbance input. By extending our previous method proposed in [10], an on-line identification method is proposed to estimate all system parameters as well as the unknown input. Firstly, the considered problem is formulated as a Stochastic Mixed Integer Programming (SMIP) problem. Then, under the assumption of known boundaries of time delays, the sliding window technique with forgetting factor is employed to cope with time resolution issue, and the Least Mean Square (LMS) method is used to obtain the optimal solution of each individual optimization problem based on different time delay assumptions. The proposed method is firstly tested through a number of numerical examples, and then it is applied to estimate a MISO TV-FOPDT model of the superheat dynamic in the considered refrigeration system. The rest of the paper is organized in the following: Section II formulates the considered model and problem; Section III introduces the proposed MISO TV-FOPDT identification method; Section IV discusses testing results of numerical examples and results based on real data from a refrigeration system; finally, we conclude the paper in Section V. II. PROBLEM FORMULATION A. Definition of MISO TV-FOPDT Model A MISO TV-FOPDT model considered in this paper is defined in the following: X(s) = Gt1 (s)U1 (s) + Gt2 (s)U2 (s),

(1)

with transfer functions K1t t e−Td s . t Tp s + 1

(2)

K2t . Tpt s + 1

(3)

y(t) = x(t) + ω (t).

(4)

Gt1 (s) = and

Gt2 (s) = The measurement is

Here u1 (t) is a known (scalar) input. u2 (t) is an unknown (scalar) input, which we define it as the system’s disturbance. x(t) is the ”noise free” (scalar) output signal, and X(s)/Ui (s), i = 1, 2 is Laplace-transform of the system output/input. y(t) is the noisy system output, with the assumption that the noise ω (t) is zero-mean white Gaussian noise with its variance Q. Tdt is the time delay, Tpt is the system time constant, and K1t , K2t are system gains for inputs

u1 (t) and u2 (t), respectively. The superscript t of variables stands for the time varying feature of the corresponding variable. We assume the time constants in Gt1 (s) and Gt2 (s) are same in this consideration, which indicts that both inputs affect the output in the same dynamic manner. We also assume that there is the time delay only to the known input u1 (t), and K2t is a known parameter as well. We model all unknown factors relevant to Gt2 (s) into the unknown input u2 (t). B. Problem Formulation The considered MISO TV-FOPDT identification problem can be formulated as: to identify system parameters K1t , Tpt and Tdt , as well as to simultaneously estimate the unknown input u2 (t) based on the sampled data of control input u1 (t) and output y(t), in an on-line optimal manner. III. IDENTIFICATION METHOD The proposed method here is an extension of the ideas in our previous work [8], [10]. In general, the proposed method consists of two steps, namely discretized problem and iterative LMS Prediction algorithm. They are summarized in the following. A. Discretized Model Since the directly available data are samples of controlled input and measured output, the continuous-time system (1) is approximated by its discrete-time equivalence, i.e., X(z) = Gk1 (z)U1 (z) + Gk2 (z)U2 (z), with

(5)

Gk1 (z) =

K1k (1 − α k ) , k zl (z − α k )

(6)

Gk2 (z) =

K2k (1 − α k ) , z − αk

(7)

and

− Tsk Tp

where α k  exp , and Ts is the sampled interval. As we stated in [10], hereby {Kik }i=1,2 and Tpk are not same as {Kit }i=1,2 and Tpt in (2) and (3): The former ones are piecewise-constant (constant during every sampling period) timed functions, while the latter ones are continuously varying timed functions. The relationships of these items are described as {Kik }i=1,2 are equal to {Kit }i=1,2 and Tpk is equal to Tpt at each sampling time, i.e., Kik = Kit , i = 1, 2 and Tpk = Tpt when t = kTs for any nonnegative integer k. Thereby, we call {Kik }i=1,2 , Tpk as the kth sampled (time-varying) system gains, the kth sampled (time-varying) time constant, respectively [10]. The l k in (6) is the discrete approximation of the kth sampled system delay Tdk (Tdk = Tdt when t = kTs for any integer k), with the property Tdk ≈ l k Ts [10]. Denote β k  K1k (1 − α k ) and γ k  u2 (k)(1 − α k ), then model (5) can be converted into x(k) = α k x(k − 1) + β k u1 (k − l k − 1) + γ k K2k ,

(8)

for k = l k + 1, l k + 2, · · · ∞.

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The measured output signal is collected at each sampled time: (9) y(k) = x(k) + ω (k). Take (8) and (9) together, then 

y(k) = α k y(k − 1) + β k u1 (k − l k − 1) + γ k K2k + ω (k), (10) 

for k = l k + 1, l k + 2, · · · ∞. Here ω (k) is a Gaussian noise  with the property ω (k)  (1 − α k )ω (k). Then, the original continuous-time model estimation problem is converted to estimate parameter sequence of α k , β k , γ k and l k for a stochastic discrete-time system (10) based on a number of sampled input and measured output. B. Extended Iterative LMS Method The considered system identification problem of (10) is formulated as a SMIP problem. After picking up the Bound and Branch strategy [3] to handle the integer estimation, the LMS method is applied to cope with each optimal parameter identification under the assumption of different time delays [10]. Suppose that system (1) is running at kth sampling step and let N as the number of latest sample pairs of the measured output and input used to make the estimation at kth step. And N is also the length of the sliding window used in each estimation step. Define θ k  [α k β k γ k ]T , then the identification/esitmation problem of (10) at the kth sampling step can be formulated as: min E{ BkN − AkN (l k )θ k 22 }, ∈L θ k ∈ Θk lk

(11)

where BkN is a vector variable consisting of N latest measured outputs with forgetting factor at the current kth sampling step, i.e., BkN  [y(k) ρ y(k − 1) · · · ρ N−2 y(k − N + 2)

(12)

ρ N−1 y(k − N + 1)]T . AkN (l k ) is a system matrix which depends on time delay parameter, and is generated by N pairs of input and measured output with a forgetting factor as: ⎛ y(k − 1) u1 (k − l k − 1) ⎜ ρ y(k − 2) ρ u1 (k − l k − 2) ⎜ ⎜ .. .. AkN (l k )  ⎜ . . ⎜ ⎝ ρ N−2 y(k − N + 1) ρ N−2 u1 (k − l k − N + 1) ρ N−1 y(k − N) ρ N−1 u1 (k − l k − N) K2k−1 ρ K2k−2 .. .

ρ N−2 K2k−N+ ρ N−1 K2k−N

⎞ ⎟ ⎟ ⎟ ⎟. ⎟ ⎠

(13) Θk represents the possible range of θ k , L stands for the boundaries of time delay, ρ is a forgetting factor, which is

θˆ k (l k ) = ((AkN (l k ))T AkN (l k ))−1 (AkN (l k ))T BkN , k ˆ ˆ −1 Ak (l k ))−1 , Cov(θ ) = ((AkN (l k ))T Q(k) N

(14)

where θˆ k (l k ) stands for the estimation of θ k at current iteration with discrete time delay l k , Cov(θˆ k ) means the covariance of θˆ k , and Q(k)=(1 ˆ − αˆ k )2 Q is the  covariance of ω (k). - (θˆ k (lˆk ), lˆk ) which leads to the minimal prediction error among all iterations with regards to l k moving from k k , denoted as (θ k∗ (l k∗ ), l k∗ ), is chosen as the lmin to lmin optimal solution for (11) at the current step. - The estimation of the kth sampled system parameters of (6) and (7) for the current sample, i.e., Tˆpk , Kˆ 1k and uˆ2 (k), can be obtained from θ k∗ (l k∗ ) = [α k∗ β k∗ γ k∗ ]T using the following relation:

and the sampled time delay l k is estimated as l k∗ . As a result, Kˆ 1k , Tˆpk , uˆ2 (k) and l k∗ Ts are set as the approximations of the original parameters and unknown input for the continuous system (1) at the current sampled step. When a new (couple) data of input and measured output is obtained, the above procedure will be repeated. Thereby the system idenficiation/estimation for the model (1) can be executed in an on-line iterative manner. IV. NUMERICAL EXAMPLES AND REAL SYSTEM DATA In the following, the proposed method in this paper and the method used in [10] are both applied and compared. For simplicity, the proposed method is noted as new method, while the other one is noted as old method. A. Tests of Numerical examples 1) Case A-I: Data generated from a system with unknown input: Consider a switching system, which system parameters are set as: when t < 30 second, there are Tpt = 2, K1t = 3, K2t = 3, Tdt = 3.05; when t ≥ 30 second, the parameters change to Tpt = 3, K1t = 4, K2t = 4, Tdt = 2.05. The noise in the measurement of the output follows the distribution N (0, 0.001). The sampling period is set as Ts = 0.1 second. The length of sliding window is selected as N = 50, and the forgetting factor ρ = 0.95. Assume we have k the pre-knowledge of the sampled time delay like lmax = 40 k = 0. In the test, the identification begins after 100th and lmin sampling time, i.e., after 10 seconds. The data is collected by simulated the considered system with a sweep signal as the control input, and a multi-step signal as the unknown input with the property ⎧ ⎪ ⎨ 1, t < 40 u2 (t) = 1.2, 40 ≤ t < 60 (18) ⎪ ⎩ 2, t ≥ 60. The known input signal and measured output obtained from this simulation are illustrated in Fig. 1. output input u1

input u1 and output 5

4

3

Value

applied to decrease the effect of the old data to the estimation at the current sampling time. It is useful especially for the case that some of system characteristics may vary according to time [2]. In the following, it is selected in the interval [0.95, 1]. If there is no time delay issue, or the time delay is known beforehand, the optimization problem (11) can be simplified to be a standard LMS problem. If some pre-knowledge about time delay in system can be obtained, such as the potential upper and lower limits at each sampling step, an iterative numerical algorithm can be constructed by searching the optimal solution in the entire possible region of time delay in an one-by-one manner. Within each iteration, a LMS problem can be solved using standard techniques [6]. Basically, this method needs to enumerate all different possible (delay) situations. The benefit of doing so is that the global optimal solution can be guaranteed in most situations, as long as the computation load is not a critical factor [10]. Assume the limitation of time delay is described by some integer numbers multiplying with sampling period, k k k , lk i.e., lmin ≤ l k ≤ lmax and lmin max are known before the procedure. The sampling interval Ts , the sliding window length N and forgetting factor ρ need to be decided as well before the main process. At the beginning, the algorithm needs to collect the sampled data until the process reaches a specific sampling step which indicates enough data to construct matrices (12) and (13). Denote this initial step kini as kini , where N + lmax ≤ kini should be satisfied. The main identification procedure starts from the kini step with k ≥ kini : - A computing loop is constructed with regard to l k k k starting from lmin and ending at lmax by taking the unit k k ≤ increment to l . For each iteration (k) of l k (lmin k ), solve the LMS problem (11) and record the l k ≤ lmax corresponding prediction error. The LMS problem has an analytical solution as:

2

1

0

Ts = − , ln α k∗ k∗ β1 Kˆ 1k = , 1 − α k∗ k∗ uˆ2 (k) = γ /(1 − α k∗ ), Tˆpk

(15) (16) (17)

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Fig. 2, Fig. 3, Fig. 4 and Fig. 5 display the results of the system estimation for time delay, system gain of known input, time constant and unknown input, respectively. Here the red line plots the real value, the blue one shows the estimated value using the proposed method and the green one is the result using the original method proposed in [10]. It is obvious that the proposed method exhibited much better results than the old one did, which is supposed to be used only for SISO TV-FOPDT case. For the proposed method, the identification algorithm starts at 10th second. Since the system is already at a steady situation, the estimations showed reasonably good approximations and precisions. This stable estimation lasted for about 20 seconds until the system had a switching at 30th second. Some deviations are clearly observed during a short period after the switch of system parameters (30 sec.). The fluctuation period is approximately equal to one window length (50∗0.1=5 seconds) before the estimated parameters started to converge to new steady-state values. The same phenomenon happened when the unknown input has jumps at 40th second and 60th second, respectively. Regarding the accuracy, the time delay estimation showed some small steady-state estimated error and they are below 2% in most cases. These offsets are mainly due to the discretization of the system model, thereby it can be reduced by increasing the sampling frequency. All results of the other three estimation showed the steady-state error are less than 1% to the real values in most steady state cases. 2) Case A-II: Data generated from a system without unknown input: In this test, the data used for estimation is generated by applying the same input as used in Case A-I except that there is no unknown input, which means u2 ≡ 0. Both identification methods are tested and compared in the following.

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The results turned out that both methods showed almost same performances except different amplitudes of fluctuations after the switching point (30 sec), where the system parameters abruptly changed. The new method led to larger fluctuations than the old method. This is because that the new method got a wrong estimation (non-zero) of the unknown input for a short while, as shown in Fig. 9, which caused further deviations to all parameter estimations. Otherwise, we can conclude that both methods can provide almost same

Parameter Kt

refrigerant temperature. The experimental data is collected from this testing system [9]. The sampling period Ts is selected as 2 second. Moreover, it has been noticed that the system time delay is no more than 400 sec., i.e., we can set up the upper limit of the time delay as 200 samples. The input data is the measurement of the openness percentage of the expansion valve, and the output data is the calculated superheat (temperature) based on two sensor measurements. The designed input signal consists of a number of asymmetrical relay cycles. One set of input and output data is illustrated in Fig. 10.

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estimation performances. B. Real Data Estimation In this part, the data generated from a real refrigeration system [9] is used to estimate a MISO TV-FOPDT model of the superheat dynamic in the considered system. The considered refrigeration system is a supermarket display case cooler. 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 Input u

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Value

−0.1

Under the assumption that the superheat (temperature) dynamic can be approximated by system model (1), we define parameter K2t as 2. It should be noticed that the value of K2t does not critically affect the estimation results, even though it could influence the estimated aptitude of the unknown input. Theoretically, it can be set as any value. A sliding window with a length of 200 samples is used. Thereby the first estimation result comes at 400 sampling step, i.e., 200 (window length) + 200 (maximal delay) =400, this means that the first estimation should start at 800 second. In this test, both the new method and old method are employed as well. The estimated system parameters are illustrated in Fig. 11, Fig. 12 and Fig. 13, respectively. At this moment, we are not always sure that the proposed new method works better than the old method did, because the validation of this test is still under going. From the so-far observed results, we can conclude that superheat model in this refrigeration system should take the disturbances into consideration, which could be due to the influences of compressor and/or the ambient thermal environment. Furthermore, since we expect a model which is suitable for modeling superheat dynamic in large operating region, there is no doubt that TV-FOPDT model should be the best candidate [7], [9], [10].

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Fig. 9. The estimated unknown input for Case A-II using the proposed method

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V. CONCLUSION AND FUTURE WORK This paper considered a TV-FOPDT system identification problem in the presence of unknown disturbance input. The SISO TV-FOPDT model is extended as a MISO model in order to model the disturbance input as well. Correspondingly, an identification algorithm to estimate the time dependent parameters as well as the unknown input is proposed. By regarding the unknown input as one extra system parameter,

the considered problem is formulated as a SMIP problem. The bound-branch method for handling the mixed integer programming, the LMS for handling the optimal parameter identification, together with the sliding window with forgetting factor for data selection, are adopted and combined to handle the formulated problem. The proposed method is firstly tested on a number of numerical examples and compared with the previous work, and then it is applied to model the superheat dynamic in a supermarket refrigeration system. There is no doubt that the proposed new method provides more flexibility to model complex systems in a more realistic manner, and thereby, hopefully provide more precise model without sacrificing the model simplicity. The validation of the proposed method using the real data is under going, and we expect to report that results in the near future.

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VI. ACKNOWLEDGMENTS The authors would like to thank Casper Andersen for realsystem testing data collection, and also thank Dr. R. IzdiZamanabadi from Danfoss A/S for providing the testing facility and many valuable discussions.

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