Identification of Fractional Order System with Scarce Measurement Data Based on Multi-Innovation Estimation Algorithm

Aiming at the modeling issues of fractional order Hammerstein system with scarce measurements, a novel multi-innovation hybrid identification algorithm is proposed to deal with them. Firstly, a multi-innovation estimation algorithm based on auxiliary model is presented to estimate the parameters of the nonlinear fractional order system, and a multi-innovation Levenberg-Marquardt algorithm are derived to confirm the fractional orders. Secondly, the convergence properties of the proposed algorithm are analyzed using the lemmas and theorems. Finally, in order to illustrate the effectiveness of the proposed algorithm, two fractional order nonlinear systems with scarce measurements are studied to prove the validity.


Introduction
In recent years, the researchers have been interested in the modeling and control of the real fractional order systems. Fractional order systems (FOSs) is more common than the traditional integer order systems, especially for the application with long memory characteristics, such as supercapacitors [1], thermal diffusion processes [2,3], and viscoelastic structures [4], etc. More and more researchers began to pay attention to fractional models, and found that fractional model has more accurate description method for many physical processes [5,6]. In order to deal with the modeling problems of fractional order system, some modeling methods are presented to deal with parameters identification and fractional orders estimation, recorders [14], electric individual-wheel system on an autonomous platform [15], etc.
For the system with scarce measurements or missing measurements, the reconstruction methods of scarce measurements or missing measurements have received widespread attention in the control field. One method is substitute for missing data points by interpolating calculation among available data points, such as linear interpolation, parabolic interpolation, cubic interpolation, spline function interpolation, or piecewise constant interpolation and so on. Another method is to use a dynamic model to reconstruct the missing measurement data. Basing on auxiliary model modeling principle, Ding et al. presented a multi-innovation gradient-descent algorithms to deal with the missing data of the scarce measurement systems [16][17]. Broersen et al. proposed a finite interval likelihood algorithm for AR model spectral estimation using the conditional log likelihoods method [18]. In [19], in order to handle the scarce measurement systems, an auxiliary model-based algorithm is proposed to deal with the identification of multi-variable output error model with scarce measurements. The effectiveness of the proposed method is proved by comparing with other methods in the literature.
In this paper, we attempt to study the identification of fractional order Hammerstein systems (FOHS) with scarce measurements. For the identification of these systems, it is difficult how to estimate the unknown parameters by traditional identification methods. The main difficult factors embody: 1) There are many parameters of FOHS to be identified by scarce measurements, including the model parameters and the corresponding fractional orders ; 2) Convergence of the identification algorithm; 3) Influence of external interference and outside noise.
For the above-mentioned issues, we proposed a new identification algorithm for FOHS with scarce measurements. The innovation and contributions of this paper are shown as follows: 1) Build a fractional order model with scarce sampling data; 2) Based on the theory of interactive estimation and the principle of hierarchical identification, a multi-innovation gradient descent (MIGD) algorithm and a multi-innovation Levenberg-Marquardt (MILM) algorithm are derived to estimate model parameters and fractional orders, respectively; 3) Give the convergence analysis of the presented algorithm using the lemmas and theorems.
The rest of our paper is organized as follows. Section 2 introduces the problem formulation of fractional order system with scarce measurements. Section 3 discusses the MIGD algorithm for fractional order system with scarce measurements. Section 4 gives the estimation of fractional order by the MILM algorithm. Section 5 proves the convergence of the proposed algorithms using the stochastic identification theorems. Section 6 studies two fractional order systems to show the effectiveness of the proposed algorithms. Finally, we offer some concluding remarks in Section 7.

Problem formulation 2.1 Calculation of fractional order system
In the past decades in system modeling and control, fractional order systems have attracted continually an increasing interesting among researchers [20][21][22]. The most commonly used in discrete cases are GL fractional calculus [20] , RL fractional calculus [21] and Caputo fractional calculus [22] , which is used in this paper is the definition of GL calculus is expressed as follows: where   is the fractional order difference operator of order  ; ) (kh x denotes a function of kh t = , which k is the k-th sampling, and h is assumed to be equal to 1. The term According to Eqs. (1)-(2), we give the following recurrence equation: According to Eq. (4), Eq. (1) can be written as the following equation: In this paper, we use Eq. (5) as the fractional calculation to study the modeling of the FOHS in subsequent sections.

Description of nonlinear fractional order model
In Fig. 1, we consider a nonlinear fractional order system which consists of a static nonlinear link in series with a linear fractional order system. are the system input and the system output, respectively, () vk is the outside noise. The nonlinear system in Fig.1 is also called a fractional order Hammerstein system (FOHS), and its linear model is shown as follows: where () uk is the output of the static nonlinear link, and it can be represented as: According to Fig. 1, the noiseless output of the system can be written as: . When the fractional orders of the denominator polynomial and the numerator polynomial in (8) are completely different, the fractional order models of Eq. (8) are generally non-identical (disproportionate) order systems; Otherwise, each fractional order is an integer multiple of the base order( is order factor), , such a model is defined as a homogeneous (proportionate) order system. In our paper, consider a proportional fractional order system. Then, Eq. (8) can be written as : By means of the discrete fractional order operator  and the derivation, Eq. (9) can be derived as follows: ... Fig.1  Herein, the overall output of FOHS with outside noise is arranged as follows, where () vk is a white noise with zeros mean and variance 2  . In our paper, the identification goal is to study the identification method of system parameters i a , i b , i c and fractional order  by measurement data points.

Fractional order systems with scarce measurements
In practice, due to some reasons, including sensor failure, hardware limitation and network transmission, etc, some sampled data in the system are lost. This system is called loss data system, such as multi-rate sampled data system. In [16], the loss data system is divided into two cases: one is missing data system, anther is scarce measurement data system. In a certain time interval, compared with all sampled data, when the lost data points account for the majority, it is called the scarce measurement data system which is described as Fig. 2. When the lost data points account for a small part, it is called the missing data system which is described as Fig. 3.
For loss data, we design a mathematical description method to deal with them. First, define an integer sequence{ : =0,1,2,...} s ks to satisfy, . The dynamic diagram of the system with scarce measurement data is shown in Fig. 2, where there is more missing data, the observable data set: . The dynamic diagram of the system with missing sampling data is shown in Fig. 3, and the missing data is less, the observable data set:

The identification of FOHS based on multi-innovation estimation algorithm
For convenience, we define the total parameter vector θ and the information vector as ( , )    1)), ( ( 2)),..., . Thus, Eqs. (11) and (12) are written as: The identification difficulty is that (14)  Here, we give the following vector definitions: ),..., Then the estimation of the parameter vector θ can be solved by Let ) ( s k  be the step-size. According to the identification model in Eq. (14), a negative gradient search is used to minimize the following cost function, We can get the following gradient descent algorithm: . In order to improve the accuracy of parameter estimation and speed up the convergence rate, we expand the term  T  T  1  1  1  1   T  T  1  1 The L-dimensional innovation vector can be also expressed as: Eq. (15) can be equivalently written as: , the fractional order  is known and a multi-innovation gradient descent algorithm (MIGD) is shown as follows:  After the estimated parameter vector θ is obtained, the estimates of the vector a elements are the first a n values of θ , b can be acquired from the a n +1 to a n + b n elements of θ . For the estimation vector θ , we notice that for j c , we have b n estimates ˆj c . Therefore, we use the mean value to be calculated as its estimate, For the performance analysis of the developed algorithm, we will give some discussions in the next section. The corresponding identification algorithms can be divided into two stages: one for order update, anther for parameter estimation. The different identification algorithms of the two stages are carried out alternately.

The fractional order estimation of FOHS
Define an error function as follows: (30) Then the identification problem can be converted into minimizing the following objective function: where N is the length of the total sampling data. For the objective function in Eq. (31), the Levenberg-Marquardt (L-M) algorithm is adopted to confirm the fractional order  as follows: The update rule is based on the calculation of the gradient and Hessian ' J and '' J with respect to fractional order  , and  is a tuning parameter for the convergence. The calculation of the gradient and Hessian with respect to the fractional order  ( 'Ĵ  and ''Ĵ  ) can be performed as follows:  In this paper, the MIGD method is proposed to deal with the parameters identification of FOHS and the MILM algorithm is derived to estimate the fractional order. According to the theory of interactive estimation and the principle of hierarchical identification, the two proposed algorithms are used to identify the parameters and estimate the fractional order alternately. In each iteration, the parameter estimation depends on the previous estimation of fractional order. In turn, the estimation of fractional order is based on the parameter estimation of the previous iteration, and both of them perform a complete hierarchical calculation process. The whole steps of the proposed method can be summarized as follows: Step Step 5: Compute 'Ĵ  and ''Ĵ  by using Eq. (35a) and Eq.(35b), and acquire the fractional order  using Eq. (37); Step 6: Update the parameter estimation () Step 7: Let 1 ss =+, if sN  return Step 6, else go to Step 8; Step 8: Compute the objective function in Eq.(31) ; Step 9: If

Performance analysis
Let us introduce some notations. The symbol X denotes the determinant of the square matrix X , i.e., X = det X .
This proves Lemma 4. Define       The estimation results of each iteration can affect the next parameter identification of fractional order system because the two proposed algorithms are alternately implemented in turn. If the fractional order estimate approximates the true value, the parameter estimates of fractional order system will also approach the true values. Thus, the upper limit of parameter estimation error is very small.
As shown in Fig. 4, under the two types of noise-signal ratios, the lower the noise-signal ratio, the smaller the parameter estimation error, indicating that the proposed algorithm can work better for low signal-to-noise ratios.  The narrower the cabinet and the closer to the 0 scale line, the smaller the parameter estimation error. It can be clearly seen that when 5000 K = and 3 L = , the parameter estimation error is the smallest, and the parameter estimation curve is shown in the Fig. 6, different color curves correspond to corresponding color coordinates. Therefore, combined with the method proposed in this paper, the estimated parameters when the number of information is 3, the data length is 5000, and the noise-to-signal ratio is 14.13% are used to model the system. The estimated output and actual output of the system obtained by the model are shown in Fig. 7, comparing the estimated output with the actual output and amplifying it partially, it can be clearly seen that the actual output data overlaps with the estimated output data and is a complete fit.
are not available. In simulation, the dual-rate coefficient q is selected as 2. It can be seen from example 1 that the method proposed in this paper has a better effect on de-noise signal ratio, so this example uses low noise signal ratio 14.13% ns  = for simulation. Taking 1,3,5 L = and applying the proposed algorithms in our paper, the parameter estimates and their relative errors Table. 3. The relative error diagram of the parameter estimation with different innovation lengths L=1, 3, 5 and data lengths 5000, 10000, 20000 K = are shown in Figs. 8, 9. . The parameter estimation curve is shown in Fig. 10. Therefore, the estimated parameters at this time are combined with the method in this paper, and the estimated output of the system and the actual output obtained are shown in Fig. 11. The partial magnification shows that the output is basically fitted, which verifies the effectiveness of the method proposed in this paper.

Conclusion
In this paper, we have studied the identification problems for FOHS with scarce measurements. The MIGD algorithm and the MILM algorithm are proposed to deal with the identification of FOHS systems with scarce measurements and fraction order, and the convergence performance of MIGD algorithm is analyzed in the random statistical frame. Choose the appropriate number of information and data length, and use the method proposed in this paper to identify system parameters with better accuracy. For the multi-rate FOHS systems with multi-variables, they can be regarded as a class of fraction order systems with scarce measurement. In the future, we will study the multi-rate FOHS with multi-variables.