Identification of Rational Systems with Logarithmic Quantized Data

This paper is concerned with the quantized identiﬁcation of rational systems, where the systems’ output is quantized by a logarithmic quantizer. Under the assumption that the systems’ input is periodic, the identiﬁcation procedure is categorized into two steps. The ﬁrst step is to identify the noise-free output of systems based on the quantized data. The second is to identify the unknown parameter based on the input and the estimation of the noise-free output. The identiﬁcation algorithm is also summarized. Asymptotic convergence of the estimators is analyzed in detail, which shows that the estimators are convergent almost everywhere. A numerical example is given to illustrate the results obtained in this paper.


Introduction
Due to sensor limitations, signal quantization, or coding for the limitation of networked bandwidth in modern control systems, the study of system identification of plants with quantized observations becomes significant in modeling systems. e quantization may affect the algorithm of system identification, the convergence of the estimates, and so on.
us, more attention should be paid when there exist quantized data in a system.
In recent years, the research of system identification with quantized data has constituted a vast body of research. For example, Wang et al. [1] presented some new methodologies for the identification of a system with quantized output observations and explored their potential in extending control capabilities for systems with limited information or networked systems. Gustafsson and Karlsson [2] studied fundamental issues in estimation that related directly to the core methods of identification for the system based on quantized observations. e identification of ARMA models was studied in [3], where the systems' outputs were subject to both quantization and packet dropout. Maximum likelihood estimation was analyzed in [4], where dithering noise was added before quantization. Widrow and Kollar [5,6] developed the statistic theory of uniform quantized effect. Identification of the system with only binary-valued sensors was investigated in [7]. Based on the Gauss-Markov theorem, a quantile-based estimator was proposed in [8]. More research about system identification using quantized data can be seen in [9][10][11][12][13][14][15].
It has to be pointed out that most of the above literatures concerning with quantized identification were based on the uniform quantized data. Besides the uniform quantizer, there is logarithmic quantizer [16][17][18][19], dynamic quantizer [20][21][22][23][24], and so on. Compared with uniform quantizer, sometimes these quantizers have better quantization effect. However, the related research literature is quite limited to the study of system identification with other types of quantized data. Since logarithmic quantizer has the advantage that it has the better efficiency on data rate for performance control, it has been widely studied and used in control theory. us, it is necessary to study the identification of systems based on logarithmic quantized data. In this paper, we will analyze the rational system identification concerning that the output data are quantized by a logarithmic quantizer.
For the identification of rational systems with quantized data, in [1], a two-step identification procedure was introduced that employs periodic signals, empirical measures, and identification features so that rational models can be identified without resorting to complicated nonlinear search algorithms. e present work of this paper extends that of [1] in the following three ways. (1) e identification of rational systems is studied based on the logarithmic quantized data.
is quantized strategy is more complex compared to [1], where the systems' output was observed via a binary-valued sensor. (2) e difficulty in treating logarithmic quantized data is that the logarithmic quantization levels are not uniform. en, the method of quantile-based estimation is used to identify the noise-free output of system.
(3) A fundamental problem of system identification with quantized data is that the convergence should be satisfied. en, the convergence almost everywhere of the estimators for both the noise-free output and the unknown parameter vector of the system are analyzed in detail.
is paper begins with a statement of the problem in Section 2, where the descriptions of the rational systems and the logarithmic quantizer are given, respectively. How to identify the unknown parameters is analyzed in Section 3. e convergence of the parameter estimates is established in Section 4. In Section 5, some simulation results are presented. Concluding remarks are given in Section 6.
Notations. roughout this paper, the superscript "T" denotes the transpose of a vector or a matrix. | · | stands for Euclidean norm. ‖ · ‖ represents the largest singular value of the corresponding matrix. Pr(·) and E(·) represent the probability and mathematical expectation of the corresponding random variables, respectively. Z denotes the set of integers. i.i.d. denotes independent identically distributed.

Problem Statement
In this paper, consider the following system which is disturbed by noise: where x k � G(q)u k , q is the one-step backward shift operator satisfying qu k � u k−1 , d k is the random sensor noise, and G(q) is a function of q described by System (1) can also be expressed in its regression form: where θ is the unknown parameter vector which needs to be identified.

Remark 1. Compared with equation disturbance structure,
which is an autoregressive moving average (ARMA) model structure, the structure of system (1) can relate the systems' output to sensor measurement errors in practical system configurations [1]. us, we adopt system (1) which is the structure of output disturbance in this paper. Due to the limitation of the network bandwidth, suppose that the output y k of the system is measured by a logarithmic quantizer, i.e., where Q(·) is a logarithmic quantizer with quantization density ρ ∈ (0, 1). In other words, the actual output of the system is s k . e structure of the system studied in this paper is illustrated in Figure 1. e quantizer Q(·) has the following quantization levels: where v 0 > 0 is a scaling parameter. Q(·) can be expressed in the following form: where i ∈ Z. is type of quantizer is depicted in Figure 2. e purpose of this paper is to identify the unknown parameter vector θ from the logarithmic quantized output data s k (k � 1, 2, . . . , ) and then establish the corresponding convergence properties. e identification procedure will be categorized into two steps. e first step is to identify the noise-free output x k based on the logarithmic quantized output data s k . e second is to identify the unknown parameter vector θ on the basis of the input u k and the estimation of x k .

Parameter Identification
For system (1), suppose that the input u k is 2m-periodic. Since G(q) is a stable function. en, after a short transient duration, the noise-free output x k � G(q)u k of system (1) will also be 2m-periodic, i.e., for any nonnegative integer l � 0, 1, . . . and positive integers j � 1, 2, . . . , 2m, x j+2ml � x j . Since x k is 2m-periodic, we can suppose that the first 2m unknown real numbers of x k are c j { } , i.e., x j � c j { } . en, the identification of x k is converted into the identification of the 2m real numbers c j { } .

Remark 2.
It should be mentioned that the number of parameters to be identified is 2m. And, we also select the input to be 2m-periodic. If the input is not 2m-periodic, then the method proposed in this paper is not feasible. Based on the above assumption and analysis, for j � 1, 2, . . . , 2m, the output of the system can be expressed as follows: en, the quantized output s k is Now, the identification procedure is converted into identifying c j { } based on s k , firstly, and then identifying θ based on the estimate of c j { } and the input u k .

3.1.
Identification of x k . e analysis of the identification of x k is based on the following assumption.

Assumption
1. For the random sensor noise d k (k � 1, 2, . . . , ) proposed in system (1), d k is a sequence of i.i.d. random variables. Moreover, the distribution function F(·) of d k is invertible and its inverse function F − 1 (·) is twice continuously differentiable.
For system (10), the quantized output s k can be described as follows: Remark 3. According to expression (11), the possible values of s k are ±ρ i v 0 , 0 with i ∈ Z. Since lim i⟶∞ ( ± ρ i v 0 ) � 0 and Pr s j+2lm � 0 � 0, then, in this paper, we will only analyze the probability of the event s j+2im ≤ ± ρ j v 0 .
In this paper, the method of quantile-based estimation which is proposed in [8] is used to estimate c j { } . For j � 1, 2, . . . , 2m, denote When s j+2lm ≥ 0, according to expression (11), Since y k can be expressed as y j+2lm � c j { } + d j+2lm , then

Mathematical Problems in Engineering
us, where F(·) is the distribution of the random sensor noise d k .
In the similar way, when s j+2lm < 0, , i ∈ Z, and define the integers a i and b i as follows: en, the probability p j { } can be expressed as For some positive integer N, let the observation length be 2mN. Given j � 1, 2, . . . , 2m, define where I · { } is the indicator function. It is described as Letting { } can be obtained as follows: en, the estimator of x k is obtained, i.e.,

Identification of θ.
e analysis of the identification of θ is based on the following assumption. A(q) and B(q) in system (1), 1 − A(q) and B(q) are coprime polynomials.

Remark 4.
It should be mentioned that Assumption 1 and Assumption 1 do not give too many restrictions on system (1). us, the method proposed in this paper is feasible for most systems.
System (1) without noise is described as Since the system has a 2m-periodic input, for some k 0 � 1, 2, . . . , we can define en, system (1) without noise can be denoted as For this system, if Π is full rank, then there exists a oneto-one mapping between the output X and the parameter vector θ. e property of the matrix Π is illustrated in the following lemma.
Lemma 1 (see [1]). Under Assumption 2, if the system's input u k is 2m-periodic and full rank, then Π is invertible for all k 0 and ‖Π −1 ‖ is independent of k 0 .
To estimate θ, we can define Let where θ is the estimator of θ. en, the following expression is obtained: us, the least-square estimator of θ is

Summary of the Identification Algorithm.
e identification steps of the parameter θ are summarized Algorithm 1.

Asymptotic Analysis
In this section, we will analyze the statistical properties of the estimators x k and θ when the sample number N tends to infinity, i.e., the convergence of the estimators x k and θ. To proceed, some definitions and lemmas are presented.
Definition 1 (convergence almost everywhere, see [25]). A sequence ξ n (ω) is said to converge almost everywhere to Mathematical Problems in Engineering Lemma 2 (Kolmogorov strong law of large numbers, see [25] e convergence of the estimator will be illustrated in the following theorem. Theorem 1. Under Assumption 1, the estimator obtained in (23) satisfies that Proof. According to the definition of the indicator function, it is easily obtained that I s j+2lm ≤ C i (j � 1, 2, . . . , m) are Bernoulli random variables. Moreover, It is obvious that According to the Kolmogorov strong law of large numbers, it follows that Furthermore, Assumption 1 implies that F − 1 (·) is continuous. en, i.e., us, i.e., e proof is completed. proof. X � Πθ can be expressed as where E N and f(E N ) are quantization errors, i.e., According to eorem 1, it is easily obtained that E N ⟶ a .s. 0. en, (30) implies that f(E N ) ⟶ 0 as E N ⟶ 0. us, f(E N ) ⟶ a .s. 0. If (30) is left multiplied by the matrix Π T , then us, e proof is completed.

Simulations
In this section, a numerical example is given to illustrate the results obtained in this paper. Consider the following system: where the disturbance d k (k � 1, 2, . . . , ) is uniformly distributed in the interval [−0.1, 0.1].
en, the regression model of this system is obtained as where Φ k � y k− 1 y k− 2 u k− 1 u k− 2 T , the parameter θ � 0.5 −0.4 0.7 0.6 T , and the noise d k � d k − 0.5d k−1 + 0.4d k−2 . Suppose that the output y k is quantized by the logarithmic quantizer Q(·) which is described as (8) with the quantization density ρ � 0.95 and the scaling parameter v 0 � 1. en, s k � Q(y k ).
For a 4-periodic input u k , using Matlab to simulate this system, it is obtained that x k � G(q)u k is also 4-periodic after a short transient duration. Moreover, generate a group of d k ; then, a group of s k is obtained. According to Algorithm 1, the estimated errors of the parameters are illustrated in Figures 3 and 4, respectively, which show that the errors converge to 0 quickly. us, the results obtained in this paper are effective.

Conclusion
is paper has addressed the quantized identification of rational systems, where the systems' output is quantized by a logarithmic quantizer. Assuming that the input is periodic, the identification procedure has been categorized into two steps. For the first step, the quantile-based estimation has been used to estimate the noise-free output of the system. For the second step, the parameter vector has been obtained according to the one-to-one mapping between the noise-free output and the unknown parameter vector. e identification algorithm has also been summarized. In the end, the asymptotic convergence of both the noise-free output and the unknown parameter has been analyzed in detail. Numerical examples have also been given to illustrate the results obtained in this paper.

Data Availability
No underlying data are used in this paper.

Conflicts of Interest
e authors declare that they have no conflicts of interest. j { } N using formula (19).