# Identification of Rational Systems with Logarithmic Quantized Data

Authors: Mingming Ji; Shengchao Su
Journal: Mathematical Problems in Engineering
 (2021)

---

## Abstract

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.


---

## Body

## 1. 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. The quantization may affect the algorithm of system identification, the convergence of the estimates, and so on. Thus, 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. The 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 [9–15].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 [16–19], dynamic quantizer [20–24], 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. Thus, 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. The present work of this paper extends that of [1] in the following three ways. (1) The identification of rational systems is studied based on the logarithmic quantized data. This quantized strategy is more complex compared to [1], where the systems’ output was observed via a binary-valued sensor. (2) The difficulty in treating logarithmic quantized data is that the logarithmic quantization levels are not uniform. Then, 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. Then, 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.This paper begins with a statement of the problem in Section2, 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. The 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. Throughout 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.


## 2. Problem Statement

In this paper, consider the following system which is disturbed by noise:(1)yk=xk+dk,k=1,2,…,where xk=Gquk, q is the one-step backward shift operator satisfying quk=uk−1, dk is the random sensor noise, and Gq is a function of q described by(2)Gq=Bq1−Aq=b1q+b2q2+⋯+bmqm1−a1q+a2q2+⋯+amqm,m=1,2,…,k.System (1) can also be expressed in its regression form:(3)yk=Aqyk+Bquk+1−Aqdk=ΦkTθ+d˜k,where(4)Φk=yk−1,yk−2,…,yk−m,uk−1,uk−2,…,uk−mT,θ=a1,a2,…,am,b1,b2,…,bmT,d˜k=1−Aqdk,where θ is the unknown parameter vector which needs to be identified.Remark 1.
Compared with equation disturbance structure,(5)yk+a1yk−1+a2yk−2+⋯+amyk−m=b1uk−1+b2uk−2+⋯+bmuk−m+dk,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]. Thus, 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 outputyk of the system is measured by a logarithmic quantizer, i.e.,(6)sk=Qyk,where Q⋅ is a logarithmic quantizer with quantization density ρ∈0,1. In other words, the actual output of the system is sk. The structure of the system studied in this paper is illustrated in Figure 1.
The quantizerQ⋅ has the following quantization levels:(7)U=±ρiv0,i=0,±1,±2,…∪0,where v0>0 is a scaling parameter. Q⋅ can be expressed in the following form:(8)Qε=ρiv0,ε∈ρ+12ρi−1v0,ρ+12ρiv0,−ρiv0,ε∈−ρ+12ρiv0,−ρ+12ρi−1v0,0,ε=0,where i∈Z. This type of quantizer is depicted in Figure 2.
The purpose of this paper is to identify the unknown parameter vectorθ from the logarithmic quantized output data skk=1,2,…, and then establish the corresponding convergence properties. The identification procedure will be categorized into two steps. The first step is to identify the noise-free output xk based on the logarithmic quantized output data sk. The second is to identify the unknown parameter vector θ on the basis of the input uk and the estimation of xk.Figure 1
The structure of the system.Figure 2
Logarithmic quantizer.


## 3. Parameter Identification

For system (1), suppose that the input uk is 2m-periodic. Since Gq is a stable function. Then, after a short transient duration, the noise-free output xk=Gquk of system (1) will also be 2m-periodic, i.e., for any nonnegative integer l=0,1,… and positive integers j=1,2,…,2m, xj+2ml=xj. Since xk is 2m-periodic, we can suppose that the first 2m unknown real numbers of xk are γj, i.e., xj=γj. Then, the identification of xk is converted into the identification of the 2m real numbers γj.Remark 2.
It should be mentioned that the number of parameters to be identified is2m. 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, forj=1,2,…,2m, the output of the system can be expressed as follows:(9)yj+2lm=xj+2lm+dj+2lm=γj+dj+2lm,l=0,1,….
Then, the quantized outputsk is(10)sj+2lm=Qyj+2lm=Qγj+dj+2lm,l=0,1,….
Now, the identification procedure is converted into identifyingγj based on sk, firstly, and then identifying θ based on the estimate of γj and the input uk.


### 3.1. Identification ofxk

The analysis of the identification ofxk is based on the following assumption.Assumption 1.
For the random sensor noisedkk=1,2,…, proposed in system (1), dk is a sequence of i.i.d. random variables. Moreover, the distribution function F⋅ of dk is invertible and its inverse function F−1⋅ is twice continuously differentiable.
For system (10), the quantized output sk can be described as follows:(11)sj+2lm=ρiv0,yj+2lm∈ρ+12ρi−1v0,ρ+12ρiv0,−ρiv0,yj+2lm∈−ρ+12ρiv0,−ρ+12ρi−1v0,0,yj+2lm=0.Remark 3.
According to expression (11), the possible values of sk are ±ρiv0, 0 with i∈Z. Since limi⟶∞±ρiv0=0 and Prsj+2lm=0=0, then, in this paper, we will only analyze the probability of the event sj+2im≤±ρjv0.
In this paper, the method of quantile-based estimation which is proposed in [8] is used to estimate γj.
Forj=1,2,…,2m, denote(12)pj=Prsj+2lm≤ρiv0,ifsj+2lm≥0,Prsj+2lm≤−ρiv0,ifsj+2lm<0.
Whensj+2lm≥0, according to expression (11),(13)pj=Prsj+2lm≤ρiv0=Pryj+2lm<ρ+12ρiv0.
Sinceyk can be expressed as yj+2lm=γj+dj+2lm, then(14)pj=Prγj+dj+2lm<ρ+12ρiv0=Prdj+2lm<ρ+12ρiv0−γj.
Thus,(15)pj=Fρ+12ρiv0−γj,where F⋅ is the distribution of the random sensor noise dk. In the similar way, when sj+2lm<0,(16)pj=F−ρ+12ρi−1v0−γj.
DenoteCi=ρiv0ifsj+2lm≥0−ρiv0ifsj+2lm<0, i∈Z, and define the integers ai and bi as follows:(17)ai=1,ifCi>0,−1,ifCi<0,(18)bi=0,ifCi>0,1,ifCi<0.
Then, the probabilitypj can be expressed as(19)pj=Prsj+2lm≤Ci=Fρ+12ρbiaiCi−γj.
For some positive integerN, let the observation length be 2mN. Given j=1,2,…,2m, define(20)p^Nj=1N∑l=0N−1Isj+2lm≤Ci,where I⋅ is the indicator function. It is described as(21)Isk∈A=1,ifsk∈A,0,otherwise.
Lettingpj=p^Nj, the estimator γ^Nj of the parameter γj can be obtained as follows:(22)γ^Nj=ρ+12ρbiaiCi−F−1p^Nj.
Then, the estimator ofxk is obtained, i.e.,(23)x^j=x^j+2lm=γ^Nj=ρ+12ρbiaiCi−F−1p^Nj.


### 3.2. Identification ofθ

The analysis of the identification ofθ is based on the following assumption.Assumption 2.
ForAq and Bq in system (1), 1−Aq and Bq are coprime polynomials.Remark 4.
It should be mentioned that Assumption1 and Assumption 1 do not give too many restrictions on system (1). Thus, the method proposed in this paper is feasible for most systems.
System (1) without noise is described as(24)yk=xk=ΦkTθ.
Since the system has a2m-periodic input, for some k0=1,2,…, we can define(25)X=xk0,xk0+1,…,xk0+2m−1T,Π=Φk0,Φk0+1,…,Φk0+2m−1T.
Then, system (1) without noise can be denoted as(26)X=Πθ.
For this system, ifΠ is full rank, then there exists a one-to-one mapping between the output X and the parameter vector θ. The property of the matrix Π is illustrated in the following lemma.Lemma 1 (see [1]).
Under Assumption2, if the system’s input uk is 2m-periodic and full rank, then Π is invertible for all k0 and Π−1 is independent of k0.
To estimateθ, we can define(27)X^=x^k,x^k+1,…,x^k+2m−1T,Φ^k=x^k−1,x^k−2,…,x^k−m,uk−1,uk−2,…,uk−mT,Π^=Φ^k,Φ^k+1,…,Φ^k+2m−1T.
Let(28)x^k=Φ^kTθ^,where θ^ is the estimator of θ. Then, the following expression is obtained:(29)X^=Π^θ^.
Thus, the least-square estimator ofθ is(30)θ^=Π^TΠ^−1Π^TX^.


### 3.3. Summary of the Identification Algorithm

The identification steps of the parameterθ are summarized Algorithm 1.


## 3.1. Identification ofxk

The analysis of the identification ofxk is based on the following assumption.Assumption 1.
For the random sensor noisedkk=1,2,…, proposed in system (1), dk is a sequence of i.i.d. random variables. Moreover, the distribution function F⋅ of dk is invertible and its inverse function F−1⋅ is twice continuously differentiable.
For system (10), the quantized output sk can be described as follows:(11)sj+2lm=ρiv0,yj+2lm∈ρ+12ρi−1v0,ρ+12ρiv0,−ρiv0,yj+2lm∈−ρ+12ρiv0,−ρ+12ρi−1v0,0,yj+2lm=0.Remark 3.
According to expression (11), the possible values of sk are ±ρiv0, 0 with i∈Z. Since limi⟶∞±ρiv0=0 and Prsj+2lm=0=0, then, in this paper, we will only analyze the probability of the event sj+2im≤±ρjv0.
In this paper, the method of quantile-based estimation which is proposed in [8] is used to estimate γj.
Forj=1,2,…,2m, denote(12)pj=Prsj+2lm≤ρiv0,ifsj+2lm≥0,Prsj+2lm≤−ρiv0,ifsj+2lm<0.
Whensj+2lm≥0, according to expression (11),(13)pj=Prsj+2lm≤ρiv0=Pryj+2lm<ρ+12ρiv0.
Sinceyk can be expressed as yj+2lm=γj+dj+2lm, then(14)pj=Prγj+dj+2lm<ρ+12ρiv0=Prdj+2lm<ρ+12ρiv0−γj.
Thus,(15)pj=Fρ+12ρiv0−γj,where F⋅ is the distribution of the random sensor noise dk. In the similar way, when sj+2lm<0,(16)pj=F−ρ+12ρi−1v0−γj.
DenoteCi=ρiv0ifsj+2lm≥0−ρiv0ifsj+2lm<0, i∈Z, and define the integers ai and bi as follows:(17)ai=1,ifCi>0,−1,ifCi<0,(18)bi=0,ifCi>0,1,ifCi<0.
Then, the probabilitypj can be expressed as(19)pj=Prsj+2lm≤Ci=Fρ+12ρbiaiCi−γj.
For some positive integerN, let the observation length be 2mN. Given j=1,2,…,2m, define(20)p^Nj=1N∑l=0N−1Isj+2lm≤Ci,where I⋅ is the indicator function. It is described as(21)Isk∈A=1,ifsk∈A,0,otherwise.
Lettingpj=p^Nj, the estimator γ^Nj of the parameter γj can be obtained as follows:(22)γ^Nj=ρ+12ρbiaiCi−F−1p^Nj.
Then, the estimator ofxk is obtained, i.e.,(23)x^j=x^j+2lm=γ^Nj=ρ+12ρbiaiCi−F−1p^Nj.


## 3.2. Identification ofθ

The analysis of the identification ofθ is based on the following assumption.Assumption 2.
ForAq and Bq in system (1), 1−Aq and Bq are coprime polynomials.Remark 4.
It should be mentioned that Assumption1 and Assumption 1 do not give too many restrictions on system (1). Thus, the method proposed in this paper is feasible for most systems.
System (1) without noise is described as(24)yk=xk=ΦkTθ.
Since the system has a2m-periodic input, for some k0=1,2,…, we can define(25)X=xk0,xk0+1,…,xk0+2m−1T,Π=Φk0,Φk0+1,…,Φk0+2m−1T.
Then, system (1) without noise can be denoted as(26)X=Πθ.
For this system, ifΠ is full rank, then there exists a one-to-one mapping between the output X and the parameter vector θ. The property of the matrix Π is illustrated in the following lemma.Lemma 1 (see [1]).
Under Assumption2, if the system’s input uk is 2m-periodic and full rank, then Π is invertible for all k0 and Π−1 is independent of k0.
To estimateθ, we can define(27)X^=x^k,x^k+1,…,x^k+2m−1T,Φ^k=x^k−1,x^k−2,…,x^k−m,uk−1,uk−2,…,uk−mT,Π^=Φ^k,Φ^k+1,…,Φ^k+2m−1T.
Let(28)x^k=Φ^kTθ^,where θ^ is the estimator of θ. Then, the following expression is obtained:(29)X^=Π^θ^.
Thus, the least-square estimator ofθ is(30)θ^=Π^TΠ^−1Π^TX^.


## 3.3. Summary of the Identification Algorithm

The identification steps of the parameterθ are summarized Algorithm 1.


## 4. Asymptotic Analysis

In this section, we will analyze the statistical properties of the estimatorsx^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 sequenceξnω is said to converge almost everywhere to ξω if Prlimn⟶∞ξnω=ξω=1. Denote it as ξnω⟶a.s.ξω.Lemma 2 (Kolmogorov strong law of large numbers, see [25]).
Letξii=1,2,…, be an i.i.d. sequence of random variables; then 1/n∑i=1nξi⟶a.s.μ if and only if Eξi=μ<∞.


### 4.1. Asymptotic Analysis ofx^k

The convergence of the estimator will be illustrated in the following theorem.Theorem 1.
Under Assumption1, the estimator obtained in (23) satisfies that(31)x^j+2lm=γ^Nj⟶a.s.γj,j=1,2,…,m.Proof.
According to the definition of the indicator function, it is easily obtained thatIsj+2lm≤Cij=1,2,…,m are Bernoulli random variables. Moreover,(32)EIsj+2lm≤Ci=1×Prsj+2lm≤Ci+0×Prsj+2lm>Ci=pj.
It is obvious that(33)Ep^Nj=pj.
According to the Kolmogorov strong law of large numbers, it follows that(34)p^Nj⟶a.s.pj.
Furthermore, Assumption1 implies that F−1⋅ is continuous. Then,(35)F−1p^Nj⟶a.s.F−1pj,i.e.,(36)F−1p^Nj⟶a.s.ρ+12ρbiaiCi−γj.
Thus,(37)ρ+12ρbiaiCi−F−1p^Nj⟶a.s.γj,i.e.,(38)x^j+2lm=γ^Nj⟶a.s.γj.
The proof is completed.


### 4.2. Asymptotic Analysis ofθ^

The convergence of the estimatorθ^ will be established in the following theorem.Theorem 2.
If Assumption1 and Assumption 2 hold, then(39)θ^⟶a.s.θ.proof.
X^=Π^θ^ can be expressed as(40)X+EN=Π+fENθ^,where EN and fEN are quantization errors, i.e.,(41)EN=X^−X,fEN=Π^−Π.
According to Theorem1, it is easily obtained that EN⟶a.s.0. Then, (30) implies that fEN⟶0 as EN⟶0. Thus, fEN⟶a.s.0. If (30) is left multiplied by the matrix ΠT, then(42)ΠTX+ΠTEN=ΠTΠ+ΠTfENθ^.
Thus,(43)θ^=ΠTΠ+ΠTfEN−1ΠTX+ΠTEN⟶a.s.θ.
The proof is completed.


## 4.1. Asymptotic Analysis ofx^k

The convergence of the estimator will be illustrated in the following theorem.Theorem 1.
Under Assumption1, the estimator obtained in (23) satisfies that(31)x^j+2lm=γ^Nj⟶a.s.γj,j=1,2,…,m.Proof.
According to the definition of the indicator function, it is easily obtained thatIsj+2lm≤Cij=1,2,…,m are Bernoulli random variables. Moreover,(32)EIsj+2lm≤Ci=1×Prsj+2lm≤Ci+0×Prsj+2lm>Ci=pj.
It is obvious that(33)Ep^Nj=pj.
According to the Kolmogorov strong law of large numbers, it follows that(34)p^Nj⟶a.s.pj.
Furthermore, Assumption1 implies that F−1⋅ is continuous. Then,(35)F−1p^Nj⟶a.s.F−1pj,i.e.,(36)F−1p^Nj⟶a.s.ρ+12ρbiaiCi−γj.
Thus,(37)ρ+12ρbiaiCi−F−1p^Nj⟶a.s.γj,i.e.,(38)x^j+2lm=γ^Nj⟶a.s.γj.
The proof is completed.


## 4.2. Asymptotic Analysis ofθ^

The convergence of the estimatorθ^ will be established in the following theorem.Theorem 2.
If Assumption1 and Assumption 2 hold, then(39)θ^⟶a.s.θ.proof.
X^=Π^θ^ can be expressed as(40)X+EN=Π+fENθ^,where EN and fEN are quantization errors, i.e.,(41)EN=X^−X,fEN=Π^−Π.
According to Theorem1, it is easily obtained that EN⟶a.s.0. Then, (30) implies that fEN⟶0 as EN⟶0. Thus, fEN⟶a.s.0. If (30) is left multiplied by the matrix ΠT, then(42)ΠTX+ΠTEN=ΠTΠ+ΠTfENθ^.
Thus,(43)θ^=ΠTΠ+ΠTfEN−1ΠTX+ΠTEN⟶a.s.θ.
The proof is completed.


## 5. Simulations

In this section, a numerical example is given to illustrate the results obtained in this paper.Consider the following system:(44)yk=xk+dk,xk=0.7q+0.6q21−0.5q+0.4q2uk,where the disturbance dkk=1,2,…, is uniformly distributed in the interval [−0.1, 0.1].Then, the regression model of this system is obtained as(45)yk=ΦkTθ+d˜k,where Φk=yk−1yk−2uk−1uk−2T, the parameter θ=0.5−0.40.70.6T, and the noise d˜k=dk−0.5dk−1+0.4dk−2. Suppose that the output yk is quantized by the logarithmic quantizer Q⋅ which is described as (8) with the quantization density ρ=0.95 and the scaling parameter v0=1. Then, sk=Qyk.For a 4-periodic inputuk, using Matlab to simulate this system, it is obtained that xk=Gquk is also 4-periodic after a short transient duration. Moreover, generate a group of dk; then, a group of sk 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. Thus, the results obtained in this paper are effective.Figure 3
Estimated errors ofa1=0.5 and a2=−0.4.Figure 4
Estimated errors ofb1=0.7 and b2=0.6.Algorithm 1: Summary of the identification steps.
Step 1: ifsj+lm≥0, let Ci=ρiv0,ifsj+lm<0,letCi=−ρiv0.Step 2: calculateai and bi using formulas (17) and (18), respectively.Step 3: calculatep^Nj using formula (19).Step 4: calculatex^j+2lm using formula (21).Step 5. calculate the estimator using formula (30).


## 6. Conclusion

This 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. The 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.



---

*Source: 6667745-2021-07-13.xml*