Abstract

This paper studies least-squares parameter estimation algorithms for input nonlinear systems, including the input nonlinear controlled autoregressive (IN-CAR) model and the input nonlinear controlled autoregressive autoregressive moving average (IN-CARARMA) model. The basic idea is to obtain linear-in-parameters models by overparameterizing such nonlinear systems and to use the least-squares algorithm to estimate the unknown parameter vectors. It is proved that the parameter estimates consistently converge to their true values under the persistent excitation condition. A simulation example is provided.

1. Introduction

Parameter estimation has received much attention in many areas such as linear and nonlinear system identification and signal processing [1–9]. Nonlinear systems can be simply divided into the input nonlinear systems, the output nonlinear systems, the feedback nonlinear systems, and the input and output nonlinear systems, and so forth. The Hammerstein models can describe a class of input nonlinear systems which consist of static nonlinear blocks followed by linear dynamical subsystems [10, 11].

Nonlinear systems are common in industrial processes, for example, the dead-zone nonlinearities and the valve saturation nonlinearities. Many estimation methods have been developed to identify the parameters of nonlinear systems, especially for Hammerstein nonlinear systems [12, 13]. For example, Ding et al. presented a least-squares-based iterative algorithm and a recursive extended least squares algorithm for Hammerstein ARMAX systems [14] and an auxiliary model-based recursive least squares algorithm for Hammerstein output error systems [15]. Wang and Ding proposed an extended stochastic gradient identification algorithm for Hammerstein-Wiener ARMAX Systems [16].

Recently, Wang et al. derived an auxiliary model-based recursive generalized least-squares parameter estimation algorithm for Hammerstein output error autoregressive systems and auxiliary model-based RELS and MI-ELS algorithms for Hammerstein output error moving average systems using the key term separation principle [17, 18]. Ding et al. presented a projection estimation algorithm and a stochastic gradient (SG) estimation algorithm for Hammerstein nonlinear systems by using the gradient search and further derived a Newton recursive estimation algorithm and a Newton iterative estimation algorithm by using the Newton method (Newton-Raphson method) [19]. Wang and Ding studied least-squares-based and gradient-based iterative identification methods for Wiener nonlinear systems [20].

Fan et al. discussed the parameter estimation problem for Hammerstein nonlinear ARX models [21]. On the basis of the work in [14, 15, 21], this paper studies the identification problems and their convergence for input nonlinear controlled autoregressive (IN-CAR) models using the martingale convergence theorem and gives the recursive generalized extended least-squares algorithm for input nonlinear controlled autoregressive autoregressive moving average (IN-CARARMA) models.

Briefly, the paper is organized as follows. Section 2 derives a linear-in-parameters identification model and gives a recursive least squares identification algorithm for input nonlinear CAR systems and analyzes the properties of the proposed algorithm. Section 4 gives the recursive generalized extended least squares algorithm for input nonlinear CARARMA systems. Section 5 provides an illustrative example to show the effectiveness of the proposed algorithms. Finally, we offer some concluding remarks in Section 6.

2. The Input Nonlinear CAR Model and Estimation Algorithm

Let us introduce some notations first. The symbol 𝐈(𝐈𝑛) stands for an identity matrix of appropriate sizes (𝑛×𝑛); the superscript 𝑇 denotes the matrix transpose; 𝟏𝑛 represents an 𝑛-dimensional column vector whose elements are 1; |𝐗|=det[𝐗] represents the determinant of the matrix 𝐗; the norm of a matrix 𝐗 is defined by ‖𝐗‖2=tr[𝐗𝐗𝑇]; 𝜆max[𝐗] and 𝜆min[𝐗] represent the maximum and minimum eigenvalues of the square matrix 𝐗, respectively; 𝑓(𝑡)=𝑜(𝑔(𝑡)) represents 𝑓(𝑡)/𝑔(𝑡)→0 as 𝑡→∞; for 𝑔(𝑡)⩾0, we write 𝑓(𝑡)=𝑂(𝑔(𝑡)) if there exists a positive constant 𝛿1 such that |𝑓(𝑡)|⩽𝛿1𝑔(𝑡).

2.1. The Input Nonlinear CAR Model

Consider the following input nonlinear controlled autoregressive (IN-CAR) systems [14, 21]: 𝐴(𝑧)𝑦(𝑡)=𝐵(𝑧)𝑢(𝑡)+𝑣(𝑡),(2.1) where 𝑦(𝑡) is the system output, 𝑣(𝑡) is a disturbance noise, the output of the nonlinear block 𝑢(𝑡) is a nonlinear function of a known basis (𝑓1,𝑓2,…,𝑓𝑚) of the system input 𝑢(𝑡) [19], 𝑢(𝑡)=𝑓(𝑢(𝑡))=𝑐1𝑓1(𝑢(𝑡))+𝑐2𝑓2(𝑢(𝑡))+⋯+𝑐𝑚𝑓𝑚(𝑢(𝑡)),(2.2)𝐴(𝑧) and 𝐵(𝑧) are polynomials in the unit backward shift operator 𝑧−1 [𝑧−1𝑦(𝑡)=𝑦(𝑡−1)], defined as 𝐴(𝑧)∶=1+𝑎1𝑧−1+𝑎2𝑧−2+⋯+𝑎𝑛𝑧−𝑛,𝐵(𝑧)∶=𝑏1𝑧−1+𝑏2𝑧−2+𝑏3𝑧−3+⋯+𝑏𝑛𝑧−𝑛.(2.3) In order to obtain the identifiability of parameters 𝑏𝑖 and 𝑐𝑖, without loss of generality, we suppose that 𝑐1=1 or 𝑏1=1 [14, 21].

Define the parameter vector 𝝑 and information vector 𝝍(𝑡) as 𝝑∶=[𝐚𝑇,𝑐1𝐛𝑇,𝑐2𝐛𝑇,…,𝑐𝑚𝐛𝑇]𝑇∈ℝ𝑛0,𝑛0∶=𝑛+𝑚𝑛,𝐚∶=[𝑎1,𝑎2,…,𝑎𝑛]𝑇∈ℝ𝑛,𝐛∶=[𝑏1,𝑏2,…,𝑏𝑛]𝑇∈ℝ𝑛,𝝍(𝑡)∶=[𝝍𝑇0(𝑡),𝝍𝑇1(𝑡),𝝍𝑇2(𝑡),…,𝝍𝑇𝑚(𝑡)]𝑇∈ℝ𝑛0,𝝍0(𝑡)∶=[−𝑦(𝑡−1),−𝑦(𝑡−2),…,−𝑦(𝑡−𝑛)]𝑇∈ℝ𝑛,𝝍𝑗𝑓(𝑡)∶=𝑗(𝑢(𝑡−1)),𝑓𝑗(𝑢(𝑡−2)),…,𝑓𝑗(𝑢(𝑡−𝑛))𝑇∈ℝ𝑛,𝑗=1,2,…,𝑚.(2.4) From (2.1), we have []𝑦(𝑡)=1−𝐴(𝑧)𝑦(𝑡)+𝐵(𝑧)𝑢(𝑡)+𝑣(𝑡)=−𝑛𝑖=1𝑎𝑖𝑦(𝑡−𝑖)+𝑛𝑖=1𝑏𝑖𝑚𝑗=1𝑐𝑗𝑓𝑗(𝑢(𝑡−𝑖))+𝑣(𝑡)=−𝑛𝑖=1𝑎𝑖𝑦(𝑡−𝑖)+𝑚𝑛𝑗=1𝑖=1𝑐𝑗𝑏𝑖𝑓𝑗(𝑢(𝑡−𝑖))+𝑣(𝑡)(2.5)=−𝑛𝑖=1𝑎𝑖𝑦(𝑡−𝑖)+𝑐1𝑏1𝑓1(𝑢(𝑡−1))+𝑐1𝑏2𝑓1(𝑢(𝑡−2))+⋯+𝑐1𝑏𝑛𝑓1(𝑢(𝑡−𝑛))+𝑐2𝑏1𝑓2(𝑢(𝑡−1))+𝑐2𝑏2𝑓2(𝑢(𝑡−2))+⋯+𝑐2𝑏𝑛𝑓2(𝑢(𝑡−𝑛))+⋯+𝑐𝑚𝑏1𝑓𝑚(𝑢(𝑡−1))+𝑐𝑚𝑏2𝑓𝑚(𝑢(𝑡−2))+⋯+𝑐𝑚𝑏𝑛𝑓𝑚(𝑢(𝑡−𝑛))+𝑣(𝑡)=𝝍𝑇(𝑡)𝝑+𝑣(𝑡).(2.6) An alternative way is to define the parameter vector 𝜽 and information vector 𝝋(𝑡) as 𝜽∶=[𝐚𝑇,𝑏1𝐜𝑇,𝑏2𝐜𝑇,…𝑇,𝑏𝑛𝐜𝑇]𝑇∈ℝ𝑛0,𝐚∶=[𝑎1,𝑎2,…,𝑎𝑛]𝑇∈ℝ𝑛,𝐜∶=[𝑐1,𝑐2,…,𝑐𝑚]𝑇∈ℝ𝑚,𝝋(𝑡)∶=[𝝋𝑇0(𝑡),𝝋𝑇1(𝑡),𝝋𝑇2(𝑡),…,𝝋𝑇𝑛(𝑡)]𝑇∈ℝ𝑛0,𝝋0(𝑡)∶=[−𝑦(𝑡−1),−𝑦(𝑡−2),…,−𝑦(𝑡−𝑛)]𝑇∈ℝ𝑛,𝝋𝑗𝑓(𝑡)∶=1(𝑢(𝑡−𝑗)),𝑓2(𝑢(𝑡−𝑗)),…,𝑓𝑚(𝑢(𝑡−𝑗))𝑇∈ℝ𝑚,𝑗=1,2,…,𝑛.(2.7) Then (2.5) can be written as 𝑦(𝑡)=−𝑛𝑖=1𝑎𝑖𝑦(𝑡−𝑖)+𝑛𝑚𝑖=1𝑗=1𝑏𝑖𝑐𝑗𝑓𝑗(𝑢(𝑡−𝑖))=−𝑛𝑖=1𝑎𝑖𝑦(𝑡−𝑖)+𝑏1𝑐1𝑓1𝑢(𝑡−1)+𝑏1𝑐2𝑓2𝑢(𝑡−1)+⋯+𝑏1𝑐𝑚𝑓𝑚𝑢(𝑡−1)+𝑏2𝑐1𝑓1𝑢(𝑡−2)+𝑏2𝑐2𝑓2𝑢(𝑡−2)+⋯+𝑏2𝑐𝑚𝑓𝑚𝑢(𝑡−2)+⋯+𝑏𝑛𝑐1𝑓1𝑢(𝑡−𝑛)+𝑏𝑛𝑐2𝑓2𝑢(𝑡−𝑛)+⋯+𝑏𝑛𝑐𝑚𝑓𝑚𝑢(𝑡−𝑛)+𝑣(𝑡)=𝝋𝑇(𝑡)𝜽+𝑣(𝑡).(2.8) Equations (2.6) and (2.8) are both linear-in-parameters identification model for Hammerstein CAR systems by using parametrization.

2.2. The Recursive Least Squares Algorithm

Minimizing the cost function 𝐽(𝜽)∶=𝑡𝑗=1𝑦(𝑗)−𝝋𝑇(𝑗)𝜽2(2.9) gives the following recursive least squares algorithm for computing the estimate 𝜽(𝑡) of 𝜽 in (2.8): 𝐏𝜽(𝑡)=𝜽(𝑡−1)+𝐏(𝑡)𝝋(𝑡)𝑦(𝑡)−𝝋(𝑡)𝜽(𝑡−1),(2.10)−1(𝑡)=𝐏−1(𝑡−1)+𝝋(𝑡)𝝋𝑇(𝑡),𝐏(0)=𝑝0𝐈.(2.11) Applying the matrix inversion formula [22] (𝐀+𝐁𝐂)−1=𝐀−1−𝐀−1𝐁𝐈+𝐂𝐀−1𝐁−1𝐂𝐀−1(2.12) to (2.11) and defining the gain vector 𝐋(𝑡)=𝐏(𝑡)𝝋(𝑡)∈ℝ𝑛0, the algorithm in (2.10)-(2.11) can be equivalently expressed as ,𝜽(𝑡)=𝜽(𝑡−1)+𝐋(𝑡)𝑦(𝑡)−𝝋(𝑡)𝜽(𝑡−1)𝐋(𝑡)=𝐏(𝑡)𝝋(𝑡)=𝐏(𝑡−1)𝝋(𝑡)1+𝝋T,(𝑡)𝐏(𝑡−1)𝝋(𝑡)𝐏(𝑡)=𝐏(𝑡−1)−𝐏(𝑡)𝝋(𝑡)𝝋𝑇(𝑡)𝐏(𝑡)1+𝝋𝑇=(𝑡)𝐏(𝑡−1)𝝋(𝑡)𝐈−𝐋(𝑡)𝝋𝑇(𝑡)𝐏(𝑡−1),𝐏(0)=𝑝0𝐈.(2.13) To initialize the algorithm, we take 𝑝0 to be a large positive real number, for example, 𝑝0=106, and 𝜽(0) to be some small real vector, for example, 𝜽(0)=10−6𝟏𝑛0.

3. The Main Convergence Theorem

The following lemmas are required to establish the main convergence results.

Lemma 3.1 (Martingale convergence theorem: Lemma D.5.3 in [23, 24]). If 𝑇𝑡, 𝛼𝑡, 𝛽𝑡 are nonnegative random variables, measurable with respect to a nondecreasing sequence of 𝜎 algebra ℱ𝑡−1, and satisfy 𝐸𝑇𝑡∣ℱ𝑡−1⩽𝑇𝑡−1+𝛼𝑡−𝛽𝑡,a.s.,(3.1) then when ∑∞𝑡=1𝛼𝑡<∞, one has ∑∞𝑡=1𝛽𝑡<∞,a.s.and 𝑇𝑡→𝑇,a.s. (a.s.: almost surely) a finite nonnegative random variable.

Lemma 3.2 (see [14, 21, 25]). For the algorithm in (2.10)-(2.11), for any 𝛾>1, the covariance matrix 𝐏(𝑡) in (2.11) satisfies the following inequality: ∞𝑡=1𝝋𝑇(𝑡)𝐏(𝑡)𝝋(𝑡)||𝐏ln−1||(𝑡)𝛾<∞,a.s.(3.2)

Theorem 3.3. For the system in (2.8) and the algorithm in (2.10)-(2.11), assume that {𝑣(𝑡),ℱ𝑡} is a martingale difference sequence defined on a probability space {Ω,ℱ,𝑃}, where {ℱ𝑡} is the 𝜎 algebra sequence generated by the observations {𝑦(𝑡),𝑦(𝑡−1),…,𝑢(𝑡),𝑢(𝑡−1),…} and the noise sequence {𝑣(𝑡)} satisfies E[𝑣(𝑡)∣ℱ𝑡−1]=0,and E[𝑣2(𝑡)∣ℱ𝑡−1]⩽𝜎2<∞,a.s [23], and [ln|𝐏−1(𝑡)|]𝛾=𝑜(𝜆min[𝐏−1(𝑡)]), 𝛾>1. Then the parameter estimation error 𝜽(𝑡) converges to zero.

Proof. Define the parameter estimation error vector 𝜽(𝑡)∶=𝜽(𝑡)−𝜽 and the stochastic Lyapunov function 𝜽𝑇(𝑡)∶=𝑇(𝑡)𝐏−1(𝑡)𝜽(𝑡). Let ̃𝑦(𝑡)∶=𝝋𝑇(𝑡)𝜽(𝑡−1)−𝝋𝑇(𝑡)𝜽=𝝋𝑇(𝑡)𝜽(𝑡−1). According to the definitions of 𝜽(𝑡) and 𝑇(𝑡) and using (2.10) and (2.11), we have 𝜽𝜽[],(𝑡)=(𝑡−1)+𝐏(𝑡)𝝋(𝑡)−̃𝑦(𝑡)+𝑣(𝑡)𝑇(𝑡)=𝑇(𝑡−1)−1−𝝋𝑇(𝑡)𝐏(𝑡)𝝋(𝑡)̃𝑦2(𝑡)+𝝋𝑇(𝑡)𝐏(𝑡)𝝋(𝑡)𝑣2(𝑡)+21−𝝋𝑇(𝑡)𝐏(𝑡)𝝋(𝑡)̃𝑦(𝑡)𝑣(𝑡)⩽𝑇(𝑡−1)+𝝋𝑇(𝑡)𝐏(𝑡)𝝋(𝑡)𝑣2(𝑡)+21−𝝋𝑇(𝑡)𝐏(𝑡)𝝋(𝑡)̃𝑦(𝑡)𝑣(𝑡).(3.3) Here, we have used the inequality 1−𝝋𝑇(𝑡)𝐏(𝑡)𝝋(𝑡)=[1+𝝋𝑇(𝑡)𝐏(𝑡−1)𝝋(𝑡)]−1⩾0. Because ̃𝑦(𝑡) and 𝝋𝑇(𝑡)𝐏(𝑡)𝝋(𝑡) are uncorrelated with 𝑣(𝑡) and are ℱ𝑡−1 measurable, taking the conditional expectation with respect to ℱ𝑡−1, we have 𝐸𝑇(𝑡)∣𝐹𝑡−1⩽𝑇(𝑡−1)+2𝝋𝑇(𝑡)𝐏(𝑡)𝝋(𝑡)𝜎2.(3.4) Since ln|𝐏−1(𝑡)| is nondecreasing, letting 𝑉(𝑡)∶=𝑇(𝑡)||𝐏ln−1||(𝑡)𝛾,𝛾>1,(3.5) we have 𝐸𝑉(𝑡)∣𝐹𝑡−1⩽𝑇(𝑡−1)||𝐏ln−1||(𝑡)𝛾+2𝝋𝑇(𝑡)𝐏(𝑡)𝝋(𝑡)||𝐏ln−1||(𝑡)𝛾𝜎2⩽𝑉(𝑡−1)+2𝝋𝑇(𝑡)𝐏(𝑡)𝝋(𝑡)||𝐏ln−1||(𝑡)𝛾𝜎2,a.s.(3.6) Using Lemma 3.2, the sum of the last term in the right-hand side for 𝑡 from 1 to ∞ is finite. Applying Lemma 3.1 to the previous inequality, we conclude that 𝑉(𝑡) converges a.s. to a finite random variable, say 𝑉0, that is: 𝑉(𝑡)=𝑇(𝑡)||𝐏ln−1||(𝑡)𝛾⟶𝑉0||𝐏<∞,a.s.,or𝑇(𝑡)=𝑂ln−1||(𝑡)𝛾,a.s.(3.7) Thus, according to the definition of 𝑇(𝑡), we have ‖‖‖‖𝜽(𝑡)2⩽𝜽tr𝑇(𝑡)𝐏−1(𝑡)𝜽(𝑡)𝜆min𝐏−1||𝐏(𝑡)=𝑂ln−1||(𝑡)𝛾𝜆min𝐏−1𝑜𝜆(𝑡)=𝑂min𝐏−1(𝑡)𝜆min𝐏−1(𝑡)⟶0,a.s.(3.8) This completes the proof of Theorem 3.3.

According to the definition of 𝜽 and the assumption 𝑏1=1, the estimates ̂𝐚(𝑡)=[̂𝑎1(𝑡),̂𝑎2(𝑡),…,̂𝑎𝑛(𝑡)]𝑇 and ̂𝐜(𝑡)=[̂𝑐1(𝑡), ̂𝑐2(𝑡), …, ̂𝑐𝑚(𝑡)]𝑇 of 𝐚 and 𝐜 can be read from the first 𝑛 and second 𝑚 entries of 𝜽, respectively. Let ̂𝜃𝑖 be the 𝑖th element of 𝜽. Referring to the definition of 𝜽, the estimates ̂𝑏𝑗(𝑡) of 𝑏𝑗, 𝑗=2,3,…,𝑛, may be computed by ̂𝑏𝑗̂𝜃(𝑡)=𝑛+(𝑗−1)𝑚+𝑖(𝑡)̂𝑐𝑖(𝑡),𝑗=2,3,…,𝑛;𝑖=1,2,…,𝑚.(3.9) Notice that there is a large amount of redundancy about ̂𝑏𝑗(𝑡) for each 𝑖=1,2,…,𝑚. Since we do not need such 𝑚 estimates ̂𝑏𝑗(𝑡), one way is to take their average as the estimate of 𝑏𝑗 [14], that is: ̂𝑏𝑗1(𝑡)=𝑚𝑚𝑖=1̂𝜃𝑛+(𝑗−1)𝑚+𝑖(𝑡)̂𝑐𝑖(𝑡),𝑗=2,3,…,𝑛.(3.10)

4. The Input Nonlinear CARARMA System and Estimation Algorithm

Consider the following input nonlinear controlled autoregressive autoregressive moving average (IN-CARARMA) systems: 𝐴(𝑧)𝑦(𝑡)=𝐵(𝑧)𝑢(𝑡)+𝐷(𝑧)𝛾(𝑧)𝑣(𝑡),(4.1)𝑢(𝑡)=𝑓(𝑢(𝑡))=𝑐1𝑓1(𝑢(𝑡))+𝑐2𝑓2(𝑢(𝑡))+⋯+𝑐𝑚𝑓𝑚(𝑢(𝑡)),𝛾(𝑧)∶=1+𝛾1𝑧−1+𝛾2𝑧−2+⋯+𝛾𝑛𝛾𝑧−𝑛𝛾,𝐷(𝑧)∶=1+𝑑1𝑧−1+𝑑2𝑧−2+⋯+𝑑𝑛𝑑𝑧−𝑛𝑑.(4.2) Let 𝑤(𝑡)∶=𝐷(𝑧)𝛾(𝑧)𝑣(𝑡),(4.3) or []𝑤(𝑡)=1−𝛾(𝑧)𝑤(𝑡)+𝐷(𝑧)𝑣(𝑡)=−𝑛𝛾𝑖=1𝛾𝑖𝑤(𝑡−𝑖)+𝑛𝑑𝑖=1𝑑𝑖𝑣(𝑡−𝑖)+𝑣(𝑡).(4.4) Define the parameter vector 𝜽 and information vector 𝝋(𝑡) as 𝜽∶=[𝜽𝑇1,𝛾1,𝛾2,…,𝛾𝑛𝛾,𝑑1,𝑑2,…,𝑑𝑛𝑑]𝑇∈ℝ𝑛+𝑚𝑛+𝑛𝛾+𝑛𝑑,𝝋(𝑡)∶=[𝝋𝑇1(𝑡),−𝑤(𝑡−1),−𝑤(𝑡−2)…,−𝑤𝑡−𝑛𝛾,𝑣(𝑡−1),𝑣(𝑡−2),…,𝑣𝑡−𝑛𝑑]𝑇∈ℝ𝑛+𝑚𝑛+𝑛𝛾+𝑛𝑑,𝜽1⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝐚𝑏∶=1𝐜𝑏2𝐜⋮𝑏𝑛𝐜⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦∈ℝ𝑛+𝑛𝑚,𝝋1⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝝋(𝑡)∶=0𝝋(𝑡)1𝝋(𝑡)2⋮𝝋(𝑡)𝑛⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦(𝑡)∈ℝ𝑛+𝑛𝑚,⎡⎢⎢⎢⎢⎢⎢⎣𝑎𝐚∶=1𝑎2⋮𝑎𝑛⎤⎥⎥⎥⎥⎥⎥⎦∈ℝ𝑛⎡⎢⎢⎢⎢⎢⎢⎣𝑐,𝐜∶=1𝑐2⋮𝑐𝑚⎤⎥⎥⎥⎥⎥⎥⎦∈ℝ𝑚,𝝋0⎡⎢⎢⎢⎢⎢⎢⎣⋮⎤⎥⎥⎥⎥⎥⎥⎦(𝑡)∶=−𝑦(𝑡−1)−𝑦(𝑡−2)−𝑦(𝑡−𝑛)∈ℝ𝑛,𝝋𝑗⎡⎢⎢⎢⎢⎢⎢⎣𝑓(𝑡)∶=1𝑓(𝑢(𝑡−𝑗))2⋮𝑓(𝑢(𝑡−𝑗))𝑚⎤⎥⎥⎥⎥⎥⎥⎦(𝑢(𝑡−𝑗))∈ℝ𝑚,𝑗=1,2,…,𝑛.(4.5) Then (4.1) can be written as []𝑦(𝑡)=1−𝐴(𝑧)𝑦(𝑡)+𝐵(𝑧)𝑢(𝑡)+𝑤(𝑡)=−𝑛𝑖=1𝑎𝑖𝑦(𝑡−𝑖)+𝑛𝑖=1𝑏𝑖𝑚𝑗=1𝑐𝑗𝑓𝑗(𝑢(𝑡−𝑖))+𝑤(𝑡)=−𝑛𝑖=1𝑎𝑖𝑦(𝑡−𝑖)+𝑛𝑚𝑖=1𝑗=1𝑏𝑖𝑐𝑗𝑓𝑗(𝑢(𝑡−𝑖))+𝑤(𝑡)=𝝋𝑇1(𝑡)𝜽1+𝑤(𝑡)=𝝋𝑇1(𝑡)𝜽1−𝑛𝛾𝑖=1𝛾𝑖𝑤(𝑡−𝑖)+𝑛𝑑𝑖=1𝑑𝑖𝑣(𝑡−𝑖)+𝑣(𝑡)=𝝋𝑇(𝑡)𝜽+𝑣(𝑡).(4.6) This is a linear-in-parameter identification model for IN-CARARMA systems.

The unknown 𝑤(𝑡−𝑖) and 𝑣(𝑡−𝑖) in the information vector 𝝋(𝑡) are replaced with their estimates 𝑤(𝑡−𝑖) and ̂𝑣(𝑡−𝑖), and then we can obtain the following recursive generalized extended least squares algorithm for estimating 𝜽 in (4.6): 𝝋𝜽(𝑡)=𝜽(𝑡−1)+𝐋(𝑡)𝑦(𝑡)−𝑇,𝝋(𝑡)𝜽(𝑡−1)𝐋(𝑡)=𝐏(𝑡−1)𝝋(𝑡)1+𝑇(𝑡)𝐏(𝑡−1)𝝋(𝑡)−1,𝝋𝐏(𝑡)=𝐈−𝐋(𝑡)𝑇(𝑡)𝐏(𝑡−1),𝐏(0)=𝑝0𝐈,𝝋(𝑡)=[𝝋𝑇1𝑤(𝑡),−𝑤(𝑡−1),−𝑤(𝑡−2),…,−𝑡−𝑛𝛾,̂̂̂𝑣𝑣(𝑡−1),𝑣(𝑡−2),…,𝑡−𝑛𝑑]𝑇,𝝋1⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝝋(𝑡)=0𝝋(𝑡)1𝝋(𝑡)2⋮𝝋(𝑡)𝑛⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦(𝑡),𝝋0⎡⎢⎢⎢⎢⎢⎢⎣⋮⎤⎥⎥⎥⎥⎥⎥⎦(𝑡)=−𝑦(𝑡−1)−𝑦(𝑡−2)−𝑦(𝑡−𝑛),𝝋𝑗⎡⎢⎢⎢⎢⎢⎢⎣𝑓(𝑡)=1𝑓(𝑢(𝑡−𝑗))2⋮𝑓(𝑢(𝑡−𝑗))𝑚(⎤⎥⎥⎥⎥⎥⎥⎦,𝝋𝑢(𝑡−𝑗))𝑤(𝑡)=𝑦(𝑡)−𝑇1(𝜽𝑡)1(̂𝑣𝝋𝑡),(𝑡)=𝑦(𝑡)−𝑇𝜽𝜽(𝑡)(𝑡),𝜽(𝑡)=[𝑇1(𝑡),̂𝛾1(𝑡),̂𝛾2(𝑡),…,̂𝛾𝑛𝛾𝑑(𝑡),1𝑑(𝑡),2𝑑(𝑡),𝑛𝑑(𝑡)]𝑇.(4.7)

This paper presents a recursive least squares algorithm for IN-CAR systems and a recursive generalized extended least squares algorithm for IN-CARARMA systems with ARMA noise disturbances, which differ not only from the input nonlinear controlled autoregressive moving average (IN-CARMA) systems in [14] but also from the input nonlinear output error systems in [15].

5. Example

Consider the following IN-CAR system: 𝐴(𝑧)𝑦(𝑡)=𝐵(𝑧)𝑢(𝑡)+𝑣(𝑡),𝐴(𝑧)=1+𝑎1𝑧−1+𝑎2𝑧−2=1−1.35𝑧−1+0.75𝑧−2,𝐵(𝑧)=𝑏1𝑧−1+𝑏2𝑧−2=𝑧−1+1.68𝑧−2,𝑢(𝑡)=𝑓(𝑢(𝑡))=𝑐1𝑢(𝑡)+𝑐2𝑢2(𝑡)+𝑐3𝑢3(𝑡)=𝑢(𝑡)+0.50𝑢2(𝑡)+0.20𝑢3(𝜃𝑡),𝜽=1,𝜃2,𝜃3,𝜃4,𝜃5,𝜃6,𝜃7,𝜃8𝑇=𝑎1,𝑎2,𝑐1,𝑐2,𝑐3,𝑏2𝑐1,𝑏2𝑐2,𝑏2𝑐3𝑇=[]−1.350,0.75,1.00,0.50,0.20,1.68,0.84,0.336𝑇,𝜽𝑠=𝑎1,𝑎2,𝑏2,𝑐1,𝑐2,𝑐3𝑇=[]−1.35,0.75,1.68,1.00,0.50,0.20𝑇.(5.1) In simulation, the input {𝑢(𝑡)} is taken as a persistent excitation signal sequence with zero mean and unit variance and {𝑣(𝑡)} as a white noise sequence with zero mean and constant variance 𝜎2. Applying the proposed algorithm in (2.10)-(2.11) to estimate the parameters of this system, the parameter estimates 𝜽 and 𝜽𝑠 and their errors with different noise variances are shown in Tables 1, 2, 3, and 4, and the parameter estimation errors 𝛿∶=‖𝜽(𝑡)−𝜽‖/‖𝜽‖ and 𝛿𝑠𝜽∶=‖𝑠(𝑡)−𝜽‖/‖𝜽𝑠‖ versus 𝑡 are shown in Figures 1 and 2. When 𝜎2=0.502 and 𝜎2=1.502, the corresponding noise-to-signal ratios are 𝛿ns=10.96% and 𝛿ns=32.87%, respectively.

Table 1 
The parameter estimates (𝜽) (𝜎2=0.502, 𝛿ns=10.96%).
Table 2 
The parameter estimates (𝜽𝑠) (𝜎2 = 0.502, 𝛿ns = 10.96%).
Table 3 
The parameter estimates (𝜽) (𝜎2 = 1.502, 𝛿ns = 32.87%).
Table 4 
The parameter estimates (𝜽𝑠) (𝜎2 = 1.502, 𝛿ns = 32.87%).

Figure 1 
The parameter estimation errors 𝛿 versus 𝑡 .

Figure 2 
The parameter estimation errors 𝛿 𝑠 versus 𝑡 .

From Tables 1–4 and Figures 1 and 2, we can draw the following conclusions.(i)The larger the data length is, the smaller the parameter estimation errors become. (ii)A lower noise level leads to smaller parameter estimation errors for the same data length. (iii)The estimation errors 𝛿 and 𝛿𝑠 become smaller (in general) as 𝑡 increases. This confirms the proposed theorem.

6. Conclusions

The recursive least-squares identification is used to estimate the unknown parameters for input nonlinear CAR and CARARMA systems. The analysis using the martingale convergence theorem indicates that the proposed recursive least squares algorithm can give consistent parameter estimation. It is worth pointing out that the multi-innovation identification theory [26–33], the gradient-based or least-squares-based identification methods [34–41], and other identification methods [42–49] can be used to study identification problem of this class of nonlinear systems with colored noises.

Acknowledgment

This work was supported by the 111 Project (B12018).