Multi-Innovation Stochastic Gradient Identification Algorithm for Hammerstein Controlled Autoregressive Autoregressive Systems Based on the Key Term Separation Principle and on the Model Decomposition

Abstract

An input nonlinear system is decomposed into two subsystems, one including the parameters of the system model and the other including the parameters of the noise model, and a multi-innovation stochastic gradient algorithm is presented for Hammerstein controlled autoregressive autoregressive (H-CARAR) systems based on the key term separation principle and on the model decomposition, in order to improve the convergence speed of the stochastic gradient algorithm. The key term separation principle can simplify the identification model of the input nonlinear system, and the decomposition technique can enhance computational efficiencies of identification algorithms. The simulation results show that the proposed algorithm is effective for estimating the parameters of IN-CARAR systems.

1. Introduction

There exist many nonlinear systems in process control [1–3]. A nonlinear system can be modeled by input nonlinear systems [4] and output nonlinear systems [5], input-output nonlinear systems [6], feedback nonlinear systems [7], and so on. Input nonlinear systems, which are called Hammerstein systems [8], include input nonlinear equation error type systems and input nonlinear output error type systems. Recently, many identification algorithms have been developed for input nonlinear systems, such as the iterative methods [9–11], the separable least squares methods [12, 13], the blind methods [14], the subspace methods [15], and the overparameterization methods [16, 17]. Some methods require paying much extra computation.

The stochastic gradient (SG) algorithm is widely applied to parameter estimation. For example, Wang and Ding presented an extended SG identification algorithm for Hammerstein-Wiener ARMAX systems [18], but it is well known that the SG algorithm has slower convergence rates. In order to improve the convergence rate of the SG algorithm, Xiao et al. presented a multi-innovation stochastic gradient parameter estimation algorithm for input nonlinear controlled autoregressive (IN-CAR) models using the over-parameterization method [19]; Chen et al. proposed a modified stochastic gradient algorithm by introducing a convergence index in order to improve the convergence rate of the parameter estimation [20]; Han and Ding developed a multi-innovation stochastic gradient algorithm for multi-input single-output systems [21]; Liu et al. studied the performance of the stochastic gradient algorithm for multivariable systems [22].

The decomposition identification techniques include matrix decomposition and model decomposition. Hu and Ding presented a least squares based iterative identification algorithm for controlled moving average systems using the matrix decomposition [23]; Ding derived an iterative least squares algorithm to estimate the parameters of output error systems, and the matrix decomposition can enhance computational efficiencies [24]. Ding also divided a Hammerstein nonlinear system into two subsystems based on the model decomposition and presented a hierarchical multi-innovation stochastic gradient algorithm for Hammerstein nonlinear systems [25].

This paper discusses identification problems of input nonlinear controlled autoregressive autoregressive (IN-CARAR) systems or Hammerstein controlled autoregressive autoregressive (H-CARAR) systems, which is one kind of input nonlinear equation error type systems. The basic idea is using the key term separation principle [26] and the decomposition technique [24] to derive a multi-innovation stochastic gradient identification algorithm, which is different from the work in [19, 21, 25].

The rest of this paper is organized as follows. Section 2 gives the identification model for the IN-CARAR systems. Section 3 introduces the SG algorithm for the IN-CARAR system. Section 4 deduces a multi-innovation SG algorithm for IN-CARAR system using the decomposition technique. Section 5 provides a numerical example to show the effectiveness of the proposed algorithm. Finally, Section 6 offers some concluding remarks.

2. The System Identification Model

The paper focuses on the parameter estimation of a Hammerstein nonlinear controlled autoregressive autoregressive (H-CARAR) system, that is, an input nonlinear controlled autoregressive autoregressive (IN-CARAR) system, which consists of a nonlinear block and a linear dynamic subsystem. It is worth noting that Xiao and Yue discussed a data filtering based recursive least squares algorithm for H-CARAR systems [27] and a multi-innovation stochastic gradient parameter estimation algorithm for input nonlinear controlled autoregressive (IN-CAR) systems.

An IN-CARAR system shown as Figure 1 is expressed as [27] 𝐴(𝑧)𝑦(𝑡)=𝐵(𝑧)𝑢(𝑡)+1𝐶(𝑧)𝑣(𝑡),𝑤(𝑡)=1𝐶(𝑧)𝑣(𝑡),𝑢(𝑡)=𝐟(𝑢(𝑡))𝛾, (1)(2)(3) where 𝑢(𝑡) and 𝑦(𝑡) are the system input and output, 𝑢(𝑡) is the output of the nonlinear part, and 𝑣(𝑡) is an uncorrelated stochastic noise with zero mean. Here, 𝑢(𝑡) is expressed as [28] 𝑢(𝑡)=𝐟(𝑢(𝑡))𝛾=^𝑛𝛾_𝑖=1𝛾_𝑖𝑓_𝑖(𝑢(𝑡))=𝛾₁𝑓₁(𝑢(𝑡))+𝛾₂𝑓₂(𝑢(𝑡))+⋯+𝛾_𝑛𝛾𝑓_𝑛𝛾(𝑢(𝑡)), (4) where 𝐟(𝑢(𝑡))∶=[𝑓₁(𝑢(𝑡)),𝑓₂(𝑢(𝑡)),…,𝑓_𝑛𝛾(𝑢(𝑡))]∈ℝ^1×𝑛𝛾 and 𝛾∶=[𝛾₁,𝛾₂,…,𝛾_𝑛𝛾]^𝑇∈ℝ^𝑛𝛾 is the parameters vector of the nonlinear part.

Figure 1 

The IN-CARAR system.

In (1), 𝐴(𝑧), 𝐵(𝑧), and 𝐶(𝑧) are the polynomials, of known orders 𝑛_𝑎, 𝑛_𝑏, and 𝑛_𝑐, in the unit backward shift operator 𝑧⁻¹ [𝑧⁻¹𝑦(𝑡)=𝑦(𝑡−1)], defined by 𝐴(𝑧)∶=1+𝑎₁𝑧⁻¹+𝑎₂𝑧⁻²+⋯+𝑎_𝑛𝑎𝑧^−𝑛𝑎,𝐵(𝑧)∶=1+𝑏₁𝑧⁻¹+𝑏₂𝑧⁻²+⋯+𝑏_𝑛𝑏𝑧^−𝑛𝑏,𝐶(𝑧)∶=1+𝑐₁𝑧⁻¹+𝑐₂𝑧⁻²+⋯+𝑐_𝑛𝑐𝑧^−𝑛𝑐. (5) Assume 𝑦(𝑡)=0, 𝑢(𝑡)=0, and 𝑣(𝑡)=0 for 𝑡⩽0. 𝑎_𝑖, 𝑏_𝑖, and 𝑐_𝑖 are the parameters to be estimated from measured input–output data {𝑢(𝑡),𝑦(𝑡)}.

Define the parameter vectors: 𝜽∶=𝜽^𝑇_𝑠,𝜽^𝑇_𝑛^𝑇∈ℝ^{𝑛𝑎+𝑛𝑏+𝑛𝛾+𝑛𝑐},𝜽_𝑠∶=𝐚^𝑇,𝐛^𝑇,𝛾^𝑇𝑇∈ℝ^{𝑛𝑎+𝑛𝑏+𝑛𝛾},𝐚∶=𝑎₁,𝑎₂,…,𝑎_𝑛𝑎^𝑇∈ℝ^𝑛𝑎,𝐛∶=𝑏₁,𝑏₂,…,𝑏_𝑛𝑏^𝑇∈ℝ^𝑛𝑏,𝜽_𝑛∶=𝑐₁,𝑐₂,…,𝑐_𝑛𝑐^𝑇∈ℝ^𝑛𝑐, (6) and the information vectors: 𝝋(𝑡)𝝋∶=^𝑠(𝑡)𝝋_𝑛(𝑡)∈ℝ^{𝑛𝑎+𝑛𝑏+𝑛𝛾+𝑛𝑐},𝝋_𝑠(𝑡)∶=−𝑦(𝑡−1),−𝑦(𝑡−2),…,−𝑦𝑡−𝑛_𝑎,𝑢(𝑡−1),…,𝑢(𝑡−𝑛_𝑏),𝐟(𝑢(𝑡))^𝑇∈ℝ^{𝑛𝑎+𝑛𝑏+𝑛𝛾},𝝋_𝑛(𝑡)∶=−𝑤(𝑡−1),−𝑤(𝑡−2),…,−𝑤𝑡−𝑛_𝑐^𝑇∈ℝ^𝑛𝑐. (7) Equation (2) can be written as 𝑤(𝑡)=[1−𝐶(𝑧)]𝑤(𝑡)+𝑣(𝑡)=−^𝑛𝑐_𝑖=1𝑐_𝑖𝑤(𝑡−𝑖)+𝑣(𝑡)=𝝋^𝑇_𝑛(𝑡)𝜽_𝑛+𝑣(𝑡). (8) Using the key term separation principle [26], (1) can be written as 𝑦(𝑡)=[1−𝐴(𝑧)]𝑦(𝑡)+[𝐵(𝑧)−1]𝑢(𝑡)+𝑢(𝑡)+𝑤(𝑡)=−^𝑛𝑎_𝑖=1𝑎_𝑖𝑦(𝑡−𝑖)+^𝑛𝑏_𝑖=1𝑏_𝑖𝑢(𝑡−𝑖)+^𝑛𝛾_𝑖=1𝛾_𝑖𝐟_𝑖(𝑢(𝑡))+𝑤(𝑡)=𝝋^𝑇_𝑠(𝑡)𝜽_𝑠+𝑤(𝑡)=𝝋^𝑇_𝑠(𝑡)𝜽_𝑠+𝝋^𝑇_𝑛(𝑡)𝜽_𝑛+𝑣(𝑡)=𝝋^𝑇(𝑡)𝜽. (9)(10)(11) This is the identification model of the IN-CARAR system.

3. The Stochastic Gradient Algorithm

According to [1] and based on the identification model in (11), we can obtain the stochastic gradient (SG) algorithm: 𝜽(𝑡)=𝜽()+𝝋(𝑡)𝑡−1𝑟(𝑡)𝑒(𝑡),𝑒(𝑡)=𝑦(𝑡)−𝝋^𝑇(𝑡)𝜽(),𝑟(𝑡)()+‖‖𝑡−1=𝑟𝑡−1𝝋(𝑡)²(0)𝝋(𝑡)=𝝋,𝑟=1,_𝑠(𝑡)𝝋_𝑛(𝑡),𝝋_𝑠(𝑡)=−𝑦(𝑡−1),−𝑦(𝑡−2),…,−𝑦𝑡−𝑛_𝑎,𝑢(𝑡−1),𝑢(𝑡−2),…,𝑢(𝑡−𝑛_𝑏),𝐟(𝑢(𝑡))^𝑇,𝝋_𝑛(𝑡)=−𝑤(𝑡−1),−𝑤(𝑡−2),…,−𝑤𝑡−𝑛_𝑐^𝑇,𝑢(𝑡)(𝑢(𝑡𝛾(𝑡),𝑤(𝑡)(𝑡)−𝝋=𝐟))=𝑦^𝑇_𝑠(𝑡)𝜽_𝑠(𝑡),𝐟(𝑢(𝑡))=𝑓₁(𝑢(𝑡)),𝑓₂(𝑢(𝑡)),…,𝑓_𝑛𝑟(𝑢(𝑡,𝜽(𝑡)=𝜽))_𝑛(𝑡)𝜽_𝑠(𝑡),𝜽_𝑠(𝑡)=𝐚^𝑇(𝑡),𝐛^𝑇(𝑡),𝛾^𝑇(𝑡)^𝑇,𝐚(𝑡)=𝑎₁(𝑡),𝑎₂(𝑡),…,𝑎_𝑛𝑎(𝑡)^𝑇,𝐛(𝑡)=𝑏̂₁(𝑡),𝑏̂₂(𝑡),…,𝑏̂_𝑛𝑏(𝑡)^𝑇,𝛾(𝑡)=𝛾₁(𝑡),𝛾₂(𝑡),…,𝛾_𝑛𝛾(𝑡)^𝑇,𝜽_𝑛(𝑡)=𝑐₁̂(𝑡),𝑐₂̂(𝑡),…,𝑐̂_𝑛𝑐(𝑡)^𝑇, (12) where 𝑋(𝑡) represents the estimate of 𝑋 at time 𝑡; for example, 𝜽(𝑡)=𝜽_𝑛(𝑡)𝜽_𝑠(𝑡)∈ℝ^{𝑛𝑎+𝑛𝑏+𝑛𝛾+𝑛𝑐} is the estimate of 𝜽=^𝜽𝑠_𝜽𝑛 at time 𝑡.

4. The Multi-Innovation Stochastic Gradient Algorithm

This section deduces the multi-innovation stochastic gradient identification algorithm for the IN-CARAR system using the decomposition technique [1].

Define two intermediate variables, 𝑦₁(𝑡)∶=𝑦(𝑡)−𝝋^𝑇_𝑛(𝑡)𝜽_𝑛,𝑦₂(𝑡)∶=𝑦(𝑡)−𝝋^𝑇_𝑠(𝑡)𝜽_𝑠. (13) From (10), we have 𝑦₁(𝑡)∶=𝝋^𝑇_𝑠(𝑡)𝜽_𝑠(𝑡),𝑦+𝑣₂(𝑡)∶=𝝋^𝑇_𝑛(𝑡)𝜽_𝑛+𝑣(𝑡). (14) These two subsystems include the parameter vectors 𝜽_𝑠 and 𝜽_𝑛, respectively. 𝜽_𝑠 contains the parameters od the system model and 𝜽_𝑛 contains the parameters od the noise model.

Define the stacked information matrices and the stacked output vectors: 𝐘𝑝,𝑡∶=𝑦(𝑡),𝑦(𝑡−1),…,𝑦𝑡−𝑝+1^𝑇∈ℝ^𝑝,𝐘₁𝑝,𝑡∶=𝑦₁(𝑡),𝑦₁(𝑡−1),…,𝑦₁𝑡−𝑝+1^𝑇∈ℝ^𝑝,𝐘₂𝑝,𝑡∶=𝑦₂(𝑡),𝑦₂(𝑡−1),…,𝑦₂𝑡−𝑝+1^𝑇∈ℝ^𝑝,Φ_𝑠𝝋𝑝,𝑡∶=_𝑠(𝑡),𝝋_𝑠(𝑡−1),…,𝝋_𝑠𝑡−𝑝+1^𝑇∈ℝ^{𝑝×(𝑛𝑎+𝑛𝑏+𝑛𝑐)},Φ_𝑛𝑝,𝑡∶=𝝋_𝑛(𝑡),𝝋_𝑛(𝑡−1),…,𝝋_𝑛𝑡−𝑝+1^𝑇∈ℝ^{𝑝×𝑛𝛾},𝐄_𝑠𝑝,𝑡∶=𝑒_𝑠(𝑡),𝑒_𝑠(𝑡−1),…,𝑒_𝑠𝑡−𝑝+1^𝑇∈ℝ^𝑝,𝐄_𝑛𝑝,𝑡∶=𝑒_𝑛(𝑡),𝑒_𝑛(𝑡−1),…,𝑒_𝑛𝑡−𝑝+1^𝑇∈ℝ^𝑝. (15) According to the multi-innovation identification theory [29–41], we expand the scalar innovations: 𝑒_𝑠(𝑡)=𝑦₁(𝑡)−𝝋^𝑇_𝑠(𝑡)𝜽_𝑠(),𝑒𝑡−1_𝑛(𝑡)=𝑦₂(𝑡)−𝝋^𝑇_𝑛(𝑡)𝜽_𝑛(𝑡−1), (16) to the innovation vectors, 𝐄_𝑠𝑝,𝑡=𝐘₁𝑝,𝑡−Φ^𝑇_𝑠𝑝,𝑡𝜽_𝑠(),𝐄𝑡−1_𝑛𝑝,𝑡=𝐘₂𝑝,𝑡−Φ^𝑇_𝑛𝑝,𝑡𝜽_𝑛(𝑡−1). (17) Define two criterion functions, 𝐽₁𝜽_𝑠∶=𝐘₁(𝑝,𝑡)−Φ^𝑇_𝑠(𝑝,𝑡)𝜽_𝑠²,𝐽₂𝜽_𝑛∶=𝐘₂(𝑝,𝑡)−Φ^𝑇_𝑛(𝑝,𝑡)𝜽_𝑛². (18) The gradients of 𝐽₁ and 𝐽₂ with respect to 𝜽_𝑠 and 𝜽_𝑛, respectively, are grad𝐽₁𝜽_𝑠=𝜕𝐽₁𝜽_𝑠𝜕𝜽_𝑛=−2Φ_𝑠𝑝,𝑡𝐘₁𝑝,𝑡−Φ^𝑇_𝑠𝑝,𝑡𝜽_𝑠,grad𝐽₂𝜽_𝑛=𝜕𝐽₂𝜽_𝑛𝜕𝜽_𝑛=−2Φ_𝑛𝑝,𝑡𝐘₂𝑝,𝑡−Φ^𝑇_𝑛𝑝,𝑡𝜽_𝑛. (19) Minimizing 𝐽₁(𝜽_𝑠) and 𝐽₂(𝜽_𝑛) using the negative gradient search, we can obtain the multi-innovation stochastic gradient algorithm (MISG) for the IN-CARAR system: 𝜽_𝑠(𝑡)=𝜽_𝑠()+Φ𝑡−1^𝑠𝑝,𝑡𝑟₁(𝑡)𝐄_𝑠,𝐄𝑝,𝑡_𝑠𝑝,𝑡=𝐘₁𝑝,𝑡−Φ^𝑇_𝑠𝑝,𝑡𝜽_𝑠()−Φ𝑡−1=𝐘𝑝,𝑡^𝑇_𝑠𝑝,𝑡𝜽_𝑠()−Φ𝑡−1_𝑇𝑛𝑝,𝑡𝜽_𝑛(),𝑟𝑡−1₁(𝑡)=𝑟₁(𝑡−1)+Φ_𝑠(𝑝,𝑡)²,𝑟₁(0)𝐘=𝑦(𝑡)()=1,𝑝,𝑡,𝑦𝑡−1,…,𝑦𝑡−𝑝+1^𝑇,Φ_𝑠𝑝,𝑡=𝝋_𝑠(𝑡),𝝋_𝑠(𝑡−1),…,𝝋_𝑠𝑡−𝑝+1^𝑇,𝜽_𝑛(𝑡)=𝜽_𝑛()+Φ𝑡−1^𝑛𝑝,𝑡𝑟₂(𝑡)𝐄_𝑛,𝐄𝑝,𝑡_𝑛𝑝,𝑡=𝐘₂𝑝,𝑡−Φ^𝑇_𝑛𝑝,𝑡𝜽_𝑛()−Φ𝑡−1=𝐘𝑝,𝑡^𝑇_𝑠𝑝,𝑡𝜽_𝑠()−Φ𝑡−1_𝑇𝑛𝑝,𝑡𝜽_𝑛(),𝑟𝑡−1₂(𝑡)=𝑟₂(𝑡−1)+Φ_𝑛(𝑝,𝑡)²,𝑟₂(0)Φ=1,_𝑛𝑝,𝑡=𝝋_𝑛(𝑡),𝝋_𝑛(𝑡−1),…,𝝋_𝑛𝑡−𝑝+1^𝑇,𝝋_𝑠(𝑡)=−𝑦(𝑡−1),−𝑦(𝑡−2),…,−𝑦𝑡−𝑛_𝑎,𝑢(𝑡−1),𝑢(𝑡−2),…,𝑢𝑡−𝑛_𝑏,𝐟(𝑢(𝑡))^𝑇,𝝋_𝑛(𝑡)=−𝑤(𝑡−1),−𝑤(𝑡−2),…,−𝑤𝑡−𝑛_𝑐^𝑇,𝑢(𝑡)(𝑢(𝑡𝛾(𝑡),𝑤(𝑡)(𝑡)−𝝋=𝐟))=𝑦^𝑇_𝑠(𝑡)𝜽_𝑠(𝑡),𝐟(𝑢(𝑡))=𝑓₁(𝑢(𝑡)),𝑓₂(𝑢(𝑡)),…,𝑓_𝑛𝛾(𝑢(𝑡,𝜽(𝑡)=𝜽))_𝑛(𝑡)𝜽_𝑠(𝑡),𝜽_𝑠(𝑡)=𝐚^𝑇(𝑡),𝐛^𝑇(𝑡),𝛾^𝑇(𝑡)^𝑇,𝐚(𝑡)=𝑎₁(𝑡),𝑎₂(𝑡),…,𝑎_𝑛𝑎(𝑡)^𝑇,𝐛(𝑡)=𝑏̂₁(𝑡),𝑏̂₂(𝑡),…,𝑏̂_𝑛𝑏(𝑡)^𝑇,𝛾(𝑡)=𝛾₁(𝑡),𝛾₂(𝑡),…,𝛾_𝑛𝛾(𝑡)^𝑇,𝜽_𝑛(𝑡)=𝑐₁̂(𝑡),𝑐₂̂(𝑡),…,𝑐̂_𝑛𝑐(𝑡)^𝑇. (20) The initial values can be taken to be 𝜽(0)=1_{𝑛𝑎+𝑛𝑏+𝑛𝛾+𝑛𝑐}/𝑝₀, 𝑤(𝑖)=1/𝑝₀, 𝑖⩽0, and 𝑝₀=10⁶.

5. Numerical Examples

Consider the following IN-CARAR system: 𝐴(𝑧)𝑦(𝑡)=𝐵(𝑧)𝑢(𝑡)+1𝐶(𝑧)𝑣(𝑡),𝐴(𝑧)=1+𝑎₁𝑧⁻¹+𝑎₂𝑧⁻²=1+1.80𝑧⁻¹+0.80𝑧⁻²,𝐵(𝑧)=1+𝑏₁𝑧⁻¹+𝑏₂𝑧⁻²=1+0.50𝑧⁻¹+0.65𝑧⁻²,𝐶(𝑧)=1+𝑐₁𝑧⁻¹+𝑐₂𝑧⁻²=1+0.30𝑧⁻¹+0.20𝑧⁻²,𝑢(𝑡)=𝐟(𝑢(𝑡))𝛾=𝛾₁𝑓₁(𝑢(𝑡))+𝛾₂𝑓₂(𝑢(𝑡))+𝛾₃𝑓₃(𝑢(𝑡(𝑡)))=1.00𝑢+0.50𝑢²(𝑡)+0.25𝑢³(𝑡),𝜽=𝑎₁,𝑎₂,𝑏₁,𝑏₂,𝛾₁,𝛾₂,𝛾₃,𝑐₁,𝑐₂^𝑇=[1.80,0.80,0.5,0.65,1.00,0.50,0.25,0.30,0.20]^𝑇. (21) In this example, 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 variance 𝜎²=0.50². Applying the SG algorithm and the MISG algorithm to estimate the parameters of this IN-CARAR system, the parameter estimates and their estimation errors are shown in Tables 1 and 2 with the data length 𝐿=3000 and the estimation error 𝛿∶=‖𝜽(𝑡)−𝜽‖/‖𝜽‖ versus 𝑡 being shown in Figures 2 and 3.

Table 1 

The SG parameter estimates and errors.

Table 2 

The MISG parameter estimates and errors.

Figure 2 

The SG estimation error 𝛿 versus 𝑡.

Figure 3 

The MISG estimation error 𝛿 versus.

From Tables 1 and 2 and Figures 2 and 3, we can draw the conclusions. The parameters estimation errors become smaller with the data length 𝑡 increasing. The estimation errors given by the MISG algorithm are much smaller than that of the SG algorithm. The convergence speed of the multi-innovation SG algorithm is faster than those of the SG algorithm. These indicate that the MISG algorithm has better performance than the SG algorithm.

6. Conclusions

The gradient and least squares algorithms are two different kinds of important identification methods. It is well known that the gradient algorithm has poor convergence rates. This paper studies the multi-innovation SG identification methods for IN-CARAR systems. The numerical examples show that the proposed MISG algorithm can estimate effectively the parameters of input nonlinear systems and indicate that increasing the innovation length can improve parameter estimation accuracy of the multi-innovation identification algorithm because the algorithm uses more information in each recursion for a large innovation length. The proposed method can be applied to nonlinear output error systems. Although the algorithm is presented for the IN-CARAR systems, the basic idea can be extended to other linear or nonlinear systems with colored noises [42–61].

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China (nos. 61273194, 61203111), the University Graduate Scientific Research Innovation Program of Jiangsu Province (CXLX13-738), the Fundamental Research Funds for the Central Universities (JUDCF13035), the PAPD of Jiangsu Higher Education Institutions.