Recursive Least-Squares Estimation for Hammerstein Nonlinear ...

Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2013, Article ID 240929, 8 pages http://dx.doi.org/10.1155/2013/240929

Research Article Recursive Least-Squares Estimation for Hammerstein Nonlinear Systems with Nonuniform Sampling Xiangli Li,1,2 Lincheng Zhou,1 Ruifeng Ding,3 and Jie Sheng4 1

Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education), Jiangnan University, Wuxi 214122, China Jiangsu College of Information Technology, Wuxi 214153, China 3 School of Internet of Things Engineering, Jiangnan University, Wuxi 214122, China 4 Institute of Technology, University of Washington, Tacoma, WA 98402-3100, USA 2

Correspondence should be addressed to Ruifeng Ding; [email protected] Received 17 June 2013; Accepted 2 September 2013 Academic Editor: Victoria Vampa Copyright © 2013 Xiangli Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper focuses on the identification problem of Hammerstein nonlinear systems with nonuniform sampling. Using the key-term separation principle, we present a discrete identification model with nonuniform sampling input and output data based on the frame period. To estimate parameters of the presented model, an auxiliary model-based recursive least-squares algorithm is derived by replacing the unmeasurable variables in the information vector with their corresponding recursive estimates. The simulation results show the effectiveness of the proposed algorithm.

1. Introduction In actual industrial processes, there exist widely nonlinear systems which are described by block-oriented nonlinear systems [1–3]. Block-oriented nonlinear models are in general divided into Hammerstein systems and Wiener systems [4]. A Hammerstein system, which consists of a static nonlinear subsystem followed by a linear dynamic subsystem, can represent some nonlinear systems [5]. Many publications have been reported for the identification of the Hammerstein systems [6, 7]. For example, Chen et al. studied identification problems for the Hammerstein systems with saturation and dead-zone nonlinearities by choosing an appropriate switching function [8]; Ding et al. presented the projection, the stochastic gradient, and the Newton recursive and the Newton iterative identification algorithms for the Hammerstein nonlinear systems, and then they analyzed and compared the performances of these approaches by numerical examples [9]. Li et al. derived a least-squares based iterative algorithm for the Hammerstein output error systems with nonuniform sampling by using the overparameterization model [10]. The different input-output updating period (or called multirate sampling) is inevitable in discrete-time systems

[11–13]. The identification of multirate sampled systems have attracted much attention of many researchers. Recently, Liu et al. proposed a novel hierarchical least-squares algorithm for a class of nonuniformly sampled systems based on the hierarchical identification principle [14]. Shi et al. presented a crosstalk identification algorithm for multirate xDSL FIR systems [15]. Han et al. gave state-space models for multirate multi-input sampled-data systems and derived an auxiliary model-based recursive least-squares algorithm for identifying the parameters of multirate systems [16]. The recursive least-squares algorithm is a class of basic parameter estimation approaches which are suitable for online applications. In this literature, Wang adopted a filtering auxiliary model-based recursive least-squares identification algorithm for output error moving average systems [17]. Differing from the work in [14, 16], this paper discusses the parameter estimation problem for nonuniformly sampled Hammerstein nonlinear systems. The basic idea is, to combine the auxiliary model identification idea [18– 24] and the key-term separation principle to derive the auxiliary model-based recursive least-squares algorithm for the Hammerstein nonlinear systems with nonuniform sampling.

2

Mathematical Problems in Engineering Suppose that 𝑃𝑐 has the following state-space representation:

𝜐(kT) u(kT + ti )

H𝜏

u(t)

u(t) P f(·) c

y(t)

ST

y(kT)

⊕

y1 (kT)

ẋ (𝑡) = A𝑐 x (𝑡) + B𝑐 𝑢 (𝑡) ,

Figure 1: Hammerstein systems with nonuniform sampling.

𝑦 (𝑡) = Cx (𝑡) + 𝐷𝑢 (𝑡) ,

The rest of this paper is organized as follows. Section 2 establishes the identification model of the Hammerstein nonlinear systems with nonuniform sampling. Section 3 derives a recursive least-squares parameter estimation algorithm based on the auxiliary model identification idea. Section 4 provides an example to illustrate the effectiveness of the proposed algorithm. The conclusions of the paper are summarized in Section 5.

(3)

where x(𝑡) ∈ R𝑛 is the state vector, 𝑢(𝑡) and 𝑦(𝑡) are the input and output of the continuous-time process, respectively, and A𝑐 , B𝑐 , C, and 𝐷 are matrices of appropriate sizes. Referring to [25] and discretizing (3) with the frame period 𝑇, we have x (𝑘𝑇 + 𝑇) = 𝑒A𝑐 𝑇 x (𝑘𝑇) 𝑟

𝜏𝑖

+ ∑𝑒A𝑐 (𝑇−𝑡𝑖 ) ∫ 𝑒A𝑐 𝑡 d𝑡B𝑐 𝑢 (𝑘𝑇 + 𝑡𝑖−1 ) 0

𝑖=1

(4)

𝑟

= Ax (𝑘𝑇) + ∑B𝑖 𝑢 (𝑘𝑇 + 𝑡𝑖−1 ) , 𝑖=1

2. The Identification Model Let us introduce some notations. The superscript 𝑇 denotes the matrix transpose; I stands for an identity matrix of appropriate sizes; 1𝑛 represents an 𝑛-dimensional column vector whose elements are 1; “X := A” stands for “A is defined as X”; and 𝑧−1 is a unit backward shift operator; that is, 𝑧−1 𝑥(𝑡) = 𝑥(𝑡 − 1). Consider a Hammerstein nonlinear system with nonuniform sampling shown in Figure 1, where 𝐻𝜏 is a nonuniform zero-order hold with irregularly updating intervals {𝜏1 , 𝜏2 , . . . , 𝜏𝑟 }, dealing with a discrete-time signal 𝑢(𝑘𝑇 + 𝑡𝑖 ) and producing the input 𝑢(𝑡) of the nonlinear subsystem 𝑓(⋅); 𝑢(𝑡) is the output of the nonlinear subsystem; 𝑃𝑐 is a continuous-time process; 𝑦(𝑡) is the true output of 𝑃𝑐 but is unmeasurable; 𝑆𝑇 is a sampler that produces a discrete-time signal 𝑦(𝑘𝑇) with period 𝑇 = 𝜏1 +𝜏2 +⋅ ⋅ ⋅+𝜏𝑟 ; and 𝑦1 (𝑘𝑇) is the system output but is corrupted by the additive noise V(𝑘𝑇). Assuming that the input 𝑢(𝑡) has the updating intervals {𝜏1 , 𝜏2 , . . . , 𝜏𝑟 }, we have [11, 25]

𝑢 (𝑘𝑇) , 𝑘𝑇 ⩽ 𝑡 < 𝑘𝑇 + 𝑡1 , { { { { { {𝑢 (𝑘𝑇 + 𝑡1 ) , 𝑘𝑇 + 𝑡1 ⩽ 𝑡 < 𝑘𝑇 + 𝑡2 , { 𝑢 (𝑡) = { { .. { { . { { { {𝑢 (𝑘𝑇 + 𝑡𝑟−1 ) , 𝑘𝑇 + 𝑡𝑟−1 ⩽ 𝑡 < (𝑘 + 1) 𝑇,

where A := 𝑒Ac 𝑇 ∈ R𝑛×𝑛 , 𝜏𝑖

B𝑖 := 𝑒Ac (𝑇−𝑡𝑖 ) ∫ 𝑒Ac 𝑡 d𝑡B𝑐 ∈ R𝑛×𝑛 . 0

The output 𝑦(𝑡) at the sampling instant 𝑡 = 𝑘𝑇 can be expressed as 𝑦 (𝑘𝑇) = Cx (𝑘𝑇) + 𝐷𝑢 (𝑘𝑇) .

𝑦1 (𝑘𝑇) = 𝑦 (𝑘𝑇) + V (𝑘𝑇) .

𝑟

𝑧−𝑛 C adj [𝑧I − A] B𝑖 −𝑛 𝑖=1 𝑧 det [𝑧I − A]

𝑦 (𝑘𝑇) = ∑

=

(8)

1 𝑟 ∑𝛽 (𝑧) 𝑢 (𝑘𝑇 + 𝑡𝑖−1 ) , 𝛼 (𝑧) 𝑖=1 𝑖

where 𝛼 (𝑧) := 𝑧−𝑛 det [𝑧I − A] = 1 + 𝛼1 𝑧−1 + 𝛼2 𝑧−2 + ⋅ ⋅ ⋅ + 𝛼𝑛 𝑧−𝑛 , 𝛽1 (𝑧) := 𝑧−𝑛 C adj [𝑧I − A] B1 + 𝐷𝛼 (𝑧) = 𝛽10 + 𝛽11 𝑧−1 + 𝛽12 𝑧−2 + ⋅ ⋅ ⋅ + 𝛽1𝑛 𝑧−𝑛 , 𝛽𝑖 (𝑧) := 𝑧−𝑛 C adj [𝑧I − A] B𝑖 = 𝛽𝑖1 𝑧−1 + 𝛽𝑖2 𝑧−2 + ⋅ ⋅ ⋅ + 𝛽𝑖𝑛 𝑧−𝑛 ,

where 𝑛𝛾 is the polynomial order.

(7)

Referring to [26] and from (4) and (6), we have

(1)

𝑢 (𝑡) = 𝑓 (𝑢 (𝑡)) = 𝛾1 𝑢 (𝑡) + 𝛾2 𝑢2 (𝑡) + ⋅ ⋅ ⋅ + 𝛾𝑛𝛾 𝑢𝑛𝛾 (𝑡) , (2)

(6)

Hence, the system output 𝑦1 (𝑘𝑇) is written as

× 𝑢 (𝑘𝑇 + 𝑡𝑖−1 ) + 𝐷𝑢 (𝑘𝑇)

where 𝑇 := 𝜏1 + 𝜏2 + ⋅ ⋅ ⋅ + 𝜏𝑟 is the frame period. The nonlinear subsystem 𝑓(⋅) in the Hammerstein nonlinear system is a polynomial of a known order:

(5)

𝑖 = 1, 2, 3, . . . , 𝑟.

(9)

Mathematical Problems in Engineering

3

Equation (8) can be transformed into

Define the information vectors and the parameter vectors as 𝜑𝑦 (𝑘𝑇)

𝑟

𝑦 (𝑘𝑇) = [1 − 𝛼 (𝑧)] 𝑦 (𝑘𝑇) + ∑𝛽𝑖 (𝑧) 𝑢 (𝑘𝑇 + 𝑡𝑖−1 ) .

(10)

𝑖=1

:= [−𝑦 (𝑘𝑇 − 𝑇) , −𝑦 (𝑘𝑇 − 2𝑇) , . . . , −𝑦 (𝑘𝑇 − 𝑛𝑇)] ∈ R𝑛 , 𝜑𝑢 (𝑘𝑇)

Substituting (10) into (7), the system output 𝑦1 (𝑘𝑇) can be expressed as

:= [𝑢 (𝑘𝑇 − 𝑇) , 𝑢 (𝑘𝑇 − 2𝑇) , . . . , 𝑢 (𝑘𝑇 − 𝑛𝑇) , 𝑢 (𝑘𝑇 − 𝑇 + 𝑡1 ) , 𝑢 (𝑘𝑇 − 2𝑇 + 𝑡1 ) , . . . , 𝑢 (𝑘𝑇 − 𝑛𝑇 + 𝑡1 ) , . . . , 𝑢 (𝑘𝑇 − 𝑇 + 𝑡𝑟−1 ) ,

𝑦1 (𝑘𝑇) = [1 − 𝛼 (𝑧)] 𝑦 (𝑘𝑇)

𝑢 (𝑘𝑇 − 2𝑇 + 𝑡𝑟−1 ) , . . . , 𝑢 (𝑘𝑇 − 𝑛𝑇 + 𝑡𝑟−1 )] ∈ R𝑟𝑛 , T

𝑟

+ ∑𝛽𝑖 (𝑧) 𝑢 (𝑘𝑇 + 𝑡𝑖−1 ) + V (𝑘𝑇) .

(11)

T

𝜑𝛾 (𝑘𝑇) := [𝑢 (𝑘𝑇) , 𝑢2 (𝑘𝑇) , . . . , 𝑢𝑛𝛾 (𝑘𝑇)] ∈ R𝑛𝛾 ,

𝑖=1

𝜑𝑦 (𝑘𝑇) 𝜑 (𝑘𝑇) := [𝜑𝑢 (𝑘𝑇)] ∈ R𝑛0 , [𝜑𝛾 (𝑘𝑇)]

Equation (11) can be rewritten equivalently as

𝑛0 := (𝑟 + 1) 𝑛 + 𝑛𝛾 ,

𝜃𝑦 := [𝛼1 , 𝛼2 , . . . , 𝛼𝑛 ] ∈ R𝑛 , T

𝑛

𝜃𝑢 := [𝛽11 , 𝛽12 , . . . , 𝛽1𝑛 , 𝛽21 , 𝛽22 , . . . , 𝛽2𝑛 , . . . ,

𝑦1 (𝑘𝑇) = ∑ 𝛼𝑗 𝑦 (𝑘𝑇 − 𝑗𝑇) + 𝛽10 𝑢 (𝑘𝑇) 𝑗=1

𝛽𝑟1 , 𝛽𝑟2 , . . . , 𝛽𝑟𝑛 ] ∈ R𝑟𝑛 , T

𝑛

(12)

𝑟

T

𝜃𝛾 := [𝛾1 , 𝛾2 , . . . , 𝛾𝑛𝛾 ] ∈ R𝑛𝛾 ,

+ ∑ ∑𝛽𝑖𝑗 𝑢 (𝑘𝑇 + 𝑡𝑖−1 − 𝑗𝑇) + V (𝑘𝑇) . 𝑗=1 𝑖=1

Here, substituting (2) into (12) results in a complex expression containing the products of parameters. To solve this problem, we use the key-term separation principle presented in [27], and let 𝛽10 = 1. Then, the identification model of the proposed system is as follows:

𝜃𝑦 𝜃 := [𝜃𝑢 ] ∈ R𝑛0 . [ 𝜃𝛾 ] (14) Equation (13) can be written in a regressive form as 𝑦1 (𝑘𝑇) = 𝜑T (𝑘𝑇) 𝜃 + V (𝑘𝑇) .

𝑛

𝑛𝛾

𝑗=1

𝑖=1

Define a quadratic criterion function as 𝑖

𝑦1 (𝑘𝑇) = ∑ 𝛼𝑗 𝑦 (𝑘𝑇 − 𝑗𝑇) + ∑𝛾𝑖 𝑢 (𝑘𝑇) 𝑛

(15)

𝑟

𝑘

(13)

+ ∑ ∑𝛽𝑖𝑗 𝑢 (𝑘𝑇 + 𝑡𝑖−1 − 𝑗𝑇) + V (𝑘𝑇) . 𝑗=1 𝑖=1

The objective of this paper is to develop a recursive leastsquares algorithm for estimating the parameters of the nonuniformly sampled Hammerstein systems by using the auxiliary model identification idea in [11].

3. The Recursive Least-Squares Algorithm In this section, we derive the recursive least-squares estimation algorithm for the Hammerstein nonlinear systems with nonuniform sampling, referring to the method in [1].

2

𝐽 (𝜃) := ∑[𝑦1 (𝑖𝑇) − 𝜑T (𝑖𝑇) 𝜃] .

(16)

𝑖=1

̂ Let 𝜃(𝑘𝑇) be the estimate of 𝜃 at time 𝑘𝑇. Minimizing 𝐽(𝜃) gives the following recursive least-squares algorithm: 𝜃̂ (𝑘𝑇) = 𝜃̂ (𝑘𝑇 − 𝑇) + P (𝑘𝑇) 𝜑 (𝑘𝑇) × [𝑦1 (𝑘𝑇) − 𝜑𝑇 (𝑘𝑇) 𝜃̂ (𝑘𝑇 − 𝑇)] ,

(17)

P (𝑘𝑇) = P (𝑘𝑇 − 𝑇) −

P (𝑘𝑇 − 𝑇) 𝜑 (𝑘𝑇) 𝜑𝑇 (𝑘𝑇) P (𝑘𝑇 − 𝑞) . 1 + 𝜑𝑇 (𝑘𝑇) P (𝑘𝑇 − 𝑇) 𝜑 (𝑘𝑇)

(18)

Note that the information vector 𝜑(𝑘𝑇) in (17) contains unknown inner variables 𝑦(𝑘𝑇 − 𝑗𝑇) and 𝑢(𝑘𝑇 + 𝑡𝑖−1 − 𝑗𝑇); the parameter vector 𝜃 cannot be estimated by the standard

4


least-squares method. The solution is based on the auxiliary model identification idea [11]: to replace the unmeasurable term 𝑦(𝑘𝑇 − 𝑗𝑇) in 𝜑(𝑘𝑇) with its estimate ̂ T (𝑘𝑇) 𝜃̂ (𝑘𝑇) . 𝑦̂ (𝑘𝑇) = 𝜑

Equations (19) to (22) form the AM-RLS algorithm for the not uniformly sampled Hammerstein nonlinear systems, which can be summarized as 𝜃̂ (𝑘𝑇) = 𝜃̂ (𝑘𝑇 − 𝑇) ̂ T (𝑘𝑇) 𝜃̂ (𝑘𝑇 − 𝑇)] , ̂ (𝑘𝑇) [𝑦1 (𝑘𝑇) − 𝜑 + P (𝑘𝑇) 𝜑

(19)

P (𝑘𝑇) = P (𝑘𝑇 − 𝑇) Replacing 𝛾𝑖 in (2) with its estimate 𝛾̂𝑖 (𝑘𝑇), we can obtain the estimate 𝑢̂(𝑘𝑇 + 𝑡𝑖−1 ) of 𝑢(𝑘𝑇 + 𝑡𝑖−1 ) as follows:

−

̂ 𝑦 (𝑘𝑇) 𝜑 [ 𝜑 ̂ 𝜑 (𝑘𝑇) = ̂ 𝑢 (𝑘𝑇)] , ̂ 𝛾 (𝑘𝑇)] [𝜑

𝑢̂ (𝑘𝑇 + 𝑡𝑖−1 ) = 𝛾̂1 (𝑘𝑇) 𝑢 (𝑘𝑇 + 𝑡𝑖−1 ) + 𝛾̂2 (𝑘𝑇) 𝑢2 (𝑘𝑇 + 𝑡𝑖−1 ) + ⋅ ⋅ ⋅ + 𝛾̂𝑛𝛾 (𝑘𝑇) 𝑢𝑛𝛾 (𝑘𝑇 + 𝑡𝑖−1 ) ,

̂ (𝑘𝑇) 𝜑 ̂ T (𝑘𝑇) P (𝑘𝑇 − 𝑇) P (𝑘𝑇 − 𝑇) 𝜑 , T ̂ (𝑘𝑇) P (𝑘𝑇 − 𝑇) 𝜑 ̂ (𝑘𝑇) 1+𝜑

(20)

̂ 𝑦 (𝑘𝑇) 𝜑 T

= [−𝑦̂ (𝑘𝑇 − 𝑇) , −𝑦̂ (𝑘𝑇 − 2𝑇) , . . . , −𝑦̂ (𝑘𝑇 − 𝑛𝑇)] ,

𝑖 = 1, 2, . . . , 𝑟.

̂ 𝑢 (𝑘𝑇) = [̂ 𝜑 𝑢 (𝑘𝑇 − 𝑇) , 𝑢̂ (𝑘𝑇 − 2𝑇) , . . . , 𝑢̂ (𝑘𝑇 − 𝑛𝑇) , 𝑢̂ (𝑘𝑇 − 𝑇 + 𝑡1 ) ,

Define the estimate of 𝜑(𝑘𝑇) as

𝑢̂ (𝑘𝑇 − 2𝑇 + 𝑡1 ) , . . . , 𝑢̂ (𝑘𝑇 − 𝑛𝑇 + 𝑡1 ) , . . . , 𝑢̂ (𝑘𝑇 − 𝑇 + 𝑡𝑟−1 ) , 𝑢̂ (𝑘𝑇 − 2𝑇 + 𝑡𝑟−1 ) , . . . ,

̂ 𝑦 (𝑘𝑇) 𝜑 ̂ 𝑢 (𝑘𝑇)] ∈ R𝑛0 , ̂ (𝑘𝑇) := [𝜑 𝜑 ̂ 𝛾 (𝑘𝑇)] [𝜑

T

𝑢̂(𝑘𝑇 − 𝑛𝑇 + 𝑡𝑟−1 )] , T

𝜑𝛾 (𝑘𝑇) = [𝑢 (𝑘𝑇) , 𝑢2 (𝑘𝑇) , . . . , 𝑢𝑛𝛾 (𝑘𝑇)] ,

̂ 𝑦 (𝑘𝑇) := [−𝑦̂ (𝑘𝑇 − 𝑇) , −𝑦̂ (𝑘𝑇 − 2𝑇) , . . . , 𝜑

𝑢̂ (𝑘𝑇 + 𝑡𝑖−1 ) = 𝛾̂1 (𝑘𝑇) 𝑢 (𝑘𝑇 + 𝑡𝑖−1 )

−𝑦̂ (𝑘𝑇 − 𝑛𝑇)] ∈ R𝑛 ,

+ 𝛾̂2 (𝑘𝑇) 𝑢2 (𝑘𝑇 + 𝑡𝑖−1 )

̂ 𝑢 (𝑘𝑇) := [̂ 𝜑 𝑢 (𝑘𝑇 − 𝑇) , 𝑢̂ (𝑘𝑇 − 2𝑇) , . . . , 𝑢̂ (𝑘𝑇 − 𝑛𝑇) ,

+ ⋅ ⋅ ⋅ + 𝛾̂𝑛𝛾 (𝑘𝑇) 𝑢𝑛𝛾 (𝑘𝑇 + 𝑡𝑖−1 ) ,

𝑢̂ (𝑘𝑇 − 𝑇 + 𝑡1 ) , 𝑢̂ (𝑘𝑇 − 2𝑇 + 𝑡1 ) , . . . ,

(23)

̂ T (𝑘𝑇) 𝜃̂ (𝑘𝑇) . 𝑦̂ (𝑘𝑇) = 𝜑

𝑢̂ (𝑘𝑇 − 𝑛𝑇 + 𝑡1 ) , . . . , 𝑢̂ (𝑘𝑇 − 𝑇 + 𝑡𝑟−1 ) ,

̂ To initialize the algorithm, we take 𝜃(0) to be a small real ̂ vector; for example, 𝜃(0) = 1𝑛0 /𝑝0 and P(0) = 𝑝0 I with 𝑝0 normally a large positive number (e.g., 𝑝0 = 106 ).

𝑢̂ (𝑘𝑇 − 2𝑇 + 𝑡𝑟−1 ) , . . . , 𝑢̂(𝑘𝑇 − 𝑛𝑇 + 𝑡𝑟−1 )] ∈ R𝑟𝑛 . T

(21)

An example is given to demonstrate the feasibility of the proposed algorithm. Assume that the dynamical linear subsystem 𝑃𝑐 has the following state-space representation:

̂ (𝑘𝑇) in place of 𝜑(𝑘𝑇) in (17) and (18), we have Using 𝜑

−0.3 −0.2 1 ẋ (𝑡) = [ ] x (𝑡) + [ ] 𝑢 (𝑡) , 1 0 0

̂ (𝑘𝑇) 𝜃̂ (𝑘𝑇) = 𝜃̂ (𝑘𝑇 − 𝑇) + P (𝑘𝑇) 𝜑

and the static nonlinear subsystem is denoted by (22)

P (𝑘𝑇) = P (𝑘𝑇 − 𝑇) −

̂ T (𝑘𝑇) P (𝑘𝑇 − 𝑇) 𝜑 ̂ (𝑘𝑇) 1+𝜑

(24)

𝑦 (𝑡) = [−0.4, 0.2] x (𝑡) + 𝑢 (𝑡) ,

̂ T (𝑘𝑇) 𝜃̂ (𝑘𝑇 − 𝑞)] , × [𝑦1 (𝑘𝑇) − 𝜑

̂ (𝑘𝑇) 𝜑 ̂ T (𝑘𝑇) P (𝑘𝑇 − 𝑇) P (𝑘𝑇 − 𝑇) 𝜑

4. Example

.

𝑢 (𝑡) = 𝑓 (𝑢 (𝑡)) = 𝛾1 𝑢 (𝑡) + 𝛾2 𝑢2 (𝑡) + 𝛾3 𝑢3 (𝑡) = 𝑢 (𝑡) + 0.5𝑢2 (𝑡) + 0.25𝑢3 (𝑡) .

(25)


5

Table 1: The parameter estimates and their errors (𝜎2 = 2.002 ). 𝑘 100 500 1000 2000 3000 4000 5000 6000 True values

𝛼1 −0.03122 −0.53614 −0.55071 −0.57852 −0.60057 −0.61945 −0.66253 −0.67482 −0.68000

𝛼2 −0.05592 0.31821 0.44816 0.42159 0.39569 0.41573 0.42857 0.46019 0.47241

𝛽1 0.02344 −0.43136 −0.41426 −0.43335 −0.44560 −0.44904 −0.49385 −0.52100 −0.52674

𝛽2 0.39192 0.63245 0.78764 0.71714 0.68205 0.68292 0.68309 0.71556 0.73948

𝛽3 −0.08169 −0.30871 −0.24332 −0.22310 −0.23745 −0.24136 −0.26702 −0.26552 −0.25070

𝛽4 0.14407 0.53689 0.62087 0.63026 0.60417 0.60799 0.62299 0.64442 0.66221

𝛾1 1.52981 1.30985 1.12565 1.04567 1.09320 1.03387 1.00550 0.99935 1.00000

𝛾2 0.68962 0.69341 0.56690 0.49863 0.49337 0.50662 0.50873 0.53599 0.50000

𝛾3 0.02254 0.10358 0.22360 0.23534 0.20626 0.24914 0.25535 0.26184 0.25000

𝛿 (%) 73.36523 26.81256 12.86475 8.87717 10.44562 7.82199 5.03493 2.87761

𝛾2 0.54701 0.54706 0.51876 0.50079 0.49893 0.50179 0.50215 0.50881 0.50000

𝛾3 0.18587 0.22836 0.25274 0.25083 0.24231 0.25233 0.25341 0.25474 0.25000

𝛿 (%) 13.23523 7.30117 4.13831 2.89926 3.07166 2.68214 2.01651 1.45453

Table 2: The parameter estimates and their errors (𝜎2 = 0.502 ). 𝑘 100 500 1000 2000 3000 4000 5000 6000 True values

𝛼1 −0.69605 −0.68503 −0.65503 −0.64233 −0.64733 −0.65262 −0.66319 −0.66697 −0.68000

𝛼2 0.48092 0.51969 0.49678 0.46233 0.45069 0.45294 0.45510 0.46155 0.47241

𝛽1 −0.59484 −0.54542 −0.50768 −0.49229 −0.49445 −0.49564 −0.50654 −0.51409 −0.52674

𝛽2 0.83269 0.81494 0.79029 0.74295 0.72782 0.72445 0.72291 0.72921 0.73948

1

𝛽3 −0.30638 −0.30331 −0.26915 −0.25140 −0.25286 −0.25257 −0.25789 −0.25706 −0.25070

𝛽4 0.59981 0.68632 0.68531 0.66911 0.65764 0.65608 0.65842 0.66243 0.66221

𝛾1 1.17677 1.05813 1.01708 1.00467 1.01820 1.00431 0.99803 0.99700 1.00000

Thus, the corresponding input-output expression is given by 𝑦1 (𝑘𝑇) = (−0.68000𝑧−1 + 0.47241𝑧−2 )

0.6

× 𝑦 (𝑘𝑇) + 𝑢 (𝑘𝑇)

𝛿

0.8

0.4

+ (−0.52674𝑧−1 + 0.73948𝑧−2 ) 𝑢 (𝑘𝑇)

𝜎2 = 0.502

+ (−0.25070𝑧−1 + 0.66221𝑧−2 )

0.2 0

(27)

𝜎2 = 2.002 0

1000

2000

3000

4000

5000

6000

k

Figure 2: The estimation errors 𝛿 versus 𝑘.

Let 𝑟 = 2, 𝜏1 = 1s, and 𝜏2 = 2s; that is, 𝑡1 = 𝜏1 = 1s and 𝑡2 = 𝑡1 + 𝜏2 = 𝑇 = 3s. Discretizing 𝑃𝑐 with the frame period 𝑇, we obtain

𝑥 (𝑘𝑇 + 𝑇) = [

0.1274 −0.2835 ] x (𝑘𝑇) 1.4177 0.5527

+[

𝑢 (𝑘𝑇) 0.2981 1.1196 ], ][ 1.2978 0.9388 𝑢 (𝑘𝑇 + 𝑡1 )

𝑦 (𝑘𝑇) = [0.50, 0.21] x (𝑘𝑇) .

(26)

× 𝑢 (𝑘𝑇 + 𝑡1 ) + V (𝑘𝑇) . In simulation, the inputs {𝑢(𝑘𝑇)} and {𝑢(𝑘𝑇 + 𝑡1 )} are taken as persistent excitation signal sequences with zero mean and unit variance; {V(𝑘𝑇)} is a white noise with zero mean and variance 𝜎2 . Applying the proposed algorithm to estimate the parameters of this system, the estimates of 𝜃 and their errors with different noise variances are shown in Tables 1 and 2, and the parameter estimation errors 𝛿 := ̂ ‖𝜃(𝑘𝑇) − 𝜃‖/‖𝜃‖ versus 𝑘 are shown in Figure 2. When 𝜎2 = 2.002 and 𝜎2 = 0.502 , the corresponding signal-tonoise ratios (the square root of the ratio of output and noise variances) are SNR = 1.0342 and SNR = 4.1367, respectively. From Tables 1 and 2 and Figure 2, we can draw the following conclusions. (i) The parameter estimation errors of the AM-RLS algorithm become (generally) smaller as 𝑘 increases; see the estimation errors of the last columns of Tables 1 and 2 and Figure 2.

6

Mathematical Problems in Engineering (ii) Under different noise levels, the parameter estimates can converge to the true value, and a lower noise level results in a faster convergence rate of the parameter estimates to the true parameters; see the error curves in Figure 2 and the estimation errors in Tables 1 and 2. (iii) The proposed recursive algorithm differs from the iterative identification approach in [10] and can be used as an online identification.

5. Conclusions In this paper, we have established the identification model of the Hammerstein nonlinear systems with nonuniform sampling by using the key-term separation principle. To estimate the parameters of the proposed model, the recursive leastsquares parameter estimation algorithm is derived based on the auxiliary model identification idea. The proposed algorithm can simultaneously estimate the parameters of the linear and nonlinear subsystems of the Hammerstein nonlinear systems with nonuniform sampling. The simulation results show that the parameters of the Hammerstein systems with nonuniform sampling can be estimated effectively by the proposed algorithm. Although the algorithm is presented for a class of nonuniformly sampled Hammerstein nonlinear systems, the basic idea can also be extended to identify other linear and nonlinear systems [28, 29] and can combine the hierarchical identification methods [30–34], the multi-innovation identification methods [35–44], and other identification methods [45–58] to present new identification algorithms for linear or nonlinear and scalar or multivariable systems [59].

Acknowledgments This work was supported by the Fundamental Research Funds for the Central Universities (JUDCF11042 and JUDCF12031) and the PAPD of Jiangsu Higher Education Institutions and the 111 Project (B12018).

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