A recursive least squares parameter estimation algorithm for output nonlinear autoregressive systems using the input–output data filtering

Accepted Manuscript

A recursive least squares parameter estimation algorithm for output nonlinear autoregressive systems using the input–output data filtering Feng Ding, Yanjiao Wang, Jiyang Dai, Qishen Li, Qijia Chen PII: DOI: Reference:

S0016-0032(17)30382-4 10.1016/j.jfranklin.2017.08.009 FI 3092

To appear in:

Journal of the Franklin Institute

Received date: Revised date: Accepted date:

18 June 2016 6 March 2017 3 August 2017

Please cite this article as: Feng Ding, Yanjiao Wang, Jiyang Dai, Qishen Li, Qijia Chen, A recursive least squares parameter estimation algorithm for output nonlinear autoregressive systems using the input–output data filtering, Journal of the Franklin Institute (2017), doi: 10.1016/j.jfranklin.2017.08.009

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ACCEPTED MANUSCRIPT

A recursive least squares parameter estimation algorithm for output nonlinear autoregressive systems using the input–output data filtering Feng Dinga,b,d,∗, Yanjiao Wanga , Jiyang Daic , Qishen Lic , Qijia Chena a School of Internet of Things Engineering, Jiangnan University, Wuxi 214122, PR China of Automation and Electronic Engineering, Qingdao University of Science and Technology, Qingdao 266042, PR China. c School of Information Engineering, Nanchang Hangkong University, Nanchang 330063, PR China d Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

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b College

Abstract

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In the revised paper, we have highlighted the changes made in blue. Nonlinear systems exist widely in industrial processes. This paper studies the parameter estimation methods of establishing the mathematical models for a class of output nonlinear systems, whose output is nonlinear about the past outputs and linear about the inputs. We use an estimated noise transfer function to filter the input-output data and obtain two identification models, one containing the parameters of the system model, and the other containing the parameters of the noise model. Based on the data filtering technique, a data filtering based recursive least squares algorithm is proposed. The simulation results show that the proposed algorithm can generate more accurate parameter estimates than the recursive generalized least squares algorithm. Keywords: Mathematical modeling, Nonlinear system, Least squares, Parameter estimation

1. Introduction

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Parameter identification and establishing the models of dynamical systems are the basis of control system analysis [1, 2, 3] and controller design [4, 5]. Linear system identification methods have been well developed [6, 7], e.g., the generalized projection algorithm [8] for time-varying systems and the auxiliary model based recursive least squares algorithm for linear-in-parameter systems [9]. Nonlinear system identification has received much research attention [10]. Xu et al studied the parameter estimation and controller design for dynamic systems from the step responses based on the Newton iteration [11] and the Newton iteration algorithm to the parameter estimation for dynamical systems [12] and presented a damping iterative parameter identification method for dynamical systems based on the sine signal measurement [13]. Raja and Chaudhary studied a twostage fractional least mean square identification algorithm for parameter estimation of CARMA systems [14]. Li discussed parameter estimation for Hammerstein CARARMA systems based on the Newton iteration [15]. Xie et al. studied the finite impulse response model identification of multirate processes with random delays using the expectation maxumum algorithm [16]. Hammerstein models [17, 18], Wiener models and their combination [19] are the most common model structures in the literature of nonlinear systems. Wang and Ding presented an interval-varying generalized extended stochastic gradient (GESG) algorithm and an interval-varying recursive generalized extended least squares (RGELS) algorithm for Hammerstein-Wiener systems [20] and a filtering based GESG algorithm and a filtering based RGELS algorithm for Hammerstein-Wiener systems with ARMA noise [21]. Wang and Zhang used the Taylor expansion and investigated an improved least squares algorithm for identifying the parameters of multivariable Hammerstein output-error moving average systems [22]. Wang et al. discussed the recasted models based hierarchical extended stochastic gradient method for MIMO nonlinear systems [23]. Information filtering has wide applications in many areas [24, 25], e.g., signal processing [26, 27], particle filtering [28] and state estimation [29]. Some filtering based methods have been used in different fields during the past decade [30, 31], e.g., signal processing [32] and parameter identification [33]. A filtering based iterative algorithm [34], a filtering based auxiliary model hierarchical gradient algorithm [35] and a filtering based auxiliary model hierarchical least squares algorithm [36] have been proposed for multivariable systems. A decomposition based least squares iterative identification algorithm has been proposed for multivariate pseudo-linear ARMA systems using the data filtering [37]. About the decomposition based identification, Xu et al. proposed the parameter estimation algorithms for dynamical response signals based on the multi-innovation theory and the ✩ This

work was supported by the National Natural Science Foundation of China (No. 61273194, 61663032). author at: School of Internet of Things Engineering, Jiangnan University, Wuxi 214122, PR China Email address: [email protected] (Feng Ding)

∗ Corresponding

Preprint submitted to Journal of the Franklin Institute

Submitted: June 18, 2016; Revised: March 6, 2017

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2. The recursive generalized least squares algorithm

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hierarchical principle [38]. On the basis of the work in [39], this paper presents a filtering based recursive least squares algorithm for output nonlinear autoregressive systems by using the hierarchical identification principle. The proposed filtering based parameter identification algorithm requires less computation cost and can give higher estimation accuracy, which can recursively estimate the system model parameters, the noise model parameters and the internal variables. Briefly, the rest of this paper is organized as follows. Section 2 gives the representation of output nonlinear systems. Section 3 derives a recursive least squares algorithm for output nonlinear systems. Two numerical examples are provided to show the effectiveness of the proposed algorithms in Section 4. Finally, some concluding remarks are offered in Section 5.

ˆ Let us define some symbols. “A =: X” or “X := A” stands for “A is defined as X”; θ(k) denotes the −1 estimate of θ at time k; z denotes a unit forward shift operator with zy(k) = y(k + 1) and z y(k) = y(k − 1); 1n denote an n-dimensional column vector whose elements are 1; In represents an n × n-dimensional identity matrix. Consider the following output nonlinear systems with colored noise:

v(k) = D(z)w(k),

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y(k) = A(z)h(y(k)) + B(z)u(k) + w(k), h(y(k)) = c1 h1 (y(k)) + c2 h2 (y(k)) + . . . + cnc hnc (y(k)) = h(y(k))c,

(1) (2) (3)

where u(k) and y(k) are the input and output of the system, respectively, w(k) is the colored noise and v(k) is a stochastic white noise with zero mean and variance σ 2 , the nonlinear part is a linear combination of a known basis h = (h1 , h2 , . . . , hnc ) with coefficients c = (c1 , c2 , . . . , cnc ), and A(z), B(z) and D(z) are polynomials in the unit backward shift operator z −1 : A(z) := a1 z −1 + a2 z −2 + . . . + ana z −na ,

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B(z) := b1 z −1 + b2 z −2 + . . . + bnb z −nb , D(z) := 1 + d1 z −1 + d2 z −2 + . . . + dnd z −nd .

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Since the disturbance w(k) = 1/D(z)v(k) in the system (1) is an infinite impulse response (IIR) filter (note that v(k) is white noise), D(z) should be stable so that the output variable y(k) is bounded and the estimation accuracy can be improved. The stability of D(z) can improve convergence and accuracy because the output yf (t) of the filtered system in (31) involves only white noise v(k). Define the parameter vectors a, b and d as       a1 b1 d1  a2   b2   d2        a :=  .  ∈ Rna , b :=  .  ∈ Rnb , d :=  .  ∈ Rnd ,  ..   ..   .. 

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ana   a θ 1 := ∈ Rna +nb , b

bnb   c θ 2 := ∈ Rnc +nd . d

dnd

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and the information matrix H(k), the input information vector ψ b (k) and the noise information vector ψ d (k) as H(k) := [h1 (k), h2 (k), . . . , hnc (k)] ∈ Rna ×nc ,

hi (k) := [hi (y(k − 1)), hi (y(k − 2)), . . . , hi (y(k − na ))]T ∈ Rna ,

ψ b (k) := [u(k − 1), u(k − 2), . . . , u(k − nb )]T ∈ Rnb ,

ψ d (k) := [−w(k − 1), −w(k − 2), . . . , −w(k − nd )]T ∈ Rnd . Following the above definitions, Equations (1)–(3) can be written as y(k) = A(z)f (y(k)) + B(z)u(k) + [1 − D(z)]w(k) + v(k) = aT H(k)c + ψ Tb (k)b + ψ Td (k)d + v(k), T

w(k) = ψ d (k)d + v(k).

(4) (5)

The objective of this paper is to use the measurement data {u(k), y(k)} and to derive novel methods for estimating the unknown parameter vector c for the nonlinear part and the unknown parameter vectors a, b and d for the linear subsystems. 2

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For model (4), referring to [39], any pair γa and c/γ for any nonzero constant γ provides the same inputoutput mapping relation. For identifiability, we have to make one assumption that kck = 1 and the first entry of the vector c ispositive,  i.e., c1 > 0.   ˆ (k) a cˆ(k) na +nb ˆ ˆ Let θ 1 (k) := ˆ ∈R and θ 2 (k) := ˆ ∈ Rnc +nd be the estimates of θ 1 and θ 2 at time k, and b(k) d(k)    T  H(k)c H (k)a define φ1 (k) := ∈ Rna +nb and φ2 (k) := ∈ Rnc +nd . ψ b (k) ψ d (k) For the identification model in (4), define two quadratic cost functions: J1 (θ 1 ) :=

k X [y(j) − ψ Td (j)d − φT1 (j)θ 1 ]2 ,

J2 (θ 2 ) :=

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j=1

k X [y(j) − ψ Tb (j)b − φT2 (j)θ 2 ]2 . j=1

According to the least squares principle, minimizing J1 (θ 1 ) and J2 (θ 2 ), we can obtain the following recursive generalized least squares relations:

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ˆ 1 (k) = θ ˆ 1 (k − 1) + L1 (k)[y(k) − ψ T (k)d(k ˆ − 1) − φT (k)θ ˆ 1 (k − 1)], θ d 1 P1 (k − 1)φ1 (k) L1 (k) = 1 + φT1 (k)P1 (k − 1)φ1 (k) P1 (k) = P1 (k − 1) − L1 (k)LT1 (k)[1 + φT1 (k)P1 (k − 1)φ1 (k)], ˆ 2 (k − 1)], ˆ ˆ 2 (k) = θ ˆ 2 (k − 1) + L2 (k)[y(k) − ψ T (k)b(k) − φT (k)θ θ 2

b

P2 (k − 1)φ2 (k) , 1 + φT2 (k)P2 (k − 1)φ2 (k) P2 (k) = P2 (k − 1) − L2 (k)LT2 (k)[1 + φT2 (k)P2 (k − 1)φ2 (k)].

L2 (k) =

(6) (7) (8) (9) (10) (11)

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ˆ (k), Replacing the unknown variables w(k), ψ d (k), φ1 (k) and φ2 (k) in (6)–(11) with their estimates w(k), ˆ ψ d ˆ ˆ φ1 (k) and φ2 (k), we can obtain the recursive generalized least squares (RGLS) algorithm for generating the ˆ 1 (k) and θ ˆ 2 (k) of θ 1 and θ 2 : estimates θ

1

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ˆ 1 (k) = θ ˆ 1 (k − 1) + L1 (k)[y(k) − ψ ˆ T (k)d(k ˆ − 1) − φ ˆ T (k)θ ˆ 1 (k − 1)], θ d 1 T ˆ (k)[1 + φ ˆ (k)P1 (k − 1)φ ˆ (k)]−1 L1 (k) = P1 (k − 1)φ 1

1

T

ˆ (k)P1 (k − 1)φ ˆ (k)], P1 (k) = P1 (k − 1) − L1 (k)L1 (k)[1 + φ 1 1   H(k)ˆ c (k − 1) ˆ (k) = φ , 1 ψ b (k)

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T

b

2

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AC d

(13) (14) (15)

ψ b (k) = [u(k − 1), u(k − 2), . . . , u(k − nb )]T , ˆ ˆ T (k)θ ˆ 2 (k − 1)], ˆ 2 (k) = ϑ(k ˆ − 1) + L2 (k)[y(k) − ψ T (k)b(k) −φ θ

ˆ (k)[1 + φ ˆ T (k)P2 (k − 1)φ ˆ (k)]−1 , L2 (k) = P2 (k − 1)φ 2 2 2 T T ˆ ˆ (k)], P2 (k) = P2 (k − 1) − L2 (k)L2 (k)[1 + φ2 (k)P2 (k − 1)φ 2  T  H (k)ˆ a (k) ˆ (k) = , φ 2 ˆ (k) ψ d ˆ (k) = [−w(k ψ ˆ − 1), −w(k ˆ − 2), . . . , −w(k ˆ − nd )]T ,

(12)

P2 (0) = p0 Inc +nd ,

(16) (17) (18) (19) (20) (21)

T

ˆ T (k)b(k), ˆ ˆ (k)H(k)ˆ w(k) ˆ = y(k) − a c(k) − ψ b

(22)

H(k) = [h1 (k), h2 (k), . . . , hnc (k)],

(23)

hi (k) = [hi (y(k − 1)), hi (y(k − 2)), . . . , hi (y(k − na ))]T ,   ˆ 1 (k) θ ˆ Θ(k) = ˆ . θ 2 (k)

(24) (25)

Theorem 1. For the identification model in (4) and the algorithm in (12)–(25), suppose that {v(k)} is a white noise sequence with zero mean and variance σ 2 , i.e., E[v(k)] = 0, E[v 2 (k)] = σ 2 , E[v(k)v(i)] = 0 (i 6= k), and that there exist positive constants αi and βi such that for large k, the following persistent excitation conditions holds, (C1)

α1 Ina +nb 6

k 1Xˆ ˆ T (j) 6 β1 In +n , a.s. φf 1 (j)φ f1 a b k j=1

3

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(C2)

α2 Inc +nd 6

k 1Xˆ ˆ T (j) 6 β2 In +n , a.s. φ (j)φ f2 c d k j=1 f 2

Then the parameter estimation error given by the RGLS algorithm in (42)–(62) converges to zero. T

ˆ − 1) − ψ ˆ (k)d(k ˆ − 1), we have ˆ T (k − 1)H(k)ˆ Proof Let e(k) := y(k) − a c(k − 1) − ψ Tb (k)b(k d ˜ 1 (k) = θ ˜ 1 (k − 1) + P1 (k)φ ˆ (k)e(k), θ 1 ˜ ˜ ˆ θ 2 (k) = θ 2 (k − 1) + P2 (k)φ (k)e(k).

(26) (27)

2

Define the nonnegative functions

2

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˜ T (k)P −1 (k)θ ˜ 1 (k), T1 (k) := θ 1 1 T ˜ (k)P −1 (k)θ ˜ 2 (k), T1 (k) := θ 2

T (k) := T1 (k) + T2 (k). Using (26) and (27), we have

˜ 1 (k − 1) + P1 (k)φ ˆ (k)e(k)]T P −1 (k)[θ ˜ 1 (k − 1) + P1 (k)φ ˆ (k)e(k)] T1 (k) = [θ 1 1 1 2 ˜ 1 (k − 1)φ ˆ (k)] + 2θ ˜ 1 (k − 1)φ ˆ (k)e(k) + φ ˆ T (k)P1 (k)φ ˆ (k)e2 (k). = T1 (k − 1) + [θ 1 1 1 1

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From the definition of e(k), we have

(28)

e(k) := aT H(k)c + ψ Tb (k)b + ψ Td (k)d + v(k) ˆ − 1) − ψˆ T (k)d(k ˆ − 1), −ˆ aT (k − 1)H(k)ˆ c(k − 1) − ψ Tb (k)b(k d = −˜ y1 (k) + ∆1 (k) + v(k),

where

ˆ T (k)θ ˜ 1 (k) + ψ ˆ T (k)d(k), ˜ y˜1 (k) := φ 1 d T ˆ (k)] θ 1 + [ψ (k) − ψ ˆ (k)]T d. ∆1 (k) := [φ1 (k) − φ 1 d d

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Thus, Equation (28) can be written as

˜ 1 (k − 1)φ ˆ (k)]2 + 2θ ˜ 1 (k − 1)φ ˆ (k)e(k) + φ ˆ T (k)P1 (k)φ ˆ (k)e2 (k). T1 (k) = T1 (k − 1) + [θ 1 1 1 1

(29)

ˆ (k)P (k)ϕ(k) ˆ ˆ (k)P (k − 1)ϕ(k)] ˆ Here, we have used the inequality 1 − ϕ = [1 + ϕ > 0. T −1 ˜ ˜ Define a non-negative definite function V (k) := E[θ (k)P (k)θ(k)]. Note that {v(k)} is white noise independent of the input signal {u(k)}. Suppose that {∆(k)} is bounded with ∆2 (k) 6 ε < ∞, a.s. Since ˜ T (k − 1)P −1 (k − 1)θ(k ˜ − 1), y˜(k), ϕ ˆ T (k)P (k)ϕ(k) ˆ θ and ∆(k) are uncorrelated with v(k), taking the expectation to both sides of (29) and referring to Equations (18)–(19) in [40], we have T

−1

PT

ED

T

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2 ˆ T (k)P (k)ϕ(k)[v ˆ V (k) 6 V (k − 1) + 0 + E{ϕ (k) + ∆2 (k)]} T 2 ˆ (k)P (k)ϕ(k)](σ ˆ 6 V (k − 1) + E[ϕ + ε)   k X ˆ T (j)P (j)ϕ(j) ˆ  (σ 2 + ε) 6 V (0) + [ln |P −1 (k)| + n ln p0 ](σ 2 + ε). 6 V (0) + E  ϕ

(30)

j=1

AC

Using Assumption (C1), we have

P −1 (k) 6 βkIn + In /p0 = (βk + 1/p0 )In , P −1 (k) > αkIn + In /p0 = (αk + 1/p0 )I,

(αk + 1/p0 )n 6 |P −1 (k)| 6 (βk + 1/p0 )n .

According to the definition of V (k), we have 2 ˜ (αk + 1/p0 )E[kθ(k)k ] 6 V (k) 6 n/p2 + [n ln(βk + 1/p0 ) + n ln p0 ](σ 2 + ε), 0

Taking the limit gives n/p20 + [n ln(βk + 1/p0 ) + n ln p0 ](σ 2 + ε) = 0. k→∞ αk + 1/p0

2 ˜ lim E[kθ(k)k ] 6 lim

k→∞

ˆ This shows that the parameter estimation error kθ(k) − θk converges to zero as k increases, and Theorem 1 is proved.  The colored noise implies that noise model w(k) = 1/D(z)v(k) contains some parameters di in θ 2 to be identified. For the white noise w(k) = v(k), no noise model parameters need be identified. Here deals with the colored noise through estimating the parameters of the noise model. 4

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3. The filtering based RGLS parameter estimation algorithm Define the filtered input uf (k) and the filtered output yf (k) as uf (k) := D(z)u(k),

yf (k) := D(z)y(k).

Multiplying both sides of (1) by D(z), we have yf (k) = A(z)D(z)h(y(k)) + B(z)uf (k) + v(k).

(31)

Define the filtered information vector ψ f (k) and the filtered information matrix F (k) as F (k) := [f1 (k), f2 (k), . . . , fnc (k)] ∈ Rna ×nc ,

fi (k) := [fi (y(k − 1)), fi (y(k − 2)), . . . , fi (y(k − na ))]T ∈ Rna ,

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ψ f (k) := [uf (k − 1), uf (k − 2), . . . , uf (k − nb )]T ∈ Rnb ,

i = 1, 2, . . . , nc ,

fj (k) := D(z)hj (y(k)), j = 1, 2, . . . , nc .

Thus, the filtered model in (31) can be rewritten as

yf (k) = A(z)(c1 f1 (k) + c2 f2 (k) + . . . + cnc fnc (k)) + B(z)uf (k) + v(k)

ˆ Let ξ(k) := cˆ(k) be the estimate of ξ := c at time k, φf 1 (k) := For the filtered model, defining two criterion functions k X J3 (θ 1 ) := [yf (j) − φTf 1 (j)θ 1 ]2 , j=1

k X [yf (j) − ψ Tf (j)b − φTf 2 (j)ξ]2 ,





F (k)ξ ∈ Rna +nb and φf 2 (k) := F T (k)a ∈ Rnc . ψ f (k)

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J4 (ξ) :=

(32)

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= aT F (k)c + ψ Tf (k)b + v(k).

j=1

ED

and minimizing J3 (θ 1 ) and J4 (ξ), we can derive the following recursive least squares relations: ˆ 1 (k) = θ ˆ 1 (k − 1) + L3 (k)[yf (k) − φT (k)θ ˆ 1 (k − 1)], θ f1

(33)

L3 (k) =

(34)

P3 (k − 1)φf 1 (k) , 1 + φTf 1 (k)P3 (k − 1)φf 1 (k)

(35)

L4 (k) =

(37)

PT

P3 (k) = P3 (k − 1) − L3 (k)LT3 (k)[1 + φTf 1 (k)P3 (k − 1)φf 1 (k)], ˆ ˆ − 1) + L4 (k)[yf (k) − ψ T (k)b(k) ˆ ˆ − 1)], ξ(k) = ξ(k − φT (k)ξ(k f

f2

CE

P4 (k − 1)φf 2 (k) , 1 + φTf 2 (k)P4 (k − 1)φf 2 (k)

P4 (k) = P4 (k − 1) − L4 (k)LT4 (k)[1 + φTf 2 (k)P4 (k − 1)φf 2 (k)].

(36)

(38)

AC

Note that the polynomial D(z) is unknown, so are the filtered variables uf (k) and yf (k), and the filtered ˆ information vectors ψ f (k), φf 1 (k), φf 2 (k) and the filtered information matrix F (k). Thus the estimates θ(k) ˆ and ξ(k) are impossible to compute. Here, we adopt the idea of replacing the unknown variables with their estimates to implement the recursive computation. For the noise model in (5), defining the criterion function J5 (d) :=

k X j=1

[w(j) ˆ − ψ Td (j)d]2 ,

and minimizing this function, we can derive the following recursive least squares relations: ˆ = d(k ˆ − 1) + L5 (k)[w(k) − ψ T (k)d(k ˆ − 1)], d(k) d P5 (k − 1)ψ d (k) L5 (k) = , 1 + ψ Td (k)P5 (k − 1)ψ d (k) P5 (k) = P5 (k − 1) − L5 (k)LT5 (k)[1 + ψ Td (k)P5 (k − 1)ψ d (k)].

5

(39) (40) (41)

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ˆ − 1) and cˆ(k − 1), respectively, ˆ (k − 1), b(k Replacing the parameter vectors a, b and c with their estimates a the estimate w(k) ˆ can be computed by ˆ − 1). ˆ T (k − 1)H(k)ˆ w(k) ˆ = y(k) − a c(k − 1) − ψ Tb (k)b(k Thus, use w(k) ˆ to construct the estimate of ψ d (k): ˆ (k) := [−w(k ψ ˆ − 1), −w(k ˆ − 2), . . . , −w(k ˆ − nd )]T ∈ Rnd . d ˆ = [dˆ1 (k), dˆ2 (k), . . . , dˆn (k)]T to construct the estimate of D(z): Using the parameter estimate d(k) d ˆ D(k, z) = 1 + dˆ1 (k)z −1 + dˆ2 (k)z −2 + . . . + dˆnd (k)z −nd .

nd X i=1

yˆf (k) = y(k) +

nd X i=1

fˆj (k) = hj (y(k)) +

dˆi (k)u(k − i), dˆi (k)y(k − i), nd X i=1

dˆi (k)hij (y(k − i)), j = 1, 2, . . . , nc .

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u ˆf (k) = u(k) +

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ˆ Filtering u(k), y(k) and h(y(k)) with D(k, z), we can get the estimates u ˆf (k), yˆf (k) and fˆj (k):

Using u ˆf (k) and fˆj (k) to construct the estimates of ψ f (k), fj (k) and F (k): ˆ (k) := [ˆ ψ uf (k − 1), u ˆf (k − 2), . . . , u ˆf (k − nb )]T ∈ Rnb , f fˆi (k) := [fˆi (y(k − 1)), fˆi (y(k − 2)), . . . , fˆi (y(k − na ))]T ∈ Rna , Fˆ (k) := [fˆ1 (k), fˆ2 (k), . . . , fˆn (k)] ∈ Rna ×nc , c

i = 1, 2, . . . , nc ,

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and replacing the unknown variables in (33)–(41) with their estimates, we can obtain the filtering based recursive generalized least squares (F-RGLS) algorithm for the output nonlinear systems: ˆ 1 (k) = θ ˆ 1 (k − 1) + L3 (k)[ˆ ˆ T (k)θ ˆ 1 (k − 1)], θ yf (k) − φ f1

(42)

ED

ˆ (k)[1 + φ ˆ T (k)P3 (k − 1)φ ˆ (k)]−1 , L3 (k) = P3 (k − 1)φ f1 f1 f1 T

ˆ (k)P3 (k − 1)φ ˆ (k)], P3 (k) = P3 (k − 1) − L3 (k)L3 (k)[1 + φ f1 f1 T

T

T

PT

ˆ ˆ − 1) + L4 (k)[ˆ ˆ (k)b(k) ˆ ˆ (k)ξ(k ˆ − 1)], ξ(k) = ξ(k yf (k) − ψ −φ f f2 T

ˆ (k)[1 + φ ˆ (k)P4 (k − 1)φ ˆ (k)] L4 (k) = P4 (k − 1)φ f2 f2 f2

−1

,

T

ˆ (k)P4 (k − 1)φ ˆ (k)], P4 (k) = P4 (k − 1) − L4 (k)LT4 (k)[1 + φ f2 f2

CE

nd X

u ˆf (k) = u(k) +

AC

yˆf (k) = y(k) +

i=1 nd X i=1

fˆj (k) = hj (y(k)) +

(43) (44) (45) (46) (47)

dˆi (k)u(k − i),

(48)

dˆi (k)y(k − i),

(49)

nd X i=1

dˆi (k)hij (y(k − i)),

(50)

Fˆ (k) = [fˆ1 (k), fˆ2 (k), . . . , fˆnc (k)], fˆi (k) = [fˆi (y(k − 1)), fˆi (y(k − 2)), . . . , fˆi (y(k − na ))]T ,

ˆ (k) = [ˆ ψ uf (k − 1), u ˆf (k − 2), . . . , u ˆf (k − nb )]T , f   ˆ ˆ ˆ (k) = F (k)ξ(k − 1) , φ f1 ˆ (k) ψ

(51) (52) (53) (54)

f

ˆ (k) = Fˆ T (k)ˆ φ a(k), f2

(55) T

ˆ = d(k ˆ − 1) + L5 (k)[w(k) ˆ (k)d(k ˆ − 1)], d(k) ˆ −ψ d T ˆ (k)[1 + ψ ˆ (k)P5 (k − 1)ψ ˆ (k)]−1 , L5 (k) = P5 (k − 1)ψ d

d

d

T

ˆ (k)P5 (k − 1)ψ ˆ (k)], P5 (k) = P5 (k − 1) − L5 (k)L5 (k)[1 + ψ d d T

6

(56) (57) (58)

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ˆ (k) = [−w(k ψ ˆ − 1), −w(k ˆ − 2), . . . , −w(k ˆ − nd )]T , d ˆ − 1), ˆ T (k − 1)H(k)ˆ w(k) ˆ = y(k) − a c(k − 1) − ψ T (k)b(k

(59) (60)

b

ψ b (k) = [u(k − 1), u(k − 2), . . . , u(k − nb )]T ,   ˆ (k) ˆ T (k), ξˆT (k), dˆT (k)]T , θ ˆ 1 (k) = a ˆ Θ(k) = [θ . 1 ˆ b(k)

(61) (62)

The procedure of computing the parameter vectors θ 1 , c and d are listed in the following.

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1. To initialize: let k = 1, and set the initial values P3 (0) = p0 Ina +nb , P4 (0) = p0 Inc , P5 (0) = p0 Ind , ˆ 1 (0) = 1n +n /p0 , kˆ ˆ = 1n /p0 , w(k θ c(0)k = 1, d(0) ˆ − i) = 1/p0 , i = 0, 1, . . . , nd , p0 = 106 . a b d 2. Collect the input-output data u(k) and y(k), construct ψ b (k) using (61). ˆ (k) by (59), compute L5 (k) using (57) and P5 (k) using (58). 3. Compute w(k) ˆ by (60), construct ψ d ˆ 4. Update the parameter estimate d(k) by (56). ˆ (k) using (53), 5. Compute u ˆf (k) by (48) and yˆf (k) by (49), construct ψ f 6. Construct fˆj (k) by (50) and form fˆi (k) using (52) and Fˆ (k) using (51). ˆ (k) in (54) and form L3 (k) using (43) and P3 (k) using (44). 7. Compute φ f1

ˆ ξ(k) ˆ , cˆ(k) = sgn[ξ(1)] ˆ kξ(k)k

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ˆ ˆ ˆ 1 (k) in (62). ˆ (k) and b(k) 8. Update the parameter estimate θ(k) using (42) and read a from θ ˆ 9. Form φf 2 (k) in (55) and compute L4 (k) using (46) and P4 (k) using (47). ˆ 10. Update the parameter estimate ξ(k) by (45), normalize cˆ(k) using ˆ ξ(k) := cˆ(k).

ˆ ˆ where sgn[ξ(1)] represents the sign of the first element of ξ(k). 11. Increase k by 1 and go to Step 2, continue the recursive calculation.

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The following studies the convergence of the presented F-RGLS algorithm.

α3 Ina +nb 6

(C4)

α4 Inc 6

k 1Xˆ ˆ T (j) 6 β4 In , a.s. φ (j)φ f2 c k j=1 f 2 k 1Xˆ ˆ T (j) 6 β5 In , a.s. ψ (j)ψ d d k j=1 d

CE

(C5)

k 1Xˆ ˆ T (j) 6 β3 In +n , a.s. φf 1 (j)φ f1 a b k j=1

PT

(C3)

ED

Theorem 2. For the identification model in (32) and (5) and the F-RGLS algorithm in (42)–(62), suppose that {v(k)} is a white noise sequence with zero mean and variance σ 2 , i.e., E[v(k)] = 0, E[v 2 (k)] = σ 2 , E[v(k)v(i)] = 0 (i 6= k), and that there exist positive constants αi and βi such that for large k, the following persistent excitation conditions hold,

α5 Ind 6

AC

Then the parameter estimation error given by the F-RGLS algorithm converges to zero. The proof can be done in a similar way in Theorem 2.

4. Examples

Example 1: Consider the following output nonlinear system: y(k) = A(z)f (y(k)) + B(z)u(k) +

v(k) , D(z)

A(z) = a1 z −1 = 0.40z −1 , B(z) = b1 z −1 + b2 z −2 = 0.60z −1 + 0.40z −2 , D(z) = 1 + d1 z −1 = 1 + 0.10z −1 , f (y(k)) = c1 y(k) + c2 cos(y(k)) = 0.80y(k) − 0.60cos(y(k)), θ = [a1 , b1 , b2 ]T = [0.40, 0.60, 0.40]T , c = [c1 , c2 ]T = [0.80, −0.60]T , 7

ACCEPTED MANUSCRIPT

d = d1 = 0.10. In simulation, the input u(k) is taken as a persistent excitation signal sequence with zero mean and unit variance, and v(k) as a white noise sequence with zero mean and variances σ 2 = 0.302 . Applying the RGLS algorithm in (12)–(25) and the F-RGLS algorithm in (42)–(62) to estimate the parameters of this system, the obtained ˆ estimates and errors are shown in Table 1, and the estimation errors δ := kΘ(k) − Θk/kΘk versus k are shown in Figure 1.

k 100 200 500 1000 2000 3000 100 200 500 1000 2000 3000

RGLS

a1 0.34225 0.37989 0.40210 0.40187 0.39042 0.39490 0.31336 0.32448 0.35475 0.37850 0.38621 0.39476 0.40000

0.6

0.5

δ (%) 13.45274 12.09980 6.92699 2.84559 2.90154 1.60978 16.55084 12.71258 9.02899 4.13110 4.59149 2.70182

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0.3

ED

δ

0.4

0.2

0.1

RGLS

500

CE

0

PT

F−RGLS 0

d1 0.12056 0.09595 0.14209 0.11438 0.11773 0.10227 0.18977 0.16757 0.18080 0.13219 0.12903 0.10631 0.10000

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True values

Table 1: The parameter estimates and errors b1 b2 c1 c2 0.62489 0.34522 0.66954 -0.52134 0.63566 0.35931 0.69518 -0.49798 0.61309 0.37926 0.75608 -0.53833 0.60538 0.39039 0.77671 -0.57777 0.60576 0.39521 0.77227 -0.61384 0.60152 0.39812 0.78590 -0.61422 0.62584 0.35369 0.63561 -0.57009 0.63730 0.38668 0.68104 -0.56361 0.61110 0.40439 0.74175 -0.55918 0.60582 0.40310 0.76649 -0.58519 0.61160 0.39041 0.76584 -0.63371 0.60586 0.39491 0.77967 -0.62632 0.60000 0.40000 0.80000 -0.60000

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Algorithms F-RGLS

1000

1500 t

2000

2500

3000

Figure 1: The estimation errors δ versus k for Example 1

From Table 1 and Figure 1, we can draw the following conclusions.

AC

• The parameter estimation errors of the RGLS and F-RGLS algorithms become generally smaller with the data length k increasing. • For the same data length, the F-RGLS algorithm has faster convergence rates than the RGLS algorithm.

Example 2: For further validation of the proposed algorithms, we consider the following output nonlinear system: y(k) = A(z)f (y(k)) + B(z)u(k) +

v(k) , D(z)

A(z) = a1 z −1 + a2 z −2 = 0.86z −1 − 0.30z −2 ,

B(z) = b1 z −1 + b2 z −2 = 0.31z −1 + 0.20z −2 ,

D(z) = 1 + d1 z −1 + d2 z −2 = 1 − 0.23z −1 + 0.28z −2 ,

f (y(k)) = c1 y(k) + c2 cos(y(k)) = 0.80y(k) + 0.60cos(y(k)), θ = [a1 , a2 , b1 , b2 ]T = [0.86, −0.30, 0.31, 0.20]T , c = [c1 , c2 ]T = [0.80, 0.60]T ,

8

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d = [d1 , d2 ] = [−0.23, 0.28]T . The simulation conditions are similar to those of Example 1 but the noise variance σ 2 = 0.102 and σ 2 = 0.202 , respectively, By applying the F-RGLS algorithm to estimate the parameters of this example system, the parameter estimates and errors are shown in Table 2 and Figure 2. For 50 sets of noise realizations, the Monte Carlo simulation results are shown in Tables 3–4.

a1 0.66399 0.77658 0.81477 0.83178 0.83604 0.85439 0.69222 0.76054 0.80490 0.80262 0.80765 0.83418 0.86000

Table a2 -0.12559 -0.23101 -0.26081 -0.27941 -0.28167 -0.29931 -0.16422 -0.22331 -0.24977 -0.25325 -0.25495 -0.28112 -0.30000

2: The F-RGLS parameter estimates and errors b1 b2 c1 c2 d1 0.31304 0.24572 0.78243 0.62274 -0.10852 0.31389 0.22477 0.78290 0.62215 -0.16219 0.31240 0.21194 0.78487 0.61966 -0.26659 0.31414 0.20409 0.78621 0.61796 -0.25076 0.31305 0.20671 0.79087 0.61198 -0.22768 0.31072 0.20511 0.79308 0.60912 -0.22497 0.31292 0.24001 0.79947 0.60071 -0.25558 0.31439 0.22286 0.79075 0.61214 -0.24471 0.31210 0.20760 0.78927 0.61405 -0.31095 0.31590 0.19980 0.78872 0.61476 -0.27704 0.31471 0.20770 0.79105 0.61175 -0.24677 0.31029 0.20693 0.79391 0.60804 -0.24135 0.31000 0.20000 0.80000 0.60000 -0.23000

0.5 0.45 0.4

δ (%) 22.64821 11.73725 5.66814 3.78511 2.71605 1.02949 18.33204 11.82485 7.84956 6.29916 5.08065 2.53814

0.3

M

δ

0.35

0.25

0.15 0.1 0.05

2

ED

0.2

σ2 = 0.202

2

σ = 0.10

PT

0

d2 0.13494 0.17436 0.24755 0.25160 0.26129 0.27730 0.13295 0.16794 0.25917 0.26419 0.27212 0.28703 0.28000

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k 100 200 500 1000 2000 3000 0.202 100 200 500 1000 2000 3000 True values

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σ2 0.102

0

500

1000

1500 t

2000

2500

3000

CE

Figure 2: The F-RGLS estimation errors δ versus k for Example 2

Table 3: The parameter estimates based on 50 Monte Carlo runs for Example 2 (σ 2 = 0.102 )

a1

AC

k

a2

b1

b2

c1

c2

d1

d2

100

0.71823±0.18723 -0.17072±0.18062 0.31085±0.01502 0.22860±0.03617 0.78522±0.04548 0.61838±0.05452 -0.13963±0.13006 0.15619±0.22755

200

0.79803±0.09727 -0.24697±0.09266 0.31379±0.01142 0.21595±0.02206 0.78897±0.03329 0.61394±0.04482 -0.19134±0.14571 0.20933±0.18367

500

0.82939±0.05495 -0.27410±0.05229 0.31178±0.00515 0.20703±0.01404 0.79232±0.02414 0.60987±0.03247 -0.23767±0.10325 0.25107±0.07384

1000

0.84307±0.04452 -0.28724±0.04473 0.31242±0.00537 0.20278±0.01093 0.79389±0.01528 0.60794±0.02036 -0.23383±0.05894 0.26051±0.03952

2000

0.84833±0.02541 -0.29104±0.02231 0.31191±0.00408 0.20320±0.00768 0.79602±0.01262 0.60520±0.01689 -0.22983±0.03123 0.27070±0.02694

3000

0.85403±0.00882 -0.29637±0.00964 0.31096±0.00275 0.20294±0.00519 0.79674±0.00688 0.60428±0.00914 -0.22851±0.01547 0.27956±0.01556

True values

0.86000

k

a1

-0.30000

0.31000

0.20000

0.80000

0.60000

-0.23000

0.28000

Table 4: The parameter estimates based on 50 Monte Carlo runs for Example 2 (σ 2 = 0.202 ) a2

b1

b2

c1

c2

d1

d2

100

0.74716±0.28447 -0.20904±0.23424 0.31852±0.04798 0.23446±0.04713 0.78795±0.06232 0.61402±0.07407 -0.18825±0.18357 0.14481±0.30660

200

0.80763±0.23371 -0.26064±0.21308 0.31462±0.02502 0.21798±0.04089 0.78700±0.04280 0.61634±0.05428 -0.21660±0.10786 0.19301±0.12053

500

0.84540±0.09493 -0.28863±0.09354 0.31081±0.01604 0.20530±0.01808 0.79099±0.03508 0.61108±0.04752 -0.26782±0.09682 0.25483±0.08238

1000

0.84435±0.08603 -0.29041±0.08690 0.31320±0.01042 0.19984±0.02298 0.79001±0.02249 0.61281±0.02985 -0.25293±0.07857 0.26388±0.05035

2000

0.83941±0.04839 -0.28383±0.05039 0.31372±0.00568 0.20510±0.01146 0.79328±0.01905 0.60869±0.02472 -0.23734±0.04202 0.27263±0.03263

3000

0.84884±0.03756 -0.29282±0.03215 0.31153±0.00395 0.20476±0.01268 0.79453±0.01579 0.60712±0.02113 -0.23564±0.03671 0.28090±0.02916

True values

0.86000

-0.30000

0.31000

0.20000

9

0.80000

0.60000

-0.23000

0.28000

ACCEPTED MANUSCRIPT

From Tables 3–4 and Figure 2, we can see that the estimation accuracy given by the proposed algorithms becomes higher with the noise-to-signal ratios decreasing and the estimation errors become generally smaller with the data length k increasing. The Monte Carlo simulation results show that the proposed algorithms are effective. 5. Conclusions

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In this paper, we present a filtering based recursive generalized least squares algorithm for output nonlinear systems using the data filtering technique. The main contribution is that the nonlinear system is identified through a recursive least squares method by means of the filtered input and output data. The presented algorithm in this paper requires less computation burden and can give more accurate parameter estimate compared with the generalized extended least squares algorithm. The proposed algorithm of this paper can be extended to study identification problems of dual-rate sampled-data systems [41, 42], other nonlinear systems with colored noises [43] and applied to other fields [44, 45, 46]. References

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