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This article presents system identification of wood drying process based on ARMAX model. Temperature and equivalent moisture content are considered as inputs, and moisture content of the wood sample during drying is taken as output of the system. The comparative study of RLS and FF-RLS to identify the system parameters is presented. Simulation results are presented to val-idate the efficacy of the ARMAX model for wood drying process.

Wood drying process plays an important role in industry of wood product [

Wood drying prediction models like [

Prediction models based on Auto Regressive Moving Average model with exogenous input (ARMAX) analysis have been widely shown in literature [

The model takes temperature and EMC as inputs, moisture content as output. ARMAX model of wood drying process is described as

A ( z ) y ( t ) = B 1 ( z ) u 1 ( t ) + B 2 ( z ) u 2 ( t ) + D ( z ) v ( t ) (1)

The constant coefficient polynomials A ( z ) , B 1 ( z ) , B 2 ( z ) in the wood drying process model are

A ( z ) = a 0 + a 1 z − 1 + a 2 z − 2 (2)

B 1 ( z ) = b 10 + b 1 z − 1 + b 2 z − 2 (3)

B 2 ( z ) = b 20 + b 3 z − 1 + b 4 z − 2 (4)

D ( z ) = d 0 + d 1 z − 1 (5)

When t ≤ 0 , y ( t ) = 0 , u j ( t ) = 0 , v ( t ) = 0 , a 0 = 1 , b 10 = 0 , b 20 = 0 , d 0 = 0 .

Substituting polynomials (2), (3), (4), (5) and the initial values into (1), gives

[ 1 + a 1 z − 1 + a 2 z − 2 ] y ( t ) = [ b 1 z − 1 + b 2 z − 2 ] u 1 ( t ) + [ b 3 z − 1 + b 4 z − 2 ] u 2 ( t ) + ( d 1 z − 1 ) v ( t ) (6)

Introducing the unit backward shift operator z^{−1} into (6) yields to

y ( t ) = − a 1 y ( t − 1 ) − a 2 y ( t − 2 ) + b 1 u 1 ( t − 1 ) + b 2 u 1 ( t − 2 ) + b 3 u 2 ( t − 1 ) + b 4 u 2 ( t − 2 ) + d 1 v ( t − 1 ) (7)

Therefore, the output y ( t ) can be expressed as

y ( t ) = φ Τ ( t ) θ + v ( t ) (8)

where θ : = [ a 1 , a 2 , b 1 , b 2 , b 3 , b 4 , d 1 ] T , φ ( t ) = [−y(t − 1), −y(t − 2), u_{1}(t − 1), u_{1}(t − 2), u_{2}(t − 1), u_{2}(t − 2),v(t − 1)]^{T}.

Let t = 1, 2, ∙∙∙, Equation (8) leads to

Y t = H t θ + V t (9)

Using the Least Squares identification principle to define the quadratic criterion function

J ( θ ) : = V t T V t = ( Y t − H t θ ) T ( Y t − H t θ ) (10)

Using recursive method, matrix P − 1 ( t ) is define as

P − 1 ( t ) = P − 1 ( 0 ) + ∑ j = 1 t φ ( j ) φ T ( j ) = P − 1 ( 0 ) + H t T H t , P ( 0 ) = p 0 I > 0 (11)

The RLS estimation of the parameter vector is

θ ^ ( t ) = ( H t T H t ) − 1 H t T Y t = P ( t ) H t T Y t = θ ^ ( t − 1 ) + P ( t ) φ T ( t ) [ y ( t ) − φ T ( t ) θ ^ ( t − 1 ) ] (12)

The estimated residual is

v ^ ( t ) = y ( t ) − φ ^ ( t ) θ ^ ( t ) (13)

where φ ^ ( t ) : = [ ϕ ( t ) Τ , v ^ ( t − 1 ) ] Τ .

RLS estimation of parameter vector θ is achieved

θ ^ ( t ) = θ ^ ( t − 1 ) + P ( t ) φ ^ ( t ) [ y ( t ) − φ ^ T ( t ) θ ^ ( t − 1 ) ] (14)

P − 1 ( t ) = P − 1 ( t − 1 ) + φ ^ ( t ) φ ^ T ( t ) , P ( 0 ) = p 0 I > 0 (15)

In experimental system, data accumulates with time, results in the failure of extracting new data information from the previous data. Especially to the time-varying parameter system, due to the characteristics of parameter, the algorithm should track the time variation parameter. Hence, forgetting factor λ is introduced into (15), an optimization algorithm to identify the parameters is obtained

P − 1 ( t ) = λ P − 1 ( t − 1 ) + φ ^ ( t ) φ ^ T ( t ) (16)

Simulation results were based on the 1000 input-output experimental data acquired during wood drying. RLS algorithm in Equations (14) and (15) is applied to estimate the parameters in the ARMAX model built in Equation (1). Parameters variation trend is shown in

Parameters variation trend with λ = 0.95 is shown in

In this paper, an ARMAX model based on the experimental data is derived to describe the wood drying model, which is adopted to predict wood moisture content during drying. RLS and FF-RLS algorithms are utilized to identify the system parameters. The proposed method is verified by simulation results. The parameters variation trend with the proposed prediction scheme is studied. Simulation results demonstrate that the FF-RLS method leads to a faster and more stable convergence compared with RLS scheme. However, the accuracy of RLS estimate is higher than FF-RLS.

This work was supported by University Nursing Program for Young Scholars with Creative Talents in Heilongjiang Province (UNPYSCT-2018083).

The authors declare no conflicts of interest regarding the publication of this paper.

Zhou, Z., Zhang, P.X., Huai, B.F. and Huang, L.P. (2019) System Identification of Wood Drying Process Based on ARMAX Model. Agricultural Sciences, 10, 241-248. https://doi.org/10.4236/as.2019.103020