System Identification of Wood Drying Process Based on ARMAX Model

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 validate the efficacy of the ARMAX model for wood drying process.


Introduction
Wood drying process plays an important role in industry of wood product [1].
The quality of wood product mainly depends on the final moisture content of wood. Temperature and equilibrium moisture content (EMC) are the two main factors influencing the drying moisture content [2]. Hence, experimental data of temperature and EMC was often used to build the prediction model of wood drying moisture in the previous literature.
Wood drying prediction models like [3] [4] [5] [6], simplified physical characteristics are used to simulate the coupled heat and mass transfer during wood drying. However, too many inputs, e.g., gravity, external pressure, capillarity, temperature gradient, water concentration, etc., used to build theoretical models has a deterioration in accuracy to solve the highly coupled equations. Mathematical method such as artificial neural networks (ANN) and support vector machines (SVM) have been used to build drying model due to their ability to capture the trend of moisture content [7] [11]. Prediction models based on Auto Regressive Moving Average model with exogenous input (ARMAX) analysis have been widely shown in literature [12] [13] [14] [15] because of the computational efficiency. To the best of authors' knowledge, ARMAX model applied in the field of wood drying have rarely been seen in the previous research. In this paper, the method of ARMAX is presented to describe the time-varying system of wood drying. Recursive Least Squares (RLS) is introduced to identify the coefficients of the ARMAX model. To improve the identifying accuracy, an optimization algorithm is also discussed. Wood sample used in this study for the drying experiment was Northeast China ash. The dying kiln and its inside structure are show in Figure 1. The temperature, EMC, and moisture content used in the simulation model were the average value collected by temperature sensors, EMC sensors, and moisture content sensors.

Mathematics Model Description
The model takes temperature and EMC as inputs, moisture content as output.
ARMAX model of wood drying process is described as ing process model are Substituting polynomials (2), (3), (4), (5) and the initial values into (1), gives Therefore, the output ( ) y t can be expressed as where

Parameter Identification
Using the Least Squares identification principle to define the quadratic crite- Using recursive method, matrix 1 ( ) P t − is define as The RLS estimation of the parameter vector is The estimated residual is

RLS estimation of parameter vector θ is achieved
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

Computation of the Model Parameters Based on RLS
Simulation results were based on the 1000 input-output experimental data acquired during wood drying. RLS algorithm in Equations (14) and (15)

Computation of the Model Parameters Based on FF-RLS
Parameters variation trend with λ = 0.95 is shown in Figure 5. It is obviously observed that the FF-RLS (Forgetting Factor Recursive Least Squares) algorithm has a faster convergence speed than RLS algorithm. From t = 200, curves of estimate parameters tend to be stable. The comparison of actual values and predicted values is shown in Figure 6. The absolute error between predicted value and actual values is shown in Figure 7. MSE is 0.3630, and RMSE is 0.7111. Convergence speed of FF-RLS algorithm is faster, however the absolute error is greater than RLS algorithm.

Conclusion
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.