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A compound control strategy is investigated for Medium Density Fiberboard (MDF) continuous hot pressing electrohydraulic servo system (EHSS) with uncertainties and input saturation. Firstly, a hyperbolic tangent function is applied to approximate saturation nonlinearity in the system. And thus the mathematical model is continuous and differentiable. Subsequently, the slab thickness tracking controller is constructed by using a dynamic surface control (DSC) method, which introduces first-order low-pass filters to calculate derivatives of virtual control input in each step. Compared with the conventional backstepping controller, complexity of the design procedure is alleviated obviously. Moreover, a composite disturbance of uncertainties and input saturation is estimated by a nonlinear disturbance observer for compensation of the control law. Finally, an appropriate Lyapunov function is chosen to prove that all signals of the closed-loop system are semiglobally uniformly ultimately bounded and the tracking error converges to zero asymptotically. Numerical simulation results are also exhibited to authenticate and validate the benefits of the proposed control scheme.

With the increasing contradiction between supply and demand of timber resource, Medium Density Fiberboard (MDF) plays a significant role in the wood based panel market, owing to its favorable physical characteristics and excellent mechanical properties [

Unfortunately, both uncertainties and input saturation, which are inherent in the MDF continuous hot pressing EHSS, have a great influence on the control performance, degrading the precision of slab thickness. On one hand, uncertainties lie in system parameter perturbations and external disturbances. Due to the existence of temperature and pressure, some system parameters such as oil bulk modulus

For the uncertainties in the EHSS, sliding mode control (SMC) attracted considerable attention [

Nevertheless, the aforementioned literatures ignored the effect of input saturation in the EHSS. To handle the saturation constraints, a number of approaches have been presented and employed into different fields so far. In [

It is supposed to point out that tackling both uncertainties and saturation for the EHSS is more challenge than taking one single problem into consideration with some specified control approach. There are still some common problems as follows: (1) When the SMC or backstepping method is used to design the controller, they need to calculate the high-order derivatives of reference signals and virtual control in each subsystems which may aggravate the calculation burden. It is unconducive to implement in practice. (2) As mentioned above, some researchers introduced an auxiliary system [

Note that the main task is simplifying the design procedure from two aspects. One is alleviating the calculation burden. In particular, the existing backstepping technique in the EHSS gives rise to “explosion of differential terms” obviously, due to its repeated derivatives of virtual control inputs and signal references. The complexity grows drastically as the order increases. Therefore, a dynamic surface control (DSC) strategy was presented by Swaroop et al. in [

The other is reducing the complexity of controller as possible. Although control algorithm with adaptive law, artificial-intelligence based method, and auxiliary systems might get perfect robustness and antisaturation properties, they also introduced quantities of extra parameters, such as adaptive law parameters and NN weights [

Except for the problem of uncertainties and input saturation, there are also some other factors which will have an impact on the MDF slab thickness, like unreliable communication links [

Inspired by the work mentioned above, a compound control strategy is investigated for the MDF continuous hot pressing electrohydraulic servo system (EHSS) with uncertainties and input saturation. The controller chooses the DSC approach to realize the tracking control and the NDO to compensate the composite disturbance term, consisting of uncertainties and saturation nonlinearities. The control motivation is used to cancel the influence which aroused the composite disturbance and guarantees a precise thickness control performance with lower calculation burden.

The contribution can be summarized as follows:

Due to the existence of input saturation in the system, the MDF slab thickness tracking control performance is impacted. A hyperbolic tangent function is introduced into the MDF continuous hot pressing EHSS system mathematic model to approximate the saturation nonlinearities. And thus a continuous differentiable model is obtained to ensure that the DSC approach can be applied.

With the help of two first-order low-pass filters, the derivatives of virtual control inputs are obtained. Compared with conventional backstepping controller, the complexity of design in the DSC method is alleviated. Meanwhile, the burdensome calculation is avoided which is conducive to the engineering implement.

We develop a NDO to estimate the composite disturbance term of uncertainties and saturation nonlinearity. It effectively compensates for the designed control law and guarantees the MDF slab thickness precision.

The organization of the rest paper is as follows. The description of hot pressing process and the EHSS model is given in Section

Hot pressing is an essential process of the flat pressing way to produce MDF. Namely, high temperature and high pressure are supplied to MDF slab after preloading process. The chemical components of fiber are degraded in the presence of high temperature. Then, the fiber activity is excited. At the same time, bonding force is formed among the fiber. The aim of high pressure is to suppress the rebounded force inside and discharge the steam. Hence, the fiber can interweave tightly [

Particularly, gauge thickness is the key to obtain the high-quality MDF. Slab thickness is determined in a fixing-thickness phase, which is mainly about the final MDF figuration. With temperature and pressure in a reasonable range, discharging the left steam in the slab is the main task to avoid the defects such as surface bubbling and lamination. Nevertheless, due to the existence of steam pressure, thickness deviation is prone to appear. In the thicker area, steam cannot be discharged under the same pressure. Therefore, precise gauge control of MDF slab thickness is particularly significant at the fixing-thickness stage to cancel the deviation immediately.

Configuration of the continuous hot pressing machine is depicted in Figure

Configuration of continuous hot pressing machine. 1: upper steel conveyor, 2: upper chain blanket, 3: rank, 4: hydraulic cylinder, 5: upper platen, 6: electric motor, 7: driven rolling wheel, 8: lower steel conveyor, 9: lower chain blanket, and 10: lower platen.

At present, MDF hot pressing control system mainly adopts electrohydraulic servo system (EHSS), shown in Figure

Configuration of the EHSS with a single cylinder. 1: piston of hydraulic cylinder, 2: upper platen, 3: MDF slab, 4: servo valve, 5: servo amplifier, and 6: signal comparator.

In practice, as the current limitation of servo valve, the control input is constrained. In the other words, the servo valve is on longer work than usual, when the control input increases, so called input saturation. Owing to the restricted control input, the desired position reference signal will not be tracked accurately, decreasing the precision of the MDF slab thickness.

Three basic equations of a four-way valve controlled hydraulic cylinder power mechanism are given as follows, where the detailed deduction refers to [

The servo amplifier is equivalent to the proportion, and the link between input current of servo valve and displacement of valve spool is also proportional; hence they can be described as follows:

State equation of EHSS is yielded from (

As mentioned above, the uncertainties and input saturation are considered in this paper. Due to the existence of temperature and pressure, some model parameters vary as time goes by, such as

It is noticed that there exists a sharp corner at point

Inspired by [

For example, choose control input

Approximation result of the smooth function.

Thus, the third equation of system model (

It is convenient for deducing the control law to define a function as follows [

Finally, the EHSS mathematic model with uncertainties and saturation is transformed into

We may have found that the composite disturbance term is made up of parameter uncertainties, external disturbance, and saturation nonlinearities. By virtue of a NDO, the term can be estimated and fed back to the control law.

The compound controller is consisting of two parts, each with its own objective. The first part is to establish a tracking controller with the DSC approach. By two first-order low-pass filters, the derivatives of virtual control inputs are obtained, avoiding the “explosion of differential terms” phenomenon in the existing backstepping. The other is to suppress the influence of uncertainties and saturation in the EHSS. NDO is introduced into the control law to estimate and compensate the composite disturbance term defined above. The configuration is shown in Figure

Configuration of the compound controller.

Similar to the backstepping technique, the design with the DSC approach is divided into three steps. In each step, a virtual input is designed for the next subsystem. However, the virtual control input is sent to first-order low-pass filters, getting a new variable and its derivative. Then, the new variable is applied to the tracking error of the next subsystem. Finally, the control law of the system is obtained in the third step. Before the procedure, tracking errors of three steps are defined as

Differentiating

The virtual control input

By a first-order low-pass filter,

Differentiating

The virtual control input

Similarly, a new variable

Differentiating

In the existing backstepping technique, the “explosion of differential term” appears in the controller design procedure owing to the calculation of the derivatives of (

Unfortunately, the assumption above is impossible because term

The basic idea of disturbance observer based control problem can be divided into two parts. Obviously, the first part is accomplished, stabilizing the system without consideration of the unknown disturbance term

The derivative of the composite disturbance term

Then, NDO is designed as follows:

The EHSS system model parameters

Ultimately, the control law (

From the control law (

In this section, the closed-loop system stability of the continuous hot pressing EHSS is given via Lyapunov theory. Due to the introduction of two first-order low-pass filters and NDO, a great challenge is imposed on the stability analysis.

Define the NDO estimation error:

Differentiating

Consider the continuous hot pressing EHSS model (

Taking the integral of (

Taking (

Considering (

Therefore, Theorem

From (

Firstly, let us do some definitions and calculations before the stability proof.

The filtering error of two first-order low-pass filters

Substituting (

Similarly,

Substituting (

Differentiating

The derivative of

Define sets as follows:

On set

In terms of the continuous hot pressing EHSS model (

A set of Lyapunov candidate are chosen as

Differentiating

Differentiating

Differentiating

Thus, from (

The parameters are designed as follows:

Substituting (

Considering

Satisfying

According to the theorem in [

Taking the integral of (

Based on (

By adjusting the DSC design parameters

For the MDF continuous hot pressing EHSS, simulation study is to validate the effectiveness of proposed strategy based on Matlab2014a/Simulink. According to the practical continuous hot pressing process in the fixing-thickness phase, with temperature and pressure in a reasonable range, the nominal parameters are given as follows:

^{−1}) = 0.01.

^{−1}) = 0.0125.

^{−3}) = 850.

^{−16} m^{5}·N^{−1}·s^{−1}) = 5.

^{6} N·s·m^{−1}) = 2.25.

^{−1}) = 2.4.

Next, the assumptions in the numerical simulation are given as follows:

The control input

The parameter perturbations are

The external load force is

The initial states are

The composite controller parameters are selected as follows:

The DSC tracking controller parameters are chosen as

The NDO design parameter is chosen as

In this section, two kinds of position tracking reference signal are applied to the continuous hot pressing EHSS which are a sinusoidal signal and a constant signal. Due to the fact that the sinusoidal signal is fluctutated as time goes by, it can testify the antisaturation ability of the proposed controller effectively. On the other hand, a constant signal is used to simulate the MDF slab thickness error properly in practice.

To illustrate the antisaturation ability and the robustness of the proposed method, the sinusoidal signal is set as

Position tracking and tracking error curve. (1 represents the desired reference signal; 2 represents sliding mode controller; 3 represents traditional DSC controller; and 4 represents the proposed strategy.)

The NDO estimation curve. (1 represents the composite disturbance term; 2 represents the estimation result of the NDO; and 3 represents the estimation error.)

Control input curve. (1 represents the traditional DSC controller and 2 represents the proposed strategy.)

Virtual control input curve and output curve of first-order low-pass filter. (1 represents the designed virtual control input and 2 represents the output of first-order low-pass filters.)

Clearly, the MDF continuous hot pressing EHSS with the composite controller can track the reference signal accurately. Compared with the DSC controller, the tracking error decreases into a small neighborhood of origin in less than 0.1 s (Figure

The estimation result of the NDO is displayed in Figure

In Figure

In the above case, we have demonstrated the robustness and resistance of the input saturation in the proposed method in detail. Taking the practical continuous hot pressing process into account, the constant reference signal should be observed to simulate the thickness error.

Assuming that initial position of hydraulic cylinder is

Control input curve in the condition of the different MDF slab thickness error.

Obviously, the existence of uncertainties and input saturation in the MDF continuous hot pressing has an undesirable impact on the tracking precision and convergence time. The overshoot phenomenon emerges in the position tracking curve from Figure

Position tracking and tracking error curve. (1 represents the desired constant signal; 2 represents the DSC controller with no input saturation and uncertainties; 3 represents the DSC in presence of uncertainties and input saturation; and 4 represents the developed compound controller.)

The control input is shown in Figure

Control input curve. (1 represents control input without limitation; 2 represents DSC controller input without NDO; and 3 represents the proposed method.)

In this paper, a compound control strategy is investigated for Medium Density Fiberboard (MDF) continuous hot pressing electrohydraulic servo system (EHSS) with uncertainties and input saturation. By dynamic surface control (DSC) approach, a tracking controller is constructed with less burdensome calculation burden and more flexible framework. Introducing a nonlinear disturbance observer (NDO), the control performance is improved, enhancing the robustness and suppressing the saturation nonlinearities. Simultaneously, fewer design parameters are required using the compound controller. Simulation results show that the proposed method may accomplish the tracking control in presence of uncertainties and input saturation successfully. The control system possesses advantages such as shorter convergence time, high control precision, and smooth control input to ensure the MDF obtained gauge thickness.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by 948 Project (2014-4-46), the National Natural Science Foundation of China (31370710 and 31370565), Fundamental Research Funds for the Central Universities (DL12EB04-02), and Postdoctoral Research Fund of Heilongjiang Province (LBH-Q13007).