A simplified winding unwinding process is illustrated in Figure 2. To simplify analysis, the following assumptions are made [14,15]: 1) the thickness of the web is very small compared with the radius of winding/unwinding rollers; 2) the strain in the web is uniform within the web span; 3) the driving motors of winding and unwinding rolls have identical specifications; 4) no slip occurs between the web and the rollers; 5) neglecting dynamics of the load cell and idler rolls. The dynamics of the winding system can be derived as

where t_w = total web tension, N;

L = web length between winding and unwinding rolls, m;

B_f = coefficient of bearing viscous friction, Nm·s/rad;

K_m = toque constant of the motors, N·m/A;

t_w0 = wound-out tension of the unwinding roll, N;

v_u, v_w = tangential velocities of the unwinding and winding rolls, m/s;

i_u, i_w = input current to the driving motors of the unwinding and winding rolls, A;

R_u, R_w = radii of unwinding and winding rolls, m;

J_u, J_w = moments of inertia of the unwinding and winding rolls, kg·m²;

a, E = cross section area (m²) and Young’s modulus (GPa) of the web material.

The wound-out tension is an initial static tension within the web roll, which is generated by the previous winding and is assumed to be zero in this work for the sake of simplicity.

With the web material transmitted from the unwinding roll to the winding roll, the roll radius and inertia vary. If the winding and unwinding rolls have the same roll cores, the variations of the radius and inertia can be approximately described as

where

= initial radii of unwinding/winding rolls, m;

R_c, J_c = radius (m) and moment of inertia (kg·m²) of unwining winding roll core;

ρ, w, h = density (kg/m³), width (m), and thickness (m) of the web material, respectively.

Taking derivative of Equations (4)-(7) yields

Substituting Equations (8)-(11) into (2) and (3) yields

Since the web thickness h is much less than its width (e.g. the paper thickness is 0.0762 mm in this work), the last terms in Equations (12) and (13) can be neglected. The real-time model becomes S

The state-space representation of the nominal winding process plant will be

where the subscript p represents plant, n stands for nominal, and

Matrix C_n is defined such that the outputs are the web tension and its line speed.

The practical connection of the closed-loop system with two weighting functions is illustrated in Figure 3(a), where P(s) is the interconnection system in the standard linear fractional transformation (LFT) framework as shown in Figure 3(b). The standard LFT framework in Figure 3(b) compactly describes the closed-loop system, whose exogenous input and output are w and, respectively.

The weighting function W_e(s) aims to limit the magnitude of the output sensitivity function. With W_e(s), the H_∞ norm of W_eS_o will be minimized by H_∞ synthesis. In general, its desired value is limited to unity [16], that is, or. W_e(s) is selected with a high gain at low frequency to reject low frequency perturbations

where M is the peak magnitude of S₀,; is the allowed steady-state error; and ω_b is the required minimum frequency bandwidth.

The control signal weighting function W_u(s) is selected to shape the frequency property of control signals

where M_u is the maximum gain of KS₀, that is, ; ω_u is the bandwidth of the controller K; and ε_u is a real value to adjust the pole location of W_u.

The selection of weighting function and their numerical realization are based on the following considerations: 1) to achieve the desired control performance; 2) to op-

timize the control effort and avoid actuator saturation; 3) to obtain the best robustness property to the closed-loop system; and 4) to get the optimal balance among different robustness properties of the closed-loop system. By simulation, the frequency weighting functions W_e(s) and W_u(s) are determined as

H_∞ synthesis is an optimization algorithm that aims to design an H_∞ controller to achieve the desired robustness of the closed-loop system. Consider a nominal system represented by a lower LFT framework in Figure 4. The closed-loop transfer matrix can be represented as . The H_∞ norm of the complex transfer matrix T_zw is defined as

where is the maximum singular value for a specific frequency ω, and represents the collection of real numbers. The commonly adopted suboptimal H_∞ control is described in [16]: Given γ > 0, find all admissible controllers if there are any, such that.

According to the selected weighting function in (18), the conditions and assumption on a standard H_∞ problem (as in [17]) are not satisfied. To tackle this problem, in this work, the initial controllers are designed based on the basic linear and time invariant model (15). One of the suboptimal H_∞ controllers is derived for the considered general H_∞ problem by solving two algebraic Riccati equations [16,18]. It can be represented by the following transfer function:

where

Control system design for web handling processes is conducted based on the basic linear and time invariant model, in which all parameters take their nominal values at the specified operating point. However, the time varying parameters (e.g., roll radius and inertia) will influence the control performance and robustness property of the corresponding closed-loop system. Within the design region where both winding and unwinding rolls are around dimensionally half web loaded, the control performance is satisfactory. However, at starting and ending stages, tension output becomes rather sensitive to line speed variations. It is generally assumed in the literature that the impact from time varying parameters is not significant since the variation is small, especially when the web roll is small [13]. In this work, a neural fuzzy (NF) approximator is developed to adaptively estimate the gain values to improve tension control performance.

In this case, five input variables are used in the developed NF approximator: the radius of winding roll, radius of the unwinding roll, tension, web speed, and armature current signals for both the winding and unwinding motors. Three membership functions (MFs), small, medium, and large, are assigned to each input variable. The reasoning processing is performed in the following form:

where are MFs; m is the number of rules.

Figure 5 schematically shows the network architecture of the developed NF approximator. Unless specified, all

the network links have unity weights. The input nodes in layer 1 transmit the monitoring indices to the next layer, successively, where n = 5 in this case. Each node in layer 2 acts as an MF, which can be either a single node that performs a simple activation function or multilayer nodes that perform a complex function. The nodes in layer 3 perform the fuzzy T-norm operations. If a max-product operator is used, the firing strength of rule will be

where denote MF grades.

After normalization in layer 4, defuzzification is performed in layer 5. The predicted gain grades to each motor will be:

where and are the number of rules associated with the decisions of and, respectively.

Correspondingly, the compensated input signals to the drive motors will be

where and are the input current signals to the unwinding motor and winding motor, respectively.

Once the NF approximator is established, the related parameters should be optimized properly in order to achieve the desired input-output mapping. In training nonlinear system parameters, the classical method is the use of gradient algorithms [9]. The classical gradient algorithm, however, is slow in convergence especially when the approximation error becomes smaller [1]. To tackle this problem, an efficient fast gradient method, recently proposed in our research group [18], will be used in this work to optimize the nonlinear parameters of the NF approximator.

A hybrid training strategy is employed in this case to train the NF approximator. Each training epoch consists of two runs: in the backward pass, the nonlinear parameters of the NF approximator are updated by the fast gradient method [18]. In the backward pass, the linear parameters of the NF approximator, , are fine-tuned using a recurrent LSE method [1].

The experimental setup used in this work is shown in Figure 6, which has been developed by the authors’ research team. It consists of four DC motors and four idler rolls. Each motor can be controlled separately to simulate different drive/load conditions. Eight encoders (2048- lines) are used to measure shaft angular velocity. Tension can be measured by a tension transducer or an alternative spring-link-idler system. In this work, only two DC motors are set up as winding and unwinding rolls, whereas

no control actions are provided in the intermediate zone.

The experiments are taken in two parts: tests in the middle stage of a winding process and tests in the starting stage of a winding process. In modeling, the radii of unwinding and wingding rolls can be approximately calculated by

where and are the initial radii of unwinding and winding rolls, respectively; and are the integrated angular displacements of the rolls; and h is the web thickness. Angular displacements are measured by two encoders mounted on the driving shafts of unwinding and winding rolls.

In this test, both the winding roll and unwinding roll are operating around dimensionally half web wounded. The performance of the developed adaptive H_∞ control is compared with a classical linear quadratic regulator (LQR) control. From our previous investigation [4], the LQR has been shown to outperform other related classical control modes (e.g., PI and PID) in web system control.

Figure 7 shows the test results of web tension control and line speed control by the developed adaptive H_∞ control using the NF approximator and without using the NF approximator (i.e., the gains are estimated with the classical method as suggested in [13]). It is seen that the NF approximator can effectively predict the gain values and process the tension and speed control operations.

Figure 8 shows the corresponding results using the

LQR control with and without using the NF approximator. Comparing the corresponding results in Figure 7, the adaptive H_∞ control outperforms the LQR in both force control and speed control, in terms of overshooting/undershooting and settling time. The proposed gain scheduling scheme can improve the control performance: specifically, 1) suppress the tension fluctuation due to line speed variations; 2) reduce the settling time for both tension and speed responses; and 3) the decrease overshot of speed response. The effectiveness of the classical gain scheduling method relies on the accuracy of the mathematical models, and the robustness to attenuate disturbances in web handling operations.

In this test, the winding process is running from the starting point, at which the winding roll is web unloaded and the unwinding roll is fully web loaded. Figure 9 illustrates the test results by the adaptive H_∞ control using the NF approximator and the classical gain scheduling methods, respectively. It is seen that the NF approximator can accommodate for more system uncertainty in gain scheduling, and provide more accurate control performance.

Figure 10 shows the control results using LQR control corresponding to different gain compensation strategies. Comparing Figures 8 and 9, it is clear the adaptive H_∞ control is superior to the classical LQR control.