Velocity-Sensorless Decentralized Tension Control for Roll-to-Roll Printing Machines

This paper presents a velocity-sensorless decentralized tension control scheme without true machine parameter dependence for roll-to-roll printing machines. Both the nonlinear nature and parameter uncertainties are taken into account the development task. As the first merit, the proposed observer eliminates the requirement of velocity feedback without any plant information even the nominal parameters. The introduction of the active-damping with the estimated state reduces the closed-loop system order to 1 by the pole-zero cancellation using the specified feedback gains. Finally, the disturbance observer (DOB) replaces the regulation error integrators with the closed-loop robustness improvement against the high-frequency disturbances. The MATLAB/Simulink-based simulations validate the effectiveness of the proposed decentralized scheme.


I. INTRODUCTION
Roll-to-roll (R2R) systems have been widely adopted for industrial manufacturing processes (e.g., in printing, drying, and coating). The unwinder and rewinder and nips provide the rolling actions to transport the web from the unwinder to the rewinder through nips to manufacture the various products, including films, metals, and textiles. There are two important control issues regulating the web velocity and tensions for each stage. The control law design task is not trivial and challenging because of the inherent dynamic nonlinearity and parameter changes according to the unexpected sudden fault and unwinder and rewinder inertia variations [1]- [4].
The two control strategies employed are called decentralized and centralized methods. For large-scale processes, preference has been to adopt the decentralized method, which regulates each agent using local measurement feedback. The interactions between local agents must be carefully considered to prevent performance degradation and instability. Traditionally, the decentralized scheme was implemented using the proportional-integral-derivative (PID) The associate editor coordinating the review of this manuscript and approving it for publication was Yonghao Gui . regulator with local measurement of web velocity and tension errors [5]. The PID gains were tuned by trial-and-error (Ziegler-Nicholas, Cohen-Coon, etc.) and Bode and Nyquist techniques using the linearized machine dynamics for a given operating condition. The resultant closed-loop performance could be limited using the fixed tuned gains resulting from changes in the operating condition [1], [5]- [7]. The web transport velocity causes dramatic winder inertia variations, resulting in performance degradation and instability. Online PID gain tuning techniques were presented using neural networks and fuzzy logic to optimize the cost function, thereby achieving significantly improved robustness despite the winder inertia variations [7]- [10]. A tension estimator based on the nonlinear observer design technique was proposed to lower implementation cost [11], [12]. The novel observer design techniques as in [13], [14] can be used as a solution to the web-velocity estimation problem by using only the lower and upper bounds of machine parameters. In [15], the recent nonlinear decentralized scheme using the output regulation method successfully showed the closed-loop robustness improvement against disturbance caused by the interactions between agents. The resultant controller forms the state-feedback and feed-forward compensation term.
Its target state and input trajectories solve the nonlinear differential-algebraic equations derived from R2R nonlinear dynamics via the proposed iteration technique. Experimental data were also included to verify the practical advantages from the analysis results [15]. The optimal state-feedback technique was used to optimize the cost index by solving an optimization problem under constraints expressed as the system-parameter-dependent linear matrix inequality online [4]. The machine dynamics are decoupled into linear and nonlinear parts. The introduced feed-forward compensation terms canceled out the disturbances from nonlinear dynamics and state-feedback, with the optimal gain robustly stabilizing the whole system to the desired equilibrium point. The issue of parameter dependence can be addressed by adopting the recent real-time parameter estimation mechanism as in [16]- [20]. Sliding-mode and adaptive controls alleviated the parameter dependence level with the inclusion of dynamic compensators dominating or canceling out the disturbances in the feed-forward loop [21]- [23].
The centralized scheme can be applied for small-and medium-scale processes with a high computational performance controller. The availability of all agent measurements makes it possible to apply novel nonlinear multivariable approaches. In the controller design task, a simple additional compensator relying on other local agent measurements was included in the feed-forward loop to ensure closed-loop stability. Studies have also proposed optimal state-feedback [24], [25] and neural-network-based schemes [26] involving parameter dependence. The desired tension and web transport velocity references are automatically obtained to meet the predetermined web longitude regulation performance based on the full-state feedback differential flatness method [27].
The previous results including the centralized and decentralized scheme suffered from parameter dependence of control and observers, at least partially, which is the problem to be solved in this study. Herein, an advanced velocity-sensorless nonlinear tension control law under a decentralized structure without the requirement of full-state feedback and true system parameter values is presented. The advantages of the proposed decentralized controller are given as follows: (a) Parameter-independent velocity derivative observers are used to eliminate the requirement of velocity feedback for tension control. (b) A proportional-derivative (PD) tension control law includes observer-based active-damping for closed-loop system order reduction through a combination of specified PD gain and pole-zero cancellation. (c) A disturbance observer (DOB) is incorporated as a replacement for regulation error integral actions in the PD-type tension and proportional (P) web velocity controller to enhance the closed-loop robustness and to ensure the two beneficial properties of performance recovery and steady-state error rejection. MATLAB/Simulink software is used to emulate the closed-loop system using the S-function in the C language to demonstrate the effectiveness of the proposed controller.
A summary of the rest of the article is given as follows: Section II briefly describes the R2R system configuration and its nonlinear dynamics. The state estimation and control algorithms are presented in Section III. Section IV provides proofs of the beneficial properties by analyzing the closed-loop dynamics. The simulation results and discussion are included in Section V. Finally, Section VI concludes this article.  This study considers the special case of N = 2 for the simplicity, which has one nip, unwinder, and rewinder. In this case, there are the three linear velocities for the unwinder, the nip, and the rewinder denoted as v 0 , v 1 , and v 2 , whose dynamics are given by [23]:

II. R2R MACHINE DYNAMICS
with coefficients of b i , i = 0, 1, 2 (friction, in kg m s/rad), ρ (web density, in kg/m 3 ), W (web width, in meters), t w (web thickness, in meters), r 1 (nip radius, in meters), and J 1 (nip inertia, in kg m 2 ). The span tensions satisfy the relationshipṡ ∀t ≥ 0, with G and T 0 being the wound-up tension of unwinding roll (in newtons) and the Young's modulus multiplied by the web cross-sectional area (in newtons), respectively. The unwinder and rewinder radii are governed byṙ The unwinder and rewinder inertias J 0 and J 2 (in kg m 2 ) vary according to the following rules: with the unwinder and rewinder bare inertias of J o0 and J o2 . This study focuses on devising the tension and web velocity control scheme for the 2-span machine for simplicity since the result for the 2-span system can be trivially expanded to this general case. See [27] for the N -span system dynamics.

III. CONTROL ALGORITHM
The first-order system in the form of low-pass filter (LPF) is adopted for the target velocity and tension performances; that isv * ∀t ≥ 0, with the constant references v 2,ref and T i,ref and cut-off frequencies ω v 2 and ω T i , i = 1, 2. This section gives the decentralized control algorithm shown in Fig. 2 to ensure the control objective: exponentially; this is referred to as performance recovery in the rest of this article. See Section IV for the closed-loop analysis. First, slight modifications of velocity and tension dynamics are made with nominal parameters (·) o . For velocity dynamics,v with r o 0 and r o 2 being the initial radii of the unwinder and the rewinder, respectively.
with lumped disturbancesd T 1 andd T 2 . The second-order tension dynamics are obtained using the modified dynamics of (11)- (15) as

A. WEB VELOCITY CONTROL ALGORITHM
To eliminate the tension feedback, rewrite the rewinder velocity dynamics of (13) aṡ with the newly defined disturbance 2 +d v 2 , whose stabilization can be obtained by the proposed control law with the DOḂ   (16) requires feeding back the tension derivatives ofṪ i that have to be estimated from the differential operation causing the high-frequency disturbance magnification. The proposed observer is given bẏ i = 1, 2, ∀t ≥ 0, with T v i :=Ṫ i and its estimateT v i , which does not involve in any plant parameters, not even nominal values. This corresponds to a main merit of this study. Section IV analyzes the state estimation error convergence.

2) CONTROLLER
The proposed tension control law for stabilizing the second-order open-loop system (16) is given by leading to the closed-loop system order reduction by stable pole-zero cancellation. See Section IV for details. The DOB updates the disturbance estimated T i aṡ Fig. 4 shows the tension loop driven by the proposed control law. For a large-scale system, the proposed controller can be generalized using the N -span system dynamics as follows: 1) Web Velocity Control with DOB:

2) Tension Controls
with observers:ż and DOBs: ♦ Proof: The output (20) giveṡ with d v :=ḋ v,AC , which is used for following analysis. Theorem 1 proves the performance recovery described in (10) on the basis of result of Lemma 1. VOLUME 8, 2020   (20) guarantees the inputto-state stability (ISS) and the inequality: with | d v | ≤ d v and x v 2 (0) being the initial value of x v 2 := ṽ * 2 e d v T . ♦ Proof: It follows from the combination of (17) and (18) thatv 2 = ω v 2ṽ 2 + e d v , which leads to (using (9)): Now consider the positive-definite function: Then, the closed-loop trajectories (27) and (29) give the time derivative of V v aṡ where λ min (Q v 2 ) denotes the minimum eigenvalue of positive-definite matrix Q v 2 := 2ω v 2 2 0 l dv . This indicates the ISS with respect to d v and the result (28) by the Comparison principle in [28].
From the result (28), it can be roughly concluded that exponentially, for sufficiently large value l d v > 0. The absence of regulation error integral actions in the proposed control law (18) leads to steady-state error issues in actual implementations, which can be cleared using the first-order disturbance estimation dynamics proved in Lemma 1. See Theorem 2 for details. Theorem 2: The web velocity loop driven by the control law (18) and DOB (19) and (20) The full-rankness of matrix A ss,v implies that x ss,v = 0, from which it can be concluded that v 2 (∞) = v 2,ref (∞).

B. TENSION LOOP
This subsection begins with an analysis of the state and disturbance estimate behavior in Lemmas 2 and 3 for a simplification of the performance recovery proof.
Lemma 2: The state estimateT v i from the observer (21) and (22) satisfieṡ ♦ Proof: The observer output (22) giveṡ with the observer dynamics (21) and definition T v i :=Ṫ i , which completes the proof.
The state variable T v i can be decomposed as with T v i :=Ṫ v i ,AC , which is used for following analysis. Lemma 3: The disturbance estimated T i from the DOB (24) and (25) satisfieṡ ♦ Proof: The output (25) giveṡ VOLUME 8, 2020 with the DOB dynamics (24), open-loop dynamics (16), and error dynamics (32), which completes the proof.
The disturbance d T i can be decomposed as d T i = d T i ,DC + d T i ,AC with d T i ,DC and d T i ,AC being its steady-state and transient components. Then, the result (33) implies thaṫ with d T i :=ḋ T i ,AC , which is used for following analysis.
A main contribution of this study is the closed-loop system order reduction by the pole-zero cancellation from the specified feedback gain. Lemma 4 handles this issue.
Lemma 4: The tension loop driven by the proposed control law (23) renders the tension response to bė with the LPFṡ for some c i > 0, i = 1, 2, 3. ♦ Proof: The combination of (16) and tension control law (23) gives and its Laplace transform is given by This can be simplified by pole-zero cancellation with the factorization of (s 2 . This completes the proof by the inverse Laplace transform. Now, Theorem 3 asserts the performance recovery described in (10) by deriving the whole control system dynamics on the basis of results of Lemmas 2-4.
Theorem 3: The tension loop driven by the proposed control law (23), observer (21) and (22), and DOB (24) and (25) guarantees the ISS and the inequality: , and x T i (0) being the initial value of ♦ Proof: It follows from the combination of (9) and (35) Now consider the positive-definite function: Then, the closed-loop trajectories (32), (34), (36), (37), (40) give the time derivative of V T i aṡ where λ min (Q T i ) denotes the minimum eigenvalue of positive-definite matrix This indicates the ISS with respect to d T i and T v i and the result (39) by the Comparison principle in [28]. From the result (39), it can be roughly concluded that exponentially, for sufficiently large values l d T i > 0 and l T v i > 0. The absence of regulation error integral actions in the proposed control law (23) leads to steady-state error issues in actual implementations, which can be cleared using the first-order state and disturbance estimation dynamics proved in Lemmas 2 and 3. For details, see Theorem 4. Theorem 4: The tension loop driven by the proposed control law (23), observer (21) and (22), and DOB (24) and (25) The full-rankness of matrix A ss,T implies that x ss,T = 0, from which it can be concluded that

V. SIMULATIONS
The nonlinear dynamics (1)-(6) with the real machine coefficients used [27] emulated the R2R machine behaviors using Simulink with the ODE45 solver. The following nominal parameters were used for controller implementation to realize the parameter variations: The active-damping injection-based integral back-stepping (IBS) controller was adopted for a comparison study in which the additional active-damping design parameters, i.e., b d,x > 0, x = v 2 , v 1 , v 0 , T 2 , and T 1 , were included to enhance the robustness against the model-plant mismatches: 1) For the velocity loop, 2) For the span-1 tension loop, 3) For the span-2 tension-loop, with the well-tuned velocity-loop cut-off frequencies and b d,T 2 = b d,T 1 = 10 for the best performance. The web velocity and tension cutoff frequency values ω v 2 and ω T i , i = 1, 2, were left to be identical to those of the proposed controller. Note that when using the active-damping injection-based IBS controller, the closed-loop system is governed by the desired LPF dynamics (9) based on pole-zero cancellation. Therefore, the two controllers (i.e., proposed and active-damping injection-based IBS controllers) share the same control objective.

A. TENSION TRACKING MODE
A pulse tension reference was applied from 50 to 100 N for three different web velocity operation modes of 0.5, 1, and 3 m/s. Figs. 6 and 7 show the comparison results, revealing a slight interference reduction from the proposed controller. A significant improvement in web velocity regulation performance was observed in Fig. 8 as a result of the improved high-frequency disturbance rejection performance of the DOB embedded in the proposed web velocity controller. As shown in Fig. 9, the tension loop observers successfully estimated the actual states, maintaining the estimation errors sufficiently small and near zero. Figs. 10 and 11 present the DOB responses that continuously estimated the undesirable disturbances to maintain the stability of the closed-loop system with the desired closed-loop performance.  The tension references were fixed to 50 N, and three difference pulse web velocity references were applied from 3 to 3.3, 3.5, and 4 m/s. The closed-loop tension responses    shown in Figs. 12 and 13 exhibit considerable over-and undershoot reduction and closed-loop stabilization from the proposed controller. However, the unwinder and rewinder inertia variations according to (7) and (8)      a fixed tension reference of 50 N. The unwinding roll wound-up tension T 0 was suddenly increased from its initial value of 70 to 170 N, and it was then restored to its initial value. As shown in Figs. 17 and 18, the proposed controller effectively suppresses the tension over-and undershoots caused by the sudden wound-up tension changes, compared with the IBS controller. The combination of novel features, observer-based active damping, and DOBs led to this practical advantage.

VI. CONCLUSIONS
In this study, an advanced observer-based decentralized tension control technique was proposed with reduced parameter and sensor dependence. There were two main innovations: (a) the first-order velocity observers without requirement of any plant information, not even nominal parameter values and (b) the combination of estimated damping injection, pole-zero cancellation technique, and DOBs offering useful closed-loop properties and performance recovery while being offset-free. The convincing numerical data from various simulations validated the practical merits of the proposed technique. This result can be applied to multi-machine synchronization applications with a systematic controller tuning technique and experimental data.