Time‐ and State‐Dependent Input Delay‐Compensated Bang‐Bang Control of a Screw Extruder for 3D Printing

Abstract

Summary In this paper, a delay‐compensated bang‐bang control design methodology for the control of the nozzle output flow rate of screw extruder‐based three‐dimensional printing processes is developed. A geometrical decomposition of the screw extruder in a partially and a fully filled regions allows to describe the material convection in the extruder chamber by a one‐dimensional hyperbolic partial differential equation (PDE) coupled with an ordinary differential equation. After solving the hyperbolic PDE by the method of characteristics, the coupled PDE–ordinary differential equation's system is transformed into a nonlinear state‐dependent input delay system. The aforementioned delay system is extended to the non‐isothermal case with the consideration of periodic fluctuations acting on the material's convection speed, which represent the process variabilities due to temperature changes in the extruder chamber, resulting to a nonlinear system with an input delay that simultaneously depends on the state and the time variable. Global exponential stability of the nonlinear delay‐free plant is established under a piecewise exponential feedback controller that is designed. By combining the nominal, piecewise exponential feedback controller with nonlinear predictor feedback, the compensation of the time‐dependent and state‐dependent input delay of the extruder model is achieved. Global asymptotic stability of the closed‐loop system under the bang‐bang predictor feedback control law is established when certain conditions related to the extruder design and the material properties, as well as to the magnitude and frequency of the materials transport speed variations, are satisfied. Simulations results are presented to illustrate the effectiveness of the proposed control design. Copyright © 2017 John Wiley & Sons, Ltd.

Control of a 2 × 2 coupled linear hyperbolic system sandwiched between 2 ODEs

“…Control of PDE‐ODE systems. Stabilization of PDE‐ODE coupled structures has drawn much attention in the last decade when consider the compensation of infinite‐dimensional states in the actuating or sensing paths of ODEs, such as time delay, vibration string, and diffusion phenomenon . Many results about control of wave PDE‐ODE, heat PDE‐ODE, and transport PDE‐ODE have been obtained so far.…”

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“…One important aspect to be considered for improving the performance of the closed-loop system is the delay of extrusion start and stop, referred as the dwelltime in [73]. This issue is tackled in [74], [75] by modeling the extrusion-on demand process via a 1D hyperbolic PDE coupled with an ordinary differential equation. Specifically, denoting by L the length of the extruder and by x(t) the moving boundary boundary delimiting the part of the extruder that is filled of material, the mass balance equation of the filled zone yields for an incompressible homogeneous material with constant density ρ 0 yields the following 1D hyperbolic PDE:…”

“…where B represents a screw geometric parameter while K d stands for the nozzle conductance. The control design of this model was investigated in [74], [75] and consists in a statedependent input delay-compensated bang-bang controller.…”

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“…The discipline of control systems engineering has a long history of dealing with time delays caused by actuation and sensing mechanisms. Throughout the past two decades, PDE backstepping has proven to be an essential tool in the design of predictor feedback controllers for linear and nonlinear finitedimensional systems with known (constant, time-dependent, or state-dependent) [1], [2], [3] and unknown time delays [4], [5]. Recently, the approach has been extended to reaction diffusion PDEs with unknown and arbitrarily large input delays [6], [7].…”