Abstract— This paper addresses the problem of compensating for the in-domain attenuation of inputs for a class of process control applications. Specific examples include the challenge of maintaining uniform temperatures or through-cure during thick- film radiative drying and curing processes in the face of the effective input’s variation with film depth due to the so–called Beer-Lambert effect. These distributed parameter control problems are modeled with parabolic PDEs for the diffusion processes along with an in-domain input with a spatial attenuation function. The approach presented in this paper involves transforming the original model to an equivalent boundary input problem to which existing output feedback backstepping boundary control design methods can be applied. The resulting compensation scheme includes a mechanism for tuning the closed-loop performance. The performance of this scheme is compared with a controller designed via modal approximation of the PDE. Keywords: distributed parameter control, in-domain actuation, boundary control, compensation of Beer-Lambert effect, curing process, stereolithography I. INTRODUCTION The choice of control design methods for distributed parameter systems modeled by partial differential equations (PDEs) often starts with categorizing whether the control inputs are applied at the boundary (boundary control) or distributed throughout or at a few locations in the spatial domain (in-domain control). These classes of problems have been studied extensively, and many PDE control approaches have been proposed and demonstrated [1-4]. A particular class of PDE control problems that has received limited attention is one where a single actuator, often physically located outside the domain, has a distributed in-domain input to the physical process under consideration. Examples can be found in various industrial applications that use radiant energy sources. These processes, include letter pressing, production of holograms, microelectronics and integrated circuits, curing of dental fillings and rapid prototyping processes such as stereolithography [5]. Additional to this list are ultraviolet (UV) curing and infrared (IR) drying processes in automotive and aerospace paint/coating applications [6-7]. In all of these processes, the single actuating input is often a lamp, laser or LED energy * Research supported by NSF CAREER Grant CMMI-1055254. A. Yebi is with the Clemson University International Center for Automotive Research, Greenville, SC 29607 (email: ayebi@clemson.edu). B. Ayalew is with the Clemson University International Center for Automotive Research, 4 Research Dr, 342 CGEC, Greenville, SC 29607 (Corresponding author: phone 864-283-7228, fax 864-283-7208, email: beshah@clemson.edu) source physically located outside the target process, and transferring energy to the target process by radiation as an in- domain input. However, this transfer is subject to the attenuation of the input intensity and thereby the reduction of its effectiveness farther in the domain of the target. The so- called Beer-Lambert law is often used to explain this attenuation of intensity with depth. Practical challenges related to this Beer-Lambert effect (also referred to as photo-absorption) are worth highlighting. The main one is a restriction it imposes on the depth of effective processing achievable using open-loop methods. In UV curing of thick-film coatings, the Beer-Lambert effect could lead to poor performance of coatings cured with open- loop methods. Typically, over-dosing, i.e., more energy than necessary, is applied to counteract the effect of depth attenuation. In stereolithography curing applications, the depth of cure limitation necessitates layered or multi-pass approaches (Figure 1). Current solutions involve offline open-loop optimization runs to determine a critical depth of cure for each curing pass [8]. If one can find compensation algorithms for the Beer-Lambert effect, it may be possible to change the stepped curing approach and speed up production (with less passes needed), improve product quality and reduce processing energy needs. Figure1. Potential Benefit of Compensating for the Beer-Lambert Effect While there appears to be no prior work on direct feedback compensation algorithms for the Beer-Lambert effect from control or actuation point of view, there have been some practical solutions proposed for the related observation/measurement problem. One is found in the field of bioengineering, where a depth compensation algorithm (DCA) is applied to compensate the decay of light propagation in a tissue so as to accurately localize absorbers in deep tissue by using depth sensitive Diffuse Optical Tomography (DOT) [9]. DCA is based on inversion to create a balancing weight matrix to compensate measurement sensitivity with depth. In combustion engine research [10], an iterative compensation algorithm is developed to compensate laser attenuation in optically dense fuel sprays (fuel image obtained by planner laser imaging). In this case, the Feedback Compensation of the In-Domain Attenuation of Inputs in Diffusion Processes Adamu Yebi and Beshah Ayalew, Member, IEEE Optimized stepped curing UV Light UV Light Z Feedback controlled one-pass curing 2013 American Control Conference (ACC) Washington, DC, USA, June 17-19, 2013 978-1-4799-0178-4/$31.00 ©2013 AACC 2092