Dynamic Observer Method Based on Modified Green’s Functions for Robust and More Stable Inverse Algorithms

This work presents a modified procedure to use the concept of dynamic observers based on Green’s functions to solve inverse problems. The original method can be divided in two distinct steps: i) obtaining a transfer function model GH and; ii) obtaining heat transfer functions GQ and GN and building an identification algorithm. The transfer function model, GH, is obtained from the equivalent dynamic systems theory using Green’s functions. The modification presented here proposes two different improvements in the original technique: i) A different method of obtaining the transfer function model, GH, using analytical functions instead of numerical procedures, and ii) Definition of a new concept of GH to allow the use of more than one response temperature. Obtaining the heat transfer functions represents an important role in the observer method and is crucial to allow the technique to be directly applied to two or three-dimensional heat conduction problems. The idea of defining the new GH function is to improve the robustness and stability of the algorithm. A new dynamic equivalent system for the thermal model is then defined in order to allow the use of two or more temperature measurements. Heat transfer function, GH can be obtained numerically or analytically using Green’s function method. The great advantage of deriving GH analytically is to simplify the procedure and minimize the estimative errors. INTRODUCTION The inverse problem can be found in a large area of science and engineering and can be applied in different ways. The great advantage of this technique is the ability of obtaining the solution of a physical problem that can not be solved directly. Different techniques can be found in literature in the solution of inverse heat conduction problem (IHCP). For instance, the mollification method [1], the conjugate gradient technique [2], the sequential function specification [3] or the use of optimization techniques such as genetic algorithms [4], simulated annealing [5] or golden section method [6]. Technique based on filter such as the Kalmn filter [7] or dynamic observers [8,9] have also been employed for the solution of the IHCP. In the dynamic observer technique, the IHCP solution algorithms are interpreted as filters passing low-frequency components of the true boundary heat flux signal while rejecting high-frequency components in order to avoid excessive amplification of measurement noise [8]. The dynamic observers technique proposed by Blum and Marquardt [8], focused on the one-dimensional linear case, was extended to solve an inverse multidimensional heat conduction problem by Sousa et al. [9]. In order to deal with multidimensional thermal models, Sousa proposed an alternative way of obtaining the heat transfer function, GH, by using the Green function concept instead of taking the Laplace transform of the spatially discretized system. This procedure allows flexibility and efficiency to solve multidimensional inverse problems. Despite the flexibility and efficiency of this procedure the way proposed in ref. [9] is complex and requires some ability to obtain the heat transfer function. In order to simplify the procedure, minimize the noise effect in GH, and give more robustness and stability to the inverse algorithms some modifications are proposed here.