Reducing domain structural complexity in PDE backstepping boundary observer design using conformal mapping

In this paper we study backstepping boundary observer design for a class of distributed parameter systems defined in a cylindrical domain with prescribed boundary conditions. We show that applying standard PDE backstepping boundary observer design results in a hyperbolic PDE with four independent variables which is difficult to solve both numerically and analytically. By introducing a modified domain structure using conformal mapping, we obtain a simplified design process which is reduced to solving a two-dimensional PDE for the kernel function. Finally, we sketch the complete treatment to solve the partial differential equation describing the kernel equation and then perform a simulation study to verify the L² performance of the designed observer. This technique has a direct application in soft-sensor design for internal temperature measurement in systems subjected to dynamic heat distribution, particularly microfluidic Lab-on-a-Chip (LOC) devices.