Proportional-integral controller modified landweber iterative method for image reconstruction in electrical capacitance tomography

While Landweber iteration is a popular image reconstruction algorithm for electrical capacitance tomography (ECT), it suffers from semi-convergence. In process control, PID controllers are commonly used, where P stands for proportional, I stands for integral, and D stands for derivative. Proper setting of the PID parameters can reduce the response time and enhance the accuracy and stability of control systems. In this paper, an extended PI controller is integrated with the Landweber method to relieve its semi-convergence, which is named the Landweber-PI method. To stabilize the iteration process, constraints are imposed on the integral term of the PI controller. In comparison with the conventional Landweber method and its modified variants, simulation was conducted to verify the effectiveness of the proposed method by imaging typical distributions using ECT sensors with different number of electrodes, 8 or 12 or 16 electrodes, and capacitance data without and with noise. The simulation results were validated by static experiment, showing that the proposed method converges within 50 iterations for the specified core, annular, stratified, and other two complex distributions and remains stable after 1000 iterations, while the other methods converge slowly and become divergent eventually. Regarding the reconstruction accuracy, the derived void fractions by the proposed method have smaller absolute deviations than the other methods in most cases, with a minimum of 0.3% and a maximum of 4.3%. The gas-solid flows in a real fluidized bed were imaged, with more image details reconstructed after five iterations. The improved performance of the Landweber-PI method was justified by singular value decomposition analysis.