STATE AND PARAMETER ESTIMATION IN NONLINEAR LUMPED AND DISTRIBUTED PARAMETER CIRCUITS USING REAL-TIME STOCHASTIC FILTERING ALGORITHMS

Abstract

This thesis documents our investigation of state and parameter estimation of lumped and distributed parameter circuits using real-time stochastic filtering algorithms. We used Kalman filter (KF) and its variants for this purpose. The complete study consists of investigation of three related problems. The state and parameter estimation of nonlinear circuits require accurate mathematical modeling of the circuit. Therefore, the first problem is to drive the mathematical model of nonlinear circuits. In the case of nonlinear system, it is difficult to obtain a closed form input-output equation. In this case, we try to obtain an approximate nonlinear input-output relation. For this purpose, we used Volterra and perturbation theory. Besides these, we also used the bipolar junction transistor (BJT) models and transmission line models to obtain mathematical expressions that include nonlinearity. The second problem is to choose an appropriate estimation algorithm that involves less mathematical computation. For example, Particle filter (PF) can also be used and it may give better results than KF, but it requires additional computations for this purpose. The H-infinity based filtering has faster convergence than KF, but the computation complexity is higher than KF. In this work, the computational complexity of extended Kalman filter (EKF), iterated extended Kalman filter (IEKF) and unscented Kalman filter (UKF) has been compared for a typical circuit. The third problem is to choose some mathematical tools to reduce the mathematical complexity. We used Kronecker product for sparse matrix representation and compact representation. In the following, we present a chapter-wise summary of the thesis. Chapter 1 begins with a literature survey. Section 1.2 and 1.3 present the literature gap and objectives. The contextual review of state estimation is mentioned in section 1.4. Theory of KF, EKF, IEKF and UKF are presented in sections 1.4.1, 1.4.2, 1.4.3, and 1.4.4 respectively. Further, a brief theory of perturbation method, stochastic differential equations (SDE), Volterra series, least mean square (LMS) algorithm and viii recursive least squares (RLS) algorithm are presented in sections 1.5, 1.6, 1.7, 1.8 and 1.9 respectively. Section 1.10 presents the organization of the thesis. Chapter 2 deals with the implementation of perturbation theory along with Ebers Moll model of BJT to derive the linear and nonlinear closed form Volterra expression between input and output of silicon controlled rectifier (SCR) circuit. It also presents the computation of the distortion occurring due to linear part only. Chapter 3 includes state estimation of the higher order RC low pass filter (LPF) and RC high pass filter (HPF) circuit using EKF and UKF methods and compared the estimation performance with LMS algorithm. Chapter 4 deals with the state estimation of single-phase rectifier circuit using EKF, IEKF and UKF methods and compared the estimation performance with LMS algorithm. Chapter 5 presents the state estimation of BJT based common emitter (CE) and Darlington amplifier (DA) circuits using EKF, IEKF and UKF methods. In the first part, we estimated the output voltage of CE BJT circuit using IEKF and compared the performance of IEKF with EKF method. In the second part, we present the application of UKF for output voltage estimation of DA circuit. This work uses Kronecker product for vector multiplication. We compared the UKF estimation results with EKF and IEKF methods. In Chapter 6, we present the modeling and real-time state and parameter estimation of nonuniform transmission lines (NTL) of single-phase and three-phase transposed and untransposed circuits. For modeling purpose, we used transmission line model, Fourier series expansion and Kronecker product. In first problem, state-space model of the single-phase NTL circuit has been derived. As Telegrapher’s equations used for modeling the NTL are a function of space and time, the Fourier series expansion of the voltage and current have been used to obtain the time-dependent equations. The measurements have been obtained by solving the eigenvector problem. The frequency-domain analysis is used to obtain the state-space equations. For this, the four distributed parameters of the line are expanded in Fourier series. We compared the estimation performance of KF, EKF and UKF with RLS method. Secondly, we present KF based state estimation and EKF and UKF based parameter estimation for three-phase NTL. For this, state space model for three-phase transposed and untransposed NTL has been obtained. Clarke transformation matrix has been utilized for phase to sequence transformation which allows to represent the three-phase trans ix mission line (TL) into fully transposed TL. Measurement model for current and voltage vectors along the line are expressed in terms of Fourier series. Also, the frequency domain analysis is used to obtain the eigenvalue and eigenvector for measurement model. The voltage and current of NTL are expanded in Fourier series to obtain the sparse matrix formulation using Kronecker product. Kronecker product representation of discrete unitary trans-forms results in computer efficient implementation. This work implements the analysis of nonlinearity effect in transmission lines using perturbation theory. For this, the nonlinearity of the transmission line is included by perturbing the voltage and current of the line. Also, we compared the estimation performances with RLS method. Finally, some concluding remarks are presented in Chapter 7 and some future work direction is also presented.