STATE ESTIMATION IN SYSTEMS MODELLED BY ORDINARY DIFFERENTIAL EQUATIONS AND PARTIAL DIFFERENTIAL EQUATIONS

Abstract

This thesis documents our investigation of state estimation in systems modelled by ordinary differential equations (ODE’s) and partial differential equations (PDE’s). The complete study can be classified as an investigation of two related problems. The first problem is to obtain the precise mathematical model that presents accurately the dynamics of the nonlinear system. Generally, a system is described using a set of differential equations. By solving these equations, we obtain the input-output relation of a system. But, nonlinear system presents difficulties in obtaining the input-output relation. Forthisnonlinearrepresentation,threedifferentapproacheshavebeenused inthisthesis. Theyare:-(i)Approximateinput-outputrelationusingperturbationtechnique. (ii) Implementation of Kronecker product to represent dynamics of a nonlinear system. (iii) Kronecker based wavelet representation of a system. After obtaining the input-outputrelationofanonlinearsystem, wehavetoimplementtheestimationalgorithm. So, the second problem is to choose the state estimation method that provides real-time estimation, simplicity, less computational complexity and compression. For state estimation, we implemented following two methods:- (i) Kalman filter (KF) and ExtendedKalmanfilter(EKF).(ii)Blockprocessingestimationbasedonwavelettransform. At first, we used perturbation theory to derive the closed form nonlinear relation of following circuits:(i) Bipolar junction transistor (BJT) based cross coupled oscillator circuit. (ii) BJT differential amplifier (DA) circuit. (iii) Metal oxide semiconductor field effect transistor (MOSFET) circuit. Inthese,wecomputedthedistortionoccurringduetotheuseoflinearpartonly,which show the importance of nonlinear expression. The second problem deals with the estimation of following two circuits:(i) BJT based DA. xi (ii) MOSFET circuit. The EKF method has been used for estimation purpose. The simulation results have been compared with the recursive least squares (RLS) method for different noisy signals. The root mean square error (RMSE) computation presents the superiority of the EKF method as compared to the RLS method. Asathirdproblem, weestimatedtheoutputvoltageofMOSFETusingEKFbymodeling the nonlinear system dynamics using Kronecker product. The proposed method has been compared with Kronecker based wavelet transform (WT) representation of thesystemandestimationusingleastmeansquare(LMS).Theproposedmethodhas the following advantages:- (i) It can be used for any mode of transistor operation. (ii) It can also be used for large amplitude input signal. (iii) The method presents the realtimeestimation. (iv)UseofKroneckerproductpresentsmoreaccuraterepresentation of the system. As a fourth problem, we estimated the electric and magnetic fields of Hertzian antenna using KF and compared the results with RLS estimator. The proposed method uses the Kronecker product for compact representation of discretized field in the form of vectors and partial differential operators in the form of matrices. Simulation results show that the KF presents better results than the RLS.