Stability, Observability and Control Analysis of DC-DC Converters Using Lie Algebra and Linear Algebra

Abstract

This thesis focusses on analysis of control and stability aspects of different topologies of DC-DC converters, such as buck converter, buck-boost converter, boost converter and flyback converter. Stability analysis of DC-DC converters is necessary to design efficient controller which can have a wide range of operating region. It is crucial to check the controllability of all the system states for effective design and control of DC-DC converters because in many industrial applications the control input of dynamical systems is found to influence only a part of the system states. The thesis work includes comparison of frequency response analysis of buck converter linearized by exact feedback linearization (EFL) and state space averaging (SSA) methods, stability analysis of buck converter using Lie Algebra, controllability analysis of flyback converter using controllability transition method, local accessibility and small time local controllability analysis of buck, boost and buck-boost converters, study of nonlinear observability rank condition (NORC) based on Lie Algebra of buck and boost converters considering both equilibrium and non-equilibrium states. It is observed from the analysis of the buck converter model linearized using EFL and SSA methods operating in both continuous conduction mode (CCM) and discontinuous conduction mode (DCM) that the phase margin of the EFL linearized model is same as that of the nonlinear model for different values of duty cycle and output resistance. On the other hand, these frequency response parameters obtained from SSA linearized model do not match with that of the nonlinear model. Lie algebraic stability analysis indicates that a nonideal buck converter is stable when the MOSFET on-state resistance and diode forward resistance are equal. The converter is found to be unstable when such resistances are unequal. On the contrary, root locus plots reveal that the buck converter is stable even when these resistances are not equal. Thus, Lie algebraic stability analysis is identified to be more conservative method for stability analysis of non-ideal buck converter and importantly, such analysis is viable for applications where asymptotic stability should be assured. It is found from small time local controllability (STLC) analysis using Sussman’s theorem that the variation in duty cycle and input voltage does not affect the controllability of DC-DC converters (buck, boost and buck-boost). The findings of Lie-algebraic analysis also show that the variation in load resistance does not influence the controllability status of the systems considered. The controllability transition based analysis of DC-DC flyback converter reveals that such converter is controllable for load resistances below a critical value for a specific input voltage. Thus, a concept regarding the range of load resistances necessary to preserve system controllability is obtained and such load resistances can be considered for designing controllers. The decrease in reciprocal condition number of switching controllability gramian below controllability transition () value indicates loss of controllability, even though the system is proved to be controllable by linear subspaces method. Thus, ill-conditioned switching controllability gramian of a switched linear system specifies uncontrollability. This finding assures that controllability transition method can distinguish between the controllable and uncontrollable status of the system. NORC based observability assessment is more accurate than Kalman rank criterion of observability (KRCO) method as it considers both equilibrium and non-equilibrium states of nonlinear system. It is observed from this study that buck converter is observable for non equilibrium states and unobservable for equilibrium states. Also, boost converter in non equilibrium states is observable for specific range of duty cycle (~from 0.2 to 0.9) and becomes unobservable for duty cycle equal to 1.0. The boost converter is found to be unobservable in equilibrium states. The compendium of the study presented in the thesis will be beneficial for estimation and improvement of the performance of DC-DC converters.