Application Of Symbolic Dynamics To Study Of The Stability In DC-DC Switching Converters

Switching power supply is the heart and power of modern electronic appliance and equipment.The common power supply,such as the raw electrical energy transmitted directly from mains supply or batteries,in a sense,is a poor quality of the coarse electricity.To satisfy with the power requirements of many kinds of instruments and electronic equipment,the coarse electricity must be converted to the precise electricity.As a result,switching power supply is power supply device,which converts the coarse electricity to the stability and precise direct-current(DC)voltage.Furthermore,the DC-DC switching converter is the core technology of the switching power supply,an important branch of power electronics,and a basic component of many other types of switching converters.Thus,it will be a direct function of guidance for the other topological structure to study the DC-DC switching converter.The prime source of nonlinearity is the switching element present in all power electronic circuits.Nonlinear component(e.g.,the power diodes)and control methods(e.g.,pulse-width modulation)are further sources of nonlinearity.Therefore,it is hardly surprising that feed-back-controlled power converters routinely exhibit various types of nonlinear phenomena.The nonlinear phenomena of interest include bifurcations(sudden changes in operating mode),coexisting attractors(alternative stable operating modes),and chaos(apparently random behavior).If reliable power converters are to be designed,an appreciation that these possibilities exist is vital,together with a knowledge of how to investigate them,use them,or avoid them.In recent decades,because the theory of nonlinear dynamics constitutes a vast body of knowledge,the related results provide new concepts for the analysis and research of such complex systems as DC-DC switching converters.Symbolic dynamics is perhaps the simplest dynamics one can ever imagine.It is a high degree of summary and abstraction of the actualdynamics system.In this paper,symbolic dynamics is applied to studying the stability in DC-DC switching converters.The nonlinear behavior of dynamical systems is characterized by simple symbolic sequences.The mechanisms that give rise to chaotic dynamics,chaos phenomena and control of chaos in DC-DC switching converters is studied.The main fruits are included:(1)Based on the * composition law,it takes a current-mode controlled of boost converter operating in continuous conduction mode(CCM)as an example to explain the self-similarity of the bifurcation diagram in the process of the DC-DC converter from the period-doubling bifurcation to chaos.In addition,the composite structure of orbits manifests itself in the fine structure of the power spectrum of the data,which is useful in telling the whereabout of periodic orbits in experiments with chaotic systems.Hence,the * composition law is specially suitable for analyzing the fine structure of the power spectrum of the composite sequences.The self-similarity and fine structure plays an important significance in the global distribution of the periodic solution in the DC-DC converters.(2)Based on the generalized composition rule,the existence of the tangent bifurcation and intermittent chaos in one-and two-dimensional boost converters is analyzing respectively.The power of symbolic sequence shows the periodic intervals occurring in the intermittent chaos.According to symbolic dynamics,the main result it is proved by the generalized composition rule,which says that if there is a periodic point with period 3,then for each integer n=1,2,3,…,there is a periodic point with period n.The proof process is simple and easy to understand.(3)Based on the invariant manifolds theory,we analyze the finely detailed structure of flows of the buck-boost converter and it is of great significance to understand deeply the characteristics of the solution of the system.Based on the Smale-Birkhoff homoclinic theorem,the structure of invariant manifolds of a fixed point indicates the presence of Smale horseshoe.The horseshoe map of the system is topologically conjugate with the shift map of the symbolic dynamics.The system can be identified with the Smale horseshoe by means of its symbolic descriptions.Furthermore,since the Smale horseshoe map implies sensitivedependence on the initial condition,homoclinic intersection of stable and unstable manifolds may be applied to illustrate the existence of chaos.(4)We will detail the application of the small parameter perturbations(OGY method)to achieve the control of chaotic dynamics in boost converter.Based on the chaotic orbit ergodicit and sensitive dependence on initial conditions of chaotic system,control is achieved through small perturbations of an accessible parameter.Thus,OGY method can be successfully applied in controlling the occurrence of chaotic behavior.According to the control strategies of OGY method,the unstable period-1 orbit embedded in the chaotic attractor of boost converter is stabilized.Furthermore,as we chose the feedback gain matrix outside the triangle,the chaotic boost converter is stabilized in different periodic orbits.As a result,the target unstable multi-periodic orbits can be obtained.