Stability Analysis of the Continuous-Conduction-Mode Buck Converter Via Filippov's Method

Abstract

Abstract-To study the stability of a nominal cyclic steady state in power electronic converters, it is necessary to obtain a linearization around the periodic orbit. In many past studies, this was achieved by explicitly deriving the Poincaré map that describes the evolution of the state from one clock instant to the next and then locally linearizing the map at the fixed point. However, in many converters, the map cannot be derived in closed form, and therefore this approach cannot directly be applied. Alternatively, the orbital stability can be worked out by studying the evolution of perturbations about a nominal periodic orbit, and some studies along this line have also been reported. In this paper, we show that Filippov's method-which has commonly been applied to mechanical switching systems-can be used fruitfully in power electronic circuits to achieve the same end by describing the behavior of the system during the switchings. By combining this and the Floquet theory, it is possible to describe the stability of power electronic converters. We demonstrate the method using the example of a voltage-mode-controlled buck converter operating in continuous conduction mode. We find that the stability of a converter is strongly dependent upon the so-called saltation matrix-the state transition matrix relating the state just after the switching to that just before. We show that the Filippov approach, especially the structure of the saltation matrix, offers some additional insights on issues related to the stability of the orbit, like the recent observation that coupling with spurious signals coming from the environment causes intermittent subharmonic windows. Based on this approach, we also propose a new controller that can significantly extend the parameter range for nominal period-1 operation.

Analysis of Bifurcation Behavior of a Piecewise Linear Vibrator with Electromagnetic Coupling for Energy Harvesting Applications

“…The eigenvalues of this matrix are called Floquet multipliers. For piecewise linear systems, as is the case for the system considered in this study, the monodromy matrix can be constructed from the product of the state transition matrices corresponding to each sub-cycle and the corresponding saltation matrix [Leine and Nijemeijer, 2004;Aizerman and Gantmakher, 1958;Fillipov, 1988;Giaouris et al, 2008]. Suppose a trajectory x(t) starts at time instant t i and is passing from the Configuration C i described by the vector field A i x+B i (t) := f i (x, t), intersects the switching boundary described by the equation σ i (x, t) = 0 at t i , and goes to Configuration C i+1 given by the vector field A i+1 x + B i+1 (t) := f i+1 (x, t).…”

“…Suppose a trajectory x(t) starts at time instant t i and is passing from the Configuration C i described by the vector field A i x+B i (t) := f i (x, t), intersects the switching boundary described by the equation σ i (x, t) = 0 at t i , and goes to Configuration C i+1 given by the vector field A i+1 x + B i+1 (t) := f i+1 (x, t). It has been shown, using the Fillipov method ( [Fillipov, 1988;Leine and Nijemeijer, 2004;Giaouris et al, 2008]), that when there is a transversal intersection, the state transition matrix across the switching boundary, called also the saltation matrix S i , is given by…”

“…The first approach will be based on a Poincaré map function [El Aroudi et al, 2007]. The second approach is the Floquet theory ( [Aizerman and Gantmakher, 1958]) together with Fillipov methods for systems with discontinuous vector field [Fillipov, 1988], [Leine and Nijemeijer, 2004], [Giaouris et al, 2008]. The third approach is based on the FDM together with Floquet theory [Nayfeh et al, 2009], [Nayfeh and Balachandran, 1995].…”

Stability Analysis and Control of Nonlinear Behavior in V2Switching Buck Converter

“…Based on theory of monodromy matrix [24], [25], if the Floquet multipliers of the monodromy matrix are all within the unit circle, then the system is stable. If the maximum Floquet multiplier equals 1, then bifurcation occurs; otherwise, it is unstable.…”

Nonlinear dynamic analysis in the V²C-mode-controlled buck converter by improved mLCE

“…Thus it is essential to measure the undesired nonlinearity, in order to make sure the stable domain of the converter. Recently, a series of tools have become available for analyzing the dynamic behaviors in power electronics circuits, including Poincaré maps [3], the Filippov method [4], and the Lyapunov method [5], where, the maximal Lyapunov character exponent (mLCE) is the most extensive way to distinguish chaotic regimes from periodic regimes in a continuous system [6,7]. However, because of the non-differentiable point on the switching surface, the previous computation method of mLCE may cause error in the switching systems.…”