Effects of non-ideal current sensing on subharmonic oscillation boundary in DC-DC switching converters under CMC

A Time-Domain Asymptotic Approach to Predict Saddle-Node and Period Doubling Bifurcations in Pulse Width Modulated Piecewise Linear Systems

“…In [6] the author applied the approach based on the Fourier series expansion to a system with A 2 A 1 only by making some approximations leading finally to A 1 = A 2 . Another different type of approximation leading to the same consequence A 1 = A 2 has been used and justified in [9] for this kind of systems.…”

“…Most of the PWL systems studied in the literature are characterized by switching among linear subsystems when certain time-varying and T − periodic boundaries in the state space are reached. This is the case of Pulse Width Modulation (PWM) systems like switching dc-dc power converters [3], [4], [5], [6], [9], [10], dc-ac inverters [11], temperature control systems [12], switched capacitor networks and chaos generators [13] and hydraulic and fluid valve drivers [14], [15]. In steady-state, during a switching period of length T , a trajectory of these systems starts at time instant nT and is described by the vector field f 1 (x) = A 1 x + B 1 u, intersects a switching boundary described by the equation σ(x(t), t) := Fx(t) − r(t) = 0 at switching instant t s = DT , and then goes to another linear system described by the vector field f 2 (x) = A 2 x + B 2 u, where r is a time-varying T −periodic external signal, x ∈ R n is the vector of the state variables, n is the order of the system A i ∈ R n×n and B i ∈ R n×m , i = 1, 2 are the system state matrices for phase i (i = 1, 2) and u ∈ R m is the vector of the system inputs in both the plant and controller, m being the number of the external inputs to the system which are supposed to be constant within a switching cycle.…”