Critical conditions for a class of switched linear systems based on harmonic balance: applications to DC-DC converters

Abstract

A general and exact critical condition of saddle-node bifurcation is derived in closed form for the buck converter. The critical condition is helpful for the converter designers to predict or prevent some jump instabilities or coexistence of multiple solutions associated with the saddle-node bifurcation. Some previously known critical conditions become special cases in this generalized framework. Given an arbitrary control scheme, a systematic procedure is proposed to derive the critical condition for that control scheme.

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

“…The 1 The author of [6] call this type of analysis a Harmonic Balance approach but one cannot see where this Harmonic Balance was performed. One can observe that equating the steady-state feedback signal Fx expressed in Fourier series to the external periodic signal at the switching instant cannot be called a Harmonic Balance as defined by experts in this field (See [7], [8] for example).…”

“…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.…”

“…After obtaining the Jacobian or monodromy matrix, critical boundary conditions for some singularities like saddlenode (SN) bifurcation or period-doubling (PD) can be obtained by imposing in the characteristic equation in the zdomain that one of the eigenvalues is equal to +1 or −1 respectively [4]. Another approach recently used in [6] for locating these boundaries which was wrongly called harmonic balance is by expanding the feedback signal into a [6]. In particular those that can be be formulated in the form of a linear system and a square-wave siganl generated by a comparator like the PWM process.…”

“…In particular those that can be be formulated in the form of a linear system and a square-wave siganl generated by a comparator like the PWM process. 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.…”

“…In [20], the Poisson sum formulae and some related Fourier series properties have been used to transform the condition for PD occurrence derived in [6] from the Fourier frequency-domain to a matrix-form state-space time-domain condition. For design purpose, these matrix-form expressions can be expressed in scalar form by using some practical assumptions [3].…”