Original article
Predicting the onset of bifurcation and stability study of a hybrid current controller for a boost converter

https://doi.org/10.1016/j.matcom.2013.03.009Get rights and content

Abstract

The proposed method develops a continuous-time averaging model of a nonlinear current controller for a continuous conduction mode (CCM) boost converter in the lack of a closed outer voltage loop. The controller associates peak-current and sliding mode controls. The advantages of the proposed approach are the capability to show the onset of fast-scale bifurcation, and the reduction of numerical calculation effort. To confirm the validity of the proposed method, simulation and experimental results are detailed.

Introduction

DC–DC converter is a piecewise-switched circuit, which might exhibit various types of nonlinear phenomena [25], [33]. Generally, the conventional peak-current-mode control, which is well-known and widely used, has various advantages like robustness dynamic responses, rejection of input voltage disturbances, and flexibility in improving small signal dynamics. However, this control technique has two disadvantages which are static error and instabilities with certain parameters. To eliminate the mentioned drawbacks, we can use slope compensation technique to increase the stability domain of the system and add integral term to cancel the static error. Hybrid current controller is a controller composed with two or more current regulators, for example, peak-current regulator and sliding mode controller [15], [18], [28]. The sliding mode controller is a nonlinear controller used to control variable structure systems. Its advantages include guaranteed stability, robustness against parameters variations such as line and/or load uncertainties, and ease to implement compared with other types of nonlinear controller.

Bifurcation and chaos are important problems when designing DC–DC converters [13], [16], [24]. They concern not only its functionality but also reliability and safety. Bifurcation is used to show a qualitative change in the properties of a system as a function of one or more parameters called bifurcation parameters like input voltage, current reference, capacitance, inductance, switching frequency, feedback gain, etc. [1], [19], [22], [31]. In this paper we consider slope compensation as the bifurcation parameter. When a system bifurcates, it means that it loses stability at a given operating point. In the literatures, many types of controller and mode of control such as voltage-mode and current-mode are investigated. Sometimes, for the sake of simplicity, the current-mode without outer loop is presented; with such controller, the converter system can exhibit bifurcation and chaos [2], [8], [32].

Various converter models are available for studying bifurcation and chaos. But three types of model of DC–DC converters are commonly used in literatures: the first is switching model [5], [23], the second is continuous-time averaging or averaged model [3], [14] and the last one is discrete-time iterative mapping model or discrete-time model [4], [9], [10], [30]. Based on the iterative map and circuit simulations, one can generate bifurcation diagrams [8]. In systems that involve two different frequencies, like DC–DC converter supplied by rectifier, there is a high switching frequency fs and another low ripple frequency fl. Since the first-order Poincaré map of such systems is time varying, the second-order Poincaré map, which allows transforming the cycle stability problem to a fixed point stability problem, may be used. In fact, the second-order Poincaré map is depending on the ratio fl/fs, which result in three different cases [20]. Therefore, developing the second-order Poincaré map is complicated.

Types of bifurcations commonly found in power electronics are Hopf (slow-scale) [14], [17], [34], [35] and periodic-doubling (fast-scale) or flip [6], [7]. Slow-scale bifurcation can be regarded as a kind of low frequency instability caused by the voltage feedback loop. The fast-scale bifurcation is generated by inner current loop instability. However, for DC–DC converters using voltage and current loops, the fast and slow scale can occur simultaneously [20]. Moreover, other types of bifurcations observed in DC–DC converter are presented in [11].

In literatures, it is shown that some instability cannot be detected with averaged model [21]. Precisely, in [20] the authors mentioned that fast-scale bifurcation cannot be predicted by using the averaged model of the presented converter. In engineering design, a stable period-1 operation is the only acceptable one. Thereby, instability often refers to failure of the circuit in maintaining its operation in the expected stable period-1 regime. Hence, the onset of bifurcation should be known. Therefore, in this paper, we propose a method to develop a continuous-time averaging model of a boost converter with nonlinear current controller which can be used to estimate the bifurcation starting point and prove the stability of the system. Generally, the discrete-time model gives more precise results than the averaged model. But in some systems, discrete-time model cannot be developed because there is only one implicit solution of the differential equation of the system in each functioning sequence [26], [27].

Section snippets

Continuous conduction mode boost converter with hybrid current controller

In this section, three approaches to analyze the dynamic behaviors of the boost converter system and its stability are presented. For the first one, a bifurcation diagram is created by using Matlab/Simulink simulation software with the switched model. With this diagram, the precise onset of the bifurcation can be evaluated. By using discrete-time and averaged model, we will obtain the onset of the bifurcation point by evaluating the eigenvalues or Floquet multipliers of the system. Assume that

Experimental results

A boost converter (12 V/50 V, 10 A, 10 kHz switching frequency, output capacitor C1 = 0.05F) has been used to verify the proposed methods. Fig. 9 shows the different part of the experimental test bench. The parameters of the system are the ones given in Table 1.

The experimental results confirmed that the current static error is canceled as shown in Fig. 10. Also, as predicted by the bifurcation diagram, with mc = 8800 A s−1 the system exhibits a stable one-periodic behavior. Furthermore if mc is reduced

Two dimensions bifurcation study

In this section, the final investigation is performed with gain Ki and compensating slope mc serving as bifurcation parameters. We will focus on the effects of varying these parameters on the bifurcation behavior of the system. The circuit parameters used in the simulations are the ones of Table 1.

Based on the two models (averaged and discrete-time models), we can obtain the boundary of stability as a function of both parameters (Fig. 12). For a stable system, when mc is kept constant, further

Conclusion

In this paper, a method to estimate the onset of flip bifurcation of CCM boost converter with a nonlinear current controller is proposed. The nonlinear controller combines peak-current with compensating slope and sliding mode controller. In fact, that onset can be found by using averaged model, iterative map, or bifurcation diagram. The first two methods are based on the calculation of Floquet multipliers while the last method uses simulations of the converter. Indeed, using iterative map is

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