applying resonant parametric perturbation to control chaos in the buck ...

International Journal of Bifurcation and Chaos, Vol. 13, No. 11 (2003) 3459–3471 c World Scientific Publishing Company

APPLYING RESONANT PARAMETRIC PERTURBATION TO CONTROL CHAOS IN THE BUCK DC/DC CONVERTER WITH PHASE SHIFT AND FREQUENCY MISMATCH CONSIDERATIONS YUFEI ZHOU∗ , CHI K. TSE† , SHUI-SHENG QIU‡ and FRANCIS C. M. LAU Department of Electronic and Information Engineering, Hong Kong Polytechnic University, Hong Kong ‡ South China University of Technology, Guangzhou, China †[email protected] Received August 13, 2002; Revised September 23, 2002 The buck converter has been known to exhibit chaotic behavior in a wide parameter range. In this paper, the resonant parametric perturbation method is applied to control chaos in a voltage-mode controlled buck converter. In particular, the effects of phase shift and frequency mismatch in the perturbing signal are studied. It is shown that the control power can be significantly reduced if the perturbation is applied with an appropriate phase shift. Moreover, when frequency mismatch is inevitable, intermittent chaos occurs, but effective control can still be accomplished at the expense of raising the control power. Analysis, simulations and experimental measurements are presented to provide theoretical and practical evidences for the proposed control method. Keywords: Chaos control; resonant parametric perturbation; phase shift; frequency mismatch; dc/dc converter.

1. Introduction Chaotic behavior has been identified in many engineering systems such as electronics, communication networks, power systems, mechanical structures, airborne systems, etc [Kim & Stringer, 1992]. Because of the unpredictable or sometimes undesirable consequences chaos causes in the systems, control of chaos has now become a topic of interest in the past. In particular, since conventional engineering designs always put “stability” and “reliability” as top priorities, especially for critical applications, much research in this topic has been directed to the suppression or prevention

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of chaotic operations [Kapitaniak, 1996; Ott et al., 1990; Ott & Spano, 1995]. Recently many chaos control methods have been proposed for various applications. Their objectives fall into two general categories. In the first category, one of the infinitely many unstable periodic orbits within a chaotic attractor is first identified as the control target, and control action is directed to stabilize the system so that it settles on the target periodic orbit. In the second category, a desired operating state is the control target, which is not necessarily one of those unstable orbits embedded in the chaotic attractor. Here,

Also with Anhui University, Hefei, China. 3459

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the control action is to achieve the desired operating state. The first type of control objective is usually achieved by feedback methods, whereas the second type can be accomplished by nonfeedback methods. Examples of feedback methods include the Ott–Grebogi–Yorke (OGY) method [Ott et al., 1990], occasional proportional feedback (OPF) method [Hunt, 1991], time-delayed feedback control method (TDFC) [Pyragas, 1992], etc. In these methods, system variables are measured, a control law implemented and some control parameters varied to achieve the required control objective. On the other hand, for the nonfeedback type of control, no system variables need to be measured and no specific periodic orbit has to be identified as the control target. Examples of nonfeedback methods include adaptive control [Huberman & Lumer, 1990], resonant parametric perturbation [Cicogna, 1990; Colet & Braiman, 1996; Fronzoni & Giocondo, 1991, 1998; Lima & Pettini, 1990; Mirus & Sprott, 1999], weak periodic perturbation [Braiman & Goldhirsch, 1991; Chac´on & D´ıaz Bejarano, 1993; Qu et al., 1995; Yang et al., 1996], entrainment and migration control [Jackson, 1990], etc. Compared to feedback methods, nonfeedback methods are simpler and equipped with the ability of anti-jamming, thus more suited for practical implementations. Our attention in this paper is focused on the nonfeedback type of control for controlling chaos. In particular, the resonant parametric perturbation method is considered. It has been shown that this technique is highly suitable for controlling chaos in periodically driven systems, despite the fact that it requires nonzero control power even when the system has been controlled to its steady state. In this paper we consider a voltage-mode controlled dc/dc buck converter and illustrate the application of resonant parametric perturbation for suppressing chaos. We further show that the control effort (perturbation power) can be reduced dramatically by introducing an appropriate phase shift. Moreover, when frequency mismatch exists in the perturbation, the system exhibits intermittent chaos and can only be stabilized by exerting a larger control effort. Analysis, simulations and experimental measurements are presented to provide theoretical and practical evidences for the proposed control method.

2. Review of Resonant Parametric Perturbation Resonant parametric perturbation can suppress chaos [Cicogna, 1990; Colet & Braiman, 1996]. In general, parametric perturbation can make a system chaotic, but applying it at appropriate frequencies and amplitudes can induce the system to stay in periodic regimes. Thus, the so-called resonant parametric perturbation is to perturb some parameters at appropriate frequencies and amplitudes, thereby converting a chaotic operation into a regular one. Usually we can choose a parameter that strongly affects the system and can be easily varied. Suppose this parameter is c. This parameter is then perturbed with the function (1 + α sin 2πf t), where α  1 and f is the perturbation frequency to be chosen. Effectively, we are replacing c by c(1 + α sin 2πf t) such that the largest Lyapunov exponent is reduced to below zero. This approach has been used by Lima and Pettini for stabilizing a chaotic Duffing–Holmes system [Lima & Pettini, 1990]. In particular, it has been shown that when the perturbation frequency f resonates with the periodic driving frequency, say f s , the largest Lyapunov exponent will approach zero from positive, and eventually chaos subsides and the periodic state emerges as the Lyapunov exponent falls further below 0. This phenomenon has been demonstrated analytically with Melnikov’s method, and verified by simulations and experiments [Cicogna, 1990; Lima & Pettini, 1990; Fronzoni & Giocondo, 1991]. Variations of the above procedure have been proposed. For instance, the weak periodic perturbation method [Braiman & Goldhirsch, 1991; Chac´on & D´ıaz Bejarano, 1993; Qu et al., 1995] makes use of external periodic forcing to weaken or control chaos. When the external forcing frequency resonates with an unstable periodic orbit of the original system, the resonating unstable periodic orbit will be inspirited while other unstable periodic orbits will be suppressed. Thus, the system settles to the resonating, now stable, periodic orbit. The resonant parametric perturbation method has been successfully applied to control chaos in a range of systems, and it has been demonstrated that resonant frequencies up to three times the driving frequency can be effectively applied to stabilize orbits of one period, half period and one-third period.

Applying Resonant Parametric Perturbation to Control Chaos in the Buck DC/DC Converter

3. Application of Resonant Parametric Perturbation to Controlling Chaos in Voltage-Mode Controlled DC/DC Buck Converters The buck converter has been shown to exhibit chaotic behavior when operated under a variety of conditions [Fossas & Olivar, 1996; Hamill et al., 1991; Tse, 1994; Tse & Di Bernardo, 2002]. In this section we review the basic operating principle of the buck converter under a common voltage-mode control, exemplify its bifurcation and chaotic behavior, and illustrate the direct application of resonant parametric perturbation for chaos suppression.

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3.1. Overview of circuit operation The buck converter consists of an inductor, a switch, a diode and a resistor load, which are connected as shown in Fig. 1(a). When switch G turns on, the inductor current ramps up almost linearly, and when switch G is turned off, the inductor current ramps down and de-energizes through the diode to the load. In the voltage-mode control scheme, the output voltage error with respect to the reference voltage is amplified to give a control voltage Vcon : Vcon (t) = A(vo − Vref )

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which is then compared with a ramp signal V ramp (t), defined as   t Vramp (t) = VL + (VU − VL ) mod 1 , (2) T where all symbols are explained in Fig. 1. The comparator output, u, gives the pulse-width-modulated signal necessary for driving the switch. Typically, the switch is turned on when Vcon (t) ≤ Vramp , and turned off when Vcon (t) > Vramp , as illustrated in Fig. 1(b). The state equation can be written as  Aon x + Bon E switch G on x˙ = (3) Aoff x + Boff E switch G off ]T ,

where x denotes the state variable, i.e. x = [v o iL the A’s and B’s are the system matrices given by   −1/RC 1/C Aon = Aoff = , −1/L 0 (4)     0 0 Bon = , and Boff = . 1/L 0

Q

M

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(b) Fig. 1. Voltage-mode controlled buck converter. Schematic diagram; (b) operation waveform.

(a)

Using the above equations, “exact” cycle-by-cycle simulation can thus be performed using the above equations. In this paper, we use SIMULINK to perform all simulations. The parameters are chosen as follows: E = 22 − 33 V,

L = 20 mH ,

C = 47 µF ,

R = 22 Ω ,

Vref = 11 V,

A = 8.4 ,

T = 400 µs ,

VL = 3.8 V,

VU = 8.2 V

3.2. Chaotic behavior The afore-described buck converter has been shown previously to exhibit period-doubling bifurcation

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(a)

(b)

(c) Fig. 2. Bifurcation of voltage-mode controlled buck converter. (a) Simulated bifurcation diagram with E as the bifurcation parameter; (b) largest Lyapunov exponent; (c) measured bifurcation diagram with inductor current versus input voltage (x-axis: 5 V/div, y-axis: 50 mA/div).

[Iu & Tse, 2001; Tse, 1994], chaos [Fossas & Olivar, 1996; Hamill et al., 1991], and coexisting attractors [Banerjee, 1997]. A typical bifurcation diagram is shown in Fig. 2(a), where E is chosen as the bifurcation parameter, and the variation of the largest Lyapunov exponent is shown in Fig. 2(b). Figure 2(c) shows the experimentally measured bifurcation diagram. This particular buck converter exhibits a complex bifurcation route, with the main bifurcation being period-doubling. When E exceeds about 32.27 V, the converter enters a chaotic region. Beyond 32.34 V, the chaotic at-

tractor encounters crisis and expands to a larger chaotic attractor. Meanwhile, in some periodic windows along the main period-doubling bifurcation route, there are other periodic or chaotic attractors coexisting. For example, chaotic attractor coexists with the periodic attractor when E is about 24 V, and a period-doubling bifurcation route beginning at period-6 coexists with a period-2 attractor when E is about 30 V. Figure 3 shows the phase portraits and Poincar´e sections, both simulated and measured, for the case of E = 33 V, which corresponds to chaotic operation.

Applying Resonant Parametric Perturbation to Control Chaos in the Buck DC/DC Converter

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(b)

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Fig. 3. Chaotic operation of voltage-mode controlled Buck converter, with L = 20 mH, C = 47 µF, R = 22 Ω, T = 400 µs, E = 33 V and Vref = 11 V. (a) Simulated phase portrait of chaotic attractor; (b) simulated Poincar´e section of chaotic attractor; (c) experimental phase portrait of chaotic attractor plotted with inductor current (y-axis: 0.1 A/div) versus output voltage (x-axis: 0.2 V/div); (d) experimental Poincar´e section of chaotic attractor plotted with inductor current (y-axis: 0.1 A/div) versus output voltage (x-axis: 0.2 V/div).

3.3. Control of chaos by resonant parametric perturbation The method of resonant parametric perturbation is now applied to stabilize the chaotic buck converter described in the foregoing. We choose V ref as the perturbation parameter. Essentially we replace V ref by Vref (1 + α sin 2πf t), where α is the perturbation amplitude, f is the perturbation frequency, and thus the term (α sin 2πf t) is the resonant perturbation applied to Vref .

The perturbation frequency f is set to the driving frequency, and the perturbation amplitude α is varied from 0 to 0.25. We examine the steadystate behavior in terms of the sampled inductor current under the variation of α. In particular, Fig. 4(a) shows the bifurcation diagram with i L sampled stroboscopically at the beginning of each switching cycle, and Fig. 4(b) shows the bifurcation diagram with iL sampled at switch-off instants. The inconsistency between the two diagrams can be attributed to a special phenomenon known as

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(a)

(b)

Fig. 4. Bifurcatioin diagrams when resonant parametric perturbation is applied. (a) iL sampled stroboscopically at the beginning of cycle; (b) iL sampled at switch-off instants.

(a)

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Fig. 5. Experimental confirmation of orbit stabilization using resonant parametric perturbation. (a) Sawtooth waveform (upper trace, 2 V/div) and perturbation signal (lower trace, 1 V/div); (b) phase portrait plotted with inductor current (y-axis: 50 mA/div) versus output voltage (x-axis: 50 mV/div) showing controlled period-1 operation.

corner collision [Di Bernardo et al., 2001], which causes multiple switchings within a cycle and hence makes the sampling at switch-off instants more frequent than the stroboscopic sampling. Since multiple switchings are not unveiled by stroboscopic sampling, one should be cautious about using stroboscopic bifurcation diagrams such as Fig. 4(a), especially in making a stability conclusion. Notwithstanding this effect, we clearly observe the stabiliza-

tion effect when resonant parametric perturbation is applied. Experimental measurements have been taken to confirm the effectiveness of the method. As shown in Fig. 5, the system is stabilized to a period-1 operation. Specifically, the sawtooth waveform and the perturbation signal are shown in Fig. 5(a), and a Poincar´e section is highlighted as a thick dot shown on the phase portrait of Fig. 5(b). Furthermore,

Applying Resonant Parametric Perturbation to Control Chaos in the Buck DC/DC Converter

we have found in our experiment that the smallest effective perturbation amplitude required is 0.7 V, i.e. α = 0.7/11.3 ≈ 0.06. This is consistent with Fig. 4(b). Note that Fig. 4(a) reveals a deceptively lower “smallest effective perturbation amplitude” because of the afore-mentioned multiple switching effect.

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However, in practice, phase shift and frequency mismatch do exist. In this section, we study the effects of phase difference and frequency mismatch in the perturbing signal on the effectiveness of the resonant parametric perturbation method, and in particular, we attempt to exploit these effects to reduce the perturbation power which has been a major concern of the method [Qu et al., 1995; Yang et al., 1996].

4. Resonant Parametric Perturbation with Phase Shift and Frequency Mismatch

4.1. Effect of phase shift

In the previous section, resonant parametric perturbation is applied with the perturbing signal synchronized perfectly to the switching frequency.

We consider the application of resonant parametric perturbation with a phase difference of θ between

(a)

(b)

(c) Fig. 6. Effects of phase shift. Bifurcation diagrams with (a) θ as bifurcation parameter for α = 0.0035 and (b) α as bifurcation parameter for θ = 2.3; (c) output voltage waveform for θ = 2.3 and α = 0.0035.

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(a)

(b)

Fig. 7. Experimental confirmation of orbit stabilization using resonant parametric perturbation with phase difference of 2.3 and reduced perturbation amplitude 0.0044. (a) Sawtooth waveform (upper trace, 2 V/div) and perturbing signal (lower trace, 50 mV/div); (b) phase portrait plotted with inductor current (y-axis: 50 mA/div) versus output voltage (x-axis: 50 mV/div) showing controlled period-1 operation.

perturbing signal and the sawtooth signal, i.e. Vref 7→ Vref [1 + α sin (2πfs t + θ)]

(5)

where fs is the switching frequency of the buck converter. In other words, the perturbing signal leads the sawtooth by a phase angle θ. We set the perturbation amplitude to a value much lower than the “smallest effective amplitude” found previously, and observe the bifurcation diagram with θ serving as the bifurcation parameter. The bifurcation diagram shown in Fig. 6(a) is plotted for α = 0.0035. Here, we clearly observe a stable period-1 region for θ between about 2.0 and 2.6, meaning that the system can be stabilized with a perturbation amplitude as small as 0.0035. It can be shown that regardless of the exact perturbation amplitude, the period-1 region is centered at around θ = 2.3. Thus, we may conclude that for this particular system, θ = 2.3 is the optimal phase shift. Let us set θ at 2.3, which gives a stable period-1 operation. Now, with the perturbation amplitude serving as the bifurcation parameter, we can identify the smallest pertubation amplitude that is needed to stabilize the system to a period-1 operation at the given phase shift. Specifically, from Fig. 6(b), we can conclude that the smallest effective resonant perturbation amplitude is about 0.0034, which is significantly lower than that required for the case without phase shift.

Experimental circuits have been built to confirm the effectiveness of the method. As shown in Fig. 7, the originally chaotic system can be stabilized to period-1 operation with a much lower perturbation amplitude when phase shift is applied.

4.2. Effect of frequency mismatch In practice, the parametric perturbation frequency may not synchronize exactly to the system’s intrinsic frequency. Suppose the perturbing frequency is differing by ∆f from the switching frequency of the buck converter. Then, the parametric perturbation becomes c(1 + α sin 2π(fs + ∆f )t). For analytical convenience, we define the fractional detuning as ∆f δf = . (6) fs For the voltage-mode controlled buck converter operating in chaotic state, we apply resonant parametric perturbation with frequency detuning to the reference voltage, i.e. Vref 7→ Vref [1 + α sin 2πfs (1 + δf )t] .

(7)

In our simulation, we use α = 0.0035, f s = 2500 Hz and δf = 0.04%. Since the frequency detuning is very small compared to the switching frequency, we may consider the frequency variation as a phase shift on a slow scale. In other words, if we look at a particular moment t or a small interval around t, the perturbation is simply phase shifted, with θ

Applying Resonant Parametric Perturbation to Control Chaos in the Buck DC/DC Converter

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Fig. 8. Intermittent chaos when resonant parametric perturbation is applied with frequency mismatch. (a) Time-bifurcation diagrams (sampled inductor current waveforms) with δf = 0.04%, α = 0.0035; (b) time-bifurcation diagrams (sampled inductor current waveforms) with δf = 0.04%, α = 0.0035; (c) inductor current waveforms with δf = 0.04%, α = 0.0035; and (d) inductor current waveforms with δf = −0.04%, α = 0.0035.

equal to 2πfs (δf )t. Thus, in this case, the system will experience a shift in phase from 0 to 2π as time elapses from 0 to 1/(fs δf ) s. Figure 8(a) shows the bifurcation diagram with time as the bifurcation parameter. From this figure, we observe that the same bifurcation pattern

as in the case of phase shift [Fig. 6(a)] repeats itself every 1 s. Note that 1/fs δf = 1 s. Figure 8(b) shows a similar situation for the case of δf = −0.04%, but the pattern is reversed in direction along the time axis. Figures 8(c) and 8(d) show the corresponding time-domain waveforms of the

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inductor current. These diagrams clearly indicate the occurrence of intermittent chaos when the reference voltage is perturbed at a slightly mismatched frequency. Such intermittency in resonant parametric perturbed systems was also studied by Qu et al. [1995] and Yang et al. [1996]. When frequency mismatch is inevitable, the only way to make the system stable for all times is to apply perturbation at a larger amplitude. It has been found that for this particular case, a perturbation amplitude of 0.077 is enough to combat the frequency mismatch and to maintain period-1 operation at all times.

5. Analysis Since the immediate bifurcation from a period-1 operation window is period-doubling, as inspired by Fig. 6, analysis may be performed to qualitatively identify the onset of this bifurcation. The approach involves constructing the relevant iterative map and deriving its Jacobian from which the stability of the system can be studied. Based on the circuit operation, we can construct the iterative map as follows [Di Bernardo & Vasca, 2000]: xn+1 = f (xn , dn ) = Non (1 −dn )Noff (dn )xn + [Non (1 −dn )Moff (dn ) + Mon (1 −dn )]E

(8)

where dn = 1 − d n ;

dn is the duty cycle

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Non (d) = eAondT

(10)

Mon (d) = A−1 on [Non (d) − I]Bon

(11)

Noff (d) = eAoffdT

(12)

Moff (d) =

A−1 off [Noff (d)

− I]Boff

To complete the model, we need to derive the defining function for the duty cycle. Essentially, we wish to find the connection between the switching instant and the state variables. From (1) and (2), we may define a switching function s(.) as s(dn ) = Vcon (dn T ) − Vramp (dn T ) .

(14)

Thus, the switch turns on when s(d n ) < 0, and off otherwise. In order words s(d n ) = 0 defines the switching instants. Expanding (14) gives " #   vo (dn T ) − V˜ref (dn T ) s(dn ) = A 0 iL (dn T ) (15) − VL − (VU − VL )dn T   = A 0 [Noff (dn )xn + Moff (dn )E] − AVref [1 + α sin(2πdn + θ)] − VL − (VU − VL )dn T

(16)

where V˜ref is the resonant perturbed parameter. The discrete-time iterative map derived above can be used to study the stability of the system. Specifically, by computing the characteristic multipliers near the equilibrium point [Di Bernardo & Vasca, 2000], we can analyze the way the system loses stability as certain parameters are varied. Precisely, the system’s stability near the equilibrium point can be inspected by tracking the movement of the characteristic multipliers on the complex number plane. Our concern now is to compute the characteristic multipliers. First, we denote the Jacobian of f (.) by J , which can be evaluated as J (xn , dn ) =

∂xn+1 ∂xn

∂f ∂f = − ∂xn ∂dn

(13)



∂s ∂dn

−1

∂s ∂xn

(17)

where ∂f = Non (1 −dn )Noff (dn ) ∂xn   ∂f ∂Non (1 −dn ) ∂Noff (dn ) = Noff (dn ) + Non (1 −dn ) xn ∂dn ∂dn ∂dn   ∂Moff (dn ) ∂Mon (1 −dn ) ∂Non (1 −dn ) Moff (dn ) + Non (1 −dn ) + E + ∂dn ∂dn ∂dn

(18)

(19)

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(c) Fig. 9. Loci of movement of characteristic multipliers. (a) As phase shift increases from 0 through 2.16, one chararacteristic multiplier enters the unit circle from outside. (b) As phase shift further increases to around 2.48, one chararacteristic multiplier leaves the unit circle from inside. (c) As perturbation amplitude increases from 0, one chararacteristic multiplier enters the unit circle from outside along the real axis and then moves off along a circle of radius less than 1.

= [−Aon T Non (1 −dn )Moff (dn ) + Non (1 −dn )Aoff T Noff (dn )]xn + [−Aon T Non (1 −dn )Moff (dn ) + Non (1 −dn )Noff (dn )Boff T − Non (1 −dn )Bon T ]E   ∂s = A 0 [Aoff T Noff (dn )xn + T Noff (dn )Boff E] ∂dn − 2πAVref α cos(2πd n + θ) − (VU − VL )T   = A 0 Noff (dn )(Aoff xn + Boff E)T − 2πAVref α cos(2πdn + θ) − (VU − VL )T

 ∂s = A ∂xn

 0 Noff (dn ) .

(20) (21)

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The characteristic multipliers, λ, can then be found by solving det[λI − J (xn , dn )]xn =xQ , dn =dQ = 0 .

(22)

where xQ and dQ are the equilibrium values. We now examine the movement of the characteristic multipliers as the phase shift θ and the perturbation amplitude α are varied. It is found that the system would experience period-doubling as these parameters are varied. Shown in Figs. 9(a) and 9(b) are the loci of characteristic multipliers when θ is varied and α = 0.0035. Also, Fig. 9(c) shows that loci when α is varied and θ = 2.3. From these figures, we observe the following. 1. As θ increases from zero, the two characteristic multipliers move along the real axis and get closer together. When θ is about 2.16, one characteristic multiplier crosses the unit circle from outside. Thus, the system becomes stable as θ increases to about 2.16. 2. On further increasing θ, the movement reverts and the two characteristic multipliers move apart. At about θ = 2.48, one characteristic multiplier leaves the unit circle and the system period-doubles at this point. 3. The above two observations are consistent with the bifurcation scenario simulated in Fig. 6(a), in which the converter has been shown to bifurcate in two directions, i.e. reducing the phase shift below 2.16 and increasing it up to 2.48. 4. From Fig. 9(c), we observe that as the perturbation amplitude α increases, the two characteristic multipliers start off on the real axis moving towards each other. One characteristic multiplier crosses the unit circle from outside when α increases to about 0.0034. Further increasing α causes the characteristic multipliers to merge and then split to track off along a circle of radius less than 1 (0.824). The system becomes stable when α exceeds about 0.0034 and remains stable afterwards. This result is consistent with the bifurcation diagram shown in Fig. 6(b).

6. Conclusion Resonant parametric perturbation is an effective nonfeedback approach for controlling chaos. Compared to the feedback chaos control methods, the nonfeedback chaos control methods are usually easier to apply and they require no prior knowledge of

the system behavior. For switching converters, such nonfeedback chaos control allows easy incorporation with the exising control circuits, as demonstrated in the voltage-mode controlled buck converter studied in this paper. The main contribution of this paper is in the study of the effects of phase shift and frequency mismatch in the application of resonant parametric perturbation for chaos control in switching power converters. It has been found that much control effort can be saved if suitable phase shift is applied to the perturbing signal. Moreover, in the practical situation of frequency mismatch, chaos control can still be accomplished at the expense of raising the control power.

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Applying Resonant Parametric Perturbation to Control Chaos in the Buck DC/DC Converter

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Ott, E., Grebogi, C. & Yorke, J. A. [1990] “Controlling chaos,” Phys. Rev. Lett. 64, 1196–1199. Ott, E. & Spano, M. [1995] “Controlling chaos,” Phys. Today 34, 34–40. Pyragas, K. [1992] “Continuous control of chaos by self-controlling feedback,” Phys. Lett. A170, 421–428. Qu, Z., Hu, G., Yang, G. & Qin, G. [1995] “Phase effect in taming nonautonomous chaos by weak harmonic perturbations,” Phys. Rev. Lett. 74, 1736–1739. Tse, C. K. [1994] “Chaos from a buck switching regulator operating in discontinuous conduction mode,” Int. J. Circuit Th. Appl. 22, 263–278. Tse, C. K. & Di Bernardo, M. [2002] “Complex behavior of switching power converters,” Proc. IEEE 90, 768–781. Yang, J., Qu, Z. & Hu, G. [1996] “Duffing equation with two periodic forcings: The phase effect,” Phys. Rev. E53, 4402–4413.