Border Collision Bifurcations in a One-Dimensional ... - CiteSeerX

International Journal of Control, Volume 75, Issue 16 & 17 November 2002 , pages 1356 - 1367

Border Collision Bifurcations in a One-Dimensional Piecewise Smooth Map for a PWM Current-Programmed H-Bridge Inverter Bruno Robert 1, * and Carl Robert 2 1

LAM-Université de Reims Champagne-Ardenne, BP 1039, 51687 Reims Cédex 2, France Department of Physics, University of California, Santa Barbara, CA 93106, U.S.A.

2

Abstract

In this article, we are studying the non-linear effects in a single-phase H-Bridge inverter. The PWM control is related to a current feedback control. We are proposing an analytical model, which is a piecewise linear map. The distinctive feature of this study lies in the investigation of the map’s properties. This investigation allows for the analytical determination of the fixed points, their domains of stability, and of the bifurcation points. More precisely, we will show that some of these bifurcations are discontinue. The analysis is performed while keeping in mind the current controller’s tuning. In this particular setting, we will show that all the bifurcations are of a certain type: Border Collision Bifurcations. Although we are treating the appearance of chaos in a converter, the work presented stays close to the preoccupations of the engineer, because the particularities of the digital control are shown as an advantage. Moreover, we have strived to comment on the different modes observed, periodic or not, by underlying their practical interest or disadvantages.

1

Introduction

The conversion of electrical energy is an important preoccupation for anyone who wishes to use it to supply energy to most diverse devices. Depending on the case, it is necessary to be able to fit any source to its receiver with a converter. The adaptation of the source to its receiver can be realized with static electronic switches that are associated to electronic controls, which are adapted to the needs of the conversion. The modelling of these converters sets new problems. Generally, the models of static converters are essentially non-linear with respect to the state. Indeed, during one period, the topology of the conversion structure changes at each active phase. Each is described by a different linear differential system. This condition is known as piecewise linear. In the concern for simplicity, the study of these systems was first considered through "average models", a priori linear. These models can be described by equivalent schemes and one can then use the powerful tools of analysis and synthesis of linear systems. The mean state space method strives to conciliate the exactness of the solving of differential systems with the simplicity of use of linear tools (Middlebrook and Cùk 1991). However, the obtained models are fundamentally not suited to take care of the non-linear properties of these converters. These models are only valid for "small variations around a active point" and limited in frequencies. Other techniques of modelling are based on solving the state equation phase by phase on a functioning period (Elbuluck et al. 1986). They lead to non-linear maps. These maps can be used to develop non-linear controls that give good results in "large signals". However, the common denominator of these techniques is the knowledge of the functioning mode of the converter, that is, the temporal sequence of different topologies. At first analysis, this sequence can be identified. The natural process is as follows: knowing the usual functioning sequence of a type of converter, the engineer scales it to ensure the usual running mode. The ulterior validity of the modelling and the synthesis of the control that ensued, depend on the respect of the formulated hypothesis. For the engineer, in most cases, only the equilibrium states or periodic functioning is acceptable. Restrained to these modes of running, a linear analysis would most often be sufficient. However, that would be neglecting much other possible behaviours. These behaviours do not necessarily need to be avoided since they do not hamper the conversion (Nagy et al. 1996). Nonetheless, it is frequent to observe

*

Tel.: (33) 3 26 91 31 05, Fax: (33) 3 26 91 31 06, E-mail: [email protected]

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aperiodic running. They are related to the variations of the model’s parameters under unusual working conditions (Mira 1990). Moreover, an inappropriate setting of the correctors, inserted in the control loop to improve the dynamical performances, may also lead to unsuspected behaviours. This paper shows how to use a non-linear map together with a non-linear dynamical approach in order to identify the appearance conditions of these unfamiliar running modes. It also shows how it is possible to precisely characterize them, to define their exact nature and to know, case by case, if they are acceptable or not.

2

Literature review

In spite of these advantages, the use of the tools, coming from non-linear dynamics in electro-technique and in electronic, is still quite rare. This is particularly true of electric drives where this type of approach is still quite infrequent (Mitobe and Adachi 1991, Hemati 1994, Santana and Marques 1996, Robert et al. 2000, Péra et al. 2000, Robert et al. 2001). On the other hand, a growing number of studies, applying chaos theory to static converters, are published. The first ones, to our best knowledge, go back to the end of the 80’s (Hamill and Jeffries 1988, Hamill et al. 1992, Conard et al. 1993). With the exception of an induction motor’s supply study (Süto et al. 1997, Süto and Nagy 1999), all these papers are about DC/DC converters or choppers (Chan and Tse 1997, Chan and Tse 1998, El Aroudi et al. 2001, Iu and Tse 2001). However, the study we are presenting is on a DC/AC converter. The converter in question is a current – programmed full bridge, i.e. a voltage inverter associated to a current control. One of the particularities of our study is precisely analysing the bifurcation diagram in function of a current controller parameter. Most of the work conducted on choppers discusses the evolution of the dynamics when the varied parameter is the supply voltage, the reference current, or the load (Di Bernardo et al. 1998, Banerjee and Chakrabarty 1998, Di Bernardo and Vasqua 2000). However, the dynamical performances require a precise tuning of the correctors, which is often delicate. That is why this study strives to show how the corrector’s tuning provokes bifurcations by moving the boundaries in a one-dimensional space. We may however mention the existence of a similar approach in "Chan and Tse 1998". Another particular aspect of our study is related to the control technique used: it is digital. The correctors and modulators used in works presented on DC/DC converters are always analog. (Poddar et al. 1998, Di Bernardo 1999). This type of modulator can generate plenty of noisy switching, which perturbs the dynamics noticeably. This phenomenon, called chattering, is most often not mentioned except by "Yuan et al. 1998". The digital modulator allows us to perfectly control the switching times. The behaviour of the inverter presented here has already been studied using various types of correctors and modulators (Carton et al. [1997, 1998, 1999]). However, the model that have been used, as well as the ones cited just before, do not allow an entire analytical treatment of the bifurcations and require the use of numerical simulations. In this article, we propose to model with a one-dimensional piecewise smooth map that is also piecewise linear. The simplifying hypotheses are realistic and few. The model also uses two boundaries instead of only one (Yuan et al. 1998, Banerjee et al. 2000). The map is three-piece. The essential idea is that because of this model, it is possible to analytically determine the fixed points and their stability, the bifurcation points, and to show that some of these bifurcations are discontinuous. The rest of the article is organized as follow: Section 3 presents in detail the modelling of the inverter associated with its current control. Section 4 specifies the bifurcation diagram parameter, as well as the numerical constants that we utilized. In following sections, we present some bifurcations and the different attractors, which appears when the corrector’s gain is varied. Section 5 studies the normal operating mode of the converter. In section 6, period two operating mode is analysed. The bifurcation toward a four-piece chaotic attractor is presented in section 7. In section 8, we conclude that the bifurcations of that inverter are border collision bifurcations.

3 3.1

Modelling the converter Model of the H-Bridge

Let us first address the converter’s running mode. It is a voltage inverter controlled by a current feedback loop. The switching frequency of the inverter is fixed. In order to impose the current to be the reference current at the beginning of each period, the current’s controller acts on the switching times. If this reference current varies as a sinusoid in time, the current measured and sampled at the beginning of each period will also vary as a sinusoid.

2

If the inverter’s period is much smaller than the main time constant of the load, the current’s ripple within a period will be minimal. Furthermore, the frequency of the current’s reference must be much smaller than the inverter’s in order to guaranty a variation of the current close to a sinusoid (see Fig. 1). To model the PWM H-bridge in current mode, we proceed with a series of steps. First, we propose a model associated to its load by taking into account the chosen PWM pattern. Secondly, we model the feedback loop for the current. Finally, the two models are joined to describe the functioning of the whole. Figure 2 illustrates the circuit of an inverter. A switch is realized in this bridge by combining a bipolar transistor and an anti-parallel diode. The four equivalent switches are named S1, S2, S3 and S4. Depending on the state of the switches, the H-bridge has two distinct topologies T1 and T2. They supply a passive load, resistive and inductive. Because the current in the load is the unique state variable, its evolution is described by a simple first order differential equation:

Figure 1: Input reference current and controlled current in the load.

S3

S1 L

S1 , S2 : off and S3 , S4 : on,  T1:  di Ri E  dt = − L − L and S1 , S2 : on and S3 , S4 : off,  T2:  di Ri E  dt = − L + L .

(1)

R

E i v S4

(2)

S2

Figure 2: H-Bridge inverter.

The control of the switches is periodic of period T and the current in the load is sampled at the beginning of each period. In addition, to avoid eventual perturbations introduced by switching at the sampling time, we chose a PWM with a centred pulse (see Figure 3). The normal running mode of the converter involves a sequence of three topologies: T1 – T2 – T1. The duration of the centred pulse, which corresponds to the T2 topology is: Dn ⋅ T (D is the duty cycle). The notion of duty cycle only makes sense when defined on one period; we notice that Dn is the duty cycle on the nth functioning period: 0 ≤ Dn ≤ 1 . From that, we deduce the functioning sequence:

E

v

in

i tn

t1

-E

t2

tn+1 t in+1

DnT

T Figure 3: Centred PWM pattern.

1 − Dn  T  nT ≤ t < t1 = nT + T1 :  , 2  i ( nT ) = in ; i ( t1 ) = i1  1 + Dn  T t1 ≤ t ≤ t2 = nT + T2 :  , 2  i t i = ( ) 2 2  t2 < t ≤ ( n + 1) T T1 :  .  i ( n + 1) T = in +1

By taking into account the continuity of the current, we have successively integrated on each of the three intervals. RT We let: δ = . 2L

3

E − 1− D δ 1 − e ( n )  , R E − 1+ D δ − 1+ D δ T2 : i2 = in e ( n ) −  2e −2 Dnδ − e ( n ) − 1 , R 2 E −δ T3 : in +1 = in e −2δ + e  2sinh ( Dnδ ) − sinh (δ )  . R T1 : i1 = in e

− (1− Dn )δ

−

To make the model independent on the numerical values of its parameter, we reduce: • the voltages with respect to E, • the currents with respect to E/R, and • the time with respect to T. For the purpose of simplicity, we will keep the same notation for the current. The reader must simply remember that it has been reduced. From now on, only the reduced quantities will be used. The bridge is being controlled in current mode. The period T is chosen to be much smaller than the time constant of the load. This is to reduce the current ripple: δ 0 . Because, in order to increase in +1 and to reduce the current error at the sampling time, the width DnT of the centred pulse (see Fig. 3) must become wider as the error gets greater. It is worth noting that the current reference is between −1 ≤ I ≤ 1 , but since it can be any gain k, un is not bounded. The modulator generates a PWM pattern devoted to the control of the switches. sat ( un ) The duty cycle being bounded, the 1 modulator introduces a saturation to limit To S1 and S2 too much wandering (see Fig. 5). It is un modelled by a saturation function sat ( un ) . un -1 1 To S3 and S4 -1

Figure 5: PWM modulator with saturation.

1 sat ( un ) Dn = + 2 2

We notice that the voltage on the load is averaged; this implies that the mean current is zero for Dn = 0.5 . With the objective of including some symmetry to the problem, we define the saturation function such that:

sat ( un ) = −1 if un ≤ −1  where: sat ( un ) = un if − 1 < un < 1 .  sat ( un ) = +1 if 1 ≤ un

4

(5)

3.3

Complete model for the converter

The converter must take into account the eventual modulator saturation. The converter is obtained by combining the bridge’s model and the model where the control parameter is the current. To simplify the writing, we introduce two new constants: 0 < α = e−2δ < 1 and 0 < β = 2δ e −δ < 1 . If 1 ≤ un , the modulator is in positive saturation and D n = 1 . Hence, the map (3) becomes: in +1 = α in + β . If −1 < un < 1 , the modulator is not saturated and the map (3) is valid, then: in +1 = α in + β un . Finally, if un ≤ −1 , the modulator negatively saturates and D n = 0 . Therefore, the map (3) leads to: in +1 = α in − β . From (4) and (5), we can express the three conditions in function reference. We obtain the complete model (6):  if  F1 ( in ) = α ⋅ in + β   in +1 = F ( in ) =  F2 ( in , k ) = in ∆ ( k ) + β kI if   if  F3 ( in ) = α ⋅ in − β 

of the corrector’s gain and of the current

in ≤ I −

1 k

1 1 < in < I + k k 1 in ≥ I + k I−

(6)

The current axis i n is divided in three intervals. Hence, there exist two one-dimensional boundaries reduced to points of the abscissa: 1 1 C1 ( k ) = I − and C2 ( k ) = I + . k k In order to simplify the notation, we define an auxiliary variable: ∆ (k ) = α − β k .

in+1

C1

C2

The recurrent model of the inverter under the current control is not only piecewise smooth system, but is also piecewise linear (see Fig. 6). in

Figure 6: Piecewise linear map F ( in ) at k = 8 .

4

Study hypotheses of the dynamics

We will study the dynamics of the converter by setting the proportional gain k > 0 . This consists of another innovative aspect of the present study. Indeed, the usual studied bifurcation parameter is the continuous input voltage of the converter, or, more rarely, a parameter linked to the PWM modulator. Thus, all modern converters are related to one or more feedback loops intended to upgrade the dynamical performances. These feedback loops include different types of analogical or digital correctors, the most elementary of them being the proportional corrector. The corrector’s tunings often turned out to be tricky from a practical point of view. Nevertheless, these tunings remain essential to obtain the required performances with the requirement specifications. Furthermore, most correctors being linear, their tuning must be frequently adapted when the load or the supply voltage vary. A maladjusted tuning often leads to an unexpected behaviour. The typical descriptions of such unexpected behaviour are often referred as "non-functioning", "instability", or an "abnormal functioning". This often reveals the lack of knowledge concerning the non-linear phenomena that frequently arise is such setting. Hence, the case we are studying here takes on a practical flavour tightly linked to the preoccupation of the engineer in the field. Previous experimental results presented in (Robert et al. 1999) shows that, when the tuning of the corrector changes, bifurcations of this type appear for this type of converter. Furthermore, it can exhibit chaotic behaviours such as the one reported on Fig. 7. In this experiment, we used an industrial converter: an inverter’s arm

5

composed of an IGBT (Insulated Gate Bipolar Transistor) of 600V and 50A. The current control is performed numerically with a 16 bits microcontroller card (Intel 80C196KC). The current is measured with a Hall probe, and then numerically converted at each end of switching period. The micro-controller card achieves both corrector and modulator functions before individually operating the four switches of the bridge. The voltage tension E is obtained by a voltage rectifier connected to the electrical power network. The experimental load R is chosen such that the power does not exceed a few kW. This is done to facilitate the realisation of the prototype. The period T must be sufficient in length to take into account the constraints linked to the computing time of the corrector and of the digital modulator. A few kHz Figure 7: Experimental waveforms of current are easily reachable, which allows us to come close to a sinusoidal current with sufficient precision. Indeed, for a range of current’s frequencies from zero to 100 Hz (largely sufficient for adjustable speed on industrial motor drives), there will be at least 50 samplings per period of the current (see Fig. 1). The coil plays the role of smoothing the current and the value of the inductance, calculated to respect δ 1 , does not cause difficulty in the realisation, due to its small value. Constants: E = 400V − R = 40Ω − L = 20mH − T = 0.2ms, δ = 0.2 − α = 670.32 ⋅10−3 − β = 327.49 ⋅10−3. We considered that the converter functions with a fixed current reference. This corresponds to the functioning mode illustrated on Fig. 1 when the current’s frequency tends to zero. This functioning is also similar to a current-programmed chopper. We arbitrarily fixed it to I = 0.5. The following bifurcation diagram shows the series of bifurcations observed when the proportional corrector’s gain increases. Figure 8 shows the computed

C2(k): higher boundary

in

C1(k): lower boundary

k: proportional gain

Figure 8: Computed bifurcation diagram with the iterative model.

6

bifurcation diagram with the iterative model. A bifurcation diagram is an efficient tool to sum up the influence of a parameter on the system and the values of this parameter, which changes the nature of the asymptotic solution (the transient dynamics is taken out). Any qualitative change of the solution type is called a bifurcation (e.g. a stable fixed point becomes a stable oscillation). The behaviour of the dynamical system is studied for different values of a parameter. Particular values of one variable of the system are chosen. For our system driven by a periodic voltage, values of the current at the same instant of successive periods are used. It is in fact a timediscrete version of the solved system and it results in a series of points ( ik )k =1, N . For each

Table 1: Some fixed points and their principal properties Limit k → ∞ Existence Stability

i2*

I−

[0

* i21

I −β α

[ k1

∞[

[ k2

∞[

F1 ( i

* 21

)

I

* F1 ( i2221 )

I+

F2  F1 ( i

) F  F  F (i ) * 2221

2

[0

k1 [

]k1

k2 [

+

I −β α

* i2221

2

∞[

1

* 2221

I

∅

−

I+

value of the parameter on the abscissa, successive ( ik )k =1, N are plotted when the asymptotic is reached. In a neighbourhood of the bifurcation points, the computation of the fixed points converges very slowly. For this reason, a sufficiently large number of points are not plotted. For our case, we chose to remove the first 5000 points. If the series ( ik ) converges to a fixed point, all the points are superimposed. The discrete system is said 1-periodic or fundamental. When q points are distinct, it means that ( ik ) takes alternatively q values. The system

is then q-periodic i.e. sub-harmonic. These points correspond to the fixed points of a composition of q functions F1 and/or F2. For this reason a q-periodic orbit can also be called a fixed point of corresponding maps composed q times. For some values of the parameter, one can see a “continuous line”, ( ik ) covering a whole range of values. The system is then aperiodic (quasi-periodic or chaotic). Such a “continuous line” could also be the hallmark of periodic orbit of a very high period, but this is not the case here. Table 1 sums up properties of fixed points of some composition of functions F1, F2, and F3. The symbols I − and

* I + means that the series converges to the fixed point from below and above respectively. For example, i2221 is a

fixed point of the function F2  F2  F2  F1 ( in ) . This fixed point exists for parameter values greater than k2, but it never can be observed experimentally or by computer because it is always unstable. On the other hand, i2* , the fixed point of F2, which is stable for parameter values smaller than k1, is observed at the left side of the bifurcation diagram. This fixed point clearly corresponds to a fundamental periodic functioning. The rest of the paper presents the study of these fixed points and their stability and presents an analysis of the bifurcations of collisions of fixed points with boundaries. On the right side of the diagram, other fixed points and collisions of fixed points with the boundaries are also present. They can be analysed with the same approach, but for reasons clarity and briefness, we will report that in a subsequent publication. In particular, we note “a one-piece chaos to three-piece chaos” bifurcation and a continuous period doubling bifurcation.

5

Periodicity T

We study the existence of fixed points of the maps F1 , F2 and F3 . More precisely, we show that only the map

F2 allows a fixed point and that there exists a stability interval for that point.

5.1

Fixed point of F1

β . 1−α We notice that fixed point i1* is independent of the current reference I applied to the control. It is therefore of no practical interest. The map F1 has the fixed point:

i1* =

7

The existence condition of the fixed point of F1 is: I < 1 , we obtain: k>

i1* < I −

1−α I (1 − α ) − β

Taking into account the numerical values:

1 . Moreover, according to our hypotheses of study, k

⇒ i1* +

1 < I 150.7 . The studied reference I = 0.5 is not in

* 1

the interval; therefore, the fixed point i does not exist.

5.2

Fixed point of F3

The fixed point of F3 is i3* = − I+

β 1 < 0 and the existence condition is i3* > I + . 1−α k

1 being positively defined on the studied interval, the fixed point i3* does not exist. k

The continuation of the studies will then be on the map F2 .

5.3

Fixed point of F2

The fixed point for the map F2 is: i2* ( k ) =

βk I. 1−α + β k

i2* is proportional to the reference I. It is the normal running mode of the whole converter-control.

α −1 I 1 . Thus, this case can never happen and the ( k1 ) ≠ F1 ( i21* ( k1 ) ) . Moreover 2 1−α 1−α 2 bifurcation is always discontinuous.

If I ≠

* = We may consider with interest the limit of the fixed point i*21 for an infinite gain: lim i21 k →∞

I −β . α

* The limit for the second point is interesting in practice, because it corresponds to the reference: lim F1 ( i21 ) = I+ . k →∞

We now show that the fixed point of F21 appears exactly when F2 becomes unstable. There are three conditions of existence: * Condition 1: i21 ( k ) ≤ C1 ( k ) . * Condition 2: F1 ( i21 ( k ) ) ≥ C1 ( k ) .

* Condition 3: F1 ( i21 ( k ) ) ≤ C2 ( k ) .

Conditions 2 and 3 are always true (See appendix B). * lim i21 ( k ) − C1 ( k )  =

I (1 − α ) − β

0 such that i21 ( k ) collides with the lower border. At infinity, condition 1 is true:

k →∞

9

6.2

* Condition 1: Collision of i21 with C1

* We show that the fixed point i21 ( k ) collides with the boundary C1 ( k ) exactly when the fixed point i2* ( k ) loses its stability. * Let the equation of the collision with the lower boundary be: i21 ( k ) = C1 ( k ) .

The resolution with k is obtained by extracting the roots of the polynomial P1 ( k ) : P1 ( k ) = ak + bk + c = 0 2

where

a = ( I − β − α I ) β ,  2  b = 2αβ + α I − I ,  2 c = 1−α .

The two roots are of opposite signs. Taking into account our hypotheses of study, we are uniquely interested in the positive root. P1 ( k ) allows k1 as a root. Indeed, it can be factorised in this way:  α + 1 P1 ( k ) =  k − [ dk + e] = 0 ⇔ β  

The two roots are:

k0 + =

1+ α >0 β

et

[ k − k1 ][ dk + e] k0 − =

where:

α −1 < 0. I (α − 1) + β

 d = β ( I − β − α I ) .   e = β (α − 1)

Thus, we obtain the point of collision with the lower boundary: k0 + = k1 .

Remark: k0 − is also solution of i2* ( k ) = C2 ( k ) .

6.3

Stability

* Naming µ 21 = α∆ ( k ) as Floquet’s characteristic multiplier (or stability coefficient). i21 ( k ) is stable if:

µ 21 < 1
1+α 2 1+α 2 * ≈ 6.6 . We call k2 the stability limit of i21 : k2 = .  ⌠ k1 < k < αβ αβ k > k1  CONCLUSION: * The stability criterion for the fixed point F1 ( i21 ( k ) ) is the same as the one for i21* ( k ) . We derived that the point of destabilization of F2 coincide with the emergence of the fixed point of F21 and that this bifurcation is discontinuous. Such a behaviour is well known in studies and is often observed when the tuning of the control exceeds the limit of stability. * The fixed point i21 exists on the interval [ k1 ∞[ . It is stable in the interval ]k1 k2 [ .

7

Four chaotic intervals

We show that new 4T-periodic fixed points * and that they appear after destabilisation of i21 are unstable on all domain of existence. The bifurcation diagram illustrates that in Fig. 10. The appearance, beyond k2, of these new unstable fixed points, generates a chaotic behaviour. The iterates are divided on four dense intervals that we call 1, 2, 3 and 4, in the increasing order of the values of the iterates belonging to these intervals (see Fig. 11). The four intervals are alternatively visited in the following order: 1-3-2-4. This can be observed during the simulation by considering a point of 1 and its next successive iterates.

C2(k) F2  F2  F1 ( i*2221 ) F1 ( i*2221 )

i*2 ( k )

F2  F1 ( i*2221 )

i i*21 ( k )

C1(k)

k1

i*2221

k2

k: proportional gain Figure 10: 2T to four-piece chaos border collision bifurcation.

10

The lower boundary C1 ( k ) divides the interval 2 in two sub-intervals. Depending on the position of the iterate in this interval, below or above the boundary, the map F 4 ( in ) has two distinct forms:  F2121 = F2  F1  F2  F1 ( in ) in + 4 = F 4 ( in ) =   F2221 = F2  F2  F2  F1 ( in )

if if

in + 2 ≤ C1 ( k ) ,

in + 2 ≥ C1 ( k ) .

We show that the fixed points of F2121 and F2221 , when they exist, are always unstable beyond k2. Concerning the fixed point of F2121 and its stability, the results are identical to the ones of F21 . Indeed: in + 4 = F2121 ( in , k ) = F212 ( in , k ) ,

* * i2121 ( k ) = i21* ( k ) and k > k2 ⇒ i2121 ( k ) is unstable.

7.1

4T-Periodicity in + 4 = F2221 ( in , k ) = ∆ 3 ( αin + β ) + βkI ( ∆ 2 + ∆ + 1) .

We now study the second form of F 4 (i n ) : The map F2221 has the following fixed points:

 kI ( ∆ 2 + ∆ + 1) + ∆3 *  i2221 (k ) = β  1 − α∆ 3  kI ( α∆ 2 + α∆ + 1) + ∆  *  F2  F1 ( i2221 ( k ) ) = β 1 − α∆ 3 

Notice that:

* * = lim i21 = lim i2221

k →∞

k →∞

* F1 ( i2221 ( k )) = β

αkI ( ∆ 2 + ∆ + 1) + 1

* F2  F2  F1 ( i2221 ( k )) = β

I −β . α

The limit of the second point corresponds to the reference:

1 − α∆ 3 kI ( α∆ 2 + ∆ + 1) + ∆ 2 1 − α∆3

* lim F1 ( i2221 ) = I+ .

k →∞

* * Limit of the third point is lim F2  F1 ( i2221 ( k ) ) = I − and limit of the fourth point is lim F2  F2  F1 ( i2221 ( k )) = I + .

k →∞

k →∞

C2(k): higher boundary

4

in

3 2 C1(k): lower boundary

1

k: proportional gain

Figure 11: Zoom of the computed diagram on the discontinuous periodical to chaos bifurcation.

11

* be (1) located under the lower border and (2) The definition of the map F2221 imposes that the fixed point i2221 that its three subsequent iterates be located beside the two borders. Then there are seven conditions of existence * : for i2221 * Condition 5: F2  F1 ( i2221 ( k ) ) ≤ C2 ( k ) .

* Condition 1: i2221 ( k ) ≤ C1 ( k ) .

* Condition 2: C1 ( k ) ≤ F1 ( i2221 ( k )) .

* Condition 6: C1 ( k ) ≤ F2  F2  F1 ( i2221 ( k )) .

* Condition 3: F1 ( i2221 ( k ) ) ≤ C2 ( k ) .

* Condition 7: F2  F2  F1 ( i2221 ( k ) ) ≤ C2 ( k ) .

* Condition 4: C1 ( k ) ≤ F2  F1 ( i2221 ( k )) .

Some proofs are given in appendix C. Now, we are especially interested in condition 4, which defines the collision of this fixed point with the lower boundary.

7.2

* Condition 4: Collision of F2  F1 ( i2221 ) with C1 * k > k2 ⇒ F2  F1 ( i2221 ) > C1 ( k ) .

Concerning condition 4, we show that:

( )

* intersects the boundary C1 ( k ) at the point k = k 2 . We show that the fixed point F2  F1 i2221

 β kI α∆ 2 + α∆ + 1 + β∆ * F  F (i ) = , Let the equation of the collision with the lower boundary be:  2 1 2221 1 − α∆ 3  F  F i*  2 1 ( 2221 ) = C1 ( k ) . The resolution in k is obtained by extracting the roots of the polynomial P4 ( k ) = ak 3 + bk 2 + ck + d = 0 . 2  a = αβ (α − 1) I + β  , Where:  4 2  c = (α − 1) I + αβ ( 3α + 1) ,

b = β ( −2α 3 + α 2 + 1) I − β ( 3α 2 + 1)  , d = 1−α 4.

P4 ( k ) admits k2 as a root. Indeed, it is factorised in the following manner:  e = αβ 2 (α − 1) I + β      1+α   2 2 2  P4 ( k ) =  k −  ( e ⋅ k + f ⋅ k + g ) = 0 ⇔ ( k − k2 ) ( e ⋅ k + f ⋅ k + g ) = 0 , where  f = αβ (1 − α ) I − 2αβ  . αβ     g = αβ (α 2 − 1) 2

This bifurcation from a 2T periodical to a chaotic one involves a discontinuity. Indeed, it's easy to show that the four fixed points of F2221 at the bifurcation point k = k2 are distinct. Then the bifurcation is discontinuous.

7.3

Stability

* Let us call µ 2221 ( k ) the stability coefficient of i2221 (k ) :

µ 2221 = α (α − β k ) = α∆ 3 ( k ) = µ 21 ( k ) ∆ 2 ( k ) .

µ 2 

3

However, ∆(k ) is monotonically decreasing on ℜ . Moreover k 2 > k1 . Since ∆(k1 ) = −1 , we infer: ∆2 (k 2 ) > 1 .  µ < −1 k > k2 ⇒  212 ⇒ µ 2221 < −1  ∆ >1 * * i2221 ( k ) exists only for k > k 2 . i2221 ( k ) is therefore never stable.

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CONCLUSION: * We may conclude that at the point k = k2 , the fixed point i 21 becomes unstable and a new unstable fixed point * * appears i 2221 . The fixed point i2221 exists on the interval [ k2

∞[ . It is always unstable. We are in the presence

of a bifurcation of 2T-periodic solution to a chaotic solution on four intervals (Nusse and Yorke 1995) and this bifurcation is discontinuous.

8

Conclusion

We have studied the dynamics of a current inverter with a piecewise smooth map. The bifurcation diagram has been obtained by varying a control parameter, not by varying an external parameter. The study of the bifurcations has all been done analytically. For that type of converter, we have shown that all the bifurcations are border collision bifurcations. We have brought to the fore a discontinuous period doubling, another one continuous, two bifurcations from a periodic orbit to chaos with and without discontinuity, and a bifurcation between two chaotic regimes. Even though the studied structure and the corrector are of the simplest kinds, we think that this method of analysis can help to understand how the constants of the system influence its dynamics. This method will also allow us to study the consequences of the motion of the boundaries under the effect of a variable reference, sinusoidal for example, as it is often the case. C.R. would like to acknowledge the financial support of the David and Lucile Packard Foundation, and NSF Grant No. DMR-9813752, and EPRI/DoD through the Program on Interactive Complex Networks. C.R. would also like to thank the hospitality of the Université de Reims.

Appendices Appendix A At infinity, both conditions are true:

lim i2* ( k ) − C1 ( k )  = lim

k →∞

k →∞

β + (α − 1) I = 0+ βk

lim i2* ( k ) − C2 ( k )  = lim −

k →∞

k →∞

β + (1 − α ) I = 0− βk

Moreover, there exists no collision of the fixed point i2* ( k ) with the boundary C1 ( k ) or C2 ( k ) . Let the equation for the lower boundary collision be: i2* ( k ) = C1 ( k )

The solution in k is unique but is not included in the studied interval: Similarly for the higher boundary: i2* ( k ) = C2 ( k ) ⇒ k =

α −1

k=

(1 − α ) I + β

1−α 0 ⇒ F1 ( i21 ( k ) ) − I > 0

* Moreover I > C1 ( k ) . Where we get: F1 ( i21 ) > C1 ( k ) . We deduce that the condition 2 is always true.

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* lim  F1 ( i21 ( k ) ) − C2 ( k )  = lim

At infinity, condition 3 is true:

k →∞

β (1 − α ) + (α 2 − 1) I

k →∞

αβ k

= 0−

* We now show that there exists, on the studied interval, no collision between the fixed point F1 ( i21 ( k ) ) and the

* boundary C2 ( k ) . The equation of the collision with the higher boundary is: F1 ( i21 ( k ) ) = C2 ( k ) The solution

does not belong to the studied interval: k =

α 2 −1 k1 ⇒ i2221 ( k ) < C1 ( k ) .

* We first show that the fixed point i2221 ( k ) cuts the boundary C1 ( k ) at the point k = k1 , then that at this point, its

derivative is negative. * Let the equation of the collision with the lower boundary be: i21 ( k ) = C1 ( k )

The resolution in k is obtained by the extraction of the roots of the polynomial P2 ( k ) :

 a = β 3 (1 − α ) I − β    b = β 2 ( 3α 2 − 2α − 1) I + 4αβ      3 2 4 3 2 P2 ( k ) = a ⋅ k + b ⋅ k + c ⋅ k + d ⋅ k + e = 0 where  c = β ( −3α + α + α + 1) I − 6α 2 β   4 3  d = (α − 1) I + 4α β  4  e = 1−α  P2 ( k ) admits k1 as a root. Indeed, it is factorised in the following manner:  1+α  3 2 3 2 P2 ( k ) =  k −  ( f ⋅ k + g ⋅ k + h ⋅ k + i ) = 0 ⇔ ( k − k1 ) ( f ⋅ k + g ⋅ k + h ⋅ k + i ) = 0 , β    g = β 2 ( 3α − 1) β + 2α I (α − 1)  f =a . where  3 2 2 i = β (α 3 − α 2 + α − 1)  h = β ( −α + α − α + 1) I + ( −3α + 2α − 1) β 

We then show that

∂i*2221 ( k ) ∂k

∂i2221 ( k ) k

0   b = αβ ( −2α 2 + α + 1) I − 3αβ  < 0  3 2   P3 ( k ) = ak + bk + ck + d = 0 where  . 4 3  c = (α − 1) I + β ( 3α + 1) > 0   d = 1 − α 4 > 0 ∂P ( k ) > 0 on ℜ . This polynomial has no real root on the studied interval. Indeed P3 ( 0 ) > 0 and 3 ∂k Condition 2 is thus always true.

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