# Fast Simulation and Chaos Investigation of a DC-DC Boost Inverter

Authors: Rachid Dhifaou; Houda Brahmi
Journal: Complexity
 (2021)

---

## Abstract

Intensive and repetitive simulations are required to study static and dynamic behaviours of systems. Particular phenomena such as bifurcation and chaos require long simulation times and analysis. To check the existence of bifurcations and chaos in a dynamic system, a fine-tuning procedure of a bifurcation parameter is to be carried out. This increases considerably the computing time, and a great amount of patience is needed to obtain adequate results. Because of the high switching frequency of a boost inverter, the integration process of the dynamic model used to describe it uses an integration step that is in general less than one microsecond. This makes the integration process time consuming even for a short simulation. Thus, a fast, but accurate, method is suitable to analyse the dynamic behaviour of the converter. This work contains two topics. First, we develop a like-discrete integration process that permits precise results in a very fast manner. For one switching period, we compute only two or a maximum of three breaking points depending on whether we treat a continuous conduction mode (CCM) or a discontinuous conduction mode (DCM) of the inductor current. Furthermore, with each segment of the dynamic trajectory, an exact analytic formula is associated. The second goal is to use this result to develop a discrete iterative map formulated as in standard discrete time series models. The Jacobian matrix of the found iterative map is defined and used to compute Lyapunov exponents to prove existence of chaos. Performance of the developed study is positively evaluated by using classical simulations and fine-tuning a bifurcation parameter to detect chaos. This parameter is the desired reference of the inductor current peak. Results show that the proposed scheme is very fast and accurate. The study can be easily extended to other switching topologies of DC-DC inverters.


---

## Body

## 1. Introduction

DC-DC boost inverter, also known as a step up inverters, are largely studied in literature [1–3]. These power electronic devices transfer electric energy from a DC input voltage source to an output load that requires a higher voltage. The efficiency of DC-DC boost inverter is good in general which makes them largely employed in various applications. Battery chargers, photovoltaic energy-based systems, and DC motor drives are typical examples [4,5]. As for all DC-DC power converter topologies, energy transfer in this device is achieved by switching the state of a power transistor at high frequencies. Output voltage or current is varied by adjusting the control variable, commonly named the duty cycle. The operating steady-state regime can be done with an open loop or closed loop configuration, by using the appropriate controller. Important advances are reached in the field of both voltage control and current control [6–8].To perform successful and reliable practical control, numerical simulations are of great importance. Intensive and repetitive simulations are also needed to study particular phenomena such as bifurcations [9,10] and chaos [10,11]. It is shown in the literature that a lot of piecewise dynamic systems exhibit chaotic behaviour [12,13]. DC-DC converters are nonlinear periodically controlled systems that operate under practically piecewise dynamic trajectories. Chaotic regimes are confirmed for various switching power converters.To check the existence of bifurcations and chaos in a dynamic system, a fine-tuning procedure of a bifurcation parameter is required. This increases considerably the computing time, and a great amount of patience is required to establish adequate results. Because of the high switching frequency of the PWM technique [14], to solve the model of a DC-DC boost inverter, an integration step smaller than one microsecond is required. This makes the integration process time consuming even for short simulations. Thus, a fast, but accurate, method is required to analyse the dynamic behaviour of the converter.The first goal of this work is to develop a type-like discrete integration process with precise and fast results. For one switching period, we compute only two or a maximum of three breaking points depending on whether we treat a continuous conduction mode (CCM) or a discontinuous conduction mode (DCM) of the inductor current [1,2]. Furthermore, with each segment of the dynamic trajectory, an exact analytic formula is associated.The second goal of the paper is to use these results in developing a discrete iterative map formulated as in standard discrete time series models [15,16]. A Jacobian matrix as found with an iterative map is defined and used to compute Lyapunov exponents [12,13] via the QR factorisation technique.In switching inverter topologies, a current peak control (CPC) is frequently used [3] to impose an adequate operating point. In such case, the feedback control block is composed of a saw tooth generator, a comparator, and a latch. A drive circuit uses latch output to generate the gating pulses. The switch is turned on at the beginning of each switching period and turned off if the inductor current becomes greater than a reference value and remains in the off state until the beginning of the next cycle. Simulations of this control system are frequently done in Matlab/Simulink environment. In this paper, we will demonstrate that there is no need for this control block. Only a simple relation is needed to compute transient duty cycle. Furthermore, the program is realised in terms of a simple and fast Matlab m-file.According to these goals, the paper is organised as follows. Section2 develops fundamentals of the inverter. It includes the equivalent electrical circuit modelling the DC-DC boost inverter, commutation equations, properties of DCM and CCM regimes, and dynamical submodels associated with ON and OFF states. Section 3 develops exact and analytical relations of dynamic variables for both ON and OFF states. This section proposes, in particular, a simple correction of the time interval associated with DCM regime. Section 4 deals with fundamental steady-state equations. For both DCM regime and CCM regime, mean values of inductor current and output voltage are detailed and expressed. Effect of inductor inductance, output load resistance, and switching period on the boundary between DCM and CCM is graphically explained. Section 5 presents the main steps of the dynamic simulation algorithm and discusses how to implement the current peak control. In Section 6, we develop the procedure of obtaining a discrete map model and discuss how to conduct the computation of Lyapunov exponents. Finally, Section 7 presents various simulation cases. Results obtained in open-loop and closed-loop control are presented and commented. The discussion is particularly focused on tuning the bifurcation parameter and analysing chaos. Computing time is also considered in the scheme evaluation.


## 2. Fundamentals of a DC-DC boost inverter

Figure1 shows an equivalent circuit diagram of a boost DC-DC inverter supplying a pure resistive loadRo. ParametersE, L, and C denote, respectively, the input voltage source, the inductance of the inductor, and the output filtering capacitor. In this work, we consider the most widely used model [2, 8] characterised by ideal components. This means that the power transistor is an ideal switch; the inductor, filtering capacitor, and diode are lossless. In the conduction mode, the voltage across the diode is zero. The power transistor works as short circuit or as an open circuit depending on whenever the signal command ut equals 1 or0. Along switching time, boost inverter equivalent circuit shown by Figure 1 takes two forms as shown by Figures 2(a) and Figure 2(b) [14]. During the ON state, the transistor behaves like a short and no current flows through the diode. Therefore, the inductor becomes in fact disconnected from the parallel part Ro−C and the positive voltage of the capacitor biases the diode. The current flowing through the inductor increases; as a result, the electromagnetic energy in the inductance L increases. At the same time, the filtering capacitor discharges into the load resistance Ro, and therefore the voltage across the capacitor decreases. During the OFF state, the power transistor behaves like an open circuit, and the diode is conducting. Therefore, the input voltage supplies directly to the load. The voltage across the capacitor increases and the current through the inductor decreases because the electromagnetic energy is transformed into electrostatic energy.Figure 1
Boost inverter equivalent circuit.Figure 2
(a) Equivalent circuit in ON state case. (b) Equivalent circuit in OFF state case.
(a)(b)Commandut is a PWM signal characterised by a switching period T and a duty cycleα. In the following, this period will be divided into three time intervals t1, t2, and t3 as indicated by equations (1) and (2). The first interval corresponds to the ON state given by ut=1. The second and third intervals are associated with the OFF state corresponding to ut=0. In these equations, coefficients α,β,γ correspond, respectively, to ratios of t1,t2,t3 with respect to the switching periodT. Coefficient αis known by duty cycle. Coefficient β is the complement of α in the case of continuous conduction mode (CCM). Coefficient γ is the complement of α+β in the case of discontinuous conduction mode (DCM).(1)t1=αT,t2=βT,t3=γT,(2)t1+t2+t3=T,α+β+γ=1.This organisation aims to take into account continuous conduction mode (CCM) and discontinuous conduction mode (DCM) of inductor current. As previously indicated, inductor currentiℓt increases during the ON state and decreases during the OFF state. Let us consider that the switching period T begins with an inductor current Iℓ0 and a capacitor voltage Vc0, and just when the OFF state takes place, these variables are, respectively, Iℓ1and Vc1. If iℓt in the OFF state remains positive until the end of the period, we have the CCM case and t3=0; therefore, t2=βT=1−αT. At the end of the period T, the current through the inductor and voltage across the capacitor are denoted Iℓ2 and Vc2, respectively. If iℓtreaches zero before the period finishes, we have the DCM case, t3>0 andt2=βT<1−αT. In this case, the current through the inductor remains null during t3 because the diode is biased. However, the voltage across the capacitor decreases exponentially from the value Vc2 to a third value Vc3. Figures 3(a) and 3(b) sketch this scenario in terms of inductor current iℓt and capacitor voltage vct. These figures correspond to the transient regime and report the possible breaking points for a DCM case. For both DCM and CCM regimes, we establish the following commutation equation:(3)vℓ=E−1−uvo,vo=vc.Figure 3
(a) Illustrative evolution of inductor current. (b) Illustrative evolution of capacitor voltage.
(a)(b)Thus, without any particular difficulty, the following dynamic model is derived:(4)diℓdt=E−1−uvcL,dvcdt=1−uRoiℓ−vcRoC.Depending on whether we are dealing with an ON state or an OFF state, we have to treat two different models. In the ON state, the model becomes equivalent to two independent submodels as given by systems (5) and (6). In the OFF state, we have the model given by systems (7) and (8).(5)diℓdt=EL,(6)dvcdt=−vcRoC,(7)diℓdt=E−vcL,(8)dvcdt=Roiℓ−vcRoC.For the OFF state case, isolatingvcfrom (7) and establishing its time derivative yields(9)vc=E−Ldiℓdt,(10)dvcdt=−Ld2iℓdt2.Thus, for the OFF state case, equations (9) and (10), combined with equation (8), result in(11)d2iℓdt2+1τcdiℓdt+iℓτℓτc−Ierτℓτc=0,(12)τc=RoC,τℓ=LRo,Ier=ERo.The second-order linear ODE (11) furnishes the following characteristic polynomial:(13)s2+sτc+1τℓτc=0.With the conditionL<4Ro2C, this polynomial admits two complex conjugate poles which means that we have a damped oscillating system. The time constant τ and the pulsation ω are expressed by(14)s=−1τ±jω,τ=2τc=2RoC,ω=1τ2ττℓ−1.


## 3. Temporal Solution of the Mathematical Model

### 3.1. Analytical Solution of ON State Case

Here, we have to solve a simple dynamic system ((5) and (6)) which is completely decoupled into two independent subsystems. The inductor current and capacitor voltage of the boost inverter are of the form(15)iℓt=Iℓ0+ELt,vct=Vc0e−t/τc.This solution is verified by the construction ofiℓ0=Iℓ0 andvc0=Vc0. Iℓ1 and Vc1 of iℓt and vct at t1=αT are expressed by(16)Iℓ1=iℓt1=Iℓ0+EαTL,Vc1=vct1=Vc0e−αT/τc.


### 3.2. Analytical Solution of OFF State Case

Solutioniℓt of (11) is naturally composed of two terms. The first term is the particular solution corresponding to the case where iℓtis constant. The second one is the homogeneous solution which is a sinusoidal damped variable. Thus, we get(17)iℓt=Ier+η1sinθ+η2cosθe−t/τ,(18)θ=ωt.Fort=0, we have iℓ0=Iℓ1 which implies(19)η2=Iℓ1−Ier.To identify parameterη1, we obtain the derivative with respect to time of equation (17) and make it equal to equation (7). Therefore, we obtain(20)diℓdt=−ωη2+η1τsinθ+η2τ−ωη1cosθe−t/τ=E−vcL.By applying this expression att=0, one deduces(21)η1=Iℓ1−Ierωτ+E−Vc1Lω.The behaviour of the voltage across the capacitor is derived from equation (20). This results in(22)vct=E+η3sinθ+η4cosθe−t/τ,(23)η3=Lωη2+η1ωτ,η4=Lωη2ωτ−η1.To compute valuesIℓ2 of iℓt and Vc2 of vct at t2, let us first assume that we are in the CCM regime by setting γ=0 andβ=1−α or equivalently by setting t3=0 andt2=1−αT. The following equations are obtained. Note that all coefficients η1,η3,η4,η5are functions ofδ=ωβT.(24)Iℓ2=Ier+η5η1sinδ+η2cosδ,(25)Vc2=E+η5η3sinδ+η4cosδ,(26)η5=e−δ/ωτ,δ=ωβT,β=1−α.IfIℓ2 computed by equation (24) is positive, the assumption of CCM holds and the computed values of Iℓ2 and Vc2 are valid. If Iℓ2 is negative, this means that the assumption is false and we are in DCM case. Consequently, coefficient β is less than 1−α and must be equal to the solution satisfying(27)Iℓ2=Ier+η5δη1δsinδ+η2cosδ=0.This relation is nonlinear in terms ofβ and needs to be solved iteratively by the Newton–Raphson method. To ovoid additional computing time, we propose to compute β by linearizing iℓt around t=0 and setting the result to zero. This approach is based on the fact that the damping time constant τ is big enough when compared with the switching period, which is true in the real world. So, applying a first-order Taylor expansion to equation (20) yields(28)iℓt=Iℓ1+diℓdt|t=0,t=Iℓ1−Vc1−ELt.Thus, we obtain(29)β=LIℓ1Vc1−ET.The voltage of the capacitorVc2 at t2=βT is recomputed by using equation (25) that continues to be valid with the new value ofβ<1−α. During the remaining time intervalt∈α+βTT, the inductor current is kept null; iℓt=Iℓ2=0. However, the voltage across the capacitor decreases exponentially from the value Vc2 to a third valueVc3. At the end of the switching period, this voltage takes the value of(30)Vc3=Vc2e−γT/τ,γ=1−α−β.Note finally that according to Figures3(a) and 3(b), the mean values of the current through the inductor and the voltage of the capacitor are calculated at each switching period by the following equations. In the case of CCM, the third term in equation (31) must be removed because γis null in this situation.(31)Iℓ=αIℓ0+Iℓ1+βIℓ1+Iℓ22,(32)Vc=αVc0+Vc1+βVc1+Vc2+γVc2+Vc32.


## 3.1. Analytical Solution of ON State Case

Here, we have to solve a simple dynamic system ((5) and (6)) which is completely decoupled into two independent subsystems. The inductor current and capacitor voltage of the boost inverter are of the form(15)iℓt=Iℓ0+ELt,vct=Vc0e−t/τc.This solution is verified by the construction ofiℓ0=Iℓ0 andvc0=Vc0. Iℓ1 and Vc1 of iℓt and vct at t1=αT are expressed by(16)Iℓ1=iℓt1=Iℓ0+EαTL,Vc1=vct1=Vc0e−αT/τc.


## 3.2. Analytical Solution of OFF State Case

Solutioniℓt of (11) is naturally composed of two terms. The first term is the particular solution corresponding to the case where iℓtis constant. The second one is the homogeneous solution which is a sinusoidal damped variable. Thus, we get(17)iℓt=Ier+η1sinθ+η2cosθe−t/τ,(18)θ=ωt.Fort=0, we have iℓ0=Iℓ1 which implies(19)η2=Iℓ1−Ier.To identify parameterη1, we obtain the derivative with respect to time of equation (17) and make it equal to equation (7). Therefore, we obtain(20)diℓdt=−ωη2+η1τsinθ+η2τ−ωη1cosθe−t/τ=E−vcL.By applying this expression att=0, one deduces(21)η1=Iℓ1−Ierωτ+E−Vc1Lω.The behaviour of the voltage across the capacitor is derived from equation (20). This results in(22)vct=E+η3sinθ+η4cosθe−t/τ,(23)η3=Lωη2+η1ωτ,η4=Lωη2ωτ−η1.To compute valuesIℓ2 of iℓt and Vc2 of vct at t2, let us first assume that we are in the CCM regime by setting γ=0 andβ=1−α or equivalently by setting t3=0 andt2=1−αT. The following equations are obtained. Note that all coefficients η1,η3,η4,η5are functions ofδ=ωβT.(24)Iℓ2=Ier+η5η1sinδ+η2cosδ,(25)Vc2=E+η5η3sinδ+η4cosδ,(26)η5=e−δ/ωτ,δ=ωβT,β=1−α.IfIℓ2 computed by equation (24) is positive, the assumption of CCM holds and the computed values of Iℓ2 and Vc2 are valid. If Iℓ2 is negative, this means that the assumption is false and we are in DCM case. Consequently, coefficient β is less than 1−α and must be equal to the solution satisfying(27)Iℓ2=Ier+η5δη1δsinδ+η2cosδ=0.This relation is nonlinear in terms ofβ and needs to be solved iteratively by the Newton–Raphson method. To ovoid additional computing time, we propose to compute β by linearizing iℓt around t=0 and setting the result to zero. This approach is based on the fact that the damping time constant τ is big enough when compared with the switching period, which is true in the real world. So, applying a first-order Taylor expansion to equation (20) yields(28)iℓt=Iℓ1+diℓdt|t=0,t=Iℓ1−Vc1−ELt.Thus, we obtain(29)β=LIℓ1Vc1−ET.The voltage of the capacitorVc2 at t2=βT is recomputed by using equation (25) that continues to be valid with the new value ofβ<1−α. During the remaining time intervalt∈α+βTT, the inductor current is kept null; iℓt=Iℓ2=0. However, the voltage across the capacitor decreases exponentially from the value Vc2 to a third valueVc3. At the end of the switching period, this voltage takes the value of(30)Vc3=Vc2e−γT/τ,γ=1−α−β.Note finally that according to Figures3(a) and 3(b), the mean values of the current through the inductor and the voltage of the capacitor are calculated at each switching period by the following equations. In the case of CCM, the third term in equation (31) must be removed because γis null in this situation.(31)Iℓ=αIℓ0+Iℓ1+βIℓ1+Iℓ22,(32)Vc=αVc0+Vc1+βVc1+Vc2+γVc2+Vc32.


## 4. Fundamental Steady-State Equations

As previously outlined, there are two possible modes for the boost converter: CCM and DCM. In CCM, the inductor current flows continuously above zero during the totality of the switching period. The output voltage can be described by a relatively simple expression making control reliable via the duty cycleα. In addition, the inductor voltage waveform is almost a constant signal that results in an inductor current ripple close to a triangular signal. In the DCM, the inductor current reaches zero before the end of the switching period. The output voltage is described by a high nonlinear equation in terms of the duty cycle making it difficult to control. Figures 4(a) and 4(b) sketch general shapes of the inductor and diode current in the case of DCM regime.Figure 4
(a) General shape of inductor current in DCM. (b) General shape of diode current in DCM.
(a)(b)To evaluate the input current and the output voltage in steady operating conditions, we use the small ripple approximation hypothesis that permits to replace instantaneous variables in minor time intervals by their mean values. Figures5(a) and 5(b) report waveforms of inductor voltage vℓt and capacitor current ict during one switching period. During the three possible time intervals forming the switching period as defined by Figures 3(a) and 3(b), from Figures 5(a) and 5(b), we establish(33)t∈t1=αT:Vℓ=E,Ic=−VoRo=−Io,t∈t2=βT:Vℓ=E−Vo,Ic=Iℓ−Io,t∈t3=γT:Vℓ=0,Ic=−Io.Figure 5
(a) Simplified waveform of inductor voltage. (b) Simplified waveform of capacitor current.
(a)(b)By applying the principle of charge-discharge balance of inductor electromagnetic energy and capacitor electrostatic energy, mean valuesVℓand Ic of vℓt and ict must be zero. So, we deduce(34)αE+βE−Vo=0,−1−βIo+βIℓ−Io=0.Solving these equations for unknownVo andIℓ, we generate the steady-state solution:(35)Vo=α+βEβ,(36)Iℓ=Ioβ=α+βIerβ2.Also, note that equations (35) and (36) established above for steady state of CCM case can be found from models (5) and (6) if we substitute variable 1−u by its mean value βand we set time derivatives to zero as in classical continuous dynamic systems.


### 4.1. Continuous Conduction Mode

In the CCM case, we haveβ=1−α, and consequently equations (35) and (36) become(37)Vo=E1−α,(38)Iℓ=Ier1−α2.With this assumption, the peak-to-peak ripple of the inductor current and capacitor voltage is approximated by the following equations:(39)Δiℓ≃EαTL,(40)ΔvC≃IerαTC1−α.


### 4.2. Discontinuous Conduction Mode

In the DCM case, the initial inductor currentIℓ0 is zero and peak current Iℓ1 is such that(41)Iℓ1=EαTL.The inductor peak currentIℓ1 and diode average current Id are defined by(42)Id=Io=βIℓ12=βEαT2L.This allows us to deduce(43)β=KVoαE,K=2LRoT.Substitutingβ into equation (35) results in(44)KVoVo−E−αE2=0.Solving equation (44) in terms of Vo allows us to define the output voltage and input current:(45)Vo=E21+1+4α2K,(46)Iℓ=Eα2+KVoRoK.Just when the steady-state operating point passes from the CCM to the DCM regime, i.e., just at the boundary separating these regimes, the voltage solution given by equation (45) becomes greater than that of equation (35). That is:(47)1+1+4α2K>21−α.Arranging this, one concludes that DCM is characterised by the following inequality:(48)fα=α1−α2>K.Functionfα of inequality (48) passes by a maximum point defined by fmax=4/27 for α=1/3 as illustrated by Figure 6. If K≥fmax, the inductor current is continuous over the complete range of the switching period and relations (37) and (38) hold for α∈01. If K<fmax, the inductor current has a discontinuous mode in the interval α∈α1α1 and a continuous mode in the remaining intervals α∈0α1 and α∈α21. Parameters α1and α2 are the solutions of equationfα=K. In the central region where we have DCM, the output voltage and input current are given by equations (45) and (46).Figure 6
DCM-CCM boost inverter curve.


## 4.1. Continuous Conduction Mode

In the CCM case, we haveβ=1−α, and consequently equations (35) and (36) become(37)Vo=E1−α,(38)Iℓ=Ier1−α2.With this assumption, the peak-to-peak ripple of the inductor current and capacitor voltage is approximated by the following equations:(39)Δiℓ≃EαTL,(40)ΔvC≃IerαTC1−α.


## 4.2. Discontinuous Conduction Mode

In the DCM case, the initial inductor currentIℓ0 is zero and peak current Iℓ1 is such that(41)Iℓ1=EαTL.The inductor peak currentIℓ1 and diode average current Id are defined by(42)Id=Io=βIℓ12=βEαT2L.This allows us to deduce(43)β=KVoαE,K=2LRoT.Substitutingβ into equation (35) results in(44)KVoVo−E−αE2=0.Solving equation (44) in terms of Vo allows us to define the output voltage and input current:(45)Vo=E21+1+4α2K,(46)Iℓ=Eα2+KVoRoK.Just when the steady-state operating point passes from the CCM to the DCM regime, i.e., just at the boundary separating these regimes, the voltage solution given by equation (45) becomes greater than that of equation (35). That is:(47)1+1+4α2K>21−α.Arranging this, one concludes that DCM is characterised by the following inequality:(48)fα=α1−α2>K.Functionfα of inequality (48) passes by a maximum point defined by fmax=4/27 for α=1/3 as illustrated by Figure 6. If K≥fmax, the inductor current is continuous over the complete range of the switching period and relations (37) and (38) hold for α∈01. If K<fmax, the inductor current has a discontinuous mode in the interval α∈α1α1 and a continuous mode in the remaining intervals α∈0α1 and α∈α21. Parameters α1and α2 are the solutions of equationfα=K. In the central region where we have DCM, the output voltage and input current are given by equations (45) and (46).Figure 6
DCM-CCM boost inverter curve.


## 5. Dynamic Simulation Algorithm and Control

Now that all necessary relations to realise the dynamic simulation of the boost converter are available, it is of importance to note that an integration step that corresponds to a one switching period is needed. We will first present how the integration process works. Second, we will discuss the possibility of implementation of the classical switching controller.


### 5.1. Dynamic Simulation Algorithm

The dynamic simulation algorithm is simple. It is based on the following main steps:(i)
Set the initial inductor current and capacitor voltageiℓ=Iℓ0=0 and vc=Vc0=0 and save these values as a first point.(ii)
ComputeIℓ1,Vc1 and save this point.(iii)
ComputeIℓ2,Vc2 using β=1−α. If Iℓ2≥0, save this point and go to step (iv); otherwise, go to step (v).(iv)
Reinitialise withIℓ0=Iℓ2 and Vc0=Vc2 and go to step (ii).(v)
Recomputeβ and Vc2 as previously explained and save the new point 0,Vc2. Then, compute Vc3, set Iℓ3=0, and save this point.(vi)
Reinitialise withIℓ0=0 and Vc0=Vc3 and go to step (ii).It is obvious that for one period, we have to save two or three points depending on whether we are in the CCM or DCM case. Furthermore, if we zoom in a plotted curve illustrating the result, we find a linear piecewise trajectory. The reason is evident.


### 5.2. Dynamic Control

Various interesting works in the field of control design of DC-DC boost inverter, such as classical control [8], sliding mode control [17], fuzzy control [18], and so on, can be found in the literature. Widely used approaches develop a second-order controller based on transfer functions for small signals. Here, the main goal is to implement the classical switching controller, which is by nature a current peak controller (CPC). The feedback control block is composed of a saw tooth generator, a comparator, and a latch. The output of the latch is used by the drive circuit that generates the gating pulses. The switch is turned on at the beginning of each switching period and turned off if the inductor current becomes greater than a programmed reference value Iref and remains in the OFF state until the beginning of the next cycle. During simulation, this is equivalent to computing the PWM duty cycle α from the initial current Iℓ0 and reference current Iref. Thus, at the beginning of the period, duty cycle α is computed by equation (49) which is equivalent to equation (41) where Iℓ1 is set equal to Iref. The simulation algorithm previously presented remains practically the same.(49)α=Iref−Iℓ0LTE.


## 5.1. Dynamic Simulation Algorithm

The dynamic simulation algorithm is simple. It is based on the following main steps:(i)
Set the initial inductor current and capacitor voltageiℓ=Iℓ0=0 and vc=Vc0=0 and save these values as a first point.(ii)
ComputeIℓ1,Vc1 and save this point.(iii)
ComputeIℓ2,Vc2 using β=1−α. If Iℓ2≥0, save this point and go to step (iv); otherwise, go to step (v).(iv)
Reinitialise withIℓ0=Iℓ2 and Vc0=Vc2 and go to step (ii).(v)
Recomputeβ and Vc2 as previously explained and save the new point 0,Vc2. Then, compute Vc3, set Iℓ3=0, and save this point.(vi)
Reinitialise withIℓ0=0 and Vc0=Vc3 and go to step (ii).It is obvious that for one period, we have to save two or three points depending on whether we are in the CCM or DCM case. Furthermore, if we zoom in a plotted curve illustrating the result, we find a linear piecewise trajectory. The reason is evident.


## 5.2. Dynamic Control

Various interesting works in the field of control design of DC-DC boost inverter, such as classical control [8], sliding mode control [17], fuzzy control [18], and so on, can be found in the literature. Widely used approaches develop a second-order controller based on transfer functions for small signals. Here, the main goal is to implement the classical switching controller, which is by nature a current peak controller (CPC). The feedback control block is composed of a saw tooth generator, a comparator, and a latch. The output of the latch is used by the drive circuit that generates the gating pulses. The switch is turned on at the beginning of each switching period and turned off if the inductor current becomes greater than a programmed reference value Iref and remains in the OFF state until the beginning of the next cycle. During simulation, this is equivalent to computing the PWM duty cycle α from the initial current Iℓ0 and reference current Iref. Thus, at the beginning of the period, duty cycle α is computed by equation (49) which is equivalent to equation (41) where Iℓ1 is set equal to Iref. The simulation algorithm previously presented remains practically the same.(49)α=Iref−Iℓ0LTE.


## 6. Discrete Map and Lyapunov Exponents of DC-DC boost inverter

In this section of the paper, the load resistance is chosen to ensure that the circuit operates theoretically in the continuous mode. The sequence of the braking points computed for a set of some consecutive switching periods is used to define a discrete iterative map process. The obtained time series added to the Jacobian matrix of the map permits the computation of Lyapunov exponents. If the largest Lyapunov exponent (LLE) is positive, the system enters in a chaotic behaviour. Points corresponding to the minimum peak current form the selected discrete time series. In this sense, let us indicate two successive switching periods byk,k+1 and denote the couple Iℓ0,Iℓ2 of inductor current and the corresponding couple Vc0,Vc2 of voltage capacitor as follows:(50)Iℓ0=xk,Iℓ2=xk+1,Vc0=yk,Vc2=yk+1.In this way, equation (50) of the duty cycle and equations (21), (23), and (26) of coefficients η1,η3,η4,η5 are reformulated as functions of the discrete iterative variables.(51)δ=δxk=ωT−LωEIref−xk,η0=η0xk=e−2ωT−δxk/ωτ,η1=η1xk,yk=η2ωτ+E−η0xkykLω,η3=η3xk,yk=Lωη2+η1xk,ykωτ,η4=η4xk,yk=Lωη2ωτ−η1xk,yk,η5=η5xk=e−δxk/ωτ.This leads to the iterative process defined by(52)δxk=ωT−LωEIref−xk,(53)fxxk,yk=Ier+η5xkη1xk,yksinδxk+η2cosδxk,(54)fyxk,yk=E+η5xkη3xk,yksinδxk+η4xk,ykcosδxk,xk+1=fxxk,yk,yk+1=fyxk,yk.LetJx,y denote the Jacobian matrix associated with the system above. The computation of this matrix at each iteration furnishes the possibility to compute Lyapunov exponents λ via the QR factorisation method. This is the most suitable method to compute these exponents for a time series process with known Jacobian. At each iteration k, the Lyapunov exponents are computed as follows:(55)λk=1k∑kLogDiagR.In this relation,DiagR denotes the diagonal of matrix R computed in Matlab environment by the following syntax. In this process, matrix Q is initialised by the identity matrix. Note also that because Jx,y is space consuming, the development of its elements is reported in the Appendix.(56)Q,Rk+1=qrJkQk.Kaplan–Yorke dimension [19] is a powerful tool that describes the complexity of chaotic attractors. This dimension is also known by the Kaplan–Yorke conjecture. First, Lyapunov exponents are arranged in decreasing order λ1>λ2>⋯λn, and index j corresponding to inequality (57) is determined. Then, the conjecture is that the dimension of the attractor is given by (53). For the studied case, equation (58) is equivalent to equation (59).(57)∑i=1jλj≥0,∑i=1j+1λj<0,(58)Dky=j+1λj+1∑i=1jλi,(59)Dky=1+λ1λ2.Reference peak currentIref will be considered a bifurcation parameter. By varying Iref, we will observe how the circuit changes its behaviour from a stable system to a chaotic system via the period-doubling mechanism.


## 7. Results and Comments

The boost inverter parameters used to illustrate this study are reported in Table1. Results are carried out in Matlab environment. Necessary software is written in a m-file code. For all simulations, the initial inductor current and capacitor voltage are set to zero.Table 1
System parameters.
ParameterNotationValueRated input voltageE10VRated resistance of the loadRo20ΩRated switching periodT100μsInductor inductanceL1000μHFiltering capacitorC12μFAccording to these parameters, the frequency and time constant characterising dynamic inverter behaviour in the OFF state aref=1.415kHz and τ=480μs, respectively. This time constant is 4.8 times greater than the switching period. Equation (48) implies that the DCM boundary coefficient is K=1>4/27. That is, the inverter works in CCM for full range of the duty cycle because the used rated resistance load is less than the critical load resistance whose value is Ro=135Ω. To simulate a DCM case, we use Ro=200Ω. In this case, we have K=0.1<4/27 that results in a DCM in the range α∈0.13300.5874.


### 7.1. Simulation No. 1

Two simulation tests are carried out. The first test corresponds toα=0.5 and Ro=20Ω. A CCM regime is expected. Using equation (38), we found Iℓ=1.967A. The second simulation is realised with α=0.3 and Ro=200Ω and a DCM regime is expected. Using equation (46), we foundIℓ=0.1237A. Figures 7(a) and 7(b) show steady-state evolution of inductor current for these tests, respectively, and confirm the predicted Iℓvalues.Figure 7
(a) CCM steady state of inductor current in open-loop case. (b) DCM steady state of inductor current in open-loop case.
(a)(b)


### 7.2. Simulation No. 2

We consider a load resistanceRo=20Ω, and we simulate two tests of inductor current peak: Iref1=2A andIref2=3A. Figures 8(a) and 8(b) show steady-state evolution of inductor current for these two tests, respectively. Iℓ2 and Vc2 of inductor current and capacitor voltage at the end of the switching period are saved. Figures 9(a) and 9(b) show phase portraits of Iℓ2 and Vc2 in dot plot form. Figures 10(a) and 10(b) show the dot plot of Iℓ2=fα and Vc2=fα, respectively. All these figures show without ambiguity the existence of chaos phenomenon in boost DC-DC dynamics when controlled in CPC type. To show areas with period doubling and those of intermittence, a fine variation of bifurcation parameter is required.Figure 8
(a) CCM chaotic steady state of inductor current in closed-loop test withIref=2A. (b) CCM chaotic steady state of inductor current in closed-loop test with Iref=3A.
(a)(b)Figure 9
(a) Phase portrait of CCM chaotic steady state of inductor current in closed-loop test withIref=3A. (b) Phase portrait of CCM chaotic steady state of voltage capacitor in closed-loop test with Iref=3A.
(a)(b)Figure 10
(a) Closed-loop CCM chaotic steady state of inductor current versus duty cycleIref=3A. (b) Closed-loop CCM chaotic steady state of capacitor voltage versus duty cycle Iref=3A.
(a)(b)


### 7.3. Simulation No. 3

The load resistanceRo=20Ω is used. We tune the bifurcation parameter Iref with a small variation in the range of 1A to 5A. For each value of Iref, we discard the transient part of the trajectory and save the remaining steady-state part. At the end, all results are plotted in dot plot form. Lyapunov exponents are calculated at the same time of dynamic trajectories. Figures 11(a) and 11(b) show bifurcation diagrams of inductor current and capacitor voltage, respectively. We observe the phenomenon of period doubling and then the entrance of the system in a chaotic behaviour. Figures 12(a) and 12(b) furnish more details on this behaviour. It is observed that period doubling begins with Iref=1.69A. Then, atIref=2.37A, Iref=2.63A, and Iref=2.7A, the system enters in 4T, 8T, and 16T type orbits, respectively. Two intermittence [20] areas are observed around Iref=4.1A andIref=4.9A. Figures 13(a) and 13(b) show Lyapunov exponents and Kaplan–Yorke dimension evolution versus the bifurcation parameter. As the system is of dimension 2, we generate two Lyapunov exponents. It is found that one (LLE) becomes positive for a bifurcation parameter Iref>2.8A. This indicates the beginning of a chaotic region. The second exponent is negative. The largest exponent is red colored while the other is blue colored. Note that LLE detects intermittence areas. Kaplan–Yorke dimension evolution is similar to LLE because LLE here is dominant. In the chaotic region, this dimension increases but remains inferior to 2, the physical dimension of the system.Figure 11
(a) Bifurcation diagram of inductor current. (b) Bifurcation diagram of capacitor voltage.
(a)(b)Figure 12
(a) First zoom on bifurcation diagram of inductor current. (b) Second zoom on bifurcation diagram of inductor current.
(a)(b)Figure 13
(a) Effect of bifurcation parameter on Lyapunov exponents. (b) Effect of bifurcation parameter on Kaplan–Yorke dimension.
(a)(b)


## 7.1. Simulation No. 1

Two simulation tests are carried out. The first test corresponds toα=0.5 and Ro=20Ω. A CCM regime is expected. Using equation (38), we found Iℓ=1.967A. The second simulation is realised with α=0.3 and Ro=200Ω and a DCM regime is expected. Using equation (46), we foundIℓ=0.1237A. Figures 7(a) and 7(b) show steady-state evolution of inductor current for these tests, respectively, and confirm the predicted Iℓvalues.Figure 7
(a) CCM steady state of inductor current in open-loop case. (b) DCM steady state of inductor current in open-loop case.
(a)(b)


## 7.2. Simulation No. 2

We consider a load resistanceRo=20Ω, and we simulate two tests of inductor current peak: Iref1=2A andIref2=3A. Figures 8(a) and 8(b) show steady-state evolution of inductor current for these two tests, respectively. Iℓ2 and Vc2 of inductor current and capacitor voltage at the end of the switching period are saved. Figures 9(a) and 9(b) show phase portraits of Iℓ2 and Vc2 in dot plot form. Figures 10(a) and 10(b) show the dot plot of Iℓ2=fα and Vc2=fα, respectively. All these figures show without ambiguity the existence of chaos phenomenon in boost DC-DC dynamics when controlled in CPC type. To show areas with period doubling and those of intermittence, a fine variation of bifurcation parameter is required.Figure 8
(a) CCM chaotic steady state of inductor current in closed-loop test withIref=2A. (b) CCM chaotic steady state of inductor current in closed-loop test with Iref=3A.
(a)(b)Figure 9
(a) Phase portrait of CCM chaotic steady state of inductor current in closed-loop test withIref=3A. (b) Phase portrait of CCM chaotic steady state of voltage capacitor in closed-loop test with Iref=3A.
(a)(b)Figure 10
(a) Closed-loop CCM chaotic steady state of inductor current versus duty cycleIref=3A. (b) Closed-loop CCM chaotic steady state of capacitor voltage versus duty cycle Iref=3A.
(a)(b)


## 7.3. Simulation No. 3

The load resistanceRo=20Ω is used. We tune the bifurcation parameter Iref with a small variation in the range of 1A to 5A. For each value of Iref, we discard the transient part of the trajectory and save the remaining steady-state part. At the end, all results are plotted in dot plot form. Lyapunov exponents are calculated at the same time of dynamic trajectories. Figures 11(a) and 11(b) show bifurcation diagrams of inductor current and capacitor voltage, respectively. We observe the phenomenon of period doubling and then the entrance of the system in a chaotic behaviour. Figures 12(a) and 12(b) furnish more details on this behaviour. It is observed that period doubling begins with Iref=1.69A. Then, atIref=2.37A, Iref=2.63A, and Iref=2.7A, the system enters in 4T, 8T, and 16T type orbits, respectively. Two intermittence [20] areas are observed around Iref=4.1A andIref=4.9A. Figures 13(a) and 13(b) show Lyapunov exponents and Kaplan–Yorke dimension evolution versus the bifurcation parameter. As the system is of dimension 2, we generate two Lyapunov exponents. It is found that one (LLE) becomes positive for a bifurcation parameter Iref>2.8A. This indicates the beginning of a chaotic region. The second exponent is negative. The largest exponent is red colored while the other is blue colored. Note that LLE detects intermittence areas. Kaplan–Yorke dimension evolution is similar to LLE because LLE here is dominant. In the chaotic region, this dimension increases but remains inferior to 2, the physical dimension of the system.Figure 11
(a) Bifurcation diagram of inductor current. (b) Bifurcation diagram of capacitor voltage.
(a)(b)Figure 12
(a) First zoom on bifurcation diagram of inductor current. (b) Second zoom on bifurcation diagram of inductor current.
(a)(b)Figure 13
(a) Effect of bifurcation parameter on Lyapunov exponents. (b) Effect of bifurcation parameter on Kaplan–Yorke dimension.
(a)(b)


## 8. Conclusion

We have developed in this paper a study on DC-DC boost inverter based on two levels. The first level of the paper develops a like-discrete integration process based on determination of precise analytical relations of breaking points characterising the switching behaviour of DC-DC boost inverter. The presented scheme gives good results in a very fast manner because for one switching period, we compute only two three breaking points depending on whether we have a continuous conduction mode (CCM) or a discontinuous conduction mode (DCM) of the inductor current. Performance of the developed analytical solution is thus successfully proved. The second level of the paper develops a discrete iterative map as in standard discrete time series models. Jacobian matrix of the found iterative map is defined and used to compute Lyapunov exponents. These exponents confirm the existence of chaos in the system behaviour when varying the reference value of the controlled inductor current. This level of the study is limited to CCM regime. On the other hand, it is obvious that established iterative map can be easily extended to other inverter topologies such as interleaved configurations [21].



---

*Source: 9162259-2021-08-14.xml*