# Chaotic Behavior in a Switched Dynamical System

Authors: Fatima El Guezar; Hassane Bouzahir
Journal: Modelling and Simulation in Engineering
 (2008)

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## Abstract

We present a numerical study of an example of piecewise linear
systems that constitute a class of hybrid systems. Precisely, we study the
chaotic dynamics of the voltage-mode controlled buck converter circuit in an
open loop. By considering the voltage input as a bifurcation parameter, we
observe that the obtained simulations show that the buck converter is prone
to have subharmonic behavior and chaos. We also present the corresponding
bifurcation diagram. Our modeling techniques are based on the new French
native modeler and simulator for hybrid systems called Scicos (Scilab connected
object simulator) which is a Scilab (scientific laboratory) package.
The followed approach takes into account the hybrid nature of the circuit.


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## Body

## 1. Introduction

Hybrid
dynamical systems (HDSs) have attracted considerable attention in recent
years. HDS arise from the interaction
between continuous variable systems (i.e., systems that can be described by a
difference or differential equation) and discrete event systems (i.e., systems
where the state transitions are initiated by events that occur at discrete time
instants). Switched piecewise linear systems are an important class of hybrid
systems that are simple and can have very rich and typical nonlinear dynamics
such as bifurcations and chaos. As example, DC-DC switching converters are
switched piecewise linear systems [1].
The three basic power electronic converters buck, boost, and buck-boost are
variable structure systems that are highly nonlinear. This kind of piecewise
model may present nonlinear phenomena such as bifurcations and chaos. The study of
nonlinear dynamics of DC-DC converters started in 1984 by Brockett's and Wood's
research [2]. Since
then, chaos and nonlinear phenomena in power electronic circuits have stolen
the spotlight and have attracted the attention of different research groups.
Different nonlinear phenomena were investigated such as flip bifurcation or
period doubling and its related route to chaos
[3–5] or quasiperiodicity route to chaos
[6, 7] as well as border collision
bifurcation [1–3, 6–16]. There are many modeling techniques, programming
languages, and design toolsets for HDS. To model and simulate our HDS, we use
Scicos (Scilab connected object simulator) which is a Scilab package for
modeling and simulation of dynamical systems including both continuous and
discrete time subsystems [17, 18].
Scilab (scientific laboratory) is a scientific software package for numerical
computations that provides a powerful open computing environment for
engineering and scientific applications [10]. It has been developed at INRIA and ENPC and is freely
available for download. This paper aims to study and analyze some dynamic
phenomena that can occur in the voltage-mode controlled buck converter. We also
show from Scicos simulations that variation of the voltage input can lead to a
particular route to chaos. In Section 2, the general equation of a hybrid
dynamical system is briefly recalled. In Section 3, we explain the operation of
the voltage-mode controlled buck converter. Then, we introduce the state
equations of
the circuit in question. In Section 4, we comment on the obtained
Scicos simulations. We end by some concluding remarks.


## 2. Hybrid Dynamical System

The evolution
of an autonomous hybrid dynamical system can be described by [19](1)x˙(t)=f(x(t),q(t)),x(t0)=x0,q(t)=e(x(t),q(t−)),q(t0)=i0,where x(t) is the continuous state vector, q(t)∈Q={1,…,nq} denotes the discrete state, and q(t−) is the previous discrete state. The state
space is H=ℝn×Q, and the initial state is supposed belonging to
the set of initial conditions (x0,i0)∈H0⊆H.
The function e:ℝn×Q→Q describes the change of the discrete state.
The change from one distinct discrete state to another is called a transition
or a switch. A transition between two states i and j occurs if x(⋅) reaches the switch set   Si,j:Si,j={x:e(x,i)=j}.
Among important classes of hybrid systems, there are piecewise linear systems
that are described by(2)x˙(t)=fq(x)=A(q)x(t)+B(q),where A(q)∈ℝn×m and B(q)∈ℝn are matrices depending on q.


## 3. Voltage-Mode Controlled Buck Converter

### 3.1. Operation of Voltage-Mode Controlled Buck Converter

A voltage
feedback buck converter is represented in Figure1. It consists of a basic RLC
circuit, a diode, and a switching element S. The aim of the circuit is to
maintain a desired voltage, across the load resistance R, lower than the input
voltage E. This can be realized by the relieve of feedback PWM control. The PWM
control of a switched converter is achieved by obtaining a control voltage vcon(t),
as a linear combination of the output capacitor voltage vC(t) and a reference signal Vref in the form
of (3)vcon(t)=a(vC(t)−Vref),where a is the gain of the error amplifier. The
control voltage is compared with an externally generated sawtooth wave Vramp(t) given by(4)vramp(t)=VL+(VU−VL)tTt∈[0,T].The output of
this comparator is used to determine the state of the switch S, in
such way that S is off when vcon(t)≥vramp(t) and S is on when vcon(t)<vramp(t).Figure 1
Voltage-mode
controlled buck converter.


### 3.2. State Equations

When operating
in continuous conduction mode (CCM), two switch states can be identified as
follows:(i)
switch off and diode on;(ii)
switch on and diode off.
Whether the
switch is on or off, the buck converter can always be described as a second-order linear system, whose states are the voltage vC across the capacitor, and the current iL along the inductor. The general equation that
models operation of the buck converter takes the form(5)x˙(t)=fq(x)=A(q)x(t)+B(q),withq∈Q={1,2}.For q=1 and q=2,
we obtain the following two systems of differential equations:(6)Soff:x˙(t)=f1(x)=Ax(t)+B1,Son:x˙(t)=f2(x)=Ax(t)+B2,where(7)A=(−1RC1C−1L0),B1=(00),B2=(0EL),and x=(vciL) is the vector of the state variables.The border function is given by(8)β(x,t)=vcon(t)−vramp(t)=avc(t)−aVref−VL−(VU−VL)tT,fort∈[0,T].Therefore, the switching
sections of each subsystem Son and Soff are given by(9)βon,off={(x,t)∈ℝ2×ℝ:β(x,t)≥0},βoff,on={(x,t)∈ℝ2×ℝ:β(x,t)<0}.The buck converter in CCM
switches between two systems Son and Soff if the state reaches the switching sections βon,off and βoff,on.
Figures 2 and 3 show the corresponding Scicos schematic diagram and the
transition diagram, respectively.Figure 2
Scicos
schematic diagram.Figure 3
Transition
diagram of the buck converter.


### 3.3. Simulation Results and Comments

We choose the
parameter values:L=30 mH, T=400 microseconcds, R=22Ω, C=47μF, a=8.4, Vref=11.3 V, VL=3.8 V, and VU=8.2 V. We consider the input voltage E=30~47 V as a parameter of bifurcation. By varying E,
the circuit changes its qualitative behavior from a stable periodic system to
another situation that exhibits chaos. At first, using Scicos we draw the
one-parameter bifurcation diagram given in Figure 4 where the input voltage E is the bifurcation parameter and the sampled vC is the variable. By increasing E,
we observe at first glance that the displayed diagram (Figure 4) shows a period
doubling route to chaos. However, after a clear 4-T periodic operation, instead of appearance of
an immediate 8-T periodic operation, the system follows a 7-T periodic operation. This means that border
collision bifurcation comes into play and interrupts the normal period doubling
cascade. Here, this type of bifurcation is characterized by the intersection of
the bifurcation diagram with the upper border line defined by(10)vc=Vref+VUa.
Figure 4 shows clearly the
occurrence of this phenomenon at around the critical value Ec=41.45 V.Figure 4
Bifurcation
diagram.For different increasing values ofE,
we give the capacitor voltage wave form vC and its corresponding phase plane vC-iL.By choosingE=30 V, we get a fundamental periodic operation.
This periodic regime is possible just for small values of E.
Figures 5 and 6 show the fundamental periodic operation.
Figure 5 displays the
capacitor voltage wave form, and Figure 6 gives the corresponding phase plane.Figure 5
Fundamental
periodic operation. (E=30 V): time waveform of the capacitor
voltage.Figure 6
Fundamental
periodic operation. (E=30 V): phase plane.ForE=37.5 V and E=41 V, subharmonic operation has been found.
Figures 7 and 8 present 2-T periodic subharmonic operation, whereas Figures 9 and
10 illustrate 4-T periodic subharmonic operation.Figure 7
2-T subharmonic operation. (E=37.5 V): time waveform of the capacitor
voltage.Figure 8
2-T subharmonic operation. (E=37.5 V): phase plane.Figure 9
4-T subharmonic operation. (E=41 V): time waveform of the capacitor
voltage.Figure 10
4-T subharmonic operation. (E=41 V): phase plane.However, the chaotic operation is given forE=46.5 V. Figure 11 indicates a chaotic signal with
infinite order, and Figure 12 shows the phase plane vC-iL that corresponds to a chaotic attractor. In order
to qualify this chaotic behavior for beyond E=42 V, we  have also computed the Lyapunov
exponents λ from time series for a voltage range 41–47 V
(see Table 1).Figure 11
Chaotic
regime. (E=46.5 V): time waveform of the capacitor
voltage.Figure 12
Chaotic
regime. (E=46.5 V): phase plane.Actually, it
was shown in [9] that
most controlled DC-DC converters like the voltage-mode controlled buck
converter can be represented by piecewise smooth maps and such type of maps
generates robust chaos, defined by the absence of periodic windows and
coexisting attractors in some neighborhood of parameter space.


## 3.1. Operation of Voltage-Mode Controlled Buck Converter

A voltage
feedback buck converter is represented in Figure1. It consists of a basic RLC
circuit, a diode, and a switching element S. The aim of the circuit is to
maintain a desired voltage, across the load resistance R, lower than the input
voltage E. This can be realized by the relieve of feedback PWM control. The PWM
control of a switched converter is achieved by obtaining a control voltage vcon(t),
as a linear combination of the output capacitor voltage vC(t) and a reference signal Vref in the form
of (3)vcon(t)=a(vC(t)−Vref),where a is the gain of the error amplifier. The
control voltage is compared with an externally generated sawtooth wave Vramp(t) given by(4)vramp(t)=VL+(VU−VL)tTt∈[0,T].The output of
this comparator is used to determine the state of the switch S, in
such way that S is off when vcon(t)≥vramp(t) and S is on when vcon(t)<vramp(t).Figure 1
Voltage-mode
controlled buck converter.


## 3.2. State Equations

When operating
in continuous conduction mode (CCM), two switch states can be identified as
follows:(i)
switch off and diode on;(ii)
switch on and diode off.
Whether the
switch is on or off, the buck converter can always be described as a second-order linear system, whose states are the voltage vC across the capacitor, and the current iL along the inductor. The general equation that
models operation of the buck converter takes the form(5)x˙(t)=fq(x)=A(q)x(t)+B(q),withq∈Q={1,2}.For q=1 and q=2,
we obtain the following two systems of differential equations:(6)Soff:x˙(t)=f1(x)=Ax(t)+B1,Son:x˙(t)=f2(x)=Ax(t)+B2,where(7)A=(−1RC1C−1L0),B1=(00),B2=(0EL),and x=(vciL) is the vector of the state variables.The border function is given by(8)β(x,t)=vcon(t)−vramp(t)=avc(t)−aVref−VL−(VU−VL)tT,fort∈[0,T].Therefore, the switching
sections of each subsystem Son and Soff are given by(9)βon,off={(x,t)∈ℝ2×ℝ:β(x,t)≥0},βoff,on={(x,t)∈ℝ2×ℝ:β(x,t)<0}.The buck converter in CCM
switches between two systems Son and Soff if the state reaches the switching sections βon,off and βoff,on.
Figures 2 and 3 show the corresponding Scicos schematic diagram and the
transition diagram, respectively.Figure 2
Scicos
schematic diagram.Figure 3
Transition
diagram of the buck converter.


## 3.3. Simulation Results and Comments

We choose the
parameter values:L=30 mH, T=400 microseconcds, R=22Ω, C=47μF, a=8.4, Vref=11.3 V, VL=3.8 V, and VU=8.2 V. We consider the input voltage E=30~47 V as a parameter of bifurcation. By varying E,
the circuit changes its qualitative behavior from a stable periodic system to
another situation that exhibits chaos. At first, using Scicos we draw the
one-parameter bifurcation diagram given in Figure 4 where the input voltage E is the bifurcation parameter and the sampled vC is the variable. By increasing E,
we observe at first glance that the displayed diagram (Figure 4) shows a period
doubling route to chaos. However, after a clear 4-T periodic operation, instead of appearance of
an immediate 8-T periodic operation, the system follows a 7-T periodic operation. This means that border
collision bifurcation comes into play and interrupts the normal period doubling
cascade. Here, this type of bifurcation is characterized by the intersection of
the bifurcation diagram with the upper border line defined by(10)vc=Vref+VUa.
Figure 4 shows clearly the
occurrence of this phenomenon at around the critical value Ec=41.45 V.Figure 4
Bifurcation
diagram.For different increasing values ofE,
we give the capacitor voltage wave form vC and its corresponding phase plane vC-iL.By choosingE=30 V, we get a fundamental periodic operation.
This periodic regime is possible just for small values of E.
Figures 5 and 6 show the fundamental periodic operation.
Figure 5 displays the
capacitor voltage wave form, and Figure 6 gives the corresponding phase plane.Figure 5
Fundamental
periodic operation. (E=30 V): time waveform of the capacitor
voltage.Figure 6
Fundamental
periodic operation. (E=30 V): phase plane.ForE=37.5 V and E=41 V, subharmonic operation has been found.
Figures 7 and 8 present 2-T periodic subharmonic operation, whereas Figures 9 and
10 illustrate 4-T periodic subharmonic operation.Figure 7
2-T subharmonic operation. (E=37.5 V): time waveform of the capacitor
voltage.Figure 8
2-T subharmonic operation. (E=37.5 V): phase plane.Figure 9
4-T subharmonic operation. (E=41 V): time waveform of the capacitor
voltage.Figure 10
4-T subharmonic operation. (E=41 V): phase plane.However, the chaotic operation is given forE=46.5 V. Figure 11 indicates a chaotic signal with
infinite order, and Figure 12 shows the phase plane vC-iL that corresponds to a chaotic attractor. In order
to qualify this chaotic behavior for beyond E=42 V, we  have also computed the Lyapunov
exponents λ from time series for a voltage range 41–47 V
(see Table 1).Figure 11
Chaotic
regime. (E=46.5 V): time waveform of the capacitor
voltage.Figure 12
Chaotic
regime. (E=46.5 V): phase plane.Actually, it
was shown in [9] that
most controlled DC-DC converters like the voltage-mode controlled buck
converter can be represented by piecewise smooth maps and such type of maps
generates robust chaos, defined by the absence of periodic windows and
coexisting attractors in some neighborhood of parameter space.


## 4. Conclusion

This article
has illustrated a Scicos numerical study of the voltage-mode controlled buck
converter that is modeled by a hybrid system. Variations of the voltage input
can lead to a particular route to chaos; the system pursues a period doubling
bifurcation that is interrupted by border collision after a4-T periodic operation.The purpose of studying the hybrid aspect of this
circuit is to interest people working on hybrid dynamical systems domain, which
may have some applications, especially, in information transmission. Also, the
paper will attract the attention of readers that work with Scilab/Scicos for
modeling and simulation of hybrid dynamical systems. It is true that [18] given in this paper is a
good reference on this matter. However, our paper is concerned with another
computational view which is the numerical study of route to chaos in a hybrid
system that is different from the one studied in [18]; displaying a bifurcation
diagram, for instance, is a more complex numerical study that is not included
in [18].Table 1E(V)λ41−0.0491421.2341430.9264441.1942451.1973461.8422473.8948



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*Source: 798395-2008-05-05.xml*