Expressions of

## 1. Introduction

A hybrid dynamical system is a system containing on the same time continuous state variables and event variables in interaction. We find hybrid systems in different fields. We cite robotic systems, chemical systems controlled by vans and pumps, biological systems (growth and division) and nonlinear electronics systems.

Because of interaction between continuous and discrete aspects, the behavior of hybrid systems can be seen as extremely complex. However, this behavior becomes relatively simple for piece-wise affine hybrid dynamical systems that can, in contrast, generate bifurcation and chaos. There are many examples such as power electronics DC-DC converters.

The common power electronics DC-DC converters are the buck converter and the boost converter. They are switching systems with time variant structure [9].

DC-DC converters are widely used in industrial, commercial, residential and aerospace environments. These circuits are typically controlled by PWM (Piece Wise Modulation) or other similar techniques to regulate the tension and the current given to the charges. The controller decides to pass from one configuration to another by considering that transitions occur cyclically or in discrete time. In order to make the analysis possible, most of mathematical treatments use some techniques that are based on averaging or discretization. Averaging can mean to wrong conclusions on operation of a system. Discrete models do not give any information on the state of the system between the sampled instants. In addition, they are difficult to obtain. In fact, in most cases, a pure analytic study is not possible. Another possible approach to analyze these converters can be done via some models of hybrid dynamical systems. DC-DC converters are particularly good candidates for this type of analysis because of their natural hybrid structure. The nature of commutations of these systems makes them strongly nonlinear. They present specific complex phenomena such as fractals structures of bifurcation and chaos.

The study of nonlinear dynamics of DC-DC converters started in 1984 by works of Brockett and Wood [4]. Since then, chaos and nonlinear dynamics in power electronics circuits have attracted different research groups around the world. Different nonlinear phenomena have been observed such as routes to chaos following the period doubling cascade [16], [5], [19], [20] and [23] or quasi-periodic phenomena [6], [7] and [8], besides border collision bifurcations [23] and [2].

Switched circuits behavior is mostly simulated by pure numerical methods where precision step is increased when the system is near a switching condition. Those numerical tools are widely used mainly because of their ease-of-use and their ability to simulate a wide range of circuits including nonlinear, time–variant, and non–autonomous systems.

Even if those simulators can reach the desired relative precision for a continuous trajectory, they can miss a switching condition and then diverge drastically from the trajectory as in figure 1. This could be annoying when one is interested by border collision bifurcations, or when local behavior is needed with a good accuracy. In those applications, an alternative is to write down analytical, or semi–analytical, trajectories and switching conditions to obtain a recurrence which is very accurate and fast to run. Building and adapting such *ad’hoc* simulators represent a lot of efforts and a risk of mistakes.

Generic and accurate simulators can be proposed if we are restricted to a certain class of systems. A simulation tool with no loss of events is proposed in [14] and [15] for PWAHSs defined on polytopes (finite regions that are bounded by hyperplanes). This class of PWA differential systems has been widely studied as a standard technique to approximate a range of nonlinear systems.

But closed polytopic partition of the state space does not allow simulation of most switching circuits where switching frontiers are mostly single affine constraints or time–dependent periodical events.

This chapter follows our previous study in [13], [12] and [11].

We focus on planar PWAHSs with such simple switching conditions which can model a family of switched planar circuits: bang–bang regulators, the Boost converter, the Charge-Pump Phase Locked Loop (CP-PLL),...

This class of systems has analytical trajectories that help to build fast algorithms with no loss of events. We propose a semi-analytical solver for hybrid systems which provides:

A pure numerical method when the system is nonlinear or non-planar;

A pure analytic method when all continuous parts of the system and switching conditions can be solved symbolically. This can be the case for the boost converter [3], [21], the second order charge-pump phase locked loop [17], [22].

A mixed method using analytical trajectories and numerical computation of the switching instant when those solutions are transcendent. This has been used for the third order CP-PLL [17]. It can also be the case for the Buck converter [10], [21],...

This chapter is organized as follows. Section 2 contains our main results. We describe the problem to be deal with, we introduce a general algorithm to solve planar HSs, we present the algorithm that detects events’ occurrence and devote a subsection to our approach efficiency. Section 3 Illustrates the current-mode controlled Boost converter example. Finally, a conclusion is stated in Section 4.

## 2. Main results

### 2.1. A HS( X , E , t ) : general definition

A general definition of HSs is presented here. This type of dynamical systems is characterized by the coexistence of two kinds of state vectors: continuous state vector

**Definition 1** *A continuous-time, autonomous HS is a system of the form:*

**Remark** — For non–autonomous HSs, the function

### 2.2. HSs class of interest

We consider a two dimensional PWAHS (

We consider two kinds of events: state dependent events and periodic events.

The state dependent event transition

Note that the set

**Remark** — We consider, without loss of generality, the case where a transition occurs at time

Both transitions can be met with the set

### 2.3. Event–driven simulation of PWAHSs

The simulation will compute the hybrid state from event to event. Knowing the states

Algorithm 1. Algorithm computing the hybrid state at

### 2.4. Event detection occurrence: description and algorithm

We consider the affine Cauchy problem in

where

If

*2.4.1. Analytical trajectories*

**Definition 2** *For any square matrix* *of order* *and* *in**, the exponential matrix* *is defined by*

where

It is well known that the analytical trajectory

When

The analytical expression of the exponential matrix

If

, then

If

, then

where

Using these expressions, we can determine the function

where

Depending on the eigenvalues

**Remark** — Coefficients

In some cases, (

In other cases, the solution can not be found with classical functions and then a numeric algorithm should be used. Using classical methods like Newton does not guaranty existence or convergence of the smallest positive root. To meet these conditions, let us use analytical roots of the derivative function

We can then compute analytically the set

**Remark** — When

Algorithm 2. Algorithm computing

will end by finding a root. In the other case, the set

## 3. Matlab modelling

Our semi-analytical solver is composed of different main programs that define the studied affine system. First, we create the affine system given with a specifically chosen name. Then, we define the matrices

### 3.1. Application: Current-mode controlled Boost converter

A current-mode controlled Boost converter in open loop consists of two parts: a converter and a switching controller. The basic circuit is given in figure 3.

This converter is a second-order circuit comprising an inductor, a capacitor, a diode, a switch and a load resistance connected in parallel with the capacitor.

The general circuit operation is driven by the switching controller. It compares the inductor current

When the inductor current

The Boost converter controlled in current mode is modeled by an affine piece-wise hybrid system defined by the same sub-systems given in equation as follows:

In the case of the Boost converter controlled in current mode, there are two types of events:

A state event defined by a fixed border of the set

and another periodic event defined by the dates

where

Before performing any study of the observed bifurcations in this circuit, a numerical simulation in the parametric plane is needed.

The following program calcule_balayage_mod.m is used to obtain the parametric plane:

After calculating the necessary points of the parametric plane saved in the file named dat_balais, the next program affiche_balayage.m plots the figure given in Fig.4

The figure 4 of the parametric plane allows to emphasize the parameters values for which there exists at least one attractor (fixed point, cycle of order

Figure 5 shows a bifurcation diagram (Feigenbaum type) in the plane

In these two figures, the voltage

For example, the blue area O1 represents the parameters’ values for which there exists an attractive fixed point (fundamental periodic regime). The red area O2 represents the existence of an attractive cycle of order

This representation of the parametric plane is not enough to establish a bifurcation structure of the hybrid model of the Boost converter, but it is useful for the initialization of programs to draw bifurcation curves..99 The simulation results (temporal domain and voltage-current plane (

## 4. Conclusion

In this chapter, we have showed an accurate and fast method to determine events’ occurrence for planar piece-wise affine hybrid systems. As a result, we have implemented our algorithm in Matlab toolbox version (free downloadable on http://felguezar.000space.com/).

This toolbox has also been completed by analysis tools such as displaying the bifurcation and parametric diagrams. The algorithm takes the advantage of the analytical form that appears in the planar case. Our approach can not be extended to a higher dimension. DC-DC converters like Boost converter are known to be simple switched circuits but very rich in nonlinear dynamics. As application, we have chosen the example of Boost converter controlled in current mode.

### Acknowledgement

The authors would like to thank Pascal Acco and Danièle Fournier–Prunaret for crucial discussions on the original version of our work on this subject.