Objetivos específicos del curso 3

3.1.1 General Mathematical Framework

Because of its multiple combinations of switch states, a power electronic converter exhibits a periodically repeated sequence of possible configurations during its operation time interval, also called switching period. Each such configuration represents in fact a unique circuit containing sources and passive elements, which can be mathematically described by a set of differential equations.

Under the assumptions already stated in Chap. 2, a generic power electronic converter, represented in Fig.3.1which switches betweenN distinct configurations,

S. Bacha et al.,Power Electronic Converters Modeling and Control: with Case Studies, Advanced Textbooks in Control and Signal Processing, DOI 10.1007/978-1-4471-5478-5_3,

© Springer-Verlag London 2014

27

is described as a dynamical system (Tymerski et al. 1989; Sun and Grotstollen 1992; Maksimovic´ et al.2001):

d

dtx tð Þ ¼ Ai x tð Þ þ Bi e tð Þ, ti t  tiþ1, (3.1) with

X^N

i¼1

ti ti1

ð Þ ¼ T,

whereT is the switching time, tiare different time points defining the switching betweenN configurations, Ai and Biare the n n state matrix and n  p input matrix respectively, corresponding to configuration i, x(t) is the n-length state vector and e(t) is the p-length vector of the independent sources of the system.

Note that in Eq. (3.1) the control input does not appear explicitly.

A more compact form of (3.1) is d

dtx tð Þ ¼X^N

i¼1

Aix tð Þ þ Bie tð Þ

ð Þ  hi, (3.2)

where hi are respective validation functions associated to configurations. These functions take values 1 or 0 depending on whether their respective configurations are activated or not.

Comments on power electronic converter classification

In general, power electronic converters are classified according to the following criteria:

• conversion mode: DC-DC, DC-AC, etc.;

i_e

v_e

v_S

i_S v_C

i_L

i₀ Fig. 3.1 Conventional

symbols for power electronic converter representation (Kassakian et al.1991)

• innermost control type (at switching level): pulse width modulation (PWM), hysteresis control, sliding-mode control, current-programmed control, etc.;

• operating regime: either natural or forced switching, either continuous or dis-continuous conduction, etc.

A classification – different from the above mentioned in the sense that it is focused on control aspects – is proposed next.

Let h¼ h½ 1   hN^T be the vector containing the various validation functions under the general representation (3.2). Based on this formulation, a classification approach is given by Krein et al. (1990) and rediscussed by Sun and Grotstollen (1992). The method is based on the functional dependencies of the elements in vector h; thus, three classes emerge:

• h does not depend on the state x;

• h depends on both the state x and the time;

• h depends only on the state x.

Note that the dependence of h(x) can either be implicit – because of the circuit operational manner – or explicit, due to the presence of state feedback.

The first class of functions in the above list is characterized by the fact that switching is controlled by a function depending solely on time, h¼ H(t), or external actions. The corresponding converters are simple to model and analyze.

For example, this is the case of buck converters with continuous conduction controlled at variable duty ratio.

As regards the second class, some functions hi may only depend on time, whereas others may depend on the system state. For example, the buck converter operating in discontinuous conduction and a thyristor-based rectifier belong to this class. Finally, the third class is illustrated by a diode-bridge rectifier connected to the grid or by a current-controlled buck converter.

One can note that a given configuration may belong to different classes according to the manner in which it operates.

It appears that analysis and control methods will be more suited to one of the above mentioned families rather than to another one. Thus, for example, as shown later in this book, the classical averaged model is a simple tool, easy to use for converters belonging to the first family, but it is unsuitable for the other two classes.

The same holds for variable-structure control and associated sliding modes.

In conclusion, to the best of our knowledge, at present there is no systematic approach able to provide a general methodology allowing a uniform analysis of converter behavior irrespective of its operation. In this respect, the action of exact modeling providing the switched model of a given studied converter is even more important.

3.1.2 Bilinear Form

The bilinear form provides a more compact representation of a power electronic converter switched model, while showing the control inputs. Instead of using

3.1 Mathematical Modeling 29

validation functions for describingN configurations by means of N models, it is possible to condense the information in a single, unified model fed withp binary functions, denoted byukand named switching functions. The number p is deter-mined as the smallest integer satisfying the relation 2^p N.

The bilinear form of the switched model is expressed by the general equation:

_x ¼ Ax þX^p

k¼1

Bkxþ bk

ð Þ  ukþ d, (3.3)

where, for every k from 1 to p, Bkare n n matrices, bk are n-length column vectors and d also are n-length column vectors. Equation (3.3) shows explicitly the control input vector u¼ u½ 1 u2   up^T. Note the presence of products between state variables and control inputs, which gives the bilinear feature of the model.

Next, one will see that every power electronic converter can be modeled by Eq. (3.3), in which particularizations are made. For example, the buck converter has Bk¼ 0, leading to a linear model; in the case of boost converter bk¼ 0.

An advantage of model (3.3) is the fact that the small-signal model can be easily obtained from it. Employing the procedure detailed in Sect.2.2.4 from Chap.2, the following relations hold:

_ex ¼ eA  ex þ eB  eu,

with ex being the vector of state variables’ variations around the steady-state point xe, established in response to vector ue¼ u½ 1e u2e   upe^T, where eu ¼ u  ue. State matrix eA and input matrix eB have the forms

Ae ¼ A þX^p

k¼1

Bkuke

eB ¼X^p

k¼1

Bkxeþ bk

ð Þ:

8>

>>

<

>>

>:

(3.4)

Model (3.4) can be used directly in designing linear control laws.