Analysis of Averaging Errors

The goal here is to perform an analysis of errors introduced by averaging. The error between the average of a given switched model state-space solution and the averaged model state-space solution (on the same time window) is easy to put into light by numerical simulation, like shown in Fig.4.5, where the error signal has been denoted byε(t). An estimation of this kind of error can also be performed analytically by referring to the sampled-data model introduced in Sect.2.2.2from Chap.2(Pe´rard et al.1979).

u(t)

Fig. 4.5 Averaged model output vs. average of switched model output

62 4 Classical Averaged Model

4.4.1 Exact Sampled-Data Model

Let us consider a linear system that switches between two configurations,α being the duty ratio; it can be described by the following set of differential equations:

dx

dt ¼ A₁ x þ B1 e for t ∈ kT, k þ α½ ð ÞTÞ A₂ x þ B2 e for t ∈ k þ α½ð ÞT, k þ 1ð ÞTÞ,



(4.16)

which is valid for each switching period [kT, (k + 1)T ), where k ∈  . It is proposed that system (4.16) be solved by integrating each configuration and postulating the state variables’ continuity. Let us introduce the general notation of the transition matrixΦ(t) ¼ exp(A  t). The transition matrices associated with the two configurations areΦ1(t)¼ exp(A1 t) and Φ2(t)¼ exp(A2 t).

For the first configuration one can write

x k½ð þ αÞT ¼ Φ1ð Þ  x kTαT ð Þ þ ð

αT

0

Φ1ð Þ  Bτ 1e dτ,

which gives, after integration

x k½ð þ αÞT ¼ Φ1ð Þ  x kTαT ð Þ þ A11 Φð 1ð Þ  IαT nÞ  B1e, (4.17) where In is the identity matrix of the same dimension as A. For the second configuration the result is given directly (algebra is the same):

x k½ð þ 1ÞT ¼ Φ2ð Þ  x k þ ααT ½ð ÞT þ A21 Φð 2½ð1 αÞT  InÞ  B2e: (4.18) By introducing the value of x[(k +α)T], given in Eq. (4.17), into Eq. (4.18) one obtains a recurrent equation of the form

xkþ1¼ Φ xð k; αk; TÞ: (4.19) Equations (4.18) and (4.19) are forms of theexact sampled-data model. Equation (4.19) is very complex and difficult to manipulate. In order to avoid cumbersome mathematical developments certain systems having simple representations will be taken into account. This can be obtained from Eq. (4.18) by employing an adequate change of variable.

dx

dt ¼ A1 x for t ∈ kT, k þ α½ ð ÞTÞ A2 x for t ∈ k þ α½ð ÞT, k þ 1ð ÞTÞ:



(4.20)

The system of Eq. (4.20) corresponds to the dynamic depicted in Fig.4.6.

Integrating the first equation of (4.20), one obtains

x k½ð þ αÞT ¼ Φ1ð Þ  x kTαT ð Þ, (4.21) whereas for the second configuration the integration gives

x k½ð þ 1ÞT ¼ Φ2½ð1 αÞT  x k þ α½ð ÞT: (4.22) By substituting Eq. (4.21) into Eq. (4.22), one obtains

x k½ð þ 1ÞT ¼ Φ2½ð1 αÞT  Φ1ð Þ  x kTαT ð Þ, (4.23) which is the output of the switched model at switching instants in a recurrent form.

Like the model described by Eq. (4.19), the model in Eq. (4.23) is the sampled-data topological model. The computation of matricesΦ1andΦ2can be simplified more or less satisfactorily, by their first-order expansions, respectively

Φ1ð Þ  I þ AαT 1 αT

Φ2½ð1 αÞT  I þ A2 1  αð ÞT,



(4.24) where I is the identity matrix of the same dimension as A1and A2. Introducing the simplified expressions (4.24) into Eq. (4.23) yields the first-order approximated sampled-data model.

4.4.2 Relation Between Exact Sampled-Data Model and Exact Averaged Model

Note that Eq. (4.23), providing the solution of the switched model, contains matrix products. In the general case a product of matrices is not commutative, i.e., the following relation generally holds between the two state-space matrices: A1 A26¼ A2 A1. Case of commutative matrices

In this subsection the exceptions where A1 A2¼ A2 A1are addressed. In such cases the matrix exponentials are also switching, i.e.,Φ1 Φ2¼ Φ2 Φ1. There-fore, the following relation holds:

exp Að Þ  exp A1 ð Þ ¼ exp A2 ð Þ  exp A2 ð Þ ¼ exp A1 ð 1þ A2Þ: (4.25) kT

t x

(k + α)T (k −1 + α)T

A₁

x(kT ) x((k + 1)T )

A₂ A₁

(k−1)T Fig. 4.6 Dynamic of

system described by Eq. (4.20)

64 4 Classical Averaged Model

This very interesting result, if applied to the exact switched model (4.23), gives x k½ð þ 1ÞT ¼ Φmðα; TÞ  x kTð Þ, (4.26) where for matrixΦmone obtains successively

Φm¼ Φ2½ð1 αÞT  Φ1ð Þ ¼ ΦαT 1ð Þ  ΦαT 2½ð1 αÞT

¼ exp Af 1 αTg  exp Af 2 1  αð ÞTg ¼ exp Af½ 1 α þ A2 1  αð Þ  Tg:

The latter result can be expressed more synthetically as

Φm¼ exp Að m TÞ, (4.27)

where

Am¼ A1 α þ A2 1  αð Þ is the state matrix of the averaged model.

Equation (4.27) represents the solution of the system (averaged model) d

dth ix 0¼ Að 1 α þ A2 1  αð ÞÞ  xh i0: (4.28) In addition, at sampling moments it holds that x[(k + 1)T]¼ hxi0[(k + 1)T].

This is why the model expressed by Eq. (4.28) is called theexact averaged model (Pe´rard et al.1979); its dynamic is represented in Fig.4.7.

General case

Unfortunately, the assumption of matrices being commutative does not hold in the quasi-totality of power converters. In the general case, the matrix product between A1and A2is not commutative; hence, their exponentials are not commutative:

exp Að Þ  exp A1 ð Þ 6¼ exp A2 ð Þ  exp A2 ð Þ:1

Therefore, relation (4.28) becomes d

dth ix 0 Am xh i0¼ A½ 1 α þ A2 1  αð Þ  xh i0: (4.29) kT

t x

(k −1+ α)T (k + α)T A₁

x(kT ) x((k + 1)T )

A₂ A₁

(k−1)T Fig. 4.7 Exact behavior

and exact averaged model

Equation (4.29) defines the so-called approximated averaged model (Pe´rard et al.1979). Its trajectory is no longer passing through the points of the sampled-data model as shown in Fig.4.7, but it will be an averaged trajectory more or less close to the sliding average of the exact trajectory.

An issue is to quantify the error introduced by the approximation in this model in relation to the exact sampled-data model. This represents an upper bound of the error between the output of the averaged model and the average of the switched model. Its absolute value is

Err¼ Φmðα; TÞ  Φ2½ð1 αÞT  Φ1ð Þ:αT (4.30) The complete computation of error expressed by (4.30) is not trivial and can only be done in numerical form. For the sake of simplicity a second-order approximation of the matrix exponential is employed,

exp Atð Þ  I þ At þA²t² 2 ,

in order to express the matrix Φm and the productΦ2[(1  α)T]  Φ1(αT). One obtains successively:

Φmðα; TÞ  I þ AmTþA²_mT²

2 ¼ I þ A½ 1α þ A2ð1 αÞT þ½A1α þ A2ð1 αÞ²T² 2

¼ I þ A½ 1α þ A2ð1 αÞT þA²₁α²

2 þA²₂ð1 αÞ²

2 T²

þ Að 1A2þ A2A1Þα 1  αð Þ 2 T²,

Φ2½ð1 αÞT  Φ1ð Þ  I þ AαT ½ 1α þ A2ð1 αÞT þ½A1α þ A2ð1 αÞ²T² 2

¼ I þ A½ 1α þ A2ð1 αÞT þA²₁α²

2 þA²₂ð1 αÞ²

2 T²

þ A1A2α 1  αð Þ 2 T²:

One can remark that the matrix error Err expressed by Eq. (4.30) between the second-order developments is reduced to the matrix E as follows:

Err E ¼ Að 1A2 A2A1Þα 1  αð Þ

2 T²: (4.31)

Equation (4.31) shows that if matrices A1and A2 are commutative then the sampled-data and averaged models are confused, as shown in the previous

66 4 Classical Averaged Model

paragraph. In the general case, the smaller the norm of the error matrix in relation to the norms of the other state matrices weighted by their associated enabling times within a switching period, αT and (1  α)T, respectively, the more precise the approximate model is. This further requires that timeT be small – i.e., the converter to operate at high frequency – and that the duty ratioα be close to one or zero. If these assumptions hold, this leads to smaller ripple of the state variables. To conclude, the averaged model can be seen as an “ideal”, operating at infinite frequency, whereas the switched model, operating at finite frequency, exhibits, besides the variables’ ripples, an average different from the averaged model solution.

Discussions on this subject will be detailed in the example presented in Sect.4.5.3.