An Iterative Decomposed Piecewise-Linear Model Description

Abstract

A model description for the representation of one-dimensional piecewise-linear characteristics is presented. The model can be denoted as a decomposed one, because the independent and dependent variables of the PWL characteristic are treated separately. It is also called iterative, because the particular representation of each segment of the PWL characteristic depends on the value of only one parameter included in the mathematical formulation, it gives the possibility of modeling both, univalued and multivalued PWL characteristics.

1. Introduction

In order to be able to simulate nonlinear networks, the nonlinear behavior of the components must be modeled first. The nonlinear behavior cannot exactly be described by familiar analytical nonlinear functions, so it must be approximated by a known function. Some approaches to overcome this problems are the polynomial approximation, table-look-up technique, and PWL approximation.

A lot of research has been done in the area of approximating nonlinear functions using polynomials as well as in the field of calculating transcendental functions. Applying such methods leads to simulators like SPICE [1].

The technique alá table-look-up consisting in storing several sample points of a nonlinear function in a table. If a function value is needed, the entry in the table is used or interpolated.

In PWL approximation, the nonlinear function is repeatedly approximated locally by a linear mapping. This results in a large collection of linear mappings which can be stored in a compact closed form model [2, 3]. These models have been applied in the so-called PWL simulators [4–6].

There are many important proposals in the area of PWL modeling, but no best model description can be found because the advantages or disadvantages of any PWL model depends on its application [7].

The main contributions of the PWL model, which in Section 2 will be exposed, are the possibility of describing all kinds of one-dimensional PWL characteristics (univalued and multivalued) having mathematical formulation which is not so heavy to be computed because it is based on an iterative system of linear equations, the advantage of obtaining the model parameters directly from the graphic coordinates and the simulation capability of the model, because it is compatible with iterative simulation techniques [8].

The model is denoted as decomposed, because the independent variable 𝑥 and the dependent variable 𝑦=𝑓(𝑥) are included separately in a system of linear equations. The model is also denoted as iterative, because there is a parameter 𝑘, which selects a specific segment from the set of 𝐿-segments that constitutes the complete PWL characteristic.

2. Model Description

The PWL characteristic, depicted in Figure 1, represents a general one-dimensional function 𝑦=𝑓(𝑥) characterized by 𝐿 linear segments and 𝐿+1 coordinates: (𝑋₁,𝑌₁),(𝑋₂,𝑌₂),…,(𝑋_𝐿+1,𝑌_𝐿+1).

Figure 1 

PWL characteristic and its graphic coordinates.

An iterative decomposed formulation for this characteristic can be obtained as follows.

Firstly, the particular problem of obtaining the line equation for any of the kth segments included in the PWL characteristic is analyzed. The graphic exposition of this problem is depicted in Figure 2.

Figure 2 

A kth segment belonging to a PWL characteristic.

The line equation for the kth segment is given as 𝑦_𝑘𝑌=(^𝑘+1−𝑌_𝑘𝑋_𝑘+1−𝑋_𝑘𝑌)𝑥+(^𝑘𝑋_𝑘+1−𝑌_𝑘+1𝑋_𝑘𝑋_𝑘+1−𝑋_𝑘). (1) Then, (1) is factored into the form 𝑦_𝑘=𝑃_𝑘𝑌_𝑘+𝑄_𝑘𝑌_𝑘+1, (2) where 𝑃_𝑘 and 𝑄_𝑘 are variables defined by the determinant relations: 𝑃_𝑘=|^{𝑥𝑋𝑘+1}₁₁||^{𝑋𝑘𝑋𝑘+1}₁₁|,𝑄_𝑘=|^𝑋𝑘𝑥₁₁||^{𝑋𝑘𝑋𝑘+1}₁₁|. (3) The relation between (2) and (3) is expressed as the system of linear equations given in 𝑦_𝑘𝑥1=𝑌_𝑘𝑌_𝑘+1𝑋_𝑘𝑋_𝑘+1𝑃11^𝑘𝑄_𝑘. (4) The above linear system represents the kth line equation included in the PWL characteristic to be modeled. Finally, a complete mathematical expression which assures a description for the PWL characteristic depicted in Figure 1 is obtained if (4) is generalized for all 𝐿 segments. Thus, the parametric model description can be written: 𝑦𝑥1=^𝐿_𝑖=1𝑌_𝑖𝑌_𝑖+1𝑋_𝑖𝑋_𝑖+1𝑃11^𝑖𝑄_𝑖𝐶_𝑖(𝑘) (5) with the condition 𝑄_𝑖(𝑥),𝑃_𝑖(𝑥),𝑄_𝑖(𝑦),𝑃_𝑖(𝑦)≥0, (6) where 𝐶_𝑖(𝑘) is denoted as activation vector and it is defined as follows: 𝐶_𝑖1(𝑘)=𝑖!(𝐿−𝑖)!^𝑖−1_𝑚=0|𝑘−𝑚|^𝐿−𝑖_𝑛=1|𝑘−(𝑖+𝑛)| (7) for 𝑖=1,…,𝐿−1, and 𝐶_𝑖1(𝑘)=𝐿!^𝐿−1_𝑟=0|𝑘−𝑟| (8) for 𝑖=𝐿.

The variables 𝑄_𝑖 and 𝑃_𝑖 in (5) have an important function because they control the interval where a kth line equation exists. This is expressed in the condition (6) and it is demonstrated as follows.

Firsty, the notations Δ𝑋_𝑖 and Δ𝑌_𝑖 are defined as Δ𝑋_𝑖=𝑋_𝑖+1−𝑋_𝑖,Δ𝑌_𝑖=𝑌_𝑖+1−𝑌_𝑖. (9) Then, mathematical expressions, for the variables 𝑃_𝑖 and 𝑄_𝑖, can be obtained if the following decomposed linear equations, which are included in (5), are algebraically manipulated: 𝑦=𝑌_𝑖𝑃_𝑖+𝑌_𝑖+1𝑄_𝑖,𝑥=𝑋_𝑖𝑃_𝑖+𝑋_𝑖+1𝑄_𝑖,1=𝑃_𝑖+𝑄_𝑖. (10)(11)(12)

Isolating 𝑃_𝑖 from (12) and substituting it into (10), (11), 𝑄_𝑖 can be expressed as 𝑄_𝑖𝑥(𝑥)=Δ𝑋_𝑖−𝑋_𝑖Δ𝑋_𝑖,𝑄_𝑖𝑦(𝑦)=Δ𝑌_𝑖−𝑌_𝑖Δ𝑌_𝑖. (13)

A similar analysis where 𝑄_𝑖 is isolated from (12) and then susbstituted into (10), (11) yields 𝑃_𝑖(𝑥)=−𝑥Δ𝑋_𝑖+𝑋_𝑖+1Δ𝑋_𝑖,𝑃_𝑖(𝑦)=−𝑦Δ𝑌_𝑖+𝑌_𝑖+1Δ𝑌_𝑖. (14)

If (13) and (14) are subjected to the exposed condition in (6) results Δ𝑋_𝑖𝑄_𝑖(𝑥)=𝑥−𝑋_𝑖≥0,𝑥≥𝑋_𝑖,Δ𝑌_𝑖𝑄_𝑖(𝑦)=𝑦−𝑌_𝑖≥0,𝑦≥𝑌_𝑖,Δ𝑋_𝑖𝑃_𝑖(𝑥)=𝑋_𝑖+1−𝑥≥0,𝑥≤𝑋_𝑖+1,Δ𝑋_𝑖𝑃_𝑖(𝑦)=𝑌_𝑖+1−𝑦≥0,𝑦≤𝑌_𝑖+1. (15) The inequations (15) indicate the intervals where a linear segment belonging to a PWL characteristic exists. The shadow area depicted in Figure 2 results by the intersection of these intervals for the kth PWL segment.

3. Electrical Model

The linear system in (5) can be electrically modeled by the circuit shown in Figure 3. In this circuit, the variables 𝑄_𝑖 and 𝑃_𝑖 have a physical meaning, and represent the current flow through the conductances whose values are (Δ𝑋_𝑖)⁻¹ and −(Δ𝑋_𝑖)⁻¹, respectively. The electrical diagram includes two switches which are controlled simultaneously by 𝐶_𝑖(𝑘). Figure 4 shows the symbol used to represent them. On the one hand, it can be seen that when the switches are open, these variables are equal to zero, it occurs when 𝐶_𝑖(𝑘)=0, on the other hand, when 𝐶_𝑖(𝑘)=1 the switches are closed and the variables (currents 𝑄_𝑖 and 𝑃_𝑖) are different to zero.

Figure 3 

Equivalent circuit for a PWL segment.

Figure 4 

Switch symbol included in the equivalent circuit diagram.

4. Example

Consider the theoretical nonlinear element and its PWL characteristic depicted in Figure 5. Determine their iterative decomposed model description.

(a)

(b)

Figure 5 

(a) Nonlinear element, (b) PWL characteristic for the nonlinear element.

An analysis from Figure 5(b) shows the existence of three linear equations labeled as (1), (2), and (3). The four graphic coordinates are designated as B1, B2, B3, and B4, and summarized in Table 1.

Table 1 

(𝑉_𝑖,𝐼_𝑖) coordinates for the PWL characteristic of Figure 5(b).

There are three segments, therefore 𝐿=3 and 𝑘=1,2,3. Substituting 𝐿 and the (𝑉_𝑖,𝐼_𝑖) coordinates of Table 1 into (5), results: 𝑗𝑢1=³_𝑖=1𝐼_𝑖𝐼_𝑖+1𝑉_𝑖𝑉_𝑖+1𝑃11^𝑖𝑄_𝑖𝐶_𝑖=𝑃(𝑘)030311¹𝑄₁𝐶₁𝑃(𝑘)+343111²𝑄₂𝐶₂+𝑃(𝑘)411411³𝑄₃𝐶₃(𝑘). (16) Substituting 𝐿 in (7) and (8), the terms of the activation vector 𝐶_𝑖(𝑘) are given by 𝐶₁(𝑘)=𝑘|𝑘−2||𝑘−3|2,𝐶₂(𝑘)=𝑘|𝑘−1||𝑘−3|2,𝐶₃(𝑘)=𝑘|𝑘−1||𝑘−2|6. (17) Figure 6 shows the electrical circuit which models the PWL characteristic analyzed in this example. Each 𝑗_𝑘th Block is a simplified representation of the circuit shown in Figure 3.

Figure 6 

Electrical circuit model for the Example.

When 𝑘=1, 𝐶₁(𝑘)=1, 𝐶₂(𝑘)=0, 𝐶₃(𝑘)=0, and the resulting linear system is 𝑗𝑢1=𝑃030311¹𝑄₁+0+0. (18) The line equation is 𝑗=𝑢 bounded by the intervals 𝑢=[0,3] and 𝑗=[0,3].

When 𝑘=2, 𝐶₁(𝑘)=0, 𝐶₂(𝑘)=1, 𝐶₃(𝑘)=0, and the resulting linear system is 𝑗𝑢1𝑃=0+343111²𝑄₂+0. (19) The line equation is 𝑗=−0.5𝑢+4.5 bounded by the intervals 𝑢=[1,3] and 𝑗=[3,4].

When 𝑘=3, 𝐶₁(𝑘)=0, 𝐶₂(𝑘)=0, 𝐶₃(𝑘)=1, and the resulting linear system is 𝑗𝑢1𝑃=0+0+411411³𝑄₃. (20)

The line equation is 𝑗=−𝑢+5 bounded by the intervals 𝑢=[1,4] and 𝑗=[1,4].

The above results are summarized in Table 2 and Figures 7, 8, and 9.

Table 2 

𝑄_𝑘 and 𝑃_𝑘 variables subjected by the conditions (6).

Figure 7 

Interval intersection for 𝑘=1.

Figure 8 

Interval intersection for 𝑘=2.

Figure 9 

Interval intersection for 𝑘=3.

5. Conclusions

The proposed iterative and decomposed model can describe any arbitrary PWL function (univalued or multivalued) by considering as model parameters its graphic coordinates. The interval in the model bounds the range where a linear segment is valid and it is determined by the variables 𝑄_𝑖 and 𝑃_𝑖. These variables have a physical meaning (currents) in the electrical model. The equivalent circuit, of the proposed PWL description, has simulation capability because it is compatible with iterative simulation techniques.

Acknowledgments

Victor Jimenez-Fernandez is holder of a scholarship from CONACyT-México by the Contract 183249. This work has been partially supported by a CONACyT-México research project under Grant 60261.