This paper shows how to obtain the values of the numerator and denominator Kharitonov polynomials of an interval plant from its value set at a given frequency. Moreover, it is proven that given a value set, all the assigned polynomials of the vertices can be determined if and only if there is a complete edge or a complete arc lying on a quadrant. This algorithm is nonconservative in the sense that if the valueset boundary of an interval plant is exactly known, and particularly its vertices, then the Kharitonov rectangles are exactly those used to obtain these value sets.
In reference to the identification problem, these have been widely motivated and analysed over recent years [
Interval plants have been widely motivated and analysed over recent years. For further engineering motivation, among the numerous papers and books, [
The identification problem using the interval plant framework, that is, to compute an interval plant from the frequency response, has not been completely solved. Interval plant identification was investigated by Bhattacharyya et al. [
A different approach was developed by Hernández et al. [
This paper improves the results obtained in [
Let us consider a linear interval plant of real coefficients, of the form
Numerator and denominator polynomial families are characterized by their respective Kharitonov polynomials, and they can be expressed in terms of their even and odd parts, at
Family
Family
As is well known, the values
As can be observed in [
As is shown [
As is well known, the Kharitonov polynomials values can be obtained from
This paper is organized as follows. Section
In order to determine the polynomials numerator and denominator associated to a vertex of the value set boundary with the minimum number of elements, the situation of a segment in a quadrant will be considered. So, let
egment and complete arcs.
Segment and no complete arcs.
As was shown, the values of
Let
It is trivial. This normalization is one of the infinite possible solutions [
This paper deals with the general case where
Given a vertex
The following Lemma shows the necessary conditions on the denominator
Let
where
The proof is obtained directly from the information of a complete segment in a quadrant and the properties of the Kharitonov rectangle. So, from the complete segment and the normalization (Lemma
If
Similarly, if
If
Finally, if
Cases where
Cases where
On the other hand, the behaviour of a segment on the complex plane when divided by a complex number is well known. The following property shows this behaviour.
Let
The following Theorem shows how to characterize and calculate the polynomials
Let
when
when
when
Vertices for the conditions of the Theorem
From the complete segment
Let
Taking into account both conditions,
Taking into account both conditions,
Let
In this case the demonstration is trivial noting that
Let
Taking into account both conditions,
As
Let
Let
As
Taking into account both conditions,
If
Therefore the condition
This theorem is used in the example of Section
The following Theorem is analogous to Theorem
Let
when
when
when
Analogous to Theorem
This theorem is used in the example of Section
In order to determine the polynomials numerator and denominator associated to a vertex of the value set boundary with the minimum number of elements, the situation of an arc in a quadrant will be considered. So, let
When these segments are completed the denominators are vertices of the Kharitonov rectangle. Figure
Arc and two complete segments.
As was shown, the values of
Let
It is trivial. This normalization is one of the infinite possible solutions for a value set. This normalization implies fitting
This paper deals with the general case where
Given a vertex
(a)
The following Lemma shows the necessary conditions on the denominator
Let
where
The proof is obtained directly from the information of a complete arc in a quadrant and the properties of the Kharitonov rectangle. So, from the complete arc and the normalization (Lemma
If
Similarly, if
If
Finally, if
On the other hand, the behaviour of an arc on the complex plane when it is divided by a complex number is well known. The following property shows this behaviour.
Let
The following Theorem shows how to characterize and calculate the polynomials
Let
when
when
when
Analogous to Theorem
This theorem is used in the example of Section
The following theorem is analogous to Theorem
Let
Then
when
when
when
Analogous to Theorem
This theorem is used in the example of Section
Finally, the following theorem points out the necessary and sufficient condition.
Given a value set, all the assigned polynomials of the vertices can be determined if and only if there is a complete edge or a complete arc lying on a quadrant when the normalized edge satisfies
It is obvious from Theorems
Given a value set with a complete segment or a complete arc in a quadrant, to obtain the Kharitonov polynomials the following.
If there is a complete segment in a quadrant,
if
if
If there is a complete arc in a quadrant,
if
if
Calculate the values of the assigned polynomials
Calculate the numerator and denominator rectangles with Kharitonov polynomial values
Figure
the vertices,
the intersections with the axis,
the shape of the boundary's elements: arc or segment.
Value set boundary information.





(a)  (b)  (c)  (a)  (b)  (c)  (a)  (b)  (c) 


0 


0 


1 


1 


1 


0 


0 

0 

0  

1 


1 


1  


0 


0 


0 


1 


1 


1 


0 


0 

0  


1 

0 


1  

1 


1 


0 
(a): Vertex (
(c): Edge (
Three value sets of an interval plant.
This example illustrates how to obtain the assigned polynomials and the numerator and denominator rectangles for each value set, and remarks the theorem used in each step.
The complete arc with vertices
In summary, the assigned polynomials are
Then
Table
Results of the algorithm for the value set at frequency








0.8000 + 10.4000 
−4.8000 + 7.6000 
−3.5862 + 1.0345 
−2.5517 + 0.6207 
−1.3443 + 1.2131 
Kharitonov rectangles calculated  

1.5676 + 2.5946 
Theorem 
Theorem 
Theorem 
Theorem 
Theorem 
Theorem 


2.0000 + 8.0000 

2.0000 + 8.0000 
10.0000 
−4.8000 + 7.6000 
−3.5862 + 1.0345 
−2.5517 + 0.6207 


0.8000 + 10.4000 

10.0000 
−3.5862 + 1.0345 
−2.5517 + 0.6207 
−1.3443 + 1.2131 
2.3336 


0 + 2.3336 

210.97  174.29  174.29  261.87  261.87 


229.40  Condition 
Theorem 
Theorem 
Theorem 
Theorem 
Theorem 


−0.6508 − 0.7592 

−0.6508 − 1.0846 
0.1084 − 1.0847 
0.1085 − 1.0846 
0.1085 − 0.7592 
0.1085 − 0.7592 


−0.3254 + 0.0542 

−0.1085 + 0.0542 
−0.1085 + 0.0542 
−0.1085 + 0.2712 
−0.1085 + 0.2712 
−0.3254 + 0.2712 


−0.1085 + 0.0542 








−0.1266 − 0.7594 


−0.6508 − 1.0846 
From these Kharitonov rectangles the value set given in Figure
The complete arc with vertices
In summary, the assigned polynomials are
Table
Results of the algorithm for the value set at frequency













Kharitonov rectangles calculated  


Theorem 
Theorem 
Theorem 
Theorem 
Theorem 
Theorem 






















8.95  127.23  127.23  210.20  186.88 


81.05  Condition 
Theorem 
Theorem 
Theorem 
Theorem 
Theorem 

































From these Kharitonov rectangles the value set given in Figure
The complete edge with vertices
Then
Then
In summary, the assigned polynomials are
Table
Results of the algorithm for the value set at frequency













Kharitonov rectangles calculated  


Theorem 
Theorem 
Theorem 
Theorem 
Theorem 
Theorem 






















308.66  308.66  23.01  352.12  352.12 


284.62  Condition 
Theorem 
Theorem 
Theorem 
Theorem 
Theorem 

































From these kharitonov rectangles the value set given in Figure
Finally, solving the equation system [
Value sets obtained at
This paper shows how to obtain the values of the numerator and denominator Kharitonov polynomials of an interval plant from its value set at a given frequency. Moreover, it is proven that given a value set, all the assigned polynomials of the vertices can be determined if and only if there is a complete edge or a complete arc lying on a quadrant, that is, if there are two vertices in a quadrant. This necessary and sufficient condition is not restrictive and practically all the value sets satisfy it. Finally, the interval plant can be identified solving the equation system between the Kharitonov rectangles and the parameters of the plant.
The algorithm has been formulated using the frequency domain properties of linear interval systems. The identification procedure of multilinear (affine, polynomial) systems will be studied using the results in [
The authors would like to express their gratitude to Dr. José Mira and to Dr. Ana Delgado for their example of ethics and professionalism, without which this work would not have been possible. Also, the authors are very grateful to the EditorinChief, Zhiwei Gao, and the referees, for their suggestions and comments that very much enhanced the presentation of this paper.