An algorithm to check the nonnegativity of singular systems

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Abstract

In the literature, an important class of generalized inverse matrices corresponds to the group inverse, that is, matrices of index 1. Recently, the nonnegativity of a singular system has been applied to different fields. In this paper, an algorithm to check the nonnegativity of a singular linear control system of index 1 is presented. To this purpose, the nonnegativity of this kind of systems is characterized using a block partition of the original matrices. In this way, we can work with matrices having smaller sizes and keeping the original information. Finally, numerical examples illustrating the obtained results are shown.

Section snippets

Introduction and background

A very important part of engineering work concerns the establishment of mathematical models. Due to its capacity to describe the dynamic and algebraic relationships between state variables simultaneously, a singular system permits to give a mathematical description for many practical dynamic systems, which do not admit a standard state–space model representation [1].

A dynamical system is said to be nonnegative if it leaves the first orthant of Rn invariant for future times when initiated in

Characterization of the nonnegativity

We start this section with a result that gives necessary and sufficient conditions on the matrices E, A, B, and C such that the system (E,A,B,C) is nonnegative.

Theorem 2.1

Let (E, A, B, C) be a discrete-time singular system such that EDEO, EA=AE, and Ker(E)Ker(A)={0}. The system (E, A, B, C) is nonnegative if and only if for each i=0,1,,ind(E)-1, the following conditions hold:

  • (a)

    EDAO,

  • (b)

    EDBO,

  • (c)

    CEDEO,

  • (d)

    -(I-EED)(EAD)iADBO, and

  • (e)

    -C(I-EED)(EAD)iADBO.

Proof

From Theorem 1.4, it is clear that the nonnegativity of the system (E, A,

Algorithm and examples

The algorithm presented below permits to decide when a singular control linear system (withind(E^)=1) is nonnegative.

Algorithm Inputs: Singular system (E, A, B, C).

  • Step 1

    Find α such that det(αE+A)0.

  • Step 2

    Compute E^, B^ and C^ as in (5) and set A^=I-αE^.

  • Step 3

    Compute ind(E^) and E^#.

  • Step 4

    If ind(E^)1 or E^O or E^E^#O then go to End.

  • Step 5

    Find a permutation P such that E^ is in the form (1).

  • Step 6

    Set E=PE^Pt and A=PA^Pt, and partition B=PB^ and C=C^Pt as in (8), (9), respectively.

  • Step 7

    Compute H=I-αXTY, J=-αXTYM, S, U and V as in (13),

Conclusions

In order to decide if a singular control system (E,A,B,C) in nonnegative we could firstly apply the Theorem 2.1. The inconvenient of this theorem is due to the fact that the Drazin inverse not always can be computed in an easy way. So, we have designed an algorithm by means of a technique that involves matrices of smaller sizes partitioning in blocks the original matrices in a suitable way. In this sense, our algorithm improves the result given in Theorem 2.1, where the whole matrices are used

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