An algorithm to check the nonnegativity of singular systems
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 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 is nonnegative. Theorem 2.1 Let (E, A, B, C) be a discrete-time singular system such that , , and . The system (E, A, B, C) is nonnegative if and only if for each , the following conditions hold: , , , , and .
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 is nonnegative.
Algorithm Inputs: Singular system (E, A, B, C).
- Step 1
Find α such that .
- Step 2
Compute , and as in (5) and set .
- Step 3
Compute and .
- Step 4
If or or then go to End.
- Step 5
Find a permutation P such that is in the form (1).
- Step 6
Set and , and partition and as in (8), (9), respectively.
- Step 7
Compute , , 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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