Reachability indices of positive linear systems

About K-Positivity Properties of Time-Invariant Linear Systems Subject to Point Delays

“…Similarly, the system is said to be internally nonnegative/positive (or, via an abbreviate notation, as nonnegative/positive) if both state and output trajectories are everywhere nonnegative/positive for any nonnegative/positive input [3,[7][8][9][10][11][12][13][14][15][16][17][18][19][20]. However, throughout this paper, the nonnegativity/positivity properties are referred to as internal (external) if they refer to the state (output) trajectory under nonnegative/positive input while no specification internal/external is given if both state and output trajectories exhibit the corresponding property.…”

“…Positive systems have an important relevance since the input, state, and output signals in many physical or biological systems are necessarily positive [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. Therefore, important attention has been paid to such systems in the last decades.…”

“…However, hyperstable systems of second and higher orders are not guaranteed to be externally positive since the impulse response kernel matrix is 2 Journal of Inequalities and Applications everywhere positive definite but not necessarily positive [19]. The properties of those systems like, for instance, stability, controllability/reachability or pole assignment through feedback become more difficult to analyze than in standard systems because those properties have to be simultaneously compatible with the nonnegativity/positivity concepts (see, e.g., [7][8][9][10][11][12][13]18]). Nonnegativity/positivity properties apply for both continuoustime and discrete-time systems and are commonly formulated on the first orthant which is an important case in applications [7-15, 18, 20].…”

An upper bound of the reachability index for a special class of positive 2-D systems

“…This index plays an important role because it enables us to determine in a finite number of steps whether a system is locally reachable or not. Moreover, once this index is obtained, several new open problems could be addressed such as the study of reachability algorithms (see [4], [11] and [24]), of a canonical form by means of an appropriated positive similarity ( [5] and [7]), of the definition of the pertaining complete sequence of invariants (see [5], [8] and [10]) as well as the analysis of this property under feedback ( [6] and [20]) and so on.…”

“…The reachability index for positive 1-D systems has been studied in [5], [9], [14], [15] and [26] among others, and it is always bounded by n (see [12] and [13]). However, despite the numerous research efforts, an upper bound on such a number for a positive 2-D system is still an unanswered question.…”