Driving variable realizations and the nonnegative realization problem for controllable behaviors

Non-negative linear systems, traditionally investigated within the state-space framework, have been recently analysed within the behavioural setting. In a couple of recent papers [J. W. Nieuwenhuis, ‘‘When to call a linear system nonnegative’’, Linear Algebra & its Appl., 281, 1998, pp. 43–58.; M. E. Valcher, ‘‘Nonnegative linear systems in the behavioural approach: the autonomous case’’, Linear Algebra and its Appl., 2000, 319, pp. 147–162.], several definitions and results about non-negative behaviours (and, in particular, about non-negative autonomous behaviours) have been derived. Moreover, the non-negative realization problem for autonomous behaviours has been fully explored in [M. E. Valcher, ‘‘Non-negative realization of autonomous systems in the behavioral approach’’, SIAM J. on Control and Opt., 2001, 40, pp. 540–556.], thus deriving an extended set of necessary and sufficient conditions for an autonomous behaviour to admit a non-negative realization. Here, we focus our attention on the non-negative realization problem for controllable behaviours. To this end, we first address the general realization problem by means of a driving variable (DV) state-space representation, and investigate under what conditions a state-space model provides a DV realization of a controllable behaviour. Based on these results, we analyse the possibility of realizing a controllable behaviour by means of a non-negative DV representation and, in particular, a reachable non-negative DV realization. Moreover, necessary and sufficient conditions for a behaviour to be the controllable part of a (complete) behaviour endowed with a non-negative realization are presented.