Ohta, Yoshito; Maeda, Hajime; Kodama, Shinzo Reachability, observability, and realizability of continuous-time positive systems. (English) Zbl 0539.93005 SIAM J. Control Optimization 22, 171-180 (1984). The reachability, observability and realizability of single-input, single-output linear time-invariant systems (A,b,c) with nonnegative inputs(outputs) and/or nonnegative states are studied. The geometrical structure of R, the reachable set from the origin with nonnegative inputs and S, the observable set of initial states which produce nonnegative zero-input responses is emphasized (R is solid iff (A,b) is controllable, S is pointed iff (A,c) is observable), as well as the duality between R and S. Also the geometric meaning of other conditions for complete controllability with nonnegative inputs and complete observability with nonnegative inputs are obtained, using Theorem 3: If (A,b) is controllable ((A,c) is observable), then R is pointed (S is solid) iff A has at least one real eigenvalue and the maximal real eigenvalue is dominant. Next, the positive realization problem for a given transfer function H(s) is examined: it is shown that H(s) has a realization (A,b,c) where A has nonnegative off-diagonal elements and b,c are nonnegative vectors (and so nonnegative inputs and initial states produce nonnegative states and outputs) iff for a minimal realization (F,g,h) there exists a polyhedral convex cone P such that \((F+\lambda I)P\subset P\) for some \(\lambda\geq 0\) and \(R\subset P\subset S\). If the degree of H(s) is two, H(s) is positive realizable iff the corresponding impulse response function is nonnegative; when the degree is larger than two, the problem of obtaining positive realizability conditions directly from the input-output relation is formulated as an open question. Reviewer: V.Prepeliţă Cited in 45 Documents MSC: 93B05 Controllability 93B07 Observability 93B15 Realizations from input-output data 93B03 Attainable sets, reachability 93C05 Linear systems in control theory 93C99 Model systems in control theory 52Bxx Polytopes and polyhedra 49N15 Duality theory (optimization) Keywords:complete controllability; complete observability; positive realization PDFBibTeX XMLCite \textit{Y. Ohta} et al., SIAM J. Control Optim. 22, 171--180 (1984; Zbl 0539.93005) Full Text: DOI