Reachability, Observability, and Realizability of Continuous-Time Positive Systems

This paper discusses reachability, observability, and realizability of single-input, single-output linear time-invariant systems, in which state variables and/or input (output) functions are restricted to be nonnegative to reflect physical constraints frequently encountered in real systems. We define a set reachable from the origin with nonnegative inputs, and also a set observable with nonnegative outputs. We investigate geometrical structures of the sets through convex analysis, and a duality relation between them is established. Next we consider positive realization of a given transfer function. Using the reachable set and the observable set, we give a necessary and sufficient condition for positive realizability. An example is given to demonstrate that a positive realizable transfer function does not in general have a jointly controllable and observable positive realization.