462 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 3, MARCH 2002 Homogeneous Dynamical Systems Theory Bijoy K. Ghosh, Fellow, IEEE, and Clyde F. Martin, Fellow, IEEE Abstract—In this paper, we study controlled homogeneous dy- where the scalar output may be considered to be the slope namical systems and derive conditions under which the system is of the line spanned by the vector and where perspective controllable. We also derive conditions under which the system is observable in the presence of a control over the com- plex base field. In the absence of any control input, we derive a nec- essary and sufficient condition for observability of a homogeneous (1.3) dynamical system over the real base field. The observability crite- rion obtained generalizes a well known Popov–Belevitch–Hautus (PBH) rank criterion to check the observability of a linear dynam- ical system. Finally, we introduce rational, exponential, interpola- For a dynamical system of the form (1.1), (1.2), one is tion problems as an important step toward solving the problem of interested in controlling only the direction of the state vector realizing homogeneous dynamical systems with minimum state di- and such problems are, therefore, of interest in mensions. gaze control. To generalize the control problem, we consider a Index Terms—Author, please supply your own keywords or send dynamical system with state variable , control variable a blank e-mail to
[email protected] to receive a list of suggested and observation function , the projective keywords. space of homogeneous lines in , where we assume that . The dynamical system is described as I.