Geometric Methods in System Theory In automatic control there are a large number of applications of a fairly simple type for which the motion of the state variables is not free to evolve in a vector space but rather must satisfy some constraints. Examples are numerous; in a switched, lossless electrical network energy is conserved and the state evolves on an ellipsoid surface defined by x'Qx equals a constant; in the control of finite state, continuous time, Markov processes the state evolves on the set x'x = 1, xi ~ O. The control of rigid body motions and trajectory control leads to problems of this type. There has been under way now for some time an effort to build up enough control theory to enable one to treat these problems in a more or less routine way. It is important to emphasise that the ordinary vector space-linear theory often gives the wrong insight and thus should not be relied upon.
Table of ContentsDynamical polysystems and control theory.- Lie algebras and lie groups in control theory.- Realization theory of bilinear systems.- An introduction to stochastic differential equations on manifolds 131.- General theory of global differential dynamics.- Two proofs of Chow’s theorem.- On necessary and sufficient conditions for localcontrollability along a reference trajectory.- The high order maximal principle.- Optimal control on manifolds.- Problems in geodesic control.- Controllability in nonlinear systems.- Control theory in transformation systems.- The imbedding problem for finite Markov chains.- Some remarks on the geometry of systems.- Minimal realizations of nonlinear systems.- Causal dynamical systems: irreducible realizations.- On the internal structure of bilinear input-output maps.- Optimal control of discrete bilinear systems.- Diffusions on manifolds arising from controllable systems.- Signal detection on lie groups.- Some estimation problems on lie groups.