serve the input-output structure of the model. Using the ideas of controllability and observability, in the early 1960's Kalman and others carried out this program for linear systems. The similar questions for nonlin-REQUENTLY, control systems of the following form ear systems were not effectively treated until the early are used to model the behavior of physical, biological, 1970' (1) Krener [ 191, [20] developed the nonlinear analog of linear controllability in terms of the Lie algebra 5 of vector where € Q , a subset of R I , x E M , a connected fields on M generated by the vector fields f(-,u) corremanifold of dimension m, y E R" and f and g are C" sponding to constant controls u E Q. The control variable 24 represents the externally applied the system Z is analytic, then there exists a unique maxicontrols or exogenous inputs to the system and the output mal submanifold M' of M through x' which carries all the variable y represents the observable parameters of the trajectories of Z passing through x ' such that any point system. The state variable x may or may not be directly on this submanifold can be reached from x' going formeasurable and is used to represent the memory of the ward and backward along the trajectories of the system. system. The past history of 2 affects its future evolution In particular if the dimension of 9(.x@) is m then M = M' only through information conveyed by this variable. and so the system is "controllable" in some sense, to be The state space M may be deficient in one of two ways. made precise in Section 11. If it is less than rn then Z can It may be too small to adequately represent the full be restricted to M' where it is "controllable." This is one variety of memory states, i.e., it may fail to distinguish half of reducing the state space M . between real states where some control exictes different For linear systems observable effects. If this is the case, then the mathematical system Z fails to adequately model the real system and hence, must be revised. The system may not be controllable, i.e., if 2 is known to known criterion that the rank of the matrix be in a given state x'E M at some time there may be other states x E M where the system cannot possibly get to ( B : A B : * . * : A " -' B ) or have come from using the given set of inputs. Or there be m. may be distinct states which are indistinguishable from an The other half of the p~o~a m of reducing the state input-output point of view, i.e., if the same input is space was supplied by Sussmann [28] for analytic or results. These problems can be caused by ignoring possi-[3] on Lie groups. Sussmann noted that indistinguishabilble real inputs or real observables. in which case these ity is an equivalence relation on and showed that for must be added to the model. If. after all the significant analytic 66controllable,, or symmetric G6controllable,9 sysinput and output variables have been incorporated into X, tems, indistinguishability is a closed regular equivalence relation so that the quotient is another manifold. He also On the other hand, the state space may be too large. where u ER', X E Rnl, JI ER", this reduces to the wellin either Of these states then the Same Output symmetric systems. This generalized the work of Brockett Manuscript received December 16, 1976; revised June 7, 1977. R. showed that the quotient inherits a system which has the
An observability analysis confirms this aspect and shows that the system is observable (in the sense of REF ) up to a global rotation in the 2D case.
Nonlinear controllability and observability
IEEE Transactions on Automatic Control
Mathematics