Controllability and Observability, Local Decompositions

In the first two sections of this chapter we will give some basic concepts and results in the study of controllability and observability for nonlinear systems. Roughly speaking we will restrict ourselves to what can be seen as the nonlinear generalizations of the Kalman rank conditions for controllability and observability of linear systems. The reason for this is that in the following chapters we will not need so much the notions of nonlinear controllability and observability per se, but only the “structural properties” as expressed by these nonlinear “controllability” and “observability” rank conditions that will be obtained. In the last section of this chapter we will show how the geometric interpretation of reachable and unobservable subspaces for linear systems as invariant subspaces enjoying some maximality or minimality properties can be generalized to the nonlinear case, using the notion of invariant distributions. In this way we make contact with the nonlinear generalization of linear geometric control theory as dealt with in later chapters, where this last notion plays a fundamental role.

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