where is the system function, and is the measurement function. It is common practice to deploy a control law that uses state feedback. In such cases, the control input is itself a function of the state vector . Before proceeding, we need to recall the concept of Lie derivative from differential geometry [22, 23]. Assuming that and are smooth vector functions (they have derivatives of all orders or they can be differentiated infinitely many times), the Lie derivative of (the th element of ) with respect to is a scalar function defined as:
(2.63)
where denotes the gradient with respect to . For simplicity, arguments of the functions have not been shown. Repeated Lie derivatives are defined as:
(2.64)
with
(2.65)
Relevance of Lie derivatives to observability of nonlinear systems becomes clear, when we consider successive derivatives of the output vector as follows:
(2.66)
The aforementioned differential equations can be rewritten in the following compact form:
