Nonlinear Filters. Simon Haykin

Continuous‐Time Nonlinear Systems

      The state‐space model of a continuous‐time nonlinear system is represented by the following system of nonlinear equations:

(2.61)

(2.62)

      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:

(2.67)

      where

      (2.68)

      and

      (2.69)