Nonlinear Filters. Simon Haykin

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СКАЧАТЬ alt="upper T"/> is finite. If the set of states in the neighborhood of a particular initial state that are indistinguishable from it includes only , then, the nonlinear system is said to be weakly observable at that initial state. A nonlinear system is called to be weakly observable if it is weakly observable at all . If the state and the output trajectories of a weakly observable nonlinear system remain close to the corresponding initial conditions, then the system that satisfies this additional constraint is called locally weakly observable [13, 20].

There is another difference between linear and nonlinear systems regarding observability and that is the role of inputs in nonlinear observability. While inputs do not affect the observability of a linear system, in nonlinear systems, some initial states may be distinguishable for some inputs and indistinguishable for others. This leads to the concept of uniform observability, which is the property of a class of systems, for which initial states are distinguishable for all inputs [12, 20]. Furthermore, in nonlinear systems, distinction between time‐invariant and time‐varying systems is not critical because by adding time as an extra state such as , the latter can be converted to the former [21].

      2.6.1 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)СКАЧАТЬ