t right-parenthesis right-parenthesis Over partial-differential bold x EndFraction EndAbsoluteValue Subscript bold x left-parenthesis t 0 right-parenthesis Baseline Subscript bold x left-parenthesis t 0 right-parenthesis Baseline left-parenthesis bold x left-parenthesis t right-parenthesis minus bold x left-parenthesis t 0 right-parenthesis right-parenthesis comma"/>
(2.75)
Then, the observability test for linear systems can be applied to the following linearized system matrices:
In this way, the nonlinear observability matrix in (2.73) can be approximated by the observability matrix, which is constructed using and in (2.76) and (2.77). Although this approach may seem simpler, observability of the linearized system may not imply the observability of the original nonlinear system [9].
2.6.2 Discrete‐Time Nonlinear Systems
The state‐space model of a discrete‐time nonlinear system is represented by the following system of nonlinear equations:
where is the system function, and is the measurement function. Similar to the discrete‐time linear case, starting from the initial cycle, system's output vectors at successive cycles till cycle can be written based on the initial state and input vectors as follows:
Functional powers of the system function can be used to simplify the notation in the aforementioned equations. Functional powers are obtained by repeated composition of a function with itself:
(2.81)
where denotes the function‐composition operator: , and is the identity map. Alternatively, the difference equations in (2.80) can be rewritten as:
