1 Subscript Baseline equals Start 5 By 1 Matrix 1st Row bold g ring bold f Superscript 0 Baseline 2nd Row bold g ring bold f Superscript 1 Baseline 3rd Row bold g ring bold f squared 4th Row vertical-ellipsis 5th Row bold g ring bold f Superscript n minus 1 Baseline EndMatrix Subscript bold x 0 comma bold u Sub Subscript 0 colon n minus 1 Subscript Baseline period"/>
Similar to the continuous‐time case, the system of nonlinear difference equations in (2.82) can be linearized about the initial state based on the Taylor series expansion to develop a linearized test for weak local observability of the nonlinear discrete‐time system (2.78) and (2.79). The nonlinear system in (2.78) and (2.79) is locally weakly observable at , if there exist a neighborhood of and an ‐tuple of integers such that [9, 25]:
1 for .
2 The following observability matrix is full rank:(2.83)
where
(2.84)
The observability matrix for discrete‐time linear systems (2.22) is a special case of the observability matrix for discrete‐time nonlinear systems (2.83). In other words, if and are linear functions, then (2.83) will be reduced to (2.22) [9, 25].
2.6.3 Discretization of Nonlinear Systems
Unlike linear systems, there is not a general functional representation for discrete‐time equivalents of continuous‐time nonlinear systems. One approach is to find a discrete‐time equivalent for the perturbed state‐space model of the nonlinear system under study [19]. In this approach, first, we need to linearize the nonlinear system in (2.61) and (2.62) about nominal values of state and input vectors, denoted by and , respectively. The perturbation terms, denoted by , , and , are defined as the difference between the actual and the nominal values of state, input, and output vectors, respectively:
(2.85)
(2.86)
(2.87)
Since input is usually derived from a feedback control law, it may be a function of the state, . In such cases, a difference between the actual and the nominal values of the state (a perturbation in the state) leads to a difference between the actual and the nominal values of the input (a perturbation in the input), and in effect therefore, . Otherwise, can be zero. Using the Taylor series expansion and neglecting the higher‐order terms, we obtain the following perturbation state‐space model:
(2.88)
(2.89)
where and , respectively, denote the Jacobian matrices obtained by taking the derivatives of with respect to
