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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 Скачать книгу