Note that $\boldsymbol{f}(\boldsymbol{x}_e, \boldsymbol{u}_e) = 0$ . Let

$\displaystyle A = \frac{\partial \boldsymbol{f}}{\partial \boldsymbol{x}} (\boldsymbol{x}_e,\boldsymbol{u}_e)$

(3)

Neglecting higher order terms, we arrive at the linear approximation

$\displaystyle \Delta \boldsymbol{x}(k+1) = A \Delta \boldsymbol{x}(k) + B \Delta \boldsymbol{u}(k)$

(4)

Similarly, if the outputs of the nonlinear system model are of the form

or in vector notation

$\displaystyle \boldsymbol{y}(k) = \boldsymbol{h}(\boldsymbol{x}(k),\boldsymbol{u}(k))$

(5)

then Taylor's series expansion can again be used to yield the linear approximation of the above output equations. Indeed, if we let

$\displaystyle \boldsymbol{y} = \boldsymbol{y}_e + \Delta \boldsymbol{y}$

(6)

then we obtain

$\displaystyle \Delta \boldsymbol{y}(k) = C \Delta \boldsymbol{x}(k) + D \Delta \boldsymbol{u}(k)$

(7)

Example: Consider a nonlinear system

8(a)

8(b)