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Forming Outputs

Any system output is some function of the state, and possibly the input (directly):

$\displaystyle \underline{y}(t) \isdef o_t[\underline{x}(t),\underline{u}(t)]

The general case of output extraction is shown in Fig.1.5.
Figure: Continuous-time state-space model with output vector $ \underline{y}(t) = o_t[\underline{x}(t),\underline{u}(t)]$.
The output signal (vector) is most typically a linear combination of state variables and possibly the current input:

$\displaystyle \underline{y}(t) \isdefs C\underline{x}(t) + D\underline{u}(t)

where $ C$ and $ D$ are constant matrices of linear-combination coefficients.
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State-Space Model of a Force-Driven Mass