31.2.4. Optimal observer for deterministic disturbances#
\[\begin{split}\begin{cases}
\dot{\mathbf{x}} = \mathbf{A} \mathbf{x} + \mathbf{B} \mathbf{u} + \mathbf{B}_d \mathbf{d} \\
\mathbf{y} = \mathbf{C} \mathbf{x} + \mathbf{D} \mathbf{u} + \mathbf{D}_d \mathbf{d} + \mathbf{D}_r \mathbf{r}
\end{cases}\end{split}\]
\[\begin{split}\begin{cases}
\dot{\mathbf{o}} = \mathbf{A} \mathbf{o} + \mathbf{B} \mathbf{u} + \mathbf{L} \left( \mathbf{y} - \mathbf{y}_o \right) \\
\mathbf{y}_o = \mathbf{C} \mathbf{o} + \mathbf{D} \mathbf{u}
\end{cases}\end{split}\]
The error is defined as \(\mathbf{e} := \mathbf{o} - \mathbf{x}\), and it’s governed by the dynamical equation
\[\dot{\mathbf{e}} = \left( \mathbf{A} - \mathbf{L} \mathbf{C} \right) \mathbf{e} + \left( \mathbf{B}_d - \mathbf{L} \mathbf{D}_d \right) \mathbf{d} - \mathbf{L} \mathbf{D}_r \mathbf{r} \ .\]
This equation shows that the dynamics is asymptotically stable if the matrix \(\mathbf{A} - \mathbf{L} \mathbf{C}\) is asymptotically stable. The matrix \(\mathbf{L}\) is involved in the dynamics from the exogenous input \(\mathbf{d}\) and measurement error \(\mathbf{r}\). The design of \(\mathbf{L}\) is driven by some requiremenets:
asymptotic stability of \(\mathbf{A} - \mathbf{L}\mathbf{C}\),
low sensitivity to exogenous input and measurement error.