The goal of mixed
-optimal state estimation is to design an observer that minimizes the
norm of the closed-loop transfer matrix from
to
, while ensuring that the
norm of the closed-loop transfer matrix from
to
is below a specified bound.
Consider the continuous-time generalized plant
with state-space realization
![{\displaystyle {\begin{aligned}{\dot {x}}&=Ax+B_{1,1}w_{1}+B_{1,2}w_{2},\\y&=C_{2}x+D_{21,1}w_{1}+D_{21,1}w_{2}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/773dfcecd8db1927504842458cb9295cfde7c938)
where it is assumed that
is detectable.
The matrices needed as input are
.
The observer gain L is to be designed to minimize the
norm of the closed-loop transfer matrix
from the exogenous input
to the performance output
while ensuring the
norm of the closed-loop transfer matrix
from the exogenous input
to the performance output
is less than
, where
![{\displaystyle {\begin{aligned}T_{11}(s)=C_{1,1}(s1-(A-LC_{2}))^{-1}(B_{1,1}-LD_{21,1})\\T_{22}(s)=C_{1,2}(s1-(A-LC_{2}))^{-1}(B_{1,2}-LD_{21,2})+D_{11,22}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e662057a4f7d37c24484ab11c915497d6f5b01ed)
is minimized.
The form of the observer would be:
![{\displaystyle {\begin{aligned}{\dot {\hat {x}}}=A{\hat {x}}+L(y-{\hat {y}}),\\{\hat {y}}=C_{2}{\hat {x}}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/163de59d85cab5fb2bed5d9144fea332b7dfa1ee)
is to be designed, where
is the observer gain.
The LMI:
Optimal Observer
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The mixed
-optimal observer gain is synthesized by solving for
, and
that minimize
subject to
,
![{\displaystyle {\begin{aligned}{\begin{bmatrix}PA+A^{T}P-GC_{2}-{C_{2}}^{T}G^{T}&&PB_{1}-GD_{21}\\\star &&-1\end{bmatrix}}<0\\{\begin{bmatrix}PA+A^{T}P-GC_{2}-{C_{2}}^{T}G^{T}&&PB_{1}-GD_{21}&&C_{1}\\\star &&-\gamma 1&&{D_{11}}^{T}\\\star &&\star &&-\gamma 1\end{bmatrix}}<0\\{\begin{bmatrix}P&&C{_{1,1}}^{T}\\\star &&Z\end{bmatrix}}>0\\trZ<\nu \end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f2fc69f3778532de78f6c6652f10ae0fe878a8d)
The mixed
-optimal observer gain is recovered by
, the
norm of
is less than
and the
norm of T(s) is less than
.
Link to the MATLAB code designing
- Optimal Observer
Code
Optimal Observer