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Minimal-order Observer

A full-order observer estimates all states whether it is measured by a sensor or not. It leads the increase of calculation cost and makes a observer design difficult. A minimal-order observer use sensor outputs as known parameters and reduces a states to be estimated.

Reduced-order system

Let z\bm{z} be measurable state by a sensor and ξ\bm{\xi} be a state to be estimated, and let us consider a system
ddt[ξz]=[A11A12A21A22][ξz]+[B1B2]u+[H1H2]vy=z+w,\begin{align} \frac{d}{dt}\begin{bmatrix} \bm{\xi} \\ \bm{z} \end{bmatrix} &= \begin{bmatrix} \bm{A}_{11} & \bm{A}_{12} \\ \bm{A}_{21} & \bm{A}_{22} \end{bmatrix}\begin{bmatrix} \bm{\xi} \\ \bm{z} \end{bmatrix} + \begin{bmatrix} \bm{B}_{1} \\ \bm{B}_{2} \end{bmatrix}\bm{u} + \begin{bmatrix} \bm{H}_{1} \\ \bm{H}_{2} \end{bmatrix}\bm{v} \\ \bm{y}&=\bm{z}+\bm{w}, \end{align}
where y\bm{y}, u\bm{u}, v\bm{v}, and w\bm{w} are an output, and input, process noise, and observation noise. This representation provides a following equation
{ξ˙=A11ξ+(A12z+B1u)+H1vz˙−A22z−B2u=A21ξ+H2v.\begin{align} & \left\{ \begin{matrix} \dot{\bm{\xi}}=\bm{A}_{11}\bm{\xi}+(\bm{A}_{12}\bm{z}+\bm{B}_{1}\bm{u})+\bm{H}_{1}\bm{v} \\ \dot{\bm{z}}-\bm{A}_{22}\bm{z}-\bm{B}_{2}\bm{u}=\bm{A}_{21}\bm{\xi} +\bm{H}_{2}\bm{v} \end{matrix} \right.. \end{align}
This equation includes two estimable parameters using sensor outputs.
A12z+B1u≡Uz˙−A22z−B2u≡Y.\begin{align} &\bm{A}_{12}\bm{z}+\bm{B}_{1}\bm{u} \equiv \bm{U} \\ &\dot{\bm{z}}-\bm{A}_{22}\bm{z}-\bm{B}_{2}\bm{u} \equiv \bm{Y}. \end{align}
These parameters simplify the equation as
{ξ˙=A11ξ+U+H1vY=A21ξ+H2v.\begin{align} & \left\{ \begin{matrix} \dot{\bm{\xi}}=\bm{A}_{11}\bm{\xi}+\bm{U} +\bm{H}_{1}\bm{v}\\ \bm{Y}=\bm{A}_{21}\bm{\xi} +\bm{H}_{2}\bm{v}\end{matrix} \right.. \\ \end{align}
The simplified representation shows that U\bm{U} and Y\bm{Y} works as an input and output.

Observer design

An estimated input and measured output of a reduced-order system is expressed as
U^=An12y+Bn1uYˉ=y˙−An22y−Bn2u.\begin{align} \hat{\bm{U}}&=\bm{A}_{{\rm n}12}\bm{y}+\bm{B}_{{\rm n}1}\bm{u} \\ \bar{\bm{Y}}&=\dot{\bm{y}}-\bm{A}_{{\rm n}22}\bm{y}-\bm{B}_{{\rm n}2}\bm{u}. \end{align}
Using these parameter, a Luenberger observer can be designed as
{ξ˙^=An11ξ^+U^+L(Yˉ−Y^)Y^=An21ξ^.\begin{align} & \left\{ \begin{matrix} \hat{\dot{\bm{\xi}}}=\bm{A}_{{\rm n}11}\hat{\bm{\xi}}+\hat{\bm{U}}+\bm{L}(\bar{\bm{Y}}-\hat{\bm{Y}})\\ \hat{\bm{Y}}=\bm{A}_{{\rm n}21}\hat{\bm{\xi}} \end{matrix} \right.. \end{align}
Expanding the equation provides an estimation formula.
ξ˙^=An11ξ^+U^+L(Yˉ−Y^)=An11ξ^+(An12y+Bn1u)+L((y˙−An22y−Bn2u)−An21ξ^)=(An11−LAn21)ξ^+(An12y+Bn1u)+L(y˙−An22y−Bn2u)=(An11−LAn21)ξ^+(An12−LAn22)y+(Bn1−LBn2)u+Ly˙\begin{align} \hat{\dot{\bm{\xi}}}&=\bm{A}_{{\rm n}11}\hat{\bm{\xi}}+\hat{\bm{U}}+\bm{L}(\bar{\bm{Y}}-\hat{\bm{Y}})\\ &=\bm{A}_{{\rm n}11}\hat{\bm{\xi}}+(\bm{A}_{{\rm n}12}\bm{y}+\bm{B}_{{\rm n}1}\bm{u})+\bm{L}((\dot{\bm{y}}-\bm{A}_{{\rm n}22}\bm{y}-\bm{B}_{{\rm n}2}\bm{u})-\bm{A}_{{\rm n} 21}\hat{\bm{\xi}}) \\ &=(\bm{A}_{{\rm n}11}-\bm{L}\bm{A}_{{\rm n} 21})\hat{\bm{\xi}}+(\bm{A}_{{\rm n}12}\bm{y}+\bm{B}_{{\rm n}1}\bm{u})+\bm{L}(\dot{\bm{y}}-\bm{A}_{{\rm n}22}\bm{y}-\bm{B}_{{\rm n}2}\bm{u}) \\ &=(\bm{A}_{{\rm n}11}-\bm{L}\bm{A}_{{\rm n} 21})\hat{\bm{\xi}}+(\bm{A}_{{\rm n}12}-\bm{L}\bm{A}_{{\rm n}22})\bm{y}+(\bm{B}_{{\rm n}1}-\bm{L}\bm{B}_{{\rm n}2})\bm{u}+\bm{L}\dot{\bm{y}} \end{align}
This formula includes a derivative of y\bm{y} and is difficlult to calculate. This problem is solved by introducing a variable transformation.
P≡ξ^−Lyξ˙^−Ly˙=(An11−LAn21)ξ^+(An12−LAn22)y+(Bn1−LBn2)u=(An11−LAn21)ξ^−(An11−LAn21)Ly+(An11−LAn21)Ly+(An12−LAn22)y+(Bn1−LBn2)u=(An11−LAn21)(ξ^−Ly)+((An11−LAn21)L+An12−LAn22)y+(Bn1−LBn2)u⇔P˙=(An11−LAn21)P+((An11−LAn21)L+An12−LAn22)y+(Bn1−LBn2)u∴P=(sI−An11+LAn21)−1{(Bn1−LBn2)u+((An11−LAn21)L+An12−LAn22)y}\begin{align} \bm{P}&\equiv\hat{\bm{\xi}}-\bm{L}\bm{y} \\ \hat{\dot{\bm{\xi}}}-\bm{L}\dot{\bm{y}}&=(\bm{A}_{{\rm n}11}-\bm{L}\bm{A}_{{\rm n} 21})\hat{\bm{\xi}}+(\bm{A}_{{\rm n}12}-\bm{L}\bm{A}_{{\rm n}22})\bm{y}+(\bm{B}_{{\rm n}1}-\bm{L}\bm{B}_{{\rm n}2})\bm{u}\\ &=(\bm{A}_{{\rm n}11}-\bm{L}\bm{A}_{{\rm n} 21})\hat{\bm{\xi}}-(\bm{A}_{{\rm n}11}-\bm{L}\bm{A}_{{\rm n} 21})\bm{L}\bm{y}+(\bm{A}_{{\rm n}11}-\bm{L}\bm{A}_{{\rm n} 21})\bm{L}\bm{y}+(\bm{A}_{{\rm n}12}-\bm{L}\bm{A}_{{\rm n}22})\bm{y}+(\bm{B}_{{\rm n}1}-\bm{L}\bm{B}_{{\rm n}2})\bm{u} \\ &=(\bm{A}_{{\rm n}11}-\bm{L}\bm{A}_{{\rm n} 21})(\hat{\bm{\xi}}-\bm{L}\bm{y})+((\bm{A}_{{\rm n}11}-\bm{L}\bm{A}_{{\rm n} 21})\bm{L}+\bm{A}_{{\rm n}12}-\bm{L}\bm{A}_{{\rm n}22})\bm{y}+(\bm{B}_{{\rm n}1}-\bm{L}\bm{B}_{{\rm n}2})\bm{u} \\ \Leftrightarrow \dot{\bm{P}} &=(\bm{A}_{{\rm n}11}-\bm{L}\bm{A}_{{\rm n} 21})\bm{P}+((\bm{A}_{{\rm n}11}-\bm{L}\bm{A}_{{\rm n} 21})\bm{L}+\bm{A}_{{\rm n}12}-\bm{L}\bm{A}_{{\rm n}22})\bm{y}+(\bm{B}_{{\rm n}1}-\bm{L}\bm{B}_{{\rm n}2})\bm{u}\\ \therefore \bm{P}&=(s\bm{I}-\bm{A}_{{\rm n}11}+\bm{L}\bm{A}_{{\rm n}21})^{-1}\{(\bm{B}_{{\rm n}1}-\bm{L}\bm{B}_{{\rm n}2})\bm{u}+((\bm{A}_{{\rm n}11}-\bm{L}\bm{A}_{{\rm n} 21})\bm{L}+\bm{A}_{{\rm n}12}-\bm{L}\bm{A}_{{\rm n}22})\bm{y}\} \end{align}
As a result, an estimated value can be obtained by following formula:
ξ^=(sI−An11+LAn21)−1{(Bn1−LBn2)u+((An11−LAn21)L+An12−LAn22)y}+Ly=(sI−An11+LAn21)−1{(Bn1−LBn2)u+(sL+An12−LAn22)y}.\begin{align} \hat{\bm{\xi}}&=(s\bm{I}-\bm{A}_{{\rm n}11}+\bm{L}\bm{A}_{{\rm n}21})^{-1}\{(\bm{B}_{{\rm n}1}-\bm{L}\bm{B}_{{\rm n}2})\bm{u}+((\bm{A}_{{\rm n}11}-\bm{L}\bm{A}_{{\rm n} 21})\bm{L}+\bm{A}_{{\rm n}12}-\bm{L}\bm{A}_{{\rm n}22})\bm{y}\}+\bm{L}\bm{y} \\ &= (s\bm{I}-\bm{A}_{{\rm n}11}+\bm{L}\bm{A}_{{\rm n}21})^{-1}\{(\bm{B}_{{\rm n}1}-\bm{L}\bm{B}_{{\rm n}2})\bm{u}+(s\bm{L}+\bm{A}_{{\rm n}12}-\bm{L}\bm{A}_{{\rm n}22})\bm{y}\}. \end{align}

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