User:Margav06/sandbox/Click here to continue/Observer synthesis/Reduced-Order State Observer
The Reduced Order State Observer design paradigm follows naturally from the design of Full Order State Observer.
![{\displaystyle {\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t),\\y(t)&=Cx(t)+Du(t)\\x(0)&=x_{0}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1dce70fd9f4a085a9999621049df937b3519deea)
where
,
,
, at any
.
- The matrices
are system matrices of appropriate dimensions and are known.
Given a State-space representation of a system given as above. First an arbitrary matrix
is chosen such that the vertical augmented matrix given as
![{\displaystyle {\begin{aligned}T={\begin{bmatrix}C\\R\\\end{bmatrix}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c84e0151d7609c91b91f406f3d933257ac439b04)
is nonsingular, then
![{\displaystyle {\begin{aligned}CT^{-1}={\begin{bmatrix}I_{m}&0\end{bmatrix}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fde6745dcabcd2411607c3da306c4d8e6ac332a1)
Furthermore, let
![{\displaystyle {\begin{aligned}TAT^{-1}={\begin{bmatrix}A_{11}&A_{12}\\A_{21}&A_{22}\\\end{bmatrix}},A_{11}\in \mathbb {R} ^{mxm}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb1e723b605344f662afbf0750e89650272d30d8)
then the matrix pair
is detectable if and only if
is detectable, then let
![{\displaystyle {\begin{aligned}Tx={\begin{bmatrix}x_{1}\\x_{2}\\\end{bmatrix}},TB={\begin{bmatrix}B_{1}\\B_{2}\\\end{bmatrix}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca73ad59663c84b2bf2a341709560979348d4978)
then a new system of the form given below can be obtained
![{\displaystyle {\begin{aligned}{\begin{bmatrix}{\dot {x}}_{1}\\{\dot {x}}_{2}\\\end{bmatrix}}={\begin{bmatrix}A_{11}&A_{12}\\A_{21}&A_{22}\\\end{bmatrix}}{\begin{bmatrix}{\dot {x}}_{1}\\{\dot {x}}_{2}\\\end{bmatrix}}+{\begin{bmatrix}B_{1}\\B_{2}\\\end{bmatrix}}u,y=x_{1}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0178f09b6697c25f32b95e0fbdd3711d2b4d8e45)
once an estimate of
is obtained the the full state estimate can be given as
![{\displaystyle {\begin{aligned}{\hat {x}}=T^{-1}{\begin{bmatrix}y\\{\hat {x}}_{2}\\\end{bmatrix}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c42fbe86babadeb524e517326b92eafa10e9b44)
the the reduced order observer can be obtained in the form.
![{\displaystyle {\begin{aligned}{\dot {z}}&=Fz+Gy+Hu,\\{\hat {x}}_{2}&=Mz+Ny\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/abeada97db322fc0b257d725d8c84aac5c8777cb)
Such that for arbitrary control and arbitrary initial system values, There holds
![{\displaystyle {\begin{aligned}lim_{t\to \infty }(x_{2}-{\hat {x}}_{2})=0\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a55858ceb25b4122c760efeaf20cf030f58a3901)
The value for
can be obtain by solving the following LMI.
The reduced-order observer exists if and only if one of the two conditions holds.
1) There exist a symmetric positive definite Matrix
and a matrix
that satisfy
![{\displaystyle A_{22}^{T}P+PA_{22}+W_{12}^{A}+A_{12}^{T}W<0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/57e7a5281f0c4fd475ec5d243d8494b0e8199d5a)
Then ![{\displaystyle L=P^{-1}W}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e834a40d23908b3b5cea6fb661bdfd624375e45)
2) There exist a symmetric positive definite Matrix
that satisfies the below Matrix inequality
Then
.
By using this value of
we can reconstruct the observer state matrices as
![{\displaystyle {\begin{aligned}F=A_{22}+LA{12},G=(A_{21}+LA_{11})-(A_{22}+LA_{12})L,H=B_{2}+LB_{1},M=I,N=-L,\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87b45c7fba72ee883e88c4565bb1e951fa751699)
Hence, we are able to form a reduced-order observer using which we can back of full state information as per the equation given at the end of the problem formulation given above.
A list of references documenting and validating the LMI.
- LMIs in Control Systems Analysis, Design and Applications - Duan and Yu
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.