LMIs in Control/Observer Synthesis/Continuous Time/Reduced-Order State Observer

LMIs in Control/Observer Synthesis/Continuous Time/Reduced-Order State Observer

Reduced Order State Observer

The Reduced Order State Observer design paradigm follows naturally from the design of Full Order State Observer.

The System

${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t),\\y(t)&=Cx(t)+Du(t)\\x(0)&=x_{0}\\\end{aligned}}$

where $x(t)\in \mathbb {R} ^{n}$, $y(t)\in \mathbb {R} ^{m}$, $u(t)\in \mathbb {R} ^{q}$, at any $t\in \mathbb {R}$.

The Data

• The matrices $A,B,C,D$ are system matrices of appropriate dimensions and are known.

The Problem Formulation

Given a State-space representation of a system given as above. First an arbitrary matrix $R\in \mathbb {R} ^{(n-m)xn}$ is chosen such that the vertical augmented matrix given as

${\begin{aligned}T={\begin{bmatrix}C\\R\\\end{bmatrix}}\end{aligned}}$

is nonsingular, then

${\begin{aligned}CT^{-1}={\begin{bmatrix}I_{m}&0\end{bmatrix}}\end{aligned}}$

Furthermore, let

${\begin{aligned}TAT^{-1}={\begin{bmatrix}A_{11}&A_{12}\\A_{21}&A_{22}\\\end{bmatrix}},A_{11}\in \mathbb {R} ^{mxm}\end{aligned}}$

then the matrix pair $(A_{22},A_{12})$ is detectable if and only if $(A,C)$ is detectable, then let

${\begin{aligned}Tx={\begin{bmatrix}x_{1}\\x_{2}\\\end{bmatrix}},TB={\begin{bmatrix}B_{1}\\B_{2}\\\end{bmatrix}}\end{aligned}}$

then a new system of the form given below can be obtained

${\begin{aligned}{\begin{bmatrix}{\dot {x}}_{1}\\{\dot {x}}_{2}\\\end{bmatrix}}={\begin{bmatrix}A_{11}&A_{12}\\A_{21}&A_{22}\\\end{bmatrix}}{\begin{bmatrix}{\dot {x}}_{1}\\{\dot {x}}_{2}\\\end{bmatrix}}+{\begin{bmatrix}B_{1}\\B_{2}\\\end{bmatrix}}u,y=x_{1}\end{aligned}}$

once an estimate of $x_{2}$ is obtained the the full state estimate can be given as

${\begin{aligned}{\hat {x}}=T^{-1}{\begin{bmatrix}y\\{\hat {x}}_{2}\\\end{bmatrix}}\end{aligned}}$

the the reduced order observer can be obtained in the form.

${\begin{aligned}{\dot {z}}&=Fz+Gy+Hu,\\{\hat {x}}_{2}&=Mz+Ny\\\end{aligned}}$

Such that for arbitrary control and arbitrary initial system values, There holds

${\begin{aligned}lim_{t\to \infty }(x_{2}-{\hat {x}}_{2})=0\end{aligned}}$

The value for $F,G,H,M,N$ can be obtain by solving the following LMI.

The LMI:

The reduced-order observer exists if and only if one of the two conditions holds.

1) There exist a symmetric positive definite Matrix $P$ and a matrix $W$ that satisfy

• $A_{22}^{T}P+PA_{22}+W_{12}^{A}+A_{12}^{T}W<0.$

Then $L=P^{-1}W$
2) There exist a symmetric positive definite Matrix $P$ that satisfies the below Matrix inequality

• $A_{22}^{T}P+PA_{22}-A_{12}^{T}A_{12}<0$
  

Then $L=-{\frac {1}{2}}P^{-1}A_{12}^{T}$.

By using this value of $L$ we can reconstruct the observer state matrices as

${\begin{aligned}F=A_{22}+LA{12},G=(A_{21}+LA_{11})-(A_{22}+LA_{12})L,H=B_{2}+LB_{1},M=I,N=-L,\end{aligned}}$

Conclusion:

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.

External Links

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.