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

$\begin{array}{cc}\dot{x}(t)& =Ax(t)+Bu(t),\\ y(t)& =Cx(t)+Du(t)\\ x(0)& =x_0\\ \end{array}$

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

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 $R\in {ℝ}^{(n-m)xn}$ is chosen such that the vertical augmented matrix given as

$\begin{array}{c}T=\left[\begin{array}{c}C\\ R\end{array}\right]\end{array}$

is nonsingular, then

$\begin{array}{c}C{T}^{-1}=\left[\begin{array}{cc}{I}_m& 0\end{array}\right]\end{array}$

Furthermore, let

$\begin{array}{c}TA{T}^{-1}=\left[\begin{array}{cc}{A}_{11}& A_{12}\\ A_{21}& A_{22}\end{array}\right],A_{11}\in {ℝ}^{mxm}\end{array}$

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

$\begin{array}{c}Tx=\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right],TB=\left[\begin{array}{c}{B}_1\\ B_2\end{array}\right]\end{array}$

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

$\begin{array}{c}\left[\begin{array}{c}{\dot{x}}_1\\ {\dot{x}}_2\end{array}\right]=\left[\begin{array}{cc}{A}_{11}& A_{12}\\ A_{21}& A_{22}\end{array}\right]\left[\begin{array}{c}{\dot{x}}_1\\ {\dot{x}}_2\end{array}\right]+\left[\begin{array}{c}{B}_1\\ B_2\end{array}\right]u,y=x_1\end{array}$

once an estimate of $x_2$ is obtained the the full state estimate can be given as

$\begin{array}{c}\hat{x}=T^{-1}\left[\begin{array}{c}y\\ {\hat{x}}_2\end{array}\right]\end{array}$

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

$\begin{array}{cc}\dot{z}& =Fz+Gy+Hu,\\ {\hat{x}}_2& =Mz+Ny\\ \end{array}$

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

$\begin{array}{c}li{m}_{t\to \mathrm{\infty }}(x_2-{\hat{x}}_2)=0\end{array}$

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

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

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

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

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

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

$\begin{array}{c}F=A_{22}+LA12,G=(A_{21}+L{A}_{11})-(A_{22}+L{A}_{12})L,H=B_2+L{B}_1,M=I,N=-L,\end{array}$

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.