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

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



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

is nonsingular, then



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

Furthermore, let



\begin{align} TAT^{-1}=\begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \\

\end{bmatrix} , A_{11} \in \R^{m x m} \end{align} $$

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

\begin{align} Tx=\begin{bmatrix} x_{1} \\ x_{2} \\ \end{bmatrix} , TB=\begin{bmatrix} B_{1} \\ B_{2} \\ \end{bmatrix} \end{align} $$ then a new system of the form given below can be obtained

\begin{align} \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{align} $$

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

\begin{align} \hat x= T^{-1}\begin{bmatrix} y \\ \hat x_{2} \\ \end{bmatrix} \end{align} $$ the the reduced order observer can be obtained in the form.



\begin{align} \dot z&=Fz+Gy+Hu,\\ \hat x_{2}&=Mz+Ny\\ \end{align} $$ Such that for arbitrary control and arbitrary initial system values, There holds

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

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
 * $$ A_{22}^TP + PA_{22} + W^A_{12} + A_{12}^TW < 0. $$

Then $$ L = P^{-1}W $$ 2) There exist a symmetric positive definite Matrix $$ P $$ that satisfies the below Matrix inequality Then $$ L = -\frac{1}{2}P^{-1}A_{12}^{T} $$.
 * $$ A_{22}^TP + PA_{22} - A_{12}^TA_{12} < 0 $$

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



\begin{align} 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{align} $$

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.