LMIs in Control/pages/HInf Optimal Filter

Optimal filtering is a means of adaptive extraction of a weak desired signal in the presence of noise and interfering signals. The goal of optimal filtering is to design a filter that acts on the output $z$ of the generalized plant and optimizes the transfer matrix from w to the filtered output.

The System:

Consider the continuous-time generalized LTI plant with minimal states-space realization

$\begin{array}{cc}\dot{x}& =Ax+B_{1}w\\ z& =C_{1}x+D_{11}w,\\ y& =C_{2}x+D_{21}w,\end{array}$

where it is assumed that $A$ is Hurwitz.

The Data

The matrices needed as inputs are $A,B_1,C_2,C_1,,D_{11},D_{21}$.

The Optimization Problem:

An $H\mathrm{\infty }$-optimal filter is designed to minimize the $H_{\mathrm{\infty }}$ norm of $\tilde{P}(s)$ in following equation.

$\begin{array}{c}\tilde{P}(s)={\tilde{C}}_1(sI-\tilde{A}{)}^{-}1{\tilde{B}}_1+{\tilde{D}}_{11},\\ \text{where}\\ \tilde{A}=\left[\begin{array}{ccc}A& & 0\\ B_f{C}_2& & A_f\end{array}\right]& <0\\ {\tilde{B}}_1=\left[\begin{array}{c}{B}_1\\ B_f{D}_{21}\end{array}\right]& <0\\ {\tilde{C}}_1=\left[\begin{array}{c}{C}_1-D_f{C}_2-C_f\end{array}\right]& <0\\ {\tilde{D}}_{11}=D_{11}-D_f{D}_{21}\\ \end{array}$

The LMI: $H_{\mathrm{\infty }}$- Optimal filter

Solve for $A_n\in {ℝ}^{n_x\times n_x},B_n\in {ℝ}^{n_x\times n_y},C_f\in {ℝ}^{n_x\times n_x}$, $X,Y\in {𝕊}^{n_x}$ and $\nu \in {ℝ}_{>0}$ that minimize $\zeta (\nu )=\nu$ subject to $X>0,Y>0$.

$\begin{array}{cc}\left[\begin{array}{ccccccc}YA+A^{T}Y+B_n{C}_2& & A_n+C_2^T{B}_n^T+A^{T}X& & Y{B}_1+B_n{D}_{21}& & {C_1}^T-{C_2}^T{D_f}^T\\ \star & & A_n+A_n^T& & X{B}_1+B_n{D}_{21}& & -{C_f}^T\\ \star & & \star & & -\gamma I& & {D_{1}1}^T-{D_{2}1}^T{D_f}^T\\ \star & & \star & & \star & & -\gamma I\end{array}\right]& <0\\ \\ Y-X>0\\ \end{array}$

Conclusion:

The filter is recovered by $A_f=X^{-1}{A}_n$ and $B_f=X^{-1}{B}_n$.

Implementation

• https://github.com/Ricky-10/coding107/blob/master/Hinfinity%20Optimal%20filter- $H\mathrm{\infty }$- Optimal Filter

External links

• [1]- Optimal Filtering
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.