LMIs in Control/pages/H2-Optimal Filter

H2-Optimal Filter

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 state-space representation

$\dot{𝐱}=𝐀𝐱+{𝐁}_1𝐰,$

$𝐳={𝐂}_1𝐱+{𝐃}_{11}𝐰,$

$𝐲={𝐂}_2𝐱+{𝐃}_{21}𝐰,$

where it is assumed that $𝐀$is Hurwitz. A continuous-time dynamic LTI filter with state-space representation

${\dot{𝐱}}_f={𝐀}_f{𝐱}_f+{𝐁}_f𝐲,$

$\widehat{𝐳}={𝐂}_f{𝐱}_f+{𝐃}_f𝐲,$

is designed to optimize the transfer function from $𝐰$to $\widetilde{𝐳}=𝐳-\widehat{𝐳}$, which is given by

$\widetilde{𝐏}(s)={\widetilde{𝐂}}_1(s𝐈-\widetilde{𝐀}{)}^{-1}{\widetilde{𝐁}}_1+{\widetilde{𝐃}}_{11},$

where

$\widetilde{𝐀}=\left[\begin{array}{cc}𝐀& \mathrm{𝟎}\\ {𝐁}_f{𝐂}_2& {𝐀}_f\end{array}\right],$

${\widetilde{𝐁}}_1=\left[\begin{array}{c}{𝐁}_1\\ {𝐁}_f{𝐃}_{21}\end{array}\right],$

${\widetilde{𝐂}}_1=\left[\begin{array}{cc}{𝐂}_1-{𝐃}_f{𝐂}_2& -{𝐂}_f\end{array}\right],$

${\widetilde{𝐃}}_{11}={𝐃}_{11}-{𝐃}_f{𝐃}_{21}.$

Optimal Filtering seeks to minimize the given norm of the transfer function $\widetilde{𝐏}(s).$There are two methods of synthesizing the H2-optimal filter.

Synthesis 1

Solve for ${𝐀}_n\in {ℝ}^{n_x\times n_x},{𝐁}_n\in {ℝ}^{n_x\times n_y},{𝐂}_f\in {ℝ}^{n_z\times n_x},{𝐃}_f\in {ℝ}^{n_z\times n_y},𝐗,𝐘\in {\mathrm{\S }}^{n_x},𝐙\in {\mathrm{\S }}^{n_z},$and $\nu \in {ℝ}_{>0}$ that minimize the objective function $J(\nu )=\nu$, subject to

$𝐗,𝐘,𝐙>0,$

$𝐘-𝐗>0,$

$tr(𝐙)<\nu ,$

${𝐃}_{11}-{𝐃}_f{𝐃}_{21}=\mathrm{𝟎},$

$\left[\begin{array}{ccc}-𝐙& {𝐂}_1-{𝐃}_f{𝐂}_2& -{𝐂}_f\\ *& -𝐘& -𝐗\\ *& *& -𝐗\end{array}\right]<0,$

$\left[\begin{array}{ccc}𝐘𝐀+{𝐀}^T𝐘+{𝐁}_n{𝐂}_2+{𝐂}_2^T{𝐁}_n^T& {𝐀}_n+{𝐂}_2^T{𝐁}_n^T+{𝐀}^T𝐗& {𝐘𝐁}_1+{𝐁}_n{𝐃}_{21}\\ *& {𝐀}_n+{𝐀}_n^T& {𝐗𝐁}_1+{𝐁}_n{𝐃}_{21}\\ *& *& -𝐈\end{array}\right]<0.$

Synthesis 2

Synthesis 2 is identical to Synthesis 1, with the exception of the final two matrix inequality constraints:

$\left[\begin{array}{ccc}-𝐙& {𝐁}_1^T{𝐘}^T+{𝐃}_{21}^T{𝐁}_n^T& {𝐁}_1^T{𝐗}^T+{𝐃}_{21}^T{𝐁}_n^T\\ *& -𝐘& -𝐗\\ *& *& -𝐗\end{array}\right]<0,$

$\left[\begin{array}{ccc}𝐘𝐀+{𝐀}^T𝐘+{𝐁}_n{𝐂}_2+{𝐂}_2^T{𝐁}_n^T& {𝐀}_n+{𝐂}_2^T{𝐁}_n^T+{𝐀}^T𝐗& {𝐂}_1^T-{𝐂}_2^T{𝐃}_f^T\\ *& {𝐀}_n+{𝐀}_n^T& -{𝐂}_f^T\\ *& *& -𝐈\end{array}\right]<0.$

Remark

In both cases, if ${𝐃}_{11}=0$and ${𝐃}_{21}\ne 0,$then it is often simplest to choose ${𝐃}_f=0$in order to satisfy the equality constraint (above).

Conclusion

In both cases, the optimal H2 filter is recovered by the state-space matrices ${𝐀}_f={𝐗}^{-1}{𝐀}_n,{𝐁}_f={𝐗}^{-1}{𝐁}_n,{𝐂}_f,$and ${𝐃}_f.$

Remark

The problem of optimal filtering can alternatively be formulated as a special case of synthesizing a dynamic output "feedback" controller for the generalized plant given by

$\dot{𝐱}=𝐀𝐱+{𝐁}_1𝐰,$

$𝐳={𝐂}_1𝐱+{𝐃}_{11}𝐰-𝐮,$

$𝐲={𝐂}_2𝐱+{𝐃}_{21}𝐰.$

The synthesis methods presented in this page take advantage of the fact that the controller in this case is not a true feedback controller, as it only appears as a feedthrough term in the performance channel.

External Links

A list of references documenting and validating the LMI.

• 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.