LMIs in Control/pages/Hinf-Optimal Filter

Hinf-Optimal Filter

The goal of optimal filtering is to design a filter that acts on the output $$ \bold{z}$$of the generalized plant and optimizes the transfer matrix from $$ \bold{w}$$to the filtered output.

The System
Consider the continuous-time generalized LTI plant, with minimal state-space representation

$$ \bold{\dot x} = \bold{Ax} + \bold{B}_1\bold{w},$$

$$ \bold{z} = \bold{C}_1\bold{x} + \bold{D}_{11}\bold{w},$$

$$ \bold{y} = \bold{C}_2\bold{x} + \bold{D}_{21}\bold{w},$$

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

$$ \bold{\dot x}_f = \bold{A}_f\bold{x}_f + \bold{B}_f\bold{y},$$

$$ \bold{\hat z} = \bold{C}_f\bold{x}_f + \bold{D}_f\bold{y},$$

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

$$ \tilde \bold P (s) = \tilde \bold C_1(s\bold{I}-\tilde \bold A)^{-1}\tilde \bold B _1+\tilde \bold D _{11},$$

where

$$ \tilde\bold A = \begin{bmatrix} \bold A & \bold 0 \\ \bold B_f \bold C_2 & \bold A_f \end{bmatrix}, $$

$$ \tilde\bold B_1 = \begin{bmatrix} \bold B_1 \\ \bold B_f \bold D_{21} \end{bmatrix}, $$

$$ \tilde\bold C_1 = \begin{bmatrix} \bold C_1 - \bold D_f\bold C_2 & -\bold C_f \end{bmatrix}, $$

$$ \tilde\bold D_{11} = \bold D_{11}-\bold D_f\bold D_{21}. $$

Optimal Filtering seeks to minimize the given norm of the transfer function $$ \tilde \bold P (s).$$

Filter Synthesis
Solve for $$ \bold A_n \in \R^{n_x\times n_x}, \bold B_n \in \R^{n_x\times n_y}, \bold C_f \in \R^{n_z\times n_x}, \bold D_f \in \R^{n_z\times n_y}, \bold{X,Y} \in \S^{n_x}, $$and $$ \gamma\in\R_{>0}$$ that minimize the objective function $$ J(\gamma)=\gamma$$, subject to

$$ \bold{X,Y} > 0,$$

$$ \bold Y-\bold X>0, $$

$$ \begin{bmatrix} \bold{YA}+\bold A^T\bold Y + \bold B_n \bold C_2 +\bold C^T_2\bold B_n^T & \bold A_n + \bold C_2^T\bold B_n^T+\bold A^T\bold X & \bold{YB}_1+\bold B_n\bold D_{21} \\ \end{bmatrix} <0.$$
 * & \bold A_n+\bold A_n^T & \bold{XB}_1+\bold B_n \bold D_{21} \\
 * & * & -\bold{I}

Conclusion
The optimal Hinf filter is recovered by the state-space matrices $$ \bold A_f=\bold X^{-1}\bold A_n $$and $$ \bold B_f=\bold X^{-1}\bold B_n. $$

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

$$ \bold{\dot x} = \bold{Ax} + \bold{B}_1\bold{w},$$

$$ \bold{z} = \bold{C}_1\bold{x} + \bold{D}_{11}\bold{w}-\bold u,$$

$$ \bold{y} = \bold{C}_2\bold{x} + \bold{D}_{21}\bold{w}.$$

The synthesis method presented in this page takes 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.