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 $$ \bold{z}$$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 $$ \tilde \bold P (s).$$There are two methods of synthesizing the H2-optimal filter.

Synthesis 1
Solve for $$ \bold A_n \in \R^{n_x\times n_x}, \bold B_n \in \R^{n_x\times n_y}, \bold C_f \in \R^{n_z\times n_x}, \bold D_f \in \R^{n_z\times n_y}, \bold{X,Y} \in \S^{n_x}, \bold Z \in \S^{n_z},$$and $$ \nu\in\R_{>0}$$ that minimize the objective function $$ J(\nu)=\nu$$, subject to

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

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

$$ tr(\bold Z)<\nu,$$

$$ \bold D_{11}-\bold D_f\bold D_{21}=\bold 0,$$

$$ \begin{bmatrix} -\bold Z & \bold C_1-\bold D_f\bold C_2 & -\bold C_f \\ \end{bmatrix} <0,$$
 * & -\bold Y & -\bold X \\
 * & * & -\bold X

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

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

$$ \begin{bmatrix} -\bold Z & \bold B_1^T\bold Y^T+\bold D_{21}^T\bold B_n^T & \bold B_1^T\bold X^T+\bold D_{21}^T\bold B_n^T \\ \end{bmatrix} <0,$$
 * & -\bold Y & -\bold X \\
 * & * & -\bold X

$$ \begin{bmatrix} \bold{YA}+\bold A^T\bold Y + \bold B_n \bold C_2 +\bold C^T_2\bold B_n^T & \bold A_n + \bold C_2^T\bold B_n^T+\bold A^T\bold X & \bold C_1^T-\bold C_2^T\bold D_f^T \\ \end{bmatrix} <0.$$
 * & \bold A_n+\bold A_n^T & -\bold C_f^T \\
 * & * & -\bold{I}

Remark
In both cases, if $$ \bold D_{11}=0$$and $$ \bold D_{21}\neq0,$$then it is often simplest to choose $$ \bold D_f=0$$in order to satisfy the equality constraint (above).

Conclusion
In both cases, the optimal H2 filter is recovered by the state-space matrices $$ \bold A_f=\bold X^{-1}\bold A_n, \bold B_f=\bold X^{-1}\bold B_n, \bold C_f, $$and $$ \bold D_f.$$

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

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

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

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

The synthesis 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.