LMIs in Control/pages/H-2 filtering

For systems that have disturbances, filtering can be used to reduce the effects of these disturbances. Described on this page is a method of attaining a filter that will reduce the effects of the disturbances as completely as possible. To do this, we look to find a set of new coefficient matrices that describe the filtered system. The process to achieve such a new system is described below. The H2-filter tries to minimize the average magnitude of error.

The System
For the application of this LMI, we will look at linear systems that can be represented in state space as



\begin{align} \dot x &= Ax + Bw, x(0) = x_{0}\\ y &= Cx + Dw\\ z &= Lx \end{align}$$

where $$x \in R^{n}, y \in R^{l}, z \in R^{m}$$ represent the state vector, the measured output vector, and the output vector of interest, respectively, $$w \in R^{p}$$ is the disturbance vector, and $$A, B, C, D$$ and $$L$$ are the system matrices of appropriate dimension. To further define: $$x$$ is $$\in R^{n}$$ and is the state vector, $$A $$ is $$\in R^{n*n}$$ and is the state matrix, $$B$$ is $$\in R^{n*r}$$ and is the input matrix, $$w$$ is $$\in R^{r}$$ and is the exogenous input, $$C $$ is $$\in R^{m*n}$$ and is the output matrix, $$D$$ and $$L$$ are $$\in R^{m*r}$$ and are feedthrough matrices, and $$y$$ and $$z$$ are $$\in R^{m}$$ and are the output and the output of interest, respectively.

The Data
The data are $$w$$ (the disturbance vector), and $$A, B, C, D$$ and $$L$$ (the system matrices). Furthermore, the $$A$$ matrix is assumed to be stable

The Optimization Problem
We need to design a filter that will eliminate the effects of the disturbances as best we can. For this, we take a filter of the following form:



\begin{align} \dot \sigma &= A_{f}\sigma + B_{f}y, \sigma(0) = \sigma_{0}\\ \hat z &= C_{f}\sigma, \end{align}$$

where $$\sigma \in R^{n}$$ is the state vector, $$\hat z \in R^{m}$$ is the estimation vector, and $$A_{f}, B_{f}, C_{f}$$ are the coefficient matrices of appropriate dimensions.

Note that the combined complete system can be represented as



\begin{align} \dot x_{e} &= \tilde{A}x_{e} + \tilde{B}w, x_{e}(0) = x_{e0}\\ \tilde{z} &= \tilde{C}x_{e}, \end{align}$$

where $$ \tilde{z} = z - \hat z \in R^{m}$$ is the estimation error,

\begin{align} x_{e} = \begin{bmatrix} x \\ \sigma\end{bmatrix}\\ \end{align}$$ is the state vector of the system, and $$\tilde{A}, \tilde{B}, \tilde{C}$$ are the coefficient matrices, defined as:



\begin{align} \tilde{A} = \begin{bmatrix} A & 0 \\ B_{f}C & A_{f} \end{bmatrix}, \tilde{B} = \begin{bmatrix} B \\ B_{f}D\end{bmatrix},\\ \tilde{C} = \begin{bmatrix} L & -C_{f} \end{bmatrix} \end{align}$$

In other words, for the system defined above we need to find $$A_{f}, B_{f}, C_{f}$$ such that



\begin{align} \end{align}$$ where $$ \gamma $$ is a positive constant, and
 * G_{\tilde{z}w}(s)||_{2} < \gamma,

\begin{align} G_{\tilde{z}w}(s) = \tilde{C}(sI-\tilde{A})^{-1}\tilde{B} \end{align}$$

The LMI: H-2 Filtering
For this LMI, the solution exists if one of the following sets of LMIs hold:

Matrices $$ R, X, M, N, Z, Q$$ exist that obey the following LMIs:



\begin{align} R - X &>0,\\ trace(Q) &< \gamma^{2},\\ \begin{bmatrix} -Q& *&*\\ L^{T} &-R & *\\ -N^{T} & -X & -X\end{bmatrix}&<0,\\ \begin{bmatrix} RA + A^{T}R + ZC + C^{T}Z^{T} & *&*\\ M^{T}+ZC+XA & M^{T}+M &*\\ B^{T}R+D^{T}Z^{T} & B^{T}X+D^{T}Z^{T} & -I\end{bmatrix}&<0.\\ \end{align}$$

or

Matrices $$ \bar{R}, \bar{X}, \bar{M}, \bar{N}, \bar{Z}, \bar{Q}$$ exist that obey the following LMIs:



\begin{align} \bar{R} - \bar{X} &>0,\\ trace(\bar{Q}) &< \gamma^{2},\\ \begin{bmatrix} -\bar{Q}& *&*\\ \bar{R}B+\bar{Z}D &-\bar{R} & *\\ \bar{X}B + \bar{Z}D & -\bar{X} & -I\end{bmatrix}&<0,\\ \begin{bmatrix} \bar{R}A + A^{T}\bar{R} + \bar{Z}C + C^{T}\bar{Z}^{T} & *&*\\ \bar{M}^{T}+\bar{Z}C+\bar{X}A & \bar{M}^{T}+\bar{M} &*\\ L & -\bar{N} & -I\end{bmatrix}&<0.\\ \end{align}$$

Conclusion:
To find the corresponding filter, use the optimized matrices from the first solution to find:



A_{f} = X^{-1}M, B_{f}=X^{-1}Z, C_{f} = N $$

Or the second solution to find:



A_{f} = \bar{X}^{-1}\bar{M}, B_{f}=\bar{X}^{-1}\bar{Z}, C_{f} = \bar{N} $$

These matrices can then be used to produce $$\tilde{A}, \tilde{B}, \tilde{C} $$ to construct the final filter below, that will best eliminate the disturbances of the system.



\begin{align} \dot x_{e} &= \tilde{A}x_{e} + \tilde{B}w, x_{e}(0) = x_{e0}\\ \tilde{z} &= \tilde{C}x_{e}, \end{align}$$

Implementation
This implementation requires Yalmip and Sedumi.

https://github.com/rezajamesahmed/LMImatlabcode/blob/master/H2_Filtering.m

Related LMIs
H-infinity filtering