# LMIs in Control/Controller Synthesis/Continuous Time/Optimal Dynamic Output Feedback/H-2

Discrete-Time H2-Optimal Dynamic Output Feedback Control
A Dynamic Output feedback controller is designed for a Continuous Time system, to minimize the H2 norm of the closed loop system with exogenous input $w$ and performance output $z$.

## The System

Continuous-Time LTI System with state space realization $(A,B,C,D)$
          ${\begin{aligned}{\dot {x}}&=A_{x}+B_{1}w+B_{2}u\\z&=C_{1}x+D_{11}w_{k}+D_{12}u\\y&=C_{2}x+D_{21}w+D_{22}u\\\end{aligned}}$

## The Data

The matrices: System $(A,B_{1},B_{2},C_{1},C_{1},D_{11},D_{12},D_{21},D_{22}),X_{1},Y_{1},Z,,X_{2},Y_{2}$
Controller $(A_{c},B_{c},C_{c},D_{c})$

## The Optimization Problem

The following feasibility problem should be optimized:
$\nu$ is minimized while obeying the LMI constraints.

## The LMI:

Solve for $A_{n}\in {R^{n_{x}\times n_{x}}},B_{n}\in {R^{n_{x}\times n_{y}}},C_{n}\in {R^{n_{u}\times n_{x}}},D_{n}\in {R^{n_{u}\times n_{y}}},X_{1},Y_{1}\in {S^{n_{x}}},Z\in {S^{n_{z}}},$ and $\nu \in {R_{>0}}$ that minimize ${\mathcal {J}}(\nu )<\nu$ subject to $X_{1}>0,Y_{1}>0,Z>0,$
          ${\begin{aligned}{\begin{bmatrix}AY_{1}+Y_{1}A^{T}+B_{2}C_{n}+C_{n}^{T}B_{2}^{T}&A+A_{n}^{T}+B_{2}D_{n}C_{2}&B_{1}+B_{2}D_{n}D_{21}\\*&X_{1}A+A^{T}X_{1}+B_{n}C_{2}+C_{2}^{T}B_{n}^{T}&X_{1}B_{1}+B_{n}D_{21}\\*&*&-\mathbf {1} \end{bmatrix}}&>0,\\{\begin{bmatrix}X_{1}&\mathbf {1} &Y_{1}C_{1}^{T}+C_{n}^{T}D_{12}^{T}\\*&Y_{1}&C_{1}^{T}+C_{2}^{T}D_{n}^{T}D_{12}^{T}\\*&*&Z\end{bmatrix}}&>0,\\D_{11}+D_{12}D_{n}D_{21}=0\\{\begin{bmatrix}X_{1}&\mathbf {1} \\*&Y_{1}\end{bmatrix}}&>0,\\trZ<\nu \end{aligned}}$
The controller is recovered by
${\begin{aligned}&A_{c}=A_{k}-B_{c}(1-D_{22}D_{c})^{-1}D_{22}C_{c}\\&B_{c}=B_{k}(1-D_{c}D_{22})\\&C_{c}=(1-D_{c}D_{22})C_{k}\\&D_{c}=(1+D_{k}D_{22})^{-1}D_{k}\\\end{aligned}}$
where, ${\begin{aligned}{\begin{bmatrix}A_{k}&B_{k}\\C_{k}&D_{k}\end{bmatrix}}&={\begin{bmatrix}X_{2}&X_{1}B_{2}\\\mathbf {0} &1\end{bmatrix}}^{-1}({\begin{bmatrix}A_{n}&B_{n}\\C_{n}&D_{n}\end{bmatrix}}-{\begin{bmatrix}X_{1}AY_{1}&\mathbf {0} \\\mathbf {0} &\mathbf {0} \end{bmatrix}}){\begin{bmatrix}Y_{2}^{T}&\mathbf {0} \\CY_{1}&\mathbf {1} \end{bmatrix}}^{-1}\end{aligned}}$ and the matrices $X_{2}$ and $Y_{2}$ satisfy $X_{2}Y_{2}^{T}=1-X_{1}Y_{1}$. If $D_{22}=0,$ then $A_{c}=A_{k},B_{c}=B_{k},C_{c}=C_{k}$ and $D_{c}=D_{k}.$
Given $X_{1}$ and $Y_{1}$, the matrices $X_{2}$ and $Y_{2}$ can be found using a matrix decomposition, such as a LU decomposition or a Cholesky decomposition.
If $D_{11}=0$, $D_{12}\neq 0,$ and $D_{21}\neq 0$, then it is often simplest to choose $D_{n}=0$ in order to satisfy the equality constraint

## Conclusion:

The Continuous-Time H2-Optimal Dynamic Output feedback controller is the system $(A_{c},B_{c},C_{c},D_{c})$

## Implementation

The LMI given above can be implemented and solved using a tool such as YALMIP, along with an LMI solver such as MOSEK.

## Related LMIs

Discrete Time H2 Optimal Dynamic Output Feedback Control
Continuous Time H∞ Optimal Dynamic Output Feedback Control

## 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.