# LMIs in Control/pages/dt mixed H2 Hinf optimal output feedback control

WIP, Description in progress

This part shows how to design dynamic outpur feedback control in mixed ${\displaystyle {\mathcal {H}}_{2}}$ and ${\displaystyle {\mathcal {H}}_{\infty }}$ sense for the continuous time.

## Problem

Consider the discrete-time generalized LTI plant ${\displaystyle {\mathcal {P}}}$  with minimal state-space realization

${\displaystyle {\dot {x}}=Ax+{\begin{bmatrix}B_{1,1}&B_{1,2}\end{bmatrix}}{\begin{bmatrix}w_{1}\\w_{2}\end{bmatrix}}+B_{2}u,}$

${\displaystyle {\begin{bmatrix}z_{1}\\z_{2}\end{bmatrix}}={\begin{bmatrix}C_{1,1}\\D_{1,2}\end{bmatrix}}x_{k}+{\begin{bmatrix}D_{11,11}&D_{11,12}\\D_{11,21}&D_{11,22}\end{bmatrix}}{\begin{bmatrix}w_{1}\\w_{2}\end{bmatrix}}+{\begin{bmatrix}D_{12,1}\\D_{12,2}\end{bmatrix}}u,}$

${\displaystyle y=C_{d2}x+{\begin{bmatrix}D_{21,1}&D_{21,2}\end{bmatrix}}{\begin{bmatrix}w_{1}\\w_{2}\end{bmatrix}}+D_{d22}u}$

## Theorem

A continuous-time dynamic output feedback LTI controllerwith state-space realization ${\displaystyle (A_{c},B_{c},C_{c},D_{c})}$  is to be designed to minimize the ${\displaystyle {\mathcal {H}}_{2}}$  norm of the closed-loop transfer matrix ${\displaystyle T_{11}(s)}$  from the exogenous input ${\displaystyle w_{1}}$  to the performance output ${\displaystyle z_{1}}$  while ensuring the H∞ norm of the closed-loop transfer matrix ${\displaystyle T_{22}(s)}$  from the exogenous input ${\displaystyle w_{2}}$  to the performance output ${\displaystyle z_{2}}$  is less than ${\displaystyle \gamma _{d}}$ , where

${\displaystyle T_{11}(s)=C_{CL1,1}(sI-A_{CL})^{-1}B_{CL1,1},}$

${\displaystyle T_{22}(s)=C_{CL1,2}(sI-A_{CL})^{-1}B_{CL1,2}+D_{CL11,22},}$

${\displaystyle A_{d_{CL}}={\begin{bmatrix}A+B_{2}D_{c}{\tilde {D}}^{-1}C_{2}&B_{2}(I+D_{c}{\tilde {D}}^{-1}D_{22})C_{c}\\B_{c}{\tilde {D}}^{-1}C_{2}&A_{c}+B_{c}{\tilde {D}}^{-1}D_{22}C_{c}\end{bmatrix}}}$ ,

${\displaystyle B_{CL1,1}={\begin{bmatrix}B_{1,1}+B_{2}D_{c}{\tilde {D}}^{-1}D_{21,1}\\B_{c}{\tilde {D}}^{-1}D_{21,1}\end{bmatrix}}}$ ,

${\displaystyle B_{CL1,2}={\begin{bmatrix}B_{1,2}+B_{2}D_{c}{\tilde {D}}^{-1}D_{21,2}\\B_{c}{\tilde {D}}^{-1}D_{21,2}\end{bmatrix}}}$ ,

${\displaystyle C_{CL1,1}={\begin{bmatrix}C_{1,1}+D_{12,1}D_{c}{\tilde {D}}^{-1}C_{2,1}&D_{12,1}(I+D_{c}{\tilde {D}}^{-1}D_{22})C_{c}\end{bmatrix}}}$ ,

${\displaystyle C_{CL1,2}={\begin{bmatrix}C_{1,2}+D_{12,2}D_{c}{\tilde {D}}^{-1}C_{2,2}&D_{12,2}(I+D_{c}{\tilde {D}}^{-1}D_{22})C_{c}\end{bmatrix}}}$ ,

${\displaystyle D_{CL11,22}=D_{11,22}+D_{12,2}D_{c}{\tilde {D}}^{-1}D_{21,2}}$ ,

and ${\displaystyle {\tilde {D}}=I-D_{22}D_{c}}$ .

## Synthesis Method

Solve for ${\displaystyle A_{n}\in \mathbb {R} ^{n_{x}\times n_{x}},B_{n}\in \mathbb {R} ^{n_{x}\times n_{x}},C_{n}\in \mathbb {R} ^{n_{u}\times n_{x}},D_{n}\in \mathbb {R} ^{n_{u}\times n_{y}},X_{1},Y_{1}\in \mathbb {S} ^{n_{x}},Z\in \mathbb {S} ^{n_{Z_{1}}},}$  and ${\displaystyle \mu \in \mathbb {R} _{>0}}$  that minimizes ${\displaystyle {\mathcal {J}}(\mu )=\mu }$  subjects to ${\displaystyle X_{1}>0,\ Y_{1}>0\ Z>0,}$

${\displaystyle {\begin{bmatrix}N_{11}&A+A_{n}^{T}+B_{2}D_{n}C_{2}&B_{1,1}+B_{2}D_{n}D{21,1}\\*&X_{1}A+A^{T}X_{1}+B_{n}C_{2}+C_{2}^{T}B_{n}^{T}&X_{1}B_{1,1}+B_{n}D_{21,1}\\*&*&-I\end{bmatrix}}<0}$ ,

${\displaystyle {\begin{bmatrix}N_{11}&A+A_{n}^{T}+B_{2}D_{n}C_{2}&B_{1,1}+B_{2}D_{n}D{21,1}&Y_{1}^{T}C_{1,2}^{T}+C_{n}^{T}D_{12,2}^{T}\\*&X_{1}A+A^{T}X_{1}+B_{n}C_{2}+C_{2}^{T}B_{n}^{T}&X_{1}B_{1,1}+B_{n}D_{21,1}&C_{1,2}^{T}+C_{2}^{T}D_{n}^{T}D_{12,2}^{T}\\*&*&-\gamma _{d}I&D_{11,22}^{T}+D_{21,2}^{T}D_{n}^{T}D_{12,2}^{T}\\*&*&*&-\gamma _{d}I\end{bmatrix}}<0}$ ,

${\displaystyle {\begin{bmatrix}Y_{1}IY_{1}C_{1,1}^{T}+C_{n}^{T}D_{12,1}^{T}\\*&X_{1}&C_{1,1}^{T}+C_{2}^{T}D_{n}^{T}D_{12,1}^{T}\\*&*&Z\end{bmatrix}}>0}$ ,

${\displaystyle D_{11,11}+D_{12,1}D_{n}D_{21,1}=0,}$

${\displaystyle {\begin{bmatrix}X_{1}&I\\*&Y_{1}\end{bmatrix}}>0,}$

tr${\displaystyle Z<\mu ,}$

where ${\displaystyle N_{11}=AY_{1}+Y_{1}A^{T}+B_{2}C_{n}+C_{n}^{T}B_{2}^{T}}$ .

The controller is recovered by

${\displaystyle A_{c}=A_{K}-B{c}(I-D_{22}D_{c})^{-1}D_{22}C_{c},}$

${\displaystyle B_{c}=B_{K}(I-D_{c}D_{22}),}$

${\displaystyle C_{c}=(I-D_{c}D_{22})C_{K},}$

${\displaystyle D_{c}=(I+D_{K}D{22})^{-1}D_{K},where[itex]{\begin{bmatrix}A_{K}&B_{K}\\C_{K}&D_{K}\end{bmatrix}}={\begin{bmatrix}X_{2}&X_{1}B_{2}\\0&I\end{bmatrix}}^{-1}({\begin{bmatrix}A_{n}&B_{n}\\C_{n}&D_{n}\end{bmatrix}}-{\begin{bmatrix}X_{1}A_{d}Y_{1}&0\\0&0\end{bmatrix}}){\begin{bmatrix}Y_{2}^{T}&0\\C_{2}Y_{1}&I\end{bmatrix}}^{-1}}$ , and the matrices ${\displaystyle X_{2}}$  and ${\displaystyle Y_{2}}$  satisfy ${\displaystyle X_{2}Y_{2}^{T}=I-X_{1}Y_{1}}$ . If ${\displaystyle D_{22}=0}$ , then ${\displaystyle A_{c}=A_{K},B_{c}=B{K},C_{c}=C_{K}}$  and ${\displaystyle D_{c}=D_{K}}$ .

Given ${\displaystyle X_{1}}$  and ${\displaystyle Y_{1}}$ , the matrices ${\displaystyle X_{2}}$  and ${\displaystyle Y_{2}}$  can be found using a matrix decomposition, such as a LU decomposition or a Cholesky decomposition.

If ${\displaystyle D_{11,11}=0,D_{12,1}\neq 0,{\text{ and }}D_{21,1}\neq 0,}$  then it is often simplest to choose ${\displaystyle D_{n}=0}$  in order to satisfy the equality constraint ${\displaystyle D_{11,11}+D_{12,1}D_{n}D_{21,1}=0,}$ .