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 ${\mathcal {H}}_{2}$ and ${\mathcal {H}}_{\infty }$ sense for the continuous time.

Problem

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

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

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

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

Theorem

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

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

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

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

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

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

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

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

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

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

Synthesis Method

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

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

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

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

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

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

tr$Z<\mu ,$

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

The controller is recovered by

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

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

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

$D_{c}=(I+D_{K}D{22})^{-1}D_{K},where<math>{\begin{bmatrix}A_{K}&B_{K}\\C_{K}&D_{K}\end{bmatrix}}={\begin{bmatrix}X_{2}&X_{1}B_{2}\\0&I\end{bmatrix}}^{-1}({\begin{bmatrix}A_{n}&B_{n}\\C_{n}&D_{n}\end{bmatrix}}-{\begin{bmatrix}X_{1}A_{d}Y_{1}&0\\0&0\end{bmatrix}}){\begin{bmatrix}Y_{2}^{T}&0\\C_{2}Y_{1}&I\end{bmatrix}}^{-1}$, and the matrices $X_{2}$ and $Y_{2}$ satisfy $X_{2}Y_{2}^{T}=I-X_{1}Y_{1}$. 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 $D_{11,11}=0,D_{12,1}\neq 0,{\text{ and }}D_{21,1}\neq 0,$ then it is often simplest to choose $D_{n}=0$ in order to satisfy the equality constraint $D_{11,11}+D_{12,1}D_{n}D_{21,1}=0,$.

WIP, additional references to be added

External Links

A list of references documenting and validating the LMI.