LMIs in Control/pages/dt mixed H2 Hinf optimal dynamic 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 discrete time .

Problem

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

${\displaystyle x_{k+1}=A_{d}x_{k}+{\begin{bmatrix}B_{d1,1}&B_{d1,2}\end{bmatrix}}{\begin{bmatrix}w_{1,k}\\w_{2,k}\end{bmatrix}}+B_{d,2}u_{k},}$

${\displaystyle {\begin{bmatrix}z_{1,k}\\z_{2,k}\end{bmatrix}}={\begin{bmatrix}C_{d1,1}\\D_{d1,2}\end{bmatrix}}x_{k}+{\begin{bmatrix}D_{d11,11}&D_{d11,12}\\D_{d11,21}&D_{d11,22}\end{bmatrix}}{\begin{bmatrix}w_{1,k}\\w_{2,k}\end{bmatrix}}+{\begin{bmatrix}D_{12,1}\\D_{12,2}\end{bmatrix}}u_{k},}$

${\displaystyle y_{k}=C_{d2}x_{k}+{\begin{bmatrix}D_{21,1}&D_{21,2}\end{bmatrix}}{\begin{bmatrix}w_{1,k}\\w_{2,k}\end{bmatrix}}+D_{d22}u_{k}}$

Theorem

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

${\displaystyle T_{11}(z)=C_{d_{CL}1,1}(zI-A_{d_{CL}})^{-1}B_{d_{CL}1,1},}$

${\displaystyle T_{22}(z)=C_{d_{CL}1,2}(zI-A_{d_{CL}})^{-1}B_{d_{CL}1,2}+D_{d_{CL}11,22},}$

${\displaystyle A_{d_{CL}}={\begin{bmatrix}A_{d}+B_{d2}D_{dc}{\tilde {D}}_{d}^{-1}C_{d2}&B_{d2}(I+D_{dc}{\tilde {D}}_{d}^{-1}D_{d22})C_{dc}\\B_{dc}{\tilde {D}}_{d}^{-1}C_{d2}&A_{dc}+B_{dc}{\tilde {D}}_{d}^{-1}D_{d22}C_{dc}\end{bmatrix}}}$ ,

${\displaystyle B_{d_{CL}1,1}={\begin{bmatrix}B_{d1,1}+B_{d2}D_{dc}{\tilde {D}}_{d}^{-1}D_{d21,1}\\B_{dc}{\tilde {D}}_{d}^{-1}D_{d21,1}\end{bmatrix}}}$ ,

${\displaystyle B_{d_{CL}1,2}={\begin{bmatrix}B_{d1,2}+B_{d2}D_{dc}{\tilde {D}}_{d}^{-1}D_{d21,2}\\B_{dc}{\tilde {D}}_{d}^{-1}D_{d21,2}\end{bmatrix}}}$ ,

${\displaystyle C_{d_{CL}1,1}={\begin{bmatrix}C_{d1,1}+D_{d12,1}D_{dc}{\tilde {D}}_{d}^{-1}C_{d2,1}&D_{d12,1}(I+D_{dc}{\tilde {D}}_{d}^{-1}D_{d22})C_{dc}\end{bmatrix}}}$ ,

${\displaystyle C_{d_{CL}1,2}={\begin{bmatrix}C_{d1,2}+D_{d12,2}D_{dc}{\tilde {D}}_{d}^{-1}C_{d2,2}&D_{d12,2}(I+D_{dc}{\tilde {D}}_{d}^{-1}D_{d22})C_{dc}\end{bmatrix}}}$ ,

${\displaystyle D_{d_{CL}11,22}=D_{d11,22}+D_{d12,2}D_{dc}{\tilde {D}}_{d}^{-1}D_{d21,2}}$ ,

and ${\displaystyle {\tilde {D}}_{d}=I-D_{d22}D_{dc}}$ .

Synthesis Method

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

${\displaystyle {\begin{bmatrix}X_{1}&I&X_{1}A_{d}+B_{d_{n}}C_{d_{2}}&A_{d_{n}}&X_{1}B_{d1,1}+B{d_{n}}D_{d21,1}\\*&Y_{1}&A_{d}+B_{d2}C_{dn}D_{d2}&A_{d}Y_{1}+B_{d2}C_{dn}&B_{d1,1}+B_{d2}D_{dn}D_{d21,1}\\*&*&X_{1}&I&0\\*&*&*&Y_{1}&0\\*&*&*&*&I\end{bmatrix}}>0,}$

${\displaystyle {\begin{bmatrix}X_{1}&I&X_{1}A_{d}+B_{d_{n}}C_{d_{2}}&A_{d_{n}}&X_{1}B_{d1,2}+B{d_{n}}D_{d21,2}&0\\*&Y_{1}&A_{d}+B_{d2}C_{dn}D_{d2}&A_{d}Y_{1}+B_{d2}C_{dn}&B_{d1,2}+B_{d2}D_{dn}D_{d21,2}&0\\*&*&X_{1}&I&0&C_{d1,2}^{T}+C_{d2}^{T}D_{dn}^{T}D_{d12,2}^{T}\\*&*&*&Y_{1}&0&Y_{1}C_{d1,2}^{T}+C_{dn}^{T}D_{d12,2}^{T}\\*&*&*&*&-\gamma _{d}I&D_{d11,22}^{T}+D_{d21,2}^{T}D_{dn}^{T}D{d12,2}^{T}\\*&*&*&*&*&-\gamma _{d}I\end{bmatrix}}>0}$

${\displaystyle {\begin{bmatrix}Z&C_{d1,1}+D_{d12,1}D_{dn}C_{d2}&C_{d1,1,}Y_{1}^{T}+D_{d12,1}C_{dn}\\*&X_{1}&I\\*&*&Y_{1}\end{bmatrix}}>0,}$

${\displaystyle D_{d11,11}+D_{d12,1}D_{dn}D_{d21,1}=0,}$

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

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

The controller is recovered by

${\displaystyle A_{dc}=A_{dK}-B{dc}(I-D_{d22}D_{dc})^{-1}D_{d22}C_{dc},}$

${\displaystyle B_{dc}=B_{dK}(I-D_{dc}D_{d22}),}$

${\displaystyle C_{dc}=(I-D_{dc}D_{d22})C_{dK},}$

${\displaystyle D_{dc}=(I+D_{dK}D{d22})^{-1}D_{dK},where[itex]{\begin{bmatrix}A_{dK}&B_{dK}\\C_{dK}&D_{dK}\end{bmatrix}}={\begin{bmatrix}X_{2}&X_{1}B_{d2}\\0&I\end{bmatrix}}^{-1}({\begin{bmatrix}A_{dn}&B_{dn}\\C_{dn}&D_{dn}\end{bmatrix}}-{\begin{bmatrix}X_{1}A_{d}Y_{1}&0\\0&0\end{bmatrix}}){\begin{bmatrix}Y_{2}^{T}&0\\C_{d2}Y_{1}&I\end{bmatrix}}^{-1}}$ , 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 ${\displaystyle A_{dc}=A_{dK},B_{dc}=B{dK},C_{dc}=C_{dK}}$  and ${\displaystyle D_{dc}=D_{dK}}$ .

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 ${\displaystyle D_{d11,11}=0,D_{d12,1}\neq 0,{\text{ and }}D{d21,1}\neq =0,}$  then it is often simplest to choose ${\displaystyle D_{dn}=0}$  in order to satisfy the equality constraint ${\displaystyle D_{d11,11}+D_{d12,1}D_{dn}D_{d21,1}=0,}$ .