# LMIs in Control/pages/mixedhinfh2desiredpole

LMI for Mixed ${\displaystyle H_{2}/H_{\infty }}$ with desired pole location Controller

The mixed ${\displaystyle H_{2}/H_{\infty }}$ output feedback control has been known as an example of a multi-objective optimal control problem. In this problem, the control feedback should respond properly to several specifications. In the ${\displaystyle H_{2}/H_{\infty }}$ controller, the ${\displaystyle H_{\infty }}$ channel is used to improve the robustness of the design while the ${\displaystyle H_{2}}$ channel guarantees good performance of the system and additional constraint is used to place poles at desired location.

## The System

We consider the following state-space representation for a linear system:

{\displaystyle {\begin{aligned}{\dot {x}}&=Ax+B_{1}u+B_{2}w\\z_{\infty }&=C_{\infty }+D_{\infty 1}u+D_{\infty 2}w\\z_{2}&=C_{2}x+D_{21}u\end{aligned}}}

where

• ${\displaystyle x\in \mathbb {R} ^{n}}$ , ${\displaystyle z_{2},z_{\infty }\in \mathbb {R} ^{m}}$ are the state vector and the output vectors, respectively
• ${\displaystyle w\in \mathbb {R} ^{p}}$ , ${\displaystyle u\in \mathbb {R} ^{r}}$  are the disturbance vector and the control vector
• ${\displaystyle A}$ , ${\displaystyle B_{1}}$ ,${\displaystyle B_{2}}$ , ${\displaystyle C_{\infty }}$ ,${\displaystyle C_{2}}$ ,${\displaystyle D_{\infty 1}}$ ,${\displaystyle D_{\infty 2}}$ , and ${\displaystyle D_{21}}$  are the system coefficient matrices of appropriate dimensions

## The Data

We assume that all the four matrices of the plant,${\displaystyle A}$ , ${\displaystyle B_{1}}$ ,${\displaystyle B_{2}}$ , ${\displaystyle C_{\infty }}$ ,${\displaystyle C_{2}}$ ,${\displaystyle D_{\infty 1}}$ ,${\displaystyle D_{\infty 2}}$ , and ${\displaystyle D_{21}}$  are given.

## The Optimization Problem

For the system with the following feedback law:
${\displaystyle u=Kx}$
The closed loop system can be obtained as:
{\displaystyle {\begin{aligned}{\dot {x}}&=(A+B_{1}K)x+B_{2}w\\z_{\infty }&=(C_{\infty }+D_{\infty 1}K)x+D_{\infty 2}w\\z_{2}&=(C_{2}+D_{21}K)u\end{aligned}}}

the transfer function matrices are ${\displaystyle G_{z\infty w}(s)}$  and ${\displaystyle G_{z2w}(s)}$
Thus the ${\displaystyle H_{\infty }}$  performance and the ${\displaystyle H_{2}}$  performance requirements for the system are, respectiverly
${\displaystyle ||G_{z\infty w}(s)||_{\infty }<\gamma _{\infty }}$
and
${\displaystyle ||G_{z2w}(s)||_{2}<\gamma _{2}}$
. For the performance of the system response, we introduce the closed-loop eigenvalue location requirement. Let
${\displaystyle D={s|s\in C,L+sM+sM^{T}<0},}$
It is a region on the complex plane, which can be used to restrain the closed-loop eigenvalue locations. Hence a state feedback control law is designed such that,

• The ${\displaystyle H_{\infty }}$  performance and the ${\displaystyle H_{2}}$  performance are satisfied.
• The closed-loop eigenvalues are all located in ${\displaystyle D}$ , that is,

${\displaystyle \,\,\,\,\,}$  ${\displaystyle \lambda (A+B_{1}K)\subset D}$ .

## The LMI: LMI for mixed ${\displaystyle H_{2}}$/${\displaystyle H_{\infty }}$ with desired Pole locations

The optimization problem discussed above has a solution if there exist two symmetric matrices ${\displaystyle X,Z}$  and a matrix ${\displaystyle W}$ , satisfying

min ${\displaystyle c_{2}\gamma _{2}^{2}+c_{\infty }\gamma _{\infty }}$
s.t
{\displaystyle {\begin{aligned}&{\begin{bmatrix}(AX+B_{1}W)^{T}+AX+B_{1}W&B_{2}&(C_{\infty }X+D_{\infty 1}W)^{T}\\B_{2}^{T}&-\gamma _{\infty }I&D_{\infty 2}^{T}\\C_{\infty }X+D_{\infty 1}W&D_{\infty 2}&-\gamma _{\infty }I\\\end{bmatrix}}<0\\&AX+B_{1}W+(AX+B_{1}W)^{T}+B_{2}B_{2}^{T}<0\\&{\begin{bmatrix}-Z&C_{2}X+D_{21}W\\(C_{2}X+D_{21}W)^{T}&-X\\\end{bmatrix}}>0\\&{\text{trace}}(Z)<\gamma _{2}^{2}\\&L\otimes +M\otimes (AX+B_{1}W)+M^{T}\otimes (AX+B_{1}W)^{T}<0\\\end{aligned}}}

where ${\displaystyle c_{2}>0}$  and ${\displaystyle c_{\infty }>0}$  are the weighting factors.

## Conclusion:

The calculated scalars ${\displaystyle \gamma _{\infty }}$  and ${\displaystyle \gamma _{2}}$  are the ${\displaystyle H_{2}}$  and ${\displaystyle H_{\infty }}$  norms of the system, respectively. The controller is extracted as ${\displaystyle K=WX^{-1}}$

## Implementation

A link to Matlab codes for this problem in the Github repository: