# LMIs in Control/Click here to continue/Applications of Linear systems/Mixed H2-H∞ Optimal Voltage Control Design for Smart Transformer Low-Voltage Inverter

LMIs in Control/Click here to continue/Applications of Linear systems/Mixed H2-H∞ Optimal Voltage Control Design for Smart Transformer Low-Voltage Inverter
Smart transformers are often used in situations with variable loads such as the integration of renewable energy sources. This section examines a system of Linear Matrix Inequalities as analyzed by Wei Hu, Yu Shen, Zhichun Yang, and Huaidong Min in their paper for Mixed $H_{2}$/$H_{\infty }$ Optimal Voltage Control Design for Smart Transformer Low-Voltage Inverter.

## The System

State space modelling is used in order to develop a system that describes the inverter behavior.
    ${\begin{aligned}{\dot {x_{p}}}&=A_{p}x+B_{p1}v_{c}+B_{p2}w_{p}\\y&=C_{p}x_{p}\\\end{aligned}}$
The system formulation for a simple inverter system draws directly from electric circuit theory. $A_{p}={\begin{bmatrix}-R/L&-1/L\\1/C&0\end{bmatrix}}$ $B_{p1}={\begin{bmatrix}1/L\\0\end{bmatrix}}$ $B_{p2}={\begin{bmatrix}0\\1/C\end{bmatrix}}$ $C_{p}={\begin{bmatrix}0&1\end{bmatrix}}$ where the above variables are defined as follows:
* $R$ is the resistance
* $L$ is the inductance
* $C$ is the capacitance
This system is further augmented by adding multiple resonant controllers to ensure the system can reach zero steady-state error when tracking sinusoidal target outputs. The augmented system is then derived as follows:
    ${\begin{aligned}{\dot {x_{p}}}&=Ax+B_{1}v_{c}+B_{2}w_{p}+Ry^{*}\\y&=Cx\\\end{aligned}}$
where
$A={\begin{bmatrix}A_{p}&0&0&&0&0\\-C_{p}&j\omega &0&...&0&0\\-C_{p}&0&-j\omega &&0&0\\&...&&...&&...\\-C_{p}&0&0&&j_{n}\omega &0\\-C_{p}&0&0&...&0&-j_{n}\omega \end{bmatrix}}$
$B_{1}={\begin{bmatrix}B_{p1}&0&0&\cdots &0&0\end{bmatrix}}^{T}$
$B_{2}={\begin{bmatrix}B_{p2}&0&0&\cdots &0&0\end{bmatrix}}^{T}$
$C={\begin{bmatrix}C_{p}&0&0&\cdots &0&0\end{bmatrix}}$
$R={\begin{bmatrix}0&1&1&\cdots &1&1\end{bmatrix}}$

## The Data

The data required to solve this problem includes the values of resistance, inductance, capacitance, switch frequency, output peak voltage, and the DC link voltage.

## The Optimization Problem

The optimization problem aims to minimize a combined $H_{2}$/$H_{\infty }$ norm of the system as defined above. In order to accomplish this task, constraints and the objective must first be defined.
* Objective: Combined $H_{2}$/$H_{\infty }$ norm
* Constraints: Closed-loop poles are restricted to a desired section of the complex plane,

## The LMI: Mixed H$_2$-H$_∞$ Optimal Voltage Control

${\begin{aligned}{\begin{cases}&{\text{min}}\quad a\gamma +bTrace(M)\\&{\text{s.t.}}\quad {\begin{bmatrix}M&I\\I&W_{1}\end{bmatrix}}>0\\\\&\quad \quad {\begin{bmatrix}(AW_{1}-B_{1}V_{1})+(AW_{1}-B_{1}V_{1})^{T}&W_{1}&V_{1}^{T}\\W_{1}&-Q_{inv}&0\\V_{1}&0&-R^{-1}\end{bmatrix}}<0\\\\&\quad \quad AX+B_{1}W+(AX+B_{1}W)^{T}+BB^{T}<0\\\\&\quad \quad L_{j}\otimes W_{3}+M_{j}\otimes (AW_{3}+B_{1}V_{3})+M_{j}^{T}\otimes (AW_{3}+B_{1}V_{3})^{T}<0,j=1,2,3\\\\&\quad \quad {\begin{bmatrix}(AW_{2}-B_{1}V_{2})+(AW_{2}-B_{1}V_{2})^{T}&B_{2}&W_{2}C^{T}\\B_{2}^{T}&-\gamma I&0\\CW_{2}&0&-\gamma I\end{bmatrix}}<0\\\\\end{cases}}\\\end{aligned}}$
In the above LMIs, the coefficients a and b represent the weighting factor applied to the $H_{2}$ and $H_{\infty }$ norms respectively. R is some given positive-definite matrix.

## Conclusion

Mixed $H_{2}$/$H_{\infty }$ optimal control can be effectively used to design efficient and useful smart transformers.

## Implementation

This LMI can be implemented using MATLAB when combined with YALMIP and an LMI solver such as SeDuMi.

## Related LMIs

* Mixed H2-H∞ Satellite Attitude Control

## External Links

* – Mixed H2/H∞ Optimal Voltage Control Design for Smart Transformer Low-Voltage Inverter – Wei Hu, Yu Shen, Zhichun Yang, and Huaidong Min
* - LMIs in Control Systems: Analysis, Design, and Applications – Guang-Ren Duan and Hai-Hua Yu
* LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.