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{align} \dot{x_p} &= A_px + B_{p1}v_c+B_{p2}w_p \\ y &= C_px_p \\ \end{align}$$

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{align} \dot{x_p} &= Ax + B_1v_c+B_2w_p+Ry^* \\ y &= Cx \\ \end{align}$$

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 H2-H∞ Optimal Voltage Control
$$\begin{align} \begin{cases} &\text{min} \quad a\gamma + bTrace(M) \\ &\text{s.t.} \quad \begin{bmatrix} M & I \\ I & W_1 \end{bmatrix} > 0 \\ \\ &\quad \quad \begin{bmatrix} (AW_1 - B_1V_1) + (AW_1 - B_1V_1)^T & W_1 & V_1^T \\ W_1 & -Q_{inv} & 0 \\ V_1 & 0 & -R^{-1} \end{bmatrix} < 0 \\ \\ &\quad \quad AX+B_1W + (AX+B_1W)^T + BB^T< 0 \\ \\ &\quad \quad L_j \otimes W_3 + M_j\otimes (AW_3+B_1V_3) + M_j^T \otimes (AW_3+B_1V_3)^T < 0, j=1,2,3 \\ \\ &\quad \quad \begin{bmatrix} (AW_2 - B_1V_2) + (AW_2 - B_1V_2)^T & B_2 & W_2C^T \\ B_2^T & -\gamma I & 0 \\ CW_2 & 0 & -\gamma I \end{bmatrix} < 0 \\ \\ \end{cases} \\ \end{align}$$

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.

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