# LMIs in Control/pages/Discrete Time Mixed H2-H∞ Optimal Full State Feedback Control

Discrete-Time Mixed H2-H∞-Optimal Full-State Feedback Control

A discrete time system operates on a discrete time signal input and produces a discrete time signal output. They are used in digital signal processing, such as digital filters for images or sound. The class of discrete time systems that are both linear and time invariant, known as discrete time LTI systems.

A full-state feedback controller ${\displaystyle K=K_{d}\in {R^{n_{u}*n_{x}}}}$ (i.e., ${\displaystyle uk=K_{d}x_{k}}$) is to be designed to minimize the H2 norm of the closed loop transfer matrix ${\displaystyle T11(z)}$ from the exogenous input ${\displaystyle w_{1,k}}$ to the performance output ${\displaystyle z_{1,k}}$ while ensuring the H∞ norm of the closed-loop transfer matrix ${\displaystyle T22(z)}$ from the exogenous input ${\displaystyle w_{2,k}}$ to the performance output ${\displaystyle z_{2,k}}$ is less than ${\displaystyle \gamma _{d}}$.

## The System

Discrete-Time LTI System with state space realization

{\displaystyle {\begin{aligned}x_{k+1}&=A_{d}x_{k}+B_{d1,1}w_{1,k}+B_{d1,2}w_{2,k}+B_{d2}u_{k},\\{\begin{bmatrix}Z_{1,k}\\Z_{2,k}\end{bmatrix}}&={\begin{bmatrix}C_{d1,1}\\C_{d1,2}\end{bmatrix}}x_{k}+{\begin{bmatrix}0&D_{d11,12}\\D_{d11,21}&D_{d11,22}\end{bmatrix}}{\begin{bmatrix}w_{1,k}\\w_{2,k}\end{bmatrix}}+{\begin{bmatrix}D_{d12,1}\\D_{d12,2}\end{bmatrix}}u_{k}\\y_{k}&=x_{k}\\\end{aligned}}}

## The Data

The matrices: System ${\displaystyle (A_{d},B_{d1,1},B_{d1,2},B_{d2},C_{d1,1},C_{d1,2},D_{d11,12},D_{d11,21},D_{d11,22},,D_{d12,1},D_{d12,2}),P,F_{d}}$ .

## The Optimization Problem

The following feasibility problem should be optimized:

Minimize the H2 norm of the closed loop transfer matrix ${\displaystyle T11(z)}$ , while ensuring the H∞ norm of the closed-loop transfer matrix ${\displaystyle T22(z)}$  is less than ${\displaystyle \gamma _{d}}$ , while obeying the LMI constraints.

## The LMI:

Discrete-Time Mixed H2-H∞-Optimal Full-State Feedback Controller is synthesized by solving for ${\displaystyle P\in S^{n_{x}},Z\in S^{n_{w}},F_{d}\in R^{n_{u}*n_{x}}}$ , and ${\displaystyle \mu \in R_{>0}}$  that minimize ${\displaystyle \mu }$  subject to ${\displaystyle P>0,Z>0}$

The LMI formulation

H∞ norm < ${\displaystyle \gamma _{d}}$

H2 norm < ${\displaystyle \mu }$

{\displaystyle {\begin{aligned}{\begin{bmatrix}P&A_{d}P-B_{d2}F_{d}&B_{d1,1}\\*&P&0\\*&*&1\end{bmatrix}}&>0\\{\begin{bmatrix}P&A_{d}P-B_{d2}F_{d}&B_{d1,2}&0\\*&P&0&PC_{d1,2}^{T}-F_{d}^{T}D_{d12,2}^{T}\\*&*&\gamma _{d}I&D_{d11,22}^{T}\\*&*&*&\gamma _{d}I\end{bmatrix}}&>0\\{\begin{bmatrix}Z&C_{d1,1}P-D_{d12,1}\\*&P\end{bmatrix}}&>0\\trace(Z)&<\mu ^{2}\end{aligned}}}

## Conclusion:

The H2-optimal full-state feedback controller gain is recovered by ${\displaystyle K_{d}=F_{d}P^{-1}}$

## Implementation

A link to CodeOcean or other online implementation of the LMI
MATLAB Code

## Related LMIs

[1] - Continuous Time Mixed H2-H∞ Optimal Full State Feedback Control