# LMIs in Control/pages/Discrete Time H2 Optimal Dynamic Output Feedback Control

Discrete-Time H2-Optimal Dynamic Output 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 Dynamic Output feedback controller is designed for a Discrete Time system, to minimize the H2 norm of the closed loop system with exogenous input ${\displaystyle w_{k}}$ and performance output ${\displaystyle z_{k}}$.

## The System

Discrete-Time LTI System with state space realization ${\displaystyle (A_{d},B_{d},C_{d},D_{d})}$
{\displaystyle {\begin{aligned}x_{k+1}&=A_{d}x_{k}+B_{d1}w_{k}+B_{d2}u_{k}\\z_{k}&=C_{d1}x_{k}+D_{d11}w_{k}+D_{d12}u_{k}\\y_{k}&=C_{d2}x_{k}+D_{d21}w_{k}+D_{d22}u_{k}\\\end{aligned}}}

## The Data

The matrices: System ${\displaystyle (A_{d},B_{d1},B_{d2},C_{d1},C_{d1},D_{d11},D_{d12},D_{d21},D_{d22}),X_{1},Y_{1},Z,,X_{2},Y_{2}}$

Controller ${\displaystyle (A_{dc},B_{dc},C_{dc},D_{dc})}$

## The Optimization Problem

The following feasibility problem should be optimized:

${\displaystyle \mu }$ is minimized while obeying the LMI constraints.

## The LMI:

Discrete-Time H2-Optimal Full-State Feedback Control

The LMI formulation

H2 norm < ${\displaystyle \mu }$

{\displaystyle {\begin{aligned}X_{1},Y_{1}\in {S^{n_{x}}};Z\in {S^{n_{z}}};\mu \in {R_{>0}}\;\\A_{dn}\in {R^{n_{x}*n_{x}}};B_{dn}\in {R^{n_{x}*n_{y}}};C_{dn}\in {R^{n_{u}*n_{x}}};D_{dn}\in {R^{n_{u}*n_{y}}};\\&X_{1}>0\\&Y_{1}>0\\&Z>0\\{\begin{bmatrix}X_{1}&1&X_{1}A_{d}+B_{dn}C_{d2}&A_{dn}&X_{1}B_{d1}+B_{dn}D_{d21}\\*&Y_{1}&A_{d}+B_{d2}D_{dn}C_{d2}&A_{d}Y_{1}+B_{d2}C_{dn}&B_{d1}+B_{d2}D_{dn}D_{d21}\\*&*&X_{1}&1&0\\*&*&*&Y_{1}&0\\*&*&*&*&1\end{bmatrix}}&>0,\\{\begin{bmatrix}Z&C_{d1}+D_{d12}D_{dn}C_{d2}&C_{d1}Y_{1}^{T}+D_{d12}C_{dn}\\*&X_{1}&1\\*&*&Y_{1}\end{bmatrix}}&>0,\\D_{d11}+D_{d12}D_{dn}D_{d21}=0\\{\begin{bmatrix}X_{1}&1\\*&Y_{1}\end{bmatrix}}&>0,\\trZ<\mu ^{2}\end{aligned}}}

The controller is recovered by

{\displaystyle {\begin{aligned}&A_{dc}=A_{dk}-B_{dc}(1-D_{d22}D_{dc})^{-1}D_{d22}C_{dc}\\&B_{dc}=B_{dk}(1-D_{d22}D_{dc})\\&C_{dc}=(1-D_{dc}D_{d22})C_{dk}\\&D_{dc}=1+D_{dk}D_{d22})^{-1}D_{dk}\\\end{aligned}}}

where, {\displaystyle {\begin{aligned}{\begin{bmatrix}A_{dk}&B_{dk}\\C_{dk}&D_{dk}\end{bmatrix}}&={\begin{bmatrix}X_{2}&X_{1}B_{d2}\\0&1\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}&1\end{bmatrix}}^{-1}\end{aligned}}}
and the matrixes ${\displaystyle X_{2}}$ and ${\displaystyle Y_{2}}$ satisfies ${\displaystyle X_{2}Y_{2}^{T}=1-X_{1}Y_{1}}$

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 ${\displaystyle D_{d11}=0}$, ${\displaystyle D_{d12}}$ ≠ 0, and ${\displaystyle D_{d21}}$ ≠ 0, then it is often simplest to choose ${\displaystyle D_{dn}=0}$ in order to satisfy the equality constraint

## Conclusion:

The Discrete-Time H2-Optimal Dynamic Output feedback controller is the system ${\displaystyle (A_{dc},B_{dc},C_{dc},D_{dc})}$

## Implementation

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

## Related LMIs

[1] - Continuous Time H2 Optimal Dynamic Output Feedback Control