# LMIs in Control/pages/Robust H inf State Feedback Control

## Robust Full State Feedback Optimal $H_{\infty }$ Control

Full State Feedback is a control technique which places a given system's closed loop system poles in locations specified by desired performance specifications. One can use $H_{\infty }$  methods to turn this into an optimization problem with the goal to minimize the impact of uncertain perturbations in a closed loop system while maintaining system stability. This is done by minimizing the $H_{\infty }$  norm of the uncertain closed loop system, which minimizes the worst case effect of the system disturbance or perturbation. This can be done for single-input single-output (SISO) or multiple-input multiple-output (MIMO) systems. Here we consider the case of a MIMO system with additive uncertainties.

## The System

Consider linear system with uncertainty below:

${\begin{bmatrix}{\dot {x}}\\z\end{bmatrix}}={\begin{bmatrix}(A+{\Delta }A)&(B_{1}+{\Delta }B_{1})&B_{2}\\C&D_{1}&D_{2}\end{bmatrix}}{\begin{bmatrix}x\\u\\w\end{bmatrix}}$

Where $x(t)\in \mathbb {R} ^{n}$  is the state, $z(t)\in \mathbb {R} ^{m}$  is the output, $w(t)\in \mathbb {R} ^{p}$  is the exogenous input or disturbance vector, and $u(t)\in \mathbb {R} ^{r}$  is the actuator input or control vector, at any $t\in \mathbb {R}$

${\Delta }A$  and ${\Delta }B_{1}$  are real-valued matrices which represent the time-varying parameter uncertainties in the form:

${\begin{bmatrix}{\Delta }A&{\Delta }B_{1}\end{bmatrix}}=HF{\begin{bmatrix}E_{1}&E_{2}\end{bmatrix}}$

Where

$H,E_{1},E_{2}$  are known matrices with appropriate dimensions and $F$  is the uncertain parameter matrix which satisfies: $F^{T}F\leq I$

For additive perturbations: ${\Delta }A={\delta }_{1}A_{1}+{\delta }_{2}A_{2}+...+{\delta }_{k}A_{k}$

Where

$A_{i},i=1,2,...k$  are the known system matrices and

${\delta }_{i},i=1,2,...k$  are the perturbation parameters which satisfy $\vert {\delta }_{i}\vert

Thus, ${\Delta }A=HFE$  with

$H={\begin{bmatrix}A_{1}&A_{2}&...&A_{k}\end{bmatrix}}$

$E=(\sum _{i=1}^{k}r_{i}^{2})^{1/2}$

$F=(\sum _{i=1}^{k}r_{i}^{2})^{-1/2}{\begin{bmatrix}{\delta }_{1}I\\{\delta }_{2}I\\\vdots \\{\delta }_{k}I\end{bmatrix}}$

## The Data

$A$ , $B_{1}$ , $B_{2}$ , $C$ , $D_{1}$ , $D_{2}$ , $E_{1}$ , $E_{2}$ , ${\gamma }$  are known.

## The LMI:Full State Feedback Optimal $H_{\infty }$ Control LMI

There exists $X>0$  and $W$  and scalar ${\alpha }$  such that

${\begin{bmatrix}{\Psi }(X,W)&B_{2}&(CX+D_{1}W)^{T}&(E_{1}X+E_{2}W)^{T}\\B_{2}^{T}&-{\gamma }I&D_{2}^{T}&0\\CX+D_{1}W&D_{2}&-{\gamma }I&0\\E_{1}X+E_{2}W&0&0&-{\alpha }I\end{bmatrix}}<0$ .

Where ${\Psi }(X,W)=(AX+B_{1}W)_{s}+{\alpha }HH^{T}$

And $K=WX^{-1}$ .

## Conclusion:

Once K is found from the optimization LMI above, it can be substituted into the state feedback control law $u(t)=Kx(t)$  to find the robustly stabilized closed loop system as shown below:

${\begin{bmatrix}{\dot {x}}\\z\end{bmatrix}}={\begin{bmatrix}(A+{\Delta }A)+(B_{1}+{\Delta }B_{1})K&B_{2}\\(C+D_{1})K&D_{2}\end{bmatrix}}{\begin{bmatrix}x\\w\end{bmatrix}}$

where $x(t)\in \mathbb {R} ^{n}$  is the state, $z(t)\in \mathbb {R} ^{m}$  is the output, $w(t)\in \mathbb {R} ^{p}$  is the exogenous input or disturbance vector, and $u(t)\in \mathbb {R} ^{r}$  is the actuator input or control vector, at any $t\in \mathbb {R}$

Finally, the transfer function of the system is denoted as follows:

$G_{zw}(s)=(C+D_{1}K)(sI-[(A+{\Delta }A)+(B_{1}+{\Delta }B_{1})K])^{-1}B_{2}+D_{2}$

## Implementation

This implementation requires Yalmip and Sedumi. https://github.com/rubindan/LMIcontrol/blob/master/HinfFilter.m