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

Robust Full State Feedback Optimal ${\displaystyle H_{\infty }}$ 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 ${\displaystyle H_{\infty }}$  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 ${\displaystyle H_{\infty }}$  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:

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

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

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

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

Where

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

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

Where

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

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

Thus, ${\displaystyle {\Delta }A=HFE}$  with

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

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

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

The Data

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

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

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

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

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

And ${\displaystyle K=WX^{-1}}$ .

Conclusion:

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

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

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

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

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

Implementation

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