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

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

Additive uncertainty

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 $\vert {\delta }_{i}\vert <r_{i},i=1,2,...,k$

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

Related LMIs

Full State Feedback Optimal H_inf LMI

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
• LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & amp; Francis Group, 2013.

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