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

$\left[\begin{array}{c}\dot{x}\\ z\end{array}\right]=\left[\begin{array}{ccc}(A+\mathrm{\Delta }A)& (B_1+\mathrm{\Delta }{B}_1)& B_2\\ C& D_1& D_2\end{array}\right]\left[\begin{array}{c}x\\ u\\ w\end{array}\right]$

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

$\mathrm{\Delta }A$ and $\mathrm{\Delta }{B}_1$ are real-valued matrices which represent the time-varying parameter uncertainties in the form:

$\left[\begin{array}{cc}\mathrm{\Delta }A& \mathrm{\Delta }{B}_1\end{array}\right]=HF\left[\begin{array}{cc}{E}_1& E_2\end{array}\right]$

Where

$H,E_1,E_2$ are known matrices with appropriate dimensions and $F$ is the uncertain parameter matrix which satisfies: $F^{T}F\le I$

For additive perturbations: $\mathrm{\Delta }A={\delta }_1{A}_1+{\delta }_2{A}_2+...+{\delta }_k{A}_k$

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 $|{\delta }_i|<r_i,i=1,2,...,k$

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

$H=\left[\begin{array}{cccc}{A}_1& A_2& ...& A_k\end{array}\right]$

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

$F=(\sum _{i=1}^k{r}_i^2{)}^{-1/2}\left[\begin{array}{c}{\delta }_{1}I\\ {\delta }_{2}I\\ ⋮\\ {\delta }_{k}I\end{array}\right]$

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 $H_{\mathrm{\infty }}$ Control LMI

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

$\left[\begin{array}{cccc}\mathrm{\Psi }(X,W)& B_2& (CX+D_{1}W{)}^T& (E_{1}X+E_{2}W{)}^T\\ B_2^T& -\gamma I& D_2^T& 0\\ CX+D_{1}W& D_2& -\gamma I& 0\\ E_{1}X+E_{2}W& 0& 0& -\alpha I\end{array}\right]<0$.

Where $\mathrm{\Psi }(X,W)=(AX+B_{1}W{)}_s+\alpha H{H}^T$

And $K=W{X}^{-1}$.

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:

$\left[\begin{array}{c}\dot{x}\\ z\end{array}\right]=\left[\begin{array}{cc}(A+\mathrm{\Delta }A)+(B_1+\mathrm{\Delta }{B}_1)K& B_2\\ (C+D_1)K& D_2\end{array}\right]\left[\begin{array}{c}x\\ w\end{array}\right]$

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

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

$G_{zw}(s)=(C+D_{1}K)(sI-[(A+\mathrm{\Delta }A)+(B_1+\mathrm{\Delta }{B}_1)K]{)}^{-1}{B}_2+D_2$

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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