User:Margav06/sandbox/Click here to continue/Robust Controls/Robust Stabilization of Hinf Optimal State Feedback 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 \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$$

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

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

Where

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

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 $$x(t)\in \R^n$$ 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:

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

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

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