# LMIs in Control/Click here to continue/Robust Controls/Robust Model Predictive Control with input and output constraints
## Model Predictive Control

Model Predictive Control is an open-loop control design procedure where at each sampling time k, plant measurements are obtained and a model of the process is used to predict future outputs of the system. Using these predictions, $m$ control moves $u(k+i|k),i=0,1,...,m-1.$ are computed by minimizing a nominal cost $J_{p}(k)$ over a prediction horizon $p$. The objective is to minimize the nominal cost function.
We consider the nominal cost function as:
       $\min _{u(k+i),i=0,1,...,m-1}J_{p}(k)$
where,
       $J_{p}(k)=\Sigma _{i=0}^{p}[x(k+i|k)^{T}Q_{1}x(k+i|k)+u(k+i|k)^{T}Ru(k+i|k)]$
       $Q_{1}>0$ and $R>0$
$Q_{1}$ and $R$ are positive definite weighting matrices.
In this case, we take $p=\infty$. This is also called infinite horizon MPC.

## Uncertainties

Here, we consider system uncertainties that are modeled as polytopic uncertainties or structured uncertainties.

#### Polytopic Uncertainty

The set $\Omega$ is the polytope $\Omega =Co{[A_{1}B_{1}],[A_{2}B_{2}],....,[A_{L}B_{L}]}$
Where, $Co$ denotes the convex hull.

#### Structured Uncertainty

The operator $\Delta$ is a block-diagonal:
       $\Delta ={\begin{bmatrix}\Delta _{1}&&&\\&\Delta _{2}&&&\\&&\ddots &\\&&&\Delta _{r}\end{bmatrix}}$
Each $\Delta$ can be a repeated scalar block or a full block.

## The System

Consider a linear time-varying(LTV) system:
       $x(k+1)=A(k)x(k)+B(k)u(k),$
       $y(k)=CX(k),$
       ${\begin{bmatrix}A(k)&B(k)\end{bmatrix}}\in \Delta$
Here, $u(k)\in \mathbb {R} ^{n}$ is the control input, $x(k)\in \mathbb {R} ^{n}$ is the state of the plant and $y(k)\in \mathbb {R} ^{n}$ is the plant output and $\Delta$ is uncertainty set that is either polytopic system or structured uncertainty.
We modify the minimization of the nominal cost function to a minimization of the worst-case objective function.
The modified objective function minimizes the robust performance objective as follows:
       $\min _{u(k+i),i=0,1,...,m-1}\max _{[A(k+i)B(k+i]\in \Delta ,i\geq 0}J_{\infty }(k)$
where,
       $J_{\infty }(k)=\Sigma _{i=0}^{\infty }[x(k+i|k)^{T}Q_{1}x(k+i|k)+u(k+i|k)^{T}Ru(k+i|k)]$

## The LMI:Robust Model Predictive Control with State Feedback and input and output constraints for polytopic uncertainty

$\min _{\gamma ,Q,Y}\gamma$
subject to
       ${\begin{bmatrix}1&x(k|k)^{T}\\x(k|k)&Q\end{bmatrix}}\geq 0$
and
       ${\begin{bmatrix}Q&QA_{j}^{T}+Y^{T}B_{j}^{T}&QQ_{1}^{1/2}&Y^{T}R^{1/2}\\A_{j}Q+B_{j}Y&Q&0&0\\Q_{q}^{1/2}&0&\gamma I&0\\R^{1/2}Y&0&0&\gamma I\end{bmatrix}}\geq 0$
Input constraint is given by the following LMI:
       ${\begin{bmatrix}u_{max}^{2}I&Y\\Y^{T}&Q\end{bmatrix}}\geq 0$
Output constraint is given by the following LMI:
       ${\begin{bmatrix}Q&(A_{j}Q+B_{j}Y)^{T}C^{T}\\C(A_{j}Q+B_{j}Y)&y_{max}^{2}I\end{bmatrix}}\geq 0$

## The LMI:Robust Model Predictive Control with input and output constraints with State Feedback for structured uncertainty

$\min _{\gamma ,Q,Y,\Lambda }\gamma$
subject to
       ${\begin{bmatrix}1&x(k|k)^{T}\\x(k|k)&Q\end{bmatrix}}\geq 0$
       ${\begin{bmatrix}Q&Y^{T}R^{1/2}&QQ_{1}^{1/2}&QC_{q}^{T}+Y^{T}D_{qu}^{T}&QA^{T}+Y^{T}B^{T}\\R^{1/2}Y&\gamma I&0&0&0\\Q_{1}^{1/2}Q&0&\gamma I&0&0\\C_{q}Q+D_{qu}Y&0&0&\Lambda &0\\AQ+BY&0&0&0&Q-B_{p}\Lambda B_{p}^{T}\end{bmatrix}}\geq 0$
where
       $\Lambda ={\begin{bmatrix}\lambda _{1}I_{n1}&&&\\&\lambda _{2}I_{n2}&&&\\&&\ddots &\\&&&\lambda _{r}I_{nr}\end{bmatrix}}>0$
Input constraint is given by the following LMI:
       ${\begin{bmatrix}u_{max}^{2}I&Y\\Y^{T}&Q\end{bmatrix}}\geq 0$
Output constraint is given by the following LMI:
       ${\begin{bmatrix}y_{max}^{2}Q&(C_{q}Q+D_{qu}Y)^{T}&(AQ+BY)^{T}C^{T}\\C_{q}Q+D_{qu}Y&T^{-1}&0\\C(AQ+BY)&0&I-CB_{p}T^{-1}B_{p}^{T}C^{T}\end{bmatrix}}\geq 0$
where,
       T = ${\begin{bmatrix}t_{1}I_{n_{1}}&&&\\&t_{2}I_{n_{2}}&&\\&&\ddots &\\&&&t_{r}I_{n_{r}}\end{bmatrix}}\geq 0$

## Conclusion:

The state feedback matrix F in the control law $u(k+i|k)=Fx(k+i|k),i\geq 0$ that minimizes the upper bound $V(x(k|k))$ on the robust performance objective function at sampling time $k$ is given by :
       $F=YQ^{-1}$
where $Q>0$ and $Y$ are obtained from the solution of the above LMI.