LMIs in Control/Click here to continue/Robust Controls/Robust Model Predictive Control

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:

$\underset{u(k+i),i=0,1,...,m-1}{\min }{J}_p(k)$

where,

$J_p(k)={\mathrm{\Sigma }}_{i=0}^p[x(k+i|k{)}^T{Q}_{1}x(k+i|k)+u(k+i|k{)}^{T}Ru(k+i|k)]$

$Q_1>0$ and $R>0$

$Q_1$ and $R$ are positive definite weighting matrices.

In this case, we take $p=\mathrm{\infty }$. 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 $\mathrm{\Omega }$ is the polytope $\mathrm{\Omega }=Co[A_1{B}_1],[A_2{B}_2],....,[A_L{B}_L]$

Where, $Co$ denotes the convex hull.

Structured Uncertainty

The operator $\mathrm{\Delta }$ is a block-diagonal:

$\mathrm{\Delta }=\left[\begin{array}{cccc}{\mathrm{\Delta }}_1& & & \\ & {\mathrm{\Delta }}_2& & & \\ & & \ddots & \\ & & & {\mathrm{\Delta }}_r\end{array}\right]$

Each $\mathrm{\Delta }$ can be a repeated scalar block or a full block.

The System

Consider a linear time-varying(LTV) system:

$x(k+1)=A(k)x(k)+B(k)u(k),$

$y(k)=CX(k),$

$\left[\begin{array}{cc}A(k)& B(k)\end{array}\right]\in \mathrm{\Delta }$

Here, $u(k)\in {ℝ}^n$ is the control input, $x(k)\in {ℝ}^n$ is the state of the plant and $y(k)\in {ℝ}^n$ is the plant output and $\mathrm{\Delta }$ 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:

$\underset{u(k+i),i=0,1,...,m-1}{\min }\underset{[A(k+i)B(k+i]\in \mathrm{\Delta },i\ge 0}{\max }{J}_{\mathrm{\infty }}(k)$

where,

$J_{\mathrm{\infty }}(k)={\mathrm{\Sigma }}_{i=0}^{\mathrm{\infty }}[x(k+i|k{)}^T{Q}_{1}x(k+i|k)+u(k+i|k{)}^{T}Ru(k+i|k)]$

The LMI:Robust Unconstrained Model Predictive Control with State Feedback for polytopic uncertainty

$\underset{\gamma ,Q,Y}{\min }\gamma$

subject to

$\left[\begin{array}{cc}1& x(k|k{)}^T\\ x(k|k)& Q\end{array}\right]\ge 0$

and

$\left[\begin{array}{cccc}Q& Q{A}_j^T+Y^T{B}_j^T& Q{Q}_1^{1/2}& Y^T{R}^{1/2}\\ A_{j}Q+B_{j}Y& Q& 0& 0\\ Q_q^{1/2}& 0& \gamma I& 0\\ R^{1/2}Y& 0& 0& \gamma I\end{array}\right]\ge 0$

The LMI:Robust Unconstrained Model Predictive Control with State Feedback for structured uncertainty

$\underset{\gamma ,Q,Y,\mathrm{\Lambda }}{\min }\gamma$

subject to

$\left[\begin{array}{cc}1& x(k|k{)}^T\\ x(k|k)& Q\end{array}\right]\ge 0$

$\left[\begin{array}{ccccc}Q& Y^T{R}^{1/2}& Q{Q}_1^{1/2}& Q{C}_q^T+Y^T{D}_{qu}^T& Q{A}^T+Y^T{B}^T\\ R^{1/2}Y& \gamma I& 0& 0& 0\\ Q_1^{1/2}Q& 0& \gamma I& 0& 0\\ C_{q}Q+D_{qu}Y& 0& 0& \mathrm{\Lambda }& 0\\ AQ+BY& 0& 0& 0& Q-B_p\mathrm{\Lambda }{B}_p^T\end{array}\right]\ge 0$

where

$\mathrm{\Lambda }=\left[\begin{array}{cccc}{\lambda }_1{I}_{n1}& & & \\ & {\lambda }_2{I}_{n2}& & & \\ & & \ddots & \\ & & & {\lambda }_r{I}_{nr}\end{array}\right]>0$

Conclusion:

The state feedback matrix F in the control law $u(k+i|k)=Fx(k+i|k),i\ge 0$ that minimizes the upper bound $V(x(k|k))$ on the robust performance objective function at sampling time $k$ is given by :

$F=Y{Q}^{-1}$

where $Q>0$ and $Y$ are obtained from the solution of the above LMI.