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,


 * $$ J_p(k) = \Sigma_{i=0}^p[x(k+i|k)^{T}Q_1x(k+i|k) + u(k+i|k)^{T}Ru(k+i|k)]$$


 * $$ Q_1 > 0$$ 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, $$ u(k)\in \R^n$$ 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 $$\Delta$$ 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,


 * $$ J_\infty(k) = \Sigma_{i=0}^\infty[x(k+i|k)^{T}Q_1x(k+i|k) + u(k+i|k)^{T}Ru(k+i|k)]$$

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^TB_{j}^T & QQ_{1}^{1/2} & Y^TR^{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^TR^{1/2} & QQ_{1}^{1/2} & QC_{q}^T+Y^TD_{qu}^T & QA^T+Y^TB^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


 * $$\Lambda = \begin{bmatrix}\lambda_1I_{n1} & & & \\ & \lambda_2I_{n2} & & & \\ & & \ddots & \\ & & & \lambda_{r}I_{nr} \end{bmatrix}>0$$

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.