Robust Unconstrained Model Predictive Control with State Feedback[edit | edit source]
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, control moves are computed by minimizing a nominal cost over a prediction horizon . The objective is to minimize the nominal cost function.
We consider the nominal cost function as:
where,
- and
and are positive definite weighting matrices.
In this case, we take . This is also called infinite horizon MPC.
Here, we consider system uncertainties that are modeled as polytopic uncertainties or structured uncertainties.
The set is the polytope
Where, denotes the convex hull.
The operator is a block-diagonal:
Each can be a repeated scalar block or a full block.
Consider a linear time-varying(LTV) system:
Here, is the control input, is the state of the plant and
is the plant output and 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:
where,
The LMI:Robust Unconstrained Model Predictive Control with State Feedback for polytopic uncertainty[edit | edit source]
subject to
and
The LMI:Robust Unconstrained Model Predictive Control with State Feedback for structured uncertainty[edit | edit source]
subject to
where
The state feedback matrix F in the control law that minimizes the upper bound on the robust performance objective function at sampling time is given by :
where and are obtained from the solution of the above LMI.