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

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, 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. Uncertainties Here, we consider system uncertainties that are modeled as polytopic uncertainties or structured uncertainties. Polytopic Uncertainty The set is the polytope Where, denotes the convex hull. Structured Uncertainty The operator is a block-diagonal: Each can be a repeated scalar block or a full block. The System 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 Data The LMI:Robust Model Predictive Control with State Feedback and input and output constraints for polytopic uncertainty subject to and Input constraint is given by the following LMI: Output constraint is given by the following LMI: The LMI:Robust Model Predictive Control with input and output constraints with State Feedback for structured uncertainty subject to where Input constraint is given by the following LMI: Output constraint is given by the following LMI: where, T = Conclusion: 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. Implementation