State space realization of model predictive controllers without active constraints

Abstract

To enable the use of traditional tools for analysis of multivariable controllers such as model predictive control (MPC), we develop a state space formulation for the resulting controller for MPC without constraints or assuming that the constraints are not active. Such a derivation was not found in the literature. The formulation includes a state estimator. The MPC algorithm used is a receding horizon controller with infinite horizon based on a state space process model. When no constraints are active, we obtain a state feedback controller, which is modified to achieve either output tracking, or a combination of input and output tracking. When the states are not available, they need to be estimated from the measurements. It is often recommended to achieve integral action in a MPC by estimating input disturbances and include their effect in the model. We show that to obtain offset free steady state the number of estimated disturbances must equal the number of measurements. The estimator is included in the controller equation, and we obtain a formulation of the overall controller with the set-points and measurements as inputs, and the manipulated variables as outputs. One application of the state space formulation is in combination with the process model to obtain a closed loop model. This can for example be used to check the steady-state solution and see whether integral action is obtained or not.