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Robust tube based MPC for tracking of piece-wise constant references

This paper presents a novel formulation of a robust model predictive controller (RMPCT) to track piecewise constant references. The real plant is assumed to be modelled as a linear system with additive bounded uncertainties on the states. Under mild assumptions, the proposed MPC can steer the uncertain system in an admissible evolution to any admissible steady state, that is, under any change of the set point. This allows us to reject constant disturbances compensating the effect of then, changing the setpoint. Feasibility of the proposed controller for any admissible setpoint is achieved by adding an artificial steady state as decision variable. Robust constraint satisfaction is guaranteed by tube-based approach and considering nominal predictions. Robust stability and convergence to (a neighborhood of) the desired steady state is ensured by considering a modified cost function and an extended terminal constraint. The cost function penalizes the tracking error with the artificial reference and the deviation between the artificial and desired steady state; the terminal constraint restricts the terminal state and the artificial steady state. The optimization problem to be solved is a quadratic programming problem, which allows explicit implementations. In order to demonstrate the benefits of the proposed controller, this has been tested on a real positioning plant consisting in a linear motor driving a cart. The fast dynamics of the systems requires an explicit calculation of the controller, which has been implemented by means of a search tree strategy. The experimental results show the robust tracking and the admissible evolution of the closed loop system.