Virtual Reference Feedback Tuning for linear discrete-time systems with robust stability guarantees based on Set Membership

In this paper we propose a novel methodology that allows to design, in a purely data-based fashion and for linear single-input and single-output systems, both robustly stable and performing control systems for tracking piecewise constant reference signals. The approach uses both (i) Virtual Reference Feedback Tuning for enforcing suitable performances and (ii) the Set Membership framework for providing a-priori robust stability guarantees. Indeed, an uncertainty set for the system parameters is obtained through Set Membership identification, where an algorithm based on the scenario approach is proposed to estimate the inflation parameter in a probabilistic way. Based on this set, robust stability conditions are enforced as Linear Matrix Inequality constraints within an optimization problem whose linear cost function relies on Virtual Reference Feedback Tuning. To show the generality and effectiveness of our approach, we apply it to two of the most widely used yet simple control schemes, i.e., where tracking is achieved thanks to (i) a static feedforward action and (ii) an integrator in closed-loop. The proposed method is not fully direct due to the Set Membership identification. However, the uncertainty set is used with the only objective of providing robust stability guarantees for the closed-loop system and it is not directly used for the performances optimization, which instead is totally based on data. The effectiveness of the developed method is demonstrated with reference to two simulation examples. A comparison with other data-driven methods is also carried out.