Robust Learning Model-Predictive Control for Linear Systems Performing Iterative Tasks

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In this article, a robust learning model-predictive controller (LMPC) for uncertain systems performing iterative tasks is presented. At each iteration of the control task, the closed-loop state, input, and cost are stored and used in the controller design. This article first illustrates how to construct robust control invariant sets and safe control policies exploiting historical data. Then, we propose an iterative LMPC design procedure, where data generated by a robust controller at iteration <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$j$</tex-math></inline-formula> are used to design a robust LMPC at the next iteration <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$j+1$</tex-math></inline-formula> . We show that this procedure allows us to iteratively enlarge the domain of the control policy, and it guarantees recursive constraints satisfaction, input-to-state stability, and performance bounds for the certainty equivalent closed-loop system. The use of different feedback policies along the horizon is the key element of the proposed design. The effectiveness of the proposed control scheme is illustrated on a linear system subject to bounded additive disturbances.