Conceitos Básicos
The authors present a novel grid-based technique called QuIFS (Quasi-Interpolation driven Feedback Synthesis) that provides strict guarantees of uniform approximation of the optimal feedback policy for nonlinear robust model predictive control problems.
Resumo
The key highlights and insights of the content are:
The QuIFS algorithm departs from conventional approximation techniques used in the MPC industry, such as multiparametric programming and kernel methods. It is driven by a particular type of grid-based quasi-interpolation scheme.
QuIFS provides one-shot uniform approximation guarantees of the optimal feedback policy, along with stability and recursive feasibility guarantees. These guarantees are robust and do not involve probabilistic (soft) bounds.
The algorithm applies to nonlinear systems and non-convex cost functions, and relies on coarse properties of the optimal feedback, such as Lipschitz continuity, rather than detailed local structural properties.
The complexity of the offline computations associated with QuIFS scales exponentially with the state dimension, unlike standard explicit MPC techniques where the complexity scales exponentially with the number of constraints.
QuIFS provides approximation error guarantees measured with respect to the uniform metric, which is necessary to ensure recursive feasibility. This is achieved without solving the associated Bellman equations/recursions.
The key technical tool used in QuIFS is a particular type of quasi-interpolation that conforms to neither parametric nor non-parametric function approximation techniques. It departs sharply from typical approximation theoretic tools that ensure asymptotic convergence.
For a prespecified uniform error margin, it is always possible to pick the quasi-interpolation parameters (discretization interval, shape parameter, and truncation parameter) such that the uniform error between the optimal feedback and the approximated one stays below the desired threshold.