Robust Tracking as Constrained Optimization by Uncertain Dynamic Plant: Mirror Descent and Average Sub-Gradient Methods -- Version of Integral Sliding Mode Control

A class of controlled plants, whose dynamics is governed by a vector system of ordinary differential equations with a partially known right-hand side, is considered. The state variables and their velocities are assumed to be measurable. The aim is to design a controller which minimizes a loss function under certain constraints which arguments is the current state of the controlled plant. The designed control action is admitted to be a function of the current sub-gradient only, which supposed to be measurable on-line. The control design is based on ASG (Average Sub-Gradient method) — version of Integral Sliding Mode (ISM) concept, aimed to minimize on average a given convex (not obligatory strongly convex) cost function of the current state under a set of given constraints. An optimization type algorithm is developed and analyzed using ideas of SDM technique. The main results consist in proving the reachability of the "desired regime" (nonstationary analogue of sliding surface) from the beginning of the process and obtaining an explicit upper bound for the averaged loss function decrement, that is, the averaged in time functional convergence is proven and the rate of such convergence is estimated.