Non-Quadratic Proxy Functions in Mirror Descent Method Applied to Designing of Robust Controllers for Nonlinear Dynamic Systems with Uncertainty

We consider a class of controlled nonlinear plants, the dynamics of which are governed by a vector system of ordinary differential equations with a right-hand side that is partially known. The study’s objective is to construct a robust tracking controller with certain constraints on the state variables, assuming that the state variables and their time derivatives can be observed. The Legendre–Fenchel transform and a chosen proxy function are utilized to develop this mathematical development using the mirror descent approach, which is frequently employed in convex optimization problems involving static objects. The Average Subgradient Method (an improved version of the Subgradient Descent Method), and the Integral Sliding Mode technique for continuous-time control systems are basically extended by the suggested unifying architecture. The primary findings include demonstrating that the “desired regime”—a non-stationary analog of the sliding surface – can be achieved from the very start of the process and getting an explicit upper bound on the cost function’s decrement.