Digital set-valued and homogeneous sliding mode control and differentiators: the implicit approach – DIGITSLID

Sliding mode control is a well-known and widely used nonlinear feedback control technique. It owes its success to its robustness with respect to large classes of disturbances, finite-time stability of the closed-loop, and the ease of tuning. Such controllers are in essence set-valued, and they yield closed-loop systems represented by differential inclusions (usually in the sense of Filippov). Two main categories exist: first-order sliding-modes (i.e.n "classical" ones), and high-order sliding-modes (the most popular ones being the twisting and super-twisting algorithms). Sliding mode state observers, for instance exact differentiators (also called Levant's differentiators), also make a promising class of state observers, because of properties like robustness and finite-time convergence. They belong to homogeneous dynamical systems, which are the object of many studies because of their very nice properties (related for instance to Lyapunov stability). However, sucg controllers and differentiators/observers have a strong darwback : the chattering phenomenon, which yields high-frequency oscillations at the output (the sliding variable), and high-frequency switcthing bang-bang inputs , which are damaging actuators. One source of chattering is a bad time-discretization, another one may be neglected dynamics. For instance it is by now well-known that the explicit discretization always yields so-called numerical (or digital) chattering, this being true even for high-order control algorithms and differentiators. Moreover the closed-loop system may become unstable because of the explicit discretization. Recent results have proved and experimentally validated, that the implicit discretization allows one to suppress the digital chattering, to guarantee the global stability, and at the top of that allows one to choose large sampling times without deteriorating too much the closed-loop system performances. In this project we will tackle therefore the analysis and experimental validation of the implicit discretization of homogeneous sliding mode high-order controllers, as well as exact differentiators. Firstly, the analytical work will consist of the study of the feasibility of the discretized system (like the existence and uniqueness of the control input in view of its implementation), stability properties (Lyapunov stability), finite-time convergence, robustness (i.e., find the classes of disturbances which are allowed). The separation principle of Automatic Control will also be studied : is it possible to use exact differentiators in the controller, preserving the above properties after the discretization ? Indeed, one of the major drawbacks of sliding mode control is, apart from chattering, the necessity of the whole state vector in order to define the sliding variable. Output feedback is usually obtained using "dirty" filters (low-pass filters) to approximate derivatives. We hope that the exact differentiators could significantly improve the performances, as well as the tuning process which can be tedious when the time-constants and gains of the low-pass filters play a role. The second objective of this project is experimental. Two partners possess several set-ups (electro-pneumatic system, electro-mechanical system, flying object, biped robot, robotic hand, inverted pendula) that will be extensively used to validate the theoretical findings. All the controllers and observers/differentiators will be tested on these set-ups. These experimental tasks are of great interest, because each set-up possesses its own dynamics as well as different types of disturbances. It will therefore be of great scientific interest to compare different controllers and differentiators on several experimental set-ups.