Euler implicit time-discretization of multivariable sliding-mode controllers

This paper aims to develop the Euler implicit time-discretization of multivariable sliding-mode controllers to solve the numerical chattering problem without modifying the continuous-time control law. To this end, a continuous-time multi-input plant under a multivariable sliding-mode control is studied, and it is shown that the implicit discretization of the continuous-time sliding-mode controller leads to a multivariable generalized equation with several set-valued terms which is not possible to be solved using the graphical interpretations. Subsequently, an algorithm is proposed to solve such a multivariable generalized equation required to synthesize the implicit sliding-mode control signal at each time step. The proposed algorithm is explained through a simple example accompanied by numerical simulations. The properties of the implicit multivariable sliding-mode controller, e.g., finite-time convergence, gain insensitivity, and chattering suppression, are studied analytically. Afterwards, a multivariable sliding-mode controller is implemented on a digital processor based on the developed algorithm to control a six-input six-output system, i.e., six-component thrust generator, and the results are compared with the case where the continuous-time sliding-mode controller is implemented using the conventional Euler explicit discretization. In the end, the related issues and drawbacks are addressed to be considered in future works.