A survey on the discrete-time differentiators in closed-loop control systems: Experiments on an electro-pneumatic system

Euler implicit time-discretization of multivariable sliding-mode controllers

“…with the condition x k+1,1 > 0 and | sgn(x k+1,2 )| ≤ 1. Subsequently, (22) can be used to obtain ζ * k and synthesize (15).…”

“…The implicit discretization was then developed for the twisting [15,16] and super-twisting controllers [17]. Furthermore, this discretization method has been used for the sliding-mode-based differentiators [6,18,19,20,21] and some useful properties, e.g., finite-time convergence, gain insensitivity and chattering reduction have been reported by [22,23,24]. The main drawback of implicit discretization is that it requires relatively higher calculation resources compared to explicit ones.…”

“…The main drawback of implicit discretization is that it requires relatively higher calculation resources compared to explicit ones. For instance, an implicitly implemented differentiator may need a solver to obtain the solutions of a polynomial equation at each time step [22,23]. However, implicit discretization can still be used to implement both controllers [15] and observers [22] in practical applications.…”

“…For instance, an implicitly implemented differentiator may need a solver to obtain the solutions of a polynomial equation at each time step [22,23]. However, implicit discretization can still be used to implement both controllers [15] and observers [22] in practical applications. In addition, the semi-implicit schemes can be used to alleviate the calculation burden while keeping some of the advantages of the implicit discretization [25,26,27].…”