Lyapunov Stability and Performance Analysis of the Implicit Discrete Sliding Mode Control

Abstract

Discrete-time sliding mode controllers with an implicit discretization of the signum function are considered. With a proper choice of the equivalent part of the control, the resulting controller is shown to be Lyapunov stable with finitetime convergence of the sliding variable to 0. The convergence of the control input, as the sampling period goes to 0, to the continuous-time one is shown. The robustness with respect to matching perturbations is also investigated. The discretization performance in terms of the error order is studied for different discretizations of the equivalent part of the input. Numerical and experimental results illustrate and support the analysis.

Set-Valued Sliding-Mode Control of Uncertain Linear Systems: Continuous and Discrete-Time Analysis

“…Now we introduce the missing term u sv k using an implicit approach, which has been studied theoretically in [1,2,24] and tested experimentally in [25,26,45] showing to be a very efficient way to deal with the chattering effect. It is clear that in a real implementation setting the selection procedure cannot be achieved directly, because if we try to mimic the same steps presented in the previous situation, we will have to impose the unreal assumption that we know perfectly the disturbance term w k + η m k , see (60).…”

“…Therefore some modification to the discrete-time controller (58) must be done. Roughly speaking, we consider the discrete-time scheme proposed in [1,2,24] by creating a virtual nominal system from where the selection process is achieved. Next, the controller computed from the virtual nominal system is applied to the original discrete-time plant.…”

“…In this context the variableσ k is called the discrete sliding variable and whenσ k+n = 0 for all n ≥ 1 and some k < +∞, we say that the system is in the discrete-time sliding phase [24].…”

“…For that reason many research efforts have been directed in the study of attenuation of the chattering effect. Among these studies we can find adaptive schemes with variable gains [42], high-order sliding modes [30], regularization techniques [46], and suitable discrete-time implementation [1,2,24,25,26,45]. Since the work of Filippov [20] sliding-mode control systems have been associated with differential inclusions.…”

Implicit numerical integration for the simulation and control of a non-smooth system with resets

“…The coefficient θ is used for the smooth part (the state) and γ for the nonsmooth component (the signums) of the ODE. The implicit discretization, a special case when θ = γ = 1, has been used to successfully discretize discontinuous systems from mechanics (system with impact and friction [4]) and control theory (sliding mode control [16], [17]). To make the presentation more accessible, we focus on the implicit case.…”