Experimental Comparisons Between Implicit and Explicit Implementations of Discrete-Time Sliding Mode Controllers: Toward Input and Output Chattering Suppression

Abstract

International audienceThis brief presents a set of experimental results concerning the sliding mode control of an electropneumatic system. Two discrete-time control strategies are considered: an explicit and an implicit (that is very easy to implement with a projection on the interval [−1, 1]) Euler discretizations. While the explicit implementation is known to generate numerical chattering , the implicit one is expected to significantly reduce chattering while keeping the accuracy. The experimental results reported in this brief remarkably confirm that the implicit discrete-time sliding mode supersedes the explicit ones, with several important features: chattering in the control input is almost eliminated (while the explicit and saturated controllers behave like high-frequency bang–bang inputs), the input magnitude depends only on the perturbation size and is independent of the controller gain and sampling time

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).…”

“…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.…”

Multivalued Robust Tracking Control of Lagrange Systems: Continuous and Discrete-Time Algorithms

“…Explicit discretizations however yield the so-called numerical chattering effect on both input and output [14,17,18,40,41].…”

“…On the other hand controllers are now most of the time implemented via microprocessors, and it is of utmost importance to analyse the time-discretization of otherwise continuous-time designed algorithms. As shown in [1,2,14,17,18,19,40,41], the time-discretization of set-valued sliding-mode control laws requires particular care, because it may yield numerical chattering (high-frequency oscillations in the output, and high-frequency bang-bang controllers) when the set-valued part of the controller is discretized in an explicit way. In fact the explicit discretization may even yield in nonlinear cases, unstable closed-loop systems [26], while the implicit method advocated in [1,2,5,19] keeps the continuous-time stability properties for the systems analysed in [11,19].…”

“…In this work we do not ask for uniqueness of solutions, however we prove that all of them (if not unique) yield a tracking error with suitable stability properties. Section 6 is dedicated to the analysis of the discrete-time controller, using the implicit Euler discretization technique introduced in [1,2,19] for linear timeinvariant systems with known parameters (see also [22] for a similar approach), and experimentally tested with success in [17,18,19,40], in which it is shown that chattering in both, the output and the input, is almost totally suppressed. Due to the nonlinearities of the Euler-Lagrange dynamics, the design of the implicit discrete-time controller is made from a non exact discretization of the continuous plant (while the zero-order-hold discretization was used in [1,2,19]).…”