Implicit Implementation of Nonsmooth Controllers to Nonsmooth Actuators

Abstract

This paper presents an approach to implement a sliding-mode position controller to a plant equipped with a nonsmooth actuator. The actuator is modeled as a set-valued function from the control input and the velocity to the actuator force, which is motivated by quasistatic characteristics of hydraulic actuators shown in a previous study. The implementation of the sliding-mode controller is performed with the implicit discretization of the nominal plant model and the controller, which copes with the difficulties caused by set-valuedness, such as numerical chattering. Stability analyses both in the continuous-time and discrete-time domains are presented. Simulation results illustrate the theoretical findings.

Remote tele‐operation using model reference discrete‐time sliding mode control

“…and 0 < 𝜌 ≤ 1, 0 < Γ < 1 are design parameters. In (31), 0 ≤ 𝜌 < 1 acts as a forgetting factor and it is clear that ε(k) will always be positive for 0 ≤ 𝜌 < 1 and the impact of the initial value diminishes and ε becomes proportional to the magnitude of the sliding variable as k → ∞ as shown below…”

“…It can be seen that due to the nature of the adaptive tuning (31) and the reaching law (29), the condition (A16) is automatically satisfied.…”

“…Another set of mentionable research are the implicit discretization of sliding mode controller. [29][30][31] The application of implicit discretization of the SMC is proposed by Brogliato and Polyakov 29,30 where the authors have shown with great detail that implicit discretization can help in alleviating chattering as compared to using the explicit discretization. However, in some cases the implicit discretization might end up being computationally expensive.…”

“…Some of the most mentionable works in this are the discrete time differentiators 27,28 designed for systems where traditional numerical differentiation methods fail to provide the desired accuracy and smoothness in the results. Another set of mentionable research are the implicit discretization of sliding mode controller 29–31 . The application of implicit discretization of the SMC is proposed by Brogliato and Polyakov 29,30 where the authors have shown with great detail that implicit discretization can help in alleviating chattering as compared to using the explicit discretization.…”

A Position Controller for Hydraulic Excavators With Deadtime and Regenerative Pipelines

“…For the use of the model in control and simulation, the function needs to be transformed into some different forms. First, as detailed in [27], we need to have the inverse function of the function with respect to its second argument, which Fig. 3.…”