Experimental results on implicit and explicit time-discretization of equivalent control-based sliding mode control

Abstract

This chapter presents a set of experimental results concerning the sliding mode control of an electropneumatic system. The controller is implemented via a microprocessor as a discrete-time input. Three discrete-time control strategies are considered for the implementation of the discontinuous part of the sliding mode controller: explicit discretizations with and without saturation, and an implicit discretization (that is very easy to implement as a projection on the interval [−1, 1]). 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 work 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. On the contrary the explicit controller shows obvious chattering for all sampling times, its magnitude increases as the controller gain increases, and it does not reduce when the sampling period augments. The tracking errors are comparable for both methods, though the implicit method keeps the precision when the control gain increases, which is not the case for the explicit one. Introducing a saturation in the explicit controller does not allow to significantly improve the explicit controller behaviour if one does not take care of the saturation width.

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

“…In words, the input obtained from the implicit scheme (61) compensates for the disturbance with a delay of one step once the discrete-time sliding phase has been reached. Moreover, it is worth to notice that in the discrete-time sliding phase the input u sv k is independent of the gain γ, a crucial fact that is experimentally verified in [25,26]. This last property becomes fundamental in the application of the control scheme (61) since it helps to drastically reduce the chattering effect of the closed-loop system.…”

“…It is a well known fact that the Yosida approximation is a Lipschitz continuous function with constant 1/µ. Hence, it follows that there exists one solution to (26) in [0, T ) for some T > 0. Next, using a Lyapunov analysis we show that the solution of (26) exists for all time t > 0.…”

“…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.…”

“…Notice that, when we regularize the control input in the conventional way there is no selection procedure which at the end is translated into the appearance of chattering in the system. Numerical chattering (i.e., the chattering due to the time-discretization) is known to be intrinsic to explicit discretizations [21,22,26]. Figure 4: Time evolution of the control input u = u nom − Kσ − γσ/( σ + 0.001) and the corresponding system trajectories and sliding variable with a sampling step h = 5ms.…”