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

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This chapter presents a set of experimental results concerning the sliding mode control of an electropneumatic system. The controller is implemented via a micro-processor 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 (i.e., 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 behavior if one does not take care of the saturation width.

Chapter Contents:

• Abstract
• 3.2.1 Introduction
• 3.2.1.1 Explicit versus implicit discrete sliding mode control
• 3.2.2 Dynamics of the plant and controllers
• 3.2.2.1 Implicit controller design
• 3.2.2.2 Plant dynamics and controllers
• 3.2.3 Experimental results
• 3.2.3.1 Comparison of the tracking errors e
• 3.2.3.2 Comparison of control inputs u (3.2.21) and v (3.2.24), (3.2.25), and (3.2.27)
• 3.2.3.3 Summary
• 3.2.4 Numerical analysis of the saturation controller
• 3.2.5 Conclusion
• Appendix 1 Some basic convex analysis tools
• Acknowledgments
• References

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Experimental results on implicit and explicit time-discretization of equivalent control-based sliding mode control, Page 1 of 2

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