have been tested on robotic systems. The super-twisting observer and sliding mode observers have been validated in silico on pancreas and diabetes control, and glucose regulation with insulin infusion in [140, 141, 142]. A parabolic differentiator is implemented on a direct current servomotor equipped with an optical encoder in [106]. A set-valued anti-sway controller is applied on an overhead crane in [143]. A super-twisting observer is applied to a magneto-rheological clutch in [144, 145], while a super-twisting controller semi-implicitly discretized is applied to an electromechanical system with time-delay in [146]. Various controllers (super-twisting, first-order) are applied on a floating wind turbine in [147, 148, 149], with extensive comparisons with other types of controllers. The super-twisting controller is applied to functional electrical stimulation of index finger's metacarpophalangeal in [150]. A first-order sliding-mode set-valued input is designed for the stabilization of a nonsmooth model of an hydraulic actuator in [36] and applied to an industrial commercial excavator [151, 152, 153]. A toolbox for automatic gain-tuning of higher-order differentiators implemented with the backward-Euler method as those introduced in section 4.2 is proposed in [154, 84] (see also [155] for a toolbox dedicated to differentiators with the forward-Euler discretization).

## 7.2 Typical Experimental Results

Let us present a few experimental results which have been obtained in [120, 156, 21, 33, 130, 157, 131] on two setups: an electropneumatic system, and an inverted pendulum. Two controllers are tested: the classical first-order SMC and the twisting algorithm. They nicely illustrate some important properties of backward-Euler schemes. The aim is to illustrate the main specific features without presenting details which are available in the cited articles. The first property is digital chattering alleviation and magnitude decrease in the input, see Fig. 8 for first-order SMC applied to the electropneumatic setup [33] (see [158] for a video showing the setup). The explicit controller (Fig. 8 top) switches between the two extremes of the sign function at each sampling step: this is a high-frequency bang-bang signal. By contrast, the values of the implicit controller are selected from the graph of the sign set-valued map at each sampling step. Consequently, when the sliding surface has been reached, the amplitude of the explicit controller depends on the feedback gain, whereas the amplitude of the implicit controller is independent of such gain and it only depends on the disturbance (in this case a square force signal, see Fig. 8 bottom): we are therefore exactly in a situation where (98) (or (77) for the passivity-based first-order SMC) hold. Fig. 9 and 10 depict experiments on the same electropneumatic system, using the twisting algorithm for a trajectory tracking task [120, 130, 102]. It is seen in Fig. 9 that the explicit implementation performance deteriorates a lot when $h$ increases, while the implicit one keeps

a very good level of performance. In Fig. 10 the gain insensitivity property (as a consequence of (21) and (98)) is shown for three different sampling times $h$, and a large range of control gains (equivalent to $K$ in (19)). Very similar results on gain insensitivity are obtained in [21] (first-order SMC and inverted pendulum) and [33] (first-order SMC and electropneumatic system). Fig. 11 and 12 show the evolution of the system's output for two different sampling times (first-order SMC, electropneumatic system [33]). Explicit and implicit controllers produce similar outputs for very small $h$ (Fig. 11 and 12, top figures). However it is striking to notice that although the explicit method displays large chattering for $h = 15$ms, the implicit one remains with very low chattering for both $h = 2$ms and $h = 15$ms. This confirms the results depicted in Fig. 9. This also demonstrates that both methods yield similar outputs for very small $h$, however the inputs are still quite different one from each other (they always look like in Fig. 8). Quite similar results are obtained with a first-order SMC applied to a six-component thrust generator installed on the head of a floating wind turbine [148, Fig. 7-10], where codimension $n \ge 2$ sliding surfaces are considered.

## 8 CONCLUSIONS AND PERSPECTIVES

The implicit (or, backward-Euler) discretization of set-valued sliding mode systems has received significant attention during the last decade, since it was shown in [31] that it applies to systems with unknown matched disturbances and is an efficient way to significantly alleviate the digital chattering at both the output and the input. This article aims to show the close link between implicit/semi-implicit discretization of set-valued sliding-mode control and differentiation, and optimization. In particular, proximal-point algorithms using the resolvents of maximal monotone operators are shown to be at the core of the analysis and of the implementation of implicit methods. The crucial role played by passivity in the design of implicit discretization of first-order sliding-mode algorithms is highlighted. Robust, time-varying, Hamiltonian and higher-order versions of the proximal-point algorithm are introduced. Computational issues, which are crucial for real-time implementations, are analyzed. Splitting and relaxation algorithms, which provide systematic ways to solve the generalized equations associated with the one-step problems, are reviewed. Some numerical experiments illustrate the presentation.

It is the authors' belief that the analysis of this class of algorithms is still in its infancy, but existing results and experiments show its great potential. The implicit discretizations of infinite-dimensional systems (maximal monotonicity being well suited to the study of some classes of partial differential equations), output feedback control, fixed-time and prescribed-time convergent systems, are still largely open problems. The problem's complexity (which can stem from the sliding surface codi-
<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /><figcaption>Figure 8: Typical control inputs for explicit (above) and implicit (below) first-order SMC with same gain and sampling period [156, Figures 4, 5], see also [33, 157]. See [158] for further illustration of chattering alleviation.</figcaption></figure>

mension, or from the controller/differentiator's structure) also deserves future studies. Finally, it seems that there exists a gap between monotonicity (extensively used in this article) and homogeneity (extensively used recently in continuous-time sliding-mode systems design). Analyzing deeply the link between both deserves future attention as well. The choice of solvers for the one-step generalized equations also represents a source of investigations for real-time implementation.

# 9 ACKNOWLEDGMENT

The work of the second author, performed during his sab-
batical leave from Cinvestav-IPN, Mexico, was supported
in part by Inria and Laboratoire Jean Kuntzman, Greno-
ble.

# References

[1] J. André and P. Seibert. Über stückweise lineare Differentialgleichungen, die bei Regelungsproble-

<figure><img src="image_15.png" /><figcaption>Figure 9: Twisting controller: Evolution of the average tracking error with respect to the sampling time for both implicit and explicit discretizations [120, Figure 4.3].</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Figure 10: Twisting controller: Evolution of the average tracking error and the implicit control magnitude when the gain varies, for three sampling periods [120, Figure 4.8] (Gain= $10^5$ in Fig. 9).</figcaption></figure>

men auftreten I. *Archiv der Mathematik*, 7:148–156, 1956.

[2] J. André and P. Seibert. Über stückweise lineare Differentialgleichungen, die bei Regelungsproblemen auftreten II. *Archiv der Mathematik*, 7:157–164, 1956.

[3] Y. Z. Tsypkin. *Theory of Relay Control Systems*. Moscow: Gostechizdat, 1955.

[4] A. Aizerman and F. R. Gantmakher. On certain switching specifics in non-linear automatic control systems with a piecewise-smooth characteristics of nonlinear element. *Automatika i Telemekhanika*, 18:1017–1028, 1957.

[5] J. J. Slotine, J. K. Hedrick, and E. A. Misawa. On sliding observers for nonlinear systems. *Journal of Dynamic Systems, Measurement, and Control*, 109(3):245–252, 1987.

[6] A. Levant. Robust exact differentiation via sliding mode technique. *Automatica*, 34(3):379–384, 1998.

[7] A. Levant. Higher-order sliding modes, differentiation and output-feedback control. *International Journal of Control*, 76(9-10):924–941, 2003.
<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /><figcaption>Figure 8: Typical control inputs for explicit (above) and implicit (below) first-order SMC with same gain and sampling period [156, Figures 4, 5], see also [33, 157]. See [158] for further illustration of chattering alleviation.</figcaption></figure>

mension, or from the controller/differentiator's structure) also deserves future studies. Finally, it seems that there exists a gap between monotonicity (extensively used in this article) and homogeneity (extensively used recently in continuous-time sliding-mode systems design). Analyzing deeply the link between both deserves future attention as well. The choice of solvers for the one-step generalized equations also represents a source of investigations for real-time implementation.

# 9 ACKNOWLEDGMENT

The work of the second author, performed during his sab-
batical leave from Cinvestav-IPN, Mexico, was supported
in part by Inria and Laboratoire Jean Kuntzman, Greno-
ble.

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<figure><img src="image_13.png" /><figcaption>Figure 9: Twisting controller: Evolution of the average tracking error with respect to the sampling time for both implicit and explicit discretizations [120, Figure 4.3].</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Figure 10: Twisting controller: Evolution of the average tracking error and the implicit control magnitude when the gain varies, for three sampling periods [120, Figure 4.8] (Gain= $10^5$ in Fig. 9).</figcaption></figure>

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<figure><img src="image_17.png" /></figure>

<figure><img src="image_18.png" /><figcaption>Figure 11: Explicit method: $h = 2$ms (top), $h = 15$ms (bottom), constant control gains [156, Figures 2, 3] and [33]. The signals $y$ and $y_{\text{ref}}$ are the position output and the desired output of the electropneumatic setup.</figcaption></figure>

<figure><img src="image_19.png" /></figure>

<figure><img src="image_20.png" /><figcaption>Figure 12: Implicit method: $h = 2$ms (top), $h = 15$ms (bottom), constant control gains [156, Figures 2, 3] and [33, 131]. The signals $y$ and $y_{\text{ref}}$ are the position output and the desired output of the electropneumatic setup.</figcaption></figure>

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<figure><img src="image_15.png" /></figure>

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<figure><img src="image_17.png" /></figure>

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