# When proximal-point algorithms meet set-valued systems. An Optimization point of view of discrete-time sliding modes

F Élix Miranda-Villatoro, Fernando Casta, Bernard Brogliato

► To cite this version:

F Élix Miranda-Villatoro, Fernando Casta, Bernard Brogliato. When proximal-point algorithms meet set-valued systems. An Optimization point of view of discrete-time sliding modes. 2024. hal-04626631

HAL Id: hal-04626631

https://inria.hal.science/hal-04626631

Preprint submitted on 27 Jun 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# An Optimization point of view of discrete-time sliding modes: When proximal-point algorithms meet set-valued systems.

Félix Miranda-Villatoro, Fernando Castanos, Bernard Brogliato

► To cite this version:

Félix Miranda-Villatoro, Fernando Castanos, Bernard Brogliato. An Optimization point of view of discrete-time sliding modes: When proximal-point algorithms meet set-valued systems.. IEEE Control Systems, 2025, 45 (5), pp.47-94. 10.1109/MCS.2025.3587510. hal-04626631v2

HAL Id: hal-04626631

https://inria.hal.science/hal-04626631v2

Submitted on 27 Jun 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# When proximal-point algorithms meet set-valued systems. An Optimization point of view of discrete-time sliding modes

FÉLIX MIRANDA-VILLATORO†, FERNANDO CASTAÑOS*, AND BERNARD BROGLIATO†

June 26, 2024

Sliding-mode control is a major field of Automatic Control that originated in Germany [1, 2] and in the former Soviet Union [3, 4], about 70 years ago during the development of relay systems. Sliding-mode principles were later incorporated in the design of differentiators and observers [5, 6, 7]. Higher-order sliding modes were introduced by Levant in [6, 7, 8, 9, 10]. They represented a significant progress for control and differentiation, the prominent algorithm being the super-twisting scheme. Overviews of sliding-mode control and its applications can be found in [11, 12, 13, 14, 15, 16]. Mathematical foundations for sliding-modes were set mainly by Filippov [17, 18] within the framework of set-valued analysis and differential inclusions. Though other mathematical settings exist (maximal monotone operators being an important one as shown in this article), Filippov's differential inclusions remain the most popular mathematical concept in the automatic control field. Later, discrete-time implementations were studied, see "Sidebar II" for a short historical summary of discrete-time sliding-modes.

The *chattering* phenomenon has long been recognized as a major obstacle to the implementation of set-valued sliding-mode systems, hampering a wider dissemination in applications. It refers to high-frequency bang-bang-like input signals which harm actuators and possibly yield closed-loop instability [19, 20, 21], as well as high-frequency oscillations of the output signal. The time-discretization of set-valued sliding-mode controllers is a major source of chattering when *explicit* (or, *forward* Euler) methods are used [22, 23, 24, 25, 26, 27, 28, 29]. This has motivated the introduction of a new discretization approach, broadly known as the *implicit* (or, the *backward* Euler) method [30, 31, 32].

The main objective of this article is to show that the implicit and semi-implicit methods for the discrete-time implementation of set-valued sliding-mode controllers, observers, and differentiators are tightly related to the subject of optimization, and that this close relationship allows us to consider the implicit/semi-implicit algorithms from a unified point of view. Within this setting, tools from convex analysis, complementarity theory, proximal-point algorithms and proximal operators, maximal monotone operators, and variational inequalities are crucial.

This was alluded to in [31, 21, 30, 33, 34, 35, 36, 37, 32, 38, 39, 40, 41], and especially in [42, 43, 44], where *resolvents* and *Yosida approximations* were used for the first time in this context. When implicitly or semi-implicitly discretized, sliding-mode controllers and differentiators yield new kinds of proximal-point algorithms, named *robust*, *time-varying*, *higher-order* and *Hamiltonian* proximal-point, all of which belong to a class of nonlinear difference equations.

Control systems are mainly designed in a continuous-time setting, their discretization being part of a subsequent step. When the discrete-time control is designed to emulate the continuous-time control — a procedure called *emulation* — it is usually expected that the continuous-time closed-loop properties (robustness, stability, passivity, etc) will be preserved if the sampling time is small enough [45, Chapter 3]. It is well-known, however, that the discretization process can deteriorate the performance and modify the system's properties (zeroes [46], passivity [47], etc).

Systems with sliding modes are particularly susceptible to the employed discretization method, some aspects of which are summarized in the flowchart of Figure 1. In fact, it has been verified analytically and experimentally that explicit emulation applied to sliding-mode set-valued systems yields ill-posed discrete-time systems that fail to approximate the set-valued control input, no matter how small the sampling time is.

This tutorial paper is organized in three main parts: the first part is dedicated to first-order sliding-mode controllers, the second part focuses on higher-order sliding-mode systems (controllers, differentiators, state observers), the third part discusses computational issues and presents experimental results. The article's main objectives are: 1) to shed new light on first-order schemes, both in continuous and discrete-time settings, using passivity as a guideline, 2) to show that proximal-point algorithms can be used as a unifying framework for the analysis and implementation of backward-Euler schemes applied to first-order and higher-order schemes, 3) to provide preliminary ideas on computational issues associated with the implementation of backward-Euler-based schemes (controllers, observers, differentiators), 4) to sur-

<footnote>†Univ. Grenoble Alpes, Inria, CNRS, LJK, Grenoble INP, Grenoble, 38000 France.</footnote>

<footnote>* Automatic Control Department, Cinvestav-IPN, 2508 Av. IPN, 07360, Mexico City, Mexico.</footnote>
# When proximal-point algorithms meet set-valued systems. An Optimization point of view of discrete-time sliding modes

FÉLIX MIRANDA-VILLATORO†, FERNANDO CASTAÑOS*, AND BERNARD BROGLIATO†

June 26, 2024

Sliding-mode control is a major field of Automatic Control that originated in Germany [1, 2] and in the former Soviet Union [3, 4], about 70 years ago during the development of relay systems. Sliding-mode principles were later incorporated in the design of differentiators and observers [5, 6, 7]. Higher-order sliding modes were introduced by Levant in [6, 7, 8, 9, 10]. They represented a significant progress for control and differentiation, the prominent algorithm being the super-twisting scheme. Overviews of sliding-mode control and its applications can be found in [11, 12, 13, 14, 15, 16]. Mathematical foundations for sliding-modes were set mainly by Filippov [17, 18] within the framework of set-valued analysis and differential inclusions. Though other mathematical settings exist (maximal monotone operators being an important one as shown in this article), Filippov's differential inclusions remain the most popular mathematical concept in the automatic control field. Later, discrete-time implementations were studied, see "Sidebar II" for a short historical summary of discrete-time sliding-modes.

The *chattering* phenomenon has long been recognized as a major obstacle to the implementation of set-valued sliding-mode systems, hampering a wider dissemination in applications. It refers to high-frequency bang-bang-like input signals which harm actuators and possibly yield closed-loop instability [19, 20, 21], as well as high-frequency oscillations of the output signal. The time-discretization of set-valued sliding-mode controllers is a major source of chattering when *explicit* (or, *forward* Euler) methods are used [22, 23, 24, 25, 26, 27, 28, 29]. This has motivated the introduction of a new discretization approach, broadly known as the *implicit* (or, the *backward* Euler) method [30, 31, 32].

The main objective of this article is to show that the implicit and semi-implicit methods for the discrete-time implementation of set-valued sliding-mode controllers, observers, and differentiators are tightly related to the subject of optimization, and that this close relationship allows us to consider the implicit/semi-implicit algorithms from a unified point of view. Within this setting, tools from convex analysis, complementarity theory, proximal-point algorithms and proximal operators, maximal monotone operators, and variational inequalities are crucial.

This was alluded to in [31, 21, 30, 33, 34, 35, 36, 37, 32, 38, 39, 40, 41], and especially in [42, 43, 44], where *resolvents* and *Yosida approximations* were used for the first time in this context. When implicitly or semi-implicitly discretized, sliding-mode controllers and differentiators yield new kinds of proximal-point algorithms, named *robust*, *time-varying*, *higher-order* and *Hamiltonian* proximal-point, all of which belong to a class of nonlinear difference equations.

Control systems are mainly designed in a continuous-time setting, their discretization being part of a subsequent step. When the discrete-time control is designed to emulate the continuous-time control — a procedure called *emulation* — it is usually expected that the continuous-time closed-loop properties (robustness, stability, passivity, etc) will be preserved if the sampling time is small enough [45, Chapter 3]. It is well-known, however, that the discretization process can deteriorate the performance and modify the system's properties (zeroes [46], passivity [47], etc).

Systems with sliding modes are particularly susceptible to the employed discretization method, some aspects of which are summarized in the flowchart of Figure 1. In fact, it has been verified analytically and experimentally that explicit emulation applied to sliding-mode set-valued systems yields ill-posed discrete-time systems that fail to approximate the set-valued control input, no matter how small the sampling time is.

This tutorial paper is organized in three main parts: the first part is dedicated to first-order sliding-mode controllers, the second part focuses on higher-order sliding-mode systems (controllers, differentiators, state observers), the third part discusses computational issues and presents experimental results. The article's main objectives are: 1) to shed new light on first-order schemes, both in continuous and discrete-time settings, using passivity as a guideline, 2) to show that proximal-point algorithms can be used as a unifying framework for the analysis and implementation of backward-Euler schemes applied to first-order and higher-order schemes, 3) to provide preliminary ideas on computational issues associated with the implementation of backward-Euler-based schemes (controllers, observers, differentiators), 4) to sur-

<footnote>†Univ. Grenoble Alpes, Inria, CNRS, LJK, Grenoble INP, Grenoble, 38000 France.</footnote>

<footnote>* Automatic Control Department, Cinvestav-IPN, 2508 Av. IPN, 07360, Mexico City, Mexico.</footnote>
Systems with sliding modes are particularly susceptible to the employed discretization method.

<figure><img src="image_2.png" /><figcaption>Figure 1: Flowchart of the implementation of control systems using sliding modes. From the point of view of traditional sliding-mode control (SMC), a discontinuous feedback law is enforced on an ordinary differential equation (ODE). A consistent notion of solution requires the conversion of the closed-loop discontinuous ODE into a differential inclusion (Filippov’s convexification). Alternatively, a set-valued feedback law can be directly formulated from the outset, yielding again a differential inclusion from which a continuous selection is uniquely chosen (thick blue line). The result is a well-posed ODE with a stable desired equilibrium. Some practitioners bypass Filippov’s convexification by replacing the discontinuous law with an arbitrary approximation (typically a saturation). The outcome may be the destabilization of the desired equilibrium. A backward-Euler discretization of the sliding-mode controller results in a difference inclusion from which, in analogy with the continuous-time case, a selection can be chosen to form a well-posed difference equation with a stable desired equilibrium (thick blue line). The formulation of the difference inclusion together with its solution techniques is usually referred to as the implicit (or backward-Euler) method.</figcaption></figure>

vey the results (theoretical and experimental) that have been obtained so far concerning the backward-Euler discretization of set-valued sliding-mode systems.

# 1 SUBGRADIENT FLOWS, PROXIMAL ALGORITHMS, FINITE-TIME CONVERGENCE

The intimate relation between optimization and the implicit discretization of set-valued sliding-mode systems is illustrated in this section (see “Sidebar III” for a definition of sliding motion). For the sake of simplicity, the presentation is restricted to the scalar case without perturbation. The disturbed case is analysed in the following section in a more general framework. Consider a scalar

continuous-time dynamical system

$$ \dot{x}(t) = ax(t) + bu(t) \quad (1) $$

with $a < 0$ and $0 < b$ constant and fixed. Suppose that the input is a feedback law $u(t) = u_{sv}(x(t))$. The set-valued feedback

$$ -u_{sv}(x) \in \kappa \operatorname{sgn}(x), \quad \kappa > 0, \quad (2) $$

where $\operatorname{sgn}(\cdot)$ is the set-valued signum function (30), achieves the finite-time convergence of the state towards the origin [51, 52]. Indeed, the closed-loop system belongs to the class of gradient systems

$$ \dot{x}(t) \in -\partial f(x(t)) \quad (3) $$

with $f(x) = -\frac{a}{2}x^2 + b\kappa|x|$. According to Proposition 2 in “Sidebar III”, the origin is finite-time stable. Due to the
Systems with sliding modes are particularly susceptible to the employed discretization method.

<figure><img src="image_2.png" /><figcaption>Figure 1: Flowchart of the implementation of control systems using sliding modes. From the point of view of traditional sliding-mode control (SMC), a discontinuous feedback law is enforced on an ordinary differential equation (ODE). A consistent notion of solution requires the conversion of the closed-loop discontinuous ODE into a differential inclusion (Filippov’s convexification). Alternatively, a set-valued feedback law can be directly formulated from the outset, yielding again a differential inclusion from which a continuous selection is uniquely chosen (thick blue line). The result is a well-posed ODE with a stable desired equilibrium. Some practitioners bypass Filippov’s convexification by replacing the discontinuous law with an arbitrary approximation (typically a saturation). The outcome may be the destabilization of the desired equilibrium. A backward-Euler discretization of the sliding-mode controller results in a difference inclusion from which, in analogy with the continuous-time case, a selection can be chosen to form a well-posed difference equation with a stable desired equilibrium (thick blue line). The formulation of the difference inclusion together with its solution techniques is usually referred to as the implicit (or backward-Euler) method.</figcaption></figure>

vey the results (theoretical and experimental) that have been obtained so far concerning the backward-Euler discretization of set-valued sliding-mode systems.

# 1 SUBGRADIENT FLOWS, PROXIMAL ALGORITHMS, FINITE-TIME CONVERGENCE

The intimate relation between optimization and the implicit discretization of set-valued sliding-mode systems is illustrated in this section (see “Sidebar III” for a definition of sliding motion). For the sake of simplicity, the presentation is restricted to the scalar case without perturbation. The disturbed case is analysed in the following section in a more general framework. Consider a scalar

continuous-time dynamical system

$$ \dot{x}(t) = ax(t) + bu(t) \quad (1) $$

with $a < 0$ and $0 < b$ constant and fixed. Suppose that the input is a feedback law $u(t) = u_{sv}(x(t))$. The set-valued feedback

$$ -u_{sv}(x) \in \kappa \operatorname{sgn}(x), \quad \kappa > 0, \quad (2) $$

where $\operatorname{sgn}(\cdot)$ is the set-valued signum function (30), achieves the finite-time convergence of the state towards the origin [51, 52]. Indeed, the closed-loop system belongs to the class of gradient systems

$$ \dot{x}(t) \in -\partial f(x(t)) \quad (3) $$

with $f(x) = -\frac{a}{2}x^2 + b\kappa|x|$. According to Proposition 2 in “Sidebar III”, the origin is finite-time stable. Due to the
This article is largely concerned with the relationships between the implicit (backward) discretization of set-valued sliding-mode systems (for control, state observation, or differentiation), and proximal-point algorithms of optimization. It is shown that using well-known notions like maximal monotonicity, proximal and resolvent operators, and Yosida approximations, it is possible to embed implicit time-discretization methods into the unified framework of proximal-point algorithms. This is illustrated with first-order and higher-order sliding-mode systems. Passivity is shown to play an important role. Extensions of the classical proximal-point algorithm are introduced. Particular attention is given to the use of splitting methods to resolve computational issues related to implementation.

computational power and flexibility of digital electronic devices, it is now common to consider the discrete-time version of the control law (2) for its implementation in a digital computer. To that end, let us consider the exact discrete-time dynamics associated with (1). That is, the exact solution of (1) evaluated at discrete times $t_k = kh$, where $k \in \mathbb{N}$ and $h > 0$, denotes the sampling time. Indeed, by the variation of constants formula it follows that

$$x(t) = e^{a(t-t_0)}x(t_0) + \int_{t_0}^{t} e^{a(t-\tau)}bu(\tau)d\tau. \quad (4)$$

Thus, assuming that the control input $u(t)$ remains constant during the intervals $[t_k, t_{k+1})$, it follows that

$$x_{k+1} = \tilde{a}x_k + \tilde{b}u_k, \quad (5)$$

where $x(t_k) = x_k$, $u_k = u(t_k)$, $\tilde{a} = e^{ah}$, and $\tilde{b} = \frac{(e^{ah}-1)b}{a}$. Let us consider first the explicit (forward-Euler) discretization of (2), that is, $u_k = u_{sv}(x_k)$, where

$$-u_{sv}(x_k) \in \kappa \operatorname{sgn}(x_k). \quad (6)$$

The closed-loop discrete-time dynamics becomes

$$x_{k+1} \in \tilde{a}x_k - \tilde{b}\kappa \operatorname{sgn}(x_k). \quad (7)$$

It is worth emphasizing that, at $x_k = 0$, the difference inclusion (7) does not have a unique solution, as the right-hand side becomes a non-singleton. Moreover, with such feedback the origin is not finite-time stable, no matter how small $h$ is. Indeed, the origin becomes unstable and an asymptotically stable limit-cycle appears, creating numerical (or discretization, or digital) chattering. The amplitude of the limit cycle is proportional to the control gain $\kappa$ and to $h$ through $\tilde{b}$: if $h$ decreases to zero, then the amplitude of the limit cycle also decreases and, in the limit, the oscillation disappears. On the other hand, if $h$ increases then $\tilde{b}$ increases and the amplitude increases. This stands in sharp contrast with the implicit method described below, which is relatively insensitive to an increase in the sampling time (see Figures 2 and 3). The existence of limit cycles in explicitly discretized sliding-mode systems has been deeply analyzed in [22, 23, 24, 25, 26, 27, 28, 29, 53]. It is inferred that

digital chattering is intrinsically linked to the explicit discretization (not to be confused with the explicit forms of the implicit (backward-Euler) discretization, as detailed below).

Let us now consider the backward-Euler discretization of (2),

$$-u_{sv}(x_k) \in \kappa \operatorname{sgn}(x_{k+1}), \quad (8)$$

which results in

$$x_{k+1} \in \tilde{a}x_k - \tilde{b}\kappa \operatorname{sgn}(x_{k+1}). \quad (9)$$

In this case the closed-loop is well-posed, as the state $x_{k+1}$ depends only on data available at time $t_k = kh$, (i.e., $\tilde{a}$, $\tilde{b}$, $\kappa$, and $x_k$). Indeed, (9) is equivalent to the following generalized equation in the unknown $x_{k+1}$:

$$\tilde{a}x_k \in (I_d + \tilde{b}\kappa \operatorname{sgn})(x_{k+1}). \quad (10)$$

where $I_d$ is the identity mapping. In view of the maximal monotonicity of the sign multifunction (see “Sidebar V”), the generalized equation (10) has a unique solution

$$x_{k+1} = \operatorname{Prox}_{\tilde{b}\kappa g}(\tilde{a}x_k), \quad (11)$$

where $\operatorname{Prox}_f$ denotes the proximal mapping of the function $f$ and $g(\cdot) = |\cdot|$ (see “Sidebar VII” for more details). In this case,

$$\begin{aligned} \operatorname{Prox}_{\tilde{b}\kappa g}(\tilde{a}x_k) &= \tilde{a}x_k - \tilde{b}\kappa \operatorname{sgn}(x_k) \min \left\{ \frac{\tilde{a}}{\tilde{b}\kappa} |x_k|, 1 \right\} \\ &= \begin{cases} 0 & \text{if } |x_k| \le \frac{\tilde{b}\kappa}{\tilde{a}} \\ \tilde{a}x_k - \tilde{b}\kappa \operatorname{sgn}(x_k) & \text{otherwise} \end{cases}. \end{aligned} \quad (12)$$

The iteration (11) is an instance of the proximal-point algorithm (see Definition 12 in “Sidebar VII”), vastly studied in the literature of Convex Optimization [54] and recognized as an effective discretization of the subgradient system (3) [55]. It is an explicit form of the algorithm implicitly defined by (9) and falls within the context of Krasnosel’skiï-Mann iterations [56]. Thus, its convergence is guaranteed by Krasnosel’skiï-Mann Theorem [56] for any proper, convex, lower semicontinuous (LSC) function $g$, whenever the set of minimizers of $g$ is nonempty [56, Theorem 1.1]. Moreover, since $g(\cdot) = |\cdot|$ implies $0 \in \partial g(0)$,
## Sidebar I: Summary

This article is largely concerned with the relationships between the implicit (backward) discretization of set-valued sliding-mode systems (for control, state observation, or differentiation), and proximal-point algorithms of optimization. It is shown that using well-known notions like maximal monotonicity, proximal and resolvent operators, and Yosida approximations, it is possible to embed implicit time-discretization methods into the unified framework of proximal-point algorithms. This is illustrated with first-order and higher-order sliding-mode systems. Passivity is shown to play an important role. Extensions of the classical proximal-point algorithm are introduced. Particular attention is given to the use of splitting methods to resolve computational issues related to implementation.

computational power and flexibility of digital electronic devices, it is now common to consider the discrete-time version of the control law (2) for its implementation in a digital computer. To that end, let us consider the exact discrete-time dynamics associated with (1). That is, the exact solution of (1) evaluated at discrete times $t_k = kh$, where $k \in \mathbb{N}$ and $h > 0$, denotes the sampling time. Indeed, by the variation of constants formula it follows that

$$x(t) = e^{a(t-t_0)}x(t_0) + \int_{t_0}^{t} e^{a(t-\tau)}bu(\tau)d\tau. \quad (4)$$

Thus, assuming that the control input $u(t)$ remains constant during the intervals $[t_k, t_{k+1})$, it follows that

$$x_{k+1} = \tilde{a}x_k + \tilde{b}u_k, \quad (5)$$

where $x(t_k) = x_k$, $u_k = u(t_k)$, $\tilde{a} = e^{ah}$, and $\tilde{b} = \frac{(e^{ah}-1)b}{a}$. Let us consider first the explicit (forward-Euler) discretization of (2), that is, $u_k = u_{sv}(x_k)$, where

$$-u_{sv}(x_k) \in \kappa \mathbf{sgn}(x_k). \quad (6)$$

The closed-loop discrete-time dynamics becomes

$$x_{k+1} \in \tilde{a}x_k - \tilde{b}\kappa \mathbf{sgn}(x_k). \quad (7)$$

It is worth emphasizing that, at $x_k = 0$, the difference inclusion (7) does not have a unique solution, as the right-hand side becomes a non-singleton. Moreover, with such feedback the origin is not finite-time stable, no matter how small $h$ is. Indeed, the origin becomes unstable and an asymptotically stable limit-cycle appears, creating numerical (or discretization, or digital) chattering. The amplitude of the limit cycle is proportional to the control gain $\kappa$ and to $h$ through $\tilde{b}$: if $h$ decreases to zero, then the amplitude of the limit cycle also decreases and, in the limit, the oscillation disappears. On the other hand, if $h$ increases then $\tilde{b}$ increases and the amplitude increases. This stands in sharp contrast with the implicit method described below, which is relatively insensitive to an increase in the sampling time (see Figures 2 and 3). The existence of limit cycles in explicitly discretized sliding-mode systems has been deeply analyzed in [22, 23, 24, 25, 26, 27, 28, 29, 53]. It is inferred that

digital chattering is intrinsically linked to the explicit discretization (not to be confused with the explicit forms of the implicit (backward-Euler) discretization, as detailed below).

Let us now consider the backward-Euler discretization of (2),

$$-u_{sv}(x_k) \in \kappa \mathbf{sgn}(x_{k+1}), \quad (8)$$

which results in

$$x_{k+1} \in \tilde{a}x_k - \tilde{b}\kappa \mathbf{sgn}(x_{k+1}). \quad (9)$$

In this case the closed-loop is well-posed, as the state $x_{k+1}$ depends only on data available at time $t_k = kh$, (i.e., $\tilde{a}$, $\tilde{b}$, $\kappa$, and $x_k$). Indeed, (9) is equivalent to the following generalized equation in the unknown $x_{k+1}$:

$$\tilde{a}x_k \in (I_d + \tilde{b}\kappa \mathbf{sgn})(x_{k+1}). \quad (10)$$

where $I_d$ is the identity mapping. In view of the maximal monotonicity of the sign multifunction (see “Sidebar V”), the generalized equation (10) has a unique solution

$$x_{k+1} = \operatorname{Prox}_{\tilde{b}\kappa g}(\tilde{a}x_k), \quad (11)$$

where $\operatorname{Prox}_f$ denotes the proximal mapping of the function $f$ and $g(\cdot) = |\cdot|$ (see “Sidebar VII” for more details). In this case,

$$\begin{aligned} \operatorname{Prox}_{\tilde{b}\kappa g}(\tilde{a}x_k) &= \tilde{a}x_k - \tilde{b}\kappa \mathbf{sgn}(x_k) \min \left\{ \frac{\tilde{a}}{\tilde{b}\kappa} |x_k|, 1 \right\} \\ &= \begin{cases} 0 & \text{if } |x_k| \le \frac{\tilde{b}\kappa}{\tilde{a}} \\ \tilde{a}x_k - \tilde{b}\kappa \mathbf{sgn}(x_k) & \text{otherwise} \end{cases}. \end{aligned} \quad (12)$$

The iteration (11) is an instance of the proximal-point algorithm (see Definition 12 in “Sidebar VII”), vastly studied in the literature of Convex Optimization [54] and recognized as an effective discretization of the subgradient system (3) [55]. It is an explicit form of the algorithm implicitly defined by (9) and falls within the context of Krasnosel’skiï-Mann iterations [56]. Thus, its convergence is guaranteed by Krasnosel’skiï-Mann Theorem [56] for any proper, convex, lower semicontinuous (LSC) function $g$, whenever the set of minimizers of $g$ is nonempty [56, Theorem 1.1]. Moreover, since $g(\cdot) = |\cdot|$ implies $0 \in \partial g(0)$,
The most widespread discrete-time algorithms for sliding-mode implementation consist of an emulation method coupled to a forward-Euler discretization (see (6)), with or without a saturation function replacing the set-valued signum function. The latter is essentially an engineering trick, and the necessary parameter tuning to find a correct trade-off between sampling time and saturation width is often uneasy. Forward-Euler has long been known to produce digital chattering [S7, S4], thus high-frequency bang-bang-like controllers (see Fig. 8). Such unwanted dynamical behavior has been deeply analyzed in [22]–[29] for various sliding-mode controllers. This has prompted researchers to design other types of discrete-time sliding-mode controllers, which should be designed directly in the discrete-time setting (should they be “pure” discrete-time systems or resulting from the time-discretization of a continuous-time plant). See, e.g., [S9] for a survey.

Let $s_k$ be the discrete-time sliding variable. Imposing $s_{k+1} = 0$ yields a dead-beat input [S2]. This should be avoided in general, see Remark 6. To that end, various strategies have been proposed in the literature. A *quasi-sliding mode* is characterized by a tube defined by an inequality $|s_k| \le \varepsilon$, $\varepsilon > 0$. Designing control laws that render it invariant and attractive in a finite number of steps is a research topic initiated in [S1, S3, S8, S10], and active until recently, see, e.g., [S5, S6, S15]. It mainly consists of setting $s_{k+1} = f(s_k, x_k, k)$ for some $f$, then looking for a controller (a reaching law) guaranteeing it. Various reaching conditions have been stated in the related literature, see [48, S11]. Most of the reaching laws use signum functions [S5, S6, S8, S15] and thus share close similarities with explicit Euler schemes. Another branch of analysis was more recently tackled by Levant [S12]–[S14], [49], concerning the discretization of higher-order and homogeneous sliding-mode schemes, with special attention paid to accuracy of differentiators.

It is noteworthy that the first-order controllers proposed in [S4] and [S7] are backward-Euler algorithms (hence similar to the algorithms described in section “3”), with the assumption that the equivalent controller is known, hence ignoring unknown disturbances. A minimum-operator scheme as in (13c) is proposed in [48] considering unknown disturbances (in the framework of so called reaching laws [S8], minimum-operator controllers merely provide another reaching law). However, no unknown disturbances were considered in a general context for backward (implicit) Euler discretization before [31], hence seriously limiting the applicability of such controllers (the link between minimum-operators and backward-Euler methods was made much later in [50]). The key idea in [31] and subsequent works cited in this article is to keep the set-valued feature of sliding-mode controllers in the discrete-time design, hence showing that digital chattering is not unavoidable (contrarily to some widely spread conclusions led in an explicit, forward-Euler context [S2]). Mathematical tools like generalized equations, variational inequalities, complementarity problems, maximal monotone operator theory, were introduced in digital sliding-mode control design for the first time in [21, 30, 31, 32, 42, 43]. The topic of this article is precisely to give a broader extent to this point of view.

## References

[S1] K. Furuta, Sliding mode control of a discrete system, *Systems and Control Letters*, 14:145-152, 1990.

[S2] S. Hui, S.H. Zak, On discrete-time variable structure sliding mode control, *Systems and Control Letters*, 38:283-288, 1999.

[S3] A. Bartosziewicz, Discrete-time quasi-sliding-mode control strategies, *IEEE Trans. Industrial Electronics*, 45(4):633-637, 1998.

[S4] G. Bartolini, A. Ferrara, V.I. Utkin, Adaptive sliding mode control in discrete-time systems, *Automatica*, 31(5):769-773, 1995.

[S5] S. Chkrabarty, B. Bandyopadhyay, A generalized reaching law for discrete time sliding mode control, *Automatica*, 52:83-86, 2015.

[S6] H. Du, X. Yu, M.Z.Q. Chen, S. Li, Chattering-free discrete-time sliding mode control, *Automatica*, 68:87-91, 2016.

[S7] S. Drakunov, V.I. Utkin, On discrete-times sliding modes, *IFAC Nonlinear Control Systems Design*, Capri, Italy, 273-278, 1989.

[S8] W. Gao, Y. Wang, A. Homaifa, Discrete-time variable structure control systems, *IEEE Trans. Ind. Electron.*, 42(4):117-122, 1995.
The most widespread discrete-time algorithms for sliding-mode implementation consist of an emulation method coupled to a forward-Euler discretization (see (6)), with or without a saturation function replacing the set-valued signum function. The latter is essentially an engineering trick, and the necessary parameter tuning to find a correct trade-off between sampling time and saturation width is often uneasy. Forward-Euler has long been known to produce digital chattering [S7, S4], thus high-frequency bang-bang-like controllers (see Fig. 8). Such unwanted dynamical behavior has been deeply analyzed in [22]–[29] for various sliding-mode controllers. This has prompted researchers to design other types of discrete-time sliding-mode controllers, which should be designed directly in the discrete-time setting (should they be “pure” discrete-time systems or resulting from the time-discretization of a continuous-time plant). See, e.g., [S9] for a survey.

Let $s_k$ be the discrete-time sliding variable. Imposing $s_{k+1} = 0$ yields a dead-beat input [S2]. This should be avoided in general, see Remark 6. To that end, various strategies have been proposed in the literature. A *quasi-sliding mode* is characterized by a tube defined by an inequality $|s_k| \le \varepsilon$, $\varepsilon > 0$. Designing control laws that render it invariant and attractive in a finite number of steps is a research topic initiated in [S1, S3, S8, S10], and active until recently, see, e.g., [S5, S6, S15]. It mainly consists of setting $s_{k+1} = f(s_k, x_k, k)$ for some $f$, then looking for a controller (a reaching law) guaranteeing it. Various reaching conditions have been stated in the related literature, see [48, S11]. Most of the reaching laws use signum functions [S5, S6, S8, S15] and thus share close similarities with explicit Euler schemes. Another branch of analysis was more recently tackled by Levant [S12]–[S14], [49], concerning the discretization of higher-order and homogeneous sliding-mode schemes, with special attention paid to accuracy of differentiators.

It is noteworthy that the first-order controllers proposed in [S4] and [S7] are backward-Euler algorithms (hence similar to the algorithms described in section “3”), with the assumption that the equivalent controller is known, hence ignoring unknown disturbances. A minimum-operator scheme as in (13c) is proposed in [48] considering unknown disturbances (in the framework of so called reaching laws [S8], minimum-operator controllers merely provide another reaching law). However, no unknown disturbances were considered in a general context for backward (implicit) Euler discretization before [31], hence seriously limiting the applicability of such controllers (the link between minimum-operators and backward-Euler methods was made much later in [50]). The key idea in [31] and subsequent works cited in this article is to keep the set-valued feature of sliding-mode controllers in the discrete-time design, hence showing that digital chattering is not unavoidable (contrarily to some widely spread conclusions led in an explicit, forward-Euler context [S2]). Mathematical tools like generalized equations, variational inequalities, complementarity problems, maximal monotone operator theory, were introduced in digital sliding-mode control design for the first time in [21, 30, 31, 32, 42, 43]. The topic of this article is precisely to give a broader extent to this point of view.

## References

[S1] K. Furuta, Sliding mode control of a discrete system, *Systems and Control Letters*, 14:145-152, 1990.

[S2] S. Hui, S.H. Zak, On discrete-time variable structure sliding mode control, *Systems and Control Letters*, 38:283-288, 1999.

[S3] A. Bartosziewicz, Discrete-time quasi-sliding-mode control strategies, *IEEE Trans. Industrial Electronics*, 45(4):633-637, 1998.

[S4] G. Bartolini, A. Ferrara, V.I. Utkin, Adaptive sliding mode control in discrete-time systems, *Automatica*, 31(5):769-773, 1995.

[S5] S. Chkrabarty, B. Bandyopadhyay, A generalized reaching law for discrete time sliding mode control, *Automatica*, 52:83-86, 2015.

[S6] H. Du, X. Yu, M.Z.Q. Chen, S. Li, Chattering-free discrete-time sliding mode control, *Automatica*, 68:87-91, 2016.

[S7] S. Drakunov, V.I. Utkin, On discrete-times sliding modes, *IFAC Nonlinear Control Systems Design*, Capri, Italy, 273-278, 1989.

[S8] W. Gao, Y. Wang, A. Homaifa, Discrete-time variable structure control systems, *IEEE Trans. Ind. Electron.*, 42(4):117-122, 1995.
[S9] K. Furuta, Y. Pan, Discrete-time variable structure control, In: Yu, X., Xu, J.X. (Eds) *Variable Structure Systems: Towards the 21st Century*. Lecture Notes in Control and Information Sciences, vol.274. Springer, Berlin, Heidelberg.

[S10] C. Milosavljevic, General conditions for the existence of a quasi-sliding mode on the switching hyperplane in discrete variable structure systems, *Automatic Remote Control*, 46:307–314, 1985.

[S11] A.J. Koshkouei, A.S.I. Zinober, Sliding mode control of discrete-time systems, *ASME. J. Dyn. Sys., Meas., Control*, 122(4):793–802, 2000.

[S12] M. Livne, A. Levant, Proper discretization of homogeneous differentiators, *Automatica*, 50(8):2007-2014, 2014.

[S13] A. Levant, M. Livne, Weighted homogeneity and robustness of sliding mode control, *Automatica*, 72(10):186-193, 2016.

[S14] A. Levant, M. Livne, D. Lunz, On discretization of high order sliding modes, in J.P. Barbot, L. Fridman, F. Plestan (Eds.) *Recent Trends in Sliding Mode Control*, 177-202, IET, 2016.

[S15] H. Ma, Z. Xiong, Y. Li and Z. Liu, Sliding mode control for uncertain discrete-time systems using an adaptive reaching law, *IEEE Transactions on Circuits and Systems II: Express Briefs*, 68(2):722-726, 2021.

the origin of (9) is globally finite-time stable for any $h > 0$ [57] and, contrary to the explicit discretization (7), it exhibits no oscillations at all. Figure 3 shows the cobweb diagram of (9).

An explicit expression for the selection of the control law in (8) can be computed with the help of (5) as shown in the following proposition.

**Proposition 1.** Consider the input implicitly defined by (8) together with the discrete-time system (5). The following statements are equivalent:

$$-u_{sv}(x_k) \in \kappa \operatorname{sgn}(x_{k+1}). \qquad (13a)$$

$$-u_{sv}(x_k) = \operatorname{Proj} \left( [-\kappa, \kappa]; \frac{\tilde{a}}{\tilde{b}} x_k \right). \qquad (13b)$$

$$-u_{sv}(x_k) = \kappa \operatorname{sgn}(x_k) \min \left\{ \frac{\tilde{a}}{\tilde{b}\kappa} |x_k|, 1 \right\}. \qquad (13c)$$

*Proof.* The proof uses various results presented in “Sidebar VI”, “Sidebar III”, “Sidebar V”, and “Sidebar VII”. It follows from (37) and (5) that (13a) is equivalent to $\tilde{a}x_k + \tilde{b}u_{sv}(x_k) \in \mathbf{N}_{[-\kappa, \kappa]}(-u_{sv}(x_k))$. Since cones are invariant under multiplication by positive scalars, the latter is equivalent to $\frac{\tilde{a}}{\tilde{b}}x_k \in (\mathbf{I}_d + \mathbf{N}_{[-\kappa, \kappa]})(-u_{sv}(x_k))$. Proposition 6 in “Sidebar VI” then establishes the equivalence between (13a) and (13b). The equivalence between (13b)

and (13c) follows from direct computations. Indeed,

$$\begin{aligned} \operatorname{Proj} \left( [-\kappa, \kappa]; \frac{\tilde{a}}{\tilde{b}} x_k \right) &= \begin{cases} \frac{\tilde{a}}{\tilde{b}} x_k & \text{if } \frac{\tilde{a}}{\tilde{b}} |x_k| \le \kappa \\ \kappa \operatorname{sgn}(x_k) & \text{otherwise.} \end{cases} \\ &= \operatorname{sgn}(x_k) \min \left\{ \frac{\tilde{a}}{\tilde{b}} |x_k|, \kappa \right\} \\ &= \kappa \operatorname{sgn}(x_k) \min \left\{ \frac{\tilde{a}}{\tilde{b}\kappa} |x_k|, 1 \right\}. \end{aligned} \qquad (14)$$

<figure><img src="image_3.png" /></figure>

The expressions in (12) and (13) can also be written using complementarity problems and variational inequalities, using equivalent formulations of normal cones (see (31) for the variational formulation, and “Sidebar IX” for the complementarity formulation of a class of nonsmooth systems which are close to the sliding-mode ones). The signum multifunction can also be rewritten using various formalisms, see [58, “Set-Valued Signum Function”]. The $\min\{\cdot, \cdot\}$ function in (13c) and in (12) is a complementarity function (or C-function) [59, 60]. The expression (13b) shows that $u_{sv}(x_k)$ is the solution of the quadratic optimization problem

$$-u_{sv}(x_k) = \arg \min_{z \in [-\kappa, \kappa]} \frac{1}{2} \left\| z - \frac{\tilde{a}}{\tilde{b}} x_k \right\|^2. \qquad (15)$$

This suggests that implicit controllers can be implemented online by solving quadratic programs (QPs) with constraints. It is noteworthy that the explicit form (13c) has been studied in [61, Example 3.1] via Lyapunov methods (see also [48], where the same algorithms were already presented, and [62, 63] for further analysis, where
[S9] K. Furuta, Y. Pan, Discrete-time variable structure control, In: Yu, X., Xu, J.X. (Eds) *Variable Structure Systems: Towards the 21st Century*. Lecture Notes in Control and Information Sciences, vol.274. Springer, Berlin, Heidelberg.

[S10] C. Milosavljevic, General conditions for the existence of a quasi-sliding mode on the switching hyperplane in discrete variable structure systems, *Automatic Remote Control*, 46:307–314, 1985.

[S11] A.J. Koshkouei, A.S.I. Zinober, Sliding mode control of discrete-time systems, *ASME. J. Dyn. Sys., Meas., Control*, 122(4):793–802, 2000.

[S12] M. Livne, A. Levant, Proper discretization of homogeneous differentiators, *Automatica*, 50(8):2007-2014, 2014.

[S13] A. Levant, M. Livne, Weighted homogeneity and robustness of sliding mode control, *Automatica*, 72(10):186-193, 2016.

[S14] A. Levant, M. Livne, D. Lunz, On discretization of high order sliding modes, in J.P. Barbot, L. Fridman, F. Plestan (Eds.) *Recent Trends in Sliding Mode Control*, 177-202, IET, 2016.

[S15] H. Ma, Z. Xiong, Y. Li and Z. Liu, Sliding mode control for uncertain discrete-time systems using an adaptive reaching law, *IEEE Transactions on Circuits and Systems II: Express Briefs*, 68(2):722-726, 2021.

the origin of (9) is globally finite-time stable for any $h > 0$ [57] and, contrary to the explicit discretization (7), it exhibits no oscillations at all. Figure 3 shows the cobweb diagram of (9).

An explicit expression for the selection of the control law in (8) can be computed with the help of (5) as shown in the following proposition.

**Proposition 1.** Consider the input implicitly defined by (8) together with the discrete-time system (5). The following statements are equivalent:

$$-u_{sv}(x_k) \in \kappa \operatorname{sgn}(x_{k+1}). \quad (13a)$$

$$-u_{sv}(x_k) = \operatorname{Proj} \left( [-\kappa, \kappa]; \frac{\tilde{a}}{\tilde{b}} x_k \right). \quad (13b)$$

$$-u_{sv}(x_k) = \kappa \operatorname{sgn}(x_k) \min \left\{ \frac{\tilde{a}}{\tilde{b}\kappa} |x_k|, 1 \right\}. \quad (13c)$$

*Proof.* The proof uses various results presented in “Sidebar VI”, “Sidebar III”, “Sidebar V”, and “Sidebar VII”. It follows from (37) and (5) that (13a) is equivalent to $\tilde{a}x_k + \tilde{b}u_{sv}(x_k) \in \mathbf{N}_{[-\kappa, \kappa]}(-u_{sv}(x_k))$. Since cones are invariant under multiplication by positive scalars, the latter is equivalent to $\frac{\tilde{a}}{\tilde{b}}x_k \in (\mathbf{I}_d + \mathbf{N}_{[-\kappa, \kappa]})(-u_{sv}(x_k))$. Proposition 6 in “Sidebar VI” then establishes the equivalence between (13a) and (13b). The equivalence between (13b)

and (13c) follows from direct computations. Indeed,

$$\begin{aligned} \operatorname{Proj} \left( [-\kappa, \kappa]; \frac{\tilde{a}}{\tilde{b}} x_k \right) &= \begin{cases} \frac{\tilde{a}}{\tilde{b}} x_k & \text{if } \frac{\tilde{a}}{\tilde{b}} |x_k| \le \kappa \\ \kappa \operatorname{sgn}(x_k) & \text{otherwise.} \end{cases} \\ &= \operatorname{sgn}(x_k) \min \left\{ \frac{\tilde{a}}{\tilde{b}} |x_k|, \kappa \right\} \\ &= \kappa \operatorname{sgn}(x_k) \min \left\{ \frac{\tilde{a}}{\tilde{b}\kappa} |x_k|, 1 \right\}. \end{aligned} \quad (14)$$

□

The expressions in (12) and (13) can also be written using complementarity problems and variational inequalities, using equivalent formulations of normal cones (see (31) for the variational formulation, and “Sidebar IX” for the complementarity formulation of a class of nonsmooth systems which are close to the sliding-mode ones). The signum multifunction can also be rewritten using various formalisms, see [58, “Set-Valued Signum Function”]. The $\min\{\cdot, \cdot\}$ function in (13c) and in (12) is a complementarity function (or C-function) [59, 60]. The expression (13b) shows that $u_{sv}(x_k)$ is the solution of the quadratic optimization problem

$$-u_{sv}(x_k) = \arg \min_{z \in [-\kappa, \kappa]} \frac{1}{2} \left\| z - \frac{\tilde{a}}{\tilde{b}} x_k \right\|^2. \quad (15)$$

This suggests that implicit controllers can be implemented online by solving quadratic programs (QPs) with constraints. It is noteworthy that the explicit form (13c) has been studied in [61, Example 3.1] via Lyapunov methods (see also [48], where the same algorithms were already presented, and [62, 63] for further analysis, where
<figure><img src="image_4.png" /><figcaption>Figure 2: Cobweb diagram of (7) with parameters: $\tilde{a} = 0.5$, $\tilde{b}\kappa = 0.75$ for two different initial conditions. The explicit discretization of the set-valued law (2) yields an iteration with a non-monotone map such that, for almost all initial conditions, the closed-loop shows an oscillatory behavior. The amplitude of the oscillation is proportional to the control gain $\kappa$.</figcaption></figure>

the form (13c) is called a minimum-operator-based input), and that the connection with proximal maps via the implicit Euler method was shown in [50]. It is worth to emphasize that, in the limit as $h \downarrow 0$, the trajectories of both schemes, (7) and (9), converge towards trajectories of the continuous time inclusion (1)-(2). However, the control input (13) is well defined at $x = 0$, whereas (6) is not. Thus, even though in the implementation of discrete-time controllers the sampling time $h > 0$ cannot be arbitrarily small because of practical limitations, it is clear that the implicitly defined system (9) is more attractive than the explicit one. Such distinction becomes more important when disturbances are considered.

To close this section, it is also worth to emphasize that arbitrary regularizations of the set-valued sign function (that is, single-valued functions sufficiently close to the graph of the sign multifunction like the saturation or the sigmoid functions) do not guarantee the global finite-time convergence property. Generically, such an approximation either breaks the global finite-time convergence towards zero (making the convergence asymptotic), or the stability of the iteration map, making the origin unstable and producing an oscillatory behavior (see “Sidebar VII”).

The following section extends the ideas presented above to the multivariable case, where non-vanishing external disturbances affecting the dynamics are also considered. Moreover, a control theoretic interpretation for the use of implicitly defined controllers is presented under the general framework of passive systems.

## 2 FIRST-ORDER SLIDING-MODE CONTROL: CONTINUOUS-TIME

In this section we argue that first-order sliding-mode control is a particular case of passivity-based control. This observation leads to a new method for designing sliding-mode controllers and establishes the grounds for the following section, which is about discrete-time sliding-mode control.

Consider the perturbed linear dynamics

$$ \dot{x}(t) = Ax(t) + B(u(t) + \delta(t)), \quad (16) $$

where $x(t) \in \mathbb{R}^n$, $u(t) \in \mathbb{R}^m$, and $\delta(t) \in \mathbb{R}^m$ are, respectively, the state, the control input, and the perturbation at time $t$. Accordingly, $A \in \mathbb{R}^{n \times n}$ and $B \in \mathbb{R}^{n \times m}$. It is assumed that $\text{rank}(B) = m$ and $\delta(t)$ is uniformly bounded.

### 2.1 The Classical Approach

In the classical approach to sliding-mode control, the robust stabilization of (16) is undertaken in two steps:

1. Choose a linear map $\sigma: x \mapsto Cx$ and define the sliding variable $s = \sigma \circ x$. The matrix $C \in \mathbb{R}^{m \times n}$ is such that $\det(CB) \neq 0$ and such that $s(t) \equiv 0$ implies $x(t) \to 0$ as $t \uparrow +\infty$.

2. Design a control law $u(t)$ such that $s(t)$ converges to zero in finite time.

Step 1 is typically accomplished by putting the system in regular form and solving a lower-dimensional stabilization problem [51, 52]. To accomplish Step 2, the time derivative $\dot{s}(t) = CAx(t) + CB(u(t) + \delta(t))$ is first computed. By setting $u(t) = u_{\text{lin}}(x(t)) + (CB)^{-1}v(t)$, where

$$ u_{\text{lin}}(x) = -(CB)^{-1}CAx \quad (17) $$

is a linear feedback and $v(t)$ is a new input, the dynamics for $s$ are simplified to

$$ \dot{s}(t) = v(t) + CB\delta(t). \quad (18) $$

The design is completed by fixing the new input as $v(t) = u_{\text{sv}}(x(t), t)$ with

$$ u_{\text{sv}}(x, t) \in -K \mathbf{Sgn}(\sigma(x)), \quad K = K^{\top} \succ 0, \quad (19) $$

for all $t \in \mathbb{R}$. The term $u_{\text{sv}}(x, t)$ is a selection of the set-valued signum function (see Definition 3 in “Sidebar III”, and (30) (40) in “Sidebar VI”). With the controls (17) and (19), the sliding variable evolves according to the differential inclusion

$$ \dot{s}(t) \in -K \mathbf{Sgn}(s(t)) + CB\delta(t). \quad (20) $$

The time derivative of the Lyapunov function $V_s(s) = \frac{1}{2}s^\top K^{-1}s$ along closed-loop trajectories is

$$ \begin{aligned} \dot{V}_s(s(t)) &= -s(t)^\top (\mathbf{Sgn}(s(t)) - K^{-1}CB\delta(t)) \\ &= -\|s(t)\|_1 + s(t)^\top K^{-1}CB\delta(t) \\ &\le -(1 - \|K^{-1}CB\delta(t)\|_\infty)\|s(t)\|_1. \end{aligned} $$
<figure><img src="image_3.png" /><figcaption>Figure 2: Cobweb diagram of (7) with parameters: $\tilde{a} = 0.5$, $\tilde{b}\kappa = 0.75$ for two different initial conditions. The explicit discretization of the set-valued law (2) yields an iteration with a non-monotone map such that, for almost all initial conditions, the closed-loop shows an oscillatory behavior. The amplitude of the oscillation is proportional to the control gain $\kappa$.</figcaption></figure>

the form (13c) is called a minimum-operator-based input), and that the connection with proximal maps via the implicit Euler method was shown in [50]. It is worth to emphasize that, in the limit as $h \downarrow 0$, the trajectories of both schemes, (7) and (9), converge towards trajectories of the continuous time inclusion (1)-(2). However, the control input (13) is well defined at $x = 0$, whereas (6) is not. Thus, even though in the implementation of discrete-time controllers the sampling time $h > 0$ cannot be arbitrarily small because of practical limitations, it is clear that the implicitly defined system (9) is more attractive than the explicit one. Such distinction becomes more important when disturbances are considered.

To close this section, it is also worth to emphasize that arbitrary regularizations of the set-valued sign function (that is, single-valued functions sufficiently close to the graph of the sign multifunction like the saturation or the sigmoid functions) do not guarantee the global finite-time convergence property. Generically, such an approximation either breaks the global finite-time convergence towards zero (making the convergence asymptotic), or the stability of the iteration map, making the origin unstable and producing an oscillatory behavior (see “Sidebar VII”).

The following section extends the ideas presented above to the multivariable case, where non-vanishing external disturbances affecting the dynamics are also considered. Moreover, a control theoretic interpretation for the use of implicitly defined controllers is presented under the general framework of passive systems.

## 2 FIRST-ORDER SLIDING-MODE CONTROL: CONTINUOUS-TIME

In this section we argue that first-order sliding-mode control is a particular case of passivity-based control. This observation leads to a new method for designing sliding-mode controllers and establishes the grounds for the following section, which is about discrete-time sliding-mode control.

Consider the perturbed linear dynamics

$$ \dot{x}(t) = Ax(t) + B(u(t) + \delta(t)), \quad (16) $$

where $x(t) \in \mathbb{R}^n$, $u(t) \in \mathbb{R}^m$, and $\delta(t) \in \mathbb{R}^m$ are, respectively, the state, the control input, and the perturbation at time $t$. Accordingly, $A \in \mathbb{R}^{n \times n}$ and $B \in \mathbb{R}^{n \times m}$. It is assumed that $\text{rank}(B) = m$ and $\delta(t)$ is uniformly bounded.

### 2.1 The Classical Approach

In the classical approach to sliding-mode control, the robust stabilization of (16) is undertaken in two steps:

1. Choose a linear map $\sigma: x \mapsto Cx$ and define the sliding variable $s = \sigma \circ x$. The matrix $C \in \mathbb{R}^{m \times n}$ is such that $\det(CB) \neq 0$ and such that $s(t) \equiv 0$ implies $x(t) \to 0$ as $t \uparrow +\infty$.

2. Design a control law $u(t)$ such that $s(t)$ converges to zero in finite time.

Step 1 is typically accomplished by putting the system in regular form and solving a lower-dimensional stabilization problem [51, 52]. To accomplish Step 2, the time derivative $\dot{s}(t) = CAx(t) + CB(u(t) + \delta(t))$ is first computed. By setting $u(t) = u_{\text{lin}}(x(t)) + (CB)^{-1}v(t)$, where

$$ u_{\text{lin}}(x) = -(CB)^{-1}CAx \quad (17) $$

is a linear feedback and $v(t)$ is a new input, the dynamics for $s$ are simplified to

$$ \dot{s}(t) = v(t) + CB\delta(t). \quad (18) $$

The design is completed by fixing the new input as $v(t) = u_{\text{sv}}(x(t), t)$ with

$$ u_{\text{sv}}(x, t) \in -K \text{Sgn}(\sigma(x)), \quad K = K^{\top} > 0, \quad (19) $$

for all $t \in \mathbb{R}$. The term $u_{\text{sv}}(x, t)$ is a selection of the set-valued signum function (see Definition 3 in “Sidebar III”, and (30) (40) in “Sidebar VI”). With the controls (17) and (19), the sliding variable evolves according to the differential inclusion

$$ \dot{s}(t) \in -K \text{Sgn}(s(t)) + CB\delta(t). \quad (20) $$

The time derivative of the Lyapunov function $V_s(s) = \frac{1}{2}s^\top K^{-1}s$ along closed-loop trajectories is

$$ \begin{aligned} \dot{V}_s(s(t)) &= -s(t)^\top (\text{Sgn}(s(t)) - K^{-1}CB\delta(t)) \\ &= -\|s(t)\|_1 + s(t)^\top K^{-1}CB\delta(t) \\ &\le -(1 - \|K^{-1}CB\delta(t)\|_\infty)\|s(t)\|_1. \end{aligned} $$
The computation of the implicit first-order SMC with a single sliding surface, boils down to projecting a scalar onto a bounded interval: it is therefore extremely simple to implement.

<figure><img src="image_5.png" /><figcaption>Figure 3: Cobweb diagram of (9) with parameters: $\tilde{a} = 0.5$, $\tilde{b}_\kappa = 0.75$. The implicit discretization of the set-valued law (2) yields an iteration with a maximal monotone map such that, for any initial condition, the state converges towards the origin after a finite number of steps.</figcaption></figure>

Using classical arguments, the finite-time convergence of $V_s(s)$ to zero is inferred provided that $\|K^{-1}CB\delta(t)\|_\infty < 1$ (using, for example, [65, Proposition 11.1.6]). Therefore, there exists some $t_{\min} < +\infty$ after which a sliding mode occurs. That is: $s(t) = 0$ and therefore

$$u_{sv}(x(t), t) = -CB\delta(t) \quad (21)$$

for all $t \ge t_{\min}$. In other words, the controller compensates exactly for the unknown perturbation. Certainly, this continuous-time property is a kind of miracle of set-valued feedback control, adding to the *miracle of feedback stabilization* [66]. The exactness (equality) in (21) is obviously due to modeling idealization. However, as demonstrated by the implicit discrete-time analysis in (98) and by experimental results, it is the idealization of an *observed behavior*. In particular, insensitivity of the dynamics with respect to $K$ during the sliding-mode is observed experimentally in both differentiation and control when the implicit method is used. This witnesses the fact that (21) is far from being just a mathematical fuss.

## 2.2 A Passivity-Based Approach

Let us ponder the classical approach through a passivity point-of-view. This will later be used to show the necessity for an implicit discretization.

**Remark 1.** Item 1) above implies that the system with output $s$ is feedback equivalent to a passive one. Indeed, a system is feedback passive if, and only if, it is of relative degree $\{1, \dots, 1\}$, i.e., $\det(CB) \ne 0$ and minimum phase, i.e., $s(t) \equiv 0$ implies $x(t) \to 0$. The linear feedback $u_{\text{lin}}$ renders (16) passive with output $s$ and input $v+CB\delta$ (see [68] for details on the unperturbed case). In this regard, first-order sliding-mode control is a particular instance of passivity-based control.

Besides passivity, a fundamental notion is implicitly used in the above analysis: maximal monotonicity (see “Sidebar V”). Following [69] the change of variable $s' = K^{-\frac{1}{2}}s$ may be performed, transforming (20) into

$$s'(t) \in -K^{\frac{1}{2}} \mathbf{Sgn}(K^{\frac{1}{2}} s'(t)) + K^{-\frac{1}{2}} CB\delta(t).$$

The differential inclusion can be written as

$$s'(t) \in -\partial f(s'(t)) + K^{-\frac{1}{2}} CB\delta(t), \quad (22)$$

where $f(\cdot) = \|K^{\frac{1}{2}} \cdot \|_1$ and where the chain rule was used (see Theorem 3 in “Sidebar V”) to compute $\partial f$, the sub-differential of $f$ (see “Sidebar VI”). The operator $\partial f$ is a maximal monotone operator, $f$ being a convex continuous proper function (see “Sidebar VI”). The differential inclusion has the form (7) so, if the eigenvalues of $K$ are large enough, it satisfies the conditions of Proposition 2 in “Sidebar III” and the origin $s' = 0$ is finite-time stable.

Let us now introduce a different, less-classical design directly based on passivity and monotonicity arguments. Recall that the pair $(A, B)$ is stabilizable if, and only if, there exists $P \in \mathbb{R}^{n \times n}$ such that $P = P^\top > 0$ and [70]

$$AP + PA^\top < BB^\top. \quad (23)$$

Similarly to [43, 71], this inequality is exploited to construct the storage function candidate

$$V(x) = \frac{1}{2} x^\top P^{-1} x. \quad (24)$$

Its time-derivative is

$$\dot{V}(x) = \frac{1}{2} x^\top (P^{-1} A + A^\top P^{-1}) x + x^\top P^{-1} B(u+\delta) \quad (25)$$
The computation of the implicit first-order SMC with a single sliding surface, boils down to projecting a scalar onto a bounded interval: it is therefore extremely simple to implement.

<figure><img src="image_4.png" /><figcaption>Figure 3: Cobweb diagram of (9) with parameters: $\tilde{a} = 0.5$, $\tilde{b}_\kappa = 0.75$. The implicit discretization of the set-valued law (2) yields an iteration with a maximal monotone map such that, for any initial condition, the state converges towards the origin after a finite number of steps.</figcaption></figure>

Using classical arguments, the finite-time convergence of $V_s(s)$ to zero is inferred provided that $\|K^{-1}CB\delta(t)\|_\infty < 1$ (using, for example, [65, Proposition 11.1.6]). Therefore, there exists some $t_{\min} < +\infty$ after which a sliding mode occurs. That is: $s(t) = 0$ and therefore

$$u_{sv}(x(t), t) = -CB\delta(t) \quad (21)$$

for all $t \ge t_{\min}$. In other words, the controller compensates exactly for the unknown perturbation. Certainly, this continuous-time property is a kind of miracle of set-valued feedback control, adding to the *miracle of feedback stabilization* [66]. The exactness (equality) in (21) is obviously due to modeling idealization. However, as demonstrated by the implicit discrete-time analysis in (98) and by experimental results, it is the idealization of an *observed behavior*. In particular, insensitivity of the dynamics with respect to K during the sliding-mode is observed experimentally in both differentiation and control when the implicit method is used. This witnesses the fact that (21) is far from being just a mathematical fuss.

## 2.2 A Passivity-Based Approach

Let us ponder the classical approach through a passivity point-of-view. This will later be used to show the necessity for an implicit discretization.

**Remark 1.** Item 1) above implies that the system with output s is feedback equivalent to a passive one. Indeed, a system is feedback passive if, and only if, it is of relative degree $\{1, \dots, 1\}$, i.e., $\det(CB) \ne 0$ and minimum phase, i.e., $s(t) \equiv 0$ implies $x(t) \to 0$. The linear feedback u$_{\text{lin}$} renders (16) passive with output s and input $v+CB\delta$ (see [68] for details on the unperturbed case). In this regard, first-order sliding-mode control is a particular instance of passivity-based control.

Besides passivity, a fundamental notion is implicitly used in the above analysis: maximal monotonicity (see “Sidebar V”). Following [69] the change of variable $s' = K^{-\frac{1}{2}}s$ may be performed, transforming (20) into

$$s'(t) \in -K^{\frac{1}{2}} \mathbf{Sgn}(K^{\frac{1}{2}} s'(t)) + K^{-\frac{1}{2}} CB\delta(t).$$

The differential inclusion can be written as

$$s'(t) \in -\partial f(s'(t)) + K^{-\frac{1}{2}} CB\delta(t), \quad (22)$$

where $f(\cdot) = \|K^{\frac{1}{2}} \cdot \|_1$ and where the chain rule was used (see Theorem 3 in “Sidebar V”) to compute $\partial f$, the sub-differential of f (see “Sidebar VI”). The operator $\partial f$ is a maximal monotone operator, f being a convex continuous proper function (see “Sidebar VI”). The differential inclusion has the form (7) so, if the eigenvalues of K are large enough, it satisfies the conditions of Proposition 2 in “Sidebar III” and the origin $s' = 0$ is finite-time stable.

Let us now introduce a different, less-classical design directly based on passivity and monotonicity arguments. Recall that the pair $(A, B)$ is stabilizable if, and only if, there exists $P \in \mathbb{R}^{n \times n}$ such that $P = P^\top > 0$ and [70]

$$AP + PA^\top < BB^\top. \quad (23)$$

Similarly to [43, 71], this inequality is exploited to construct the storage function candidate

$$V(x) = \frac{1}{2} x^\top P^{-1} x. \quad (24)$$

Its time-derivative is

$$\dot{V}(x) = \frac{1}{2} x^\top (P^{-1} A + A^\top P^{-1}) x + x^\top P^{-1} B(u+\delta) \quad (25)$$
A set-valued operator or multifunction $\mathbf{F}: \mathbb{R}^n \rightrightarrows \mathbb{R}^m$ is a map that associates, with any $x \in \mathbb{R}^n$, a subset $\mathbf{F}(x) \subset \mathbb{R}^m$. The domain of $\mathbf{F}$ is given by

$$ \mathrm{dom} \mathbf{F} = \{ \xi \in \mathbb{R}^n | \mathbf{F}(\xi) \neq \emptyset \}. \qquad (1) $$

The inverse mapping (set-valued or single-valued) of $\mathbf{F}$ is defined as $\mathbf{F}^{-1}: \mathbb{R}^m \rightrightarrows \mathbb{R}^n$ with $\mathbf{F}^{-1}(y) = \{\xi \in \mathbb{R}^n | y \in \mathbf{F}(\xi)\}$. Given a set $E \subset \mathbb{R}^m$, its inverse image by $\mathbf{F}$ is defined as $\mathbf{F}^{-1}(E) = \bigcup_{y \in E} \mathbf{F}^{-1}(y)$. It will be useful to consider rint dom $\mathbf{F}$, the relative interior of dom $\mathbf{F}$. The range of the operator $\mathbf{F}$ is given by

$$ \mathrm{rge} \mathbf{F} = \{ \eta \in \mathbb{R}^m | \eta \in \mathbf{F}(\xi) \text{ for some } \xi \in \mathbb{R}^n \} \qquad (2) $$

and its graph by

$$ \mathrm{gph} \mathbf{F} = \{ (\xi, \eta) \in \mathbb{R}^n \times \mathbb{R}^m | \eta \in \mathbf{F}(\xi) \}. \qquad (3) $$

Semicontinuity properties (inner, upper, outer) are fundamental for set-valued maps analysis. The material below is taken from [S16, S17, S18, S19].

Differential inclusions provide a mathematical framework for the study of systems with discontinuous models. Namely, for a discontinuous ODE of the form

$$ \dot{x}(t) = f(t, x(t)), \quad x(t_0) = x_0, \qquad (4) $$

where the single-valued map $f: \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n$ is measurable in both arguments and locally bounded, it is well-known that, in general, a solution in the sense of Carathéodory fails to exist (i.e., an absolutely continuous function that satisfies (4) a.e.). Thus, in order to make sense of (4), the Filippov's convexification method considers instead the differential inclusion

$$ \dot{x}(t) \in \mathbf{F}(t, x(t)), \quad x(t_0) = x_0, \qquad (5) $$

where $\mathbf{F}(t, x) = \bigcap_{\varepsilon>0} \bigcap_{\mu(\Omega)=0} \overline{\mathrm{co}} \{f(t, (x + \varepsilon\mathbf{B}) \setminus \Omega)\}$, $\overline{\mathrm{co}}A$ denotes the closed convex-hull of $A$, and $\mu(\Omega)$ denotes the Lebesgue measure of the set $\Omega \subset \mathbb{R}^n$. Thus, absolutely continuous functions that satisfy (5) a.e., are solutions of (4) in the sense of Filippov, see e.g., [17, 18]. On the other hand, let instead of (5) the following differential inclusion be considered

$$ \dot{x}(t) \in \mathbf{K}(t, x(t)), \quad x(t_0) = x_0, \qquad (6) $$

where $\mathbf{K}(t, x) = \bigcap_{\varepsilon>0} \overline{\mathrm{co}}\{f(t, x + \varepsilon\mathbf{B})\}$. Then, absolutely continuous functions that satisfy (6) a.e. are solutions of (4) in the sense of Krasovskii. In general, Filippov and Krasovskii solutions are different, see, e.g., [64].

**Definition 1.** The set-valued map $\mathbf{F}: \mathbb{R}^n \rightrightarrows \mathbb{R}^m$ is outer semicontinuous (OSC) if, for any $x \in \mathrm{dom} \mathbf{F}$ and any sequence $\{(x_k, y_k)\}$ such that $(x_k, y_k) \in \mathrm{gph} \mathbf{F}$ for all $k$, and $(x_k, y_k) \to (x, y)$, one has $(x, y) \in \mathrm{gph} \mathbf{F}$.

**Fact 1.** If $\mathbf{F}$ is OSC at $x$, then $\mathbf{F}(x)$ is a closed set (hence, an OSC set-valued map $\mathbf{F}$ is closed valued). Moreover, $\mathbf{F}$ is OSC if and only if $\mathrm{gph} \mathbf{F}$ is closed.

**Theorem 1.** [S17, Theorem 5.1] Let $\mathbf{F}: \mathbb{R}^n \times [0, T] \rightrightarrows \mathbb{R}^m$ be an OSC map with closed and convex images. Assume there exists a function $c: [0, T] \to \mathbb{R}$, $c \in L_1([0, T])$, such that for all $t \in [0, T]$,

$$ \sup_{\eta \in F(x,t)} \|\eta\| \le c(t)(1 + \|x\|). $$

Then, the differential inclusion $\dot{x}(t) \in \mathbf{F}(x(t), t)$ has an absolutely continuous solution on $[0, T]$.

**Definition 2.** A solution $x: [0, T] \to \mathbb{R}^n$, $T > 0$ of the differential inclusion $\dot{x}(t) \in \mathbf{F}(x(t), t)$ is said to be a sliding motion if $\mathrm{int} \mathbf{F}(x(t), t) \neq \emptyset$ for all $t \in [0, T]$.

**Definition 3.** [S19, p. 35] Let $\mathbf{F}: \mathbb{R}^n \rightrightarrows \mathbb{R}^m$ be a set-valued mapping. Then, a single-valued mapping $\lambda: \mathbb{R}^n \to \mathbb{R}^m$ satisfying $\lambda(x) \in \mathbf{F}(x)$ for all $x \in \mathbb{R}^n$ is a selection.
A set-valued operator or multifunction $\mathbf{F}: \mathbb{R}^n \rightrightarrows \mathbb{R}^m$ is a map that associates, with any $x \in \mathbb{R}^n$, a subset $\mathbf{F}(x) \subset \mathbb{R}^m$. The domain of $\mathbf{F}$ is given by

$$ \mathrm{dom} \mathbf{F} = \{ \xi \in \mathbb{R}^n | \mathbf{F}(\xi) \neq \emptyset \}. \qquad (1) $$

The inverse mapping (set-valued or single-valued) of $\mathbf{F}$ is defined as $\mathbf{F}^{-1}: \mathbb{R}^m \rightrightarrows \mathbb{R}^n$ with $\mathbf{F}^{-1}(y) = \{\xi \in \mathbb{R}^n | y \in \mathbf{F}(\xi)\}$. Given a set $E \subset \mathbb{R}^m$, its inverse image by $\mathbf{F}$ is defined as $\mathbf{F}^{-1}(E) = \bigcup_{y \in E} \mathbf{F}^{-1}(y)$. It will be useful to consider $\mathrm{rint}(\mathrm{dom} \mathbf{F})$, the relative interior of $\mathrm{dom} \mathbf{F}$. The range of the operator $\mathbf{F}$ is given by

$$ \mathrm{rge} \mathbf{F} = \{ \eta \in \mathbb{R}^m | \eta \in \mathbf{F}(\xi) \text{ for some } \xi \in \mathbb{R}^n \} \qquad (2) $$

and its graph by

$$ \mathrm{gph} \mathbf{F} = \{ (\xi, \eta) \in \mathbb{R}^n \times \mathbb{R}^m | \eta \in \mathbf{F}(\xi) \}. \qquad (3) $$

Semicontinuity properties (inner, upper, outer) are fundamental for set-valued maps analysis. The material below is taken from [S16, S17, S18, S19].

Differential inclusions provide a mathematical framework for the study of systems with discontinuous models. Namely, for a discontinuous ODE of the form

$$ \dot{x}(t) = f(t, x(t)), \quad x(t_0) = x_0, \qquad (4) $$

where the single-valued map $f: \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n$ is measurable in both arguments and locally bounded, it is well-known that, in general, a solution in the sense of Carathéodory fails to exist (i.e., an absolutely continuous function that satisfies (4) a.e.). Thus, in order to make sense of (4), the Filippov's convexification method considers instead the differential inclusion

$$ \dot{x}(t) \in \mathbf{F}(t, x(t)), \quad x(t_0) = x_0, \qquad (5) $$

where $\mathbf{F}(t, x) = \bigcap_{\varepsilon>0} \bigcap_{\mu(\Omega)=0} \overline{\mathrm{co}}\{f(t, (x + \varepsilon\mathbf{B}) \setminus \Omega)\}$, $\overline{\mathrm{co}}A$ denotes the closed convex-hull of A, and $\mu(\Omega)$ denotes the Lebesgue measure of the set $\Omega \subset \mathbb{R}^n$. Thus, absolutely continuous functions that satisfy (5) a.e., are solutions of (4) in the sense of Filippov, see e.g., [17, 18]. On the other hand, let instead of (5) the following differential inclusion be considered

$$ \dot{x}(t) \in \mathbf{K}(t, x(t)), \quad x(t_0) = x_0, \qquad (6) $$

where $\mathbf{K}(t, x) = \bigcap_{\varepsilon>0} \overline{\mathrm{co}}\{f(t, x + \varepsilon\mathbf{B})\}$. Then, absolutely continuous functions that satisfy (6) a.e. are solutions of (4) in the sense of Krasovskii. In general, Filippov and Krasovskii solutions are different, see, e.g., [64].

**Definition 1.** The set-valued map $\mathbf{F}: \mathbb{R}^n \rightrightarrows \mathbb{R}^m$ is outer semicontinuous (OSC) if, for any $x \in \mathrm{dom} \mathbf{F}$ and any sequence $\{(x_k, y_k)\}$ such that $(x_k, y_k) \in \mathrm{gph} \mathbf{F}$ for all $k$, and $(x_k, y_k) \to (x, y)$, one has $(x, y) \in \mathrm{gph} \mathbf{F}$.

**Fact 1.** If $\mathbf{F}$ is OSC at $x$, then $\mathbf{F}(x)$ is a closed set (hence, an OSC set-valued map $\mathbf{F}$ is closed valued). Moreover, $\mathbf{F}$ is OSC if and only if $\mathrm{gph} \mathbf{F}$ is closed.

**Theorem 1.** [S17, Theorem 5.1] Let $\mathbf{F}: \mathbb{R}^n \times [0, T] \rightrightarrows \mathbb{R}^m$ be an OSC map with closed and convex images. Assume there exists a function $c: [0, T] \to \mathbb{R}$, $c \in L_1([0, T])$, such that for all $t \in [0, T]$,

$$ \sup_{\eta \in F(x,t)} \|\eta\| \le c(t)(1 + \|x\|). $$

Then, the differential inclusion $\dot{x}(t) \in \mathbf{F}(x(t), t)$ has an absolutely continuous solution on $[0, T]$.

**Definition 2.** A solution $x: [0, T] \to \mathbb{R}^n, T > 0$ of the differential inclusion $\dot{x}(t) \in \mathbf{F}(x(t), t)$ is said to be a sliding motion if $\mathrm{int} \mathbf{F}(x(t), t) \neq \emptyset$ for all $t \in [0, T]$.

**Definition 3.** [S19, p. 35] Let $\mathbf{F}: \mathbb{R}^n \rightrightarrows \mathbb{R}^m$ be a set-valued mapping. Then, a single-valued mapping $\lambda: \mathbb{R}^n \to \mathbb{R}^m$ satisfying $\lambda(x) \in \mathbf{F}(x)$ for all $x \in \mathbb{R}^n$ is a selection.
In the time-varying case with $F: \mathbb{R}^n \times \mathbb{R} \rightrightarrows \mathbb{R}^n$ one can augment the state with time as $\dot{\tau} = 1$, $\tau(0) = t_0$ and define a selection accordingly as $\lambda(x, \tau) \in F(x, \tau)$ for all $x$ and $\tau$. Differential inclusions of the form

$$ \dot{s}(t) \in -\partial f(s(t)) + \delta(t) \qquad (7) $$

are of interest, where $\partial f$ is the subdifferential of $f$ in the sense of convex analysis (see “Sidebar VI”) and $\delta: \mathbb{R} \to \mathbb{R}^m$ is a measurable and uniformly bounded function accounting for external disturbances affecting the system. Indeed they appear in (22) and (36).

**Proposition 2.** Consider the perturbed subgradient system (7) with $f: \mathbb{R}^m \to \mathbb{R} \cup \{+\infty\}$ a proper, convex, and LSC function such that $\text{dom } f = \mathbb{R}^m$. Suppose that there exists a fixed $\varepsilon > 0$ such that

$$ \delta(t) + \varepsilon \mathcal{B}_2 \subseteq \partial f(0) \qquad (8) $$

for all $t \ge 0$ (see (38) in “Sidebar VI”). The origin is finite-time stable.

Note that the condition (8) implies that $\partial f$ is not a singleton at $s = 0$ and that $\delta(t)$ is uniformly bounded. See Definition 6 for LSC functions.

*Proof.* The differential inclusion (7) is of the form $\dot{s}(t) \in -M(s(t), t)$, where the operator $M(\cdot, t)$ is maximal monotone for each $t$. Under mild conditions on $\delta$, the differential inclusion possesses unique absolutely continuous solutions on $\mathbb{R}_+$, for any initial condition [S20, Theorem 3.4]. Consider the Lyapunov function candidate $V(s) = \frac{1}{2}\|s\|^2$. The set-valued derivative [S21] of $V$ along trajectories of (7) is

$$ \dot{V}(s) = \{-s^\top \vartheta + s^\top \delta \mid \vartheta \in \partial f(s)\}. $$

Since $(s, \vartheta) \in \text{gph} \partial f$ and $f$ is proper, convex and LSC, it follows that

$$ s^\top \vartheta = f(s) + f^*(\vartheta), \qquad (9) $$

where $f^*$ denotes the conjugate function to $f$ (see Definition 11 and Theorem 5 in “Sidebar VI”). It follows from (8) that $\delta(t) + \varepsilon\rho \in \partial f(0)$ for any $\rho \in \mathcal{B}_2$ and any $t \in \mathbb{R}_+$. By the definition of subdifferential (see Definition 7 in “Sidebar VI”),

$$ \delta^\top s \le -\varepsilon \rho^\top s + f(s) - f(0) \qquad (10) $$

for any $s \in \mathbb{R}^n$ and for any $\rho \in \mathcal{B}_2$. Since $\partial f(s)$ is compact for any $s \in \mathbb{R}^m$ [S22, Theorem 23.4], $\max \dot{V}(s)$ is well defined. Combining (9) and (10), it follows that

$$ \max \dot{V}(s) \le -\varepsilon \rho^\top s - f^*(\vartheta^*) - f(0) \qquad (11) $$

for any $\rho \in \mathcal{B}_2$ and some $\vartheta^* \in \partial f(x)$. Suppose that $s \ne 0$ and set $\rho = s/\|s\| \in \mathcal{B}_2$. Recalling that $f^*(\vartheta) + f(0) \ge 0$ for any $\vartheta \in \mathbb{R}^n$ gives

$$ \max \dot{V}(s) \le -\varepsilon \|s\| = -\varepsilon \sqrt{2V(s)}. \qquad (12) $$

For $s = 0$ the inequality follows trivially. $\square$

## References

[S16] J.B. Hiriart Urruty, C. Lemaréchal, Fundamentals of Convex Analysis, Grundlehren Text Editions, Springer-Verlag Berlin Heidelberg, 2001.

[S17] K. Deimling, Multivalued Differential Equations, De Gruyter Series in Nonlinear Analysis and Applications, Berlin, New York, 1992.

[S18] R.T. Rockafellar and R.J.B. Wets, Variational Analysis, Springer-Verlag Berlin Heidelberg, Grundlehren der mathematischen Wissenschaften, vol.317, 1998.

[S19] G.V. Smirnov, Introduction to the Theory of Differential Inclusions, Graduate Studies in Mathematics vol. 41, American Mathematical Society, 2002.
In the time-varying case with $F: \mathbb{R}^n \times \mathbb{R} \rightrightarrows \mathbb{R}^n$ one can augment the state with time as $\dot{t} = 1$, $\tau(0) = t_0$ and define a selection accordingly as $\lambda(x, \tau) \in F(x, \tau)$ for all $x$ and $\tau$. Differential inclusions of the form

$$ \dot{s}(t) \in -\partial f(s(t)) + \delta(t) \qquad (7) $$

are of interest, where $\partial f$ is the subdifferential of $f$ in the sense of convex analysis (see “Sidebar VI”) and $\delta: \mathbb{R} \to \mathbb{R}^m$ is a measurable and uniformly bounded function accounting for external disturbances affecting the system. Indeed they appear in (22) and (36).

**Proposition 2.** Consider the perturbed subgradient system (7) with $f: \mathbb{R}^m \to \mathbb{R} \cup \{+\infty\}$ a proper, convex, and LSC function such that $\text{dom } f = \mathbb{R}^m$. Suppose that there exists a fixed $\varepsilon > 0$ such that

$$ \delta(t) + \varepsilon B_2 \subseteq \partial f(0) \qquad (8) $$

for all $t \ge 0$ (see (38) in “Sidebar VI”). The origin is finite-time stable.

Note that the condition (8) implies that $\partial f$ is not a singleton at $s = 0$ and that $\delta(t)$ is uniformly bounded. See Definition 6 for LSC functions.

*Proof.* The differential inclusion (7) is of the form $\dot{s}(t) \in -M(s(t), t)$, where the operator $M(\cdot, t)$ is maximal monotone for each $t$. Under mild conditions on $\delta$, the differential inclusion possesses unique absolutely continuous solutions on $\mathbb{R}_+$, for any initial condition [S20, Theorem 3.4]. Consider the Lyapunov function candidate $V(s) = \frac{1}{2}\|s\|^2$. The set-valued derivative [S21] of $V$ along trajectories of (7) is

$$ \dot{V}(s) = \{-s^\top \vartheta + s^\top \delta \mid \vartheta \in \partial f(s)\}. $$

Since $(s, \vartheta) \in \text{gph} \partial f$ and $f$ is proper, convex and LSC, it follows that

$$ s^\top \vartheta = f(s) + f^*(\vartheta), \qquad (9) $$

where $f^*$ denotes the conjugate function to $f$ (see Definition 11 and Theorem 5 in “Sidebar VI”). It follows from (8) that $\delta(t) + \varepsilon\rho \in \partial f(0)$ for any $\rho \in B_2$ and any $t \in \mathbb{R}_+$. By the definition of subdifferential (see Definition 7 in “Sidebar VI”),

$$ \delta^\top s \le -\varepsilon \rho^\top s + f(s) - f(0) \qquad (10) $$

for any $s \in \mathbb{R}^n$ and for any $\rho \in B_2$. Since $\partial f(s)$ is compact for any $s \in \mathbb{R}^m$ [S22, Theorem 23.4], $\max \dot{V}(s)$ is well defined. Combining (9) and (10), it follows that

$$ \max \dot{V}(s) \le -\varepsilon \rho^\top s - f^*(\vartheta^*) - f(0) \qquad (11) $$

for any $\rho \in B_2$ and some $\vartheta^* \in \partial f(x)$. Suppose that $s \ne 0$ and set $\rho = s/\|s\| \in B_2$. Recalling that $f^*(\vartheta) + f(0) \ge 0$ for any $\vartheta \in \mathbb{R}^n$ gives

$$ \max \dot{V}(s) \le -\varepsilon \|s\| = -\varepsilon \sqrt{2V(s)}. \qquad (12) $$

For $s = 0$ the inequality follows trivially.

□

## References

[S16] J.B. Hiriart Urruty, C. Lemaréchal, Fundamentals of Convex Analysis, Grundlehren Text Editions, Springer-Verlag Berlin Heidelberg, 2001.

[S17] K. Deimling, Multivalued Differential Equations, De Gruyter Series in Nonlinear Analysis and Applications, Berlin, New York, 1992.

[S18] R.T. Rockafellar and R.J.B. Wets, Variational Analysis, Springer-Verlag Berlin Heidelberg, Grundlehren der mathematischen Wissenschaften, vol.317, 1998.

[S19] G.V. Smirnov, Introduction to the Theory of Differential Inclusions, Graduate Studies in Mathematics vol. 41, American Mathematical Society, 2002.
[S20] H. Brézis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hibert, North-Holland, Mathematics Studies 5,1973.

[S21] A. Bacciotti and F. Ceragioli, Stability and stabilization of discontinuous systems and non-smooth Lyapunov function, ESAIM: Control, Optimisation and Calculus of Variations, 4:361-376, 1999.

[S22] R.T. Rockafellar, Convex Analysis, Princeton Landmarks in Mathematics, 1970.

(when clear from context, explicit dependence on $t$ is omitted). The only candidate passive output of relative degree equal to one and associated with the storage function (24) is $s(t) = \sigma(x(t))$ with

$$ \sigma(x) = B^{\top} P^{-1} x, \qquad (26) $$

which corresponds to the passivity of the quadruple $(A, B, C, D)$ with $C = B^{\top} P^{-1}$, $D = 0$, $P^{-1}$ a solution to (16), supply rate $u^{\top}y$, see "Sidebar IV". This gives the energy balance

$$ \dot{V}(x) = \frac{1}{2} x^{\top} (P^{-1} A + A^{\top} P^{-1}) x + s^{\top} (u + \delta). \quad (27) $$

Since (23) can be equivalently written as $Q = P^{-1}BB^{\top}P^{-1} - P^{-1}A - A^{\top}P^{-1} > 0$, and since we have $x^{\top}P^{-1}BB^{\top}P^{-1}x = s^{\top}s$, the energy balance is

$$ \dot{V}(x) = s^{\top} \left( \frac{1}{2}s + u + \delta \right) - \frac{1}{2}x^{\top}Qx. \quad (28) $$

This form suggests that, to render the system passive, a control law of the form

$$ u(t) = u_{\text{lin}}(x(t)) + v(t) \quad (29) $$

with

$$ u_{\text{lin}}(x) = -\frac{1}{2}\sigma(x) \quad (30) $$

can be used, since it gives

$$ \dot{V}(x) = s^{\top}(v + \delta) - \frac{1}{2}x^{\top}Qx. \quad (31) $$

It can be readily seen that the system is passive with storage function (24), output (26), and new input $v + \delta$. Now let us set $v(t) = u_{\text{sv}}(x(t), t)$ with

$$ u_{\text{sv}}(x, t) \in -K \mathbf{Sgn}(Ks), \quad K = K^{\top} > 0, \quad (32) $$

for all $t \in \mathbb{R}$. Substituting this control law in (31) gives

$$ \begin{aligned} \dot{V}(x) = & -s^{\top} K (\mathbf{Sgn}(Ks) - K^{-1}\delta) - \frac{1}{2}x^{\top}Qx \\ & \le -\|Ks\|_{1}(1 - \|K^{-1}\delta\|_{\infty}) - \frac{1}{2}x^{\top}Qx, \end{aligned} \quad (33) $$

where, as above, the right-hand side is single-valued for all $x$. For suitable gains $K$, the origin of the closed-loop system is exponentially stable, regardless of the time-varying

perturbation $\delta(t)$. Let us now prove that $s(t)$ converges to zero in finite time. The sliding variable evolves according to

$$ \dot{s} \in B^{\top} P^{-1} \left( \left( A - \frac{1}{2} BB^{\top} P^{-1} \right) x - BK \mathbf{Sgn}(Ks) + B\delta \right). \quad (34) $$

Consider the quadratic Lyapunov function $V_s(s) = \frac{1}{2}s^{\top}(B^{\top}P^{-1}B)^{-1}s$. Its time derivative satisfies

$$ \dot{V}_s(s) = -s^{\top} (K \mathbf{Sgn}(Ks) - \delta - Lx), \quad (35) $$

where $L = (B^{\top}P^{-1}B)^{-1}B^{\top}P^{-1}(A - \frac{1}{2}BB^{\top}P^{-1})$. In (34), the state $x(t)$ can be considered as an exogenous signal, uniformly bounded according to the above analysis. Furthermore, it has already been established that $x$ converges to zero exponentially fast. Using classical arguments, it is inferred that $s$ converges to zero in finite time for suitable $K$. Alternatively, the dynamics (34) can be analyzed within the framework of differential inclusions with maximal monotone right-hand sides, similarly to (22). To reveal the maximal monotone structure in (34), let us perform the change of variables $s' = (B^{\top}P^{-1}B)^{-\frac{1}{2}}s$. The following is obtained:

$$ \dot{s}' \in -\partial g(s') + (B^{\top}P^{-1}B)^{\frac{1}{2}}(\delta + Lx) \quad (36) $$

with $g(\cdot) = \|\cdot\|_{1} \circ K(B^{\top}P^{-1}B)^{\frac{1}{2}}(\cdot)$ a proper, convex, continuous function. Again, the differential inclusion (36) has the structure (7) and, if the eigenvalues of $K$ are large enough, it satisfies the conditions of Proposition 2 in "Sidebar III", and the origin $s' = 0$ is finite-time stable. During the sliding mode, $s(t) = 0$ for all $t$, which means that there exists a selection of $\partial g(0)$ (see Definition 3 in "Sidebar III") which compensates exactly for the disturbance $(B^{\top}P^{-1}B)^{\frac{1}{2}}(\delta(t) + Lx(t))$ in (36). Equivalently, consider (32) and note that there is a selection $u_{\text{sv}}(x, t)$ in $K \mathbf{Sgn}(0)$ which compensates exactly for the equivalent disturbance in (34). That is, similarly to (21),

$$ \begin{aligned} u_{\text{sv}}(x(t), t) = & - (B^{\top} P^{-1} B)^{-1} B^{\top} P^{-1} \left( A - \frac{1}{2} B B^{\top} P^{-1} \right) x(t) - \delta(t). \end{aligned} \quad (37) $$

Let us stress once again that neither (21) nor (37) mean that the disturbance $\delta(t)$ is known. Finally, mark that
[S20] H. Brézis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hibert, North-Holland, Mathematics Studies 5,1973.

[S21] A. Bacciotti and F. Ceragioli, Stability and stabilization of discontinuous systems and non-smooth Lyapunov function, ESAIM: Control, Optimisation and Calculus of Variations, 4:361-376, 1999.

[S22] R.T. Rockafellar, Convex Analysis, Princeton Landmarks in Mathematics, 1970.

(when clear from context, explicit dependence on $t$ is omitted). The only candidate passive output of relative degree equal to one and associated with the storage function (24) is $s(t) = \sigma(x(t))$ with

$$ \sigma(x) = B^{\top} P^{-1} x, \qquad (26) $$

which corresponds to the passivity of the quadruple $(A, B, C, D)$ with $C = B^{\top} P^{-1}$, $D = 0$, $P^{-1}$ a solution to (16), supply rate $u^{\top}y$, see "Sidebar IV". This gives the energy balance

$$ \dot{V}(x) = \frac{1}{2} x^{\top} (P^{-1} A + A^{\top} P^{-1}) x + s^{\top}(u + \delta). \quad (27) $$

Since (23) can be equivalently written as $Q = P^{-1}BB^{\top}P^{-1} - P^{-1}A - A^{\top}P^{-1} \succ 0$, and since we have $x^{\top}P^{-1}BB^{\top}P^{-1}x = s^{\top}s$, the energy balance is

$$ \dot{V}(x) = s^{\top} \left( \frac{1}{2}s + u + \delta \right) - \frac{1}{2}x^{\top}Qx. \quad (28) $$

This form suggests that, to render the system passive, a control law of the form

$$ u(t) = u_{\text{lin}}(x(t)) + v(t) \quad (29) $$

with

$$ u_{\text{lin}}(x) = -\frac{1}{2}\sigma(x) \quad (30) $$

can be used, since it gives

$$ \dot{V}(x) = s^{\top}(v + \delta) - \frac{1}{2}x^{\top}Qx. \quad (31) $$

It can be readily seen that the system is passive with storage function (24), output (26), and new input $v + \delta$. Now let us set $v(t) = u_{\text{sv}}(x(t), t)$ with

$$ u_{\text{sv}}(x, t) \in -K \mathbf{Sgn}(Ks), \quad K = K^{\top} \succ 0, \quad (32) $$

for all $t \in \mathbb{R}$. Substituting this control law in (31) gives

$$ \begin{aligned} \dot{V}(x) = & -s^{\top} K (\mathbf{Sgn}(Ks) - K^{-1}\delta) - \frac{1}{2}x^{\top}Qx \\ & \le -\|Ks\|_{1}(1 - \|K^{-1}\delta\|_{\infty}) - \frac{1}{2}x^{\top}Qx, \end{aligned} \quad (33) $$

where, as above, the right-hand side is single-valued for all $x$. For suitable gains $K$, the origin of the closed-loop system is exponentially stable, regardless of the time-varying

perturbation $\delta(t)$. Let us now prove that $s(t)$ converges to zero in finite time. The sliding variable evolves according to

$$ \dot{s} \in B^{\top} P^{-1} \left( \left( A - \frac{1}{2} BB^{\top} P^{-1} \right) x - BK \mathbf{Sgn}(Ks) + B\delta \right). \quad (34) $$

Consider the quadratic Lyapunov function $V_s(s) = \frac{1}{2}s^{\top}(B^{\top}P^{-1}B)^{-1}s$. Its time derivative satisfies

$$ \dot{V}_s(s) = -s^{\top} (K \mathbf{Sgn}(Ks) - \delta - Lx), \quad (35) $$

where $L = (B^{\top}P^{-1}B)^{-1}B^{\top}P^{-1}(A - \frac{1}{2}BB^{\top}P^{-1})$. In (34), the state $x(t)$ can be considered as an exogenous signal, uniformly bounded according to the above analysis. Furthermore, it has already been established that $x$ converges to zero exponentially fast. Using classical arguments, it is inferred that $s$ converges to zero in finite time for suitable $K$. Alternatively, the dynamics (34) can be analyzed within the framework of differential inclusions with maximal monotone right-hand sides, similarly to (22). To reveal the maximal monotone structure in (34), let us perform the change of variables $s' = (B^{\top}P^{-1}B)^{-\frac{1}{2}}s$. The following is obtained:

$$ \dot{s}' \in -\partial g(s') + (B^{\top}P^{-1}B)^{\frac{1}{2}}(\delta + Lx) \quad (36) $$

with $g(\cdot) = \|\cdot\|_{1} \circ K(B^{\top}P^{-1}B)^{\frac{1}{2}}(\cdot)$ a proper, convex, continuous function. Again, the differential inclusion (36) has the structure (7) and, if the eigenvalues of $K$ are large enough, it satisfies the conditions of Proposition 2 in "Sidebar III", and the origin $s' = 0$ is finite-time stable. During the sliding mode, $s(t) = 0$ for all $t$, which means that there exists a selection of $\partial g(0)$ (see Definition 3 in "Sidebar III") which compensates exactly for the disturbance $(B^{\top}P^{-1}B)^{\frac{1}{2}}(\delta(t) + Lx(t))$ in (36). Equivalently, consider (32) and note that there is a selection $u_{\text{sv}}(x, t)$ in $K \mathbf{Sgn}(0)$ which compensates exactly for the equivalent disturbance in (34). That is, similarly to (21),

$$ \begin{aligned} u_{\text{sv}}(x(t), t) = & - (B^{\top} P^{-1} B)^{-1} B^{\top} P^{-1} \left( A - \frac{1}{2} B B^{\top} P^{-1} \right) x(t) - \delta(t). \end{aligned} \quad (37) $$

Let us stress once again that neither (21) nor (37) mean that the disturbance $\delta(t)$ is known. Finally, mark that
Consider a linear time-invariant system

$$ 
\begin{aligned}
\dot{x}(t) &= Ax(t) + Bu(t) \\
y(t) &= Cx(t) + Du(t)
\end{aligned}
\qquad (13) $$

with state $x(t) \in \mathbb{R}^n$, input $u(t) \in \mathbb{R}^m$, and output $y(t) \in \mathbb{R}^m$. The system is passive in Willems' sense [S23, S24] if there exists a symmetric matrix $P \in \mathbb{R}^{n \times n}$, $P \succ 0$ such that the storage function

$$ V(x) = \frac{1}{2} x^\top P x \qquad (14) $$

satisfies the dissipation inequality

$$ V(x(t_1)) - V(x(t_0)) \le \int_{t_0}^{t_1} u^\top(t)y(t)dt \qquad (15) $$

for all $t_1 \ge t_0$ along the continuous-time system trajectories. The system is passive if and only if the LMI

$$ M_{\text{cont}} := \begin{pmatrix} -A^\top P - PA & -PB + C^\top \\ -B^\top P + C & D + D^\top \end{pmatrix} \succ 0 \qquad (16) $$

has a positive-definite solution $P = P^\top$ [67]. Indeed, if the inequality (16) holds, then

$$ V(x(t_1)) - V(x(t_0)) = -\frac{1}{2} \int_{t_0}^{t_1} \begin{pmatrix} x(t) \\ u(t) \end{pmatrix}^\top M_c \begin{pmatrix} x(t) \\ u(t) \end{pmatrix} dt + \int_{t_0}^{t_1} u^\top(t)y(t)dt. \qquad (17) $$

The LMI in (16) implies $-A^\top P - PA \succ 0$, $D+D^\top \succ 0$, and $\ker(D+D^\top) \subseteq \ker(PB-C^\top)$. The latter further implies that $PB = C^\top$ when $D+D^\top = 0$ [65, Corollary 10.2.3, Proposition 10.2.5]. Similarly, a discrete-time system

$$ 
\begin{aligned}
x_{k+1} &= Ax_k + Bu_k \\
y_k &= Cx_k + Du_k
\end{aligned}
\qquad (18) $$

is passive if there exists a storage function (14) such that

$$ V(x_{k+1}) - V(x_k) \le u_k^\top y_k \qquad (19) $$

holds for all $k \ge 0$ along the discrete-time system trajectories. The system is passive if and only if the LMI

$$ M_{\text{disc}} := \begin{pmatrix} -A^\top PA + P & -A^\top PB + C^\top \\ -B^\top PA + C & D + D^\top - B^\top PB \end{pmatrix} \succ 0 \qquad (20) $$

possesses a solution $P = P^\top \succ 0$, in which case we have

$$ V(x_{k+1}) - V(x_k) = -\frac{1}{2} \begin{pmatrix} x_k \\ u_k \end{pmatrix}^\top M_{\text{disc}} \begin{pmatrix} x_k \\ u_k \end{pmatrix} + u_k^\top y_k. \qquad (21) $$

The LMI in (20) implies $-A^\top PA + P \succ 0$, $D+D^\top \succ B^\top PB$, and $\ker(D+D^\top - B^\top PB) \subseteq \ker(-A^\top PB + C^\top)$ (hence $C = B^\top PA$ if $D+D^\top = B^\top PB$),

If (13) or (18) are passive, we say that the quadruple $(A, B, C, D)$ is passive. The product $u^\top y$ is the supply rate of the system. Strict state-passivity holds for the continuous and discrete-time systems when $-A^\top P - PA \succ 0$ and $-A^\top PA + P \succ 0$, respectively. Strong passivity holds when $M_{\text{cont}} \succ 0$ and $M_{\text{disc}} \succ 0$. Incremental passivity holds if, for any pair of admissible inputs $(u_1, u_2)$, corresponding trajectories $(x_1, x_2)$, and outputs $(y_1, y_2)$, the auxiliary system with state $x^\top = (x_1^\top, x_2^\top)$ is passive with supply rate $(u_1 - u_2)^\top(y_1 - y_2)$ and storage function $V(x_1, x_2) = \frac{1}{2}(x_1 - x_2)^\top P(x_1 - x_2)$.

The celebrated Kalman-Yakubovich-Popov Lemma allows us to establish relationships between positive-real transfer functions and passive systems as characterized by the above LMIs [67, Chapter 3].
Consider a linear time-invariant system

$$ 
\begin{aligned}
\dot{x}(t) &= Ax(t) + Bu(t) \\
y(t) &= Cx(t) + Du(t)
\end{aligned}
\qquad (13) $$

with state $x(t) \in \mathbb{R}^n$, input $u(t) \in \mathbb{R}^m$, and output $y(t) \in \mathbb{R}^m$. The system is passive in Willems' sense [S23, S24] if there exists a symmetric matrix $P \in \mathbb{R}^{n \times n}$, $P \succ 0$ such that the storage function

$$ V(x) = \frac{1}{2} x^{\top} P x \qquad (14) $$

satisfies the dissipation inequality

$$ V(x(t_1)) - V(x(t_0)) \le \int_{t_0}^{t_1} u^{\top}(t)y(t)dt \qquad (15) $$

for all $t_1 \ge t_0$ along the continuous-time system trajectories. The system is passive if and only if the LMI

$$ M_{\text{cont}} := \begin{pmatrix} -A^{\top}P - PA & -PB + C^{\top} \\ -B^{\top}P + C & D + D^{\top} \end{pmatrix} \succ 0 \qquad (16) $$

has a positive-definite solution $P = P^{\top}$ [67]. Indeed, if the inequality (16) holds, then

$$ V(x(t_1)) - V(x(t_0)) = -\frac{1}{2} \int_{t_0}^{t_1} \begin{pmatrix} x(t) \\ u(t) \end{pmatrix}^{\top} M_c \begin{pmatrix} x(t) \\ u(t) \end{pmatrix} dt + \int_{t_0}^{t_1} u^{\top}(t)y(t)dt. \qquad (17) $$

The LMI in (16) implies $-A^{\top}P - PA \succ 0$, $D+D^{\top} \succ 0$, and $\ker(D+D^{\top}) \subseteq \ker(PB-C^{\top})$. The latter further implies that $PB = C^{\top}$ when $D+D^{\top} = 0$ [65, Corollary 10.2.3, Proposition 10.2.5]. Similarly, a discrete-time system

$$ 
\begin{aligned}
x_{k+1} &= Ax_k + Bu_k \\
y_k &= Cx_k + Du_k
\end{aligned}
\qquad (18) $$

is passive if there exists a storage function (14) such that

$$ V(x_{k+1}) - V(x_k) \le u_k^{\top} y_k \qquad (19) $$

holds for all $k \ge 0$ along the discrete-time system trajectories. The system is passive if and only if the LMI

$$ M_{\text{disc}} := \begin{pmatrix} -A^{\top}PA + P & -A^{\top}PB + C^{\top} \\ -B^{\top}PA + C & D + D^{\top} - B^{\top}PB \end{pmatrix} \succ 0 \qquad (20) $$

possesses a solution $P = P^{\top} \succ 0$, in which case we have

$$ V(x_{k+1}) - V(x_k) = -\frac{1}{2} \begin{pmatrix} x_k \\ u_k \end{pmatrix}^{\top} M_{\text{disc}} \begin{pmatrix} x_k \\ u_k \end{pmatrix} + u_k^{\top} y_k. \qquad (21) $$

The LMI in (20) implies $-A^{\top}PA + P \succ 0$, $D+D^{\top} \succ B^{\top}PB$, and $\ker(D+D^{\top} - B^{\top}PB) \subseteq \ker(-A^{\top}PB + C^{\top})$ (hence $C = B^{\top}PA$ if $D+D^{\top} = B^{\top}PB$).

If (13) or (18) are passive, we say that the quadruple $(A, B, C, D)$ is passive. The product $u^{\top}y$ is the supply rate of the system. Strict state-passivity holds for the continuous and discrete-time systems when $-A^{\top}P - PA \succ 0$ and $-A^{\top}PA + P \succ 0$, respectively. Strong passivity holds when $M_{\text{cont}} \succ 0$ and $M_{\text{disc}} \succ 0$. Incremental passivity holds if, for any pair of admissible inputs $(u_1, u_2)$, corresponding trajectories $(x_1, x_2)$, and outputs $(y_1, y_2)$, the auxiliary system with state $x^{\top} = (x_1^{\top}, x_2^{\top})$ is passive with supply rate $(u_1 - u_2)^{\top}(y_1 - y_2)$ and storage function $V(x_1, x_2) = \frac{1}{2}(x_1 - x_2)^{\top}P(x_1 - x_2)$.

The celebrated Kalman-Yakubovich-Popov Lemma allows us to establish relationships between positive-real transfer functions and passive systems as characterized by the above LMIs [67, Chapter 3].
## References

[S23] J.C. Willems, Dissipative dynamical systems, part I: General theory, *Arch. Ration. Mech. Anal.*, 45(5):21-351, 1972, doi: 10.1007/BF00276493.

[S24] J.C. Willems, Dissipative dynamical systems, part II: Linear systems with quadratic supply rates, *Arch. Ration. Mech. Anal.*, 45(5):352-393, 1972.

the well-posedness of (36) holds under mild measurability conditions on the perturbation, and from the analysis of the closed-loop dynamics

$$ \dot{x}(t) \in Ax(t) + B \left( -\frac{1}{2}\sigma(x(t)) - K\mathbf{Sgn}(\sigma(x(t))) \right) + B\delta(t), \quad (38) $$

using the fact that the set-valued right-hand side has compact and convex images, and that it is outer semicontinuous. It is then inferred from Theorem 1 (see “Sidebar III”) that (38) has absolutely continuous solutions.

## 2.3 Recapitulation

Classical sliding-mode control is achieved by first choosing an output for which the system is of relative degree one and minimum phase, and then assigning a set-valued maximal monotone structure to the dynamics of the output variable. The main features of the resulting closed-loop system are robustness with respect to matched perturbations and finite-time convergence of the output to zero. Once the output reaches the origin, it stays at the origin in a *sliding motion*. During such motion, it follows from (34) that $u_{sv}$ in (32) is selected as

$$ u_{sv}(x, t) = -Lx - \delta(t). \quad (39) $$

Note that, during the sliding motion, the input is independent of the control gain $K$, as is well-known. A similar conclusion holds for the feedbacks (17) and (19) (see (21)). Incidentally, the differential inclusions in (22) and (36) possess incremental passivity properties. Indeed, they have the generic form $\dot{s}'(t) \in -M(s'(t)) + Nw(t)$ with $M$ a maximal monotone mapping. The system is incrementally passive with storage function $V(s'_1, s'_2) = \frac{1}{2}(s'_1 - s'_2)^{\top}(s'_1 - s'_2)$ and output $N^{\top}s'$ [67].

It is worth stressing that (32) can be written as $u_{sv}(x) \in -\partial g(s)$ with $g(s) = \|Ks\|_p$ and $p = 1$. Other norms can be used. In general:

$$ \partial \|x\|_p = \frac{1}{\|x\|_p^{p-1}} \begin{pmatrix} |x_1|^{p-1} \mathbf{sgn}(x_1) \\ \vdots \\ |x_n|^{p-1} \mathbf{sgn}(x_n) \end{pmatrix} \quad (40) $$

if $x \neq 0$ and $\partial \|x\|_p = \mathcal{B}_{p*}$ if $x = 0$, see (37) in “Sidebar

VI”. For example, the norm $p = 2$ yields the *unit control*

$$ u_{sv}(x, t) \in \begin{cases} -\frac{1}{\|Ks\|_2} K^2 s & \text{if } s \neq 0 \\ -K\mathcal{B}_2 & \text{otherwise} \end{cases}. \quad (41) $$

To close this section, the reader’s attention is brought to the fact that, in the foregoing developments, no mention was made about the celebrated Filippov’s convexification method, which guarantees the existence of absolutely continuous solutions by transforming a discontinuous ODE into a specific differential inclusion. In fact, the above closed-loop systems are introduced directly in a set-valued setting (see Figure 1 and Theorem 1 in “Sidebar III”).

## 3 FIRST-ORDER SLIDING-MODE CONTROL: DISCRETE-TIME

In this section we exploit the passive properties of the sliding variable to derive a new design method for discrete-time sliding-mode control. We also show the necessity of defining any discrete-time sliding-mode control implicitly. Consider a sampled-data model of (16),

$$ x_{k+1} = \tilde{A}x_k + \tilde{B}(u_k + \tilde{\delta}_k), \quad (42) $$

where $\tilde{A} \in \mathbb{R}^{n \times n}$, $\tilde{B} \in \mathbb{R}^{n \times m}$, and it is assumed that $\text{rank } \tilde{B} = m$. In the spirit of Section “2.2”, an output $s_k$ is constructed for which the system is feedback passive. In contrast with the continuous-time scenario, a necessary condition for a system to be feedback passive is that it has relative degree $\{0, 0, \dots, 0\}$ [72, 73], which ultimately results in the necessity for implicitly defined control laws.

### 3.1 A Discrete-Time Passivity-Based Approach

The discrete-time counterpart of (23) is

$$ \tilde{A}\tilde{P}\tilde{A}^{\top} - \tilde{P} \prec \tilde{B}\tilde{B}^{\top}. \quad (43) $$

That is, the pair $(\tilde{A}, \tilde{B})$ is stabilizable if, and only if, inequality (43) holds for some $\tilde{P} = \tilde{P}^{\top} \succ 0$ [70]. The forward difference $V(x_{k+1}) - V(x_k)$ of the storage function
## References

[S23] J.C. Willems, Dissipative dynamical systems, part I: General theory, *Arch. Ration. Mech. Anal.*, 45(5):21-351, 1972, doi: 10.1007/BF00276493.

[S24] J.C. Willems, Dissipative dynamical systems, part II: Linear systems with quadratic supply rates, *Arch. Ration. Mech. Anal.*, 45(5):352-393, 1972.

the well-posedness of (36) holds under mild measurability conditions on the perturbation, and from the analysis of the closed-loop dynamics

$$ \dot{x}(t) \in Ax(t) + B \left( -\frac{1}{2}\sigma(x(t)) - K\mathbf{Sgn}(\sigma(x(t))) \right) + B\delta(t), \quad (38) $$

using the fact that the set-valued right-hand side has compact and convex images, and that it is outer semicontinuous. It is then inferred from Theorem 1 (see “Sidebar III”) that (38) has absolutely continuous solutions.

## 2.3 Recapitulation

Classical sliding-mode control is achieved by first choosing an output for which the system is of relative degree one and minimum phase, and then assigning a set-valued maximal monotone structure to the dynamics of the output variable. The main features of the resulting closed-loop system are robustness with respect to matched perturbations and finite-time convergence of the output to zero. Once the output reaches the origin, it stays at the origin in a *sliding motion*. During such motion, it follows from (34) that $u_{sv}$ in (32) is selected as

$$ u_{sv}(x, t) = -Lx - \delta(t). \quad (39) $$

Note that, during the sliding motion, the input is independent of the control gain $K$, as is well-known. A similar conclusion holds for the feedbacks (17) and (19) (see (21)). Incidentally, the differential inclusions in (22) and (36) possess incremental passivity properties. Indeed, they have the generic form $\dot{s}'(t) \in -M(s'(t)) + Nw(t)$ with $M$ a maximal monotone mapping. The system is incrementally passive with storage function $V(s'_1, s'_2) = \frac{1}{2}(s'_1 - s'_2)^{\top}(s'_1 - s'_2)$ and output $N^{\top}s'$ [67].

It is worth stressing that (32) can be written as $u_{sv}(x) \in -\partial g(s)$ with $g(s) = \|Ks\|_p$ and $p = 1$. Other norms can be used. In general:

$$ \partial \|x\|_p = \frac{1}{\|x\|_p^{p-1}} \begin{pmatrix} |x_1|^{p-1} \mathbf{sgn}(x_1) \\ \vdots \\ |x_n|^{p-1} \mathbf{sgn}(x_n) \end{pmatrix} \quad (40) $$

if $x \neq 0$ and $\partial \|x\|_p = \mathcal{B}_{p*}$ if $x = 0$, see (37) in “Sidebar

VI”. For example, the norm $p = 2$ yields the *unit control*

$$ u_{sv}(x, t) \in \begin{cases} -\frac{1}{\|Ks\|_2} K^2 s & \text{if } s \neq 0 \\ -K\mathcal{B}_2 & \text{otherwise} \end{cases}. \quad (41) $$

To close this section, the reader’s attention is brought to the fact that, in the foregoing developments, no mention was made about the celebrated Filippov’s convexification method, which guarantees the existence of absolutely continuous solutions by transforming a discontinuous ODE into a specific differential inclusion. In fact, the above closed-loop systems are introduced directly in a set-valued setting (see Figure 1 and Theorem 1 in “Sidebar III”).

## 3 FIRST-ORDER SLIDING-MODE CONTROL: DISCRETE-TIME

In this section we exploit the passive properties of the sliding variable to derive a new design method for discrete-time sliding-mode control. We also show the necessity of defining any discrete-time sliding-mode control implicitly. Consider a sampled-data model of (16),

$$ x_{k+1} = \tilde{A}x_k + \tilde{B}(u_k + \tilde{\delta}_k), \quad (42) $$

where $\tilde{A} \in \mathbb{R}^{n \times n}$, $\tilde{B} \in \mathbb{R}^{n \times m}$, and it is assumed that $\text{rank } \tilde{B} = m$. In the spirit of Section “2.2”, an output $s_k$ is constructed for which the system is feedback passive. In contrast with the continuous-time scenario, a necessary condition for a system to be feedback passive is that it has relative degree $\{0, 0, \dots, 0\}$ [72, 73], which ultimately results in the necessity for implicitly defined control laws.

### 3.1 A Discrete-Time Passivity-Based Approach

The discrete-time counterpart of (23) is

$$ \tilde{A}\tilde{P}\tilde{A}^{\top} - \tilde{P} \prec \tilde{B}\tilde{B}^{\top}. \quad (43) $$

That is, the pair $(\tilde{A}, \tilde{B})$ is stabilizable if, and only if, inequality (43) holds for some $\tilde{P} = \tilde{P}^{\top} \succ 0$ [70]. The forward difference $V(x_{k+1}) - V(x_k)$ of the storage function
Since their introduction in 1960 by Minty [S25], maximal monotone operators have found application in several domains. From optimization to partial differential equations, maximal monotonicity appears as a fundamental property in the study of convergence of numerical schemes via fixed-point methods.

**Definition 4.** A set-valued operator $\mathbf{M}: \mathbb{R}^n \rightrightarrows \mathbb{R}^n$ is said to be monotone (resp. strongly monotone) if, for any two pairs $(x_i, y_i) \in \text{gph } \mathbf{M}$, $i = 1, 2$,

$$ \langle x_1 - x_2, y_1 - y_2 \rangle \geq 0 \quad (\text{resp, } \geq \alpha \|x_1 - x_2\|^2, \alpha > 0). \quad (22) $$

The operator $\mathbf{M}$ is maximal monotone if it is monotone and its graph is not strictly contained inside the graph of any other monotone operator.

The Euclidean scalar product $\langle \cdot, \cdot \rangle$ is used in (22). However, in the framework of this article (finite-dimensional systems defined on $\mathbb{R}^n$) it is possible to extend the definition to weighted inner products $\langle x, y \rangle_M = x^\top My$ with $M = M^\top > 0$.

**Fact 2.** Let $\mathbf{M}: \mathbb{R} \rightrightarrows \mathbb{R}$ be maximal monotone and let $x \in \mathbb{R}$. Then $\mathbf{M}(x)$ is closed and convex.

The following follows from Fact 1 in "Sidebar III".

**Fact 3.** A maximal monotone operator $\mathbf{M}$ has a closed graph, hence it is OSC.

Maximal monotone maps can be considered as OSC maps with an additional monotonicity (or incremental passivity) property. Maximal monotonicity is preserved by elementary operations. Consider two maximal monotone operators $\mathbf{M}_i: \mathbb{R}^n \rightrightarrows \mathbb{R}^n$, $i = 1, 2$, and let $L \in \mathbb{R}^{n \times m}$ and $\gamma \ge 0$. Then:

1. $\mathbf{M}_i^{-1}$ and $\gamma \mathbf{M}_i$ are maximal monotone.

2. $\mathbf{M}_1 + \mathbf{M}_2$ is maximal monotone whenever $\text{rint}(\text{dom } \mathbf{M}_1) \cap \text{rint}(\text{dom } \mathbf{M}_2) \neq \emptyset$.

3. $\mathbf{T}_i(x) = L^\top \mathbf{M}_i(Lx)$ are maximal monotone whenever $\text{rg}(L) \cap \text{rint}(\text{dom } \mathbf{M}_i) \neq \emptyset$.

4. The mapping $x \mapsto -Ax+B(D+\mathbf{M}^{-1})^{-1}(Cx)$ is maximal monotone whenever $\text{Im}(C) \cap \text{rint}(\text{Im}(D+\mathbf{M}^{-1})) \neq \emptyset$, $\mathbf{M}$ is maximal monotone, and the quadruple $(A, B, C, D)$ is passive.

Items 1, 2 and 3 are classical [S18]. Inverse mappings are defined in "Sidebar III". Item 4 follows from [S26], building upon results in [S27] and [69] (there, monotonicity is understood with the weighted inner product defined by the solution $P = P^\top > 0$ of the linear matrix inequality characterizing the passivity of the quadruple). It means that the negative feedback interconnection of a passive linear invariant system with a maximal monotone static nonlinearity $\mathbf{M}$ preserves the maximal monotonicity. This is a kind of extension of [S18, Example 12.45 (b)]. One of the most important properties of maximal monotonicity is stated in the following theorem.

**Theorem 2.** Let $\mathbf{M}: \mathbb{R}^n \rightrightarrows \mathbb{R}^n$ be a monotone operator. Then, $\mathbf{M}$ is maximal monotone if, and only if, for any $\gamma > 0$, $\text{rg}(\text{Id} + \gamma \mathbf{M}) = \mathbb{R}^n$.

Given a maximal monotone operator $\mathbf{M}$ let us construct

$$ \mathcal{J}_{\gamma \mathbf{M}} := (\text{Id} + \gamma \mathbf{M})^{-1}, \quad (23) $$

the resolvent of $\mathbf{M}$ of index $\gamma > 0$. Thus, an alternative statement of Theorem 2 is that $\text{dom } \mathcal{J}_{\gamma \mathbf{M}} = \mathbb{R}^n$. An immediate corollary is that, for any $y \in \mathbb{R}^n$, the generalized equation

$$ y \in x + \gamma \mathbf{M}(x) \quad (24) $$

always admits a solution $x \in \text{dom } \mathbf{M}$. It can be shown that $\mathcal{J}_{\gamma \mathbf{M}}$ is single-valued [S28, Corollary 23.10]. In other words, that the solutions to (24) are unique.

The archetypal example of a maximal monotone map is the convex subdifferential of a proper, convex, and LSC function (see Definitions 6 and 7 in "Sidebar VI").

**Theorem 3.** [S28, Theorem 20.40] Let $f: \mathbb{R}^n \to \mathbb{R} \cup \{+\infty\}$ be a proper, convex, and LSC function. Then, $\partial f: \mathbb{R}^n \rightrightarrows \mathbb{R}^n$ is maximal monotone.
Since their introduction in 1960 by Minty [S25], maximal monotone operators have found application in several domains. From optimization to partial differential equations, maximal monotonicity appears as a fundamental property in the study of convergence of numerical schemes via fixed-point methods.

**Definition 4.** A set-valued operator $\mathbf{M}: \mathbb{R}^n \rightrightarrows \mathbb{R}^n$ is said to be monotone (resp. strongly monotone) if, for any two pairs $(x_i, y_i) \in \text{gph } \mathbf{M}$, $i = 1, 2$,

$$ \langle x_1 - x_2, y_1 - y_2 \rangle \geq 0 \quad (\text{resp, } \geq \alpha \|x_1 - x_2\|^2, \alpha > 0). \qquad (22) $$

The operator $\mathbf{M}$ is maximal monotone if it is monotone and its graph is not strictly contained inside the graph of any other monotone operator.

The Euclidean scalar product $\langle \cdot, \cdot \rangle$ is used in (22). However, in the framework of this article (finite-dimensional systems defined on $\mathbb{R}^n$) it is possible to extend the definition to weighted inner products $\langle x, y \rangle_M = x^\top My$ with $M = M^\top > 0$.

**Fact 2.** Let $\mathbf{M}: \mathbb{R} \rightrightarrows \mathbb{R}$ be maximal monotone and let $x \in \mathbb{R}$. Then $\mathbf{M}(x)$ is closed and convex.

The following follows from Fact 1 in "Sidebar III".

**Fact 3.** A maximal monotone operator $\mathbf{M}$ has a closed graph, hence it is OSC.

Maximal monotone maps can be considered as OSC maps with an additional monotonicity (or incremental passivity) property. Maximal monotonicity is preserved by elementary operations. Consider two maximal monotone operators $\mathbf{M}_i: \mathbb{R}^n \rightrightarrows \mathbb{R}^n$, $i = 1, 2$, and let $L \in \mathbb{R}^{n \times m}$ and $\gamma \ge 0$. Then:

1. $\mathbf{M}_i^{-1}$ and $\gamma \mathbf{M}_i$ are maximal monotone.

2. $\mathbf{M}_1 + \mathbf{M}_2$ is maximal monotone whenever $\text{rint}(\text{dom } \mathbf{M}_1) \cap \text{rint}(\text{dom } \mathbf{M}_2) \neq \emptyset$.

3. $\mathbf{T}_i(x) = L^\top \mathbf{M}_i(Lx)$ are maximal monotone whenever $\text{rg}(L) \cap \text{rint}(\text{dom } \mathbf{M}_i) \neq \emptyset$.

4. The mapping $x \mapsto -Ax+B(D+\mathbf{M}^{-1})^{-1}(Cx)$ is maximal monotone whenever $\text{Im}(C) \cap \text{rint}(\text{Im}(D+\mathbf{M}^{-1})) \neq \emptyset$, $\mathbf{M}$ is maximal monotone, and the quadruple $(A, B, C, D)$ is passive.

Items 1, 2 and 3 are classical [S18]. Inverse mappings are defined in "Sidebar III". Item 4 follows from [S26], building upon results in [S27] and [69] (there, monotonicity is understood with the weighted inner product defined by the solution $P = P^\top > 0$ of the linear matrix inequality characterizing the passivity of the quadruple). It means that the negative feedback interconnection of a passive linear invariant system with a maximal monotone static nonlinearity $\mathbf{M}$ preserves the maximal monotonicity. This is a kind of extension of [S18, Example 12.45 (b)]. One of the most important properties of maximal monotonicity is stated in the following theorem.

**Theorem 2.** Let $\mathbf{M}: \mathbb{R}^n \rightrightarrows \mathbb{R}^n$ be a monotone operator. Then, $\mathbf{M}$ is maximal monotone if, and only if, for any $\gamma > 0$, $\text{rg}(\text{Id} + \gamma \mathbf{M}) = \mathbb{R}^n$.

Given a maximal monotone operator $\mathbf{M}$ let us construct

$$ \mathcal{J}_{\gamma \mathbf{M}} := (\text{Id} + \gamma \mathbf{M})^{-1}, \qquad (23) $$

the resolvent of $\mathbf{M}$ of index $\gamma > 0$. Thus, an alternative statement of Theorem 2 is that $\text{dom } \mathcal{J}_{\gamma \mathbf{M}} = \mathbb{R}^n$. An immediate corollary is that, for any $y \in \mathbb{R}^n$, the generalized equation

$$ y \in x + \gamma \mathbf{M}(x) \qquad (24) $$

always admits a solution $x \in \text{dom } \mathbf{M}$. It can be shown that $\mathcal{J}_{\gamma \mathbf{M}}$ is single-valued [S28, Corollary 23.10]. In other words, that the solutions to (24) are unique.

The archetypal example of a maximal monotone map is the convex subdifferential of a proper, convex, and LSC function (see Definitions 6 and 7 in "Sidebar VI").

**Theorem 3.** [S28, Theorem 20.40] Let $f: \mathbb{R}^n \to \mathbb{R} \cup \{+\infty\}$ be a proper, convex, and LSC function. Then, $\partial f: \mathbb{R}^n \rightrightarrows \mathbb{R}^n$ is maximal monotone.
## Sidebar (cont.)

It follows from the previous theorem that, if $f$ is proper, convex, and LSC, the generalized equation

$$y \in x + \gamma \partial f(x) \qquad (25)$$

has a unique solution $x = \mathcal{J}_{\gamma \partial f}(y)$ for any $y \in \mathbb{R}^n$. Such generalized equations appear mainly in convex optimization problems and the explicit computation of a control selection.

Another important single-valued map is the *Yosida approximation* of $\mathbf{M}$ of index $\gamma$,

$$\mathcal{Y}_{\gamma \mathbf{M}} := \frac{1}{\gamma}(\mathbf{I}_d - \mathcal{J}_{\gamma \mathbf{M}}), \qquad (26)$$

which is Lipschitz continuous with Lipschitz constant equal to $\frac{1}{\gamma}$. It is known that

$$\mathcal{Y}_{\gamma \mathbf{M}} = (\gamma \mathbf{I}_d + \mathbf{M}^{-1})^{-1}, \qquad (27)$$

i.e., $\mathcal{Y}_{\gamma \mathbf{M}}(x) = \mathcal{J}_{\frac{1}{\gamma} \mathbf{M}^{-1}}(\frac{1}{\gamma}x)$ [S18, Lemma 12.14]. Moreover, for any $x \in \mathbb{R}^n$, we have [S29, Theorem 2, Chapter 3]

$$(\mathcal{J}_{\gamma \mathbf{M}}(x), \mathcal{Y}_{\gamma \mathbf{M}}(x)) \in \text{gph} \mathbf{M}. \qquad (28)$$

## References

[S25] G.J. Minty, Monotone Networks, Proc. R. Soc. Lond. A 257:194-212, 1960.

[S26] M.K. Camlibel, J.M. Schumacher, Linear passive systems and maximal monotone mappings, Mathematical Programming, Ser. B, 157(2):397-420, 2016.

[S27] B. Brogliato, D. Goeleven, Well-posedness, stability and invariance results for a class of multivalued Lur'e dynamical systems, Nonlinear Analysis: Theory, Methods and Applications, 74(1):195-212, 2011.

[S28] H.H. Bauschke, P.L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Canadian Mathematical Society, Société mathématique du Canada, Springer Science+Business Media, 2011.

[S29] J.-P. Aubin, A. Cellina, Differential Inclusions: Set-Valued Maps and Viability Theory, Springer Berlin, Heidelberg, 2012.
## Sidebar (cont.)

It follows from the previous theorem that, if $f$ is proper, convex, and LSC, the generalized equation

$$y \in x + \gamma \partial f(x) \qquad (25)$$

has a unique solution $x = \mathcal{J}_{\gamma \partial f}(y)$ for any $y \in \mathbb{R}^n$. Such generalized equations appear mainly in convex optimization problems and the explicit computation of a control selection.

Another important single-valued map is the *Yosida approximation* of $\mathbf{M}$ of index $\gamma$,

$$\mathcal{Y}_{\gamma \mathbf{M}} := \frac{1}{\gamma} (I_d - \mathcal{J}_{\gamma \mathbf{M}}), \qquad (26)$$

which is Lipschitz continuous with Lipschitz constant equal to $\frac{1}{\gamma}$. It is known that

$$\mathcal{Y}_{\gamma \mathbf{M}} = (\gamma I_d + \mathbf{M}^{-1})^{-1}, \qquad (27)$$

i.e., $\mathcal{Y}_{\gamma \mathbf{M}}(x) = \mathcal{J}_{\frac{1}{\gamma} \mathbf{M}^{-1}}(\frac{1}{\gamma}x)$ [S18, Lemma 12.14]. Moreover, for any $x \in \mathbb{R}^n$, we have [S29, Theorem 2, Chapter 3]

$$(\mathcal{J}_{\gamma \mathbf{M}}(x), \mathcal{Y}_{\gamma \mathbf{M}}(x)) \in \text{gph} \mathbf{M}. \qquad (28)$$

## References

[S25] G.J. Minty, Monotone Networks, Proc. R. Soc. Lond. A 257:194-212, 1960.

[S26] M.K. Camlibel, J.M. Schumacher, Linear passive systems and maximal monotone mappings, Mathematical Programming, Ser. B, 157(2):397-420, 2016.

[S27] B. Brogliato, D. Goeleven, Well-posedness, stability and invariance results for a class of multivalued Lur'e dynamical systems, Nonlinear Analysis: Theory, Methods and Applications, 74(1):195-212, 2011.

[S28] H.H. Bauschke, P.L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Canadian Mathematical Society, Société mathématique du Canada, Springer Science+Business Media, 2011.

[S29] J.-P. Aubin, A. Cellina, Differential Inclusions: Set-Valued Maps and Viability Theory, Springer Berlin, Heidelberg, 2012.
$V(x_k) = \frac{1}{2} x_k^\top \tilde{P} x_k$ is

$$
\Delta V(x_k) = \frac{1}{2} (\tilde{A}x_k + \tilde{B}(u_k + \tilde{\delta}_k))^\top \tilde{P}^{-1} (\tilde{A}x_k + \tilde{B}(u_k + \tilde{\delta}_k)) \\ - \frac{1}{2} x_k^\top \tilde{P}^{-1} x_k. \quad (44)
$$

By isolating the terms that do not depend on the inputs,
the counterpart of (25) is obtained as

$$
\Delta V(x_k) = \frac{1}{2} x_k^\top (\tilde{A}^\top \tilde{P}^{-1} \tilde{A} - \tilde{P}^{-1}) x_k \\
+ (u_k + \tilde{\delta}_k)^\top \tilde{B}^\top \tilde{P}^{-1} (\tilde{A}x_k + \tilde{B}(u_k + \tilde{\delta}_k)) \\
- \frac{1}{2} (u_k + \tilde{\delta}_k)^\top \tilde{B}^\top \tilde{P}^{-1} \tilde{B}(u_k + \tilde{\delta}_k). \quad (45)
$$

The passive output candidate

$$
s_{k+1} := \sigma(x_k, u_k + \delta_k) \quad (46)
$$

with

$$
\sigma(x, w) = \tilde{B}^{\top} \tilde{P}^{-1} (\tilde{A}x + \tilde{B}w) \quad (47)
$$

is unveiled and the energy balance becomes

$$
\Delta V(x_k) = \frac{1}{2} x_k^\top (\tilde{A}^\top \tilde{P}^{-1} \tilde{A} - \tilde{P}^{-1}) x_k + s_{k+1}^\top (u_k + \tilde{\delta}_k) \\ - \frac{1}{2} (u_k + \tilde{\delta}_k)^\top \tilde{B}^\top \tilde{P}^{-1} \tilde{B} (u_k + \tilde{\delta}_k), \quad (48)
$$

which corresponds to the dissipation equality (20) of the
quadruple $(\tilde{A}, \tilde{B}, \tilde{C}, \tilde{D})$ with $\tilde{C} = \tilde{B}^\top \tilde{P}^{-1} \tilde{A}$, $\tilde{D} + \tilde{D}^\top = 2 \tilde{B} \tilde{P}^{-1} \tilde{B}$, $\tilde{P}^{-1}$ solution to the LMI (20), supply rate $u^\top y$
(see "Sidebar IV").

**Remark 2.** The equations (46), (47) have an implicit (backward-Euler) flavour since

$$
s_{k+1} = \tilde{B}^{\top} \tilde{P}^{-1} x_{k+1}. \quad (49)
$$

Note that (47) is the counterpart of (26). In this context,
the key difference is that discrete-time passive systems
have relative degree zero whereas continuous-time passive
systems have relative degree either one or zero [67].

**Remark 3.** Neither $s_{k+1} = B^\top P^{-1} x_k$ nor even $s_{k+1} = B^\top P^{-1} x_{k+1}$ were used, as it would be the case in a pure emulation method. The sliding variable (hence the sliding surface) is designed in the discrete-time context to preserve passivity. A similar remark holds for the linear controller (30), which can be discretized in various ways (by emulation, $u_{\text{lin}}(x_k) = -\frac{1}{2} B^\top P^{-1} x_k$, or directly in discrete time, as in (55) and (56) below). Not all discretized controllers share similar closed-loop properties [21]. The same comments hold for the case of the classical first-order control in (91) and (92) below. See also Section "5.2".

The following theorem shows that stabilizability implies
feedback equivalence to passivity.

**Theorem 4.** Suppose that system (42) is stabilizable.
Consider the output (46) with $\tilde{P} = \tilde{P}^\top > 0$ a solution of (43). The control

$$
u_k = u_{\text{lin}}(x_k) + v_k, \quad (50)
$$

with

$$
u_{\text{lin}}(x_k) = -\frac{1}{2}\sigma(x_k, u_{\text{lin}}(x_k)) \quad (51)
$$

renders the system passive with storage function (24), out-
put $s_{k+1}$ given by (46), and new input $v_k + \tilde{\delta}_k$.

Notice that (51) is the counterpart of (30). The proof
of this theorem requires the following lemma, which is
inspired by [70, Chapter 6].

**Lemma 1.** The inequality (43) implies that

$$
\tilde{A}^{\top} \left( \tilde{P} + \frac{1}{2} \tilde{B} \tilde{B}^{\top} \right)^{-1} \tilde{P} \left( \tilde{A} + \frac{1}{2} \tilde{B} \tilde{B}^{\top} \right)^{-1} \tilde{A} - \tilde{P}^{-1} \asymp 0. \tag{52}
$$

Proof. Using Schur complements, it can be seen that (43)
is equivalent to

$$
\begin{pmatrix} \tilde{P} & \tilde{P} \tilde{A}^{\top} \\ \tilde{A} \tilde{P} & \tilde{P} + \tilde{B} \tilde{B}^{\top} \end{pmatrix} \succ 0, \qquad (53)
$$

which implies

$$
\begin{pmatrix} \tilde{P} & \tilde{P}\tilde{A}^{\top} \\ \tilde{A}\tilde{P} & \tilde{P} + \tilde{B}\tilde{B}^{\top} + \frac{1}{4}\tilde{B}\tilde{B}^{\top}\tilde{P}^{-1}\tilde{B}\tilde{B}^{\top} \end{pmatrix} \succ 0, \quad (54)
$$

that is, $\tilde{A}^{\top} (\tilde{P} + \tilde{B}\tilde{B}^{\top} + \frac{1}{4}\tilde{B}\tilde{B}^{\top}\tilde{P}^{-1}\tilde{B}\tilde{B}^{\top})^{-1} \tilde{A} - \tilde{P}^{-1} \asymp 0$. Inequality (52) is recovered by noting that $\tilde{P} + \tilde{B}\tilde{B}^{\top} + \frac{1}{4}\tilde{B}\tilde{B}^{\top}\tilde{P}^{-1}\tilde{B}\tilde{B}^{\top} = (\tilde{P} + \frac{1}{2}\tilde{B}\tilde{B}^{\top})\tilde{P}^{-1}(\tilde{P} + \frac{1}{2}\tilde{B}\tilde{B}^{\top})$. $\square$

Proof of Theorem 4. According to (47), the control (51)
is implicitly defined by the equation

$$
u_{\text{lin}}(x_k) = -\frac{1}{2} \tilde{B}^{\top} \tilde{P}^{-1} (\tilde{A}x_k + \tilde{B}u_{\text{lin}}(x_k)). \quad (55)
$$

It can be solved explicitly as

$$
u_{\text{lin}}(x_k) = - \left( 2I + \tilde{B}^{\top} \tilde{P}^{-1} \tilde{B} \right)^{-1} \tilde{B}^{\top} \tilde{P}^{-1} \tilde{A} x_k, \quad (56)
$$

which gives

$$
\begin{align*}
\tilde{A}x_k + \tilde{B}u_{\text{lin}}(x_k) &= \tilde{A}x_k - \tilde{B} (2I + \tilde{B}^\top \tilde{P}^{-1} \tilde{B})^{-1} \tilde{B}^\top \tilde{P}^{-1} \tilde{A} x_k \\
&= \tilde{P} (\tilde{P}^{-1} - \tilde{P}^{-1} \tilde{B} (2I + \tilde{B}^\top \tilde{P}^{-1} \tilde{B})^{-1} \tilde{B}^\top \tilde{P}^{-1}) \tilde{A} x_k.
\end{align*}
$$

By the Matrix Inversion Lemma [65, Corollary 3.9.8],

$$
\tilde{A}x_k + \tilde{B}u_{\text{lin}}(x_k) = \tilde{P} \left( \tilde{P} + \frac{1}{2} \tilde{B} \tilde{B}^{\top} \right)^{-1} \tilde{A} x_k. \quad (57)
$$
$V(x_k) = \frac{1}{2} x_k^\top \tilde{P} x_k$ is

$$ \Delta V(x_k) = \frac{1}{2} (\tilde{A}x_k + \tilde{B}(u_k + \tilde{\delta}_k))^\top \tilde{P}^{-1} (\tilde{A}x_k + \tilde{B}(u_k + \tilde{\delta}_k)) - \frac{1}{2} x_k^\top \tilde{P}^{-1} x_k. \quad (44) $$

By isolating the terms that do not depend on the inputs, the counterpart of (25) is obtained as

$$ \Delta V(x_k) = \frac{1}{2} x_k^\top (\tilde{A}^\top \tilde{P}^{-1} \tilde{A} - \tilde{P}^{-1}) x_k + (u_k + \tilde{\delta}_k)^\top \tilde{B}^\top \tilde{P}^{-1} (\tilde{A}x_k + \tilde{B}(u_k + \tilde{\delta}_k)) - \frac{1}{2} (u_k + \tilde{\delta}_k)^\top \tilde{B}^\top \tilde{P}^{-1} \tilde{B}(u_k + \tilde{\delta}_k). \quad (45) $$

The passive output candidate

$$ s_{k+1} := \sigma(x_k, u_k + \tilde{\delta}_k) \quad (46) $$

with

$$ \sigma(x, w) = \tilde{B}^\top \tilde{P}^{-1} (\tilde{A}x + \tilde{B}w) \quad (47) $$

is unveiled and the energy balance becomes

$$ \Delta V(x_k) = \frac{1}{2} x_k^\top (\tilde{A}^\top \tilde{P}^{-1} \tilde{A} - \tilde{P}^{-1}) x_k + s_{k+1}^\top (u_k + \tilde{\delta}_k) - \frac{1}{2} (u_k + \tilde{\delta}_k)^\top \tilde{B}^\top \tilde{P}^{-1} \tilde{B} (u_k + \tilde{\delta}_k), \quad (48) $$

which corresponds to the dissipation equality (20) of the quadruple $(\tilde{A}, \tilde{B}, \tilde{C}, \tilde{D})$ with $\tilde{C} = \tilde{B}^\top \tilde{P}^{-1} \tilde{A}$, $\tilde{D} + \tilde{D}^\top = 2\tilde{B}^\top \tilde{P}^{-1} \tilde{B}$, $\tilde{P}^{-1}$ solution to the LMI (20), supply rate $u^\top y$ (see "Sidebar IV").

**Remark 2.** The equations (46), (47) have an implicit (backward-Euler) flavour since

$$ s_{k+1} = \tilde{B}^\top \tilde{P}^{-1} x_{k+1}. \quad (49) $$

Note that (47) is the counterpart of (26). In this context, the key difference is that discrete-time passive systems have relative degree zero whereas continuous-time passive systems have relative degree either one or zero [67].

**Remark 3.** Neither $s_{k+1} = \tilde{B}^\top \tilde{P}^{-1} x_k$ nor even $s_{k+1} = \tilde{B}^\top \tilde{P}^{-1} x_{k+1}$ were used, as it would be the case in a pure emulation method. The sliding variable (hence the sliding surface) is designed in the discrete-time context to preserve passivity. A similar remark holds for the linear controller (30), which can be discretized in various ways (by emulation, $u_{\text{lin}}(x_k) = -\frac{1}{2} \tilde{B}^\top \tilde{P}^{-1} x_k$, or directly in discrete time, as in (55) and (56) below). Not all discretized controllers share similar closed-loop properties [21]. The same comments hold for the case of the classical first-order control in (91) and (92) below. See also Section "5.2".

The following theorem shows that stabilizability implies feedback equivalence to passivity.

**Theorem 4.** Suppose that system (42) is stabilizable. Consider the output (46) with $\tilde{P} = \tilde{P}^\top > 0$ a solution of (43). The control

$$ u_k = u_{\text{lin}}(x_k) + v_k, \quad (50) $$

with

$$ u_{\text{lin}}(x_k) = -\frac{1}{2}\sigma(x_k, u_{\text{lin}}(x_k)) \quad (51) $$

represents the system passive with storage function (24), output $s_{k+1}$ given by (46), and new input $v_k + \tilde{\delta}_k$.

Notice that (51) is the counterpart of (30). The proof of this theorem requires the following lemma, which is inspired by [70, Chapter 6].

**Lemma 1.** The inequality (43) implies that

$$ \tilde{A}^\top (\tilde{P} + \frac{1}{2} \tilde{B} \tilde{B}^\top)^{-1} \tilde{P} (\tilde{A} + \frac{1}{2} \tilde{B} \tilde{B}^\top)^{-1} \tilde{A} - \tilde{P}^{-1} \lesssim 0. \quad (52) $$

*Proof.* Using Schur complements, it can be seen that (43) is equivalent to

$$ \begin{pmatrix} \tilde{P} & \tilde{P} \tilde{A}^\top \\ \tilde{A} \tilde{P} & \tilde{P} + \tilde{B} \tilde{B}^\top \end{pmatrix} \succ 0, \quad (53) $$

which implies

$$ \begin{pmatrix} \tilde{P} & \tilde{P} \tilde{A}^\top \\ \tilde{A} \tilde{P} & \tilde{P} + \tilde{B} \tilde{B}^\top + \frac{1}{4} \tilde{B} \tilde{B}^\top \tilde{P}^{-1} \tilde{B} \tilde{B}^\top \end{pmatrix} \succ 0, \quad (54) $$

that is, $\tilde{A}^\top (\tilde{P} + \tilde{B} \tilde{B}^\top + \frac{1}{4} \tilde{B} \tilde{B}^\top \tilde{P}^{-1} \tilde{B} \tilde{B}^\top)^{-1} \tilde{A} - \tilde{P}^{-1} \lesssim 0$. Inequality (52) is recovered by noting that $\tilde{P} + \tilde{B} \tilde{B}^\top + \frac{1}{4} \tilde{B} \tilde{B}^\top \tilde{P}^{-1} \tilde{B} \tilde{B}^\top = (\tilde{P} + \frac{1}{2} \tilde{B} \tilde{B}^\top) \tilde{P}^{-1} (\tilde{P} + \frac{1}{2} \tilde{B} \tilde{B}^\top)$. $\square$

*Proof of Theorem 4.* According to (47), the control (51) is implicitly defined by the equation

$$ u_{\text{lin}}(x_k) = -\frac{1}{2} \tilde{B}^\top \tilde{P}^{-1} (\tilde{A}x_k + \tilde{B}u_{\text{lin}}(x_k)). \quad (55) $$

It can be solved explicitly as

$$ u_{\text{lin}}(x_k) = - \left( 2I + \tilde{B}^\top \tilde{P}^{-1} \tilde{B} \right)^{-1} \tilde{B}^\top \tilde{P}^{-1} \tilde{A} x_k, \quad (56) $$

which gives

$$ \begin{aligned} \tilde{A}x_k + \tilde{B}u_{\text{lin}}(x_k) &= \tilde{A}x_k - \tilde{B} (2I + \tilde{B}^\top \tilde{P}^{-1} \tilde{B})^{-1} \tilde{B}^\top \tilde{P}^{-1} \tilde{A} x_k \\ &= \tilde{P} (\tilde{P}^{-1} - \tilde{P}^{-1} \tilde{B} (2I + \tilde{B}^\top \tilde{P}^{-1} \tilde{B})^{-1} \tilde{B}^\top \tilde{P}^{-1}) \tilde{A} x_k. \end{aligned} $$

By the Matrix Inversion Lemma [65, Corollary 3.9.8],

$$ \tilde{A}x_k + \tilde{B}u_{\text{lin}}(x_k) = \tilde{P} \left( \tilde{P} + \frac{1}{2} \tilde{B} \tilde{B}^\top \right)^{-1} \tilde{A} x_k. \quad (57) $$
It now follows from Lemma 1 that

$$ (\tilde{A}x_k + \tilde{B}u_{\text{lin}}(x_k))^{\top} \tilde{P}^{-1} (\tilde{A}x_k + \tilde{B}u_{\text{lin}}(x_k)) - x_k^{\top} \tilde{P}^{-1} x_k \\ = -x_k^{\top} \tilde{Q} x_k \quad (58) $$

for some positive definite matrix $\tilde{Q}$. The dissipation equality (48) can be written as

$$ \Delta V(x_k) = s_{k+1}^{\top} (v_k + \tilde{\delta}_k) - \frac{1}{2} (v_k + \tilde{\delta}_k)^{\top} \tilde{B}^{\top} \tilde{P}^{-1} \tilde{B} (v_k + \tilde{\delta}_k) \\ - \frac{1}{2} x_k^{\top} \tilde{Q} x_k, \quad (59) $$

which is the counterpart of (31). This shows passivity with the required ingredients. □

### 3.1.1 Stability Analysis

It follows from (59) that the origin is exponentially stable if $v_k$ is such that

$$ s_{k+1}^{\top} (v_k + \tilde{\delta}_k) \le 0. \quad (60) $$

By continuing the analogy with the continuous-time case, we may be tempted to write

$$ v_k = u_{sv}(x_k) \quad (61) $$

with $u_{sv}(x_k) \in -\tilde{K} \text{Sgn}(\tilde{K} s_{k+1})$. Such control does satisfy (60) but, unfortunately, the passive output (46) is not available to the controller, because it depends directly on the unknown perturbation. To avoid this problem let us follow [31] and define the nominal output

$$ \tilde{s}_{k+1} := \sigma(x_k, u_k) \quad (62) $$

instead of (46). In what follows it is shown that, the full state of the closed-loop (42), (50)-(51), with the proposed nominal output $\tilde{s}_{k+1}$ is ultimately bounded.

**Proposition 3.** Consider the perturbed dynamics (42) with feedback law given by (50)-(51) where

$$ v_k = u_{sv}(x_k) \in -\tilde{K} \text{Sgn}(\tilde{K} \tilde{s}_{k+1}). \quad (63) $$

where $\tilde{s}_{k+1}$ is given by (62). Then, the trajectories of the closed-loop are ultimately bounded.

**Proof.** Since $s_{k+1} = \tilde{s}_{k+1} + \tilde{B}^{\top} \tilde{P}^{-1} \tilde{B} \tilde{\delta}_k$, the forward difference (59) becomes

$$ \Delta V(x_k) = (\tilde{s}_{k+1} + \tilde{B}^{\top} \tilde{P}^{-1} \tilde{B} \tilde{\delta}_k)^{\top} (v_k + \tilde{\delta}_k) \\ - \frac{1}{2} (v_k + \tilde{\delta}_k)^{\top} \tilde{B}^{\top} \tilde{P}^{-1} \tilde{B} (v_k + \tilde{\delta}_k) - \frac{1}{2} x_k^{\top} \tilde{Q} x_k. \quad (64) $$

By developing the right-hand side one obtains

$$ \Delta V(x_k) = \tilde{s}_{k+1}^{\top} (v_k + \tilde{\delta}_k) + \tilde{\delta}_k^{\top} \tilde{B}^{\top} \tilde{P}^{-1} \tilde{B} v_k \\ + \tilde{\delta}_k^{\top} \tilde{B}^{\top} \tilde{P}^{-1} \tilde{B} \tilde{\delta}_k - \frac{1}{2} v_k^{\top} \tilde{B}^{\top} \tilde{P}^{-1} \tilde{B} v_k - v_k^{\top} \tilde{B}^{\top} \tilde{P}^{-1} \tilde{B} \tilde{\delta}_k \\ - \frac{1}{2} \tilde{\delta}_k^{\top} \tilde{B}^{\top} \tilde{P}^{-1} \tilde{B} \tilde{\delta}_k - \frac{1}{2} x_k^{\top} \tilde{Q} x_k, \quad (65) $$

which can be simplified as

$$ \begin{aligned} \Delta V(x_k) = \tilde{s}_{k+1}^{\top} (v_k + \tilde{\delta}_k) &+ \frac{1}{2} \tilde{\delta}_k^{\top} \tilde{B}^{\top} \tilde{P}^{-1} \tilde{B} \tilde{\delta}_k \\ &- \frac{1}{2} v_k^{\top} \tilde{B}^{\top} \tilde{P}^{-1} \tilde{B} v_k - \frac{1}{2} x_k^{\top} \tilde{Q} x_k. \end{aligned} \quad (66) $$

Define the set-valued control

$$ u_{sv}(x_k) \in -\tilde{K} \text{Sgn}(\tilde{K} \tilde{s}_{k+1}), \quad \tilde{K} = \tilde{K}^{\top} \succ 0, \quad (67) $$

as a discrete-time counterpart of (32). It gives the inequality

$$ \begin{aligned} \Delta V(x_k) \le & -\|\tilde{K} \tilde{s}_{k+1}\|_1 (1 - \|\tilde{K}^{-1} \tilde{\delta}_k\|_{\infty}) - \frac{1}{2} v_k^{\top} \tilde{B}^{\top} \tilde{P}^{-1} \tilde{B} v_k \\ & - \frac{1}{2} x_k^{\top} \tilde{Q} x_k + \frac{1}{2} \tilde{\delta}_k^{\top} \tilde{B}^{\top} \tilde{P}^{-1} \tilde{B} \tilde{\delta}_k. \end{aligned} \quad (68) $$

Notice that, since $\tilde{\delta}_k$ is uniformly bounded, $\Delta V(x_k)$ is negative for $\|x_k\|$ sufficiently large and $\tilde{K}$ such that $\|\tilde{K}^{-1} \tilde{\delta}_k\|_{\infty} < 1$ for all $k \ge 0$. This establishes that the state is ultimately bounded (exponential stability is achieved only in the nominal case) with a bound proportional to the bound on $\tilde{\delta}_k$. □

### 3.1.2 Discrete-time Sliding Dynamics

In what follows, we show that the nominal output $\tilde{s}_{k+1}$ converges to zero in a finite number of steps. Once it reaches zero, it stays at zero in a discrete-time sliding-motion. To that end, notice that the nominal output satisfies

$$ \tilde{s}_{k+1} \in \tilde{B}^{\top} \tilde{P}^{-1} (\tilde{A}x_k + \tilde{B}u_{\text{lin}}(x_k) - \tilde{B}\tilde{K} \text{Sgn}(\tilde{K}\tilde{s}_{k+1})) \quad (69) $$

or, using (57),

$$ \tilde{s}_{k+1} \in \tilde{B}^{\top} \left( \left( \tilde{P} + \frac{1}{2} \tilde{B} \tilde{B}^{\top} \right)^{-1} \tilde{A} x_k - \tilde{P}^{-1} \tilde{B} \tilde{K} \text{Sgn}(\tilde{K} \tilde{s}_{k+1}) \right). \quad (70) $$

Defining $\tilde{L} = (\tilde{B}^{\top} \tilde{P}^{-1} \tilde{B})^{-\frac{1}{2}} \tilde{B}^{\top} (\tilde{P} + \frac{1}{2} \tilde{B} \tilde{B}^{\top})^{-1} \tilde{A}$ and considering the change of coordinates $\tilde{s}'_k = (\tilde{B}^{\top} \tilde{P}^{-1} \tilde{B})^{-\frac{1}{2}} \tilde{s}_k$ yields

$$ \tilde{s}'_{k+1} \in -(\tilde{B}^{\top} \tilde{P}^{-1} \tilde{B})^{\frac{1}{2}} \tilde{K} \text{Sgn}(\tilde{K}(\tilde{B}^{\top} \tilde{P}^{-1} \tilde{B})^{\frac{1}{2}} \tilde{s}'_{k+1}) + \tilde{L}x_k. \quad (71) $$

The generalized equation (71) shares the same structure as the differential inclusions (22) and (36). Namely, using once again the chain rule of convex analysis (see Fact 8 in “Sidebar VI”) one can see that

$$ \tilde{s}'_{k+1} \in -\partial \tilde{g}(\tilde{s}'_{k+1}) + \tilde{L}x_k \quad (72) $$

with $\tilde{g}(\cdot) = \| \cdot \|_1 \circ \tilde{K}(\tilde{B}^{\top} \tilde{P}^{-1} \tilde{B})^{\frac{1}{2}}(\cdot)$ (formally the same function as $g(\cdot)$ in (36)). This difference inclusion, which is of the implicit type since $\tilde{s}'_{k+1}$ appears on both sides, is equivalently rewritten as the equality

$$ \tilde{s}'_{k+1} = (I_d + \partial \tilde{g})^{-1} (\tilde{L} x_k). \quad (73) $$
It now follows from Lemma 1 that

$$ (\tilde{A}x_k + \tilde{B}u_{\text{lin}}(x_k))^{\top} \tilde{P}^{-1} (\tilde{A}x_k + \tilde{B}u_{\text{lin}}(x_k)) - x_k^{\top} \tilde{P}^{-1} x_k \\ = -x_k^{\top} \tilde{Q} x_k \quad (58) $$

for some positive definite matrix $\tilde{Q}$. The dissipation equality (48) can be written as

$$ \Delta V(x_k) = s_{k+1}^{\top} (v_k + \tilde{\delta}_k) - \frac{1}{2} (v_k + \tilde{\delta}_k)^{\top} \tilde{B}^{\top} \tilde{P}^{-1} \tilde{B} (v_k + \tilde{\delta}_k) \\ - \frac{1}{2} x_k^{\top} \tilde{Q} x_k, \quad (59) $$

which is the counterpart of (31). This shows passivity with the required ingredients. □

### 3.1.1 Stability Analysis

It follows from (59) that the origin is exponentially stable if $v_k$ is such that

$$ s_{k+1}^{\top} (v_k + \tilde{\delta}_k) \le 0. \quad (60) $$

By continuing the analogy with the continuous-time case, we may be tempted to write

$$ v_k = u_{sv}(x_k) \quad (61) $$

with $u_{sv}(x_k) \in -\tilde{K} \text{Sgn}(\tilde{K} s_{k+1})$. Such control does satisfy (60) but, unfortunately, the passive output (46) is not available to the controller, because it depends directly on the unknown perturbation. To avoid this problem let us follow [31] and define the nominal output

$$ \tilde{s}_{k+1} := \sigma(x_k, u_k) \quad (62) $$

instead of (46). In what follows it is shown that, the full state of the closed-loop (42), (50)-(51), with the proposed nominal output $\tilde{s}_{k+1}$ is ultimately bounded.

**Proposition 3.** Consider the perturbed dynamics (42) with feedback law given by (50)-(51) where

$$ v_k = u_{sv}(x_k) \in -\tilde{K} \text{Sgn}(\tilde{K} \tilde{s}_{k+1}). \quad (63) $$

where $\tilde{s}_{k+1}$ is given by (62). Then, the trajectories of the closed-loop are ultimately bounded.

**Proof.** Since $s_{k+1} = \tilde{s}_{k+1} + \tilde{B}^{\top} \tilde{P}^{-1} \tilde{B} \tilde{\delta}_k$, the forward difference (59) becomes

$$ \Delta V(x_k) = (\tilde{s}_{k+1} + \tilde{B}^{\top} \tilde{P}^{-1} \tilde{B} \tilde{\delta}_k)^{\top} (v_k + \tilde{\delta}_k) \\ - \frac{1}{2} (v_k + \tilde{\delta}_k)^{\top} \tilde{B}^{\top} \tilde{P}^{-1} \tilde{B} (v_k + \tilde{\delta}_k) - \frac{1}{2} x_k^{\top} \tilde{Q} x_k. \quad (64) $$

By developing the right-hand side one obtains

$$ \Delta V(x_k) = \tilde{s}_{k+1}^{\top} (v_k + \tilde{\delta}_k) + \tilde{\delta}_k^{\top} \tilde{B}^{\top} \tilde{P}^{-1} \tilde{B} v_k \\ + \tilde{\delta}_k^{\top} \tilde{B}^{\top} \tilde{P}^{-1} \tilde{B} \tilde{\delta}_k - \frac{1}{2} v_k^{\top} \tilde{B}^{\top} \tilde{P}^{-1} \tilde{B} v_k - v_k^{\top} \tilde{B}^{\top} \tilde{P}^{-1} \tilde{B} \tilde{\delta}_k \\ - \frac{1}{2} \tilde{\delta}_k^{\top} \tilde{B}^{\top} \tilde{P}^{-1} \tilde{B} \tilde{\delta}_k - \frac{1}{2} x_k^{\top} \tilde{Q} x_k, \quad (65) $$

which can be simplified as

$$ \begin{aligned} \Delta V(x_k) = \tilde{s}_{k+1}^{\top} (v_k + \tilde{\delta}_k) &+ \frac{1}{2} \tilde{\delta}_k^{\top} \tilde{B}^{\top} \tilde{P}^{-1} \tilde{B} \tilde{\delta}_k \\ &- \frac{1}{2} v_k^{\top} \tilde{B}^{\top} \tilde{P}^{-1} \tilde{B} v_k - \frac{1}{2} x_k^{\top} \tilde{Q} x_k. \end{aligned} \quad (66) $$

Define the set-valued control

$$ u_{sv}(x_k) \in -\tilde{K} \text{Sgn}(\tilde{K} \tilde{s}_{k+1}), \quad \tilde{K} = \tilde{K}^{\top} > 0, \quad (67) $$

as a discrete-time counterpart of (32). It gives the inequality

$$ \begin{aligned} \Delta V(x_k) \le & -\|\tilde{K} \tilde{s}_{k+1}\|_1 (1 - \|\tilde{K}^{-1} \tilde{\delta}_k\|_{\infty}) - \frac{1}{2} v_k^{\top} \tilde{B}^{\top} \tilde{P}^{-1} \tilde{B} v_k \\ & - \frac{1}{2} x_k^{\top} \tilde{Q} x_k + \frac{1}{2} \tilde{\delta}_k^{\top} \tilde{B}^{\top} \tilde{P}^{-1} \tilde{B} \tilde{\delta}_k. \end{aligned} \quad (68) $$

Notice that, since $\tilde{\delta}_k$ is uniformly bounded, $\Delta V(x_k)$ is negative for $\|x_k\|$ sufficiently large and $\tilde{K}$ such that $\|\tilde{K}^{-1} \tilde{\delta}_k\|_{\infty} < 1$ for all $k \ge 0$. This establishes that the state is ultimately bounded (exponential stability is achieved only in the nominal case) with a bound proportional to the bound on $\tilde{\delta}_k$. □

### 3.1.2 Discrete-time Sliding Dynamics

In what follows, we show that the nominal output $\tilde{s}_{k+1}$ converges to zero in a finite number of steps. Once it reaches zero, it stays at zero in a discrete-time sliding-motion. To that end, notice that the nominal output satisfies

$$ \tilde{s}_{k+1} \in \tilde{B}^{\top} \tilde{P}^{-1} (\tilde{A}x_k + \tilde{B}u_{\text{lin}}(x_k) - \tilde{B}\tilde{K} \text{Sgn}(\tilde{K}\tilde{s}_{k+1})) \quad (69) $$

or, using (57),

$$ \tilde{s}_{k+1} \in \tilde{B}^{\top} \left( \left( \tilde{P} + \frac{1}{2} \tilde{B} \tilde{B}^{\top} \right)^{-1} \tilde{A} x_k - \tilde{P}^{-1} \tilde{B} \tilde{K} \text{Sgn}(\tilde{K} \tilde{s}_{k+1}) \right). \quad (70) $$

Defining $\tilde{L} = (\tilde{B}^{\top} \tilde{P}^{-1} \tilde{B})^{-\frac{1}{2}} \tilde{B}^{\top} (\tilde{P} + \frac{1}{2} \tilde{B} \tilde{B}^{\top})^{-1} \tilde{A}$ and considering the change of coordinates $\tilde{s}'_k = (\tilde{B}^{\top} \tilde{P}^{-1} \tilde{B})^{-\frac{1}{2}} \tilde{s}_k$ yields

$$ \tilde{s}'_{k+1} \in -(\tilde{B}^{\top} \tilde{P}^{-1} \tilde{B})^{\frac{1}{2}} \tilde{K} \text{Sgn}(\tilde{K}(\tilde{B}^{\top} \tilde{P}^{-1} \tilde{B})^{\frac{1}{2}} \tilde{s}'_{k+1}) + \tilde{L}x_k. \quad (71) $$

The generalized equation (71) shares the same structure as the differential inclusions (22) and (36). Namely, using once again the chain rule of convex analysis (see Fact 8 in “Sidebar VI”) one can see that

$$ \tilde{s}'_{k+1} \in -\partial \tilde{g}(\tilde{s}'_{k+1}) + \tilde{L}x_k \quad (72) $$

with $\tilde{g}(\cdot) = \| \cdot \|_1 \circ \tilde{K} (\tilde{B}^{\top} \tilde{P}^{-1} \tilde{B})^{\frac{1}{2}} (\cdot)$ (formally the same function as $g(\cdot)$ in (36)). This difference inclusion, which is of the implicit type since $\tilde{s}'_{k+1}$ appears on both sides, is equivalently rewritten as the equality

$$ \tilde{s}'_{k+1} = (I_d + \partial \tilde{g})^{-1} (\tilde{L} x_k). \quad (73) $$
Let us stop and explain the meaning of the right-hand side of (73) using the material in the “Sidebar VII”, “Sidebar V”, and “Sidebar VI” sidebars. Unquestionably, the structure of this difference equation will be encountered again in all the discretizations presented in the following sections (for both controllers and differentiators). Notice first that the right-hand side of (73) is in fact a proximal operator,

$$ \tilde{s}'_{k+1} = \operatorname{Prox}_{\tilde{g}}(\tilde{L}x_k) \quad (74) $$

(see (50) in “Sidebar VII”).

**Proposition 4.** Consider the virtual sliding variable (74). It follows that $\tilde{s}'_{k+1} = 0$ if, and only if,

$$ \tilde{K}^{-1}(\tilde{B}^{\top}\tilde{P}^{-1}\tilde{B})^{-\frac{1}{2}}\tilde{L}x_k \in \mathcal{B}_{\infty}, \quad (75) $$

where the unit ball $\mathcal{B}_{\infty}$ is defined in (38).

*Proof.* The claim follows from (54) in “Sidebar VII”. Indeed zero $\operatorname{Prox}_{\tilde{g}} = \partial \tilde{g}(0) = (\tilde{B}^{\top}\tilde{P}^{-1}\tilde{B})^{\frac{1}{2}}\tilde{K}\mathcal{B}_{\infty}$, where the equality follows from Fact 8 and from (40) in “Sidebar VI”, and the set of zeroes is defined in “Sidebar VII”. Thus, it follows from (75) that $\tilde{L}x_k \in \partial \tilde{g}(0)$ and the proof is complete. □

If the eigenvalues of $\tilde{K}$ are chosen sufficiently large, then the inclusion (75) holds once the ultimate bound in $x_k$ is attained. This implies that the condition $\tilde{s}'_{k+1} = 0$ is achieved and maintained after a finite number of steps.

**Remark 4.** The difference inclusion (69) is the counterpart of the differential inclusion (34), while (71) or (72) are the counterparts of (36). The equality in (73) is the explicit form of the implicit (backward-Euler) algorithm (72).

Following [21] we define the discrete-time sliding motion in terms of the nominal variable $\tilde{s}_{k+1}$.

**Definition 5.** The closed-loop system (42), (50), (51), (63) reaches a discrete-time sliding motion whenever there is a finite $k^* \in \mathbb{N}$ such that

$$ \tilde{s}_k = 0, \quad (76) $$

holds for all $k \ge k^*$.

**Proposition 5.** During the discrete-time sliding mode, the controller $u_k = u_{\text{lin}}(x_k) + u_{\text{sv}}(x_k)$ satisfies

$$ u_k = -\tilde{\delta}_{k-1} - (\tilde{B}^{\top}\tilde{P}^{-1}\tilde{B})^{-1}\tilde{B}\tilde{P}^{-1}(\tilde{A} - I)x_k. \quad (77) $$

*Proof.* It follows from (69) and (49) that

$$ \tilde{s}_{k+1} = s_k + \tilde{B}^{\top}\tilde{P}^{-1}(\tilde{A} - I)x_k + \tilde{B}^{\top}\tilde{P}^{-1}\tilde{B}u_k, \quad (78) $$

whereas the definition of $\tilde{s}_{k+1}$ yields

$$ s_{k+1} = \tilde{s}_{k+1} + \tilde{B}^{\top}\tilde{P}^{-1}\tilde{B}\tilde{\delta}_k. \quad (79) $$

Thus, the substitution of (79) back into (78), with a one-step delay, leads us to the expression

$$ \tilde{s}_{k+1} = \tilde{s}_k + \tilde{B}^{\top}\tilde{P}^{-1}\tilde{B}\tilde{\delta}_{k-1} + \tilde{B}^{\top}\tilde{P}^{-1}(\tilde{A} - I)x_k + \tilde{B}^{\top}\tilde{P}^{-1}\tilde{B}u_k \quad (80) $$

The conclusion follows directly by setting $\tilde{s}_{k+1} = \tilde{s}_k = 0$. □

This last result shows that the exact compensation in continuous-time (39) can be well approximated by the implicit algorithm. The controller during the sliding phase is independent of $\tilde{K}$, as predicted by the continuous-time analysis.

### 3.1.3 Control Calculation and Input–Sliding-Variable Conjugacy

The feedback control and future sliding-variable are implicitly defined by (62), (67). Such expressions are useful for stability analysis of closed-loop trajectories, but not for implementation. For the latter, an explicit expression is needed. To obtain such expression, the duality between $\tilde{s}_{k+1}$ and $-u_{\text{sv}}(x_k)$ can be exploited. Proposition 7 provides different descriptions of the sliding variable $\tilde{s}_{k+1}$ and the control input $u_{\text{sv}}(x_k)$. Depending on the problem at hand, some expressions may result simpler than others.

**Proposition 7.** Let $\Lambda_1 \in \mathbb{R}^{m \times n}$, $\Lambda_2 = \Lambda_2^\top \in \mathbb{R}^{m \times m}$ be non-singular, and let $\mathbf{M}: \mathbb{R}^m \to \mathbb{R}^m$ be maximal monitone. Then, the control input and sliding output expressions shown in Figure 4 are equivalent.

*Proof.* The equivalence between the implicit expressions i) and ii) is immediate. It suffices to invert $\Lambda_1$ and $\mathbf{M}$. Let us show the equivalence between i) and iii). To that end, let $\tilde{\mathbf{M}} = \Lambda_2^{\frac{1}{2}} \circ \mathbf{M} \circ \Lambda_2^{\frac{1}{2}}$. The following inclusions are equivalent:

$$ \begin{aligned} \tilde{s}_{k+1} &\in \Lambda_1 x_k - \Lambda_2 \mathbf{M}(\tilde{s}_{k+1}) \\ \tilde{s}_{k+1} &\in (\Lambda_2^{-1} + \mathbf{M})^{-1} (\Lambda_2^{-1} \Lambda_1 x_k) \\ \tilde{s}_{k+1} &\in (\Lambda_2^{-\frac{1}{2}} \circ (\mathbf{I}_d + \tilde{\mathbf{M}}) \circ \Lambda_2^{-\frac{1}{2}})^{-1} (\Lambda_2^{-1} \Lambda_1 x_k) \\ \tilde{s}_{k+1} &\in (\Lambda_2^{\frac{1}{2}} \circ (\mathbf{I}_d + \tilde{\mathbf{M}})^{-1} \circ \Lambda_2^{-\frac{1}{2}})^{-1} (\Lambda_1 x_k) \\ \tilde{s}_{k+1} &= \Lambda_2^{\frac{1}{2}} \mathcal{J}_{\tilde{\mathbf{M}}} (\Lambda_2^{-\frac{1}{2}} \Lambda_1 x_k), \end{aligned} \quad (81) $$

where the fact that the resolvent $\mathcal{J}_{\tilde{\mathbf{M}}}$ is single-valued in the last expression is used. Now, the explicit expression for the control input is given as

$$ \begin{aligned} -u_{\text{sv}}(x_k) &= \Lambda_2^{-1} (\Lambda_1 x_k - \tilde{s}_{k+1}) \\ &= \Lambda_2^{-\frac{1}{2}} (\mathbf{I}_d - \mathcal{J}_{\tilde{\mathbf{M}}}) (\Lambda_2^{-\frac{1}{2}} \Lambda_1 x_k) \\ &= \Lambda_2^{-\frac{1}{2}} \mathcal{Y}_{\tilde{\mathbf{M}}} (\Lambda_2^{-\frac{1}{2}} \Lambda_1 x_k). \end{aligned} \quad (82) $$
Let us stop and explain the meaning of the right-hand side of (73) using the material in the “Sidebar VII”, “Sidebar V”, and “Sidebar VI” sidebars. Unquestionably, the structure of this difference equation will be encountered again in all the discretizations presented in the following sections (for both controllers and differentiators). Notice first that the right-hand side of (73) is in fact a proximal operator,

$$ \tilde{s}'_{k+1} = \operatorname{Prox}_{\tilde{g}}(\tilde{L}x_k) \quad (74) $$

(see (50) in “Sidebar VII”).

**Proposition 4.** Consider the virtual sliding variable (74). It follows that $\tilde{s}'_{k+1} = 0$ if, and only if,

$$ \tilde{K}^{-1}(\tilde{B}^{\top}\tilde{P}^{-1}\tilde{B})^{-\frac{1}{2}}\tilde{L}x_k \in \mathcal{B}_{\infty}, \quad (75) $$

where the unit ball $\mathcal{B}_{\infty}$ is defined in (38).

*Proof.* The claim follows from (54) in “Sidebar VII”. Indeed zero $\operatorname{Prox}_{\tilde{g}} = \partial \tilde{g}(0) = (\tilde{B}^{\top}\tilde{P}^{-1}\tilde{B})^{\frac{1}{2}}\tilde{K}\mathcal{B}_{\infty}$, where the equality follows from Fact 8 and from (40) in “Sidebar VI”, and the set of zeroes is defined in “Sidebar VII”. Thus, it follows from (75) that $\tilde{L}x_k \in \partial \tilde{g}(0)$ and the proof is complete. □

If the eigenvalues of $\tilde{K}$ are chosen sufficiently large, then the inclusion (75) holds once the ultimate bound in $x_k$ is attained. This implies that the condition $\tilde{s}'_{k+1} = 0$ is achieved and maintained after a finite number of steps.

**Remark 4.** The difference inclusion (69) is the counterpart of the differential inclusion (34), while (71) or (72) are the counterparts of (36). The equality in (73) is the explicit form of the implicit (backward-Euler) algorithm (72).

Following [21] we define the discrete-time sliding motion in terms of the nominal variable $\tilde{s}_{k+1}$.

**Definition 5.** The closed-loop system (42), (50), (51), (63) reaches a discrete-time sliding motion whenever there is a finite $k^* \in \mathbb{N}$ such that

$$ \tilde{s}_k = 0, \quad (76) $$

holds for all $k \ge k^*$.

**Proposition 5.** During the discrete-time sliding mode, the controller $u_k = u_{\text{lin}}(x_k) + u_{\text{sv}}(x_k)$ satisfies

$$ u_k = -\tilde{\delta}_{k-1} - (\tilde{B}^{\top}\tilde{P}^{-1}\tilde{B})^{-1}\tilde{B}\tilde{P}^{-1}(\tilde{A} - I)x_k. \quad (77) $$

*Proof.* It follows from (69) and (49) that

$$ \tilde{s}_{k+1} = s_k + \tilde{B}^{\top}\tilde{P}^{-1}(\tilde{A} - I)x_k + \tilde{B}^{\top}\tilde{P}^{-1}\tilde{B}u_k, \quad (78) $$

whereas the definition of $\tilde{s}_{k+1}$ yields

$$ s_{k+1} = \tilde{s}_{k+1} + \tilde{B}^{\top}\tilde{P}^{-1}\tilde{B}\tilde{\delta}_k. \quad (79) $$

Thus, the substitution of (79) back into (78), with a one-step delay, leads us to the expression

$$ \tilde{s}_{k+1} = \tilde{s}_k + \tilde{B}^{\top}\tilde{P}^{-1}\tilde{B}\tilde{\delta}_{k-1} + \tilde{B}^{\top}\tilde{P}^{-1}(\tilde{A} - I)x_k + \tilde{B}^{\top}\tilde{P}^{-1}\tilde{B}u_k \quad (80) $$

The conclusion follows directly by setting $\tilde{s}_{k+1} = \tilde{s}_k = 0$. □

This last result shows that the exact compensation in continuous-time (39) can be well approximated by the implicit algorithm. The controller during the sliding phase is independent of $\tilde{K}$, as predicted by the continuous-time analysis.

### 3.1.3 Control Calculation and Input–Sliding-Variable Conjugacy

The feedback control and future sliding-variable are implicitly defined by (62), (67). Such expressions are useful for stability analysis of closed-loop trajectories, but not for implementation. For the latter, an explicit expression is needed. To obtain such expression, the duality between $\tilde{s}_{k+1}$ and $-u_{\text{sv}}(x_k)$ can be exploited. Proposition 7 provides different descriptions of the sliding variable $\tilde{s}_{k+1}$ and the control input $u_{\text{sv}}(x_k)$. Depending on the problem at hand, some expressions may result simpler than others.

**Proposition 7.** Let $\Lambda_1 \in \mathbb{R}^{m \times n}$, $\Lambda_2 = \Lambda_2^\top \in \mathbb{R}^{m \times m}$ be non-singular, and let $\mathbf{M}: \mathbb{R}^m \to \mathbb{R}^m$ be maximal monitone. Then, the control input and sliding output expressions shown in Figure 4 are equivalent.

*Proof.* The equivalence between the implicit expressions i) and ii) is immediate. It suffices to invert $\Lambda_1$ and $\mathbf{M}$. Let us show the equivalence between i) and iii). To that end, let $\tilde{\mathbf{M}} = \Lambda_2^{\frac{1}{2}} \circ \mathbf{M} \circ \Lambda_2^{\frac{1}{2}}$. The following inclusions are equivalent:

$$ \begin{aligned} \tilde{s}_{k+1} &\in \Lambda_1 x_k - \Lambda_2 \mathbf{M}(\tilde{s}_{k+1}) \\ \tilde{s}_{k+1} &\in (\Lambda_2^{-1} + \mathbf{M})^{-1} (\Lambda_2^{-1} \Lambda_1 x_k) \\ \tilde{s}_{k+1} &\in (\Lambda_2^{-\frac{1}{2}} \circ (\mathbf{I}_d + \tilde{\mathbf{M}}) \circ \Lambda_2^{-\frac{1}{2}})^{-1} (\Lambda_2^{-1} \Lambda_1 x_k) \\ \tilde{s}_{k+1} &\in (\Lambda_2^{\frac{1}{2}} \circ (\mathbf{I}_d + \tilde{\mathbf{M}})^{-1} \circ \Lambda_2^{-\frac{1}{2}})^{-1} (\Lambda_1 x_k) \\ \tilde{s}_{k+1} &= \Lambda_2^{\frac{1}{2}} \mathcal{J}_{\tilde{\mathbf{M}}} (\Lambda_2^{-\frac{1}{2}} \Lambda_1 x_k), \end{aligned} \quad (81) $$

where the fact that the resolvent $\mathcal{J}_{\tilde{\mathbf{M}}}$ is single-valued in the last expression is used. Now, the explicit expression for the control input is given as

$$ \begin{aligned} -u_{\text{sv}}(x_k) &= \Lambda_2^{-1} (\Lambda_1 x_k - \tilde{s}_{k+1}) \\ &= \Lambda_2^{-\frac{1}{2}} (\mathbf{I}_d - \mathcal{J}_{\tilde{\mathbf{M}}}) (\Lambda_2^{-\frac{1}{2}} \Lambda_1 x_k) \\ &= \Lambda_2^{-\frac{1}{2}} \mathcal{Y}_{\tilde{\mathbf{M}}} (\Lambda_2^{-\frac{1}{2}} \Lambda_1 x_k). \end{aligned} \quad (82) $$
The following results and definitions are taken from classical references [S16, S18, S22, S28, 60]. Most of the tools below extend to non-convex sets and functions. However, it is essentially the convex case that is of interest to us in this article. In convex analysis, it is useful to extend the space $\mathbb{R}$ by allowing a function to take the value $+\infty$. A function $f: \mathbb{R}^n \to \mathbb{R} \cup \{+\infty\}$ is said to be proper if it is not identically infinite.

**Definition 6.** A proper convex function $f: \mathbb{R}^n \to \mathbb{R} \cup \{+\infty\}$ is lower semicontinuous (LSC) if its epigraph is closed.

**Definition 7.** Let $f: \mathbb{R}^n \to \mathbb{R} \cup \{+\infty\}$ be a proper, convex, and LSC function. The convex subdifferential of $f$ at the point $x$ is given by

$$ \partial f(x) := \{\eta \in \mathbb{R}^n \mid \langle \eta, \xi - x \rangle \le f(\xi) - f(x) \text{ for all } \xi \in \text{dom } f\}. \quad (29) $$

Note that $\partial f: \mathbb{R}^n \to \mathbb{R}^n$ is in general a set-valued mapping. Consider, for example the real-valued function $f(x) = \kappa|x|$ with $\kappa > 0$. Then, $\partial f(x) = \kappa \text{sgn}(x)$ with

$$ \text{sgn}(x) = \begin{cases} -1 & \text{if } x < 0 \\ [-1, 1] & \text{if } x = 0. \\ 1 & \text{if } x > 0 \end{cases} \quad (30) $$

**Definition 8.** Let $C \subseteq \mathbb{R}^n$ be a closed convex set. Its normal cone is the set-valued mapping $N_C: \mathbb{R}^n \to \mathbb{R}^n$ defined by

$$ N_C(x) = \{\eta \in \mathbb{R}^n \mid \langle \eta, \xi - x \rangle \le 0 \text{ for all } \xi \in C\} \quad (31) $$

if $x \in C$ and $N_K(C) = \emptyset$ if $x \notin C$.

The normal cone to a closed convex set is convex-valued. Suppose that $C$ is finitely represented, that is, $C = \{\xi \in \mathbb{R}^n \mid g(\xi) \ge 0\}$ for some differentiable function $g: \mathbb{R}^n \to \mathbb{R}^m$. Assume further that the Mangasarian-Fromovitz constraint qualification holds [S18]. Then, $N_C(x)$ is generated by the outward normals $\nabla g_i(x)$ to $C$ at the active constraints $g_i(x) = 0, i \in \{1, \dots, m\}$. In other words, $N_C(x) = \{\eta \in \mathbb{R}^n \mid \eta = -\nabla g(x)\lambda, 0 \le \lambda \perp g(x) \ge 0\}$. It is noteworthy that weighted inner products can be used in the normal-cone definition. For instance, $\langle \eta, \xi - x \rangle_M$ with $M = M^\top > 0$ can be used in (31), in which case the normals are calculated as $M^{-1}\nabla g_i(x)$.

**Definition 9.** Let $C \subseteq \mathbb{R}^n$ be a closed convex set. Its tangent cone is the set-valued mapping $T_C: \mathbb{R}^n \to \mathbb{R}^n$ defined by

$$ T_C(x) = \{\xi \in \mathbb{R}^n \mid \langle \eta, \xi \rangle \le 0, \text{ for all } \eta \in N_C(x)\}. \quad (32) $$

This means that the normal and tangent cones are polar to each other.

**Fact 4.** The normal cone to a closed convex set defines a maximal monotone mapping.

**Definition 10.** Let $C \subseteq \mathbb{R}^n$ be a set. Its indicator function is defined as $\psi_C(x) = 0$ if $x \in C$ and $\Psi_C(x) = +\infty$ if $x \notin C$.

**Fact 5.** Let $C \subseteq \mathbb{R}^n$ be a closed convex set. Then, $\Psi_C$ is proper, convex, LSC, and $\partial \Psi_C(x) = N_C(x)$ for all $x \in \mathbb{R}^n$.

**Definition 11.** Let $f: \mathbb{R}^n \to \mathbb{R} \cup \{+\infty\}$ be a proper, convex, and LSC function. Its convex conjugate, $f^*: \mathbb{R}^n \to \mathbb{R}$, is defined as

$$ f^*(y) = \sup_{\xi \in \mathbb{R}^n} \{\xi^\top y - f(\xi)\}. \quad (33) $$
The following results and definitions are taken from classical references [S16, S18, S22, S28, 60]. Most of the tools below extend to non-convex sets and functions. However, it is essentially the convex case that is of interest to us in this article. In convex analysis, it is useful to extend the space $\mathbb{R}$ by allowing a function to take the value $+\infty$. A function $f: \mathbb{R}^n \to \mathbb{R} \cup \{+\infty\}$ is said to be proper if it is not identically infinite.

**Definition 6.** A proper convex function $f: \mathbb{R}^n \to \mathbb{R} \cup \{+\infty\}$ is lower semicontinuous (LSC) if its epigraph is closed.

**Definition 7.** Let $f: \mathbb{R}^n \to \mathbb{R} \cup \{+\infty\}$ be a proper, convex, and LSC function. The convex subdifferential of $f$ at the point $x$ is given by

$$ \partial f(x) := \{\eta \in \mathbb{R}^n \mid \langle \eta, \xi - x \rangle \le f(\xi) - f(x) \text{ for all } \xi \in \text{dom } f\}. \quad (29) $$

Note that $\partial f: \mathbb{R}^n \to \mathbb{R}^n$ is in general a set-valued mapping. Consider, for example the real-valued function $f(x) = \kappa|x|$ with $\kappa > 0$. Then, $\partial f(x) = \kappa \text{sgn}(x)$ with

$$ \text{sgn}(x) = \begin{cases} -1 & \text{if } x < 0 \\ [-1, 1] & \text{if } x = 0. \\ 1 & \text{if } x > 0 \end{cases} \quad (30) $$

**Definition 8.** Let $C \subseteq \mathbb{R}^n$ be a closed convex set. Its normal cone is the set-valued mapping $N_C: \mathbb{R}^n \to \mathbb{R}^n$ defined by

$$ N_C(x) = \{\eta \in \mathbb{R}^n \mid \langle \eta, \xi - x \rangle \le 0 \text{ for all } \xi \in C\} \quad (31) $$

if $x \in C$ and $N_K(C) = \emptyset$ if $x \notin C$.

The normal cone to a closed convex set is convex-valued. Suppose that $C$ is finitely represented, that is, $C = \{\xi \in \mathbb{R}^n \mid g(\xi) \ge 0\}$ for some differentiable function $g: \mathbb{R}^n \to \mathbb{R}^m$. Assume further that the Mangasarian-Fromovitz constraint qualification holds [S18]. Then, $N_C(x)$ is generated by the outward normals $\nabla g_i(x)$ to $C$ at the active constraints $g_i(x) = 0, i \in \{1, \dots, m\}$. In other words, $N_C(x) = \{\eta \in \mathbb{R}^n \mid \eta = -\nabla g(x)\lambda, 0 \le \lambda \perp g(x) \ge 0\}$. It is noteworthy that weighted inner products can be used in the normal-cone definition. For instance, $\langle \eta, \xi - x \rangle_M$ with $M = M^\top > 0$ can be used in (31), in which case the normals are calculated as $M^{-1}\nabla g_i(x)$.

**Definition 9.** Let $C \subseteq \mathbb{R}^n$ be a closed convex set. Its tangent cone is the set-valued mapping $T_C: \mathbb{R}^n \to \mathbb{R}^n$ defined by

$$ T_C(x) = \{\xi \in \mathbb{R}^n \mid \langle \eta, \xi \rangle \le 0, \text{ for all } \eta \in N_C(x)\}. \quad (32) $$

This means that the normal and tangent cones are polar to each other.

**Fact 4.** The normal cone to a closed convex set defines a maximal monotone mapping.

**Definition 10.** Let $C \subseteq \mathbb{R}^n$ be a set. Its indicator function is defined as $\psi_C(x) = 0$ if $x \in C$ and $\Psi_C(x) = +\infty$ if $x \notin C$.

**Fact 5.** Let $C \subseteq \mathbb{R}^n$ be a closed convex set. Then, $\Psi_C$ is proper, convex, LSC, and $\partial \Psi_C(x) = N_C(x)$ for all $x \in \mathbb{R}^n$.

**Definition 11.** Let $f: \mathbb{R}^n \to \mathbb{R} \cup \{+\infty\}$ be a proper, convex, and LSC function. Its convex conjugate, $f^*: \mathbb{R}^n \to \mathbb{R}$, is defined as

$$ f^*(y) = \sup_{\xi \in \mathbb{R}^n} \{\xi^\top y - f(\xi)\}. \quad (33) $$
**Fact 6.** The convex conjugate of a proper convex LSC function, is also proper, convex and LSC. Moreover, $(f^*)^* = f$.

**Fact 7.** Let $f: \mathbb{R}^n \to \mathbb{R} \cup \{+\infty\}$ be proper, convex, and LSC. Then, $(\partial f)^{-1} = \partial f^*$ and $\partial f = (\partial f^*)^{-1}$. In other words: $y \in \partial f(x)$ if, and only if, $x \in \partial f^*(y)$.

**Theorem 5.** [S28, Theorem 16.23] Let $f: \mathbb{R}^n \to \mathbb{R} \cup \{+\infty\}$ be proper, convex, and LSC. Then, the following statements are equivalent:

$$ 
\begin{align} 
(x, y) \in \text{gph} \partial f \tag{34} \\ 
(y, x) \in \text{gph} \partial f^* \notag \\ 
x^\top y = f(x) + f^*(y) \notag 
\end{align} 
$$

Consider again the real-valued function $f(x) = \kappa|x|$ with $\kappa > 0$. Its convex conjugate is

$$ f^*(y) = \sup_{\xi \in \mathbb{R}} \{\xi^\top y - \kappa|\xi|\}. \quad (35) $$

By concavity, the supremum is actually a maximum whenever the subdifferential of $\xi^\top y - \kappa|\xi|$ contains the origin, that is, whenever $y \in [\kappa, \kappa]$ a maximum exists and $f^*(y) = 0$. On the other hand, when $y \notin [-\kappa, \kappa]$ there is no maximum and, by piecewise linearity, $f^*(y) = +\infty$. In other words, the convex conjugate equals the indicator function of a closed interval, $f^* = \Psi_{[-\kappa, \kappa]}$. Fact 5 is easy to verify in this example:

$$ \partial \Psi_{[-\kappa, \kappa]}(y) = \mathbf{N}_{[-\kappa, \kappa]}(y) = \begin{cases} (-\infty, 0] & \text{if } y = -\kappa \\ 0 & \text{if } y \in (-\kappa, \kappa) \\ [0, +\infty) & \text{if } y = \kappa \\ \emptyset & \text{otherwise} \end{cases}. \quad (36) $$

Verifying Fact 7 is also straightforward,

$$ \mathbf{N}_{[-\kappa, \kappa]} = (\kappa \operatorname{sgn})^{-1}. \quad (37) $$

More generally, let $\|x\|_p$ be the $p$ norm $(\sum_{i=1}^n |x_i|^p)^{1/p}$, $1 \le p < +\infty$, and $\|x\|_\infty = \max_{1 \le i \le n} |x_i|$. The dual norm is $\|x\|_{p^*}$ with $1/p + 1/p^* = 1$. It is known that, if $f = \kappa \| \cdot \|_{p^*}$, $\kappa > 0$, then $f^* = \Psi_{\kappa B_{p^*}}$, where

$$ B_{p^*} = \{\xi \in \mathbb{R}^n \mid \|\xi\|_{p^*} \le 1\} \quad (38) $$

is the unit ball in the $p^*$ norm. Thus, by Facts 7 and 5,

$$ \partial \kappa \| \cdot \|_p = (\partial \Psi_{\kappa B_{p^*}})^{-1} = (\mathbf{N}_{\kappa B_{p^*}})^{-1}. \quad (39) $$

Setting, for example $p = 1$ gives $p^* = \infty$, $B_\infty = [-1, 1]^n$ and

$$ \kappa \operatorname{Sgn}(x) = \mathbf{N}_{\kappa B_\infty}^{-1}(x) \quad (40) $$

with $\operatorname{Sgn}(x) = (\operatorname{sgn}(x_1), \operatorname{sgn}(x_2), \dots, \operatorname{sgn}(x_m))^\top$. For $p = 2$ we have

$$ \mathbf{N}_{\kappa B_2}^{-1}(x) = \kappa \partial \|x\|_2 = \begin{cases} \kappa x / \|x\|_2 & \text{if } x \neq 0 \\ \kappa B_2 & \text{if } x = 0. \end{cases} \quad (41) $$

The following result, known as the chain rule of convex analysis, is used at several places in this article.

**Fact 8.** Let $f: \mathbb{R}^n \to \mathbb{R}$ be a proper convex LSC function, and let $A: \mathbb{R}^m \to \mathbb{R}^n$ be a linear mapping. Assume that $f$ is a polyhedral function (that is, its epigraph is a polyhedral set), or that there exists $x_0$ with $Ax_0 \in \text{dom}(f)$ such that $\text{rg}(A) - \mathbb{R}_+(\text{dom}(f) - Ax_0)$ is a vector subspace of $\mathbb{R}^n$. Then, the subdifferential in the sense of Convex Analysis of the composite function $f \circ A: \mathbb{R}^m \to \mathbb{R} \cup \{+\infty\}$ is given by $\partial(f \circ A)(x) = A^\top \partial f(Ax)$ for all $x \in \mathbb{R}^m$.
**Fact 6.** The convex conjugate of a proper convex LSC function, is also proper, convex and LSC. Moreover, $(f^*)^* = f$.

**Fact 7.** Let $f: \mathbb{R}^n \to \mathbb{R} \cup \{+\infty\}$ be proper, convex, and LSC. Then, $(\partial f)^{-1} = \partial f^*$ and $\partial f = (\partial f^*)^{-1}$. In other words: $y \in \partial f(x)$ if, and only if, $x \in \partial f^*(y)$.

**Theorem 5.** [S28, Theorem 16.23] Let $f: \mathbb{R}^n \to \mathbb{R} \cup \{+\infty\}$ be proper, convex, and LSC. Then, the following statements are equivalent:

$$ (x, y) \in \text{gph} \partial f \\ (y, x) \in \text{gph} \partial f^* \\ x^\top y = f(x) + f^*(y) \tag{34} $$

Consider again the real-valued function $f(x) = \kappa|x|$ with $\kappa > 0$. Its convex conjugate is

$$ f^*(y) = \sup_{\xi \in \mathbb{R}} \{\xi^\top y - \kappa|\xi|\}. \tag{35} $$

By concavity, the supremum is actually a maximum whenever the subdifferential of $\xi^\top y - \kappa|\xi|$ contains the origin, that is, whenever $y \in [\kappa, \kappa]$ a maximum exists and $f^*(y) = 0$. On the other hand, when $y \notin [-\kappa, \kappa]$ there is no maximum and, by piecewise linearity, $f^*(y) = +\infty$. In other words, the convex conjugate equals the indicator function of a closed interval, $f^* = \Psi_{[-\kappa, \kappa]}$. Fact 5 is easy to verify in this example:

$$ \partial \Psi_{[-\kappa, \kappa]}(y) = \mathbf{N}_{[-\kappa, \kappa]}(y) = \begin{cases} (-\infty, 0] & \text{if } y = -\kappa \\ 0 & \text{if } y \in (-\kappa, \kappa) \\ [0, +\infty) & \text{if } y = \kappa \\ \emptyset & \text{otherwise} \end{cases}. \tag{36} $$

Verifying Fact 7 is also straightforward,

$$ \mathbf{N}_{[-\kappa, \kappa]} = (\kappa \operatorname{sgn})^{-1}. \tag{37} $$

More generally, let $\|x\|_p$ be the $p$ norm $(\sum_{i=1}^n |x_i|^p)^{1/p}$, $1 \le p < +\infty$, and $\|x\|_\infty = \max_{1 \le i \le n} |x_i|$. The dual norm is $\|x\|_{p^*}$ with $1/p + 1/p^* = 1$. It is known that, if $f = \kappa \|\cdot\|_{p^*}$, $\kappa > 0$, then $f^* = \Psi_{\kappa \mathcal{B}_{p^*}}$, where

$$ \mathcal{B}_{p^*} = \{\xi \in \mathbb{R}^n \mid \|\xi\|_{p^*} \le 1\} \tag{38} $$

is the unit ball in the $p^*$ norm. Thus, by Facts 7 and 5,

$$ \partial \kappa \| \cdot \|_p = (\partial \Psi_{\kappa \mathcal{B}_{p^*}})^{-1} = (\mathbf{N}_{\kappa \mathcal{B}_{p^*}})^{-1}. \tag{39} $$

Setting, for example $p = 1$ gives $p^* = \infty$, $\mathcal{B}_\infty = [-1, 1]^n$ and

$$ \kappa \operatorname{Sgn}(x) = \mathbf{N}_{\kappa \mathcal{B}_\infty}^{-1}(x) \tag{40} $$

with $\operatorname{Sgn}(x) = (\operatorname{sgn}(x_1), \operatorname{sgn}(x_2), \dots, \operatorname{sgn}(x_m))^\top$. For $p = 2$ we have

$$ \mathbf{N}_{\kappa \mathcal{B}_2}^{-1}(x) = \kappa \partial \|x\|_2 = \begin{cases} \kappa x / \|x\|_2 & \text{if } x \neq 0 \\ \kappa \mathcal{B}_2 & \text{if } x = 0. \end{cases} \tag{41} $$

The following result, known as the chain rule of convex analysis, is used at several places in this article.

**Fact 8.** Let $f: \mathbb{R}^n \to \mathbb{R}$ be a proper convex LSC function, and let $A: \mathbb{R}^m \to \mathbb{R}^n$ be a linear mapping. Assume that $f$ is a polyhedral function (that is, its epigraph is a polyhedral set), or that there exists $x_0$ with $Ax_0 \in \text{dom}(f)$ such that $\text{rg}(A) - \mathbb{R}_+(\text{dom}(f) - Ax_0)$ is a vector subspace of $\mathbb{R}^n$. Then, the subdifferential in the sense of Convex Analysis of the composite function $f \circ A: \mathbb{R}^m \to \mathbb{R} \cup \{+\infty\}$ is given by $\partial(f \circ A)(x) = A^\top \partial f(Ax)$ for all $x \in \mathbb{R}^m$.
The projection of a point $x \in \mathbb{R}^n$ onto the nonempty closed convex set $C \subseteq \mathbb{R}^n$ with inner product weighted by $M = M^\top > 0$ is defined by

$$ \mathrm{Proj}_M[C; x] = \arg\min_{\xi \in C} \frac{1}{2} \langle \xi - x, \xi - x \rangle_M. \qquad (42) $$

For simplicity let us write $\mathrm{Proj}_I[C; x]$ in place of $\mathrm{Proj}_I[C; x]$. The projection onto $\mathcal{B}_2$ is straightforward:

$$ \mathrm{Proj}[\mathcal{B}_2, x] = \begin{cases} \frac{x}{\|x\|_2} & \text{if } \|x\|_2 \ge 1 \\ x & \text{otherwise} \end{cases}. \qquad (43) $$

For $\mathcal{B}_\infty$, the projection is easily carried out component-wise:

$$ (\mathrm{Proj}[\mathcal{B}_\infty, x])_i = \min\{|x_i|, 1\} \mathrm{sgn}(x_i), \quad i = 1, \dots, n. \qquad (44) $$

**Proposition 6.** Consider a vector $x \in \mathbb{R}^n$, a closed convex set: $C \subseteq \mathbb{R}^n$, and a weighted inner product $\langle \cdot, \cdot \rangle_M$. The following statements are equivalent:

$$ y = \mathrm{Proj}_M[C; x], \qquad (45a) $$

$$ M(x - y) \in \mathbf{N}_C(y). \qquad (45b) $$

$$ y = (M + \mathbf{N}_C)^{-1}(Mx) \qquad (45c) $$

*Proof.* The proof can be found in [60, p.79] inside another proof. It is reproduced here for convenience. The optimization problem (42) can be equivalently written as $\mathrm{Proj}_M[C; x] = \arg\min_{\xi \in \mathbb{R}^n} \frac{1}{2}\langle \xi - x, \xi - x \rangle_M + \Psi_C(\xi)$. By the convexity of the weighted norm and of the indicator function, it follows that $y = \mathrm{Proj}_M[C; x]$ if, and only if, $0 \in M(y - x) + \partial\Psi_C(y)$, that is, if, and only if, $M(x - y) \in \mathbf{N}_C(y)$ or equivalently, if and only if $Mx \in (M + \mathbf{N}_C)(y)$. Hence, $y \in (M + \mathbf{N}_C)^{-1}(Mx)$. □

In addition if $C$ is a cone, then the statements (45) are equivalent to [S22, Corollary 23.5.4]

$$ C^* \ni M(y - x) \perp y \in C, \qquad (46) $$

where

$$ C^* = \{\eta \in \mathbb{R}^n \mid \langle \eta, \xi \rangle \ge 0 \text{ for all } \xi \in C\} \qquad (47) $$

is the dual cone to $C$.

If $C$ is a polyhedral set, then there exists a matrix $G \in \mathbb{R}^{m \times n}$ and a vector $b \in \mathbb{R}^m$ such that

$$ C = \{\xi \in \mathbb{R}^n \mid G\xi \le b\}. \qquad (48) $$

For instance, if $C = \mathcal{B}_\infty$, then $G = [I_n, -I_n]^\top$ and $b = \mathbf{1}_{2n}$. For polyhedral sets, the projection (45a) can be computed solving a conventional quadratic program. The code below shows an implementation in Python 3 using the open source modeling language for convex optimization CVXPY [74].

```python
import cvxpy as cvx
xi = cvx.Variable(n)
objective = 0.5*cvx.quad_form(xi, M) - x.T@M@xi
cvx.Problem(cvx.Minimize(objective), G@xi <= b).solve()
y = xi.value
```

If $C = \mathcal{B}_1$, it is possible to use the above code with $G \in \mathbb{R}^{2n \times n}$ with the rows of $G$ covering all possible vectors with entries in $\{-1, 1\}$ and $b = \mathbf{1}_n$. For large problems, a more efficient solution consists in splitting $\xi$ into its positive and negative components, that is $\xi = \xi^+ - \xi^-$, where $\xi^+, \xi^- \in \mathbb{R}_+^n$.
The projection of a point $x \in \mathbb{R}^n$ onto the nonempty closed convex set $C \subseteq \mathbb{R}^n$ with inner product weighted by $M = M^\top > 0$ is defined by

$$ \mathrm{Proj}_M[C; x] = \arg \min_{\xi \in C} \frac{1}{2} \langle \xi - x, \xi - x \rangle_M. \qquad (42) $$

For simplicity let us write $\mathrm{Proj}_I[C; x]$ in place of $\mathrm{Proj}_I[C; x]$. The projection onto $\mathcal{B}_2$ is straightforward:

$$ \mathrm{Proj}[\mathcal{B}_2, x] = \begin{cases} \frac{x}{\|x\|_2} & \text{if } \|x\|_2 \ge 1 \\ x & \text{otherwise} \end{cases}. \qquad (43) $$

For $\mathcal{B}_\infty$, the projection is easily carried out component-wise:

$$ (\mathrm{Proj}[\mathcal{B}_\infty, x])_i = \min\{|x_i|, 1\} \mathrm{sgn}(x_i), \quad i = 1, \dots, n. \qquad (44) $$

**Proposition 6.** Consider a vector $x \in \mathbb{R}^n$, a closed convex set: $C \subseteq \mathbb{R}^n$, and a weighted inner product $\langle \cdot, \cdot \rangle_M$. The following statements are equivalent:

$$ y = \mathrm{Proj}_M[C; x], \qquad (45a) $$

$$ M(x - y) \in \mathbf{N}_C(y). \qquad (45b) $$

$$ y = (M + \mathbf{N}_C)^{-1}(Mx) \qquad (45c) $$

*Proof.* The proof can be found in [60, p.79] inside another proof. It is reproduced here for convenience. The optimization problem (42) can be equivalently written as $\mathrm{Proj}_M[C; x] = \arg \min_{\xi \in \mathbb{R}^n} \frac{1}{2} \langle \xi - x, \xi - x \rangle_M + \Psi_C(\xi)$. By the convexity of the weighted norm and of the indicator function, it follows that $y = \mathrm{Proj}_M[C; x]$ if, and only if, $0 \in M(y - x) + \partial \Psi_C(y)$, that is, if, and only if, $M(x - y) \in \mathbf{N}_C(y)$ or equivalently, if and only if $Mx \in (M + \mathbf{N}_C)(y)$. Hence, $y \in (M + \mathbf{N}_C)^{-1}(Mx)$. □

In addition if $C$ is a cone, then the statements (45) are equivalent to [S22, Corollary 23.5.4]

$$ C^* \ni M(y - x) \perp y \in C, \qquad (46) $$

where

$$ C^* = \{ \eta \in \mathbb{R}^n \mid \langle \eta, \xi \rangle \ge 0 \text{ for all } \xi \in C \} \qquad (47) $$

is the dual cone to $C$.

If $C$ is a polyhedral set, then there exists a matrix $G \in \mathbb{R}^{m \times n}$ and a vector $b \in \mathbb{R}^m$ such that

$$ C = \{ \xi \in \mathbb{R}^n \mid G\xi \le b \}. \qquad (48) $$

For instance, if $C = \mathcal{B}_\infty$, then $G = [I_n, -I_n]^\top$ and $b = \mathbf{1}_{2n}$. For polyhedral sets, the projection (45a) can be computed solving a conventional quadratic program. The code below shows an implementation in Python 3 using the open source modeling language for convex optimization CVXPY [74].

```python
import cvxpy as cvx
xi = cvx.Variable(n)
objective = 0.5*cvx.quad_form(xi, M) - x.T@M@xi
cvx.Problem(cvx.Minimize(objective), G@xi <= b).solve()
y = xi.value
```

If $C = \mathcal{B}_1$, it is possible to use the above code with $G \in \mathbb{R}^{2n \times n}$ with the rows of $G$ covering all possible vectors with entries in $\{-1, 1\}$ and $b = \mathbf{1}_n$. For large problems, a more efficient solution consists in splitting $\xi$ into its positive and negative components, that is $\xi = \xi^+ - \xi^-$, where $\xi^+, \xi^- \in \mathbb{R}_+^n$.
## Sidebar (cont.)

In such a case the code below solves the projection problem by setting

$$H = \begin{bmatrix} \mathbf{1}_n^\top & \mathbf{1}_n^\top \\ -I_n & 0_{n \times n} \\ 0_{n \times n} & -I_n \end{bmatrix}, c = \begin{bmatrix} 1 \\ 0_{n \times 1} \\ 0_{n \times 1} \end{bmatrix}.$$

```python
import cvxpy as cvx
xi_p = cvx.Variable(n)
xi_n = cvx.Variable(n)
objective = 0.5*cvx.quad_form(xi_p - xi_n, M) - x.T@M@(xi_p - xi_n)
cvx.Problem(cvx.Minimize(objective), H@cvx.bmat([xi_p, xi_n]) <= c).solve()
y = xi_p.value - xi_n.value
```

$$ \begin{array}{l} \left( \begin{array}{l} \tilde{i} \\ \tilde{i} \\ \tilde{i} \end{array} \right) \left[ \begin{array}{l} \tilde{s}_{k+1} = \Lambda_1 x_k + \Lambda_2 u_{sv}(x_k) \\ -u_{sv}(x_k) \in \mathbf{M}(\tilde{s}_{k+1}) \\ \Downarrow \\ \Downarrow \\ \Downarrow \end{array} \right] \iff \left( \begin{array}{l} \tilde{i} \tilde{i} \\ \tilde{i} \tilde{i} \\ \tilde{i} \tilde{i} \end{array} \right) \left[ \begin{array}{l} \tilde{s}_{k+1} = \Lambda_2^{\frac{1}{2}} \mathcal{J}_{\mathbf{M}} \left( \Lambda_2^{-\frac{1}{2}} \Lambda_1 x_k \right) \\ -u_{sv}(x_k) = \Lambda_2^{-\frac{1}{2}} \mathcal{Y}_{\mathbf{M}} \left( \Lambda_2^{-\frac{1}{2}} \Lambda_1 x_k \right) \\ \Downarrow \\ \Downarrow \\ \Downarrow \end{array} \right] \\ \left( \begin{array}{l} \tilde{i} \tilde{i} \\ \tilde{i} \tilde{i} \\ \tilde{i} \tilde{i} \end{array} \right) \left[ \begin{array}{l} \tilde{s}_{k+1} \in \mathbf{M}^{-1}(-u_{sv}(x_k)) \\ -u_{sv}(x_k) = \Lambda_2^{-1} \Lambda_1 x_k - \Lambda_2^{-1} \tilde{s}_{k+1} \\ \Downarrow \\ \Downarrow \\ \Downarrow \end{array} \right] \iff \left( \begin{array}{l} \tilde{i} \tilde{i} \\ \tilde{i} \tilde{i} \\ \tilde{i} \tilde{i} \end{array} \right) \left[ \begin{array}{l} \tilde{s}_{k+1} = \Lambda_2^{\frac{1}{2}} \mathcal{Y}_{\mathbf{M}^{-1}} \left( \Lambda_2^{-\frac{1}{2}} \Lambda_1 x_k \right) \\ -u_{sv}(x_k) = \Lambda_2^{-\frac{1}{2}} \mathcal{J}_{\mathbf{M}^{-1}} \left( \Lambda_2^{-\frac{1}{2}} \Lambda_1 x_k \right) \\ \Downarrow \\ \Downarrow \\ \Downarrow \end{array} \right] \end{array} $$

<caption>Figure 4: Implicit (left-hand side) and explicit (right-hand side) equivalent descriptions of the closed-loop sliding dynamics. Implicit expressions are useful for stability analysis, whereas the explicit ones are used for the numerical implementation of the controller.</caption>
## Sidebar (cont.)

In such a case the code below solves the projection problem by setting

$$H = \begin{bmatrix} \mathbf{1}_n^\top & \mathbf{1}_n^\top \\ -I_n & 0_{n \times n} \\ 0_{n \times n} & -I_n \end{bmatrix}, c = \begin{bmatrix} 1 \\ 0_{n \times 1} \\ 0_{n \times 1} \end{bmatrix}.$$

```python
import cvxpy as cvx
xi_p = cvx.Variable(n)
xi_n = cvx.Variable(n)
objective = 0.5*cvx.quad_form(xi_p - xi_n, M) - x.T@M@(xi_p - xi_n)
cvx.Problem(cvx.Minimize(objective), H@cvx.bmat([xi_p, xi_n]) <= c).solve()
y = xi_p.value - xi_n.value
```

$$ \begin{array}{l} \left( \begin{array}{l} \bar{\tilde{i}} \\ \tilde{s}_{k+1} = \Lambda_1 x_k + \Lambda_2 u_{sv}(x_k) \\ -u_{sv}(x_k) \in \mathbf{M}(\tilde{s}_{k+1}) \end{array} \right) \\ \Downarrow \\ \left( \begin{array}{l} \bar{\tilde{ii}} \\ \tilde{s}_{k+1} \in \mathbf{M}^{-1}(-u_{sv}(x_k)) \\ -u_{sv}(x_k) = \Lambda_2^{-1} \Lambda_1 x_k - \Lambda_2^{-1} \tilde{s}_{k+1} \end{array} \right) \\ \Downarrow \\ \left( \begin{array}{l} \bar{\tilde{iv}} \\ \tilde{s}_{k+1} = \Lambda_2^{\frac{1}{2}} \mathcal{J}_{\mathbf{M}} \left( \Lambda_2^{-\frac{1}{2}} \Lambda_1 x_k \right) \\ -u_{sv}(x_k) = \Lambda_2^{-\frac{1}{2}} \mathcal{Y}_{\mathbf{M}} \left( \Lambda_2^{-\frac{1}{2}} \Lambda_1 x_k \right) \end{array} \right) \end{array} $$

<caption>Figure 4: Implicit (left-hand side) and explicit (right-hand side) equivalent descriptions of the closed-loop sliding dynamics. Implicit expressions are useful for stability analysis, whereas the explicit ones are used for the numerical implementation of the controller.</caption>
To complete the proof let us show the equivalence between ii) and iv). To this end, let us proceed in a similar manner as above, i.e., the following inclusions are equivalent:

$$
\begin{align}
& -u_{\text{sv}}(x_k) \in \Lambda_2^{-1} \Lambda_1 x_k - \Lambda_2^{-1} \mathbf{M}^{-1} (-u_{\text{sv}}(x_k)) \nonumber \\
& -u_{\text{sv}}(x_k) \in (\Lambda_2 + \mathbf{M}^{-1})^{-1} (\Lambda_1 x_k) \nonumber \\
& -u_{\text{sv}}(x_k) \in (\Lambda_2^{1/2} \circ (\mathbf{I}_d + \tilde{\mathbf{M}}^{-1}) \circ \Lambda_2^{1/2})^{-1} (\Lambda_1 x_k) \nonumber \\
& -u_{\text{sv}}(x_k) \in \Lambda_2^{-1/2} \circ (\mathbf{I}_d + \tilde{\mathbf{M}}^{-1})^{-1} \circ \Lambda_2^{-1/2} (\Lambda_1 x_k) \nonumber \\
& -u_{\text{sv}}(x_k) = \Lambda_2^{-1/2} \mathcal{J}_{\tilde{\mathbf{M}}^{-1}} (\Lambda_2^{-1/2} \Lambda_1 x_k). \tag{83}
\end{align}
$$

Finally, the substitution of (83) into the sliding output yields

$$
\begin{align}
\tilde{s}_{k+1} &= \Lambda_1 x_k - \Lambda_2^{1/2} \mathcal{J}_{\tilde{\mathbf{M}}^{-1}} \left( \Lambda_2^{-1/2} \Lambda_1 x_k \right) \nonumber \\
&= \Lambda_2^{1/2} \left( \mathbf{I}_d - \mathcal{J}_{\tilde{\mathbf{M}}^{-1}} \right) \left( \Lambda_2^{-1/2} \Lambda_1 x_k \right) \nonumber \\
&= \Lambda_2^{1/2} \mathcal{Y}_{\tilde{\mathbf{M}}^{-1}} \left( \Lambda_2^{-1/2} \Lambda_1 x_k \right). \tag{84}
\end{align}
$$

<figure><img src="image_6.png" /></figure>

The controller $u_{sv}(x_k)$ in (62), (67), is calculated with the help of Proposition 7 by setting $\mathbf{M}^{-1} = \partial(\Psi_{\mathcal{B}_{\infty}} \circ \tilde{\mathbf{K}}^{-1})$, $\Lambda_1 = \tilde{\mathbf{B}}^{\top}(\tilde{\mathbf{P}} + \frac{1}{2}\tilde{\mathbf{B}}\tilde{\mathbf{B}}^{\top})^{-1}\tilde{\mathbf{A}}$ and $\Lambda_2 = \tilde{\mathbf{B}}^{\top}\tilde{\mathbf{P}}^{-1}\tilde{\mathbf{B}}$, where the inversion of set-valued mappings and (40) were used to compute $\mathbf{M}^{-1}$ (see “Sidebar VI”). To write the explicit expression for $-u_{sv}(x_k)$ we make use of the implicit expression (ii) in the lower part of Figure 4 by pre-multiplying by $\tilde{\mathbf{K}}^{-1}$. Thus, after rearranging terms, it follows that

$$
\tilde{\mathbf{K}}^{-1} \Lambda_2^{-1} \Lambda_1 x_k + \tilde{\mathbf{K}}^{-1} u_{\text{sv}}(x_k) \in (\tilde{\mathbf{K}} \Lambda_2 \tilde{\mathbf{K}})^{-1} \mathbf{N}_{\mathcal{B}_{\infty}} (-\tilde{\mathbf{K}}^{-1} u_{\text{sv}}(x_k)). \quad (85)
$$

It thus follows from (45) that

$$
\begin{align}
-u_{sv}(x_k) &= \tilde{K} (\mathbf{I}_d + M^{-1} \mathbf{N}_{\mathcal{B}_{\infty}})^{-1} (\tilde{K}^{-1} \varphi(x_k)) \nonumber \\
&= \tilde{K} \operatorname{Proj}_M (\mathcal{B}_{\infty}; \varphi(x_k)), \tag{86}
\end{align}
$$

where

$$
M = M^{\top} := \tilde{K}(\tilde{B}^{\top}\tilde{P}^{-1}\tilde{B})\tilde{K}, \quad (87)
$$

$$
\begin{align}
\varphi(x_k) &:= \tilde{K}^{-1} (\tilde{B}^{\top} \tilde{P}^{-1} \tilde{B})^{-1} \tilde{B}^{\top} \left( \tilde{P} + \frac{1}{2} \tilde{B} \tilde{B}^{\top} \right)^{-1} \tilde{A} x_k \nonumber \\
&= \tilde{K}^{-1} (\tilde{B}^{\top} \tilde{P}^{-1} \tilde{B})^{-1/2} \tilde{L} x_k. \tag{88}
\end{align}
$$

This extends the scalar case in (13). A simple choice for the gain is $\tilde{K} = \gamma(\tilde{B}^{\top}\tilde{P}^{-1}\tilde{B})^{-1/2}$, $\gamma > 0$, in which case $M = \gamma^2 I$ and (44) can be applied to obtain

$$
\begin{align}
w_i &= \min \{ \varphi(x_k)_i, 1 \} \mathbf{sgn}(\varphi(x_k)_i) \quad i = 1, \dots, n \nonumber \\
u_{\text{sv}}(x_k) &= -\gamma (\tilde{B}^{\top} \tilde{P}^{-1} \tilde{B})^{-1/2} w \tag{89}
\end{align}
$$

Otherwise, the controller can be computed as the solution of a quadratic program under constraints, similarly to (15),

$$
\begin{align}
\operatorname{Proj}_M(\mathcal{B}_\infty; \tilde{\mathbf{K}}^{-1} \varphi(x_k)) & \nonumber \\
&= \arg \min_{\xi \in \mathcal{B}_\infty} \frac{1}{2} (\xi - \tilde{\mathbf{K}}^{-1} \varphi(x_k))^{\top} M (\xi - \tilde{\mathbf{K}}^{-1} \varphi(x_k)), \tag{90}
\end{align}
$$

see “Sidebar VI” for $\mathcal{B}_{\infty}$. (45a) and the associated piece of code for the computation of the projection.

## 3.2 Discrete-Time Classical Approach

Let us pass to the discretization of the first controller in (17), (19). To start with, consider (42) and set $\tilde{s}_{k+1} = \sigma(x_k, u_k)$ as in (62), but with $\sigma(x, w) = \tilde{C}(\tilde{A}x + \tilde{B}w)$, so that $s_{k+1} = \sigma(x_k, u_k + \tilde{\delta}_k) = \tilde{C}x_{k+1}$. Let us impose the condition

$$
\tilde{C}(\tilde{A}x_k + \tilde{B}u_{\text{lin}}(x_k)) = \tilde{C}x_k = s_k \quad (91)
$$

and assume as usual that $\tilde{C}\tilde{B} \in \mathbb{R}^{m \times m}$ has rank $m$. In such a case, $u_{\text{lin}}$ can be computed as

$$
u_{\text{lin}}(x_k) = (\tilde{C}\tilde{B})^{-1}\tilde{C}(I_n - \tilde{A})x_k. \quad (92)
$$

The rationale behind the choice for $u_{\text{lin}}$ is to subsequently ensure that $\tilde{s}_{k+1}$ only depends on $s_k$ and $u_k$. Further, setting $u_k = u_{\text{lin}}(x_k) + (\tilde{C}\tilde{B})^{-1}v_k$ gives

$$
s_{k+1} = s_k + v_k + \tilde{C}\tilde{B}\tilde{\delta}_k \quad (93a)
$$

$$
\tilde{s}_{k+1} = s_k + v_k. \tag{93b}
$$

From (93b) it is natural to define the control action $v_k = u_{sv}(x_k) \in -\tilde{K} \mathbf{Sgn}(\tilde{s}_{k+1})$. It gives

$$
s_{k+1} \in s_k - \tilde{K} \mathbf{Sgn}(\tilde{s}_{k+1}) + \tilde{C}\tilde{B}\tilde{\delta}_k \quad (94a)
$$

$$
\tilde{s}_{k+1} \in s_k - \tilde{K} \mathbf{Sgn}(\tilde{s}_{k+1}). \quad (94b)
$$

A generalized equation with unknown $\tilde{s}_{k+1}$ is obtained in (94b). The resemblance between equalities in (94b) and (70) is clear. Define $\tilde{s}'_k = \tilde{K}^{-1/2} \tilde{s}_k$ and note that (94b) is equivalent to

$$
\tilde{s}'_{k+1} \in \tilde{K}^{-\frac{1}{2}} s_k - \tilde{K}^{\frac{1}{2}} \mathbf{Sgn}(\tilde{K}^{\frac{1}{2}} \tilde{s}'_{k+1}) = \tilde{K}^{-\frac{1}{2}} s_k - \partial f(\tilde{s}'_{k+1}) \quad (95)
$$

with $f(\cdot) = \| \cdot \|_1 \circ \tilde{K}^{1/2}$. Once again, the chain rule has been used (see Fact 8 in “Sidebar VI”). Notice that the functions $f$ in (95) and (22) share the same structure and are both proper, convex, and LSC. Thus, the expression (95) is also a generalized equation with unknown $\tilde{s}'_{k+1}$, rewritten equivalently as

$$
\begin{align}
\tilde{s}'_{k+1} &= \operatorname{Prox}_f (\tilde{K}^{-\frac{1}{2}} s_k) \nonumber \\
&= \operatorname{Prox}_f (\tilde{K}^{-\frac{1}{2}} (\tilde{s}_k + \tilde{C}\tilde{B}\tilde{\delta}_{k-1})) \nonumber \\
&= \operatorname{Prox}_{\|\tilde{K}^{\frac{1}{2}}(\cdot)\|_1} (\tilde{s}'_k + \tilde{C}\tilde{B}\tilde{\delta}_{k-1}). \tag{96}
\end{align}
$$

This algorithm may be named a *robust* or *perturbed proximal-point algorithm*. Using (93) it is also inferred that

$$
s_{k+1} = \tilde{K}^{\frac{1}{2}} \operatorname{Prox}_{\|\tilde{K}^{\frac{1}{2}}(\cdot)\|_1} (s_k) + \tilde{C}\tilde{B}\tilde{\delta}_k. \quad (97)
$$
To complete the proof let us show the equivalence between ii) and iv). To this end, let us proceed in a similar manner as above, i.e., the following inclusions are equivalent:

$$
\begin{align}
-u_{sv}(x_k) &\in \Lambda_2^{-1} \Lambda_1 x_k - \Lambda_2^{-1} \mathbf{M}^{-1} (-u_{sv}(x_k)) \nonumber \\
-u_{sv}(x_k) &\in (\Lambda_2 + \mathbf{M}^{-1})^{-1} (\Lambda_1 x_k) \nonumber \\
-u_{sv}(x_k) &\in (\Lambda_2^{\frac{1}{2}} \circ (\mathbf{I}_d + \tilde{\mathbf{M}}^{-1}) \circ \Lambda_2^{\frac{1}{2}})^{-1} (\Lambda_1 x_k) \nonumber \\
-u_{sv}(x_k) &\in \Lambda_2^{-\frac{1}{2}} \circ (\mathbf{I}_d + \tilde{\mathbf{M}}^{-1})^{-1} \circ \Lambda_2^{-\frac{1}{2}} (\Lambda_1 x_k) \nonumber \\
-u_{sv}(x_k) &= \Lambda_2^{-\frac{1}{2}} \mathcal{J}_{\tilde{\mathbf{M}}^{-1}} (\Lambda_2^{-\frac{1}{2}} \Lambda_1 x_k). \tag{83}
\end{align}
$$

Finally, the substitution of (83) into the sliding output yields

$$
\begin{align}
\tilde{s}_{k+1} &= \Lambda_1 x_k - \Lambda_2^{\frac{1}{2}} \mathcal{J}_{\tilde{\mathbf{M}}^{-1}} \left( \Lambda_2^{-\frac{1}{2}} \Lambda_1 x_k \right) \nonumber \\
&= \Lambda_2^{\frac{1}{2}} (\mathbf{I}_d - \mathcal{J}_{\tilde{\mathbf{M}}^{-1}}) \left( \Lambda_2^{-\frac{1}{2}} \Lambda_1 x_k \right) \nonumber \\
&= \Lambda_2^{\frac{1}{2}} \mathcal{Y}_{\tilde{\mathbf{M}}^{-1}} \left( \Lambda_2^{-\frac{1}{2}} \Lambda_1 x_k \right). \tag{84}
\end{align}
$$

<figure><img src="image_5.png" /></figure>

The controller $u_{sv}(x_k)$ in (62), (67), is calculated with the help of Proposition 7 by setting $\mathbf{M}^{-1} = \partial(\Psi_{\mathcal{B}_{\infty}} \circ \tilde{\mathbf{K}}^{-1})$, $\Lambda_1 = \tilde{\mathbf{B}}^{\top}(\tilde{\mathbf{P}} + \frac{1}{2}\tilde{\mathbf{B}}\tilde{\mathbf{B}}^{\top})^{-1}\tilde{\mathbf{A}}$ and $\Lambda_2 = \tilde{\mathbf{B}}^{\top}\tilde{\mathbf{P}}^{-1}\tilde{\mathbf{B}}$, where the inversion of set-valued mappings and (40) were used to compute $\mathbf{M}^{-1}$ (see “Sidebar VI”). To write the explicit expression for $-u_{sv}(x_k)$ we make use of the implicit expression (ii) in the lower part of Figure 4 by pre-multiplying by $\tilde{\mathbf{K}}^{-1}$. Thus, after rearranging terms, it follows that

$$
\tilde{\mathbf{K}}^{-1} \Lambda_2^{-1} \Lambda_1 x_k + \tilde{\mathbf{K}}^{-1} u_{sv}(x_k) \in (\tilde{\mathbf{K}} \Lambda_2 \tilde{\mathbf{K}})^{-1} \mathbf{N}_{\mathcal{B}_{\infty}} (-\tilde{\mathbf{K}}^{-1} u_{sv}(x_k)). \quad (85)
$$

It thus follows from (45) that

$$
\begin{align}
-u_{sv}(x_k) &= \tilde{K} (\mathbf{I}_d + M^{-1} \mathbf{N}_{\mathcal{B}_{\infty}})^{-1} (\tilde{K}^{-1} \varphi(x_k)) \nonumber \\
&= \tilde{K} \operatorname{Proj}_M (\mathcal{B}_{\infty}; \varphi(x_k)), \tag{86}
\end{align}
$$

where

$$
M = M^{\top} := \tilde{K}(\tilde{B}^{\top}\tilde{P}^{-1}\tilde{B})\tilde{K}, \quad (87)
$$

$$
\begin{align}
\varphi(x_k) &:= \tilde{K}^{-1} (\tilde{B}^{\top} \tilde{P}^{-1} \tilde{B})^{-1} \tilde{B}^{\top} \left( \tilde{P} + \frac{1}{2} \tilde{B} \tilde{B}^{\top} \right)^{-1} \tilde{A} x_k \nonumber \\
&= \tilde{K}^{-1} (\tilde{B}^{\top} \tilde{P}^{-1} \tilde{B})^{-\frac{1}{2}} \tilde{L} x_k. \tag{88}
\end{align}
$$

This extends the scalar case in (13). A simple choice for the gain is $\tilde{K} = \gamma(\tilde{B}^{\top}\tilde{P}^{-1}\tilde{B})^{-1/2}$, $\gamma > 0$, in which case $M = \gamma^2 I$ and (44) can be applied to obtain

$$
\begin{align}
w_i &= \min \{ \varphi(x_k)_i, 1 \} \mathbf{sgn}(\varphi(x_k)_i) \quad i = 1, \dots, n \nonumber \\
u_{sv}(x_k) &= -\gamma (\tilde{B}^{\top} \tilde{P}^{-1} \tilde{B})^{-1/2} w \tag{89}
\end{align}
$$

Otherwise, the controller can be computed as the solution of a quadratic program under constraints, similarly to (15),

$$
\begin{align}
\operatorname{Proj}_M(\mathcal{B}_\infty; \tilde{\mathbf{K}}^{-1} \varphi(x_k)) & \nonumber \\
&= \arg \min_{\xi \in \mathcal{B}_\infty} \frac{1}{2} (\xi - \tilde{\mathbf{K}}^{-1} \varphi(x_k))^\top M (\xi - \tilde{\mathbf{K}}^{-1} \varphi(x_k)), \tag{90}
\end{align}
$$

see “Sidebar VI” for $\mathcal{B}_{\infty}$. (45a) and the associated piece of code for the computation of the projection.

## 3.2 Discrete-Time Classical Approach

Let us pass to the discretization of the first controller in (17), (19). To start with, consider (42) and set $\tilde{s}_{k+1} = \sigma(x_k, u_k)$ as in (62), but with $\sigma(x, w) = \tilde{C}(\tilde{A}x + \tilde{B}w)$, so that $s_{k+1} = \sigma(x_k, u_k + \tilde{\delta}_k) = \tilde{C}x_{k+1}$. Let us impose the condition

$$
\tilde{C}(\tilde{A}x_k + \tilde{B}u_{\text{lin}}(x_k)) = \tilde{C}x_k = s_k \quad (91)
$$

and assume as usual that $\tilde{C}\tilde{B} \in \mathbb{R}^{m \times m}$ has rank $m$. In such a case, $u_{\text{lin}}$ can be computed as

$$
u_{\text{lin}}(x_k) = (\tilde{C}\tilde{B})^{-1}\tilde{C}(I_n - \tilde{A})x_k. \quad (92)
$$

The rationale behind the choice for $u_{\text{lin}}$ is to subsequently ensure that $\tilde{s}_{k+1}$ only depends on $s_k$ and $u_k$. Further, setting $u_k = u_{\text{lin}}(x_k) + (\tilde{C}\tilde{B})^{-1}v_k$ gives

$$
s_{k+1} = s_k + v_k + \tilde{C}\tilde{B}\tilde{\delta}_k \quad (93a)
$$

$$
\tilde{s}_{k+1} = s_k + v_k. \tag{93b}
$$

From (93b) it is natural to define the control action $v_k = u_{sv}(x_k) \in -\tilde{K} \mathbf{Sgn}(\tilde{s}_{k+1})$. It gives

$$
s_{k+1} \in s_k - \tilde{K} \mathbf{Sgn}(\tilde{s}_{k+1}) + \tilde{C}\tilde{B}\tilde{\delta}_k \quad (94a)
$$

$$
\tilde{s}_{k+1} \in s_k - \tilde{K} \mathbf{Sgn}(\tilde{s}_{k+1}). \quad (94b)
$$

A generalized equation with unknown $\tilde{s}_{k+1}$ is obtained in (94b). The resemblance between equalities in (94b) and (70) is clear. Define $\tilde{s}'_k = \tilde{K}^{-\frac{1}{2}}\tilde{s}_k$ and note that (94b) is equivalent to

$$
\tilde{s}'_{k+1} \in \tilde{K}^{-\frac{1}{2}} s_k - \tilde{K}^{\frac{1}{2}} \mathbf{Sgn}(\tilde{K}^{\frac{1}{2}} \tilde{s}'_{k+1}) = \tilde{K}^{-\frac{1}{2}} s_k - \partial f(\tilde{s}'_{k+1}) \quad (95)
$$

with $f(\cdot) = \| \cdot \|_1 \circ \tilde{K}^{\frac{1}{2}}$. Once again, the chain rule has been used (see Fact 8 in “Sidebar VI”). Notice that the functions $f$ in (95) and (22) share the same structure and are both proper, convex, and LSC. Thus, the expression (95) is also a generalized equation with unknown $\tilde{s}'_{k+1}$, rewritten equivalently as

$$
\begin{align}
\tilde{s}'_{k+1} &= \operatorname{Prox}_f (\tilde{K}^{-\frac{1}{2}} s_k) \nonumber \\
&= \operatorname{Prox}_f (\tilde{K}^{-\frac{1}{2}} (\tilde{s}_k + \tilde{C}\tilde{B}\tilde{\delta}_{k-1})) \nonumber \\
&= \operatorname{Prox}_{\|\tilde{K}^{\frac{1}{2}}(\cdot)\|_1} (\tilde{s}'_k + \tilde{C}\tilde{B}\tilde{\delta}_{k-1}). \tag{96}
\end{align}
$$

This algorithm may be named a *robust* or *perturbed proximal-point algorithm*. Using (93) it is also inferred that

$$
s_{k+1} = \tilde{K}^{\frac{1}{2}} \operatorname{Prox}_{\|\tilde{K}^{\frac{1}{2}}(\cdot)\|_1} (s_k) + \tilde{C}\tilde{B}\tilde{\delta}_k. \quad (97)
$$
**Remark 5.** The above control strategy has been proposed in [21, 31]. For instance, equation (94b) is exactly [21, Equation (9)].

### 3.2.1 Stability Analysis and Sliding Mode

Let us briefly recall the main stability properties of (94). It is proved in [21, Proposition 2] that {$\tilde{s}_k$}$_{k \in \mathbb{N}}$ solution of (94b) converges to zero in a finite number $k_{\min} < +\infty$ of steps (cf. Proposition 8 in "Sidebar VII" as well), while {$s_k$}$_{k \in \mathbb{N}}$ solution of (94a) is bounded. During the sliding mode ($\tilde{s}_k = 0$ for $k \ge k_{\min}$) we have $s_{k+1} = \tilde{C}\tilde{B}\tilde{\delta}_k$, since $s_k + u_{sv}(x_k) = 0$ is equivalent to

$$u_{sv}(x_k) = -s_k = -\tilde{C}\tilde{B}\tilde{\delta}_{k-1}. \quad (98)$$

As a fundamental result, the control (21) is approximated by (98): in discrete time, the implicit method allows us to design an input which compensates for the perturbation with a one-step delay, in a similar way to (39). Once again the set-valued controller does not depend on the gain $\tilde{K}$ during the sliding phase.

### 3.2.2 Control-Input Calculation

Using Proposition 6 in "Sidebar VI", equation (94), and mapping inversion (see "Sidebar III"), we can see that

$$\tilde{K} (\tilde{K}^{-1} s_k - (-\tilde{K}^{-1} u_{sv}(x_k)) \in \mathbf{N}_{\mathcal{B}_{\infty}} (-\tilde{K} u_{sv}(x_k)) \quad (99)$$

is equivalent to $-u_{sv}(x_k) = \tilde{K}(I_d + \tilde{K}^{-1} \mathbf{N}_{\mathcal{B}_{\infty}})^{-1}(\tilde{K}^{-1} s_k)$, which in turn is equivalent to

$$-u_{sv}(x_k) = \tilde{K} \operatorname{Proj}_{\tilde{K}} (\mathcal{B}_{\infty}; \tilde{K}^{-1} s_k). \quad (100)$$

Again, the projection can be computed by solving a quadratic program or, similarly to (89), conveniently choose $\tilde{K} = \gamma I$ so that

$$(u_{sv}(x_k))_i = -\gamma \min \left\{ \frac{|(s_k)_i|}{\gamma}, 1 \right\} \operatorname{sgn}((s_k)_i), \quad (101)$$

see "Sidebar VI" for $\mathcal{B}_{\infty}$, (45a) and the associated piece of code for the computation of the projection in a general setting. The controller in (101) is in the minimum-operator formalism [61, 48, 50, 62, 63].

**Remark 6.** Dead-beat controller. The linear controllers in (17) and in (92) have different structures. Suppose that $u_{\text{lin}}(x_k)$ is computed as $u_{\text{lin}}(x_k) = -(\tilde{C}\tilde{B})^{-1}\tilde{C}\tilde{A}x_k$, which would be a naive implementation of (17). Then,

$$\tilde{C}x_{k+1} = \tilde{C}\tilde{A}x_k + \tilde{C}\tilde{B}(u_{\text{lin}}(x_k) + \tilde{\delta}_k) + v_k = v_k + \tilde{C}\tilde{B}\tilde{\delta}_k, \quad (102)$$

which is different from (93). Applying the above $u_{sv}(x_k)$ yields $\tilde{s}_{k+1} \in -\tilde{K} \operatorname{sgn}(\tilde{s}_{k+1})$, which is equivalent to $\tilde{s}'_{k+1} = (I_d + \partial f)^{-1}(0)$. Since the operator $(I_d + \partial f)^{-1}$ is single-valued and its graph contains $(0, 0)$, it follows that $\tilde{s}_{k+1} = 0$. Therefore, this algorithm is a one-step

dead-beat controller. The implicit algorithms introduced in [30, 31, 21] are not of the dead-beat type. It is noteworthy that a major discrepancy exists between (92) and the dead-beat input in the framework of discretization of continuous systems. Consider for instance the exact ZOH discretization with constant sampling time $h > 0$, where $\tilde{A} = e^{Ah}$, $\tilde{B} = e^{Ah} \int_0^h e^{-A\tau} B d\tau$. Then, as $h \to 0$ we have $(\tilde{C}\tilde{B})^{-1}\tilde{C}(I_n - \tilde{A}) = \mathcal{O}(1)$ while $(\tilde{C}\tilde{B})^{-1}\tilde{C}\tilde{A} = \mathcal{O}(h^{-1})$. Clearly, the dead-beat input grows unbounded as $h \to 0$ and cannot be a good candidate for convergence towards its continuous-time counterpart. This shows that such dead-beat controller is well-suited to purely discrete-time systems, but not for discretized continuous-time systems.

## 3.3 Recapitulation

Allow us to summarize some of the connections between sliding-mode control, passivity, and optimization.

### 3.3.1 Link with Optimization

It is interesting to note that operators of the form $(I_d + M)^{-1}$ are ubiquitous in the above discrete-time algorithms: they are the well-known resolvent mappings, very close to Yosida approximations (see (23) and (26) in "Sidebar V"). They have been exhibited in (10), (11), (73), (86), (96), and (100). They are a direct consequence of the presence of a maximal monotone mapping $M$ on the right-hand side of the differential inclusions in (22), (36). These operators are resolvents of mappings of the form $M = \partial f$ for some proper convex lower semi-continuous function $f$. This shows the strong connection between the implicit discretization and Optimization (see also (86)). Thus, a common feature between model-predictive control (MPC) and set-valued sliding-mode implicit discretization is that an optimization problem has to be solved to compute the controller. This is formalized in "Sidebar VIII", where the implicitly discretized scheme is reinterpreted as a one-step MPC. The explicit discretization (that is, choosing $u_{sv}(x_k) \in -\tilde{K} \operatorname{sgn}(s_k)$) yields operators of the form $(I_d - \partial f)$, which are neither monotone nor possess interesting properties, neither in terms of stability nor robustness.

The above analysis is closely related to the analysis of proximal-point algorithms (Definition 12 in "Sidebar VII"). However, a peculiarity of discrete sliding-mode control is the notion of a robust proximal-point algorithm, formulated in (57) in "Sidebar VII" and shown to converge in a finite number of steps, given sufficiently large gains. It is worth emphasizing that (67) has the form $u_{sv}(x_k) \in -\partial \|\tilde{K} \tilde{s}_{k+1}\|_p$ with $p = 1$. Just as with the continuous-time case, other norms can be used (or other proper LSC convex functions for that matter). For arbitrary $p$, it suffices to replace $\mathcal{B}_{\infty}$ in (86) by $\mathcal{B}_{p*}$. For