Observer-based robust integral of the sign of the error control of class I of underactuated mechanical systems: Theory and real-time experiments

Abstract

In this paper, the control problem of a class I of underactuated mechanical systems (UMSs) is addressed. The considered class includes nonlinear UMSs with two degrees of freedom and one control input. Firstly, we propose the design of a robust integral of the sign of the error (RISE) control law, adequate for this special class. Based on a change of coordinates, the dynamics is transformed into a strict-feedback (SF) form. A Lyapunov-based technique is then employed to prove the asymptotic stability of the resulting closed-loop system. Numerical simulation results show the robustness and performance of the original RISE toward parametric uncertainties and disturbance rejection. A comparative study with a conventional sliding mode control reveals a significant robustness improvement with the proposed original RISE controller. However, in real-time experiments, the amplification of the measurement noise is a major problem. It has an impact on the behaviour of the motor and reduces the performance of the system. To deal with this issue, we propose to estimate the velocity using the robust Levant differentiator instead of the numerical derivative. Real-time experiments were performed on the testbed of the inertia wheel inverted pendulum to demonstrate the relevance of the proposed observer-based RISE control scheme. The obtained real-time experimental results and the obtained evaluation indices show clearly a better performance of the proposed observer-based RISE approach compared to the sliding mode and the original RISE controllers.

Keywords

Underactuated mechanical systems robust integral of the sign of the error feedback control strict-feedback form Levant differentiator inertia wheel inverted pendulum

Introduction

Control of underactuated mechanical systems (UMSs) has gained over the years a growing attention within the automatic control community (Fantoni and Lozano, 2002). The presence of underactuation in mechanical systems can be due to various reasons (Choukchou-Braham et al., 2014). It may be intentionally specified during the design stage to minimize the weight of the system. Otherwise, it may be unintentional, as a result of the damage of one actuator or more of the system. The underactuation in mechanical systems gives rise to various challenging control issues (Krafes et al., 2018), including:

Many difficulties are often exhibited by such systems, such as their complex dynamics and their nonlinear coupling between actuated and non-actuated coordinates.

Non-integrable and second-order non-holonomic constraints that may exist in the dynamics of UMSs. For instance, the dynamic model of the pendubot is subject to second-order non-integrable differential constraints.

UMSs are often considered as a non-minimum phase, as the internal dynamics of such systems is often unstable. Hence, the application of input–output feedback linearization is not efficient to control such systems (Guemghar, 2005).

One of the control challenges is the lack of feedback linearizability. For instance, the dynamic model of the acrobot (Spong, (1995) and the pendubot (Zhang and Tarn, 2001) systems are not feedback linearizable with a static state feedback. However, only a partial feedback linearization is possible in such cases.

Due to their different structural properties, UMSs are studied in a separate way and there is no general approach to control all variants of them. Some classifications of these systems have been reported in the literature. For instance, in Seto and Baillieul (1994) the authors proposed three representations called tree, isolated vertex and chain, classified using the method of the control flow diagram. It is a graphical representation of the dynamics which represents the interaction forces through the degrees of freedom (DOF). Another classification was proposed in Olfati-Saber (2001), using some structural properties of the system such as kinetic symmetry, interacting inputs, etc. The author proposed three normal representations for two-DOF systems where a backstepping procedure and a forwarding scheme were developed for the feedback form and the feed-forward form, respectively. Besides, a wide range of control techniques have been proposed in the literature, such as partial feedback linearization (Spong and Praly, 1997), passivity-based-control (Donaire et al., 2017; Ortega et al., 2002), adaptive neural network-based control (Ghommam and Chemori, 2017; Moreno-Valenzuela et al., 2017), optimization-based-control (Andary et al., 2009), and observer-based super-twisting control (Hfaiedh et al., 2020a), to name a few.

One particular control approach, mostly used to address the control issue of perturbed UMSs, is the sliding mode control (SMC). It is a robust control strategy characterized by its invariance toward some model uncertainties and external disturbances (Utkin, 2008). Various SMC approaches were proposed in the literature for UMSs. After representation of the system model into a quasi-chained form, a nonlinear robust SMC controller was proposed in Lu et al. (2016) and Sun et al. (2015) to make the system states driven to reach the sliding manifold, and guarantee their convergence to the equilibrium point. In Khalid and Memon (2016), a first-order SMC with a high-gain observer has been proposed for the stabilization of the inertia wheel inverted pendulum (IWIP). In Cheng and Ho (2017) an adaptive sliding mode controller was designed for a class of nonlinear UMSs with matched and mismatched disturbances. Another SMC scheme based on an extended disturbance observer was proposed in Ding et al. (2017) to control a second-order UMS. Thakar et al. (2013) propose a sliding mode approach to control an underactuated system based on the nonlinear model of lateral slosh-container. The main drawback of the first order SMC is the so-called chattering phenomenon. From a practical point of view, this effect is undesirable, since it involves a high control activity generated by the high-frequency of unmodelled dynamics (Slotine and Li, 1991).

To overcome this problem, we are interested in this paper, to redesign another nonlinear robust controller, namely the robust integral of the sign of the error (RISE) controller (Xian et al., 2004). Compared to the SMC, RISE feedback control generates a continuous control law and prevents the chattering phenomenon due to the integral of the discontinuous term in the expression of the control law (Fischer et al., 2014). It has also the advantage to compensate for sufficiently smooth nonlinear disturbances and system uncertainties. Various theoretical extensions of this control scheme have been proposed in the literature, and were applied to a wide range of systems, including autonomous underwater vehicles (Fischer et al., 2014), multi-link flexible manipulators (Jian et al., 2014), parallel robots (Bennehar et al., 2018), hard disc drives (Taktak-Meziou et al., 2014), UMSs (Hfaiedh et al., 2018) and exoskeletons (Sherwani et al., 2020).

The main contributions of the present paper can be summarized as follows:

This work is an extension of the conference paper of Hfaiedh et al. (2018), where a RISE controller was applied to a two-DOF underactuated IWIP. The new findings of this paper include: (a) the proof of the stability analysis of the resulting closed-loop system; (b) new simulations and real-time experimental results; and (c) quantification of the performance through various performance evaluation indices.

The first contribution lies in the redesign of the RISE controller for UMSs. To the best of the authors’ knowledge, no previous work in the literature has dealt with the RISE control to stabilize UMSs whose control design is more difficult than fully-actuated systems. The choice of designing a RISE controller is motivated by the advantage of generating a continuous control law with a guaranteed closed-loop stability based on some trivial properties of the system dynamics. Compared to fully actuated systems, the proposed RISE control could not be straightforwardly applied to UMSs. Referring to the classification of Olfati-Saber (2001), we transform the system into a strict-feedback (SF) form, which decouples the original system into two cascaded nonlinear and linear subsystems. The new control input is included in the actuated subsystem, which is not the case for the unactuated subsystem. Then, new desired trajectories and tracking errors are defined according to the Lyapunov concept of the nonlinear subsystem.

Numerical simulations were conducted to demonstrate the robustness of the proposed RISE controller on the IWIP. Two scenarios have been performed in simulation test: the first one deals with robustness towards parametric uncertainties; and the second one concerns external disturbance rejection. A comparative analysis between the first order sliding mode controller and the proposed original RISE controller is presented and discussed. Moreover, robustness evaluation indices are used to evaluate and compare each controller. This proposed extended version includes a confirmed simulation with a rigorous stability analysis of the resulting closed-loop system.

The second contribution focuses on improving the experimental results of the original RISE controller with a differentiator. It is worth noting that the RISE controller can compensate for uncertainties. This is of a considerable importance for an underactuated mechanical system, since uncertainties are extremely abundant in their dynamics. However, by computing the velocity signal using the numerical derivative of the measured position, an amplification of the measurement noise has been noticed. This noise effect has an impact on the behaviour of the motor, so that it may reduce the system performance. Accordingly, we propose to estimate the angular velocity of the pendulum using a Levant differentiator, to reduce the effect of the measurement noise.

A comparative study between a first-order sliding mode controller, the original RISE controller and the observer-based RISE controller is validated experimentally on the benchmark of the IWIP. Two different scenarios are conducted: in the first one, no additional mass is added to the pendulum body; and in the second, the system is exposed to more challenging external disturbances, compared to the previous work (Hfaiedh et al., 2018), where only punctual disturbances were considered. The disturbances, in the second scenario, consist of an additional mass added to the pendulum body in two successive and close periods. It is an important scenario, as we can evaluate the performance of both the original RISE control and the proposed observer-based RISE control. A comparative analysis is also presented based on various evaluation indices to quantify the performance of the SMC and the original RISE controller with respect to the proposed approach in terms of convergence and energy consumption.

This paper is organized as follows: in the second section, we present a general background on class I of nonlinear UMSs, represented in SF form, and give more details about the application of RISE feedback control; the third section is devoted to an example of application, provided to stabilize a second-order UMS with analytical proof of Lyapunov stability of the resulting closed-loop system; numerical simulation results with different scenarios are presented and discussed in the fourth section; in the fifth section, we present real-time experimental results to show the effectiveness of the proposed control approach combined with the Levant differentiator; and in the sixth section we provide a conclusion and suggestions for future work.

Proposed control solution

Class I of UMSs

Consider an UMS described by Equation (1):

D (q) \overset{\cdot\cdot}{q} + C (q, \overset{\cdot}{q}) \overset{\cdot}{q} + G (q) = F_{e} (q) τ

(1)

where $τ = [τ_{1}, τ_{2}]^{T} \in R^{m}$ is the control input, $D (q) \in R^{n \times n}$ is the inertia matrix, $C (q) \in R^{n \times n}$ is the Coriolis and centrifugal matrix, $G (q) \in R^{n \times 1}$ is the gravity vector and $F_{e} (q) = [0, I_{m}]^{T} \in R^{n \times m}$ ) denotes a non-square matrix of external forces with $m < n$ . Assume that the configuration is partitioned into $q = (q_{1}, q_{2})$ where $q_{1} \in R^{n - m \times 1}$ is the non-actuated coordinates and $q_{2} \in R^{m \times 1}$ is the actuated coordinates. UMSs with two degrees of freedom can be described as given by Equation (2):

[\begin{matrix} m_{11} (q) & m_{12} (q) \\ m_{21} (q) & m_{22} (q) \end{matrix}] [\begin{matrix} {\overset{\cdot\cdot}{q}}_{1} \\ {\overset{\cdot\cdot}{q}}_{2} \end{matrix}] + [\begin{matrix} h_{1} (q, \overset{\cdot}{q}) \\ h_{2} (q, \overset{\cdot}{q}) \end{matrix}] = [\begin{matrix} 0 \\ τ \end{matrix}]

(2)

where $m_{11} (q), m_{12} (q), m_{21} (q), m_{22} (q)$ represent the components of the inertia matrix and $h_{1} (q, \overset{\cdot}{q}), h_{2} (q, \overset{\cdot}{q})$ contain the Coriolis, centrifugal and gravity terms. Generally, UMSs in the form of Equation (2) are not fully feedback linearizable and conventional control approaches could not be straightforwardly applied to such systems. Based on a global change of coordinates, the system Equation (2) can be transormed into a cascade form. With this transformation, the control input can act on both actuated and non-actuated coordinates.

Due to the lack of actuation in the first part of Equation (2), the system belongs to the class I, otherwise, it belongs to the class II according to the classification of Olfati-Saber (2001). For instance, the translational oscillator with rotational actuator system, the acrobot (Zhao and Yi, 2006) and the IWIP belong to the class I, whereas the cart pole inverted pendulum and the pendubot belong to the class II. The main motivation of this classification is to obtain a general transformation of the system model into other simplified representations which make the control problem more simplified. For instance, the class I can be partially linearized and then transformed into a SF form, leading to a double integrator thanks to the lack of control input in the first part of Equation (2). This linearization method is known as collocated partial feedback linearization (Spong, 1994). In next subsection, we present a brief background on the transformation of class I of UMSs into a SF form.

Definition 1: A nonlinear system presented into a SF form (Krstic et al., 1995) can be described by the following triangular structure as givn by Equations (3) to (6):

\overset{\cdot}{z} = f (z, ξ_{1})

(3)

{\overset{\cdot}{ξ}}_{1} = ξ_{2}

(4)

{\overset{\cdot}{ξ}}_{n - 1} = ξ_{n}

(5)

{\overset{\cdot}{ξ}}_{n} = u

(6)

where $(z, ξ_{1}, ξ_{2}, . . ., ξ_{n})$ is the composite state, $f$ is a nonlinear function and $u$ is the control input. $\overset{\cdot}{z}, {\overset{\cdot}{ξ}}_{1}, . . ., {\overset{\cdot}{ξ}}_{n - 1}$ equations only depend on states $z, ξ_{1}, . . ., ξ_{i}$ that are fed back to that subsystem.

Background on RISE controller for class I of UMSs

The main motivation behind this subsection is to give a background about the design of the RISE controller using the global change of coordinates which simplifies the control problem by transforming the system into a SF form.

SF linearization

The UMSs of class I with two-DOF and only one control input can be represented into a cascaded nonlinear system in the SF representation if the following assumption is satisfied.

Assumption 1: Using the Lagrangian of the system, its configuration is partitioned into $q = (q_{1}, q_{2})$ and if the quantity $m_{11}^{- 1} (q_{2}) m_{12} (q_{2})$ is integrable, a global change of coordinates can be obtained as given by Equation (7):

\begin{matrix} z_{1} = q_{1} + γ (q_{2}) \\ z_{2} = m_{11} (q_{2}) p_{1} + m_{12} (q_{2}) p_{2} \\ ξ_{1} = q_{2} \\ ξ_{2} = p_{2} \end{matrix}

(7)

where $p_{1} = {\overset{\cdot}{q}}_{1}$ , $p_{2} = {\overset{\cdot}{q}}_{2}$ , $γ (q_{2}) = \int_{0}^{q_{2}} m_{11}^{(- 1)} (θ) m_{12} (θ) \partial θ$ and $[z_{1}, z_{2}, ξ_{1}, ξ_{2}]^{T}$ is the novel state vector. Using the new states, we obtain a new representation of the system composed of a linear (double integrator) subsystem and a nonlinear core subsystem (Olfati-Saber, 2001). The new system representation is then described as given by Equations (8) to (11):

{\overset{\cdot}{z}}_{1} = m_{11}^{- 1} (ξ_{1}) z_{2}

(8)

{\overset{\cdot}{z}}_{2} = g_{1} (z_{1} - γ (ξ_{1}), ξ_{1})

(9)

{\overset{\cdot}{ξ}}_{1} = ξ_{2}

(10)

{\overset{\cdot}{ξ}}_{2} = u

(11)

where $g_{1} (q_{1}, q_{2}) = - \frac{\partial V (q)}{\partial q_{1}}$ , $V (q)$ denotes the potential energy of the system and $u$ is the new control input from collocated partial feedback linearization. The relation between the torque and the new control term is described as given by Equations (12) to (14):

τ = Θ (q) u + ϒ (q, \overset{\cdot}{q})

(12)

Θ (q) = m_{22} (q) - m_{21} (q) m_{11}^{- 1} (q) m_{12} (q)

(13)

ϒ (q, \overset{\cdot}{q}) = h_{2} (q, \overset{\cdot}{q}) - m_{21} m_{11}^{- 1} (q) h_{1} (q, \overset{\cdot}{q})

(14)

Proposed RISE control design approach for class I of UMSs

The RISE feedback control was successfully applied to fully actuated systems, and was validated as a promising control technique. The main motivation of this paper is to propose a new redesign of the RISE feedback control for a more complex and challenging system such as the class I of UMSs. The following hypotheses are essential to ensure the design of the proposed nonlinear RISE controller. Consider the dynamics of a nonlinear system described by Equation (15):

M (x, \overset{\cdot}{x}, . . ., x^{n - 1}) x^{n} + F (x, \overset{\cdot}{x}, . . ., x^{n - 1}) = U (t)

(15)

where $(.)^{(i)} (t)$ denotes the $i^{th}$ time derivative of (.), $U$ represents the control input vector, $X = [x, \overset{\cdot}{x}, . . ., x^{n - 1}, x^{n}] \in R^{n}$ is the state vector, $M$ and $F \in R^{n}$ are uncertain nonlinear functions.

Property 1: The uncertain nonlinear function matrix $M$ satisfies the following inequality Equation (16):

\underline{m} ∥ ξ ∥^{2} \leq ξ^{T} M (.) ξ \leq \bar{m} (q) ∥ ξ ∥^{2}

(16)

$M (.)$ is bounded from below by a positive constant $\underline{m}$ and from above by a non-decreasing positive function $\bar{m} (q)$ .

Property 2: The two functions $M (.)$ and $F (.)$ of the nonlinear system Equation (15) are second-order differentiable.

Property 3: The desired reference trajectory is continuously differentiable with respect to time, up to the $(n + 2)^{th}$ derivative.

The expression of the RISE control law requires some auxiliary error signals, denoted by $e_{i} (t), i = 1, 2$ and defined as given by Equations (17) to (20):

e_{1} \overset{Δ}{=} q_{d} - q

(17)

{\overset{\cdot}{e}}_{1} \overset{Δ}{=} {\overset{\cdot}{q}}_{d} - \overset{\cdot}{q}

(18)

e_{2} \overset{Δ}{=} {\overset{\cdot}{e}}_{1} + α_{1} e_{1}

(19)

r \overset{Δ}{=} {\overset{\cdot}{e}}_{2} + α_{2} e_{2}

(20)

where the two constant gains $α_{1}$ and $α_{2}$ are positive. $q_{d}$ and ${\overset{\cdot}{q}}_{d}$ are the desired trajectory and its first time derivative, respectively. Some systems are not equipped with sensors required to measure acceleration forces due to their very high cost. So this quantity can be derived by differentiating the velocity signal. However, the limitation of this method is its high sensitivity to noise, as a low level of noise in the velocity signal can lead to an amplified noise in the estimated acceleration signal. For this reason, in this case, the auxiliary error signal $r (t)$ is not used in the control design since it requires a numerical differentiation through the computational algorithm to obtain the second derivative of the position signal which is foremost a noisy signal. One practical solution to attenuate the noise level in the velocity estimation is proposed in the next subsection. The expression of the RISE control law is expressed as given by Equation (21) (Fischer et al., 2014; Xian et al., 2004):

\begin{matrix} u = (k_{s} + 1) e_{2} - (k_{s} + 1) e_{2} (0) + ν_{f} \\ {\overset{\cdot}{ν}}_{f} = (k_{s} + 1) α_{2} e_{2} + β sign (e_{2}) \\ ν_{f} (0) = 0 \end{matrix}

(21)

where the two constant gains $k_{s}$ and $β$ are positive and $sign$ function denotes the mathematical Signum function described as

sign (x) = {\begin{matrix} 0 if x = 0 \\ \frac{x}{| x |} if x / = 0 \end{matrix}

State estimation with robust Levant differentiator

The presence of noise in measured signals is inevitable in any real-time experimental application. It certainly affects the performance of the system, especially when computing the first and second time derivatives of those noisy measured signals. To overcome this problem, we propose to use the differentiator of Levant (2003) to provide the derivatives required to implement the RISE controller. The choice of the differentiator is motivated by its high precision and robustness in the presence of noise and parametric uncertainties (Imine et al., 2011). The differentiator of Levant (2003) is described as given by Equations (22) to (26):

{\overset{\cdot}{w}}_{0} = v_{0}

(22)

v_{0} = w_{0} - λ_{2} L^{\frac{1}{3}} | w_{0} - y |^{\frac{2}{3}} sign (w_{0} - y)

(23)

{\overset{\cdot}{w}}_{1} = v_{1}

(24)

v_{1} = w_{1} - λ_{1} L^{\frac{1}{2}} | w_{1} - v_{0} |^{\frac{1}{2}} sign (w_{1} - v_{0})

(25)

{\overset{\cdot}{w}}_{2} = - λ_{0} Lsign (w_{2} - v_{1})

(26)

If $t = 0$ , $w_{0} = y$ and $w_{1} (0) = w_{2} (0) = 0$ . $y$ is the output of the system, $w_{0}$ is the estimated output. $w_{1}$ is its first time derivative and $w_{2}$ is its second time derivative. The gains $L$ , $λ_{0}$ , $λ_{1}$ and $λ_{2}$ are positive. In next section, we propose to design the RISE controller using a global change of coordinates for the IWIP and present the Lyapunov stability analysis of the closed-loop system. We also demonstrate the use of the resulting Levant differentiator in real-time experiments.

Application example: IWIP

The design of the proposed RISE controller using a global change of coordinates combined with the Levant differentiator is presented in this section. An application example of this control approach is illustrated on a second-order UMS. A brief description of the system, the control design and the stability analysis are introduced in this section.

Description of the IWIP

The IWIP (see Figure 1) has attracted the attention of many researchers within the control community (Freidovich et al., 2009; Zhang et al., 2016). It was considered a lot as a benchmark to study new nonlinear approaches (Aguilar-Avelar et al., 2017; Andary et al., 2012; Estrada et al., 2012; Gritli et al., 2017; Haddad et al., 2018; Moreno-Valenzuela et al., 2017). It has the flatness property with kinetic symmetry (Olfati-Saber, 2001). The pendulum angle $θ_{1}$ with respect to vertical axis is unactuated and the angle between the body and the inertia wheel $θ_{2}$ is actuated. The configuration in which the pendulum is pointed upwards corresponds to the unstable equilibrium point. Suppose that the torque $τ_{a}$ is the sum of the torque acting at $O$ and generated by the force $F^{+}$ and $τ_{b}$ is the sum of the torques generated by both the gravity $P$ and $F^{-}$ forces.

To straighten up the pendulum, the torque $τ_{a}$ must be greater than $τ_{b}$ . An angular acceleration is produced by the motor torque acting on the rotating wheel, and consequently, due to the dynamic coupling, a torque acting on the passive joint is generated. Therefore, the objective is to design a control acting on the inertia wheel to steer and maintain the pendulum in its unstable equilibrium point. The equation of motion of the system is obtained using the Euler–Lagrange method. The Lagrangian expression of this system is given by Equation (27):

L = \frac{1}{2} (I_{0} {\overset{\cdot}{θ}}_{1}^{2} + i_{2} ({\overset{\cdot}{θ}}_{1} + {\overset{\cdot}{θ}}_{2})^{2}) - m_{0} g \cos (θ_{1})

(27)

where $θ_{1}$ and $θ_{2}$ denote, respectively, the angular position of the pendulum and the angular position of the inertia wheel, ${\overset{\cdot}{θ}}_{1}$ and ${\overset{\cdot}{θ}}_{2}$ represent their corresponding velocities. The application of the Lagrange method leads to the following dynamic model of the IWIP (Fantoni and Lozano, 2002) as given by Equation (28):

[\begin{matrix} I_{0} + i_{2} i_{2} \\ i_{2} i_{2} \end{matrix}] [\begin{matrix} {\overset{\cdot\cdot}{θ}}_{1} \\ {\overset{\cdot\cdot}{θ}}_{2} \end{matrix}] - [\begin{matrix} m_{0} g \sin (θ_{1}) \\ 0 \end{matrix}] = [\begin{matrix} 0 \\ τ \end{matrix}]

(28)

where $τ$ is the torque applied on the active joint of the system. The parameters of this second-order UMS are summarized in Table 1. The expressions of the two constants $I_{0}$ and $m_{0}$ are given by Equations (29) and (30):

I_{0} = m l^{2} + M L^{2} + i_{1}

(29)

m_{0} = ml + ML

(30)

Table 1.

Summary of the dynamic parameters of the inertia wheel inverted pendulum.

Parameter	Description	Value
$i_{1}$	Body inertia	$0.031468 [kg . m^{2}]$
$i_{2}$	Wheel inertia	$4 . 17610^{- 4} [kg . m^{2}]$
$l$	Body centre of mass position	$0.06 [m]$
$L$	Wheel centre of mass position	$0.044 [m]$
$m$	Body mass	$3.228 [kg]$
$M$	Wheel mass	$0.86422 [kg]$
$g$	Constant of gravitational acceleration	$9.8 [m . s^{- 2}]$

The control design

The dynamic model Equation (28) satisfies Assumption 1, since the inertia matrix is constant. Therefore, the global change of coordinates can be defined as given by Equations (31) to (33):

z_{1} = \frac{\partial L}{\partial {\overset{\cdot}{θ}}_{1}} = (I_{0} + i_{2}) {\hat{\overset{\cdot}{θ}}}_{1} + i_{2} {\overset{\cdot}{θ}}_{2}

(31)

z_{2} = {\hat{θ}}_{1}

(32)

z_{3} = {\overset{\cdot}{θ}}_{2}

(33)

where $Z = [z_{1}, z_{2}, z_{3}]$ is the novel state vector, ${\hat{θ}}_{1}, {\hat{\overset{\cdot}{θ}}}_{1}$ are, respectively, the estimated angular position and velocity of the pendulum through the proposed Levant differentiator.

Remark 1: The observer is used to estimate the angular velocity of the pendulum. In fact, by computing the velocity signal using the numerical derivative of the measured position of the pendulum, an amplification of the measurement noise has been noticed. This noise effect has an impact on the behaviour of the actuator, so that it may reduce the performance of the closed-loop system. Accordingly, we propose to estimate both the angular position and the velocity of the pendulum using a differentiator, to obtain a more smooth velocity and control input signals.

Using the change of coordinates, the system is transformed into a SF form as given by Equations (34) to (36):

{\overset{\cdot}{z}}_{1} = \frac{\partial L}{\partial θ_{1}} = m_{0} g \sin (z_{2})

(34)

{\overset{\cdot}{z}}_{2} = \frac{z_{1}}{I_{0} + i_{2}} - \frac{i_{2} z_{3}}{I_{0} + i_{2}}

(35)

{\overset{\cdot}{z}}_{3} = u

(36)

With this new SF representation, the control input can act on both actuated and unactuated coordinates. The resulting system is presented as a cascade connection between a linear and a nonlinear subsystem where $z_{2}$ is considered as a virtual control input of the nonlinear $z_{1}$ -subsystem. Lyapunov analysis is adopted to make the nonlinear-subsystem represented by Equation (34) globally asymptotically stable. Assume that the desired trajectory for the $z_{1}$ -subsystem is described as a sigmoidal function and its first derivative are expressed as given by Equations (37) and (38):

θ_{1 d} = z_{2 d} = - \arctan (z_{1})

(37)

{\overset{\cdot}{θ}}_{1 d} = \frac{m_{0} g \sin (z_{2})}{(1 + (z_{1}^{2}))}

(38)

Let a valid Lyapunov function be expressed as as given by Equation (39):

V (z_{1}) = \frac{z_{1}^{2}}{2}

(39)

Taking the time derivative of the Lyapunov candidate Equation (39), we obtain Equation (40):

\overset{\cdot}{V} (z_{1}) = z_{1} {\overset{\cdot}{z}}_{1}

(40)

we substitute ${\overset{\cdot}{z}}_{1}$ by its expression Equation (34) into Equation (40), then we add the expression of Equation (37). After some trigonometric simplifications, we obtain Equations (41) to (43):

\overset{\cdot}{V} (z_{1}) = z_{1} m_{0} g \sin (z_{2})

(41)

\overset{\cdot}{V} (z_{1}) = z_{1} m_{0} g \sin (- \arctan (z_{1}))

(42)

\overset{\cdot}{V} (z_{1}) = - z_{1} m_{0} g \frac{z_{1}}{\sqrt{1 + z_{1}^{2}}}

(43)

Based on Lyapunov function Equation (39), we can conclude that the choice of the desired sigmoid function Equation (37) can globally stabilize the nonlinear subsystem Equation (34):

if $z_{1} > 0$ , then, $\overset{\cdot}{V} (z_{1}) < 0$

if $z_{1} = 0$ , then, $\overset{\cdot}{V} (z_{1}) = 0$

if $z_{1} < 0$ , then, $\overset{\cdot}{V} (z_{1}) < 0$

The proposed desired trajectories can now be used for the design of the RISE controller. The control law expressed in Equation (21) depends on the auxiliary errors ${\hat{e}}_{1}$ and ${\hat{e}}_{2}$ . Replacing Equations (37) and (38) in Equation (17) and substituting $\hat{θ}$ and $\overset{\cdot}{\hat{θ}}$ by their estimates $\hat{θ}$ and $\hat{\overset{\cdot}{θ}}$ , gives Equations (44) to (46):

{\hat{e}}_{1} = {\hat{θ}}_{1} - θ_{1 d}

(44)

{\hat{\overset{\cdot}{e}}}_{1} = {\hat{\overset{\cdot}{θ}}}_{1} - {\overset{\cdot}{θ}}_{1 d}

(45)

{\hat{e}}_{2} = {\hat{\overset{\cdot}{e}}}_{1} - α_{1} {\hat{e}}_{1}

(46)

where $θ_{1 d}$ denotes the desired trajectory and its first derivative is ${\overset{\cdot}{θ}}_{1 d}$ . $\hat{θ}$ and $\hat{\overset{\cdot}{θ}}$ are respectively the estimated pendulum position and velocity generated by the proposed Levant differentiator.

Using Equations (44) and (21), the control input designed, based on the estimated states, can be expressed by Equations (47) and (48):

u = (k_{s} + 1) {\hat{e}}_{2} - (k_{s} + 1) {\hat{e}}_{2} (0) + ν_{f}

(47)

{\overset{\cdot}{ν}}_{f} = (k_{s} + 1) α_{2} {\hat{e}}_{2} + β sign ({\hat{e}}_{2})

(48)

Figure 2 illustrates the block diagram of the proposed control approach based on the Levant differentiator.

Closed-loop stability analysis

By replacing the expression ${\overset{\cdot\cdot}{θ}}_{2}$ in the first part of Equation (28), we obtain the same form as in Equation (15) $m (q) {\overset{\cdot\cdot}{θ}}_{1} + f (θ_{1}) = u$ , where $m (q) = I_{0} + i_{2}$ and $f (θ_{1}) = - m_{0} gsin (θ_{1})$ . Using properties 1–3 and following the same reasoning as in Xian et al. (2004), the stability of the RISE controller is demonstrated. We consider that the desired joint positions, velocities and accelerations are denoted, respectively, by $q_{d} (t)$ , ${\overset{\cdot}{q}}_{d} (t)$ and ${\overset{\cdot\cdot}{q}}_{d} (t)$ . Taking the expressions of the tracking errors in Equation (17), the filtered tracking error $r (t$ ) is expressed by Equation (49):

r = {\overset{\cdot}{e}}_{2} + α_{2} e_{2}

(49)

where $α_{2} > 0$ is a positive design constant parameter. Multiplying the first derivative of Equation (49) by $M (q)$ , we obtain the open-loop error system as given by Equation (50):

m (q) \overset{\cdot}{r} = - \frac{1}{2} \overset{\cdot}{m} (q) r - e_{2} - \overset{\cdot}{u} + N

(50)

In our case, the inertia matrix is constant, then $\overset{\cdot}{m} (q) = 0$ , the expression Equatin (50) becomes Equation (51):

m (q) \overset{\cdot}{r} = - e_{2} - \overset{\cdot}{u} + N

(51)

where $N (.)$ denotes a nonlinear immeasurable auxiliary function expressed by Equation (52):

N (.) = m (q) (q_{d}^{(3)} + α_{2} {\overset{\cdot}{e}}_{2}) + e_{2} + \overset{\cdot}{f} (q)

(52)

After taking the time derivative of the control input, we obtain Equation (53):

\overset{\cdot}{u} = (k_{s} + 1) r + β sign (e_{2})

(53)

By substituting Equation (53) into Equation (51), we obtain the following closed-loop system Equation (54):

m (q) \overset{\cdot}{r} = - e_{2} - (k_{s} + 1) r - β sign (e_{2}) + N

(54)

Let us now consider an auxiliary function $N_{d} (.)$ expressed by Equation (55):

N_{d} (.) = m (q) q_{d}^{(3)} + \overset{\cdot}{f} (q_{d})

(55)

after we add and subtract the function $N_{d} (t)$ Equation (55) to the right-hand side of the closed-loop error system Equation (54), the following expression is obtained as given by Equation (56):

m (q) \overset{\cdot}{r} = - e_{2} - (k_{s} + 1) r - β sign (e_{2}) + \tilde{N} + N_{d}

(56)

where $\tilde{N}$ is defined by Equation (57):

\tilde{N} : = N - N_{d}

(57)

with the function $N$ defined in Equation (52) continuously differentiable, the function Equation (57) can be upper bounded as given by Equation (58):

∥ \tilde{N} ∥ < ρ (∥ z_{ρ} ∥) ∥ z_{ρ} ∥

(58)

where $∥ z_{ρ} ∥$ represents the Euclidean norm and $z_{ρ} (t)$ is defined as given by Equation (59):

z_{ρ} : = [e_{1}, e_{2}, r]^{T}

(59)

where $ρ : R_{\geq 0} \to R_{\geq 0}$ is some non-decreasing and globally invertible function.

Lemma 1: Consider the expression of the auxiliary function $L (t) \in R$ defined as given by Equation (60):

L = r (N_{d} - β sign (e_{2}))

(60)

If the control gain $β$ is chosen according to the following sufficient inequality expressed by Equation (61):

β > ∥ N_{d} (t) ∥_{L_{\infty}} + \frac{1}{α_{2}} ∥ {\overset{\cdot}{N}}_{d} (t) ∥_{L_{\infty}}

(61)

then we obtain Equation (62):

\int_{0}^{t} L (τ) d τ \leq ζ_{b}

(62)

where $∥ ∥_{L_{\infty}}$ is the $L_{\infty}$ norm and $ζ_{b}$ is a positive constant defined as given by Equation (63):

ζ_{b} = β | e_{2} (0) | - e_{2} (0) N_{d} (0)

(63)

Theorem 1: The control law ensures that all system signals are bounded under closed-loop operation and expressed as given by Equation (64):

∥ e (t) ∥ \to 0 as t \to \infty,

(64)

provided that the control gain $β$ is selected according to Equation (61), $α_{2} > \frac{1}{2}$ and the control gain $k_{s}$ is chosen sufficiently large relative to the system initial conditions.

Proof: Let us consider the auxiliary function $P (t) \in R$ expressed as given by Equation (65):

P (t) = ζ_{b} - \int_{0}^{t} L (τ) d τ

(65)

where $ζ_{b}$ and $L$ are defined in Equations (63) and (60), respectively. We define now the Lyapunov candidate $V : R^{n + 1} \times R_{\geq 0} \to R_{\geq 0}$ as given by Equation (66):

V (y, t) = \frac{1}{2} \sum_{j = 1}^{n = 2} e_{j}^{2} + \frac{1}{2} m r^{2} + P

(66)

where $y$ is defined as given by Equation (67):

y : = [z_{ρ}^{T} \sqrt{P}]^{T}

(67)

and $z_{ρ} (t)$ is defined in Equation (59). Since $m (q) > 0$ by deduction we obtain Equation (68):

\underline{m} \leq m (q) \leq (\bar{m} (∥ y ∥))

(68)

where $\underline{m}$ is a positive constant and $\bar{m} (q) : R_{\geq 0} \to R_{\geq 0}$ is some nondecreasing function (see Property 1). Therefore, we can bound Equation (66) as as given by Equation (69):

λ_{1} ∥ y ∥ \leq V \leq λ_{2} (∥ y ∥) ∥ y ∥^{2}

(69)

where $λ_{1} = (1 / 2) \min {1, \underline{m}}$ and $λ_{2} (∥ y ∥) : = \max {(1 / 2) \bar{m} (∥ y ∥), 1}$ .

After taking the time derivative of Equation (66), we obtain Equation (70):

\overset{\cdot}{V} = - α_{2} e_{2}^{2} + e_{1} e_{2} - r^{2} + r \tilde{N} - K_{s} r^{2} + [r (N_{d} - Bsign (e_{2}) - L]

(70)

where $\overset{\cdot}{P} = r (N_{d} - Bsign (e_{2}))$ . By using the fact that $e_{1} e_{2} \geq (1 / 2) (e_{1}^{2} + e_{2}^{2})$ and Equation (58) we can obtain an upper bound of Equation (70) as given by Equation (71):

\begin{matrix} \overset{\cdot}{V} \leq - λ_{3} ∥ z_{ρ} ∥^{2} + | r | ρ (∥ z_{ρ} ∥) ∥ z_{ρ} ∥ - k_{s} r^{2} \\ \overset{\cdot}{V} \leq - (λ_{3} - \frac{ρ^{2} (∥ z_{ρ} ∥)}{4 k_{s}}) ∥ z_{ρ} ∥^{2} \end{matrix}

(71)

where $α_{2} > 1 / 2$ and $λ_{3} = \min {(1 / 2), α_{2} - (1 / 2)}$

We can conclude from Equation (71) that we obtain Equation (72):

\overset{\cdot}{V} \leq - γ ∥ z_{ρ} ∥^{2}

(72)

with $k_{s} > \frac{1}{4 λ_{3}} ρ^{2} (∥ z_{ρ} ∥)$ and $γ$ some positive constant. Based on the lemma 2 of Xian et al. (2004), it can be verified that the closed-loop system states asymptotically converge to zero, thereby, $e_{1} \to 0$ , $e_{2} \to 0, r \to 0$ as $t \to \infty$ .

Numerical simulation results

In this section, we present simulation-based results applied to the IWIP to analyse the performance of the proposed control method without using the Levant differentiator. Two different simulation scenarios are proposed. In the first one, parametric uncertainties are introduced in some physical parameters of the system, whereas in the second one, a sawtooth and sinusoidal disturbing signals are considered.

Scenario 1: Robustness towards parametric uncertainties

In this scenario, we propose to evaluate the robustness of the proposed RISE control by considering an uncertainty term $Δ I_{0}$ on the parameter $I_{0}$ and an uncertainty term $Δ m_{0}$ on the parameter $m_{0}$ . $I_{0}$ and $m_{0}$ represent a combination of several physical parameters as described in Equations (29) and (30). We propose to compare the results with those of the first order SMC proposed in Khalid and Memon (2016). The physical parameters of the real benchmark are summarized in Table 1. The control gains are summarized in Table 2. To analyse the robustness of the RISE control with respect to the SMC, in case of parametric uncertainties, let us consider the following robustness evaluation indices:

(a) Integral Square error (ISE) as given by Equation (73):

ISE = \int e_{θ_{1}}^{2} dt

(73)

(b) Integral absolute error (IAE) as given by Equation (74):

IAE = \int | e_{θ_{1}} | dt

(74)

where $e_{θ_{1}}$ denotes the tracking error of the unactuated joint $θ_{1}$ .

Table 2.

Summary of the control design parameters.

Parameters	Value	Parameters	Value
$α_{1}$	$15.3$	$I_{0}$	$0.0130$
$α_{2}$	$1.5$	$m_{0}$	$0.213$
$β$	$14.05$	$K_{s}$	$125.0$
$ϒ_{0}$	1.4800	$Θ_{0}$	0.0092

In this subsection the obtained simulation results of the first scenario are presented and discussed. We chose the initial configuration as $[θ_{1}, {\overset{\cdot}{θ}}_{1}, {\overset{\cdot}{θ}}_{2}, v_{f}]^{T} = [0.2, 0, 0, 0]^{T}$ . The evolution of the system states versus time is depicted in Figure 3.

$Δ I_{0} = 0 %$ and $Δ m_{0} = 0 %$ : In this case, no uncertainties are introduced, thus the simulation is performed in nominal case. For the case of the RISE controller, we can clearly observe the convergence of the pendulum angle, the pendulum velocity and the inertia wheel velocity to the equilibrium point $[θ_{1}, {\overset{\cdot}{θ}}_{1}, {\overset{\cdot}{θ}}_{2}] = [0, 0, 0]$ in a smaller interval of time than the SMC approach. The input torque converges to a zero steady-state value as shown in Figure 3(d).

$Δ I_{0} = 20 %$ and $Δ m_{0} = 20 %$ : Despite the introduced uncertainties, no oscillations are produced by the proposed RISE controller compared to the standard SMC. A significant improvement of 79.2% and 83.2% are noticed when evaluating the ISE and IAE robustness indices, respectively, as reported in Table 3.

$Δ I_{0} = 50 %$ and $Δ m_{0} = 50 %$ : By introducing large variations of 50% in the model’s parameters $m_{0}$ and $I_{0}$ , we can clearly notice from simulation results a remarkable difference between the RISE and SMC controllers. The SMC is not able to compensate for the introduced parametric uncertainties. More oscillations around the equilibrium point are observed when increasing the percentage of uncertainties in the case of the SMC approach. However, it is clear from the results of the proposed RISE that the control approach can compensate for parametric uncertainties. To sum up, we can conclude that the proposed standard RISE controller is more robust towards parametric uncertainties than the SMC one.

Table 3.

Control robustness evaluation of the sliding mode control (SMC) and robust integral of the sign of the error (RISE) controllers for scenario 1.

	Integral absolute error	Integral square error
SMC $Δ I_{0} = 20 %$ , $Δ m_{0} = 20 %$	0.0965	0.0086
RISE $Δ I_{0} = 20 %$ , $Δ m_{0} = 20 %$	0.02	0.0014
Improvements	79.2%	83.2%

Scenario 2: External disturbances rejection

The first considered perturbation is a non-sinusoidal sawtooth disturbing torque equal to [0.3 Nm] applied to the pendulum at time $t = 4 s$ and the remaining $3$ seconds. The second applied disturbing signal is expressed by $D = 0.1 \sin (15 t) + 0.15$ . It has been applied during the interval [10 13] seconds.

The simulation results of the second scenario are displayed in Figure 4, where we can see the effect of the disturbances in the evolution of the states. The proposed original RISE controller reacts immediately to these disturbances and brings back the states to the desired equilibrium point. The deviation due to the external disturbances is compensated for by the control action. The numerical simulation shows the effectiveness of the proposed control approach compared to the SMC especially when the system is subject to the first non-sinusoidal and periodic disturbance. The performance of the proposed control approach is improved by more than 90% as illustrated in Table 4. In the next section, we discuss the performance of the proposed controller through real-time experiments.

Table 4.

Control performance evaluation of the sliding mode control (SMC) and robust integral of the sign of the error (RISE) controllers for scenario 2.

	Integral absolute error	Integral square error
SMC	0.6228	0.0728
RISE	0.0533	0.0017
Improvements	91.4%	97.7%

Real-time experimental results

To validate that the performance of the proposed RISE control approach complied with the Levant differentiator, real-time experiments have been performed on the testbed of the IWIP designed at the Montpellier Laboratory of Computer Science, Robotics, and Microelectronics of the University of Montpellier and the National Center for Scientific Research, France (www.lirmm.fr). The electrical and mechanical components of the experimental testbed are shown in Figure 5, for more details about the experimental platform, the readers can refer to Andary et al. (2009), Touati and Chemori (2013), Haddad et al. (2018) and Hfaiedh et al. (2020b).

Remark 2: The dynamic model of the IWIP in Equation (28) and the one of the experimental model in Figure 5 are considered the same. Indeed, there may still be a difference between the dynamics of the experimental plant and the mathematical model even with a good identification. For instance, the dynamic model of the actuator and the frictional forces of the system are neglected in the proposed dynamic model of the system. Two experimental scenarios have been performed. In the first one, no external disturbances are considered, that is the experimental validation is achieved in the nominal case.

The proposed observer-based RISE control, the original RISE and the SMC are compared in nominal case.

In the second scenario, we consider persistent external disturbances. The control design parameters are summarized in Table 5. In order to analyse the control performance and to highlight the improvement gained by using the Levant differentiator in terms of tracking error, let us consider the following evaluation indices:

The root mean square of the tracking error (RMSE) as given by Equation (75):

RMSE = {(\frac{1}{N} \sum_{i = 1}^{N} (e_{θ_{1}} (i))^{2})}^{\frac{1}{2}}

(75)

where $N$ denotes the number of the recorded samples and $e_{θ_{1}}$ denotes the tracking error of the unactuated joint $θ_{1}$ .

Table 5.

Summary of the control design parameters.

Parameters	Value	Parameters	Value
$α_{1}$	$7.3$	${\hat{θ}}_{1} (0)$	$0.16$
$α_{2}$	$3.2$	$λ_{0}$	$110$
$β$	$14.05$	$λ_{2}$	$50$
$K_{s}$	$19.09$	$λ_{1}$	$80$

We consider the following input-torque-based criterion in order to analyse the performance of the system in terms of energy consumption as given by Equation (76):

E_{τ} = \sum_{i = 1}^{N} | τ_{θ_{2}} (i) |

(76)

where $τ_{θ_{2}} = u$ denotes the torque generated by the actuator of the inertia wheel.

Scenario 1: Nominal case

In this subsection, we present the results of both the SMC, the RISE controller using numerical derivative as well as the obtained results of the proposed extended RISE controller combined with the Levant differentiator. The evolution of the angular position $θ_{1}$ and velocity of the pendulum ${\overset{\cdot}{θ}}_{1}$ are depicted in Figure 6. The results show that all states converge to the unstable equilibrium point. However, the response of the pendulum and the wheel are fast for the case of the original RISE and the proposed approach compared to the first-order SMC approach. For more clarity, the plots in Figure 6 are zoomed within the intervals [6.5,7] seconds and [6,7], respectively. The measured states of the standard RISE controller are presented with a dashed line and the estimated signals obtained from the differentiator are presented with a solid line. We observe a good stabilization of the pendulum around its unstable equilibrium. However, a noticeable difference is highlighted in the zoomed plots between the numerical derivative of the pendulum position and the estimated one generated by the differentiator. We can easily find out that the use of the differentiator attenuates significantly the noise as opposed to the numerical derivative.

The evolution of the control input for the three controllers and the auxiliary error $e_{2}$ are displayed in Figure 7. The plots are zoomed within the interval [10,15] seconds where the solid line represents the results of the proposed control approach and the dashed line represents the results of the standard RISE without the differentiator. We can also observe an improvement in the signal quality of the estimated auxiliary errors $e_{2}$ with respect to the measured one.

Although the standard RISE controller ensures a good stabilization of the system, the use of the Levant differentiator leads to an attenuation of noise in the velocity estimation resulting in a smoother control input signal, opposed to the use of a numerical derivative which amplifies any embedded noise in the measurement signals. This noise is due, for instance, to some interferences from other electrical sources, hardware, sensors or environment.

To sum up, the real-time experimental results illustrate clearly that the original RISE and the proposed observer-based RISE approach outperform the SMC in terms of convergence. On the other hand, the Levant differentiator provides a better real-time estimation of the velocity than the numerical derivative.

To compare the effectiveness of the three approaches, we suggest computing the above performance-evaluation indices. The obtained results are summarized in Table 6. The improvements are computed with respect to the SMC approach. Note that, the smallest value of the performance criteria reflects the best performance. This scenario shows clearly that both the proposed observer-based RISE control scheme and the original RISE control one overcome the SMC in terms of convergence and energy consumption. It can be concluded from the obtained results that the RISE controller without differentiator provides satisfactory results. However, the proposed control scheme has the best performance evaluation indices and improvement compared to the standard RISE approach, thanks to the Levant differentiator.

Table 6.

Quantification of the performance through different evaluation criteria for scenario 1.

Criteria	Sliding mode control	Original robust integral of the sign of the error (RISE)	Improvements	RISE with differentiator	Improvements
Root mean square error [rad]	0.0051	0.0012	76.4%	8.643 $e^{- 4}$	82.35%
Integral square error	0.0239	0.0098	59%	0.0089	62.760%
Integral absolute error	0.3961	0.165	58.34%	0.1548	60.92%
$E_{τ} [N . m]$	189.8981	163.96	13.66%	134.09	18.22%

Scenario 2: Persistent disturbances rejection

The main motivation behind the second scenario is to show the effectiveness of the control approach when the system is under a more complex and more challenging type of disturbances. It consists of two persistent forces generated through an additional mass attached to the pendulum body. The mass was attached during the interval $[10.5, 16.5]$ seconds and $[21, 26]$ seconds. An illustration of added persistent mass is shown in Figure 8. It is an interesting scenario as the added mass creates a variation in the dynamic model. Indeed the pendulum mass becomes ( $m_{*} = m + m_{dis}$ ) where $m_{dis}$ is the added mass. On the other hand, the applied disturbances create an external torque disturbing the pendulum. Compared to the previous work in Hfaiedh et al. (2018), the disturbances are not punctual but applied in two consecutive and different periods. It is a more challenging scenario, to evaluate the performance and energy consumption of the proposed control scheme. In this scenario, the original standard RISE controller and the proposed observer-based RISE controller are tested and compared. For the purpose of clarity, a zoomed-in view of the evolution of the states within the two intervals is depicted in Figure 9. According to these results, we can observe that the persistent disturbances are compensated for by both control approaches and the measured position and velocity converge to equilibrium steady states. However, it can be seen from the zoom plots that the differentiator-based RISE controller provides better results than the standard RISE controller. We can also point out that the estimation of the angular position of the pendulum and its derivative has an impact on the control input signal and therefore improves the disturbances rejection in the controlled system. These results are verified also by the above proposed quantification of the performance through different evaluation criteria which are summarized in Table 7.

Table 7.

Quantification of the performance through different evaluation criteria for scenario 2.

Criteria	Without differentiator	With differentiator	Improvement
Root mean square error [rad]	9.265 $e^{- 3}$	5.853 $e^{- 3}$	36.83%
Integral square error	14.4905 $e^{- 3}$	14.35766 $e^{- 3}$	0.92%
Integral absolute error	368.0434 $e^{- 3}$	322.1330 $e^{- 3}$	12.47%
$E_{τ} [N . m]$	218.7362	206.8575	5.43%

Conclusion and future work

This paper proposes two robust RISE control approaches for the stabilization of class I of UMSs. For the design of the RISE controller, the model of the system is first transformed into a SF form. With a Lyapunov-based analysis, two new desired trajectories are defined. Then the stability analysis of the resulting closed-loop system and simulation results for different operating conditions were addressed. They attest clearly that the first proposed RISE control approach ensures better stabilization and robustness towards different parametric uncertainties and external disturbances, compared to the SMC approach. However, the limitation of this approach was experimentally noticed through the amplified noise, resulting from the numerical derivative action in the control law. To overcome this problem, a Levant differentiator was proposed and implemented to estimate the system states, including the time derivative of the angular position. Based on the obtained experimental results and further analysis based on some performance-evaluation indices, the combination of Levant differentiator with RISE controller has significantly attenuated the effect of the noise included in the measured signals and improved the performance in terms of tracking error and energy consumption, compared to original RISE and SMC schemes. In future work, various possible perspectives of this work can be investigated. At first, we can combine an adaptation law with the RISE controller to estimate the gains. Furthermore, the automatic optimal tuning of the control design parameters of the controller can also be considered. Finally, further discussions can be investigated about the generalization of this study to the case of other classes of UMSs.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Afef Hfaiedh

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