A revisited adaptive super-twisting control of underactuated mechanical systems: Design and experiments

Abstract

This paper presents a real-time implementation of an adaptive super-twisting (ASTW) controller. This work deals with the stabilization challenging problems of underactuated mechanical system subject to parametric variations and state- and time-dependent disturbances. Due to the low number of control inputs compared to the number of DOFs, the proposed approach has been designed based on an explicit global change of coordinates. The stability analysis of the resulting closed-loop system was performed based on Lyapunov’s theory. Extensive experimental investigations were conducted on the inertia wheel inverted pendulum (IWIP) to demonstrate the robustness of the proposed solution. These experiments include a real-time comparison with existing control solutions from the literature, providing a detailed analysis of performance, effectiveness, and robustness under various conditions. Quantitative evaluations were carried out based on different performance criteria, highlighting the practical applicability and superiority of the proposed control strategy in real-time scenarios. The results from both numerical simulations and physical experiments demonstrate the controller’s capability to maintain stability and performance despite the presence of disturbances and parametric variations.

Keywords

Underactuated mechanical systems robust control strict-feedback form Lyapunov analysis inertia wheel inverted pendulum real-time experiments

Introduction

Sliding mode control (SMC) is a well-known robust control widely used for nonlinear systems (Utkin, 1992) due to its robustness against matched uncertainties and disturbances. The first-order SMC is characterized by an undesirable finite frequency and amplitude oscillations around a predefined switching manifold, which is known as the chattering phenomenon. From a practical point of view, the chattering involves high control activity by generating the high-frequency of non-modeled dynamics (Slotine and Li, 1991) and will result in unnecessary/rapid wear of the actuation components (Andrzej, 2000). The high-frequency oscillation can be caused, for instance, by the presence of a parasitic dynamics in series with the control system or by switching time delays. More details regarding the causes of this phenomenon are provided in Yu and Kaynak (2009). To overcome the chattering problem, various solutions include, but not limited to, continuous approximation of the discontinuous control law (Slotine and Li, 1991), observer-based approaches (Bondarev et al., 1985), second-order SMC (Gonzalez et al., 2012), and intelligent approximation algorithms (Nafa et al., 2021). The super-twisting algorithm (STA) introduced in Levant (1993) is a second-order sliding mode controller that is a powerful tool for minimizing the effect of chattering. STA has shown promising results in terms of robustness and fast response. It has the ability to generate a continuous control signal and is appropriate for systems with Lipschitz continuous matched uncertainties/disturbances with bounded gradients (Ramesh-Kumar and Bandyopadhyay, 2014). Various stability analysis techniques based on Lyapunov’s theory have been proposed in the literature to ensure finite-time convergence with an estimation of the convergence time for unperturbed dynamics (Sanchez and Moreno, 2012), systems with perturbations dependent only on time (Levant, 1993), and systems with perturbations depending on both state and time (Castillo et al., 2018).

Several research efforts have been made to address modeling, stability, and control issues of nonlinear second-order underactuated mechanical systems (UMSs) (Krafes et al., 2018). The super-twisting (STW) approach has been widely employed for controlling various UMSs as underactuated crane (Sun et al., 2015), Pendubot (Ramos-Paz et al., 2017), beam-and-ball (Din et al., 2018), and IWIP (Hfaiedh et al., 2020a). Adaptive and variable gain sliding mode approaches are both strategies used in control systems to improve system performance, particularly in dealing with uncertainties and disturbances. In Nafa et al. (2021), an adaptive controller using SMC and a wavelet network (WN) was proposed for a class of second-order UMSs with two degrees of freedom (DOFs). Theoretical and numerical analyses demonstrated that the proposed approach ensures the asymptotic stability and convergence of the closed-loop system. In Lu and Fang (2021), a gain-adapting coupling controller was designed by incorporating several unactuated state-related information into the control law. A real-time comparative study with a proportional–integral–derivative (PID) controller was performed to demonstrate the efficiency of the proposed method. In Hfaiedh and Abdelkrim (2022), another real-time implementation of an adaptive SMC was proposed, where the switching gain was dynamically estimated using adaptive laws. It is based on the approximation of the signum function using a hyperbolic tangent function. The main drawback of this control approach lies in the chattering, which still represents the main issue for the SMC in terms of real-time implementation. Existing works can also be found in the literature, based on adaptive super-twisting (ASTW) control (El-Sousy et al., 2022; Mobayen, 2019; Zhang et al., 2019). Various other control strategies are based on feedback robust integral of the sign of the error (RISE) control (Hfaiedh et al., 2021), third-order discontinuous integral algorithm (Gutierrez-Oribio et al., 2021), interconnection and damping assignment-passivity-based control (IDA-PBC) approaches (Gritli et al., 2017; Hfaiedh et al., 2020b) and passivity-based approaches (Zhai et al., 2022). Other intelligent algorithms used to improve robustness and capture unknown dynamics of underactuated systems with constraints were proposed in Yang et al. (2023; 2021). These algorithms are designed to learn from training and can adapt to changing system dynamics; however, they may require a large amount of data to train and specific hardware architectures to be implemented.

In this study, we focus on the control problem of nonlinear second-order UMSs, while dealing with external disturbances, parametric uncertainties, and perturbations depending on both states and time. The proposed control solution is an ASTW algorithm, based on collocated partial-feedback linearization. It is validated through real-time experiments on the IWIP and compared to the standard STW and other existing methods from the literature including RISE (Hfaiedh et al., 2021) and adaptive SMC (Hfaiedh and Abdelkrim, 2022).

We propose in this work to benefit from the advantages of the STW with the important feature of the adaptation law to guarantee the establishment of a real second-sliding mode. Furthermore, it is worth noting that the proposed controller is appropriate for the case in which the bounds of the uncertainties and perturbations are unknown and cannot be estimated for real applications. It does not require any information regarding the bounds of disturbances and their gradients, except for their existence. To deal with the nonlinear coupling between coordinates, we propose to transform the system into a strict-feedback form in order to decouple the system into (inner nonlinear and linear subsystems). Then, we propose two new desired trajectories, required for the design of the sliding surface. The choice behind the desired trajectory is to ensure the asymptotic stability of the internal dynamics represented by the nonlinear subsystem. The stability analysis of the whole resulting closed-loop system is provided based on Lyapunov’s theory. Moreover, in real-time experiments, control gains need to be calibrated; we proposed a control tuning algorithm to select the adjustable gains in real process.

To validate the proposed approach, numerical simulation and experimental comparative studies were conducted. Three main simulation scenarios are considered. The first one is the nominal case, where the system operates under ideal conditions without any disturbances or uncertainties. In this case, the effect of control parameters has been studied. The second scenario is robustness towards parametric uncertainties, which evaluates the controller’s performance when there are variations in system parameters, and the third one is state- and time-dependent disturbance rejection, assessing the ability of the proposed approach to handle disturbances that vary with both the state and time. In this case, a comparison is conducted with respect to the standard STW controller. In real-time validation, two experiments are performed—one under nominal conditions and the second under external disturbances. In the experiments, the proposed approach is compared with four controllers from the literature namely: SMC, adaptive sliding mode control (ASMC) (Hfaiedh and Abdelkrim, 2022), STW (Hfaiedh et al., 2020a), and RISE (Hfaiedh et al., 2021). The results were also validated by computing various performance indices.

ASTW control has already been used in the literature to control fully actuated systems in Shtessel et al. (2012) and Rajappa et al. (2016). However, it is not straightforwardly applicable to UMSs because of (1) underactuation, which represents a source of dynamic constraints; (2) the complexity and diversity of the nonlinear coupling between coordinates, which differs from one system to another; (3) their nonlinear dynamics; (4) their unstable internal dynamics; and (5) external disturbances and parametric uncertainties. For fully actuated systems, the system can be directly controlled and the control problem is often well-defined; however, for UMSs, the control problem involves designing a controller that can exploit the inherent dynamics of the system to achieve the desired output behavior. Nevertheless, it is possible to indirectly control the coordinates of the internal dynamics using appropriate techniques. Control design is intended to allow such systems to perform complex tasks with fewer actuators. If the aforementioned aspects are not considered in the control design, the system will fail to provide the intended response, which may result in instability or degraded performance of the control system. Moreover, ASTW control has already been used in the literature to control underactuated systems (El-Sousy et al., 2022; Mobayen, 2019). Compared to the proposed approach, Mobayen (2019) introduce an adaptive global SMC approach based on the STW scheme, aimed at stabilizing a specific class of underactuated systems. This method employs a different sliding surface expression and control law expression. Furthermore, El-Sousy et al. (2022) present an adaptive PID SMC technique combined with the STA. However, disturbances dependent on time and state have not been addressed in the above references. In addition, the systems under study differ from the IWIP. In Mobayen (2019), the control law is implemented for a cart-inverted pendulum system, while in El-Sousy et al. (2022), it is applied to a rotating pendulum. Both systems belong to class II of UMSs, as classified in Olfati-Saber (2001). In this work, we focus only on UMSs belonging to class I, which can be transformed into strict-feedback form. The proposed approach can potentially be extended to higher-order UMS, provided that the dynamic model can be expressed in this form. It is worth noting that Hfaiedh and Abdelkrim (2022) proposed a similar transformation to decouple the system. The difference compared to this work lies in both the proposed desired trajectories and control approach. Similarly, for the case of Hfaiedh et al. (2021), which proposed an RISE control with a strict-feedback transformation. However, disturbances depending on time and state had not been addressed in the aforementioned published studies. In this work, we have validated through numerical simulation the effectiveness of ASTW with respect to disturbances depending on time and state. In addition, in the real-time comparative study, we have conducted an additional experimental scenario involving punctual perturbation, different from the operating conditions proposed in Hfaiedh and Abdelkrim (2022) and Hfaiedh et al. (2021).

The rest of this paper is organized as follows. Section “Class I of UMSs” presents a brief background on class I UMSs. Section “Control design” presents a brief review of the STW and ASTW controllers. Section “An illustrative application example” describes the application of the proposed approach to the testbed of the IWIP. Section “Numerical simulation results” discusses the simulation results, and section “Implementation issues and real-time experimental results” presents the implementation issues and real-time experimental results. The conclusion and future directions are summarized in section “Conclusion and future work.”

Class I of UMSs

Using an invertible change of control, UMSs with two DOFs and one control input can be partially linearized by collocated partial-feedback linearization.

Collocated partial-feedback linearization

The Euler–Lagrange equations of motion of a UMS, with $n$ configuration variables and $m$ control inputs, can be written in the form: $\frac{d}{dt} (\frac{\partial L}{\partial \overset{\cdot}{q}}) - \frac{\partial L}{\partial q} = F (q) τ$ , where $L$ is the Lagrangian, $q$ denotes the generalized coordinates, $F (q) \in R^{n \times m}$ is a full-column rank non-square matrix of the distribution of control inputs on the DOFs with $m < n$ , and $τ \in R^{m}$ denotes the control input vector. The general matrix form of the equation of motion for a UMS can be expressed as follows

M (q) \overset{\cdot\cdot}{q} + C (q, \overset{\cdot}{q}) \overset{\cdot}{q} + G (q) = F (q) τ

(1)

where $M (q)$ is the inertia matrix, $C (q, \overset{\cdot}{q})$ denotes the Coriolis and centrifugal force matrix, and $G (q)$ represents the gravity vector, and $τ$ is the control input vector. Variables $q$ , $\overset{\cdot}{q}$ , and $\overset{\cdot\cdot}{q}$ denote the position, velocity, and acceleration vectors, respectively. Consider $F (q) = [0, I_{m}]^{T}$ , and the first configuration variable $q_{1} \in R^{n - m}$ is the unactuated coordinate and the second configuration variable $q_{2} \in R^{m}$ represents the actuated coordinate. The dynamics of a second-order UMS ( $m = 1, n = 2)$ is expressed in the following form

[\begin{matrix} m_{11} & m_{12} \\ m_{21} & m_{22} \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 $m_{ij}$ $i$ , for $j = 1, 2$ represent the inertial terms. $h_{i} (q, \overset{\cdot}{q})$ for $i = 1, 2$ include the Coriolis, centrifugal, and gravity terms.

In this study, we adopted the method of partial-feedback linearization proposed in Spong (1994), where the system (2) is partially linearized.

Lemma 1. Consider a global invertible change of control of the form $τ = Θ (q) u + ϒ (q, \overset{\cdot}{q})$ , for which the dynamics (2) is partially linearized as follows

\begin{matrix} {\overset{\cdot}{q}}_{1} = p_{1} \\ {\overset{\cdot}{p}}_{1} = f_{0} (q, p) + g_{0} (q) u \\ {\overset{\cdot}{q}}_{2} = p_{2} \\ {\overset{\cdot}{p}}_{2} = u \end{matrix}

(3)

where $x = [q_{1}, p_{1}, q_{2}, p_{2}]^{T}$ is the new state vector after partial linearization, $f_{0} (q, p)$ and $g_{0} (q)$ are expressed by $f_{0} (q, p) = - m_{11}^{- 1} (h_{1} (q, \overset{\cdot}{q}))$ and $g_{0} (q) = - m_{11}^{- 1} (q) m_{12} (q)$ . The expressions for $Θ (q)$ and $ϒ (q, \overset{\cdot}{q})$ of the invertible change of control are described by $Θ (q) = m_{22} (q) - m_{21} (q) m_{11}^{- 1} (q) m_{12} (q)$ and $ϒ (q, \overset{\cdot}{q}) = h_{2} (q, \overset{\cdot}{q}) - m_{21} m_{11}^{- 1} (q) h_{1} (q, \overset{\cdot}{q})$ .

Proof. By isolating the expression of ${\overset{\cdot\cdot}{q}}_{1}$ from the first equation of equation (2), we obtain

{\overset{\cdot\cdot}{q}}_{1} = - m_{11}^{- 1} (h_{1} (q, \overset{\cdot}{q}) - m_{12} (q) {\overset{\cdot\cdot}{q}}_{2})

(4)

Then, substituting equation (4) in the second equation of equation (2), we obtain

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

(5)

Replacing $τ$ by $τ = Θ (q) u + ϒ (q, \overset{\cdot}{q})$ , we obtain the expressions of $Θ (q)$ and $ϒ (q, \overset{\cdot}{q})$

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

(6)

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

(7)

After the partial-feedback linearization, the system (3) is represented in the form of two subsystems. The control input $u$ appears in both the nonlinear ( $q_{1}, p_{1}$ )- and the linear ( $q_{2}, p_{2}$ )-subsystems. These two subsystems can be decoupled through a global change of coordinates. This transformation is described in detail in the next section.

Strict-feedback form

The explicit change of coordinates was determined, which, from the partial-feedback linearization of Spong and Praly (1997), transforms the system (3) into a strict-feedback representation, where the control input appears only in the linear subsystem (Olfati-Saber, 2001). This change in coordinates is expressed as follows

\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}

(8)

where $γ (q_{2}) = \int_{0}^{q_{2}} m_{11}^{(- 1)} (θ) m_{12} (θ) d θ$ and $x_{z} = [z_{1}, z_{2}, ξ_{1}, ξ_{2}]^{T}$ denotes the state vector of the new representation. Using these new states, a new representation of the system composed of a linear (double integrator) subsystem and a nonlinear core subsystem was obtained (Olfati-Saber, 2001). The new system representation is described as follows

\begin{matrix} {\overset{\cdot}{z}}_{1} = m_{11}^{- 1} (ξ_{1}) z_{2} \\ {\overset{\cdot}{z}}_{2} = g_{1} (z_{1} - γ (ξ_{1}), ξ_{1}) \\ {\overset{\cdot}{ξ}}_{1} = ξ_{2} \\ {\overset{\cdot}{ξ}}_{2} = u \end{matrix}

(9)

where $g_{1} (q_{1}, q_{2}) = - \frac{\partial V_{p} (q)}{\partial q_{1}}$ , $V_{p} (q)$ denotes the potential energy of the system, and $u$ denotes the control input obtained based on the collocated partial-feedback linearization (Spong, 1994).

To sum up step-by-step, the dynamics of equation (2) belongs to the class I of UMSs based on the classification of Olfati-Saber (2001). By changing the control, the dynamics (2) were partially linearized to obtain the normal form (3). Through a global change in coordinates described in equation (8), the system (3) is decoupled and transformed into a strict-feedback representation (9).

Remark 1. Without loss of generality, the proposed approach can easily be generalized to other systems of class I, such as the translational oscillator with rotational actuator (TORA) and Acrobot. Furthermore, it can be extended to higher-order systems that can be transformed into a strict-feedback form, such as the vertical take-off and landing aircraft (Olfati-Saber, 2001) and underactuated surface vessel (Ding et al., 2013).

Remark 2. A system may be controllable but still not stabilizable for various reasons such as unmeasurable states. To address these issues in underactuated systems, we propose to use the partial-feedback linearization to transform the system into a controllable and stabilizable form and design a control law to achieve the desired performance.

In the following section, an ASTW scheme is proposed for a second-order UMS, and the stability analysis is addressed using Lyapunov’s theory.

Control design

After the transformation of the equation of motion (1) into a strict-feedback form, a sliding mode controller was designed in the presence of bounded disturbances for the single-input cascade nonlinear system (9).

A brief background on STW SMC

Consider that the dynamics (9) has the general form $\overset{\cdot}{x} = f (x) + g (x) u$ , where $f$ and $g$ are the nonlinear functions and $u$ is the control input. The STW control law (Moreno and Osorio, 2008) can be described as follows

u = - k_{1} {| σ |}^{\frac{1}{2}} sign (σ) + w

(10)

\overset{\cdot}{w} = - k_{2} sign (σ)

(11)

where $σ$ is the sliding variable, $k_{1}$ and $k_{2}$ are the positive constant design gains, and $w = - k_{2} \int sign (σ) d t$ . Now, considering an uncertain nonlinear scalar system (Rivera et al., 2011)

\overset{\cdot}{σ} = f (t, σ) + u

(12)

where $u$ is the control input (10) and the function $f (t, σ)$ is a perturbation term globally bounded as follows: $f (t, σ) \leq δ {| σ |}^{\frac{1}{2}}$ , for some positive constant $δ > 0$ . The closed-loop system resulting from replacing equations (10) and (11) in equation (12) is expressed as

\overset{\cdot}{σ} = - k_{1} {| σ |}^{\frac{1}{2}} sign (σ) + w + f (t, σ)

(13)

\overset{\cdot}{w} = - k_{2} sign (σ)

(14)

For stability analysis, let us consider the vector $ζ^{T} = [ζ_{1}, ζ_{2}]^{T} = [{| σ |}^{\frac{1}{2}} sign (σ), w]^{T}$ and propose the following quadratic Lyapunov candidate

V = 2 k_{2} | σ | + \frac{1}{2} w^{2} + \frac{1}{2} {(k_{1} {| σ |}^{\frac{1}{2}} sign (σ) - w)}^{2}

(15)

ζ^{T} P ζ

(16)

where $P$ is a positive, symmetric matrix given by $P = \frac{1}{2} (\begin{matrix} 4 k_{2} + k_{1}^{2} & - k_{1} \\ - k_{1} & 2 \end{matrix})$ .

Remark 3. It is worth noting that $V (x)$ appears to be discontinuous because it contains the term $sign (σ)$ ; however, it can be considered a continuous function everywhere, except at $x_{1} = 0$ , where it is not differentiable (Moreno and Osorio, 2008).

The time derivative of Lyapunov’s function (15) along the solutions of the system (13) leads to the following

\overset{\cdot}{V} = - \frac{k_{1} ζ^{T} Q ζ}{2 \sqrt{| σ |}}

(17)

where $Q = (\begin{matrix} 2 k_{2} + k_{2}^{1} - (\frac{4 k_{2}}{k_{1}} + k_{1}) δ & - k_{1} + 2 δ \\ - k_{1} + 2 δ & 1 \end{matrix})$ .

It is worth noting that $\overset{\cdot}{V}$ is negative definite for any $Q > 0$ and $k_{1}$ and $k_{2}$ chosen such that $k_{1} > 2 δ$ and $k_{2} > k_{1} \frac{5 δ k_{1} + 4 δ^{2}}{2 (k_{1} - 2 δ)}$ .

Proposed ASTW controller

The system (9) may be reformulated in the general form $\overset{\cdot}{x} = F (x, t) + G (x) u$ , where $u \in R$ is the control input and $x \in R^{n}$ is the state vector.

Assumption 1. The sliding dynamics of the system (9) described in the form $\overset{\cdot}{x} = F (x, t) + G (x) u$ , represents an uncertain nonlinear system that can be expressed as follows

\overset{\cdot}{σ} = Ψ (x, t) + Φ (x, t) u

(18)

where $Ψ (x, t) = \frac{\partial σ}{\partial t} + \frac{\partial σ}{\partial x} F (x, t)$ and $Φ (x, t) = \frac{\partial σ}{\partial x} G (x) u$ . Furthermore, assume that the function $Φ \in R$ can be expressed as a sum of a known function $Φ_{0}$ and a bounded perturbation $Δ Φ$ with unknown boundaries

Φ (x, t) = Φ_{0} (x, t) + Δ Φ (x, t)

(19)

where $\frac{| Δ Φ |}{Φ_{0}} \leq φ_{1} < 1$ .

Assumption 2. The function $Ψ$ can be expressed as a sum of two bounded terms

Ψ (x, t) = Ψ_{1} (x, t) + Ψ_{2} (x, t)

(20)

Suppose that $δ_{1}$ , $δ_{2} > 0$ exist but are not known; therefore, $| Ψ_{1} (x, t) |$ and $Ψ_{2} (x, t)$ can be bounded as follows

\begin{matrix} | Ψ_{1} (x, t) | \leq δ_{1} {| σ |}^{\frac{1}{2}} \\ {\overset{\cdot}{Ψ}}_{2} (x, t) \leq δ_{2} \end{matrix}

(21)

Assumption 3. The system (18) can be rewritten in the following form

\overset{\cdot}{σ} = Ψ + (1 + \frac{Δ Φ}{Φ_{0}}) Φ_{0} u

(22)

The function $Φ_{1} = (1 + \frac{Δ Φ}{Φ_{0}})$ is bounded, that is

1 - φ_{1} \leq (1 + \frac{Δ Φ}{Φ_{0}}) \leq 1 + φ_{1}

(23)

Remark 4. The main goal of the control law is to drive $σ$ and $\overset{\cdot}{σ}$ to zero in the presence of the bounded perturbations considered in equations (19) and (21).

Assumptions about the unknown and unmodeled parts in the system’s dynamics have already been reported in the literature (Shtessel et al., 2012). Bounded uncertainties can arise from a variety of sources, such as modeling errors, measurement noise, and external disturbances. If the system is subject to bounded perturbations with unknown boundaries, as stated in those assumptions, its inputs or parameters can vary within an unknown range of values. In this case, it is important to develop adaptive-based control strategies using online estimation techniques to handle the uncertainty and ensure stable and satisfactory performance. In addition, the perturbations can depend on both state and time. They are often the most challenging to handle in control systems because they can lead to highly complex and unpredictable system behavior. The effect of the perturbation on the system can also vary depending on the current state of the system as well as how the perturbation changes over time.

Remark 5. The ASTW controller was designed for fully actuated systems, as in Shtessel et al. (2012) for an electro-pneumatic actuator and in Rajappa et al. (2016) for a quadrotor UAV. This control scheme was also applied to a floating wind turbine in Zhang and Plestan (2021). It produces a continuous control signal, enabling the maintenance of control accuracy in the presence of perturbations and uncertainties, even if the system model knowledge is limited. However, to the best of our knowledge, this controller has never been generalized to the challenging case of class I of UMSs.

In this study, we considered the extension of this controller to the challenging case of class I UMSs described by the form (9), while considering unknown bounds of uncertainties and perturbations. In the next section, the design of the proposed controller is illustrated on a second-order UMS, and the stability analysis is addressed using Lyapunov’s theory.

Remark 6. When the bounds of uncertainties and perturbations are known, the control problem can be resolved with an STW controller (Castillo et al., 2018; Moreno and Osorio, 2008). However, in this case, the bounds of uncertainties and perturbations $δ_{1}$ , $δ_{2}$ , $φ_{1}$ are unknown and accordingly cannot be estimated, particularly for real-time systems.

Considering that the bounds on the uncertainties and perturbations are unknown, and the functions $Φ_{1}$ , $Ψ_{1} (x, t)$ , $Ψ_{2} (x, t)$ satisfy Assumptions 1–3, the ASTW control guarantees a non-overestimation of the control gains $k_{1}$ and $k_{2}$ and ensures in a finite time $t_{F} \leq \frac{2 V {(t_{0})}^{1 / 2}}{η_{0}}$ the establishment of a real second-sliding mode, with $| σ | \leq η_{1}$ , $η_{1} > μ$ , $| \overset{\cdot}{σ} | \leq η_{2}$ , and $η_{2} > 0$ $\forall t > t_{F}$ . Then, we consider that the gains are time-varying and described by $k_{1} = k_{1} (σ, \overset{\cdot}{σ}, t)$ and $k_{2} = k_{2} (σ, \overset{\cdot}{σ}, t)$ . These can be estimated online according to the adaptation law proposed in Shtessel et al. (2012). The first gain $k_{1}$ is adapted as follows

{\overset{\cdot}{k}}_{1} = {\begin{matrix} ω_{1} \sqrt{\frac{γ_{1}}{2}} sign (| σ | - μ) & if k_{1} > k_{m} \\ η & if k_{1} \leq k_{m} \end{matrix}

(24)

and the second gain $k_{2}$ varies proportionally to $k_{1}$ , according to the following expression

k_{2} = 2 ε k_{1}

(25)

where

$ω_{1}$ , $γ_{1}$ , $ε$ , and $η$ are the arbitrary positive constants and $k_{1} (0) > k_{m}$ .

$k_{m}$ is a small positive parameter introduced in the control law to maintain the positive gains.

$μ$ is a positive constant that defines the boundary layer for the real sliding mode.

As mentioned in Rajappa et al. (2016), a wrong choice of the parameter $μ$ could lead to either instability or overestimation of the control gains.

An illustrative application example

UMSs are typically studied on a case-by-case basis. The proposed ASTW control scheme for the general case of a class I UMS is presented in this section. The entire block diagram of the designed approach is shown in Figure 1.

The inertia wheel inverted pendulum

In this study, an example of a system from class I UMSs, namely the inertia wheel inverted pendulum (IWIP), was considered. This plant is an interesting typical benchmark, because, with a reduced-order model, it inherits the same features of much more complex systems (e.g. human balance support devices (Wojtara et al., 2012), gyrostabilizers, and stabilizers for self-balancing electric motorcycles (Ho and Pham, 2018)). It is characterized by two DOFs and one control input. This system has attracted considerable interest from researchers. It was considered a benchmark for various recent studies, as in Gritli et al. (2017). The system includes an unactuated joint between the frame and pendulum body and an actuated joint between the body and inertia wheel, as illustrated in Figure 2.

The mechanical design of the system differs from those presented in Belascuen and Aguilar (2018) and Tavakol Aghaei and Komurcu (2021). However, both operating principle and number of DOFs remain the same. The position of the actuator and the dimension of pendulum were selected to minimize the overall resistance moment (moment of the pendulum $+$ moment of the inertia wheel). This has been accomplished in Garabedian et al. (2007) by adjusting the positions of the centers of mass $G_{1}$ and $G_{2}$ .

With the actuator positioned close to the center of rotation, the lever arm, which is the perpendicular distance between the force (applied by the actuator) and the pivot point, is reduced. It makes the control design more challenging. A shorter lever arm results in reduced torque generation for a given force applied by the actuator. This limitation may compromise the control system to compensate for disturbances or maintain stability. Moreover, if the actuator were positioned farther from the center of rotation, stabilizing the pendulum would require a higher torque. Yet such adjustments are constrained by the maximum torque capacity of the actuator.

The use of a simplified low-order system with the same features and basic functionality to address the common problems in high-order complex systems can be very useful. Low-order systems require fewer computational resources for simulations and a low cost for real-time experiments. They are often easier to use and understand the major control problems and challenges of high-order systems while enabling experimental validation of the control design at an affordable cost in a laboratory (e.g. case of the IWIP), instead of an excessively high cost in real conditions (e.g. case of underwater vehicles).

The aim of the control problem was to design an ASTW algorithm to solve the stabilization problem of the system in the presence of unknown model uncertainties and external disturbances.

The dynamic model of the system can be computed following the Lagrange method. To this end, the Lagrangian of this system can be expressed by $L = \frac{1}{2} (I {\overset{\cdot}{θ}}_{1}^{2} + i_{2} {({\overset{\cdot}{θ}}_{1} + {\overset{\cdot}{θ}}_{2})}^{2}) - m_{0} g \cos (θ_{1})$ .

The application of the Lagrange equation leads to the following dynamic model

[\begin{matrix} I + 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}]

(26)

where $θ = [θ_{1}, θ_{2}]^{T}$ is the vector of the generalized coordinates, and $τ$ is the torque generated by the actuator of the inertia wheel. Through identification with equation (2), the terms $h_{1}$ and $h_{2}$ are expressed by $h_{1} (q, \overset{\cdot}{q}) = m_{0} g \sin (θ_{1})$ and $h_{2} (q, \overset{\cdot}{q}) = 0$ , respectively. The dynamic parameters of the system can be found in Hfaiedh et al. (2021) and Hfaiedh and Abdelkrim (2022) are summarized in Table 1.

Table 1.

Summary of the dynamical parameters of the IWIP.

Parameter	Description	Value
$i_{1}$	The inertia of the unactuated pendulum	$4.176 e^{- 4} (kg . m^{2})$
$i_{2}$	The inertia of the actuated wheel	$0.031468 (kg . m^{2})$
$L_{1}$	Length from the pendulum base to the pendulum center of mass	$0.06 (m)$
$l$	Length from the pendulum base to the wheel center of mass	$0.044 (m)$
$M$	Mass of the pendulum	$3.30 (kg)$
$m$	Mass of the wheel	$0.33 (kg)$
$g$	Constant of gravitational acceleration	$9.81 (m . s^{- 2})$

The parameters $I$ and $m_{0}$ of the model are expressed by $I = m l^{2} + {ML}_{1}^{2} + i_{1}$ and $m_{0} = ml + M L_{1}$ , respectively.

Control strategy

Using the method described in equation (9), the global change of coordinates is defined by: $z_{1} = \frac{\partial L}{\partial {\overset{\cdot}{θ}}_{1}} = (I + i_{2}) {\overset{\cdot}{θ}}_{1} + i_{2} {\overset{\cdot}{θ}}_{2}$ , $z_{2} = θ_{1}$ , $z_{3} = {\overset{\cdot}{θ}}_{2}$ , where $z = [z_{1}, z_{2}, z_{3}]^{T}$ denotes the state vector of the strict-feedback model. Using the Lagrangian formulation, the system is transformed into a cascade form as follows

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

(27)

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

(28)

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

(29)

The resulting system is presented as a cascade connection between the linear and nonlinear subsystems Olfati-Saber (2001), where $z_{2}$ is considered as a virtual control input of the $z_{1}$ nonlinear subsystem. Different approaches in the literature (Aguilar-Ibanez et al. (2008); Haddad et al. (2018); Tavakol Aghaei and Komurcu (2021)) considered the angular position of the inertial wheel $θ_{2}$ as a state variable. It is worth noting that because it does not play a significant role in the dynamic model (26) and does not appear in the Lagrangian equation, $θ_{2}$ is not considered as a state in the new model (27–29). Using a sigmoidal function $- (z_{1})$ for $z_{2}$ in the $z_{1}$ -subsystem can globally asymptotically stabilize equation (27), by choosing Lyapunov’s function $V (z_{1}) = \frac{1}{2} z_{1}^{2}$ . Let us now consider the desired trajectory and its time derivative as follows

θ_{1 d} = z_{2 d} = - \arctan (z_{1})

(30)

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

(31)

The time derivative of Lyapunov’s function leads to the following

\begin{matrix} \overset{\cdot}{V} (z_{1}) = z_{1} {\overset{\cdot}{z}}_{1} \\ \overset{\cdot}{V} (z_{1}) = z_{1} m_{0} g \sin (z_{2}) \\ \overset{\cdot}{V} (z_{1}) = z_{1} m_{0} g \sin (- \arctan (z_{1})) \\ \overset{\cdot}{V} (z_{1}) = - z_{1} m_{0} g \frac{z_{1}}{\sqrt{1 + z_{1}^{2}}} \end{matrix}

(32)

It can be seen that equation (32) is negative definite by the choice of the desired trajectory (30) that globally asymptotically stabilizes $z 1 = 0$ . The stability analysis of the entire closed-loop system will be demonstrated in the sequel.

Let us now reconsider the control laws (10) and (11) for the systems (27–29) with adaptive feedback gains as follows

u = - k_{1} (t) {| σ |}^{\frac{1}{2}} sign (σ) + w

(33)

\overset{\cdot}{w} = - k_{2} (t) sign (σ)

(34)

where the control feedback gains $k_{1} (t)$ and $k_{2} (t)$ are adapted from equations (24) and (25), respectively. Then, consider the proposed sliding surface $σ$ described as follows

σ = (θ_{1 d} - θ_{1}) + α_{1} ({\overset{\cdot}{θ}}_{1 d} - {\overset{\cdot}{θ}}_{1})

(35)

where $θ_{1 d}$ and ${\overset{\cdot}{θ}}_{1 d}$ are the desired pendulum angular position (30) and its first time derivative (31), respectively, and $α_{1}$ is a positive constant gain.

By applying the time derivative of the sliding surface (35), we obtain the expression of the sliding variable dynamics

\overset{\cdot}{σ} = ({\overset{\cdot}{θ}}_{1 d} - {\overset{\cdot}{θ}}_{1}) + α_{1} ({\overset{\cdot\cdot}{θ}}_{1 d} - {\overset{\cdot\cdot}{θ}}_{1})

(36)

If we derive the expression of ${\overset{\cdot\cdot}{θ}}_{1}$ from equations (28) and (29), we obtain

{\overset{\cdot\cdot}{θ}}_{1} = - \frac{i_{2}}{I + i_{2}} u + \frac{m_{0} g \sin z_{2}}{I + i_{2}}

(37)

Computing the time derivative of ${\overset{\cdot}{θ}}_{1 d}$ , leads to

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

(38)

Replacing equations (37) and (38) in equation (36) leads to the following system

\overset{\cdot}{σ} = [- \frac{m_{0} g \sin (z_{2})}{(1 + (z_{1}^{2}))} - {\overset{\cdot}{z}}_{2}] + α_{1} [\frac{2 m_{0}^{2} g^{2} z_{1} \sin {(z_{2})}^{2}}{{(1 + z_{1}^{2})}^{2}} - \frac{m_{0} g {\overset{\cdot}{z}}_{2} \cos (z_{2})}{1 + z_{1}^{2}} + \frac{i_{2}}{I + i_{2}} u - \frac{m_{0} g \sin (z_{2})}{I + i_{2}}]

(39)

By rearranging the terms, we obtain the following form

\overset{\cdot}{σ} = Ψ (z) + Φ u

(40)

Based on Assumptions 1–3, the control system (40) with equation (33) can be rewritten as follows

\begin{matrix} \overset{\cdot}{σ} = - k_{1} Φ_{1} {| σ |}^{\frac{1}{2}} sign (σ) + ω_{0} + Ψ_{1} \\ {\overset{\cdot}{ω}}_{0} = - k_{2} Φ_{1} sign (σ) + {\overset{\cdot}{Ψ}}_{2} + {\overset{\cdot}{Φ}}_{1} w \\ ω_{0} (0) = 0 \end{matrix}

(41)

where $Φ_{1} = (1 + \frac{Δ Φ}{Φ_{0}})$ and $ω_{0} = Ψ_{2} + Φ_{1} w$ . A new state vector $Z = [Z_{1}, Z_{2}]^{T}$ can be defined as

Z = [Z_{1}, Z_{2}]^{T} = {[\begin{matrix} {| σ |}^{\frac{1}{2}} sign (σ), ω_{0} \end{matrix}]}^{T}

(42)

and accordingly, the system (41) can be rewritten as follows

(\begin{matrix} {\overset{\cdot}{Z}}_{1} \\ {\overset{\cdot}{Z}}_{2} \end{matrix}) = \frac{1}{2 | Z_{1} |} (\begin{matrix} - α Φ_{1} & 1 \\ - β Φ_{1} & 0 \end{matrix}) (\begin{matrix} Z_{1} \\ Z_{2} \end{matrix}) + \frac{1}{2 | Z_{1} |} (\begin{matrix} 1 & 0 \\ 0 & 2 | Z_{1} | \end{matrix}) (\begin{matrix} Ψ_{1} \\ \overset{\cdot}{ℏ} \end{matrix})

(43)

where $\overset{\cdot}{ℏ} = {\overset{\cdot}{Ψ}}_{2} + {\overset{\cdot}{Φ}}_{1} w$ and $α = k_{1}$ , $β = 2 k_{2}$ . Then, assume that $Ψ_{1} = ρ_{1} Z_{1}$ and $\overset{\cdot}{ℏ} = \frac{ρ_{2}}{2} sign (σ) = \frac{ρ_{2}}{2} \frac{Z_{1}}{| Z_{1} |}$ , where $0 < ρ_{1} < δ_{1}$ and $0 < ρ_{2} < δ_{5}$ , leads to

(\begin{matrix} {\overset{\cdot}{Z}}_{1} \\ {\overset{\cdot}{Z}}_{2} \end{matrix}) = \frac{1}{2 | Z_{1} |} (\begin{matrix} - (α Φ_{1} - ρ_{1}) & 1 \\ - (β Φ_{1} - ρ_{2}) & 0 \end{matrix}) (\begin{matrix} Z_{1} \\ Z_{2} \end{matrix})

(44)

(\begin{matrix} {\overset{\cdot}{Z}}_{1} \\ {\overset{\cdot}{Z}}_{2} \end{matrix}) = A (Z_{1}) (\begin{matrix} Z_{1} \\ Z_{2} \end{matrix})

(45)

Now, let us consider the following Lyapunov function candidate Shtessel et al. (2012) for the system (45)

V (Z /, α, β) = Z^{T} PZ + \frac{1}{2 φ_{1}} (α - α *)^{2} + \frac{1}{2 φ_{2}} {(β - β_{*})}^{2}

(46)

where $α_{*}$ and $β_{*}$ are positive constants, and $P$ is given by $P = [\begin{matrix} λ + 4 ε^{2} & - 2 ε \\ - 2 ε & 1 \end{matrix}]$ . The positive definiteness of $P$ is guaranteed for $λ > 0$ . The first time derivative of $V$ leads to the following (Shtessel et al., 2012)

\overset{\cdot}{V} (Z, α, β) = Z^{T} [A^{T} P + PA] Z + \frac{1}{φ_{1}} (α - α *) \overset{\cdot}{α} + \frac{1}{φ_{2}} (β - β *) \overset{\cdot}{β}

(47)

where the first term is bounded as follows

Z^{T} [A^{T} P + PA] Z \leq - \frac{1}{2 | Z_{1} |} Z^{T} QZ

(48)

where $Q = [\begin{matrix} Q_{1} & Q_{3} \\ Q_{3} & Q_{4} \end{matrix}] = [\begin{matrix} 2 λ α Φ_{1} + 4 ε Φ_{1} (2 ε α - β) - 2 (λ + 4 ε^{2}) ρ_{1} + 4 ε ρ_{2} & Q_{3} \\ (β Φ_{1} - 2 ε α Φ_{1} - λ - 4 ε^{2}) + 2 ε ρ_{1} - ρ_{2} & 4 ε \end{matrix}]$ .

If we assume that $β = 2 ε α$ and $α > \frac{δ_{1} (λ + 4 ε^{2}) - ε (2 δ_{5} + 1)}{λ (1 - Φ_{1}} + \frac{{(2 ε δ_{1} - δ_{5} - λ - 4 ε^{2})}^{2}}{12 ε (1 - Φ_{1})}$ , we guarantee that the matrix $Q$ is positive definite.

Following the reasoning in Shtessel et al. (2012), it can be shown that the time derivative of $V$ along the solutions of the system leads to the following

\begin{matrix} \overset{\cdot}{V} (Z, α, β) = Z^{T} [A^{T} P + PA] Z + \frac{1}{φ_{1}} (α - α_{*}) \overset{\cdot}{α} + \frac{1}{φ_{2}} (β - β_{*}) \overset{\cdot}{β} \\ = {\overset{\cdot}{Z}}^{T} PZ + Z^{T} P \overset{\cdot}{Z} + \frac{1}{φ_{1}} (α - α_{*}) \overset{\cdot}{α} + \frac{1}{φ_{2}} (β - β_{*}) \overset{\cdot}{β} \end{matrix}

(49)

\overset{\cdot}{V} (Z, α, β) \leq - \frac{1}{| Z_{1} |} Z^{T} QZ + \frac{1}{φ_{1}} (η_{α}) \overset{\cdot}{α} + \frac{1}{φ_{2}} (η_{β}) \overset{\cdot}{β}

(50)

\overset{\cdot}{V} (Z, α, β) \leq - r {(Z^{T} PZ)}^{\frac{1}{2}} + \frac{1}{φ_{1}} (η_{α}) \overset{\cdot}{α} + \frac{1}{φ_{2}} (η_{β}) \overset{\cdot}{β}

(51)

\overset{\cdot}{V} (Z, α, β) \leq - η_{0} {[V (Z, α, β)]}^{\frac{1}{2}} + ℵ

(52)

with ℵ = - | η_{α} | (\frac{1}{φ_{1}} \overset{\cdot}{α} - Ω_{1}) - | η_{β} | (\frac{1}{φ_{2}} \overset{\cdot}{β} - Ω_{2})

(53)

where $η_{α} = α - α_{*} < 0$ and $η_{β} = β - β_{*} < 0$ . $r = \frac{ε λ_{\min}^{\frac{1}{2}} (P)}{λ_{\max} (P)}$ , $Ω_{1} = \frac{ω_{1}}{\sqrt{2 φ_{1}}}$ and $Ω_{2} = \frac{ω_{2}}{\sqrt{2 φ_{2}}}$ , $η_{0} = Min (r, ω_{1}, ω_{2})$ .

The first time derivative of $β = 2 ε α$ , leads to $\overset{\cdot}{β} = 2 ε \overset{\cdot}{α}$ . Altogether with $ε = \frac{ω_{2}}{2 ω_{1}} \sqrt{\frac{φ_{2}}{φ_{1}}}$ replaced in $ℵ$ , leads to $ℵ = 0$ ; accordingly, $\overset{\cdot}{V}$ can be reduced and upper bounded as follows

\overset{\cdot}{V} (Z, α, β) \leq - η_{0} \sqrt{[V (Z, α, β)]}

(54)

Numerical simulation results

In this study, the control objective was to stabilize the IWIP around its unstable equilibrium point. Both the STW and ASTW approaches were implemented using MATLAB/SIMULINK software (MathWorks). The following three main simulation scenarios are considered:

Scenario 1: Nominal case.

Scenario 2: Robustness towards parametric uncertainties.

Scenario 3: State and time-dependent disturbance rejection.

In addition, the following initial conditions for the ASTW approach were considered: $x_{ASTW} (0) = [θ_{1}, {\overset{\cdot}{θ}}_{1}, {\overset{\cdot}{θ}}_{2}, w (0) K_{1} (0)] = [π / 2, 0, 0, 0, 50]$ and $x_{STW} (0) = [θ_{1}, {\overset{\cdot}{θ}}_{1}, {\overset{\cdot}{θ}}_{2}, w (0)] = [π / 2, 0, 0, 0]$ for the STW controller.

Scenario 1: Nominal case

In the first scenario, we considered two case studies. The first case study showed the performance of the proposed ASTW controller for different values of the key parameters $γ_{1}$ and $ω_{1}$ . The results obtained for this case are shown in Figure 3, including the evolution versus time of the states, control input, sliding variable, and estimated gains.

From the results, the best response was obtained for the case of ( $ω_{1} = 105$ , $γ_{1} = 57$ ) compared to the other selected values of $ω_{1}$ and $γ_{1}$ . These curves show the stabilization of the system states around the unstable equilibrium point $[θ_{1}, {\overset{\cdot}{θ}}_{1}, {\overset{\cdot}{θ}}_{2}] = [0, 0, 0]$ within a short time interval. The generated control input torque remained within the admissible limits of the actuator and converged to a zero steady-state value once the equilibrium point was reached.

Remark 7. The control input applied to the system is based on the model parameters, expressed as $τ = Θ (q) u + ϒ (q, \overset{\cdot}{q})$ (cf. Lemma 1), where $u$ is the expression of the ASTW controller, $Θ (q) = m_{22} - m_{21} m_{11}^{- 1} m_{12} = i_{2} - i_{2} {(I + i_{2})}^{- 1} i_{2}$ and $ϒ (q, \overset{\cdot}{q}) = h_{2} (q, \overset{\cdot}{q}) - m_{21} m_{11}^{- 1} h_{1} (q, \overset{\cdot}{q}) = - i_{2} {(I + i_{2})}^{- 1} m_{0} \sin (θ_{1})$ . In this case, the obtained value of $Θ$ was a small constant equal to $0.0092$ . For this reason, gains $ω_{1}$ and $γ_{1}$ have been selected to obtain robust results.

In the second case study, we have fixed the gains $ω_{1} = 105$ and $γ_{1} = 57$ and simulated the behavior of the system for different values of $μ$ and $ε$ .

The simulation results are presented in Figure 4 which highlights the importance of selecting appropriate parameter values to avoid system instability, illustrated by cases where incorrect parameter choices may lead to instability. The best results were observed for the cases of ( $μ = 10^{- 3}, ε = 10^{- 2}$ ) and ( $μ = 10^{- 3}, ε = 10^{- 1}$ ). It is worth noting that the system becomes unstable for larger parameter values. Indeed, the wrong choice of these parameters can lead to instability of the system, as shown for the case of $μ = 5, ε = 1$ and $μ = 0.5, ε = 10^{- 2}$ .

Scenario 2: Robustness towards parametric uncertainties

The second scenario aimed to study the robustness of the proposed control scheme. Indeed, it allows us to check whether the applied control can adapt to uncertainties in the system parameters, which may be due to modeling errors, sensor inaccuracies, frictional forces, or other external factors. We introduced an uncertainty $Δ_{I}$ on the parameter $I$ and an uncertainty $Δ_{m_{0}}$ on the parameter $m_{0}$ . The parameters are selected because they represent a combination of several dynamic parameters. The simulation results are presented in Figure 5.

The proposed approach is able to compensate for these introduced additive parametric uncertainties in the dynamic model even for large percentages of $Δ_{I} = 200 %$ and $Δ_{m_{0}} = 200 %$ , where the estimated parameters remain bounded. Furthermore, if the considered uncertainty increases, the reaching time also increases. This implies that the system’s response time is influenced by the degree of uncertainty, with higher uncertainty levels requiring more time for controller to ensure the stabilization.

Scenario 3: External disturbances rejection

The main motivation behind the third scenario is to demonstrate the effectiveness of the proposed control scheme when the controlled system is subjected to state- and time-dependent disturbances. The applied disturbing signal for this case is expressed by $D = \frac{1}{1 + {(1500 z_{2})}^{2}} + \frac{1}{1 + {(20 \cos (t))}^{2}}$ and was applied during the interval $[0, 60]$ seconds. The standard STW and the proposed ASTW controllers were compared. The selected design parameters used for the STW controller are set to $k_{1} = 195$ and $k_{2} = 2.43$ . The simulation results for this scenario are presented in Figure 6. For clarity, the plots are zoomed, where we can clearly see the effect of the applied external disturbance on the evolution of the states in Figure 6(a)–(c) and the sliding variable $σ$ in Figure 6(e) for the case of the STW controller.

However, based on the obtained results, we can notice that the proposed ASTW controller manages to compensate the applied external disturbance and brings back the states around the desired equilibrium point. The introduced external disturbance was compensated by the control action. A noticeable difference can be observed in the zoomed plots of the behavior of the STW and proposed ASTW controllers.

To sum up, the proposed ASTW controller provides better results than the standard STW controller and ensures a better compensation of time- and state-dependent disturbances.

Implementation issues and real-time experimental results

In numerical simulations, the environment is controlled, and uncertainties/ external disturbances can be modeled precisely. In real-time experiments, there are unpredictable and unmodeled disturbances such as friction, sensor noise, and environmental variations that may affect the system and lead to unexpected behavior. Subsequently, real-time validation is performed in order to observe the real behavior and discuss reliable end precise results regarding the performance of the proposed control approach. The experimental setup and some implementation issues are discussed in this section. Two experimental scenarios were conducted to validate the proposed control scheme. In the first scenario, no external disturbance is considered; however, the objective of the second scenario is to test the robustness of the proposed control scheme towards external disturbances. These scenarios tested the proposed controller under different dynamic operating conditions and compared its performance with that of other existing controllers from the literature. The results of each test are presented and discussed in this section.

Description of the experimental setup

Real-time experiments were performed using an experimental testbed of the IWIP, as shown in Figure 7(a), and designed at LIRMM Laboratory (Andary et al., 2009; Haddad et al., 2018; Hfaiedh et al., 2018).

The system is equipped with the following:

Underactuated mechanical IWIP: A pendulum connected to a fixed support by an unactuated pivot link and controlled by an inertial wheel.

Sensors: The system is equipped with two sensors to measure the angular positions of the pendulum and the inertial wheel in real-time, respectively. The angle of the inertial wheel was measured by an encoder mounted on its motor’s shaft, and the pendulum body was equipped with an inclinometer to measure the pendulum’s angle with respect to the vertical. Their corresponding velocities were calculated in real-time by numerical derivatives.

Power supply: The entire system was supplied with a voltage equal to $12$ V.

Computer: The PC control is equipped with a real-time operating system known as RTX (Real-time Extension for Windows) by Ardence. In addition, Visual C Studio software is utilized for programming the control law, while MATLAB/SIMULINK software serves for visualizing the results. RTX provides the capabilities to Windows environment to handle real-time tasks with precise timing and reliability. The configured sampling time is set at $6$ ms, ensuring precise and timely data acquisition. The RTX environment is implemented as a real-time subsystem, operating as a kernel device driver for Windows. It provides libraries that are compatible with Visual Studio, facilitating the development process. Once a project is built in Visual studio C++, it generates an executable (real-time subsystem) .RTSS file. Clicking on this file initiates the experimentation process. Upon completion of the execution time, the data from variables and sensors are recorded in .MAT files. They can be easily visualized using MATLAB, allowing for thorough analysis and interpretation of the experiment results.

Interface card: This is a hardware component that ensures the communication between a computer and the physical system. It also enables the data exchange between the computer and other components of the testbed.

Conducted experiments

Two experiments were conducted to fairly compare the performance of the proposed ASTW scheme with existing controllers from the literature, including the (1) standard STW, (2) first-order SMC (Hfaiedh and Abdelkrim, 2022), (3) ASMC (Hfaiedh and Abdelkrim, 2022), and (4) RISE (Hfaiedh et al., 2021) controllers. In the first scenario, namely nominal case, no external disturbances were considered. However, in the second case, we considered punctual external disturbances applied to the controlled system.

These disturbances were generated by a pendular system suspended from a fixed support, as illustrated in Figure 7(b). The same initial releasing position of the disturbing mass of the pendular system (e.g. $40^{°}$ ) was set for fair comparison. Accordingly, once the disturbing mass (with the same weight for all controllers) was launched at a fixed time instant (e.g. 20 seconds after initiation), the same disturbing force $F_{ext}$ at the impact position will be approximately achieved.

Remark 8. The second scenario is different from the experiments conducted in Hfaiedh and Abdelkrim (2022), where the introduced disturbance is a persistent perturbation. It consisted of a mass attached to the pendulum’s body at the onset of the experiment. However, in this paper, we are addressing the case of punctual external perturbation.

Remark 9. In the physical setup, the movement of the pendulum axis is restricted by mechanical stops, forcing it to stay within the range of $- 10^{°}$ to $10^{°}$ . In simulation results provided in section “Numerical simulation results,” we opted to expand this operational range to $90^{°}$ , allowing the pendulum to move beyond the $\pm 10^{°}$ limits, in order to evaluate the performance of the controllers under more challenging conditions and a broader range of motion scenarios associated with the IWIP.

Performance evaluation criteria

To quantify the relevance of the control algorithms, we proposed computing the following criteria for all the implemented controllers:

The Integral Square Error (ISE)

ISE = \int e_{θ_{1}}^{2} dt

(55)

The Integral Absolute Error (IAE)

IAE = \int | e_{θ_{1}} | dt

(56)

where $e_{θ_{1}}$ denotes the tracking error of the pendulum joint $θ_{1}$ with respect to its desired value (i.e. the equilibrium point).

The input torque–based criterion

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

(57)

where $τ_{θ_{2}}$ represents the torque of the inertia wheel.

ISE is a performance metric used in control theory to evaluate the performance of a control system. It measures the square of the difference between the desired value and the actual output of the system and is integrated over a period of time. IAE is also a performance metric computed to measure the cumulative absolute error $| e_{θ_{1}} |$ between the desired and actual system output. In our case, we have only computed the unactuated coordinate as we are dealing with a stabilization problem around an unstable equilibrium point. It is worth noting that the angular position of the inertial wheel $θ_{2}$ is irrelevant and is not considered for the performance evaluation criteria because it does not appear in the Lagrangian equation and dynamic model (26) and it does not influence the system’s dynamic behavior and the stabilization of the pendulum. It is not considered important for evaluating the performance of the system.

The $E_{τ}$ metric sums up the absolute values of the torque inputs. It serves as an indicator of the efficiency of the control strategy as higher value indicates higher torque and increased energy consumption.

Parameters’ tuning

To select the adjustable gains in real tuning process, we started with small values to avoid any high and abrupt changes in the control input that may lead to a dangerous behavior or damage in the system’s actuators. Subsequently, to obtain better performances, the parameters were progressively increased and the behavior of the closed-loop system was observed until the expected stabilization is reached. To sum up, the following eight-step algorithm is proposed.

Parameters’ tuning algorithm

Step 1: Compute the expressions of $Θ$ and $ϒ$ , taking into account the invertible change of control.

Step 2: Set the values of $k_{1}$ and $k_{m}$ such that $k_{1} (0) > 1$ and $k_{m} = 1$ .

Step 3: Initialization of $ω_{1}$ , $γ_{1}$ , $ε$ , and $η$ with small values.

Step 4: Increase the value of $Θ (q)$ until the pendulum is able to move from the initial condition.

Step 5: For fixed values of $γ_{1}$ , $ε$ , and $η$ , increase $ω_{1}$ .

Step 6: For fixed values of $ω_{1}$ , $ε$ , and $η$ , increase $γ_{1}$ .

Step 7: Set the values of $ω_{1}$ and $γ_{1}$ and increase the value of $ε$ and $η$ .

Step 8: Increase the value of $k_{m}$ until obtaining a satisfactory behavior.

Experiment 1: Nominal case

In the first experiment, no external disturbance was considered. The following design parameters were used for the STW control scheme: $α_{1} = 0.057$ , $K_{2} = 0.00103$ , $K_{1} = 1.6$ , $Θ = 0.795$ , and $ϒ = 1.48 \sin (θ_{1})$ . The following design parameters were used for the ASTW control scheme: $α_{1} = 0.4$ , $γ_{1} = 5$ , $ω_{1} = 20$ , $η = 0.0010$ , $μ = 0.3382$ , $K_{m} = 17.5$ , and $ε = 0.1506$ . For the RISE, SMC, and ASMC controllers, the design parameters were set as in Hfaiedh et al. (2021) and Hfaiedh and Abdelkrim (2022), respectively. The obtained experimental results for all controllers are depicted in Figure 8. The convergence of state variables was established for all controllers. The state variables (Figure 8(a), (c), (e)) in the case of the ASTW and ASMC controller as well the control input (Figure 8(b)) present less oscillations and ensure a faster convergence towards the equilibrium point than the standard STW, SMC, and RISE controllers. However, ASMC subsequently resulted in oscillations with low amplitude around the equilibrium point, especially for the case of $θ_{1}$ and ${\overset{\cdot}{θ}}_{2}$ . The effectiveness of the proposed control scheme can also be observed in the phase profile shown in Figure 8(d). More chattering in the voltage signal was observed for SMC and ASMC than for the proposed ASTW approaches (see the zoom plot in Figure 8(f)). In comparing the results of STW and ASTW, we observe the advantages of the adaptation law used in improving the convergence and reducing the oscillations of the classic STW control. The main drawback of the ASTW controller is the high torque required to ensure stabilization at the beginning of the experiment.

Experiment 2: External disturbances rejection

In this scenario, external disturbances were applied by the above-mentioned disturbances generator. An external disturbance was introduced at approximately $t_{1} = 20$ seconds. Throughout the experiments, we ensured that all controllers were subjected to identical disturbances. The experimental results are presented in Figure 9. For a better view, the plots are magnified within the interval $[19, 23]$ seconds.

It is notable that there are differences in the system’s reactions for different controllers. It can be attributed to variations in the responses of controllers. In addition, the states of the IWIP at the time of disturbance influence how each controller reacts and compensates for this perturbation.

Both the proposed ASTW and the STW controllers reacted better to the applied disturbances than the RISE, ASMC, and SMC approaches and maintain the system around the desired equilibrium point as we can see in the evolution of the states and control signal.

The exerted disturbance induces a high deviation from the desired equilibrium point of the pendulum angle position and velocity in the case of the first-order SMC and ASMC, which results in poor behavior. However, the disturbance is better compensated by the proposed ASTW controller. More oscillations were observed in the standard STW control scheme. An improvement in the robustness of the proposed ASTW controller was observed when the controller compensated for the external punctual disturbance. These results were also confirmed by the proposed performance indices defined in equations (55) and (56) and summarized in Table 2.

Table 2.

Quantification of the performance through different evaluation criteria.

Criteria	SM	ASM	RISE	STW	ASTW
ISE	$0.0043$	0.0028	$2.09 e^{- 4}$	$1.73 e^{- 4}$	$4.47 e^{- 6}$
Improvements%		$34.88 %$	$95.1395 %$	$95.97 %$	$99.8 %$
IAE	$0.2127$	0.1622	$0.0460$	$0.0422$	$0.0066$
Improvements %		$23.74 %$	$78.37 %$	$80.15 %$	$96.8 %$
$E_{τ}$	$246.7$	220.6	$125.6$	$263.0$	$175.6$
Improvements %		$10.5 %$	$49.09 %$	$6.61 %$	$28.82 %$

The improvement in the IAE and ISE criteria was recorded in comparison to the standard sliding mode controller. The performance indices of SMC and ASMC are nearly identical. Similar enhancements were obtained for the RISE and STW approaches. Superior results were noted for the proposed ASTW controller. It is evident that both indices were reduced by up to $90 %$ with respect to the standard SMC scheme. By squaring the error values $e_{θ_{1}}^{2}$ , the ISE emphasizes larger errors more than smaller ones. This implies that the smaller value of ISE corresponds to the control approach minimizing large errors. However, this can lead to a rapid response but may result in higher control effort. In contrast, the IAE treats both small and large errors equally. As a result, the control system tends to respond more smoothly, as it works to minimize all errors uniformly rather than focusing primarily on larger errors. In this case, small value obtained for ASTW results in a smoother response compared to other implemented controllers.

By comparing the values of the input torque–based criterion across all controllers, we observe that RISE exhibits the lowest energy consumption, followed by ASTW, ASMC, SMC, and finally STW. The obtained metrics validate the benefits of adaptive control laws, showing an improvement of $28 %$ compared to SMC and $33.27 %$ compared to STW approach. However, further improvement is still necessary to optimize the proposed approach to achieve minimal energy consumption compared to the RISE control approach.

Conclusion and future work

In this paper, an implementation of a revisited ASTW algorithm is proposed for the case of the IWIP. The methodology involves defining the sliding surface through two necessary desired trajectories derived from transforming the system into a strict-feedback form. The stability of the resulting closed-loop system is rigorously analyzed using Lyapunov’s theory. Numerical simulations under various operating conditions, including nominal scenarios and robustness tests against uncertainties and state-/time-dependent disturbances, convincingly demonstrate the superior performance of the proposed adaptive control approach compared to the conventional STW algorithm. Specifically, the ASTW demonstrates superior performance in terms of disturbance rejection. Furthermore, real-time experimental results and performance-evaluation criteria validate these findings, highlighting its effectiveness in practical applications compared to several existing controllers. The main drawback identified is the energy consumption, which requires improvement, especially for systems with input constraints. Future works may include the extension of the proposed control scheme with a prediction-based optimal gain selection of the STW controller, as well as its generalization to 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 iDs

Afef Hfaiedh

Ahmed Chemori

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