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Abstract

Stabilization of underactuated mechanical systems is one of the fundamental benchmark problem in the field of control theory. In this article, a hierarchical sliding mode control based on state-dependent switching gain is proposed for stabilization of underactuated mechanical systems. This controller is based on the so called first-level and second-level sliding surfaces. The asymptotic stability of these surfaces is proved by the Lyapunov stability theory. The proposed control technique is applied to two nonlinear underactuated mechanical systems and its feasibility is verified by numerical simulation. The proposed controller efficiently tackles the bounded external disturbance and shows robust performance. The convergence rate of the proposed controller is much faster as compared with the conventional decoupled sliding mode control and the integral sliding mode control.

Keywords

Sliding mode control underactuated systems state-dependent switching gain

Introduction

The design of nonlinear controllers for mechanical systems has become an active field of research from the last few decades.¹ From a theoretical point of view, this attention can be attributed to their interesting dynamical behavior, which is well-suited for the validation and practical application of ideas emerging in control theory. However, recent technological advances have produced many real-world engineering applications that require the automatic control of mechanical systems.²

An underactuated mechanical system is a system that has a fewer number of control input actuators than the degrees of freedom of the system. These systems are widely used in spacecrafts, underwater vehicles, mobile robots, surface vessels, and many other systems. Due to numerous practical applications of these systems, many researchers have worked to find reliable mechanisms for the control of underactuated mechanical systems. Stabilization of these systems is a fundamental benchmark problem. These systems cannot be stabilized by smooth feedback because their dynamics are governed by differential equations in the presence of some non-integrable differential conditions.^1,3–5 The development of new control techniques and numerical tools have played a significant role in recognition and explaining the concept of stabilization. However, it is still an open and challenging task for researchers.

Many techniques have been proposed to solve the stabilization problem for underactuated mechanical systems.^6–13 Most of the suggested control approaches undergo from either the oscillatory behavior, slow convergence rate, or lack of robustness against the external disturbances. The problem of robustness and slow convergence can be solved using sliding mode control (SMC). Due to inherent robustness properties of SMC, it has been widely used for stabilization of nonlinear systems. It has been used to design a controller for robotics,^14–16 under-actuated cranes,^17–20 unmanned aerial vehicle (UAV) quadrotors,²¹ and under-actuated vessels.²² SMC technique, which is a powerful approach to handle internal and external disturbances, was introduced in the literature.^23–25 The results developed by these techniques were very interesting. The control designed by these techniques had the ability to handle a class of uncertainties. However, these techniques require the use of virtual disturbances to satisfy some conditions. These controllers also require a lot of calculations that increase the computational burden.

Recently, many researchers have been showing their interest in combination of fuzzy logic control (FLC) and SMC which is referred to fuzzy SMC.^26–28 Fuzzy SMC has advantages of both FLC and SMC. However, there are some drawbacks of this technique. For higher order systems, the number of fuzzy sets and fuzzy rules becomes incredibly large, which compromises the applicability of this technique. An often remarked drawback of methods based on fuzzy logic is the lack of appropriate tools for analyzing the controller performance, such as stability, optimality, and so on. The SMC technique consists of two distinct phases. One is reaching phase and the other is sliding phase. In the reaching phase, the system states are forced toward the sliding surface while in the sliding phase, the system trajectories drive along the sliding surface.

A known limitation of SMC technique is chattering. To overcome this problem, Slotine proposed adopting thin boundary layer neighboring switching surfaces, by replacing the sign function with a saturation function.²⁹ The chattering phenomenon can be removed completely by utilizing observer. The main idea of using observers to prevent chattering is to generate an ideal sliding mode in the auxiliary loop including the observer.³⁰ Another way to reduce chattering without designing any asymptotic observer is to use state-dependent switching gain.³¹ With the choice of state-dependent switching gain and saturation function, the chattering effect becomes negligible. The main goal of this article is to design nonlinear SMC for a class of mechanical systems based on state-dependent switching gain which is robust for bounded external disturbances, chattering free, and have low complexity and computational burden.

Using switching gain which are functions of the system states, the SMC guarantees the robustness and faster convergence. We design the proposed controller for stabilization of inverted pendulum (IP) and ball-beam systems, and simulate it using MATLAB. The simulation results show effectiveness of our proposed controller.

Problem statement

The general form of a class of underactuated mechanical systems is given by

\begin{matrix} {\overset{\cdot}{x}}_{1} & = & x_{2} \\ {\overset{\cdot}{x}}_{2} & = & g_{1} (x) + b_{1} (x) τ + Δ \\ {\overset{\cdot}{x}}_{3} & = & x_{4} \\ {\overset{\cdot}{x}}_{4} & = & g_{2} (x) + b_{2} (x) τ + Δ \\ ⋮ & ⋮ & ⋮ \\ {\overset{\cdot}{x}}_{2 m - 1} & = & x_{2 m} \\ {\overset{\cdot}{x}}_{2 m} & = & g_{m} (x) + b_{m} (x) τ + Δ \end{matrix}}

(1)

where $x_{1}, x_{2}, x_{3}, \dots, x_{2 m}$ are state variables, $g_{m} (x)$ and $b_{m} (x)$ are defined as nonlinear functions of state variables, $τ$ is control input, and $Δ$ is bounded external disturbance.³² Different nonlinear underactuated mechanical systems can be presented in the form of system (1) with different nonlinear functions $g_{m} (x)$ and $b_{m} (x)$ . If $m = 2$ , then system (1) represents overhead crane system, IP system, and so on. Similarly, for $m = 3$ , system (1) represents the double rotary IP system, the double IP system, and so on. We can also write system (1) as follows

\begin{matrix} {\overset{\cdot}{x}}_{2 k - 1} & = & x_{2 k} \\ {\overset{\cdot}{x}}_{2 k} & = & g_{k} (x) + b_{k} (x) τ + Δ \end{matrix}}

(2)

where $k = 1, 2, 3, \dots, m$ . The control objective is to design a $τ$ such that as states of the system quickly converge to the equilibrium point in presence of external disturbances.

Control design

SMC due to simplicity of design has been successfully employed for solving nonlinear control problems.^33–36 It was first used in 1960s and its basic formulations are due to the work of Utkin.³⁷ Utkin provided the definition of sliding surface from which equivalent control is derived. In 1978, Utkin and Yang continued their work and derived the term nonlinear switching from linear state space derivation which ensures the robustness of SMC.³⁸ SMC is based on the concept of the sliding surface, and it switches the system trajectories on this sliding surface. The total control input is a combination of switching control and equivalent control, that is

τ = τ_{eq} + τ_{sw}

(3)

where $τ_{sw}$ is designed to switch system trajectories onto sliding surface and $τ_{eq}$ is designed to keep system trajectories in closed neighborhood of sliding surface. Stabilization of underactuated mechanical systems has become a benchmark to verify the effectiveness of new control techniques. These systems are distinguished by the fact that they have fewer actuators. There are many examples of these systems such as manipulators, robots, and industrial equipment. The dynamics of these systems mostly contain feedback nonlinearities, couplings, nonholonomic constraints, and non-minimum phase zero dynamics, which makes it difficult to design controllers for these systems.

In this article, we are focusing on a class of nonlinear underactuated mechanical systems. SMC is a powerful and robust nonlinear feedback control method. However, for underactuated systems, designing a conventional single layer sliding surface is not appropriate because the parameters of the sliding mode surface can not be obtained directly according to the Hurwitz condition. The physical structure of the considered class also matters as it can be divided into several subsystems. Based on this structure, a simple way to design the sliding mode surface for this class was needed. To overcome this problem, Qian et al.³⁹ proposed a hierarchical sliding mode control (HSMC) for such a class of underactuated systems.

HSMC is a systematic and effective design procedure, which has both theoretical and practical significance. It fills the gap between SMC and its applications to underactuated mechanical systems.⁴⁰ It is composed of two types of sliding surfaces: the first-level sliding surfaces and a second-level sliding surface. The number of first-level sliding surfaces depends upon the number of subsystems and the second-level sliding surface is a linear combination of all first-level sliding surfaces. The schematic structure of the surfaces of HSMC is illustrated in Figure 1.

Figure 1.

Schematic structure of sliding surfaces for HSMC.

In order to reduce the response time of convergence and decrease chattering phenomenon, we proposed to design state-dependent switching gain instead of a constant switching gain. Both, first-level and second-level sliding surfaces, are asymptotically stable in the proposed control design scheme. Numerical simulations validate the effectiveness of the proposed HSMC. To design this controller, we start by defining the first-level sliding surfaces as follows

σ_{k} = ϱ_{k} x_{2 k - 1} + x_{2 k}

(4)

where $ϱ_{k}$ is a positive constant. Differentiating equation (4) with respect to $t$ and using equations given in system (2), we get

{\overset{\cdot}{σ}}_{k} = ϱ_{k} {\overset{\cdot}{x}}_{2 k - 1} + {\overset{\cdot}{x}}_{2 k}

(5)

{\overset{\cdot}{σ}}_{k} = ϱ_{k} x_{2 k} + g_{k} + b_{k} τ + Δ

(6)

The equivalent control of the $k^{th}$ subsystem can be obtained by assuming ${\overset{\cdot}{σ}}_{k} = 0$ and it is given by

τ_{eqk} = - \frac{ϱ_{k} x_{2 k} + g_{k} + Δ}{b_{k}}

(7)

Now, we define the second-level sliding surface as follows

Ω = \sum_{k = 1}^{m} λ_{k} σ_{k}

(8)

where $λ_{k}$ is a real constant. Differentiating equation (8) with respect to $t$ , we have

\overset{\cdot}{Ω} = \sum_{k = 1}^{m} λ_{k} {\overset{\cdot}{σ}}_{k}

(9)

In order to derive a control law which drives the trajectories of the system to the sliding surface $Ω$ , we define a Lyapunov function as follows

V = \frac{Ω^{2}}{2}

(10)

\overset{\cdot}{V} = Ω \overset{\cdot}{Ω}

(11)

\overset{\cdot}{V} = Ω [\sum_{k = 1}^{m} λ_{k} {\overset{\cdot}{σ}}_{k}]

(12)

\overset{\cdot}{V} = Ω [\sum_{k = 1}^{m} λ_{k} (ϱ_{k} x_{2 k} + g_{k} + b_{k} τ + Δ)]

(13)

\overset{\cdot}{V} = Ω [\sum_{k = 1}^{m} λ_{k} (ϱ_{k} x_{2 k} + g_{k} + b_{k} (τ_{eq} + τ_{sw}) + Δ)]

(14)

\overset{\cdot}{V} = Ω [\sum_{k = 1}^{m} λ_{k} \cdot b_{k} \cdot τ_{sw}]

(15)

We design $\overset{\cdot}{Ω}$ as

\overset{\cdot}{Ω} = - ϵ \cdot sat (Ω)

(16)

where “sat” is the saturation function and $ϵ$ is a switching gain. The saturation function in the form of sign function is defined as⁴¹

sat (Ω) = {\begin{matrix} sgn (Ω), & if | Ω | ⩾ 1 \\ Ω, & if | Ω | < 1 \end{matrix}

and

sgn (Ω) = {\begin{matrix} + 1, & if Ω > 0 \\ - 1, & if Ω < 0 \end{matrix}

The switching gain can be selected in two ways: one is the switching gain with a constant value $ϵ$ and the other is the switching gain $ϵ (x)$ which is a function of the state variables. If the switching gain is selected as a constant, then it must be a positive constant to fulfill the stability condition for second-level sliding surface. We proposed a state-dependent switching gain for fast convergence of the system states in presence of bounded external disturbances, that is

ϵ (x) = β (Ω^{2} + γ)

(17)

where $β$ and $γ$ are positive constants. The switching control law obtained from equations (15) and (16) is given by

τ_{sw} = \frac{- ϵ (x) \cdot sat (Ω)}{\sum_{k = 1}^{m} λ_{k - 1} \cdot b_{k}}

(18)

Theoretical stability

This section consists of two theorems which deal with the asymptotic stability of the second-level sliding surface and the first-level sliding surfaces. To prove these theorems, we use the Lyapunov stability theory and the Barbalat lemma.²⁹

Theorem 1 (Asymptotic stability of second-level sliding surface)

Consider the class of underactuated systems given in system (2). If we use the control law given in equation (3) along with sliding surfaces designed in equations (4) and (8), then the second-level sliding surface is asymptotically stable.

Proof

Consider the Lyapunov function defined in equation (10)

V = \frac{1}{2} Ω^{2}

Differentiating both sides with respect to $t$ , we get

\overset{\cdot}{V} = Ω \cdot \overset{\cdot}{Ω}

\overset{\cdot}{V} = - ϵ (x) \cdot Ω \cdot sat (Ω)

Integrating both sides from $0$ to $t$ , we have

\begin{matrix} \int_{0}^{t} \overset{\cdot}{V} d χ = \int_{0}^{t} - ϵ (x) \cdot Ω \cdot sat (Ω) d χ \\ V (t) - V (0) = \int_{0}^{t} - ϵ (x) \cdot Ω \cdot sat (Ω) d χ \\ V (0) = V (t) + \int_{0}^{t} ϵ (x) \cdot Ω \cdot sat (Ω) d χ \\ V (0) ⩾ \int_{0}^{t} ϵ (x) \cdot Ω \cdot sat (Ω) d χ \\ \Rightarrow \lim_{t \to \infty} \int_{0}^{t} ϵ (x) \cdot Ω \cdot sat (Ω) d χ ⩽ V (0) < \infty \end{matrix}

(19)

According to the Barbalat lemma²⁹

\lim_{t \to \infty} ϵ (x) \cdot Ω \cdot sat (Ω) = 0

(20)

Therefore, $\lim_{t \to \infty} Ω = 0$ . Hence second-level sliding surface is asymptotically stable.

Theorem 2 (Asymptotic stability of first-level sliding surfaces)

Consider the underactuated system given in system (2). If the control law is adopted as given in equation (3) with sliding surfaces designed in equations (4) and (8), then the first-level sliding surfaces are asymptotically stable.

Proof

The proof of this theorem is quite simple. From equation (15) and considering that $\overset{\cdot}{V} = Ω \overset{\cdot}{Ω} < \infty$ , we can write

Ω, \overset{\cdot}{Ω} \in L^{\infty}

where $L^{\infty}$ denotes the space of all bounded functions. It follows from equations (8) and (9) that $\sum_{k = 1}^{m} λ_{k} σ_{k}$ and $\sum_{k = 1}^{m} λ_{k} {\overset{\cdot}{σ}}_{k}$ both represent linear combinations of both bounded functions. It means that $σ_{k}, {\overset{\cdot}{σ}}_{k} \in L^{\infty}$ . Therefore

\sup_{t ⩾ 0} | σ_{k} | = ∥ σ_{k} ∥_{\infty} < \infty \Rightarrow \lim_{t \to \infty} σ_{k} = 0

Simulation results and discussion

The proposed controller is simulated in MATLAB and results are compared with the decoupled sliding mode controller (DSMC) and the integral sliding mode controller (ISMC). Consider a linear time-invariant (LTI) system of the form

\overset{\cdot}{x} = Ax + B τ

(21)

where $A \in R^{n \times n}$ and $B \in R^{n \times m}$ . The integral surface is given by

σ = Cx - \int_{0}^{t} C (A - BK) x (τ) d τ

(22)

where $C \in R^{m \times n}$ is selected to guarantee that $CB$ is nonsingular and $K \in R^{m \times n}$ is designed via pole placement such that all eigenvalues of the matrix $A - BK$ lie in the left half plane. The control input for ISMC is given by

τ = τ_{eq} + τ_{sw}

(23)

where $τ_{eq}$ and $τ_{sw}$ are defined as

\begin{matrix} τ_{eq} = - Kx \\ τ_{sw} = - {(CB)}^{- 1} [k σ + η sgn (σ)] \end{matrix}

where $k$ and $η$ are positive constants.

Application to IP

IP system is a strongly nonlinear underactuated mechanical system which is dynamically suitable to test new control strategies. The structure of IP system is shown in Figure 2. In case of IP model, system (1) can be represented as

\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = g_{1} (x) + b_{1} (x) τ + Δ \\ {\overset{\cdot}{x}}_{3} = x_{4} \\ {\overset{\cdot}{x}}_{4} = g_{2} (x) + b_{2} (x) τ + Δ \end{matrix}

Figure 2.

Structure of the inverted pendulum system.

The nonlinear functions $g_{1} (x)$ , $g_{2} (x)$ , $b_{1} (x)$ , and $b_{2} (x)$ are defined as

\begin{matrix} g_{1} (x) = \frac{m_{t} g \sin x_{1} - m_{p} L \sin x_{1} \cos x_{1} x_{2}^{2}}{L \cdot (\frac{4}{3} m_{t} - m_{p} \overset{2}{\cos} x_{1})} \\ g_{2} (x) = \frac{- \frac{4}{3} m_{p} {Lx}_{2}^{2} \sin x_{1} + m_{p} g \sin x_{1} \cos x_{1}}{\frac{4}{3} m_{t} - m_{p} \overset{2}{\cos} x_{1}} \\ b_{1} (x) = \frac{\cos x_{1}}{L \cdot (\frac{4}{3} m_{t} - m_{p} \overset{2}{\cos} x_{1})} \\ b_{2} (x) = \frac{4}{3 (\frac{4}{3} m_{t} - m_{p} \overset{2}{\cos} x_{1})} \end{matrix}

where $m_{t} = m_{c} + m_{p}$ . The description of variables and parameters used in this model is given in Table 1. The first-level and second-level sliding surfaces for IP system are defined as

σ_{1} = ϱ_{1} x_{1} + x_{2}

(24)

σ_{2} = ϱ_{2} x_{3} + x_{4}

(25)

Ω = λ_{1} σ_{1} + λ_{2} σ_{2}

(26)

Table 1.

Description of variables and parameters used in inverted pendulum model.

Symbols	Description
$x_{1}$	Angle of the pole with vertical axis
$x_{2}$	Angular velocity of the pole
$x_{3}$	Position of the cart
$x_{4}$	Velocity of the cart
$τ$	Control input
$Δ$	External disturbances
$L$	Length of the pole
$m_{p}$	Mass of the pole
$m_{c}$	Mass of the cart

The control input is given by

τ = τ_{eq 1} + τ_{eq 2} + τ_{sw}

(27)

where

τ_{eq 1} = \frac{- c_{1} x_{2} - g_{1} - Δ}{b_{1}}

(28)

τ_{eq 2} = \frac{- c_{2} x_{4} - g_{2} - Δ}{b_{2}}

(29)

τ_{sw} = \frac{- ϵ (x) \cdot sat (Ω)}{λ_{1} b_{1} + λ_{2} b_{2}}

(30)

and $ϵ (x)$ is defined in equation (17). The values of parameters used for simulations are given in Table 3. We define a bounded external disturbance as $Δ = rand () . \sin (t)$ , where MATLAB command “rand()” is used to describe the fluctuation of external disturbances. Stabilization of the angle of inverted pendulum is shown in Figure 3 and the position of the cart is shown in Figure 4. Both figures demonstrate that the proposed controller is robust and more efficient as compared with DSMC and ISMC.

Figure 3.

The angle of pendulum with time when system starts from $x_{0} = [\frac{π}{3} 0 0 0]^{T}$ .

Figure 4.

The position of cart with time when system starts from $x_{0} = [\frac{π}{3} 0 0 0]^{T}$ .

Application to ball-beam system

Ball-beam (BB) system is another nonlinear underactuated system. Its structure is illustrated in Figure 5 and mathematical equations are given as

\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = τ + Δ \\ {\overset{\cdot}{x}}_{3} = x_{4} \\ {\overset{\cdot}{x}}_{4} = \frac{m r^{2}}{j_{b} + m r^{2}} (x_{3} x_{2}^{2} - g \sin x_{1}) \end{matrix}

Figure 5.

Structure of the ball-beam system.

The description of different variables and parameters of ball-beam model is given in Table 2. The sliding surfaces are given by

σ_{1} = ϱ_{1} x_{1} + x_{2}

(31)

σ_{2} = ϱ_{2} x_{3} + x_{4}

(32)

Ω = λ_{1} σ_{1} + λ_{2} σ_{2}

(33)

Table 2.

Description of variables and parameters used in ball-beam model.

Symbols	Description
$x_{1}$	Angle of the beam with horizontal axis
$x_{2}$	Angular velocity of the beam
$x_{3}$	Position of the ball
$x_{4}$	Velocity of the ball
$τ$	Control input
$Δ$	External disturbances
$m$	Mass of the ball
$r$	Radius of the ball
$j_{b}$	Moment of inertia of the ball

The control input is defined as

τ = τ_{eq} + τ_{sw}

(34)

where

τ_{eq} = - ϱ_{1} x_{2} - Δ

(35)

τ_{sw} = - ϵ (x) \cdot sat (Ω) - λ_{1} ϱ_{2} x_{4} - \frac{λ_{1} m r^{2}}{j_{b} + m r^{2}} (x_{3} x_{2}^{2} - g \sin x_{1})

(36)

The values of parameters used in ball-beam model are given in Table 3. We use the same external disturbance as described in the inverted pendulum model. The change in the dynamics of angle of beam and position of the ball are shown in Figures 6 and 7. The simulation results show that the proposed controller has a faster convergence as compared with DSMC and ISMC in presence of bounded external disturbances.

Table 3.

The values of parameters and constants used in the simulations of model.

Parameters	Values
$L$	0.5 m
$m$	0.5 kg
$m_{c}$	1 kg
$m_{p}$	0.05 kg
$r$	0.031 m
$g$	9.8 ms⁻²
$j_{b}$	2×10⁻⁶
$ϱ_{1}$	5
$ϱ_{2}$	0.5
$λ_{1}$	1
$λ_{2}$	−0.388
$β$	5
$γ$	4

Figure 6.

The angle of beam with time when system starts from $x_{0} = [\frac{π}{3} 0 0.8 0]^{T}$ .

Figure 7.

The position of ball with time when system starts from $x_{0} = [\frac{π}{3} 0 0.8 0]^{T}$ .

We also compare HSMC with the state-dependent switching gain to HSMC with a constant switching gain. It can be seen that response time of the proposed HSMC is much smaller as compared with the conventional constant gain HSMC. Simulation results are shown in Figures 8 and 9. These results demonstrate the significance of proposed HSMC over existing HSMC, DSMC, and ISMC techniques. We also fluctuate the external disturbance to validate the robustness of proposed HSMC.

Figure 8.

Comparison of proposed HSMC with existing HSMC in case of IP system.

Figure 9.

Comparison of proposed HSMC with existing HSMC in case of BB system.

Conclusion

In this article, we discussed the general form of a class of underactuated mechanical systems and presented a nonlinear control strategy for stabilization of these systems. We focused on the potential uses of SMC methods for stabilization of underactuated mechanical systems. We proposed HSMC based on state-dependent switching gain. The proposed controller consists of two first-level sliding surfaces and one second-level sliding surface which is essential to reduce chattering phenomenon.

The designed control technique is implemented on two nonlinear underactuated systems and simulated in MATLAB. The simulation results demonstrate that the proposed control technique is much efficient as compared with the existing SMC techniques such as HSMC, DSMC, and ISMC. The proposed control technique is robust and has a faster convergence as compared with these techniques. Furthermore, asymptotic stability of first-level and second-level sliding surfaces, used in the proposed technique, is also proved by the Lyapunov stability theory.

Footnotes

Acknowledgements

The authors are thankful to the editor and reviewers for their precious comments, which have led to the improvement in the clarity and quality of this article.

Handling Editor: Yueying Wang

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

Muhammad Idrees

Saif Ullah

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Sliding mode control design for stabilization of underactuated mechanical systems

Abstract

Keywords

Introduction

Problem statement

Control design

Theoretical stability

Theorem 1 (Asymptotic stability of second-level sliding surface)

Proof

Theorem 2 (Asymptotic stability of first-level sliding surfaces)

Proof

Simulation results and discussion

Application to IP

Application to ball-beam system

Conclusion

Footnotes

Acknowledgements

Declaration of conflicting interests

Funding

ORCID iDs

References