# Second-order sliding mode approaches for the control of a class of underactuated systems

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## Abstract

In this paper, first-order and second-order sliding mode controllers for underactuated manipulators are proposed. Sliding mode control (SMC) is considered as an effective tool in different studies for control systems. However, the associated chattering phenomenon degrades the system performance. To overcome this phenomenon and track a desired trajectory, a twisting, a super-twisting and a modified super-twisting algorithms are presented respectively. The stability analysis is performed using a Lyapunov function for the proposed controllers. Further, the four different controllers are compared with each other. As an illustration, an example of an inverted pendulum is considered. Simulation results are given to demonstrate the effectiveness of the proposed approaches.

## Keywords

Underactuated manipulator sliding mode control twisting algorithm super-twisting algorithm inverted pendulum## 1 Introduction

Underactuated mechanical systems (UMS) are increasingly present in the robotic field. They have less actuators than their degrees of freedom. In these systems, we find manipulators, vehicles and humanoids with several passive joints. Underactuations arise by deliberate design for the purpose of reducing the weight of the manipulator or might be caused by actuator failures. The difficulty in controlling underactuated mechanisms is due to the fact that techniques developed for fully actuated systems cannot be directly used. These systems are not feedback linearizable, yet they exhibit nonholonomic constraints and nonminimum phase characteristics[1]. Moreover, it has been shown that it is not difficult to stabilize this class of systems by continuous controllers. Because of this, the class of under-actuated mechanical systems presents challenging control problems. One of the common methods used to control underactuated systems is the SMC based on Lyapunov design.

The SMC has always been considered as an efficient approach in control systems, due to its high accuracy and robustness against internal and external disturbances. The SMC approach consists of two steps. The first is to choose a manifold in the state space that forces the state trajectories to remain along it. The second is to design a discontinuous state-feedback capable of forcing the system to reach the state on the manifold in a finite time. However, the drawback of the SMC is the presence of the chattering effect, caused by the switching frequency of the control[2]. The high frequency components of the control propagating on the system can excite the unmodeled fast dynamics and therefore cause undesired oscillations. In fact, this can degrade the system performance or may even lead to instability. In the literature, three main approaches have been presented which help to reduce the chattering effects. The class of methods consist in the use of the saturation control instead of the discontinuous one. It ensures the convergence to a boundary layer of the sliding manifold. Moreover, a switching function, inside the boundary layer of the sliding manifold, was approximated by a linear feedback gain[3, 4]. However, the accuracy and the robustness of the sliding mode were partially lost.

The second class of methods consist in the use of a system observer-based approach[5]. It can reduce the problem of robust control to the problem of exact robust estimation. This can lead to the deterioration of the robustness with respect to the plant uncertainties or disturbances.

Using high-order sliding mode controllers given by Levant[6, 7] as a way to reduce the chartering phenomenon and keep the main advantages of the original approach of the SMC is another way to eliminate chattering. On the other hand, the second-order sliding mode control is relatively simple to implement and it gives good robustness to external disturbances. The second-order sliding mode control (SSMC) approach can reduce the number of differentiator stages in the controller. However, the stability proofs are based usually on a geometrical or homogeneity method since the Lyapunov function is a difficult task to define[7]. The stability and the convergence using SSMC is challenging and several trials were made to deal with those difficulties. Recently, Moreno and Osorio[8] constructed a Lyapunov function that provides a finite time convergence, a robustness and an estimate of the convergence time for super-twisting algorithm. In [9], a multivariable super-twisting structure was proposed, which analyzed the stability using the ideas of Lyapunov function given in [8].

*u*that moves the cart horizontally and the output is the angular position of the pendulum 6. Therefore, the inverted pendulum has been a popular candidate to illustrate different control methods. However, despite of its simple mechanical structure, this prototype is not easy to control and requires sufficiently sophisticated control designs. Indeed, it is proven that the system is not feedback linearizable and has no corresponding constant relative degree[11]. Moreover, Zhao and Spong[12] have shown that several geometric properties of the system are lost when the pendulum moves through horizontal positions. The objective consists of moving the cart from an initial position to a desired one and keeping the pendulum in the vertical position with a minimum of its oscillations around this position. In general, the main difficulty is to swing up the pendulum from the downward vertical position and to keep the cart stable. Numerous control techniques have been employed to stabilize the inverted pendulum such as proportional-integral-derivative (PID) controllers where the control gains are adjustable and updated online with a stable adaptation mechanism[13].

The objective of this paper is to develop a robust position tracking controller based on the first-order and the second-order sliding mode approaches applied to an inverted pendulum. Stability of the closed loop system is carried out using candidate Lyapunov functions for the proposed controllers. The contributions of this paper are presenting the stability analysis of twisting and super-twisting controllers. Further, a modified super-twisting algorithm with simple stability analysis is proposed. This controller has almost the same propriety as the super-twisting algorithm. The paper is organized as follows. Section 2 describes the model of the inverted pendulum and the first sliding mode controller. Section 3 deals with the sliding mode controllers and the design of second-order sliding mode controllers. Section 4 discusses the simulation results of the proposed controllers.

## 2 Dynamic model and control approach for an inverted pendulum

### 2.1 Dynamic model

*l*is the length of the pendulum,

*m*is the pendulum mass,

*M*is the cart mass,

*τ*is the horizontal force action,

*θ*is the angular deviation, and

*y*is the position of the cart which is moving horizontally.

*x*

_{1}=

*y*, \({x_2} = \dot y\),

*x*

_{3}=

*θ*and \({x_4} = \dot \theta\). According to the canonical form of a class of underactuated systems, we can transform (1) into the following state space representation

*x*= [

*x*

_{1},

*x*

_{2},

*x*

_{3},

*x*

_{4}]

^{T}is the state variable vector,

*τ*is the control input,

*ƒ*

_{1},

*ƒ*

_{2},

*b*

_{1}and

*b*

_{2}are nominal nonlinear functions, expressed as

*T*:

*x*↦

*ξ*defined by

*ξ*= (

*ξ*

_{1}

*ξ*

_{2}

*ξ*

_{3}

*ξ*

_{4})

^{T}.

*T*is defined as a local diffeomorphism with

*T*(0) = 0.

*L*

_{ ƒ }

*h*(

*x*) is the Lie derivative of

*h*(

*x*) along the vector

*f*(

*x*).

*ξ*=

*T*(

*x*) and

*T*

_{1}(

*x*) =

*h*(

*x*)=

*ξ*

_{1}. Define the transformation

*T*:

*x*↦

*ξ*by

*ξ*, we obtain

*x*

_{4}tan(

*x*

_{3}) because this term is

*o*(

*x*

_{3},

*x*

_{4})

^{2}, we obtain a feedback linearizable nonlinear system in the state

*ξ*with

### 2.2 First-order sliding mode controller

*s*= {

*ξ*∈

**R**

^{4}| s(

*ξ*)=0}, for

*λ*> 0,

*z*

_{ d }= [2 0 0 0]

^{T}. The time derivative of

*s*along the system trajectory

*ξ*is equal to

*u*

_{ sw }is the switching control,

*u*

_{ eq }is the equivalent control yielded from \(\dot s(\xi ) = 0\), and

*η*and

*K*are positive constants.

*V*along the trajectories of (14) yields

Then, the system is stable and the convergence of the sliding mode is guaranteed.

## 3 Second-order sliding mode controller

*r*defined by \(s = \dot s = \ddot s = \dddot s = \cdots = s^{(r - 1)} = 0\).

*s*defines the sliding variable with the

*r*-th order sliding and with its (

*r*− 1)-th first time derivatives depending only on the state

*x*. The first-order sliding mode tries to keep

*s*= 0. In the case of second-order sliding mode control, which only needs its measurement or evaluation of

*s*, the relation should be verified as

### 3.1 Twisting controller

#### 3.1.1 Controller approach

*ξ*) =

*ƒ*

_{ e }(

*ξ*) + 2λ

*z*

^{(3)}+ λ

^{2}

*z*

^{(2)}and

*ϕ*(

*ξ*) =

*g*

_{ e }(

*ξ*).

*ϕ*are bounded such that

_{ d },

*ϕ*

_{ m },

*ϕ*

_{ M }and

*ϕ*

_{ d }are positive scalars. Then, we have

By imposing \({\ddot s_1} = 0\), the equivalent control can be expressed as \({u_{eq}} = - {\textstyle{\Psi \over \varphi}}\)

#### 3.1.2 Stability analysis

*β*> 0, 0 <

*K*≤

*K*

_{ M }and \({K_M} > {\textstyle{1 \over {1 - \beta}}}{\textstyle{{{\Psi _d}} \over {{\varphi _M}}}}\). The total control is defined by

*λ*

_{2}−

*K*< 0.

### 3.2 Super twisting controller

SSMC controllers require the of values of the derivatives except for the super twisting algorithm (STW). The STW is a continuous sliding mode algorithm ensuring main properties of the first-order sliding mode control for systems with Lipschitz continuous matched uncertainties or disturbances with bounded gradients[7]. It has been developed to control systems with a relative degree equal to one in order to avoid chattering.

Trajectories on the two sliding planes are characterized by twisting around the origin, but the continuous control law *u*(*t*) is constituted by two terms. The first is defined by the discontinuous time derivative and the second is a continuous function of the available sliding variable[2].

#### 3.2.1 Controller approach

_{1}(

*ξ*) =

*f*

_{ e }(

*ξ*) + 3

*λz*

^{(3)}+ 3

*λ*

^{2}

*z*

^{(2)}+

*λ*

^{3}

*z*

^{(1)}.The control law can be expressed by[16]

*σ*are described by

*k*

_{1},⋯,

*k*

_{4}are positive scalars.

#### 3.2.2 Stability analysis

*k*

_{5}is a positive scalar and

*k*

_{5}for more flexible stability conditions and for more generalization of the expression of

*V*

_{2}. Substituting (37) into (36) gives

*Ψ*and Υ are positives definite if

So matrices Ψ and Υ are positive definite and consciously \({\dot V_2}(s,\sigma) \leq 0\). Thus, we can conclude that the system is stable.

### 3.3 A modified super twisting controller

To simplify the stability analysis, we propose a modified super twisting controller (MSTW).

#### 3.3.1 Controller approach

*σ*

_{1}are given by

*k*

_{6},

*k*

_{7},

*k*

_{8}and

*k*

_{9}are positive scalars.

#### 3.3.2 Stability analysis

## 4 Simulation results and discussions

Parameters of the inverted pendulum system are set as *M* = 20 kg, *m*_{0} = 1.8 kg, *l* = 0.3m, *g* = 9.8N/kg. The initial conditions of the cart pendulum are \(({y_0},{\dot y_0}) = (0,0),\;({\theta _0},{\dot \theta _0}) = (0.1,0)\) and the desired position is set as *y*_{ d } = 2, *θ*_{ d } = 0 and \({\dot y_d} = {\dot \theta _d} = 0\).

Simulations are done using *λ* =1 and *k* =20 for the SMC, *k*_{1} =40 and *k*_{2} = 90 for the twisting controller.

Comparison of the energy criterion (×10^{4} J)

SMC | TW | STW | MSTW |
---|---|---|---|

3.6954 | 1.5798 | 1.2624 | 1.3056 |

## 5 Conclusions

In this paper, a second-order sliding mode controller has been designed for underactuated manipulators. This controller keep the main advantages of the original sliding mode approach and removes the chattering caused by the sliding mode approach. Simulation results of the twisting, the super-twisting and the modified super-twisting controllers show that these controllers give better performance compared to the first-order sliding mode controller. It has been shown that the proposed modified super-twisting controller have almost the same properties of the super-twisting with a simple stability analysis. Moreover, the second-order sliding-mode controller is an effective tool for the control of uncertain nonlinear systems since it overcomes the main drawback of the classical sliding-mode control approach.

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