# Fixed time terminal sliding mode trajectory tracking design for a class of nonlinear dynamical model of air cushion vehicle

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## Abstract

This manuscript proposes a robust fixed time terminal sliding mode prototype for trajectory tracking of the nonlinear dynamics of an under-actuated air cushion vehicle. Nonlinearity, external disturbances, internal uncertainties and unmodeled dynamics are the main difficulties that an amphibious vehicle is faced with in its maneuver. The main contribution of the proposed methodology is to overcome these problems based on both the guaranteed stability in sense of Lyapunov and the fixed time tracking error even if the initial values are changed. Robustness against uncertainties and disturbances, fixed time convergence of tracking error to zero are other merits of the proposed approach. The simulation results demonstrate the effectiveness and superiority of suggested scheme.

## Keywords

Sliding mode control Fixed time Nonlinear control Trajectory tracking Hovercraft Air cushion vehicle## 1 Introduction

### 1.1 Background and motivations

Air cushion vehicle (ACV) is an under-actuated electromechanical system. The amphibious hovercraft is located on air cushion therefore, it can move on every surface and consequently its used in many applications [1], but the control of hovercraft is so complex, because of low friction, high speed, second order nonholonomic constraint which makes its motion restrict, actuators generate forces only in heading path, dynamic coupling exist among states, external disturbances can affect its motion and directly compensate for side-external disturbance is impossible [1, 2]. Therefore, controller must be designed so that overcome aforementioned challenges. Sliding mode control (SMC) has been widely used to control various systems in recent decades [3, 4, 5, 6, 7], this methodology of control has some notable merits for control engineers such as robustness against disturbances, uncertainties and parameter variations, high accuracy, fast dynamic response, simplicity of computations, significant transient performance and guaranteed stability [8, 9]. The main disadvantages of SMC are chattering effect due to discontinues control action and infinity settling time. Chattering phenomena can hurt the system actuators and sensors. These problems can be solved by high order SMC [10, 11, 12, 13]. Trajectory tracking control is one of the most attractive and challenging tasks of control for a long time [14, 15, 16].

The question may be inspired control engineers, how can do tracking control and stabilizing hovercraft in a fixed time, whether initial values are changed despite all of aforementioned SMC merits.

### 1.2 Brief survey

ACV is a high performance and high usage vehicle, but control of ACV is difficult, the main problems of ACV control are point stabilization, path following and trajectory tracking [17]. Two control methods have been applied on hovercraft so far, classic and intelligent.

In [18, 19, 20, 21] fuzzy controllers derived on hovercraft, although the methods are simple, fuzzy method has problems with systematic analysis and design. In [22] the controller has been designed based on a neural network and applied to a hovercraft dynamic in which the actuator has not been modeled, and the use of the neural network has not led to a globally stability, it can’t follow the desired path completely. In [23] a radial basis function neural networks controller has been implemented in a number of hovercraft coordinates. The effect of external disturbances and friction have not taken into account and the dynamic model is a simple model, and also the time delay of communication between devices has been ignored. Therefore, it can generally be noted that intelligent controllers, are very complex decision making processes and fuzzy rules are based on the knowledge of experts, which is not always available. In addition, great set of rules need great time to compute and adjust.

The classical control methods that carried out on hovercraft so far include a Proportional, Integrated, Derivative(PID), adaptive, backstepping, open loop control, linear and nonlinear Lyapunov-based, linear regulator controllers, in which the effect of external disturbances and frictions are ignored, and also they are not robust against them [24, 25, 26, 27, 28]. Other controllers such as feedback linearization, state feedback, etc. don’t have high accuracy for this kind of extreemly nonlinear dynamics and may not be applicable because of the existance of discountinus functions in the dynamics [29, 30, 31, 32, 33].

Human-centered tracking system for a hovercraft based on terminal sliding mode control is applied in [34] the chattering-free and full order TSM has solved chattering and singularity problems then its combined with radial basis function neural networks to deal with the nonlinearity and uncertainty of hovercraft’s model and it can convergs velocities and position tracking errors at finite-time.

The path following and trajectory tracking tasks in a hovercraft physical model are controlled by the backstepping controller in [35], the friction coefficients and disturbance dynamic estimators are introduced, global practical stability is achieved and actuations are remained bounded with respect to the position error.

In [36, 37, 38, 39, 40, 41, 42], some combinations of sliding mode with other controllers on hovercraft and other systems are presented. Although they are robust against disturbances, they haven’t been able to stabilize system at a fixed time.

### 1.3 Contribution

A robust fixed time terminal nonlinear SMC scheme is designed for tracking control of the ACV, which isn’t designed in this methodology for the ACV before.

To get the fixed time stability even if the initial values are changed, the sliding surface is designed in new form, which isn’t used in sliding mode control algorithms of the hovercraft.

The chattering reduction is done by using the sigmoid function.

The dynamical model structure which is used in this paper induces lateral forces on the ACV depending on the torque. In other models, thrust and torque are independent. Therefore, the nonlinearity properties of hovercraft dynamic are fully modeled based on its interaction.

### 1.4 Paper organization

The remainder of this paper is structured as follows, Sect. 2 describes some preliminaries such as notations which are used in this paper, problem formulation which demonstrates vehicle modeling contains a description of nonlinear model for hovercraft and some lemmas which are required for controller design and express the rest of the paper. Section 3 presents fixed time terminal sliding mode control design and its structure. Simulation results are presented in Sect. 4. Finally, Sect. 5 provides a brief conclusion.

## 2 Preliminaries

### 2.1 Notations

Symbol table

Symbol | Description | Symbol | Description |
---|---|---|---|

\(u\) | Longitudinal velocity | \(b_{T}\) | Force scaling coefficient |

\(v\) | Lateral velocity | \(k_{1} ,k_{2}\) | Switching gain |

\(\theta\) | Rudder angle | \(e_{u} ,e_{v}\) | Velocity errors |

\(r\) | Angular velocity | \(u_{d} ,v_{d} ,x_{d} ,y_{d}\) | Desired parameters |

\(T\) | Trust force | \(\left\{ {d_{{u_{0} }} ,d_{{v_{0} }} ,d_{{r_{0} }} ,d_{u} ,d_{v} ,d_{r} } \right\}\) | Friction coefficients |

\(m\) | Mass of hovercraft | \(l_{x} ,l_{y}\) | Saturation constants |

\(J\) | Inertia moment | \(k_{x} ,k_{y}\) | Controller gains |

\(a\) | The arm length from the center of mass to the rudder surface | \(\left[ {\begin{array}{*{20}c} {cos\,{\kern 1pt} \theta } & {sin\,{\kern 1pt} \theta } \\ { - sin\,{\kern 1pt} \theta } & {cos\,{\kern 1pt} \theta } \\ \end{array} } \right]\) | Rotational matrix |

\(x_{e} ,y_{e}\) | Position tracking error |

### 2.2 Problem formulation

To better comprehension of the problem first of all, the following lemma [43] should be expressed.

### **Lemma 1**

*Consider a system such as*

*where*\(\alpha ,\beta\)

*are positive,*\(m,n,p,q > 0\)

*and odd which*\(m > n,p < q\)

*, therefore, the equilibrium point of*(1)

*is fixed time stable and fixed time upper bound is as follows*

*And it is independent of initial states*.

_{d}and v

_{d}are desired longitudinal and lateral velocity, and the time derivative of (4) by substituting (3) is obtained as

_{d}and v

_{d}are considered as

### **Theorem 1**

*If the errors of velocity e*_{u} *and e*_{v} *in* (6) *converge to zero, then it is ensured that the errors of position tracking (x*_{e}*,y*_{e}*) asymptotically converge to origin* [15].

### *Proof*

_{1}is

As \(tanh\) is an odd function and controller and saturation coefficients are positive therefore,\(\dot{V}_{1}\) is negative and position errors converge to zero.

The model and the control conditions make the following assumptions.□

### **Assumption 1**

For simplicity, the explicit time dependence of position and uncertainties is omitted.

### **Assumption 2**

The states of the system are fully measureable.

## 3 Controller design methodology

### 3.1 Controller structure

From Lemma 1 and the Eqs. (13) and (14), the convergence conditions of sliding surfaces and errors to the origin in a fixed time are obtained.

### 3.2 Stability and convergence time analysis

### **Theorem 2**

*Consider hovercraft dynamics given in* (3) *satisfying Assumptions* 1 *and* 2*. Then the sliding surface mentioned in Eq.* (12) *and control inputs proposed as* (17) *to* (20) *makes the closed loop system asymptotically stable, tracking errors and sliding surface converge to zero furthermore all signals in the closed loop system will be bounded.*

### *Proof*

It’s clear that (26) is negative and an \(\eta\) is exists which can satisfy inequality (23), therefore stability of fixed time sliding mode control is proven. This completes the proof.□

### **Theorem 3**

*Consider the ACV model* (3)*, with the control inputs selected as* (15) *and* (16)*, then the states converge to origin in fixed time and for each sliding surfaces upper bound of settling time (T*_{s}*) obtain as equation* [43, 44].

### *Proof*

Therefore, the system reaches the sliding surfaces in a fixed time which the upper bound of it is \(T_{max}^{\prime }\).

The Eq. (32) presents that states of system (3) converge in a fixed time which it’s upper bound is \(2T_{max}^{\prime }\). This time can be set based on problem requirements.□

## 4 Simulation results

^{2}). Table 2 presents the controller parameters.

The controller parameters

Parameter | Value | Parameter | Value |
---|---|---|---|

| 2 |
| 2 |

| 3 |
| 10 |

| 3 |
| 0.1 |

| 2 |
| 0.9 |

| 4 |
| 1 |

| 16 |
| 1 |

| 64 |
| 8 |

| 32 |

As it is explained before \(m > n,0.5 < \frac{p}{q} < 1\) and \(m,n,p,q\) are integers which are positive, \(\alpha ,\beta ,\lambda\) \(k_{x} ,k_{y} ,l_{x} ,l_{y} ,\)\(\psi ,\sigma\) are positive constant therefore, the best values for better controller behavior are obtained using the trial and error method.

Control laws which are mentioned before, applied to the model and following figures are the results. The time value of simulation is around 100 s for the ordinary PCs. It should be noticed that a linear system with sinusoidal input is given to the controller as the desired tracking system.

As it is expressed before, a linear system with sinusoidal input considered as the desired tracking system of controller, then at the split time the controller could make system asymptotically stable and sliding surfaces and states tracking errors converge to the origin.

## 5 Conclusion

In this paper, fixed time terminal sliding mode controller is designed ACV to track the predefined trajectories. This technique is robust against disturbances and uncertainties but the main target of the paper is to persuade the ACV to track the desired reference and stabilizing at the fixed time even if the initial conditions are changed. The settling time can be estimated in advance and the chattering is reduced easily. Asymptotic stability of overall close loop system is proven and by making velocity error to zero, ACV tracks desired path and position tracking error converges to zero too. Simulation results demonstrate efficiency and superiority of proposed method. In this research states are assumed accessible, in future works it is better to design a neural observer for estimating inaccessible states, disturbance and uncertainties. The boundedness of stability time may cause the enhancement of control effort therefore, it is better to have a trade-off between settling time and energy. The extension of this method to higher order systems can be considered for future studies.

## Notes

### Compliance with the ethical standards

### Conflict of interest

The authors (Hemede Karami, Reza Ghasemi) declare that they have no conflict of interest.

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