Elsevier

Mechatronics

Volume 61, August 2019, Pages 96-105
Mechatronics

Saturation based nonlinear PID control for underwater vehicles: Design, stability analysis and experiments

https://doi.org/10.1016/j.mechatronics.2019.06.006Get rights and content

Abstract

During sea missions, underwater vehicles are often exposed to changes in the parameters of their control systems and subject to external disturbances due to the influences of ocean currents. These issues make the design of a robust controller quite a challenging task. This paper focuses on the design of a nonlinear PID controller, based on a set of saturation functions for trajectory tracking on an underwater vehicle. The main feature of the proposed control law is that it preserves the advantages of robust control and remains easy to fine-tune in real applications. Using the Lyapunov concept, we prove the asymptotic stability of the closed-loop tracking system. The effectiveness and robustness of our proposed controller for trajectory tracking in depth and yaw dynamics is demonstrated through real-time experiments.

Introduction

There are two main classes of Unmanned Underwater Vehicles (UUVs): Remotely Operated Vehicles (ROVs), and Autonomous Underwater Vehicles (AUVs). Both require advanced controllers, as their dynamics are highly non linear and they have to deal with unpredictable external disturbances, such as the ones generated by ocean currents or by the tether [1]. In the case of AUVs, all of the degrees of freedom (DoF) are controlled, while in the case of ROVs a part of the DoF are piloted by a human (shared control). Both classes require controllers and this paper will refer to Underwater Vehicles in general.

Proportional-Derivative (PD) and Proportional Integral Derivative (PID) controllers are the most commonly used techniques to control the position and orientation of commercial underwater vehicles, this is due to their design simplicity and their good performance, especially when some system parameters are unknown [2], [3], [4]. However, it is well-known that when the plant’s dynamics is highly nonlinear, time-varying, or with significant time delays the PID controls performance is often degraded. The impact of these drawbacks can be reduced by using adaptive, saturated or nonlinear PD/PID strategies. Inspired by this problem, several advanced PD/PID control schemes for underwater vehicles have been proposed in previous literature and some of them are summarized below.

It is acknowledged that the PID control tuning process to obtain the best controller behavior can be time-consuming. Consequently, intelligent tuning and self-adjusting control parameter methodologies have been developed in recent years. In [5], a genetic algorithm was used to tune the gains of a fractional order PID for setpoint regulation in depth and steering subsystems of an AUV. Following the same lines, a PID control was tuned using the Particle Swarm Optimization (PSO) method for setpoint regulation and trajectory tracking in diving and steering subsystems [6]. A Fuzzy Logic Controller (FLC) was used with the PID algorithm to tune its gains adaptively. For example, in [7], a decoupled Adaptive Fuzzy PID Controller (AFPIDC) for trajectory tracking in heading and depth of an AUV was proposed. In this work, the adaption law is composed of two elements, the initial constant control gains, given by the designer, and the time-varying incremental gains which depend on the tracking error and its ratio. The incremental gain is adjusted by fuzzy rules derived from the expert’s knowledge. Based on simulation results, the performance of the AFPIDC is superior to nominal PID design during tracking trajectory tests. Similar methodologies, using fuzzy logic to improve the PID controller for path following or to demonstrate its robustness with respect to external disturbances can be found in [8], [9]. Finally, inverse optimal PID control applied to a self-tuning controller for an AUV, modeled as a nonlinear autoregressive moving average model with exogenous inputs was proposed in [10].

The Active Disturbance Rejection Controller (ADRC) can estimate the influence of the external disturbances such as ocean currents or wave effects over an AUV. On the one hand, in paper [1], an adaptive DOB control (ADOB) for set point regulation and trajectory tracking problems on the 6 degrees of freedom of ODIN AUV was proposed. In this work, the proposed controller was designed for a known nominal model, where the external disturbances and modeling errors were estimated through the DOB method. Then, a regressor-free adaptive control law was adopted to provide robustness to the DOB control towards uncertainties in the system model. The effectiveness of the proposed methodology was shown through real-time experiments on the x-y-z dynamics, while the AUV’s orientation was kept stable (i.e., ϕ=θ=ψ=0). From these results, we can observe that the ADOB algorithm improves the performance of the PID controller considerably under constant external disturbances and parameter uncertainties. On the other hand, in [11], the DOB method was applied to the PID control of an AUV based on the frequency analysis approach. In [12], a diving ADRC has been proposed to deal with the high nonlinearity, strong coupling and time-varying features in the AUV system.

It is worth noting that, during sea missions, an AUV can be disturbed by ocean currents or subject to unknown objects sticking to the submarine body which suddenly changes its physical parameters. To overcome this problem, adaptive controllers can be used as a suitable solution to control AUVs. The main feature of an adaptive controller lies in its ability to update the control gains based on the changes in vehicle dynamics and external disturbances. As an example of this methodology, an adaptive PD controller for setpoint regulation was proposed in [13]. The designed controller needed only the knowledge of the vector of gravitational and buoyancy forces. The control law consists of a PD plus buoyancy compensation (PD+) with an adaptive term that estimates and compensates parameter uncertainties and external disturbances. The behavior of the adaptive controller was validated through simulations and real-time experiments for setpoint regulation in (x, y, z, ψ) dynamics. Based on the obtained experimental results, it can be observed that the adaptive control has a better performance in depth dynamics than the PD, but the behavior of both methodologies is almost the same for (x, y, ψ) dynamics. Also, following the same methodology, an adaptive PD controller for a region reaching controller was proposed in [14]. We can compare this to [13], which is based on a saturated PD control instead of a linear PD law. Although the effectiveness of the proposed controller was only shown in simulations.

In practical applications, it can be observed that a standard PID control design can be improved by bounding its signal [15]. Consequently, several nonlinear bounded PID controllers have been proposed. For instance, in [16], a model reference adaptive (MRAC) PID control structure with an anti-windup (AW) compensator for pitch trajectory tracking of the REMUS AUV was proposed. It was demonstrated that adding the AW compensator improves the nominal adaptive control. The AW term is obtained by solving a linear Ricatti equation. Simulation results show the improvements of the proposed controller over the nominal MRAC in terms of external disturbances rejection and saturation of the AUV’s actuators. Another version taking into account the saturation of the AUV’s actuators, resulting in a μ modified adaptive controller was proposed in [17]. Also, a dual-loop variable-structure PID (VSPID) controller with AW term for controlling the surge and sway dynamics of an AUV is proposed in [18]. Experimental results show that the VSPID with AW reduces the overshoot as well as the settling time compared to the nominal VSPID. Finally, inspired by the works of [19] and [20], a nonlinear PD and PD+ controllers for trajectory tracking on depth and yaw dynamics of an AUV has been proposed in [15]. In this work, the authors introduced a whole set of nonlinear functions to improve the PD controller. Real-time experiments on the L2ROV vehicle demonstrate the effectiveness and robustness of the proposed control law. Indeed, an improved performance of the proposed controller for trajectory tracking in yaw dynamics is demonstrated. However, the performance of the controller for depth trajectory tracking is reduced when the system’s parameters are subject to uncertainties.

In summary, on the one hand, fuzzy approaches [7] and intelligent algorithms such as PSO and AG (see [5], [6]), which are used to tune the PID control, can be useful to obtain good performance from the controller. On the other hand, the disturbance estimation made by ADRC can provide robustness to the PID, as seen in the experiments shown in references [1], [11], [12]. However, based on the experimental results of these works, the robustness improvement of this methodology towards parameter uncertainties or external disturbances is not clear. Furthermore, the main ADRC drawback is the tedious task of tuning numerous parameters. Concerning the adaptive controllers [13], [14], their main advantages are the self-adjustment of gains and the fact that only partial information about the vehicle’s mathematical model is required. However, the low rate of gains adjustment and the overestimation of feedback gains remains a drawback to this method. Continuing, the MRAC with the AW term can improve PID performance, as shown in the simulation results of study [16]. Nevertheless, the proposed methodology requires computing the Ricatti equation online, which could be difficult. Finally, the introduction of saturation functions in the gains of the PD controller improves its performance, as one can see in the experimental results of work [15]. However, this methodology is not robust enough to encompass large and persistent parameter uncertainties. Taking into account this drawback, in this paper, a nonlinear PID controller is proposed to overcome the shortcoming of the previous algorithm introduced in [15]. The main contributions of the actual work are summarized as follows:

  • (i)

    A whole range of nonlinear functions to improve the PID controller are proposed.

  • (ii)

    The stability analysis of the proposed controller is formalized based on Lyapunov design.

  • (iii)

    External disturbance rejection and robustness towards parameter uncertainties are demonstrated through real-time experiments.

  • (iv)

    Compared to previous work [15], the speed in the time-varying yaw trajectory is increased twice during experiments.

This paper is organized as follows: a description of the dynamic model of the underwater vehicle is given in Section 2. The proposed control technique is described in Section 3. Real-time experimental results for trajectory tracking of two DoF of the submarine are presented and, discussed in Section 4. Finally, some concluding remarks, and future work on the proposed controller are presented in Section 5.

Section snippets

Dynamic model

The dynamic model of underwater vehicles has been described in several references (see [3], [21], [22] for examples).

The dynamics of an underwater vehicle involves two frames of reference: the body-fixed frame and the earth-fixed frame (as illustrated in Fig. 1). Considering the generalized inertial forces, hydrodynamic effects, gravity and buoyancy contributions as well as the forces of actuators (i.e., thrusters), the dynamic model of an underwater vehicle in matrix form, using the SNAME

Proposed nonlinear PID controller

In this section, a nonlinear PID controller based on saturation functions with variable parameters is introduced. The design of the controller is focused on both setpoint regulation, as well as trajectory tracking. The stability analysis of the resulting closed-loop system for both cases is explained in detail. Let us consider the underwater vehicle mathematical model (2), and the following control lawτ=JT[Mη(η)η¨d+Cη(ν,η)η˙d+Dη(ν,η)η˙d+g(η)τPID]where the PID controller is defined as follows:τP

Real-Time experimental results

To demonstrate the feasibility and efficiency of our proposed control solution, we applied the control algorithm to Leonard (Fig. 3), an underwater vehicle developed at the LIRMM (CNRS/University of Montpellier, France). Leonard is a tethered underwater vehicle that measures 75 × 55 × 45 cm and weighs 28 kg. The propulsion system of this vehicle consists of six thrusters to obtain a fully actuated system.

The underwater robot is controlled by a laptop computer, with CPU Intel Core i7-3520M

Conclusion

In this paper, a decoupled nonlinear PID (NLPID) control has been developed for trajectory tracking control of an underwater vehicle. The NLPID controller has been improved through the introduction of a whole range of nonlinear functions to replace the constant feedback gains. Then, a Lyapunov design has been proposed to prove the stability of the closed-loop system. The main advantages of the proposed control law are: 1) it improves robustness with respect to classical PID controllers by

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgment

This work was supported by Conacyt, grant 490978. The Leonard underwater vehicle has been financed by the European Union (FEDER grant 49793) and the Region Occitanie (ARPE Pilot Plus project). The authors would like to express their gratitude to the anonymous reviewers for their comments for the improvement of the manuscript.

Jesus Guerrero received his Ph.D. degree in 2019 in automatic control from the Center for Research and Advanced Studies of the National Polytechnic Institute (CINVESTAV), Mexico. His research interests include nonlinear, adaptive and time-delay control and their applications in underactuated systems, ground, aerial, and underwater vehicles.

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    Jesus Guerrero received his Ph.D. degree in 2019 in automatic control from the Center for Research and Advanced Studies of the National Polytechnic Institute (CINVESTAV), Mexico. His research interests include nonlinear, adaptive and time-delay control and their applications in underactuated systems, ground, aerial, and underwater vehicles.

    Jorge Torres was born in Mexico City, on May 13, 1960. He received his Ph.D. degree in Automatic Control from LAG, INPG, France, in 1990. He joined the Department of Electrical Engineering at the CINVESTAV, Mexico, in 1990. His research interest lies in the structural approach of linear systems, stability of multivariate polynomials, and the control of bioprocess for waste water treatment and the control of mini-submarines.

    Vincent Creuze received his Ph.D. degree in 2002 in robotics from the University Montpellier 2, France. He is currently an associate professor at the University Montpellier 2, attached to the Robotics Department of the LIRMM (Montpellier Laboratory of Computer Science, Robotics, and Microelectronics). His research interests include design, modelling, and control of underwater robots, as well as underwater computer vision.

    Ahmed Chemori received his M.Sc. and Ph.D. degrees respectively in 2001 and 2005, both in automatic control from the Grenoble Institute of Technology. Then, he has been a Post-doctoral fellow for one year with the Automatic control laboratory of Grenoble. He is currently a tenured research scientist in Automatic control and Robotics at the Montpellier Laboratory of Informatics, Robotics, and Microelectronics. His research interests include nonlinear, adaptive and predictive control and their applications in humanoid robotics, underactuated systems, parallel robots, and underwater vehicles.

    Eduardo Campos received his B.S. degree in electromechanical engineering from the ITZ (Instituto Tecnolgico de Zacatepec) in 2008, and the M.S. degree in automatic control from the CINVESTAV (Centro de Investigación y de Estudios Avanzados del IPN), México, in 2010. He received his Ph.D. degree in 2014 from CINVESTAV and LIRMM (Laboratoire dInformatique, de Robotique et de Microlectronique de Montpellier). Currently he is working on the development of the AUV (Autonomous Underwater Vehicle) and artificial vision applications in underwater robots.

    This paper was recommended for publication by Associate Editor Dr. Y.-Q. Chen

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