Elsevier

Mechatronics

Volume 47, November 2017, Pages 49-66
Mechatronics

Nonlinear internal model control applied to VTOL multi-rotors UAV

https://doi.org/10.1016/j.mechatronics.2017.08.002Get rights and content

Abstract

This paper describes a novel Nonlinear Internal Model Control (NLIMC) approach. We show that if we change, in a new manner, the control based on flatness property in order to get a closed loop structure, we can raise the performance and the robustness level with regards to structured or unstructured disturbances. The control performance is evaluated through a Vertical Take-Off and Landing (VTOL) Unmanned Aerial Vehicle (UAV) where numerical simulations are performed demonstrating the effectiveness of this novel controller form. The obtained results are confirmed by series of experimental tests showing also the robustness level of this technique in presence of wind.

Introduction

Unmanned Aerial Vehicles (UAVs) are very popular systems. They are used by the military for surveillance, reconnaissance and rescue. However, the civilian sector has also given an important interest for UAVs. The multi-rotors UAV layout topology has been widely selected by many researchers as a very promising drone for indoor/outdoor manipulations. In the last years, many types, with several shapes of multi-rotors are introduced as for instance Hexa-Copters [1] and Octo-Copters [2]. Thus, they have begun to gain popularity in the commercial and academic sectors.

The multi-rotors are dynamically unstable, multi-variable systems and highly coupled, so a nonlinear control strategy should be used. Moreover, it is necessary to design a controller such that the system will be able to follow a 3D predefined trajectory with good performance. For this reason, a broad range of nonlinear control strategies have been proposed. Reference [3] proposed a nonlinear H controller and backstepping approach in order to stabilize the attitude and to solve the path-tracking problem of the quadrotor respectively. References [4] and [5] present a combined backstepping technique for the sake of stabilization, regulation and trajectory tracking. The reader may refer to monograph [6] for a large list of control strategies. However, these control techniques, initially designed in the ideal case, behave with less efficiency in the presence of some disturbances and may lead to unstable system.

To alleviate the above problem, robust control techniques have been proposed particularly in the presence of disturbances. In reference [7], a robust adaptive control is proposed for the attitude stabilization under external disturbances and validated by real-time implementation. Another adaptive controller for steering a quadrotor vehicle along a trajectory, while rejecting constant force disturbances is addressed in [8]. An Active Disturbance Rejection Control (ADRC) algorithm is introduced in [9] by means of extended state observer (ESO). A Model-Free Control (MFC) scheme inspired from the Passivity-Based Sliding Mode Control (PBSMC) is proposed in [10]. A revisited model free control is designed in [11]. Other approaches are applied as for instance direct adaptive sliding mode controller [12] and SMC-neural networks controller [13]. Overall, most of these published papers do not pay attention to the performance of control.

In fact, almost all the controllers explored in the previous paragraphs may fail to deal with the raised issues (i.e. control performance and robustness) jointly. Many controllers allow good set-point tracking but, for many processes control, disturbance rejection is much more significant. Hence, controller design that emphasizes disturbance rejection with a good set point tracking is a real design problem. Our focus is consequently pushed on this point of view.

The aim of this research work is to stabilize the vehicle while ensuring the tracking of prescribed 3D trajectories with good performance even in presence of disturbances. The idea is to employ a reference model based control technique in order to meet readily the desired specifications, and which lead to a simplified controller. Moreover, the controller has to ensure a good level of robustness against disturbances.

In our previous works, some reference model based control strategies were proposed. Immersion and Invariance (I&I) based control strategy is detailed in [14] to resolve the problem of under-actuation of VTOL multi-rotors. In reference [15], a damped time response with less consumed energy is ensured using the well-known Interconnection Damping Assignment (IDA) technique. Unfortunately, the solutions that have been given feature poor robustness capabilities.

Among the large variety of control techniques available in the literature, the well-known reference model based technique namely the so-called Internal Model Control (IMC), is popular in industrial process control applications due to its good trade-off between the control performance and disturbance rejection capability [16]. There are many approaches developed and based upon IMC principle as for instance: the adaptive IMC [17], Neural Network based IMC [18] and Support Vector Machine approximate based IMC (SVM-IMC) [19]. However, these controllers are designed to be effective with linear plants. Therefore, a modified form and extended Nonlinear IMC is herein applied by taking care of having an adequate control structure. Doing so, the technique uses the IMC basic principle to synthesize a nonlinear controller that is involved in an equivalent classic closed loop. The controller employs the flatness property instead of the classic dynamic inversion technique in order to overcome some limitations of the feedback linearization. The performance of the control is ensured by fixing a reference model for the tracking error dynamic.

Throughout this paper, a performance assessment is presented via results of several illustrations, scenarios and numerical simulations, with complementary comments of the proposed strategy of control with respect to other techniques. Particular attention is paid to the tracking accuracy and energy consumption of each control strategy considering some performance criteria, such as the Integral Squared Error (ISE) and the Integral Squared Control Input (ISCI).

This paper is structured as follows. The second section concerns the dynamics of the VTOL multi-rotors. Section 3 explains the design of a novel nonlinear control approach, namely Nonlinear Internal Model Control (NLIMC). This approach is applied on the multi-rotors UAV in Section 4 where a stability study is performed. In Section 5, the results from numerical simulations test the effectiveness of the proposed control strategy under different operating conditions and confirmed by experimentation tests for a quadrotor. The final section gives some conclusions.

  • Notations

The gradient vector of a mappingV:DxRnR is denoted as:V(x)=[V(x)x1V(x)xn]T where n is the dimension ofx=[x1xn]TDxRn. diag(x) denotes the square diagonal matrix where its diagonal is formed by the vector x. P=Tr(PTP) denotes the Frobenius norm of matrix P where Tr(.) is the trace of square matrix and (.)T denotes the transpose. |x|denotes the absolute value of scalar x.The spectrum eig(A)is the set of eigenvalues of square matrixARn×n. xmax  denotes the maximal scalar in vector x. ddt is a time derivative operator acts onwhith(ddt)ididti.

  • Definitions

Definition 1

Let f(x) be a locally Lipschitz function defined over a domainDxRn that contains the origin. Let V(x) be a continuously differentiable function such thatσ1xαV(x)σ2xαV˙(x)σ3xαfor all x ∈ Dx, where σ1, σ2, σ3 and α are positive constants. Then, the origin is an exponentially stable equilibrium point ofx˙=f(x). If the assumptions hold globally, the origin will be globally exponentially stable.

Section snippets

Multi-rotors dynamics background

This section contains a brief explanation of the classic process used to obtain the complete model, which describes the VTOL UAV’s in-flight behavior. As shown in Fig. 1, the system operates in two coordinate frames: the earth fixed frame R0(O, X, Y, Z) and the body fixed frame R1(O1, X1, Y1, Z1). Let χ=(x,y,z)TR3 represent the absolute position of the system and η=(φ,θ,ψ)Tare the Euler angles (roll, pitch, and yaw). Many dynamics for multi-rotors are presented as for instance [20], [21], etc.

General description

For the reader’s convenience, let us recall quite briefly the IMC basic principle for linear system, using Laplace variable. In Fig. 2, the input U(s) is introduced to both the process G(s) and its model G˜(s). D(s) is an unknown disturbance affecting the system. The output, Y(s) is compared with the output of the model Ym(s), resulting in a signal Y˜(s), which is compared to the reference signal Yr(s). That isY˜(s)=(G(s)G˜(s))U(s)+D(s)The closed loop expression is thenY(s)=(Yr(s)D(s))C(s)G(s)

VTOL multi-rotors application

Our goal is to allow the vehicle to track a reference trajectory described by the absolute position with a given yaw angle (xr, yr, zr, ψr). Obviously, from systems (4) and (5), the inputs uφ, uθ and uψ act directly on the rotational subsystem. However, they do not have any direct effect on the translational subsystem since they act through the attitude angles. Therefore, the autopilot is based on the decomposition into two sub-systems. The first is related to the position control, which

Results and discussion

In this section, we check the effectiveness of the proposed approach, applied on X-Shaped quadrotor, through numerical simulations and experimental results. The available parameters of the system are displayed in Table 1.

Conclusion

A novel way to design a NLIMC controller is described. It uses as basis the flatness property and by bringing some changes, it operates in closed loop form. This controller differs from the one that can be derived by using feedback linearization and the asymptotic output problem based approach. It raises the performance with respect to structured and unstructured uncertainties. This proposed controller has been applied to a multi-rotors model. Numerical simulations have been performed on the

Y. Bouzid is currently a Ph.D. student at IBISC Laboratory, Universite d Evry Val dEssonne, Universit e Paris Saclay in France. He received the Master degree in Automatic Control from the University Paris-Sud, France in 2015. His research interests include guidance, robust and nonlinear control, with application to autonomous vehicles.

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    Y. Bouzid is currently a Ph.D. student at IBISC Laboratory, Universite d Evry Val dEssonne, Universit e Paris Saclay in France. He received the Master degree in Automatic Control from the University Paris-Sud, France in 2015. His research interests include guidance, robust and nonlinear control, with application to autonomous vehicles.

    H. Siguerdidjane is currently full Professor and Deputy Dean of Research at CentraleSupelec, in charge of the projects related to the new University of Paris-Saclay. Her teaching activity at the Automatic Control Department and research interests at the Signals and Systems Laboratory, include linear and nonlinear control systems and applications to aerospace, mechanical and power systems problems. She is an Associate Editor of the IFAC Journal Control Engineering Practice (CEP) and for the actual triennial (2014–2017), she is the Vice-Chair of the IFAC Aerospace Technical Committee and nominated as member of the IFAC Policy Committee. Houria Siguerdidjane, obtained her Engineering degree from Supelec (now CentraleSup elec). She received the Doctorate degree in Automatic Control and Signal Processing in 1985 and the Habilitation degree in Physics Sciences from University Paris XI (in 1998). She held an AVH Fellowship for a post-doctoral position (in 1989) in the Department of Mathematics at Munich Technical University where her main project concerned the optimization of aircrafts trajectories. In 1994–1995, she was on sabbatical leave at the industrial company AlstomT&D, where her main interest was the application of new concepts to improve the relaying protection performance in high voltage electrical networks. In 2013, she was also Director Deputy of Research and Industrial Partnerships of Metz Campus. Prof. Siguerdidjane has been serving, as the Chair of the IFAC Aerospace Technical Committee from 2006 to 2014. She is recipient of the 2012 IFAC-France Award.

    Y. Bestaoui Professor at IBISC laboratory, UFR Sciences and Technologies, Universite dEvry, FRANCE. She get a Ph.D.in Control and Computer engineering from Ecole Nationale Superieure de M ecanique, Nantes, in 1989 (Currently Ecole Centrale de Nantes) and a Habilitation to Direct Research in Robotics, from University of Evry in 2000. She is the author of 3 books: Lighter than air robots, Springer, ISCA 58, 2012; Planning and decision making for aerial robots, Springer, ISCA 71, 2014; Smart Autonomous Aircraft, CRC Press, 2016. She has also written 10 book chapters, 30 journal papers, 90 fully refereed conference papers. She is the Coordinator of the Master SAAS : Smart Aerospace and Autonomous Systems (with Polytechnic University of Poznan (Poland). She is an AIAA Senior member, IEEE Senior member. From Jan. 2008 - Dec. 2012, she has been an Associate Editor of the IEEE Transactions on Control Systems Technology. She has also supervised 10 Ph.D.theses.

    This paper was recommended for publication by Associate Editor Dr Radhakant Padhi.

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