A probabilistic approach for designing nonlinear optimal robust tracking controllers for unmanned aerial vehicles

Highlights

• The subjects covered in the article, such as evolutionary computing, unmanned aerial vehicles, and their control forms attract a lot of interest nowadays.

• A extensive literature review presents several advantages for the proposed approach.

• The controller performance is notable even considering significant robustness requirements.

• Difficult topics, such as optimality, robustness, stability, nonlinearity, and correctness are addressed with relative simplicity.

Abstract

In this study, we propose a probabilistic approach for designing nonlinear optimal robust tracking controllers for unmanned aerial vehicles. The controller design is formulated in terms of a multi-objective optimization problem that is solved by using a bio-inspired optimization algorithm, offering high likelihood of finding an optimal or near-optimal global solution. The process of tuning the controller minimizes differences between system outputs and optimal specifications given in terms of rising time, overshoot and steady-state error, and the controller succeed in fitting the performance requirements even considering parametric uncertainties and the nonlinearities of the aircraft. The stability of the controller is proved for the nominal case and its robustness is carefully verified by means of Monte Carlo simulations.

Introduction

Unmanned aerial vehicles and their control forms are an actual and important research subject. This technology can be used to solve many social problems, and this explains the great interest that the area has received from different research groups and organizations throughout the world. Examples of research related to unmanned aerial vehicles are path planning [27], [52], squadrons formation reconfiguration [16], wireless networks [28], autopilot design [5], multiple unmanned aircraft coordination [30], formation [22], task assignment [35], trajectories generation [34], patrolling or surveillance [2], searching [11], tracking [48], flight control [24] and source seeking [53].

Despite using unmanned aerial vehicles, the aircraft has to be capable of following references, which are commands that determine the motion of the aircraft. These references can be created by defining trajectories [20] that an unmanned aerial vehicle has to follow. When trajectories are defined, performance requirements can be established by directly considering these trajectories. For instance, an aircraft controller can be designed so that the aircraft is able to follow a trajectory with optimal values for rising time, overshoot and steady-state error for its controlled variables.

In this work, we are interested in developing an unmanned aerial vehicle that is capable of following a set of trajectories such as those illustrated in Fig. 1. These trajectories exhibit interesting properties. For instance, the trajectories shown in Fig. 1a and b focus on testing latitudinal and longitudinal movements, while angle of heading is considerably modified in both cases. The trajectory shown in Fig. 1c illustrates a movement that emphasizes variations for height and angle of attack, while control of velocity is prioritized in Fig. 1d. As a consequence, all state variables of the dynamic model are substantially stressed in one way or another. Besides, optimal values for rising time, overshoot and steady-state error are desired features of response for the cases considered. There is a challenging problem to determine optimal values for these metrics by taking into account all trajectories, considering nonlinearities of the aircraft dynamic model, stability assurance, and parametric uncertainties of the plant. For example, the task of analytically designing a controller by taking into account all worst-case parametric uncertainties of the aircraft dynamic model becomes very hard, since the problem is NP-hard [49].

The problem considered in this study can be posed as follows: Design a tracking controller for unmanned aerial vehicles by taking into account the set of trajectories shown in Fig. 1. The performance of the controller for each aircraft controlled variable must be global near-optimal in terms of a weighted average among rising time, overshoot and steady-state error for all trajectories illustrated in Fig. 1, and it must be capable of dealing with nonlinearities of the aircraft dynamic model even considering parametric uncertainties of the plant. The stability must be proved, and the robustness has to be carefully analyzed.

Aircraft dynamics are typically nonlinear, and there are many ways of designing tracking controllers for nonlinear systems. Linearization is one of the possible alternatives. The linearized controller is designed for several points of operation, and a gain schedule is used to complete interpolation between each pair of these points. Although this technique has been broadly used, it exhibits some problems. For instance, it requires a lot of work because several designs must be completed for each point of operation [49], while scheduling the gains from point to point during regime operation is a difficult task [17]. The stability cannot be formally proved when considering this approach [40]. On the other hand, the proposed approach considerably simplifies the design because it is based on dynamic inversion, which acts as a universal gain scheduler [39], whose nonlinearities are canceled without approximations, thereby enabling its proof of stability by using linear control theory.

Several approaches have also been proposed for dealing with robust path tracking controllers for nonlinear systems. Common approaches along this line of research are backstepping [31], adaptive control [47], model predictive control [37], H-2/H-∞ control [32], Lyapunov's direct method [18], sliding mode control [50], and linear quadratic Gaussian control [41].

With regard to the backstepping approach, it is used in [31] for designing a tracking controller for underactuated unmanned surface vessels. Although good dynamic performance and robustness can be achieved by using this approach, it has some drawbacks. For instance, repeated differentiations and recursive design [7] can significantly increase the complexity of the approach and make it difficult to apply to multiple state control systems [13]. Otherwise, the proposed probabilistic approach does not present such complexity because the gains of the controller are determined by using an optimization algorithm capable of solving non-convex and multi-objective optimization problems, whose fact significantly simplifies the design.

In [47], adaptive control is used for tracking nonholonomic mechanical systems. By using this approach, it is possible to asymptotically track the desired trajectory and the tracking error can be bounded within a controllable bound. However, this approach is criticized because the robustness of its transient response cannot be ensured [12], and it is quite sensitive to changeable uncertainties of the system [29]. Besides, adaptive control requires quite comprehensive theoretical background [8]. These are some reasons why adaptive control has still been subject of research. On the other hand, the proof of stability of the proposed approach is made by using linear control theory, which is simple and well-known. Besides, the design is formulated in terms of a multi-objective optimization problem, and this helps to reduce the conservatism by using a bio-inspired optimization algorithm that has very high likelihood of finding global optimal or near-optimal results for problems that can even be non-convex.

Model predictive control is another relevant control design. An example can be found in [37], where this approach is used to control an autonomous ground vehicle that has to track trajectories. Considering advantages [37], model predictive control is capable of dealing with performance criteria, and it is able to generate trajectories that are optimal according to these criteria. Besides, constraints can be explicitly considered and tuning of the parameters can be intuitively performed. However, model predictive control depends on the precise knowledge of the system parameters [36], and it may require significant computational effort to be implemented [36], whose fact may be prohibitive for systems that require fast sampling rates [1]. As a consequence, model predictive control can require some special treatment in order to avoid these problems. Otherwise, the proposed approach is not so dependent on the precise knowledge of the system parameters because the designing is performed by taking into account a significant number of parametric uncertainties of the plant. Besides, the gains of the controller do not change during operation, thereby reducing the required computational effort, and making the proposed approach interesting for real-time implementations.

In [32], robust H-2/H-∞ control is used to design an unmanned helicopter controller for tracking trajectories. As some benefits [32], robust H-2/H-∞ approach can run very fast in modern embedded systems, and its stability and performance criteria can be ensured even considering disturbances. Nevertheless, H-∞ approaches have been criticized for some reasons. Although weighting functions are crucial in order that H-∞ controllers reach some of their control objectives, there is no direct way of choosing weighting functions and some trial and error iterations may be necessary to determine them [51]. In addition to this fact, the order of H-∞ controllers is usually high [33]. This helps to explain why robust H-2/H-∞ control has still been subject of research. In contrast, the proposed approach does not present any of these problems. Although the optimization algorithm used in this study can require little adjustments while tuning the controller, it does not need trial and error iterations, and all analytical expressions can be directly determined since dynamic inversion is applicable. Besides, dynamic inversion does not result in high order controllers.

Another approach to solve the problem is the Lyapunov's method, as in [18], where a micro aerial vehicle is capable of tracking roads by using a control switching mechanism. Lyapunov's approach play an important role in nonlinear systems control theory because it is able to readily ensure the closed-loop system stability [10], without solving ordinary differential equations [23], thereby reducing its dependence on the problem. Although this fact makes Lyapunov's approach applicable to the solution of a wide range of linear and nonlinear problems, one of its main drawbacks is the practical difficulty in finding a Lyapunov's function [43]. In contrast, the proposed approach is straightforward since dynamic inversion is applicable.

As examples of sliding mode control, unmanned aerial vehicles track trajectories in [50] by using this approach, which has important advantages, such as stability, invariance, robustness, and good precision [42]. However, the main drawback of sliding mode control is the chattering phenomenon [3], which is common and difficult to avoid or attenuate [45]. Chattering is inevitable in sliding mode control when control is discontinuous [9], and special treatment should be given to reduce or to avoid it [45]. However, as control is continuous in the proposed approach, chattering is very rare or does not occur when this approach is used.

According to linear quadratic Gaussian (LQG) control, matrices R and G must typically be determined before optimizing the control so that a given performance criterion is indirectly specified by them. As advantages, linear quadratic approaches result in constant gains, which exhibit some robustness against model uncertainties [6], [41]. However, a usual drawback of LQG approaches is the fact that it is not quite intuitive to specify performance by means of matrices R, and G. Otherwise, performance metrics considered in this study are relatively intuitive because they are specified in terms of the desired output of the system.

A probabilistic robust method is used by Wang and Stengel [49] for designing an aircraft controller whose model is quite nonlinear and considerably different from the one considered here. As the proposed approach, the method of Wang and Stengel [49] is based on dynamic inversion and probabilistic analysis. Although dynamic inversion is quite dependent on the precise knowledge of the model [19], and cannot deal with parametric uncertainties of the plant by itself [38], Wang and Stengel [49] mitigated these drawbacks by considering probabilistic analysis during the design phase, thereby enlarging the range of applications where dynamic inversion can be applied. Despite the similarity with the work of Wang and Stengel [49], the approach proposed in this study has significant differences, such as: (a) instead of minimizing the likelihood of having unacceptable stability or performance when considering parametric uncertainties of possibly manned aerial vehicles [49], the proposed approach focus on minimizing the norm between optimal specifications and desired outputs for controlled variables of unmanned aerial vehicles by taking into account parametric uncertainties of the plant, (b) instead of using genetic algorithms [49], the proposed approach uses a hybrid partitioned-population optimization algorithm (HPPOA) [21], which has very high likelihood of finding optimal or near-optimal global solutions for functions with noisy parameters, and (c) instead of using the multi-objective optimization approach called scalarization, which defines a weighted average among several optimization criteria, the proposed approach optimizes in a multi-objective multi-level fashion, which can better deal with solution candidates that have different components but are equal in module [15].

The main contributions of this work are as follows:

• A probabilistic approach for designing nonlinear optimal robust path tracking controllers for unmanned aerial vehicles, which does not present any of the problems previously described above, such as chattering, gain scheduling, trial and error iterations, conservatism, and real-time limitations. Besides, the optimization process exhibits very high likelihood of finding optimal or near-optimal global solutions and can better deal with multi-objective solution candidates that have different components but are equal in module.

• A nonlinear near-optimal robust path tracking controller for unmanned aerial vehicles, resulted from the proposed approach. It is capable of dealing with the tracking problem, where the process of tuning the controller minimizes differences between system outputs and optimal specifications given in terms of rising time, overshoot and steady-state error, by considering all trajectories shown in Fig. 1, and by taking into account parametric uncertainties of the plant. The stability is proved for the nominal case and the robustness is carefully verified by means of Monte Carlo simulations.

This paper is organized as follows: Section 2 focuses on the idea of the probabilistic approach, where the fundamentals of the proposed approach are explained. Section 3 presents a proof of stability, where the stability of the controller is proved for the nominal case. Section 4 covers simulation results, where optimal specifications and near-optimal robust results are graphically compared. Section 5 covers robustness analysis, where robustness against parametric uncertainties of the plant is verified by means of Monte Carlo simulations, and main results and conclusions are included in Section 6.