Fast finite-time backstepping controller for a quadrotor UAV under state constraints

. Quadrotors have gained popularity in a wide range of applications. In this paper, a new approach for solving the tracking control problem of quadrotors with full-state constraints is presented. The proposed method involves a backstepping control scheme integrated with a fast finite-time filter. First, necessary state transformations are performed to support the design of the finite-time filter and controller. Next, the controller is formulated based on the backstepping technique. All the state constraints are taken into consideration in the controller. However, it is well-known that the backstepping control design can lead to the “explosion of complexity” when calculating time derivatives of certain nonlinear functions. Therefore, the proposed filter comes to provide a solution for estimating the time derivatives with the estimation errors converging to zero in finite time. The closed-loop system's finite-time stability is rigorously proved using the Lyapunov theory, despite the state constraints. Simulation results demonstrate the feasibility and efficacy of the proposed method.


INTRODUCTION
Quadrotor unmanned aerial vehicles (UAVs) have become a subject of great interest in the research community, primarily because of their unique capabilities, such as vertical taking off and landing, a wide range of flight capabilities, from hovering to cruising, the ability to fly at low altitudes, and their agility in tightly constrained environments [1], [2].In practical applications, it is often necessary for these quadrotors to accurately track a given trajectory in a timely manner.However, due to their coupled nonlinear dynamics, parameterized uncertainties, and external disturbances [3], controlling quadrotors remains a challenging task.
Linear control methods, including the proportional-integral-derivative (PID) technique [4], linear quadratic regulation (LQR) [5], and robust H-infinity control [6], have been widely used in quadrotor control system design and have advantages in project realization.However, their performance degrades when quadrotors leave their designed trim points or perform aggressive maneuvers.To address these shortcomings, various nonlinear control approaches have been introduced, such as dynamic inverse [7], backstepping (BS) [8], model predictive control [9], singular perturbation theory [10], and sliding mode (SM) control [11].Recursive BS, which is a powerful Lyapunov-based tool, is typically used as a baseline controller for quadrotors due to the cascaded structure of their dynamics.The BS framework provides two significant benefits: 1) it accommodates nonlinearities and avoids wasteful cancelations [12], and 2) it allows several flight modes, such as position hold, loiter, and stabilized flight modes, to be integrated into the vehicle's control system due to the hierarchical scheme of BS.
The traditional BS method is known to have several issues.One such problem is the "explosion of complexity" caused by repeated differentiation of certain nonlinear functions, which is especially evident in high-order nonlinear systems.Moreover, traditional BS lacks a guarantee of robustness against perturbations.To address these limitations, researchers have proposed several adaptive control techniques for quadrotors.For instance, in [13], a radial basis function neural approximator was introduced to estimate and compensate for perturbations.Meanwhile, an immersion and invariance-based adaptive controller was developed in [14] for quadrotors that can handle uncertain inertial parameters, albeit in a relatively slow and indirect manner.More recently, the work in [15] proposed an active disturbance rejection control that employs a disturbance observer (DO) to estimate and compensate for lumped unknown disturbances in real-time.This approach has been further refined in [16], which combines command filtering with a DO to accurately track virtual control signals and attenuate the effect of disturbing forces and moments.To achieve highly precise and fast tracking, under the presence of uncertainties and disturbances, the use of dynamic surface control and an extended state observer within a DO-based control framework was presented in [17].Overall, the application of these adaptive techniques can help achieve accurate tracking for quadrotors.
Our work aims to enhance the performance of the BS control by introducing finite-time convergence properties.Finite-time controllers provide a faster transient response and higher precision compared to asymptotic stability controllers when states are close to the equilibrium point [18].We apply fraction powers of the tracking errors to achieve finite-time convergence and employ finite-time filters to estimate the time derivatives of virtual control inputs, thereby avoiding the "explosion of complexity" phenomenon.Simulation results demonstrate the effectiveness of the finite-time controller, which, to the best of our knowledge, has not been tested for quadrotor tracking control before.

Our work presents several contributions:
 First, we propose a multivariable composite finite-time BS framework that introduces fractional powers of tracking errors to improve convergence near the trim point.We estimate derivatives of virtual control inputs (virtual commands) using finite-time filters to avoid the "explosion of complexity." Second, we solve the tracking control problem of quadrotors that are subject to state constraints using our proposed finite-time BS.Our method guarantees finite-time convergence of tracking errors. Third, we prove the finite-time stability of the closed-loop system based on the finitetime Lyapunov theory.We demonstrate the effectiveness of our proposed control law through numerical simulations.
The organization of this paper is as follows.In Section 2, the dynamic model of the quadrotor is presented, and the control problem is defined.Section 3 introduces the finite-time controller design and provides a stability analysis of the closed-loop system.Simulation results are presented and discussed in Section 4. Finally, Section 5 concludes the paper.

Quadrotor dynamics model
In this section, we provide a brief summary of the quadrotor dynamics model, which was previously presented in many existing studies [1 -4].

Let ,
, and  denote three Euler angles roll, pitch, and yaw, respectively, in an earthfixed frame {E} (Figure 1 (2) where,   ,, denotes the control inputs which are generated by the quadrotor's motor thrust forces, () where, It is seen in ( 4) that the dynamics of the roll, pitch, and yaw angles are in similar form to each other.Therefore, the following representative system (6) will be considered, and the control design and verification process of this representative system can be applied directly to that of the roll, pitch, and yaw angle dynamics.where, 1 ,, ; and i y is the output.Here, it is worth noting that all the states suffer from state constraints, that is: Fast finite-time backstepping controller for a quadrotor UAV under state constraints For a typical quadrotor, the values of the upper and lower limits of the attitude are 11 kk  for the yaw angle.

Control Objective
The overall control scheme consists of position and attitude controllers (Figure 2).The quadrotor tracking control problem involves position tracking and attitude tracking.However, attitude tracking control plays a key role in improving the whole system's tracking performance due to two major reasons.First, as can be seen in Figure 2, the position controller generates the desired roll and pitch angles which can only be tracked if the attitude tracking controllers are designed properly.Second, every unexpected change in the system states can be quickly compensated by the attitude tracking controller as the attitude dynamics is much faster than the position dynamics.
From the above observation, this paper aims to propose a novel fast finite-time backstepping attitude control algorithm to drive the quadrotor, under state constraints (7), to track the desired attitude ,,  The following assumption, lemmas, and proposition are used in our control design and stability analysis, which will be presented in Section 3.   . Lemma 2. [20] (Finite-Time Uniformly Ultimately Boundedness): Consider the following system.
Then, the state () xt is uniformly ultimately finite-time stable.It means that the state can converge to a region of the equilibrium point in a finite time.
 .Then, the system is stable in a finite-time, and the convergence time c T satisfies:  and tunable parameters 12 ,0  , there always exist positive numbers 1  and 2  such that 12 22 12

METHODOLOGY
The control system is designed in this section by integrating a fast finite-time filter into the backstepping control technique.The filter aims to circumvent the "explosion of complexity" problem.Before moving on to the design, some state transformations are conducted.For the sake of simplicity in control design and closed-loop system's stability analysis, the following state transformations are performed.

State Transformations
and where, ) With the above state transformations, the tracking errors are defined as follows.
and where, 0 and 1 ˆi v is the filtered signal provided by the filter, which will be described in next subsection.

Fast Finite-time Backstepping Controller
Step 1. From ( 6), ( 8), ( 9) and ( 10), the time derivative of 1i e can be obtained as: A virtual control input, 1i v , for ( 12) is designed as: , ii Then, the derivative of 2i e is where, 1 ˆi v denotes the estimated value of 1i v , which is obtained through the following finite-time filter to avoid the "explosion of complexity" in the calculation of 1i v .
with i  being the filter parameter, sig ( ) sgn( )

, and 3i
 is defined as where, is the estimate error.Thus, we have Then, from (12), (13), and (15), we have Step 2. From ( 6) and ( 14), we have The actual control input is designed as Substituting (20)

into (19) yields
Theorem 1, Consider the quadrotor system (6) with full states being constrained as in (7), for a given time-varying desired attitude id y , under the Assumption 1, the tracking errors 1i e and 2i e are ultimately bounded and asymptotically approach the origin in finite-time if the control input i u is designed as in (20) and the virtual control law is formed as in (13) with the use of the filter in (15).
Proof of Theorem 1.
The inequality (28) indicates all the errors, including 12 ,, ii ee and fi e , are uniformly ultimately bounded (Lemma 2) and asymptotically approach the very small region close to the origin.This completes the proof.
Remark 1.The filter in (15) contributes to avoiding the "explosion of complexity" phenomenon in calculating the time derivative, 1 ˆi v , of the virtual control input 1i v in (13), which is required to compute the actual control input i u in (20).
Remark 2. From (28), according to Lemma 3, it can be concluded that all the system states converge to the desired values in finite time c T , which is bounded as follows.
Remark 3. The scheme of the attitude controller is summarized in the following block diagram (Figure 3).  , with 0i v being calculated from the desired attitude (bounded), and 1 ˆi v being the bounded output of the filter (15).Therefore, the denominators in (8), i.e., , do not reach zeros.In other words, 1i x does not reach its limits, i.e., 11i k  and 12i k ; and 2i x does not reach its limits, i.e., 21i k  and 22i k .Hence, that is the capability of our proposed method of dealing with the constraints.
Remark 5.The controllers and the filter in ( 13), (20), and ( 15) are designed based on model (4), which is obtained from (1) through a linearization near the hovering point, i.e., 0   and 0   .Therefore, the quadrotor should be operated in an attitude range near this point to ensure the proposed controller functions properly.The further the quadrotor attitude is away from the hovering point, the more the control performance is degraded.

SIMULATION RESULTS AND DISCUSSIONS
This section demonstrates how our proposed method is effective in the tracking control of a quadrotor subject to full-state constraints.It is worth noting that, while the controller was designed based on (4), we utilized the dynamic model in (1) and the kinematic model in (2) to assess the closed-loop system's stability and confirm that our controller delivers satisfactory tracking control performance.

Simulation Assumptions
The simulation was conducted assuming that the quadrotor's attitude is determined by an inertial navigation system (INS).The controllers were implemented in Matlab/Simulink with a sampling time of 0.0025 s, which corresponds to an operating frequency of 400 Hz.The quadrotor parameters and its state constraints used in the simulation (Table 1) were collected based on the parameters of a real F450 quadrotor platform [22], which is equipped with four 2312E motors and four 9450 propellers [23].Table 2 lists the controller gains, initial conditions, and desired state values.By selecting fast, time-varying desired state values (Table 2), we illustrate the accuracy and rapidness of our new controller to track complex and fast-varying desired trajectories.

Simulation Results and Discussions
To evaluate the effectiveness of the proposed method, two test scenarios were conducted, namely the tracking of (i) step and (ii) complex and time-varying attitude references.The system performance corresponding to each scenario is examined and discussed in the following subsections.

Tracking of step attitude references
In this scenario, the desired attitude is set as [ 0.3, 0.3, 0.1]  rad (Table 2).To generalize this test case, the quadrotor's initial states are set to non-zero, that is, the tracking flight is triggered at a non-hovering status.At 0 t  s, the quadrotor's attitude tracking errors rapidly start converging to zero (Figure 4a).Simultaneously, the angular rate errors also see a quick convergence (Figure 4b).It is seen that the settling time (2%) all attitude angle tracking is about 2.5 seconds.Even though having a rapid convergence speed, the proposed controller does not require much control effort.As per Figure 4c, all the control inputs do not exceed 1.0 N.m, which is already a pretty small value of control torque for a quadrotor like F450.Besides, to have such a fast response, a backstepping controller usually results in the chattering phenomenon in the control input.However, our controller input can be seen as chattering-free.Examining the filter's performance, as expected, the proposed filter provides a highly accurate and fast-converged estimate which is proved through the performance of the filter estimate errors (Figure 4d).Another aspect of being validated is the state constraints (the dashed black lines in Figures 5a-b).It is clear that all states lie in the range of the constraints while exhibiting fast responses.The second scenario is conducted to ensure the capability of the proposed controller in tracking time-varying references, which are usually fast and complex in most real-world quadrotor missions.As per Table 2, each desired attitude angle is composed of sine waves at different frequencies and amplitudes, which makes it complex and time-varying.

Tracking of complex and time-varying attitude references
Even though the references vary in a relatively large range of amplitudes, the controller steers the quadrotor to track them rapidly (Figures 6a-b).Depending on the references, the attitude and angular rate tracking errors can converge to zero slightly faster or slower than each other.However, it is seen (Figure 6) that the settling time (2%) does not exceed 2.5 seconds.Compared to the previous scenario, the control inputs see marked increases (Figure 6c).These changes are due to the system requiring larger torques to be able to track the fast-varying references.Nevertheless, the control inputs remain chattering-free.Meanwhile, the filter still works effectively, as its estimation errors take a very short time to reach zero (Figure 6d).This timely estimation allows the backstepping controller properly delivers a fast system response and maintains the system's states not to reach the constraints (Figure 7).

CONCLUSIONS
This paper addresses the tracking control problem of a quadrotor subject to full-state constraints.By integrating a backstepping control scheme with a fast finite-time filter, the advantages of the backstepping control technique, including rapid response, simple design, and straightforward implementation, are taken while its most significant shortcoming, the "explosion of complexity," is overcome.The simulation results demonstrate the feasibility and effectiveness of the proposed method in tracking step attitude references and complex time-varying attitude references.The rapid convergence speed and low control effort demand of the controller indicate its applicability in many real-world quadrotor missions.Future work is dedicated to implementing this method into an actual quadrotor and the demonstration in actual flight conditions, in which uncertainties and external disturbances will be considered.

Figure 1 .
Figure 1.Quadrotor configuration.Motors 1 and 3 rotate counterclockwise while the other two motors rotate clockwise.
drag coefficients, respectively; l the arm length of the quadrotor.

Figure 2 .
Figure 2. Full quadrotor controller scheme.The proposed algorithm is applied to the attitude controllers, which play key roles in the system's tracking control performance.

Assumption 1 .
The desired attitude id y and its time derivative id y are continuous and bounded.Lemma 1. [19] Let 12 , , . . . ,q      , if p > 0, then the following holds.

Remark 4 .
With the proposed controller, it is proved that the tracking errors, 1i e and 2i e , converge to zeros in finite-time.From the definitions of the tracking errors in(10) and (11), we have 10 ii v   and 21 îi

Figure 4 .
Figure 4. Flight performance of the quadrotor tracking step attitude references.a) The attitude tracking errors rapidly converge to zero without any overshoot; b) The angular rate tracking errors also exhibit fast convergence; c) The controller requires small control effort; d) the filter delivers fast and accurate estimations.

Figure 5 .
Figure 5. States of the quadrotor tracking step attitude references.The quadrotor's attitude angles (a) and angular rates (b) all lie in the range of the constraints despite having a rapid response.

Figure 6 .
Figure 6.Flight performance of the quadrotor tracking complex and time-varying attitude references.a) The attitude tracking errors rapidly converge to zero; b) The angular rate tracking errors also exhibit fast convergence; c) The controller requires more control effort, but the control inputs are still chatteringfree; d) the filter delivers fast and accurate estimations.

Figure 7 .
Figure 7. States of the quadrotor tracking complex and time-varying attitude references.The quadrotor's attitude angles (a) and angular rates (b) all lie in the range of the constraints despite having a rapid response.

Table 2 .
Desired and initial attitude and parameters of the filters and controllers.