Nonsingular Fast Terminal Sliding Mode Control Based on Nonlinear Disturbance Observer for a Quadrotor

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1.College of Automation & College of Artificial Intelligence，Nanjing University of Posts and Telecommunications，Nanjing 210023，P.R.China；

2.Jiangsu Engineering Lab for IOT Intelligent Robots，Nanjing University of Posts and Telecommunications，Nanjing 210023，P.R.China；

3.School of Electrical and Automation Engineering，Nanjing Normal University，Nanjing 210023，P.R.China；

4.Nanjing Zhongke Raycham Laser Technology Co.，Ltd.，Nanjing 210023，P.R.China

Abstract: Given external disturbances and system uncertainties，a nonsingular fast terminal sliding mode control（NFTSMC）method integrated a nonlinear disturbance observer（NDO）is put forward for quadrotor aircraft.First，a NDO is proposed to estimate the actual values of uncertainties and disturbances.Second，the NFTSM controller based on the reaching law is designed for the attitude subsystem（inner loop），and the control strategy can ensure Euler angles’fast convergence and stability of the attitude subsystem.Moreover，the NFTSMC strategy combined with backstepping is proposed for the position subsystem（outer loop），which can ensure subsystem tracking performance.Finally，comparative simulations show the trajectory tracking performance of the proposed method is superior to that of the traditional sliding mode control（SMC）and the SM integral backstepping control under uncertainties and disturbances.

Key words：quadrotor aircraft；nonlinear disturbance observer（NDO）；nonsingular fast terminal sliding mode control（NFTSMC）；disturbances

0 Introduction

During the past decade，quadrotor unmanned aerial vehicles（UAVs）have attracted extensive research interest of many scientists all over the world.Compared with conventional aircraft，the quadrotors have many unique advantages，such as low cost，easy operation，hover control and so on.Therefore，the quadrotors are used to accomplish various tasks，such as military surveillance，agricultural investigation，and payload transport［1-4］.In practice，they are susceptible to external disturbances，which is easy to cause a relatively large discrepancy in attitude calculations.Consequently，the effective control is bound to decrease.Therefore，the influence of external disturbances on flight control stability should be concerned and solved in the design of controllers.

Various advanced control methods are proposed not only to stabilize the control system but also to improve system tracking performance，such as backstepping control［5］，sliding mode control（SMC）［6-8］，adaptive control［9］，fuzzy control［10］，etc.As we all know，SMC is regarded as an effective technique that results in a high degree of robustness to tackle system uncertainties and external disturbances.Mu et al.［11］studied an integral SMC to weaken the impact on the bounded external disturbances and model uncertainties.Wang et al.［12］used a terminal SMC strategy for controlling the quadcopter rotation and translation subsystem considering external distur-bances.A hierarchical SMC strategy was proposed to copy with continuous disturbance and parameter uncertainty by Li et al［13］.Zhang et al.［14］designed an improved robust controller for the sliding mode and the controller not only stabilized the quadcopter system，but also performed well against time-varying disturbances.

However，the above approaches only utilize the robustness of SMC to tackle system uncertainties and unknown external disturbances.To improve robustness，an observer is usually used to suppress and compensate for unknown system uncertainties and external disturbances.For example，Nuradeen et al.［15］used a method—combining SMC with a robust disturbance observer to stabilize the system and track the command trajectory.Additionally，Shi et al.［16］proposed an attenuation-control strategy based on a generalized extended state observer，which can meet the high-precision control target.Benallegue et al.［17］presented a high-order SMC based on disturbance observer strategy that is capable of dealing the effect of external disturbances.

Further，the backstepping technology is excellent for dealing with under-actuated nonlinear systems.Due to the under-actuated subsystem in the quadrotor system，the backstepping technology is widely adopted in the design of controllers.Chen et al.［18］applied the backstepping technology to the quadrotor position loop.Shao et al.［19］studied a stable trajectory tracking strategy combining the extended state and the backstepping technology to tackle the quadrotor system involving external disturbances and parameter uncertainties.Djamel et al.［20］designed an optimal control method for a quadrotor’s attitude control based on the backstepping and nonlinear approach.

Inspired by the above approaches，we propose a nonsingular fast terminal sliding mode control（NFTSMC）combined with a nonlinear disturbance observe（NDO）for a quadrotor subjected to system uncertainties and external disturbances.The main innovations of this paper are summarized as follows：

（1）The designed observer can efficiently estimate the actual values of compound unknown disturbances.Compared with Ref.［21］，it does not require the condition that the time derivative of the disturbance needs to be close to zero.

（2）Compared with the sliding mode control integral backstepping（IBS-SMC）［22］，the classical SMC［23］，the proposed NFTSM controller shows great advantages on a quadrotor system，such as accurate tracking performance，fast convergence and ability of avoiding the singularity problems.

（3）The robustness of a quadrotor system is strong by introducing NDO design.Meanwhile，Gaussian white noises can be effectively suppressed.

The rest of the paper is organized as follows.The disturbed model of quadrotors is introduced in Section 1.Then，NDO is used to estimate the uncertainties and disturbances in Section 2.Further，a backstepping NFTSMC based on the reaching law for the attitude and the position subsystems is constructed in Section 3.Some contrast results are shown in Section 4.Finally，some conclusions are drawn in Section 5.

1 Problem Formulation

1.1 Dynamic model of quadrotors

The structure of a quadrotor is illustrated in Fig.1.Let us employ two reference frames：The axes of the inertial coordinate are denoted as(Oe，xe，ye，ze) and the axes of the body coordinate are denoted as (Ob，xb，yb，zb).The vectorη=[x，y，z]Tdenotes the position subsystem in the inertial coordinate.The vectorξ=[?，θ，ψ]Texpresses Euler angles in the inertial coordinate and the three Euler angles are roll (-π 2 ＜?＜π 2 )rad，pitch( -π 2 ＜θ＜π 2)rad and yaw( -π ＜ψ＜π)rad.The vectorω=[p，q，r]Tdenotes the angular velocities in the body coordinate，that is，roll，pitch，and yaw，respectively.

Fig.1 A quadrotor’s structure

The dynamic model of a quadrotor is described in the form［19］

wheregis the gravitational force；U1，U2，andU3represent the control input for the three attitude angle channels，U4is the control input for the position subsystem，andU1，U2，U3，U4satisfy the relationships as follows

whereΩi(i=1，2，3，4) is the speed of theith rotor；I=diag(Ix，Iy，Iz) denotes the moment of the inertial.ai(i=1，…，9) in Eq.（1）is some known constants，and their detailed expressions are as follows

wheremis the total mass of a quadrotor；J?，Jθ，Jψ，Jx，JyandJzare corresponding drag coefficients.

1.2 Disturbed quadrotor dynamic model

Considering system’s uncertainties and external disturbances as an integration item，we namedi，dj.Eq.（1）can be decomposed into the following forms

wherei=1，2，3；j=4，5，6.

f(X2i) andf(X2j) are nonlinear terms，and the concrete expressions are as follows

Assumption 1For each subsystem， the lumped uncertaintiesdi djare not measurable and unknown，and differentiable with bounded derivatives，i.e.

For some positive constantsDi，Dj，i=1，2，3，j=4，5，6.

Remark 1The disturbance for the system cannot be infinite，which means that the disturbance is bounded.It is unreasonable to only consider the constant disturbance in the system，that is，the derivative of the disturbance converges to zero，so the bounded derivative of the external disturbance is logical and reasonable［15］.

1.3 Control strategy

The main control objectives of this paper are to design a control strategy combining NFTSMC and NDO，which can ensure the stability and tracking performance of a quadrotor system subjected to system uncertainties and external disturbances.A NDO is utilized to estimate lumped uncertainty，which compensate for lumped uncertainty.Then，via dividing a quadrotor system into two subsystems：Attitude and position，the controllers of the two subsystems are given.Based on that，the precise tracking and stability of the system are realized.The control diagram is shown in Fig.2.

Fig.2 Control block diagram

2 Nonlinear Disturbance Observer Design

In this section，a NDO is designed to estimate the actual value of the lumped uncertainty.For both subsystems of the position and the attitude，NDO can take the same form

whereZi，Zjare the states of NDO；Li，Ljthe observer gains；andthe estimation of lumped uncertainties.

Define the estimation errors of NDO as

Substituting Eqs.（2—4）into the derivative of，we obtain

Then，the derivative of

Hence，we obtain

Lemma 1For smooth multivariate dynamic systems，Vis assumed to be a strictly positive and continuously differentiable Lyapunov function.LetC?Rnbe an arbitrarily given set of initial conditions for a connected and compact dynamic system.Further，suppose that along the trajectory system of any system，x∶R+→Rn，starting atC，differential inequality

for allt＞0 withx(0)∈Cis satisfied withω＞0，a fixed positive constant，and?，a positive and tunable parameter.In a sufficiently large time，all trajectories of the dynamic system are bounded starting inCfor a sufficiently large time［15］.

Theorem 1If the system（2）satisfies Assumption 1，the observer gainsLi，Ljare selected as appropriate positive numbers.The disturbance estimationsof Eq.（3）can asymptotically track the actual lumped uncertaintiesdi，djof the system［21］.

ProofSelect the first Lyapunov candidate function as follows Taking the derivative of Eq.（8），we obatain

Similarly，one obtains

According to Lemma 1，the proof of Theorem 1 has been finished.

3 Controller Design

In this section，to control the system better，the double loop control method is adopted to deal with the underactuated property.

3.1 Attitude subsystem controller design

The NFTSMC method is proposed owing to its advantage of fast convergence of Euler angles.Firstly，attitude tracking errors are defined as fol-lows：e1=?-?d，e2=θ-θd，e3=ψ-ψd.The NFTSM surface is designed as

wherei=1，2，3；λi＞0；1 ＜γ＜2.

An improved double power reaching law is proposed as

whereΔ=|si|-1，α＞1， 0 ＜β＜1，k1＞0，k2＞0，c1＞0 andc2＞0.Then the control laws are given by Eqs.（13—15）.

Theorem 2Given the lumped uncertainty in the attitude subsystem，if NFTSM controllers（13—15）are applied to system（2）under Assumption 1，the closed loop subsystem could guarantee the fast convergence and the stability.

ProofThe roll angle is taken as an example，and the lumped uncertainty isd1.The Lyapunov function is designed as

Substituting Eq.（16）into the derivative ofV1，we obtain

whereδ1=min{2ρ1，2L1-1}.

The result of Theorem 2 comes from Lemma 1.Therefore，the control laws（13—15）are designed to ensure tracking performance of the attitude subsystem.

3.2 Position subsystem controller design

A backstepping NFTSM scheme is proposed for the position subsystem in this section.Here we takexposition as an example to introduce the design process.

The position tracking error is

The time derivative ofxand it follows that

whereα1is a virtual control input.

The first Lyapunov function is defined as

The derivative ofV2is as follows

α1is designed as

wheres4denotes the sliding surface andcis a constant.

Then，the NFTSM surfaces4is designed as

wherev1，v2are positive constants，andp，q(q＜p)the positive odd numbers.

The time derivative of Eq.（23）follows that

The second Lyapunov function is defined as

The derivative of Eq.（25）obtains

By substituting Eq.（3）in Eq.（26），we get

The virtual control inputUxcan be defined as

Similarly，they，zposition tracking errors are defined as

Virtual control inputsα2，α3are designed as

And the sliding surfaces5，s6are defined as

The virtual control inputUy，Uzcan be defined as

Theorem 3For the quadrotor’s position subsystem（2），given the design of NFTSM surface（23） combined with the backstepping technology and the estimated value of the lumped uncertainty obtained from observer（3），the designed control laws（28，35，36）can stabilize the position subsystem.

ProofTakexposition for an example.The lumped uncertainty is denoted asd4and we select the Lyapunov function

Substituting Eq.（27）into the derivative ofwe obtain

whereδ4=min{2c，2τ1，2L4-1}.

Similarly，the result of Theorem 3 can be obtained from Lemma 1.Therefore，the stability and tracking performance of the position subsystem are ensured under the control laws（28，35，36）.

As mentioned above，the yaw angleψcould track the given desired valueψdbased on the attitude controllers.Therefore，θd，?dandU4can be calculated from Eqs.（28，35，36）with theψconsidered as known.The solutions are as follows

Remark 2It is worth noting that the yaw commandψdshould satisfyψd∈(-π 2，π 2 )，such that the solvability ofU4is guaranteed.Further，the condition also accords with the range of a yaw angle in practice.

Remark 3To reduce the chatting characteristic of SMC，we substitute sat(s)for sgn(s)in this paper.The sat(s)function is defined as follows

whereΔis a positive constant.

4 Simulation and Discussion

4.1 Parameter selection

To fully verify the efficiency and superiority of the proposed control technology，the closed-loop quadrotor system under NFTSMC controllers（13—15）and（28，35，36）and Eq.（3）are tested by simulations.

Quadrotor parametersThe physical param-eters of a quadrotor are chosen in Table 1.

Table 1 Parameter setting of quadrotors

Observer parametersThe nonlinear observer gains are chosen as：Li=Lj=15(i=1，2，3；j=4，5，6).

Controller parametersThe attitude subsystem：λi=0.000 1，γ=1.1，α=1.02，β=0.9，k1=1.5，k2=1.2，c1=1.2，c2=1.4，ρ1=ρ2=ρ2=5；the position subsystem：c=3，p=5，q=3，v1=v3=v5=20，v2=v4=v6=5，τ1=τ2=τ3=5.

4.2 Simulation results

The desired position trajectory is set as

The desired signal of the yaw angle is set asψd=rad，and the initial states arex0=y0=z0=0 m，?0=θ0=ψ0=0.The objective of the simulation results is to compare and confirm the advantages of the proposed approach compared with the different approaches.Three simulation scenarios will be set to verify the effectiveness of the algorithm.

4.2.1 Scenario 1：Trajectory tracking under ideal case

In this scenario，we focus on the stability and performance of the proposed strategy without disturbances.To demonstrate the capabilities of the provided controllers，they were compared with SM integral backstepping control［23］（IBS-SMC）and conventional SMC［24］（SMC）approaches.

The tracking responses of attitude angles in Scenario 1 are presented in Fig.3，and the actual yaw angle is nearly fully tracked the desired signal.Here，the NFTSMC is compared with IBS-SMC and SMC approaches.According to Fig.3，IBSSMC and SMC performance is worse than that of the NFTSMC.Further，F(xiàn)ig.4 shows that the tracking errors of the attitude subsystem in NFTSMC are relatively smaller than that in IBS-SMC and SMC.

Fig.3 Attitude tracking in Scenario 1

Fig.4 Attitude tracking errors in Scenario 1

In Scenario 1，for the position subsystem，simulation results for the backstepping NFTSM controller are shown in Figs.5—7.The tracking responses ofx，y，zin Fig.5 show satisfactory tracking performances，and illustrate that the proposed strategy can accurately and effectively ensure the desired position signals.Here，the NFTSMC is compared with IBSSMC and SMC.According to Figs.5，7，the convergence speed of NFTSMC of position tracking is relatively faster than those of IBS-SMC and SMC.Further，position errors in NFTSMC are less than that in IBS-SMC and SMC in Fig.6.

Fig.5 Position tracking in Scenario 1

Fig.6 Position tracking errors in Scenario 1

Fig.7 Position trajectory tracking in Scenario 1

4.2.2 Scenario 2：Trajectory tracking under stochastic disturbance

This scenario is set to validate the performance and characterizations of the control strategy in this paper under stochastic disturbances.The lumped uncertainties are considered as a“gust of wind”proposed in Ref.［15］，then the similar form given by

The information of the compounded uncertainties in controllers comes from NDO.Fig.8 shows the lumped uncertainty estimation curves of the attitude and position subsystems.According to the simulation results，NDO can ensure that the lumped uncertainty estimates quickly and accurately converge to the actual value.When the lumped uncertainties change，NDO can still track the change trend of the upper lumped uncertainties in real time，which proves the effectiveness of NDO for the estimation of lumped uncertainties.Furthermore，the attitude tracking trajectories are given in Fig.9.The tracking responses of roll，pitch and yaw angles under the lumped uncertainty also have good tracking of the attitude reference trajectory.It is clear from Figs.10，12 that the proposed backstepping NFTSMC con-troller with the quadrotor provides an excellent tracking of trajectories with a very small error of tracking.Corresponding signal inputs are depicted in Fig.11.

Fig.8 Estimations of the lumped uncertainties

Fig.9 Attitude tracking in Scenario 2

Fig.11 Control inputs in Scenario 2

Fig.10 Position tracking in Scenario 2

Fig.12 Position trajectory tracking in Scenario 2

In Scenario 2，the proposed NFTSMC with reaching law for attitude subsystem and the backstepping NFTSMC for position subsystem are able to provide a high level of accuracy and robustness against the lumped uncertainty.

4.2.3 Scenario 3：Trajectory tracking under random disturbances

In addition to the influence of stochastic disturbances considered，the influence of Gaussian random disturbances on the quadrotor dynamics is also considered in this scenario.Gaussian random disturbances are displayed in Fig.13.

Fig.13 Gaussian random disturbances

The tracking performances are presented in Figs.14—16，which are shown the responses of the states and controllers with disturbances and Gaussian white noises applied.It is shown that the proposed approach demonstrates good performance.Fig.16 illustrates the trajectories of control inputs.It can be found that the NFTSM controller can track the desired signals faster even when there is random disturbance.In Fig.17，the 3D trajectories of the proposed controller are presented.

Fig.14 Attitude tracking in Scenario 3

Fig.15 Position tracking in Scenario 3

Fig.16 Control inputs in Scenario 3

Fig.17 Position trajectory tracking in Scenario 3

5 Conclusions

A strategy of NFSMC based on a NDO is designed for quadrotors considering external disturbances and system uncertainties.A model for a quadrotor system considering the lumped uncertainty is proposed.Next，a NDO is constructed to estimate the actual values of the lumped uncertainties.Then，NFTSMC combining the reaching law is proposed for the attitude subsystem and a backstepping NFTSMC strategy is proposed for the position subsystem.The comparative study established between IBS-SMC，SMC and the proposed controller highlighted the merits of the aforementioned control schemes.Furthermore，simulation results also indicated its superior performance coping with the lumped uncertainty.Based on existing work［24-25］，we will further carry out the fault prediction work for actuator faults.