# Antidisturbance Control for Helicopter Stochastic Systems

Authors: Dongping Li; Yankai Li
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, an antidisturbance controller is presented for helicopter stochastic systems under disturbances. To enhance the antidisturbance abilities, the nonlinear disturbance observer method is applied to reject the time-varying disturbances. Then, the antidisturbance nonlinear controller is designed by combining the backstepping control scheme. And the stochastic theory is used to guarantee that the closed-loop system is asymptotically bounded in mean square while the proposed control method is shown via some traditional nonlinear control techniques, which still show some common issues such as “dimension explosion” or others. The result of this paper can be regarded as a typical case of the nonlinear control method to help and promote the generation of advanced methods.


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## Body

## 1. Introduction

For actual systems, the nonlinearities of the model are probably one of the most noticeable characteristics, which are issued from the physical laws, material science, mathematical derivation, and so on. Hence, nonlinear systems are typical research models in various control fields and can be used to describe the dynamics of the system states. In order to handle nonlinear control problems, many classical control methods were proposed based on the nonlinear control theory [1–3], such as the baskstepping control method, feedback linearization technique, and nonlinear disturbance observer-based control scheme. Based on these nonlinear control theories, a number of advanced control techniques were proposed for various control systems. In [4], a class of nonlinear systems was studied via the event-triggered robust adaptive fuzzy control method. In [5], the adaptive neural control scheme was adopted for the nonlinear multiple output systems under the time-varying output constraints. Combining the nonlinear disturbance observer control with sliding-mode fuzzy neural network methods, a new control scheme was presented to deal with the nonlinear systems with disturbances in [6]. In [7, 8], the issue of adaptive fuzzy tracking control was discussed for a class of strict-feedback nonlinear systems. In [9], the periodic event-triggered control method was used to design the controller for nonlinear networked control systems. In [10], the robust adaptive control problem was investigated for state-constrained nonlinear systems under the input saturation and unknown control direction. The observer-based H∞ control was used for discrete-time T-S fuzzy systems in [11]. In [12], the switched-observer-based adaptive output-feedback control scheme was designed for pure-feedback switched nonlinear systems with unknown gains. The model-based adaptive event-triggered control method was discussed for nonlinear continuous-time systems in [13]. From the above discussion, the nonlinear control methods and theories are effective for disposing many troublesome nonlinear control problems. Therefore, the nonlinear control system theories are useful approaches to solve the issues of flight control for helicopters.The flight control is a crucial issue of helicopters. An excellent flight controller guarantees the well flight performances of flying helicopters and prevents the crash, breakdown, and so on. In many books involving helicopters [14, 15], linear control methods were used for designing the helicopter flight controller and obtained expected control performances while, in fact, helicopter models are complicated nonlinear systems, which are constructed according to the relationships of the components of helicopters. In some outstanding results, the helicopter flight controllers are designed by using the linear models, which are evolved from the nonlinear helicopter models under the linearization techniques. In [16], the tracking control issue was discussed for small-scale unmanned helicopter linear systems. In [17], the flight controller was designed for helicopters via the probabilistic robust linear parameter-varying control method using the iterative scenario approach. In [18], the model reference resilient control methods were designed for linear helicopter systems with time-varying disturbances. In [19], tracking control issues were discussed for the linear helicopter systems under time-varying disturbances and input stochastic perturbation. However, the dynamics of the helicopter states are complicated with many nonlinear characteristics. With development of the helicopter flight control techniques, the nonlinear flight controllers are designed for various helicopter systems. In [20], the adaptive trajectory tracking control approach was proposed for model-scaled nonlinear helicopter systems. The trajectory tracking control problems were discussed for small-scale nonlinear unmanned helicopters under model uncertainties in [21]. In [22], the composite block backstepping trajectory tracking control scheme was presented for disturbed nonlinear unmanned helicopter systems. In [23], the fixed-time autonomous shipboard landing control issues were studied for nonlinear helicopter systems under external disturbances. The finite-time control issue was discussed for small-scale nonlinear unmanned helicopter systems with disturbances in [24]. In [25], the sliding-mode control and extended disturbance observer methods were used to solve the tracking control issue of unmanned nonlinear helicopter systems. The dynamic decoupling control optimization methods were adopted for small-scale nonlinear unmanned helicopter systems in [26]. While helicopters fly in full noise surroundings, various random disturbances exist in helicopter control system models. In [27], the antidisturbance control was proposed for the attitude and altitude nonlinear helicopter systems under random disturbances. The random disturbances are considered for flight control, which improves the control performances and precision of flying helicopters.In this paper, in order to enhance the control precision and robustness of the helicopter models, the random disturbances are considered to construct the fight controller. The helicopter stochastic systems are better to describe the flying dynamics states of the helicopter. While with the random disturbances introducing in the models, many advanced control methods, such as dynamic surface control, fuzzy control, and other intelligent control methods, cannot be used to design the flight controller directly because many control variables will lose some good characteristics, such as the continuity and derivability, under the random disturbances. Therefore, in this manuscript, the backstepping control method and nonlinear disturbance observer control scheme are used to establish the nonlinear flight controller. Due to the characteristics of underactuation, strong nonlinearity, and high-order of helicopter stochastic system models, the phenomenon of “dimension explosion” is inevitable for nonlinear controller designed, and the controller solved steps are presented in this paper. Our results present an approach to design the strong robust flight controller.The organization of this paper is standard. The problem statement is stated in Section2. The position loop control issues are discussed in Section 3. The attitude loop control problems are studied in Section 4. The stability analysis of the main result is given in Section 5. The conclusions are presented in Section 6.


## 2. Problem Statement

Consider the following helicopter dynamic system:(1)dp=vdt,dv=−ge3+1mRfdt+δ1dt+G1dβ1,dR=Rw×dt,Jdw=−w×Jw+τdt+δ2dt+G2dβ2,where p and v denote the position and velocity in inertial frame, respectively, g is the gravitational acceleration, e3=001T is a unit vector, and R is the rotation matrix from body frame to inertial frame and defined by(2)R=CθCψSθSϕCψ−CϕSψCϕSθCψ+SϕSψCθSψSϕSθSψ+CϕCψCϕSθSψ−SϕCψ−SθSϕCθCϕCθ=R11R12R13R21R22R23R31R32R33=R1R2R3,where C⋅ and S⋅denote cos⋅ and sin⋅ with respective variables. ϕt,θt, and φt are roll, pitch, and yaw angles in body frame, respectively. w=ptqtrt is the angular velocity. p, q, and r are the roll, pitch, and yaw angular rates, respectively. J is the inertia matrix and denoted by J=diagJxxJyyJzz. f and τ are the sum of the external forces and moments. Consider the characteristics of these forces and moments, which are denoted as follows:(3)f=00TmT,τu=Cm+TmLz−HzCm+TmLz00−TmLyTmLxHxabTt+−TmLy−TmLx−Qm=ATmρ+BTm,with Qm=CkTm1.5+Dk. Cm, Lz, Hz, Ly, Lx, Hx, and Dk are known constants. β1 and β2 are independent one-dimensional standard Wiener processes, dβ1/dt=ξ1 and dβ2/dt=ξ2, where ξ1 and ξ2 are independent one-dimensional stochastic noises. G1ξ1 and G2ξ2 are the stochastic accelerated velocity and stochastic angular acceleration generated by the stochastic force and stochastic moment, G1 and G2 are known and bounded weight parameters, and δ1 and δ2 are disturbances with bounded derivatives, which are δ˙1≤μ1 and δ˙2≤μ2.The following definitions and lemmas are crucial important to further analyze the main results of this paper. In order to state those definitions and lemmas, consider the following system:(4)dx=ft,xdt+gt,xdβ,where x is the system state, f∈ℒ1ℝ+;ℝn and g∈ℒ2ℝ+;ℝn×m are known functions, and β is the one-dimensional standard Wiener process.ForVt,x∈C2,1ℝ;ℝn×ℝ+, the infinitesimal generator along (4) is defined by(5)ℒVt,x=Vtt,x+Vxt,xft,x+12TrgTt,xVxxt,xgt,x,where Tr is the trace of a matrix, and(6)Vtt,x=∂Vt,x∂t,Vxt,x=∂Vt,x∂ϑ1,…,∂Vt,x∂ϑn,Vxxt,x=∂2Vt,x∂ϑi∂ϑjn×n.Definition 1 (see [28]).
Letp>0. System (4) is said to be asymptotically bounded in p-th moment if there is a positive constant H such that(7)limt⟶∞supExt;x0p≤H,for all t0,x0∈ℝ+×ℝn. When p=2, we say system (4) is asymptotically bounded in mean square.Lemma 1.
For system (4), assume that there exists a function V∈C2,1ℝn×ℝ+,ℝ+, positive constants ki,ki′,pi,pi′,c,dc, such that(8)∑i=1nkixipi≤Vxt,t≤∑i=1nki′xipi′,(9)ℒVxt,t≤−λVxt,t+dc.Then, there exists a unique strong solutionxt=xt;x0,t0 of system (4) for each xt0=x0∈ℝn, and system (4) is p-th moment exponentially practically stable, where p=minp1,…,pn.Lemma 2.
For any vectorsx,y∈ℝn and any scalars ϵ>0,p>1, there holds xTy≤ϵp/pxp+1/qϵqyq, where q=p/p−1.


## 3. Position Loop Control

In this section, the DOBC method and backsetpping method are used to construct the flight controller for helicopters under stochastic disturbances. We first build disturbance observers to estimate the common disturbancesδ1 and δ2.


### 3.1. Disturbance Observer Designed for Position Control

In what follows, the disturbance observer is designed to estimate the disturbanceδ1,(10)δ^1=σ1+ρ1v,dσ1=−∂ρ1v∂v−ge3+1mRfdt+δ^1dt+σ11dt,where σ1 is an auxiliary variable, σ11 is an auxiliary function to be designed, and ρ1v is a nonlinear function to be designed.Estimate error is defined asδ˜1=δ^1−δ1. Then, the dynamic of the estimate error is given by(11)dδ˜1=−∂ρ1v∂vδ˜1dt+G1dβ1−δ˙1dt+σ11dt.The disturbance observer gain is selected as∂ρ1v/∂v=L1, with L1=−diagl1l1l1, where l1>0.Choose the Lyapunov function as follows:(12)V01=14δ˜1Tδ˜12.Then, the infinitesimal generator ofV01 along with (11) is shown as follows:(13)ℒV01=−L1δ˜1Tδ˜12−δ˜1Tδ˜1δ˜1Tδ˙1+δ˜1Tδ˜1δ˜1Tσ11+12TrG1TL12δ˜1δ˜1T+δ˜1Tδ˜1IL1G1.From Lemma2, there exist parameters ε01>0 and ε02>0 such that(14)−δ˜1Tδ˜1δ˜1Tδ˙1≤3ε014/34δ˜1Tδ˜12+14ε014δ˙14,12TrG1TL12δ˜1δ˜1T+δ˜1Tδ˜1IL1G1≤32δ˜12L1G1F2≤3ε0224L1G1F4δ˜1Tδ˜12+34ε022.Designσ11 as(15)σ11=−3ε014/34+3ε0224L1G1F4δ˜1+σ12,where σ12 is introduced in the following steps. Hence, from (15) and δ˙1≤μ1, we have(16)ℒV01≤−L1δ˜1Tδ˜12+14ε014μ14+34ε022+δ˜1Tδ˜1δ˜1Tσ12=−L1δ˜1Tδ˜12+δ˜1Tδ˜1δ˜1Tσ12+H01,where(17)H01=14ε014μ14+34ε022.


### 3.2. Position Loop Controller Designed

Consider the position loop system as follows:(18)dp=vdt,dv=−ge3+1mRfdt+δ1dt+G1dβ1.The predefined position ispr=xr,yr,zr, define the tracking error as ep=p−pr. The dynamic of the error system is given by(19)dep=vdt−dpr=evdt,dev=−ge3+TmmR3e+TmmR3c−p¨r+δ˜1+δ^1dt+G1dβ1.Define the controlleru1=Tm/mR3c and R3c=R13cR23cR33cT, and(20)u1=ge3+p¨r−δ^1−k1tanhn1ep+n2ev−k2tanhn2ev,where k1>0, k2>0, n1>0, and n2>0 are parameters to be designed. At this point, R3c and Tm can be constructed as(21)R3c=mTmu1,Tm=mge3+p¨r−δ^1−k1tanhn1ep+n2ev−k2tanhn2ev.Then, the closed-loop position loop system is written as(22)dep=evdt,(23)dev=−k1tanhn1ep+n2ev−k2tanhn2ev+TmmR3e+δ˜1dt+G1dβ1.Remark 1.
In fact, the helicopter should not be overturn under controller (21), for the continuity of cosϕcosθ. So that, we guarantee cosϕcosθ>0 to ensure that the helicopter will not be overturn. Meanwhile, the defined signal R33c should also satisfy R33c>0. Thus, the parameters k1 and k2 are designed to satisfy g+z¨r+δ^1e−k1−k2>0. In fact, the acceleration of disturbance in z axis is far less than acceleration of gravity.
ConsiderR3=1, R3e=R3−R3c, R¯3=R13R23, and R¯3c=R13cR23c. Thus,(24)R3=R13R23R33=R13R231−R132−R232,R3e=R13R23R33−R13cR23cR33c=R13eR23e1−R132−R232−1−R13c2−R23c2.
Due toR33>0 and R33c>0, we have(25)1−R132−R232−1−R13c2−R23c2=α1R¯3e1−R132−R232+1−R13c2−R23c2,where α1=−R13+R13c,R23+R23c. Hence,(26)R3e=R¯3eα1R¯3e1−R132−R232+1−R13c2−R23c2=I2α11−R132−R232+1−R13c2−R23c2R¯3e=αR¯3e=αR¯3−R¯3c,where α=I2α1/1−R132−R232+1−R13c2−R23c2.
The velocity tracking error equation (23) can be rewritten as(27)dev=−k1tanhn1ep+n2ev−k2tanhn2ev+TmmαR¯3e+δ˜1dt+G1dβ1.
Choose Lyapunov functionV11t as follows:(28)V11t=k1∫0n1ep+n2evtanhμdμ+k2∫0n2evtanhμdμ+n12evTev.
Then, the infinitesimal generator ofV11 is represented as(29)ℒV11t=−n2k1tanhn1ep+n2ev+k2tanhn2evT×k1tanhn1ep+n2ev+k2tanhn2ev−n1k2evTtanhn2ev+FtTTmmαR¯3e+δ˜1+12Tr0G1T∂2V11∂ep2∂2V11∂ep∂ev∂2V11∂ep∂ev∂2V11∂ev20G1,where(30)Ft=n1ev+n2k1tanhn1ep+n2ev+n2k2tanhn2ev,∂2V11∂ep2=diagn12k11+tanh2n1ep1+n2ev1,n12k11+tanh2n1ep2+n2ev2,n12k11+tanh2n1ep3+n2ev3,∂2V11∂ep∂ev=diagn1n2k11+tanh2n1ep1+n2ev1,n1n2k11+tanh2n1ep2+n2ev2,n1n2k11+tanh2n1ep3+n2ev3,∂2V11∂ev2=diagn22k11+tanh2n1ep1+n2ev1+n22k21+tanh2n2ev1,n22k11+tanh2n1ep2+n2ev2+n22k21+tanh2n2ev2,n22k11+tanh2n1ep3+n2ev3+n22k21+tanh2n2ev3+diag1,1,1.
For the functiontanh⋅≤1 and the bounded parameter G1, there exists a parameter H11>0, such that(31)12Tr0G1T∂2V11∂ep2∂2V11∂ep∂ev∂2V11∂ep∂ev∂2V11∂ev20G1≤H11.
Notice that−n1k2evTtanhn2ev≤−n1/n2k2tanhn2evTtanhn2ev. Then, we have(32)ℒV11t≤−n2k1tanhn1ep+n2ev+k2tanhn2evTk1tanhn1ep+n2ev+k2tanhn2ev−n1n2k2tanhn2evTtanhn2ev+FtTTmmαR¯3e+δ˜1+H11=−tanhn1ep+n2evtanhn2evn2k12n2k1k2n2k1k2n2k22+n1n2k2tanhn1ep+n2evtanhn2ev+FtTδ˜1+FtTTmmαR¯3e+H11.
In fact, according to Lemma2, there exist parameters ε11>0 and ε12>0, such that(33)FtTδ˜1≤ε113δ˜1Tδ˜1δ˜1TFt2Ft+23ε111/2,FtTTmmαR¯3e≤ε12Tm3α2Ft23m3R¯3eTR¯3eR¯3eTαTFt+23ε121/2.
Letk3=mineign2k12n2k1k2n2k1k2n2k22+n1/n2k2, we obtain(34)ℒV11t≤−k3tanhn1ep+n2evtanhn2evTtanhn1ep+n2evtanhn2ev+ε12Tm3α2Ft23m3R¯3eTR¯3eR¯3eTαTFt+ε113δ˜1Tδ˜1δ˜1TFt2Ft+23ε111/2+23ε121/2+H11.
Sincehx=tanhx/x is a continuous function, for any x¯>0, 0≤x≤x¯, there exists a positive number βx¯ such that βx≤tanhx≤x. Thus,(35)−tanhn1ep+n2evTtanhn1ep+n2ev≤−β2n1ep+n2evTn1ep+n2ev,−tanhn2evTtanhn2ev≤−β2n22evTev.
Define Lyapunov function as follows:(36)V1t=V01t+V11t.
Based on (16) and (34), we obtain(37)ℒV1t≤−k3β2n1ep+n2evTn1ep+n2ev−k3β2n22evTev+ε12Tm3α2Ft23m3R¯3eTR¯3eR¯3eTαTFt+ε113δ˜1Tδ˜1δ˜1TFt2Ft+23ε111/2+23ε121/2+H11−L1δ˜1Tδ˜12+δ˜1Tδ˜1δ˜1Tσ12+H01.
Hence,σ12 is designed as(38)σ12=−ε113Ft2Ft.
Then,(39)ℒV1t≤−l1δ˜1Tδ˜12−k4epTep−k4evTev+ε12Tm3α2Ft23m3R¯3eTR¯3eR¯3eTαTFt+H1,where k4=mineigk3β2n12k3β2n1n2k3β2n1n22k3β2n22>0 and H1=H11+H01+2/3ε111/2+2/3ε121/2.
Moreover, the error dynamic of disturbanceδ˜1 can be rewritten as(40)dδ˜1=−L1+3ε014/34+3ε0224L1G1F4δ˜1dt−L1G1dβ1−δ˙1dt−ε113Ft2Ftdt.


## 3.1. Disturbance Observer Designed for Position Control

In what follows, the disturbance observer is designed to estimate the disturbanceδ1,(10)δ^1=σ1+ρ1v,dσ1=−∂ρ1v∂v−ge3+1mRfdt+δ^1dt+σ11dt,where σ1 is an auxiliary variable, σ11 is an auxiliary function to be designed, and ρ1v is a nonlinear function to be designed.Estimate error is defined asδ˜1=δ^1−δ1. Then, the dynamic of the estimate error is given by(11)dδ˜1=−∂ρ1v∂vδ˜1dt+G1dβ1−δ˙1dt+σ11dt.The disturbance observer gain is selected as∂ρ1v/∂v=L1, with L1=−diagl1l1l1, where l1>0.Choose the Lyapunov function as follows:(12)V01=14δ˜1Tδ˜12.Then, the infinitesimal generator ofV01 along with (11) is shown as follows:(13)ℒV01=−L1δ˜1Tδ˜12−δ˜1Tδ˜1δ˜1Tδ˙1+δ˜1Tδ˜1δ˜1Tσ11+12TrG1TL12δ˜1δ˜1T+δ˜1Tδ˜1IL1G1.From Lemma2, there exist parameters ε01>0 and ε02>0 such that(14)−δ˜1Tδ˜1δ˜1Tδ˙1≤3ε014/34δ˜1Tδ˜12+14ε014δ˙14,12TrG1TL12δ˜1δ˜1T+δ˜1Tδ˜1IL1G1≤32δ˜12L1G1F2≤3ε0224L1G1F4δ˜1Tδ˜12+34ε022.Designσ11 as(15)σ11=−3ε014/34+3ε0224L1G1F4δ˜1+σ12,where σ12 is introduced in the following steps. Hence, from (15) and δ˙1≤μ1, we have(16)ℒV01≤−L1δ˜1Tδ˜12+14ε014μ14+34ε022+δ˜1Tδ˜1δ˜1Tσ12=−L1δ˜1Tδ˜12+δ˜1Tδ˜1δ˜1Tσ12+H01,where(17)H01=14ε014μ14+34ε022.


## 3.2. Position Loop Controller Designed

Consider the position loop system as follows:(18)dp=vdt,dv=−ge3+1mRfdt+δ1dt+G1dβ1.The predefined position ispr=xr,yr,zr, define the tracking error as ep=p−pr. The dynamic of the error system is given by(19)dep=vdt−dpr=evdt,dev=−ge3+TmmR3e+TmmR3c−p¨r+δ˜1+δ^1dt+G1dβ1.Define the controlleru1=Tm/mR3c and R3c=R13cR23cR33cT, and(20)u1=ge3+p¨r−δ^1−k1tanhn1ep+n2ev−k2tanhn2ev,where k1>0, k2>0, n1>0, and n2>0 are parameters to be designed. At this point, R3c and Tm can be constructed as(21)R3c=mTmu1,Tm=mge3+p¨r−δ^1−k1tanhn1ep+n2ev−k2tanhn2ev.Then, the closed-loop position loop system is written as(22)dep=evdt,(23)dev=−k1tanhn1ep+n2ev−k2tanhn2ev+TmmR3e+δ˜1dt+G1dβ1.Remark 1.
In fact, the helicopter should not be overturn under controller (21), for the continuity of cosϕcosθ. So that, we guarantee cosϕcosθ>0 to ensure that the helicopter will not be overturn. Meanwhile, the defined signal R33c should also satisfy R33c>0. Thus, the parameters k1 and k2 are designed to satisfy g+z¨r+δ^1e−k1−k2>0. In fact, the acceleration of disturbance in z axis is far less than acceleration of gravity.
ConsiderR3=1, R3e=R3−R3c, R¯3=R13R23, and R¯3c=R13cR23c. Thus,(24)R3=R13R23R33=R13R231−R132−R232,R3e=R13R23R33−R13cR23cR33c=R13eR23e1−R132−R232−1−R13c2−R23c2.
Due toR33>0 and R33c>0, we have(25)1−R132−R232−1−R13c2−R23c2=α1R¯3e1−R132−R232+1−R13c2−R23c2,where α1=−R13+R13c,R23+R23c. Hence,(26)R3e=R¯3eα1R¯3e1−R132−R232+1−R13c2−R23c2=I2α11−R132−R232+1−R13c2−R23c2R¯3e=αR¯3e=αR¯3−R¯3c,where α=I2α1/1−R132−R232+1−R13c2−R23c2.
The velocity tracking error equation (23) can be rewritten as(27)dev=−k1tanhn1ep+n2ev−k2tanhn2ev+TmmαR¯3e+δ˜1dt+G1dβ1.
Choose Lyapunov functionV11t as follows:(28)V11t=k1∫0n1ep+n2evtanhμdμ+k2∫0n2evtanhμdμ+n12evTev.
Then, the infinitesimal generator ofV11 is represented as(29)ℒV11t=−n2k1tanhn1ep+n2ev+k2tanhn2evT×k1tanhn1ep+n2ev+k2tanhn2ev−n1k2evTtanhn2ev+FtTTmmαR¯3e+δ˜1+12Tr0G1T∂2V11∂ep2∂2V11∂ep∂ev∂2V11∂ep∂ev∂2V11∂ev20G1,where(30)Ft=n1ev+n2k1tanhn1ep+n2ev+n2k2tanhn2ev,∂2V11∂ep2=diagn12k11+tanh2n1ep1+n2ev1,n12k11+tanh2n1ep2+n2ev2,n12k11+tanh2n1ep3+n2ev3,∂2V11∂ep∂ev=diagn1n2k11+tanh2n1ep1+n2ev1,n1n2k11+tanh2n1ep2+n2ev2,n1n2k11+tanh2n1ep3+n2ev3,∂2V11∂ev2=diagn22k11+tanh2n1ep1+n2ev1+n22k21+tanh2n2ev1,n22k11+tanh2n1ep2+n2ev2+n22k21+tanh2n2ev2,n22k11+tanh2n1ep3+n2ev3+n22k21+tanh2n2ev3+diag1,1,1.
For the functiontanh⋅≤1 and the bounded parameter G1, there exists a parameter H11>0, such that(31)12Tr0G1T∂2V11∂ep2∂2V11∂ep∂ev∂2V11∂ep∂ev∂2V11∂ev20G1≤H11.
Notice that−n1k2evTtanhn2ev≤−n1/n2k2tanhn2evTtanhn2ev. Then, we have(32)ℒV11t≤−n2k1tanhn1ep+n2ev+k2tanhn2evTk1tanhn1ep+n2ev+k2tanhn2ev−n1n2k2tanhn2evTtanhn2ev+FtTTmmαR¯3e+δ˜1+H11=−tanhn1ep+n2evtanhn2evn2k12n2k1k2n2k1k2n2k22+n1n2k2tanhn1ep+n2evtanhn2ev+FtTδ˜1+FtTTmmαR¯3e+H11.
In fact, according to Lemma2, there exist parameters ε11>0 and ε12>0, such that(33)FtTδ˜1≤ε113δ˜1Tδ˜1δ˜1TFt2Ft+23ε111/2,FtTTmmαR¯3e≤ε12Tm3α2Ft23m3R¯3eTR¯3eR¯3eTαTFt+23ε121/2.
Letk3=mineign2k12n2k1k2n2k1k2n2k22+n1/n2k2, we obtain(34)ℒV11t≤−k3tanhn1ep+n2evtanhn2evTtanhn1ep+n2evtanhn2ev+ε12Tm3α2Ft23m3R¯3eTR¯3eR¯3eTαTFt+ε113δ˜1Tδ˜1δ˜1TFt2Ft+23ε111/2+23ε121/2+H11.
Sincehx=tanhx/x is a continuous function, for any x¯>0, 0≤x≤x¯, there exists a positive number βx¯ such that βx≤tanhx≤x. Thus,(35)−tanhn1ep+n2evTtanhn1ep+n2ev≤−β2n1ep+n2evTn1ep+n2ev,−tanhn2evTtanhn2ev≤−β2n22evTev.
Define Lyapunov function as follows:(36)V1t=V01t+V11t.
Based on (16) and (34), we obtain(37)ℒV1t≤−k3β2n1ep+n2evTn1ep+n2ev−k3β2n22evTev+ε12Tm3α2Ft23m3R¯3eTR¯3eR¯3eTαTFt+ε113δ˜1Tδ˜1δ˜1TFt2Ft+23ε111/2+23ε121/2+H11−L1δ˜1Tδ˜12+δ˜1Tδ˜1δ˜1Tσ12+H01.
Hence,σ12 is designed as(38)σ12=−ε113Ft2Ft.
Then,(39)ℒV1t≤−l1δ˜1Tδ˜12−k4epTep−k4evTev+ε12Tm3α2Ft23m3R¯3eTR¯3eR¯3eTαTFt+H1,where k4=mineigk3β2n12k3β2n1n2k3β2n1n22k3β2n22>0 and H1=H11+H01+2/3ε111/2+2/3ε121/2.
Moreover, the error dynamic of disturbanceδ˜1 can be rewritten as(40)dδ˜1=−L1+3ε014/34+3ε0224L1G1F4δ˜1dt−L1G1dβ1−δ˙1dt−ε113Ft2Ftdt.


## 4. Attitude Loop Control

In this section, the DOBC method and backsetpping method are used to construct the flight controller for helicopters under stochastic disturbances. We first build disturbance observers to estimate the common disturbancesδ1 and δ2.


### 4.1. Disturbance Observer Designed for Attitude Control

In order to estimate the disturbanceδ2, we design the following disturbance observers:(41)δ^2=σ2+Jρ2w,dσ2=−∂ρ2w∂w−w×Jw+τdt+δ^2dt+σ02dt,where σ2 is an auxiliary variable, σ02 is an auxiliary function to be designed, and ρ2v is a nonlinear function to be designed.Define estimate error asδ˜2=δ^2−δ2. Then, the dynamic of the estimate error can be shown as(42)dδ˜2=−∂ρ2w∂wδ˜2dt+G2dβ2−δ˙2dt+σ02dt.The disturbance observer gain is selected as∂ρ2w/∂w=L2, with L2=−diagl2l2l2, where l2>0.Choose Lyapunov function as follows:(43)V02=14δ˜2Tδ˜22.Similarly withV01, for V02, there exist ε03>0 and ε04>0, such that(44)ℒV02≤−l2δ˜2Tδ˜22+H02,where(45)H02=14ε034μ24+34ε042.Design the auxiliary variableσ02 as follows:(46)σ02=−3ε034/34+3ε0424L2G2F4δ˜2.The error dynamics of disturbance (42) can be rewritten as(47)dδ˜2=−l2+3ε034/34+3ε0424L2G2F4δ˜2dt−δ˙2dt−L2G2dβ2.


### 4.2. Attitude Loop Controller Designed

The predefined tracking signals are position and yaw angle, which are denoted bypr and ψr. After designing the position loop control, the reasonable attitude tracking signals are obtained, which are given as R¯3c and ψr. Moreover, defining γr=R¯3Tψ and γ=θϕψ, we can confirm that ∂γr/∂γ=cosθ≠0, for θ<π/2. Hence, the mapping that from γ to γr is a local topological homeomorphism, which means that the dynamic of γr can represent the dynamic of γ for θ<π/2. The error dynamic of γr is given by(48)dR¯3e=R^3w1dt−dR¯3c,(49)deψ=sinϕcosθq+cosϕcosθrdt−dψr,where w1=pqT and R^3=−R12R11−R22R21.For (48), dR3c can be written as(50)dR3c=du1u1Tu1=1u1Tu1I3−u1u1Tu1Tu1du1.Fordu1, we have(51)du1=dp¨r−dδ^1−k1dtanhn1ep+n2ev−k2dtanhn2ev.From (10), dδ^1 is written as follows:(52)dδ^1=−l¯1δ˜1dt−L1G1dβ1−ε113Ft2Ftdt,where l¯1=l1+3ε014/3/4+3ε022/4L1G1F4.Hence, we get(53)du1=Δ01tdt+Δ02tδ˜1dt+Δ03tdβ1,for x=x1,x2,x3, Tanh2x=diagtanh2x1,tanh2x2y,tanh2x3, where(54)Δ01t=p⋯r−k1n1I+Tanh2n1ep+n2evev−k2n2I+Tanh2n2ev+k1n2I+Tanh2n1ep+n2ev−k1tanhn1ep+n2ev−k2tanhn2ev+TmmαR¯3e+ε113Ft2Ft,Δ02t=l¯1I−k2n2I+Tanh2n2ev+k1n2I+Tanh2n1ep+n2ev,Δ03t=L1G1−k2n2I+Tanh2n2ev+k1n2I+Tanh2n1ep+n2evG1.From (50), (55)dR3c=1u1Δ101t+Δ102tev+Δ103tΔ104t+ε113Ft2Ftdt+1u1Δ12tδ˜1dt+1u1Δ13tdβ1,where(56)Δ101t=I3−u1u1Tu1Tu1p⋯r,Δ102t=−k2I3−u1u1Tu1Tu1I+Tanh2ev,Δ103t=−I3−u1u1Tu1Tu1k2I+Tanh2ev+k1I+Tanh2ep+ev,Δ104t=−k1tanhep+ev−k2tanhev+TmmαR¯3e,Δ12t=I3−u1u1Tu1Tu1l¯1I−k2I+Tanh2ev+k1I+Tanh2ep+ev,Δ13t=I3−u1u1Tu1Tu1L1G1−k2I+Tanh2ev+k1I+Tanh2ep+evG1.Foru1u1TF≤u1Fu1TF=u12, one has(57)I3−u1u1Tu1Tu1F≤I3F+u1u1TFu12≤3+1.Then, we confirm that(58)Δ101t≤3+1p⋯r,Δ102tF≤k23+1I+Tanh2evF≤k232+6,Δ103tF≤3+1k2I+Tanh2ev+k1I+Tanh2ep+evF≤6+23k1+k2,Δ12t≤3+1l¯1+23k1+k2,Δ13t≤3+1L1G1+23k1+k2G1.Thus, (48) can be rewritten as(59)dR¯3e=R^3ew1dt+R^3w1cdt−E01u1Δ101t+Δ102tev+Δ103tΔ104t+ε113Ft2Ftdt−E01u1Δ12tδ˜1dt−E01u1Δ13tdβ1,where E0=100010.Choose Lyapunov functionV21t as(60)V21t=14R¯3eTR¯3e2.The infinitesimal generator ofV21 along with (60) is shown as(61)ℒV21t=R¯3eTR¯3eR¯3eTR^3ew1+R¯3eTR¯3eR¯3eTR^3w1c−R¯3eTR¯3eR¯3eTE01u1Δ101t−R¯3eTR¯3eR¯3eTE01u1Δ102tev−R¯3eTR¯3eR¯3eTE01u1Δ103tΔ104t−ε113R¯3eTR¯3eR¯3eTE01u1Ft2Ft−R¯3eTR¯3eR¯3eTE01u1Δ12tδ˜1+12u12TrΔ13TtE0T2R¯3eR¯3eT+R¯3eTR¯3eIE0Δ13t.From Lemma2, there exist parameters ε21>0, ε22>0, ε231>0, ε232>0, ε24>0, and ε25>0, such that(62)R¯3eTR¯3eR¯3eTR^3ew1≤34ε211/3R¯3eTR¯3e2+ε214R^34ew1Tew12,−R¯3eTR¯3eR¯3eTE01u1Δ101t≤3ε221/34u14/3R¯3eTR¯3e2+14ε22E04Δ101t4,−R¯3eTR¯3eR¯3eTE01u1Δ102tev≤ε2312R¯3eTR¯3e2+ε2322u14R¯3eTR¯3e2evTev2+E04Δ102t48ε232ε231,−R¯3eTR¯3eR¯3eTE01u1Δ12tδ˜1≤3ε241/34u14/3R¯3eTR¯3e2+14ε24E0F4Δ12t4δ˜1Tδ˜12,12u12TrΔ13TtE0T2R¯3eR¯3eT+R¯3eTR¯3eIE0Δ13t≤3ε254u12R¯3eTR¯3e2+12ε25E0Δ13tF4.Furthermore, forTm=mu1, there exist parameters ε26>0 and ε27>0, such that(63)−R¯3eTR¯3eR¯3eTE01u1Δ103tΔ104t≤3ε261/34u14/3R¯3eTR¯3e2+14ε26E0F4Δ103t4k1tanhep+ev+k2tanhev4+3ε271/34E0F4/3Δ103tF4/3R¯3eTR¯3e2+14ε27αF4R¯3eTR¯3e2.Then,ℒV21t satisfies(64)ℒV21t≤H25+H244u14/3+3ε254u12+14ε27αF4R¯3eTR¯3e2+R¯3eTR¯3eR¯3eTR^3w1c+H23δ˜1Tδ˜12−ε113R¯3eTR¯3eR¯3eTE01u1Ft2Ft+ε2322u14R¯3eTR¯3e2evTev2+ε214R^34ew1Tew12+H21,where(65)H21=12ε25E0F43+1L1G1+23k1+k2G1F4+14ε22E0F43+14p⋯r4+14ε26E0F26+23k1+k24k1+k24+E04k232+648ε232ε231,H22=14ε23E0F4k232+62,H23=14ε243+1l¯1F+23k1+k24,H24=3ε221/3+3ε241/3+3ε261/3,H25=34ε211/3+3ε271/34E0F4/36+23k1+k24/3+ε2312.DefineV2t=V1+V21. Based on the above discussion and according to (39) and (64), we have(66)ℒV2t≤−l1−H23δ˜1Tδ˜12−k4epTep−k4evTev+ε214R^34ew1Tew12+R¯3eTR¯3eR¯3eTR^3w1c+H25+H244u14/3+3ε254u12+14ε27αF4R¯3eTR¯3e2+ε12Tm3α2Ft23m3R¯3eTR¯3eR¯3eTαTFt+ε2322u14R¯3eTR¯3e2evTev2−ε113R¯3eTR¯3eR¯3eTE01u1Ft2Ft+H2,where H2=H1+H21. Hence, we design the virtual control law w1c as(67)w1c=−k5+H25+H244u14/3+3ε254u12+14ε27αF4R^3−1R¯3e−ε12Tm3α2Ft23m3R^3−1αTFt−ε2322u14evTev2R^3−1R¯3e+ε11Ft23u1R^3−1E0Ft,where k5>0. Then, we obtain(68)ℒV2t≤−L1−H23δ˜1Tδ˜12−k4epTep−k4evTev−k5R¯3eTR¯3e2+ε214R^34ew1Tew12+H2.From (61), the dynamic of R¯3e can be rewritten as(69)dR¯3e=R¯3e1tdt+R¯3e2tδ˜1dt+R¯3e3tdβ1,where(70)R¯3e1t=R^3ew1−k5+H25+H244u14/3+3ε254u12+14ε27αF4R¯3e−ε12Tm3α2Ft23m3R^3−1αTFt−ε2322u14evTev2R^3−1R¯3e−E0u1Δ101t+Δ102tev+Δ103tΔ104t,R¯3e2t=E0u1Δ12t,R¯3e3t=−E0u1Δ13t.For (37), we have(71)deψ=E11w1+cosϕcosθer+cosϕcosθrc−ψ˙rdt,where E11=0sinϕ/cosθ and rc is the virtual control law to be designed.Choose Lyapunov functionV31 as(72)V31=14eψ4.The derivative ofV31t along with (71) is derived as follows:(73)V˙31t=eψ3E11w1+cosϕcosθer+cosϕcosθrc−ψ˙r.Then, the virtual control lawrc is designed as(74)rc=−cosθcosϕE11w1c−ψ˙r+k3eψ.At this point, the dynamic ofeψ can be written as(75)deψ=−k6eψ+E11ew1+cosϕcosθerdt=−k6eψ+E1ewdt,where E1=0sinϕ/cosθcosϕ/cosθ and ew=p,q,rT−pc,qc,rcT. Thus, based on Lemma 2, there exists ε28>0, such that(76)V˙31t=eψ3−k6eψ+E1ew=−k6eψ4+eψ3E1ew≤−k6−3ε281/34eψ4+14ε28E14ewTew2.DefineV3t=V2t+V31t; combining (68) and (76) yields(77)ℒV3t≤−l1−H23δ˜1Tδ˜12−k4epTep−k4evTev−k5R¯3eTR¯3e+ε214R^34E04ewTew2+H2−k6−3ε281/34eψ4+14ε28E14ewTew2.In what follows, consider the dynamic of angular velocity:(78)Jdw=−w×Jw+ATmρ+BTm+δ2dt+G2dβ2.Definingρ=ATm−1u2−BTm+w×Jw−δ^2, the error dynamic of ew is(79)Jdew=u2+δ˜2dt+G2dβ2−dwc,where dwc is obtained in Appendix. Then, according to (79), (80)Jdew=u2+δ˜2dt+G2dβ2−Wc1tdt−Wc2tδ˜1dt−Wc20t,δ˜1dt−Wc3tdβ1,which implies that(81)dew=J−1u2−J−1Wc1t−J−1Wc2tδ˜1−J−1Wc20t,δ˜1+J−1δ˜2dt−J−1Wc3tdβ1+J−1G2dβ2.Choose Lyapunov function as follows:(82)V41t=14ewTew2.The infinitesimal generator ofV41 along with (81) is obtained as follows:(83)ℒV41t=ewTewewTJ−1u2−ewTewewTJ−1Wc1t−ewTewewTJ−1Wc2tδ˜1−ewTewewTJ−1Wc20t,δ˜1+ewTewewTJ−1δ˜2+12TrWc3TtJ−T2ewewT+ewTewIJ−1Wc3t+12TrG2TJ−T2ewewT+ewTewIJ−1G2.From Lemma2, there exist parameters ε31>0, ε32>0, ε33>0, ε34>0, and ε35>0 such that(84)−ewTewewTJ−1Wc2tδ˜1≤34ε311/3J−1F4/3Wc2tF4/3ewTew2+ε314δ˜1Tδ˜12,ewTewewTJ−1δ˜2≤34ε321/3J−1F4/3ewTew2+ε324δ˜2Tδ˜22,12TrWc3TtJ−T2ewewT+ewTewIJ−1Wc3t≤3ε334J−1F4Wc3t4ewTew2+34ε33,12TrG2TJ−T2ewewT+ewTewIJ−1G2≤3ε344J−1F4G24ewTew2+34ε34,−ewTewewTJ−1Wc20t,δ˜1≤34ε351/3ΠewTew2+ε352δ˜1Tδ˜12,where Π=J−1Wc21t4/3Δ502t4/3α314/3Ft4/3+J−1Wc21t4/3Δ512t4/3α324/3Ft4/3. Hence, based on (83), we get(85)ℒV41t≤ewTewewTJ−1u2−ewTewewTJ−1Wc1t+34ε311/3J−1F4/3Wc2tF4/3ewTew2+ε314δ˜1Tδ˜12+34ε321/3J−1F4/3ewTew2+ε324δ˜2Tδ˜22+3ε334J−1F4Wc3t4ewTew2+34ε33+3ε344J−1F4G24ewTew2+34ε34+34ε351/3ΠewTew2+ε354δ˜1Tδ˜12.Design the controlleru2 as(86)u2=−k7+34ε311/3J−1F4/3Wc2tF4/3+34ε321/3J−1F4/3+3ε334J−1F4Wc3t4+3ε344J−1F4G24+34ε351/3Πew+Wc1t+u3,where u3 is designed in the following steps.ℒV41t can be written by(87)ℒV41t≤−k7ewTew2+ε314+ε354δ˜1Tδ˜12+ε324δ˜2Tδ˜22+ewTewewTJ−1u2+34ε33+34ε34.Define Lyapunov function as follows:(88)Vt=V02t+V3t+V41.Combining (44), (77), and (87), the infinitesimal generator of (88) is represented as(89)ℒVt≤−l1−H23−ε314−ε354δ˜1Tδ˜12−l2−ε324δ˜2Tδ˜22−k4epTep−k4evTev−k5R¯3eTR¯3e2−k6−3ε281/34eψ4−k7ewTew2+ewTewewTJ−1u3+ε214R^3F4E04ew1Tew12+14ε28E14ewTew2+H3,where(90)H3=H2+34ε33+34ε34+H02.Recalling thatew1Tew1≤ewTew, u3 is designed as(91)u3=−ε214R^3F4E04+14ε28E14Jew.Then, from (89), (92)ℒVt≤−l1−H23−ε314−ε354δ˜1Tδ˜12−l2−ε324δ˜2Tδ˜22−k4epTep−k4evTev−k5R¯3eTR¯3e2−k6−3ε281/34eψ4−k7ewTew2+H3.Based on (86) and (91), (81) can be written as(93)Jdew=−k7+34ε311/3J−1F4/3Wc2tF4/3+34ε321/3J−1F4/3+3ε334J−1F4Wc3t4+3ε344J−1F4G24+34ε351/3Πewdt−ε214R^3F4E04+14ε28E14Jewdt−Wc20t,δ˜1dt+δ˜2dt−Wc3tdβ1+G2dβ2.By using (40), (41), (22), (27), (68), (75), and (93), we get(94)dδ˜1=−L1+3ε014/34+3ε0224L1G1F4δ˜1dt−L1G1dβ1−δ˙1dt−ε113Ft2FtTdt,dδ˜2=−l2+3ε034/34+3ε0424L2G2F4δ˜2dt−δ˙2dt−L2G2dβ2,dep=evdt,dev=−k1tanhep+ev−k2tanhev+TmmαR¯3e+δ˜1dt+G1dβ1,deψ=−k6eψ+E1ewdt,dR¯3e=R¯3e1tdt+R¯3e2tδ˜1+R¯3e3tdβ1,Jdew=−k7+34ε311/3J−1F4/3Wc2tF4/3+34ε321/3J−1F4/3+3ε334J−1F4Wc3t4+3ε344J−1F4G24+34ε351/3Πewdt−ε214R^3F4E04+14ε28E14Jewdt−Wc20t,δ˜1dt+δ˜2dt−Wc3tdβ1+G2dβ2.Remark 2.
In this paper, the dynamics of helicopter are modeled as the stochastic systems due to the disturbances including the random disturbances. Many control variables of the helicopter stochastic systems are discontinuous. Hence, some advanced control methods cannot be used to design the flight controller, directly, such as the dynamic surface control, fuzzy control, and other control methods. In order to construct the antidisturbance flight controller for the helicopter stochastic systems, the disturbance observer method and backstepping control scheme are applied in our paper.Remark 3.
The control method proposed in this paper shows some weaknesses, such as the complicated controller structure and tedious mathematical derivation while the helicopter stochastic system models are first discussed in this paper and the nonlinear control method is feasible in theory. In the future, we will construct the more advanced antidisturbance flight controller with the simple structure based on the proposed control scheme of this paper.Remark 4.
The structure block diagram of the antidisturbance control scheme is shown in Figure1. The noises from inside and outside of the helicopter are considered to construct the helicopter models and divided as to different parts: the random disturbances and other times-varying disturbances. Then, the stochastic control theory is applied to analyze the stability of the closed-loop systems. And the stochastic antidisturbance flight controller is constructed.Figure 1
The structure block diagram of the antidisturbance control scheme.


## 4.1. Disturbance Observer Designed for Attitude Control

In order to estimate the disturbanceδ2, we design the following disturbance observers:(41)δ^2=σ2+Jρ2w,dσ2=−∂ρ2w∂w−w×Jw+τdt+δ^2dt+σ02dt,where σ2 is an auxiliary variable, σ02 is an auxiliary function to be designed, and ρ2v is a nonlinear function to be designed.Define estimate error asδ˜2=δ^2−δ2. Then, the dynamic of the estimate error can be shown as(42)dδ˜2=−∂ρ2w∂wδ˜2dt+G2dβ2−δ˙2dt+σ02dt.The disturbance observer gain is selected as∂ρ2w/∂w=L2, with L2=−diagl2l2l2, where l2>0.Choose Lyapunov function as follows:(43)V02=14δ˜2Tδ˜22.Similarly withV01, for V02, there exist ε03>0 and ε04>0, such that(44)ℒV02≤−l2δ˜2Tδ˜22+H02,where(45)H02=14ε034μ24+34ε042.Design the auxiliary variableσ02 as follows:(46)σ02=−3ε034/34+3ε0424L2G2F4δ˜2.The error dynamics of disturbance (42) can be rewritten as(47)dδ˜2=−l2+3ε034/34+3ε0424L2G2F4δ˜2dt−δ˙2dt−L2G2dβ2.


## 4.2. Attitude Loop Controller Designed

The predefined tracking signals are position and yaw angle, which are denoted bypr and ψr. After designing the position loop control, the reasonable attitude tracking signals are obtained, which are given as R¯3c and ψr. Moreover, defining γr=R¯3Tψ and γ=θϕψ, we can confirm that ∂γr/∂γ=cosθ≠0, for θ<π/2. Hence, the mapping that from γ to γr is a local topological homeomorphism, which means that the dynamic of γr can represent the dynamic of γ for θ<π/2. The error dynamic of γr is given by(48)dR¯3e=R^3w1dt−dR¯3c,(49)deψ=sinϕcosθq+cosϕcosθrdt−dψr,where w1=pqT and R^3=−R12R11−R22R21.For (48), dR3c can be written as(50)dR3c=du1u1Tu1=1u1Tu1I3−u1u1Tu1Tu1du1.Fordu1, we have(51)du1=dp¨r−dδ^1−k1dtanhn1ep+n2ev−k2dtanhn2ev.From (10), dδ^1 is written as follows:(52)dδ^1=−l¯1δ˜1dt−L1G1dβ1−ε113Ft2Ftdt,where l¯1=l1+3ε014/3/4+3ε022/4L1G1F4.Hence, we get(53)du1=Δ01tdt+Δ02tδ˜1dt+Δ03tdβ1,for x=x1,x2,x3, Tanh2x=diagtanh2x1,tanh2x2y,tanh2x3, where(54)Δ01t=p⋯r−k1n1I+Tanh2n1ep+n2evev−k2n2I+Tanh2n2ev+k1n2I+Tanh2n1ep+n2ev−k1tanhn1ep+n2ev−k2tanhn2ev+TmmαR¯3e+ε113Ft2Ft,Δ02t=l¯1I−k2n2I+Tanh2n2ev+k1n2I+Tanh2n1ep+n2ev,Δ03t=L1G1−k2n2I+Tanh2n2ev+k1n2I+Tanh2n1ep+n2evG1.From (50), (55)dR3c=1u1Δ101t+Δ102tev+Δ103tΔ104t+ε113Ft2Ftdt+1u1Δ12tδ˜1dt+1u1Δ13tdβ1,where(56)Δ101t=I3−u1u1Tu1Tu1p⋯r,Δ102t=−k2I3−u1u1Tu1Tu1I+Tanh2ev,Δ103t=−I3−u1u1Tu1Tu1k2I+Tanh2ev+k1I+Tanh2ep+ev,Δ104t=−k1tanhep+ev−k2tanhev+TmmαR¯3e,Δ12t=I3−u1u1Tu1Tu1l¯1I−k2I+Tanh2ev+k1I+Tanh2ep+ev,Δ13t=I3−u1u1Tu1Tu1L1G1−k2I+Tanh2ev+k1I+Tanh2ep+evG1.Foru1u1TF≤u1Fu1TF=u12, one has(57)I3−u1u1Tu1Tu1F≤I3F+u1u1TFu12≤3+1.Then, we confirm that(58)Δ101t≤3+1p⋯r,Δ102tF≤k23+1I+Tanh2evF≤k232+6,Δ103tF≤3+1k2I+Tanh2ev+k1I+Tanh2ep+evF≤6+23k1+k2,Δ12t≤3+1l¯1+23k1+k2,Δ13t≤3+1L1G1+23k1+k2G1.Thus, (48) can be rewritten as(59)dR¯3e=R^3ew1dt+R^3w1cdt−E01u1Δ101t+Δ102tev+Δ103tΔ104t+ε113Ft2Ftdt−E01u1Δ12tδ˜1dt−E01u1Δ13tdβ1,where E0=100010.Choose Lyapunov functionV21t as(60)V21t=14R¯3eTR¯3e2.The infinitesimal generator ofV21 along with (60) is shown as(61)ℒV21t=R¯3eTR¯3eR¯3eTR^3ew1+R¯3eTR¯3eR¯3eTR^3w1c−R¯3eTR¯3eR¯3eTE01u1Δ101t−R¯3eTR¯3eR¯3eTE01u1Δ102tev−R¯3eTR¯3eR¯3eTE01u1Δ103tΔ104t−ε113R¯3eTR¯3eR¯3eTE01u1Ft2Ft−R¯3eTR¯3eR¯3eTE01u1Δ12tδ˜1+12u12TrΔ13TtE0T2R¯3eR¯3eT+R¯3eTR¯3eIE0Δ13t.From Lemma2, there exist parameters ε21>0, ε22>0, ε231>0, ε232>0, ε24>0, and ε25>0, such that(62)R¯3eTR¯3eR¯3eTR^3ew1≤34ε211/3R¯3eTR¯3e2+ε214R^34ew1Tew12,−R¯3eTR¯3eR¯3eTE01u1Δ101t≤3ε221/34u14/3R¯3eTR¯3e2+14ε22E04Δ101t4,−R¯3eTR¯3eR¯3eTE01u1Δ102tev≤ε2312R¯3eTR¯3e2+ε2322u14R¯3eTR¯3e2evTev2+E04Δ102t48ε232ε231,−R¯3eTR¯3eR¯3eTE01u1Δ12tδ˜1≤3ε241/34u14/3R¯3eTR¯3e2+14ε24E0F4Δ12t4δ˜1Tδ˜12,12u12TrΔ13TtE0T2R¯3eR¯3eT+R¯3eTR¯3eIE0Δ13t≤3ε254u12R¯3eTR¯3e2+12ε25E0Δ13tF4.Furthermore, forTm=mu1, there exist parameters ε26>0 and ε27>0, such that(63)−R¯3eTR¯3eR¯3eTE01u1Δ103tΔ104t≤3ε261/34u14/3R¯3eTR¯3e2+14ε26E0F4Δ103t4k1tanhep+ev+k2tanhev4+3ε271/34E0F4/3Δ103tF4/3R¯3eTR¯3e2+14ε27αF4R¯3eTR¯3e2.Then,ℒV21t satisfies(64)ℒV21t≤H25+H244u14/3+3ε254u12+14ε27αF4R¯3eTR¯3e2+R¯3eTR¯3eR¯3eTR^3w1c+H23δ˜1Tδ˜12−ε113R¯3eTR¯3eR¯3eTE01u1Ft2Ft+ε2322u14R¯3eTR¯3e2evTev2+ε214R^34ew1Tew12+H21,where(65)H21=12ε25E0F43+1L1G1+23k1+k2G1F4+14ε22E0F43+14p⋯r4+14ε26E0F26+23k1+k24k1+k24+E04k232+648ε232ε231,H22=14ε23E0F4k232+62,H23=14ε243+1l¯1F+23k1+k24,H24=3ε221/3+3ε241/3+3ε261/3,H25=34ε211/3+3ε271/34E0F4/36+23k1+k24/3+ε2312.DefineV2t=V1+V21. Based on the above discussion and according to (39) and (64), we have(66)ℒV2t≤−l1−H23δ˜1Tδ˜12−k4epTep−k4evTev+ε214R^34ew1Tew12+R¯3eTR¯3eR¯3eTR^3w1c+H25+H244u14/3+3ε254u12+14ε27αF4R¯3eTR¯3e2+ε12Tm3α2Ft23m3R¯3eTR¯3eR¯3eTαTFt+ε2322u14R¯3eTR¯3e2evTev2−ε113R¯3eTR¯3eR¯3eTE01u1Ft2Ft+H2,where H2=H1+H21. Hence, we design the virtual control law w1c as(67)w1c=−k5+H25+H244u14/3+3ε254u12+14ε27αF4R^3−1R¯3e−ε12Tm3α2Ft23m3R^3−1αTFt−ε2322u14evTev2R^3−1R¯3e+ε11Ft23u1R^3−1E0Ft,where k5>0. Then, we obtain(68)ℒV2t≤−L1−H23δ˜1Tδ˜12−k4epTep−k4evTev−k5R¯3eTR¯3e2+ε214R^34ew1Tew12+H2.From (61), the dynamic of R¯3e can be rewritten as(69)dR¯3e=R¯3e1tdt+R¯3e2tδ˜1dt+R¯3e3tdβ1,where(70)R¯3e1t=R^3ew1−k5+H25+H244u14/3+3ε254u12+14ε27αF4R¯3e−ε12Tm3α2Ft23m3R^3−1αTFt−ε2322u14evTev2R^3−1R¯3e−E0u1Δ101t+Δ102tev+Δ103tΔ104t,R¯3e2t=E0u1Δ12t,R¯3e3t=−E0u1Δ13t.For (37), we have(71)deψ=E11w1+cosϕcosθer+cosϕcosθrc−ψ˙rdt,where E11=0sinϕ/cosθ and rc is the virtual control law to be designed.Choose Lyapunov functionV31 as(72)V31=14eψ4.The derivative ofV31t along with (71) is derived as follows:(73)V˙31t=eψ3E11w1+cosϕcosθer+cosϕcosθrc−ψ˙r.Then, the virtual control lawrc is designed as(74)rc=−cosθcosϕE11w1c−ψ˙r+k3eψ.At this point, the dynamic ofeψ can be written as(75)deψ=−k6eψ+E11ew1+cosϕcosθerdt=−k6eψ+E1ewdt,where E1=0sinϕ/cosθcosϕ/cosθ and ew=p,q,rT−pc,qc,rcT. Thus, based on Lemma 2, there exists ε28>0, such that(76)V˙31t=eψ3−k6eψ+E1ew=−k6eψ4+eψ3E1ew≤−k6−3ε281/34eψ4+14ε28E14ewTew2.DefineV3t=V2t+V31t; combining (68) and (76) yields(77)ℒV3t≤−l1−H23δ˜1Tδ˜12−k4epTep−k4evTev−k5R¯3eTR¯3e+ε214R^34E04ewTew2+H2−k6−3ε281/34eψ4+14ε28E14ewTew2.In what follows, consider the dynamic of angular velocity:(78)Jdw=−w×Jw+ATmρ+BTm+δ2dt+G2dβ2.Definingρ=ATm−1u2−BTm+w×Jw−δ^2, the error dynamic of ew is(79)Jdew=u2+δ˜2dt+G2dβ2−dwc,where dwc is obtained in Appendix. Then, according to (79), (80)Jdew=u2+δ˜2dt+G2dβ2−Wc1tdt−Wc2tδ˜1dt−Wc20t,δ˜1dt−Wc3tdβ1,which implies that(81)dew=J−1u2−J−1Wc1t−J−1Wc2tδ˜1−J−1Wc20t,δ˜1+J−1δ˜2dt−J−1Wc3tdβ1+J−1G2dβ2.Choose Lyapunov function as follows:(82)V41t=14ewTew2.The infinitesimal generator ofV41 along with (81) is obtained as follows:(83)ℒV41t=ewTewewTJ−1u2−ewTewewTJ−1Wc1t−ewTewewTJ−1Wc2tδ˜1−ewTewewTJ−1Wc20t,δ˜1+ewTewewTJ−1δ˜2+12TrWc3TtJ−T2ewewT+ewTewIJ−1Wc3t+12TrG2TJ−T2ewewT+ewTewIJ−1G2.From Lemma2, there exist parameters ε31>0, ε32>0, ε33>0, ε34>0, and ε35>0 such that(84)−ewTewewTJ−1Wc2tδ˜1≤34ε311/3J−1F4/3Wc2tF4/3ewTew2+ε314δ˜1Tδ˜12,ewTewewTJ−1δ˜2≤34ε321/3J−1F4/3ewTew2+ε324δ˜2Tδ˜22,12TrWc3TtJ−T2ewewT+ewTewIJ−1Wc3t≤3ε334J−1F4Wc3t4ewTew2+34ε33,12TrG2TJ−T2ewewT+ewTewIJ−1G2≤3ε344J−1F4G24ewTew2+34ε34,−ewTewewTJ−1Wc20t,δ˜1≤34ε351/3ΠewTew2+ε352δ˜1Tδ˜12,where Π=J−1Wc21t4/3Δ502t4/3α314/3Ft4/3+J−1Wc21t4/3Δ512t4/3α324/3Ft4/3. Hence, based on (83), we get(85)ℒV41t≤ewTewewTJ−1u2−ewTewewTJ−1Wc1t+34ε311/3J−1F4/3Wc2tF4/3ewTew2+ε314δ˜1Tδ˜12+34ε321/3J−1F4/3ewTew2+ε324δ˜2Tδ˜22+3ε334J−1F4Wc3t4ewTew2+34ε33+3ε344J−1F4G24ewTew2+34ε34+34ε351/3ΠewTew2+ε354δ˜1Tδ˜12.Design the controlleru2 as(86)u2=−k7+34ε311/3J−1F4/3Wc2tF4/3+34ε321/3J−1F4/3+3ε334J−1F4Wc3t4+3ε344J−1F4G24+34ε351/3Πew+Wc1t+u3,where u3 is designed in the following steps.ℒV41t can be written by(87)ℒV41t≤−k7ewTew2+ε314+ε354δ˜1Tδ˜12+ε324δ˜2Tδ˜22+ewTewewTJ−1u2+34ε33+34ε34.Define Lyapunov function as follows:(88)Vt=V02t+V3t+V41.Combining (44), (77), and (87), the infinitesimal generator of (88) is represented as(89)ℒVt≤−l1−H23−ε314−ε354δ˜1Tδ˜12−l2−ε324δ˜2Tδ˜22−k4epTep−k4evTev−k5R¯3eTR¯3e2−k6−3ε281/34eψ4−k7ewTew2+ewTewewTJ−1u3+ε214R^3F4E04ew1Tew12+14ε28E14ewTew2+H3,where(90)H3=H2+34ε33+34ε34+H02.Recalling thatew1Tew1≤ewTew, u3 is designed as(91)u3=−ε214R^3F4E04+14ε28E14Jew.Then, from (89), (92)ℒVt≤−l1−H23−ε314−ε354δ˜1Tδ˜12−l2−ε324δ˜2Tδ˜22−k4epTep−k4evTev−k5R¯3eTR¯3e2−k6−3ε281/34eψ4−k7ewTew2+H3.Based on (86) and (91), (81) can be written as(93)Jdew=−k7+34ε311/3J−1F4/3Wc2tF4/3+34ε321/3J−1F4/3+3ε334J−1F4Wc3t4+3ε344J−1F4G24+34ε351/3Πewdt−ε214R^3F4E04+14ε28E14Jewdt−Wc20t,δ˜1dt+δ˜2dt−Wc3tdβ1+G2dβ2.By using (40), (41), (22), (27), (68), (75), and (93), we get(94)dδ˜1=−L1+3ε014/34+3ε0224L1G1F4δ˜1dt−L1G1dβ1−δ˙1dt−ε113Ft2FtTdt,dδ˜2=−l2+3ε034/34+3ε0424L2G2F4δ˜2dt−δ˙2dt−L2G2dβ2,dep=evdt,dev=−k1tanhep+ev−k2tanhev+TmmαR¯3e+δ˜1dt+G1dβ1,deψ=−k6eψ+E1ewdt,dR¯3e=R¯3e1tdt+R¯3e2tδ˜1+R¯3e3tdβ1,Jdew=−k7+34ε311/3J−1F4/3Wc2tF4/3+34ε321/3J−1F4/3+3ε334J−1F4Wc3t4+3ε344J−1F4G24+34ε351/3Πewdt−ε214R^3F4E04+14ε28E14Jewdt−Wc20t,δ˜1dt+δ˜2dt−Wc3tdβ1+G2dβ2.Remark 2.
In this paper, the dynamics of helicopter are modeled as the stochastic systems due to the disturbances including the random disturbances. Many control variables of the helicopter stochastic systems are discontinuous. Hence, some advanced control methods cannot be used to design the flight controller, directly, such as the dynamic surface control, fuzzy control, and other control methods. In order to construct the antidisturbance flight controller for the helicopter stochastic systems, the disturbance observer method and backstepping control scheme are applied in our paper.Remark 3.
The control method proposed in this paper shows some weaknesses, such as the complicated controller structure and tedious mathematical derivation while the helicopter stochastic system models are first discussed in this paper and the nonlinear control method is feasible in theory. In the future, we will construct the more advanced antidisturbance flight controller with the simple structure based on the proposed control scheme of this paper.Remark 4.
The structure block diagram of the antidisturbance control scheme is shown in Figure1. The noises from inside and outside of the helicopter are considered to construct the helicopter models and divided as to different parts: the random disturbances and other times-varying disturbances. Then, the stochastic control theory is applied to analyze the stability of the closed-loop systems. And the stochastic antidisturbance flight controller is constructed.Figure 1
The structure block diagram of the antidisturbance control scheme.


## 5. Stability Analysis

In this section, the asymptotic boundedness in mean square of the composite closed-loop system (94) is guaranteed under the controllers u1 and u2.Theorem 1.
Considering helicopter system (1) with disturbances, design the disturbance observers as (10) and (41) and the controllers as (20) and (86). The composite closed-loop system (94) is asymptotically bounded in mean square. If there exist parameters k1>0, k2>0, k3>0,k4>0, k5>0, k6>0, k7>0, k8>0, k9>0, l1>0, l2>0, ε01>0, ε02>0, ε03>0, ε04>0, ε11>0, ε12>0, ε21>0, ε22>0, ε23>0, ε231>0, ε232>0, ε24>0, ε25>0, ε26>0, ε27>0, ε28>0, ε31>0, ε32>0, ε33>0, ε34>0, and ε35>0, such that l1−H23−ε31/4−ε35/4>0, l2−ε32/4>0, k4−H22>0, k6−3ε281/3/4>0 hold.Proof.
Consider the Lyapunov functionVt as follows:(95)Vt=V02t+V3t+V41t=14δ˜1Tδ˜12+14δ˜2Tδ˜22+k1∫0ep+evtanhμdμ+k2∫0evtanhμdμ+12evTev+14R¯3eTR¯3e2+14eψ4+14ewTew2.
Sincehx=tanhx/x is a continuous function, for any x¯>0 and 0≤x≤x¯, there exists a positive number β1=βx¯, such that β1x≤tanhx≤x. Hence, we have(96)k1∫0ep+evtanhμdμ+k2∫0evtanhμdμ≥k8epTep+k8evTev,where k8=maxeig1/2k1β11/2k1β11/2k1β11/2k1β1+1/2k2β1>0. Moreover,(97)k1∫0ep+evtanhμdμ+k2∫0evtanhμdμ≤k9epTep+k9evTev,where k9=mineig1/2k11/2k11/2k11/2k1+1/2k2>0.
Then, we confirm that(98)14δ˜1Tδ˜12+14δ˜2Tδ˜22+k8epTep+k8+12evTev+14R¯3eTR¯3e2+14eψ4+14ewTew2≤Vt≤14δ˜1Tδ˜12+14δ˜2Tδ˜22+k9epTep+k9+12evTev+14R¯3eTR¯3e2+14eψ4+14ewTew2,which satisfies (8) in Lemma 1 with p=2.
From the discussion in Sections3 and 4, we have(99)ℒVt≤−l1−H23−ε314−ε354δ˜1Tδ˜12−l2−ε324δ˜2Tδ˜22−k4epTep−k4evTev−k5R¯3eTR¯3e2−k6−3ε281/34eψ4−k7ewTew2+H3≤−λVt+H3,where(100)λ=min4l1−H23−ε314−ε354,4l2−ε324,k4k9,k4k9+1/2,4k5,4k6−3ε281/34,4k7,which implies that Vt satisfies (9) in Lemma 1 with p=2.
Consequently, based on the discussion above, the composite closed-loop system (94) is asymptotically bounded in mean square.


## 6. Conclusion

This paper studies the tracking control issue for nonlinear helicopter systems under stochastic disturbances and time-varying disturbances. These disturbances are attenuated and rejected via the nonlinear disturbance observer control method and stochastic control theory, and the backstepping control approach is adopted for nonlinear helicopter systems. The problems of “dimension explosion” are analyzed for complicated nonlinear control systems.Since the helicopter systems are modeled as the stochastic systems, many advanced nonlinear control methods can be directly used. The phenomenon of “dimension explosion” is inevitable in nonlinear controller constructing process. In the future, we will combine some advanced nonlinear control methods and stochastic control theory and propose the more advanced antidisturbance flight control scheme with simple structure.



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