Antidisturbance Control for Helicopter Stochastic Systems

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. )en, 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. )e 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.


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][2][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. e 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. e 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. erefore, the nonlinear control system theories are useful approaches to solve the issues of flight control for helicopters.
e 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 parametervarying 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 timevarying 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. e 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. e finitetime 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. e 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. e 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. e 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. erefore, 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. e organization of this paper is standard. e problem statement is stated in Section 2. e position loop control issues are discussed in Section 3. e attitude loop control problems are studied in Section 4. e stability analysis of the main result is given in Section 5. e conclusions are presented in Section 6.

Problem Statement
Consider the following helicopter dynamic system: where p and v denote the position and velocity in inertial frame, respectively, g is the gravitational acceleration, e 3 � 0 0 1 T is a unit vector, and R is the rotation matrix from body frame to inertial frame and defined by where C· and S · denote cos(·) and sin(·) with respective variables. ϕ(t), θ(t), and φ(t) are roll, pitch, and yaw angles in body frame, respectively. w � p(t) q(t) r(t) 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 � diag J xx J yy J zz . 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: 2 Mathematical Problems in Engineering with Q m � C k T 1.5 m + D k . C m , L z , H z , L y , L x , H x , and D k are known constants. β 1 and β 2 are independent one-dimensional standard Wiener processes, (dβ 1 /dt) � ξ 1 and (dβ 2 /dt) � ξ 2 , where ξ 1 and ξ 2 are independent one-dimensional stochastic noises. G 1 ξ 1 and G 2 ξ 2 are the stochastic accelerated velocity and stochastic angular acceleration generated by the stochastic force and stochastic moment, G 1 and G 2 are known and bounded weight parameters, and δ 1 and δ 2 are disturbances with bounded derivatives, which are ‖ _ δ 1 ‖ ≤ μ 1 and ‖ _ δ 2 ‖ ≤ μ 2 . e 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: where x is the system state, f ∈ L 1 (R + ; R n ) and g ∈ L 2 (R + ; R n×m ) are known functions, and β is the onedimensional standard Wiener process. For V(t, x) ∈ C 2,1 (R; R n × R + ), the infinitesimal generator along (4) is defined by where Tr is the trace of a matrix, and Definition 1 (see [28]). Let p > 0. System (4) is said to be asymptotically bounded in p-th moment if there is a positive constant H such that for all (t 0 , x 0 ) ∈ R + × R n . When p � 2, we say system (4) is asymptotically bounded in mean square.

Lemma 1.
For system (4), assume that there exists a function en, there exists a unique strong solution x(t) � x(t; x 0 , t 0 ) of system (4) for each x(t 0 ) � x 0 ∈ R n , and system (4) is p-th moment exponentially practically stable, where p � min p 1 , . . . , p n .

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 disturbances δ 1 and δ 2 .

Disturbance Observer Designed for Position Control.
In what follows, the disturbance observer is designed to estimate the disturbance δ 1 , where σ 1 is an auxiliary variable, σ 11 is an auxiliary function to be designed, and ρ 1 (v) is a nonlinear function to be designed. Estimate error is defined as δ 1 � δ 1 − δ 1 . en, the dynamic of the estimate error is given by e disturbance observer gain is selected as (zρ 1 (v)/zv) � L 1 , with L 1 � − diag l 1 l 1 l 1 , where l 1 > 0.
Choose the Lyapunov function as follows: Mathematical Problems in Engineering (12) en, the infinitesimal generator of V 01 along with (11) is shown as follows: From Lemma 2, there exist parameters ε 01 > 0 and ε 02 > 0 such that Design σ 11 as where σ 12 is introduced in the following steps. Hence, from (15) and ‖ _ δ 1 ‖ ≤ μ 1 , we have where

Position Loop Controller Designed.
Consider the position loop system as follows: e predefined position is p r � (x r , y r , z r ), define the tracking error as e p � p − p r . e dynamic of the error system is given by Define the controller T , and where k 1 > 0, k 2 > 0, n 1 > 0, and n 2 > 0 are parameters to be designed. At this point, R 3c and T m can be constructed as en, the closed-loop position loop system is written as Remark 1. In fact, the helicopter should not be overturn under controller (21), for the continuity of cos ϕ cos θ. So that, we guarantee cos ϕ cos θ > 0 to ensure that the helicopter will not be overturn. Meanwhile, the defined signal R 33c should also satisfy R 33c > 0. us, the parameters k 1 and k 2 are designed to satisfy g + € z r + δ 1 e − k 1 − k 2 > 0. In fact, the acceleration of disturbance in z axis is far less than acceleration of gravity.
Due to R 33 > 0 and R 33c > 0, we have where e velocity tracking error equation (23) can be rewritten as Choose Lyapunov function V 11 (t) as follows: en, the infinitesimal generator of V 11 is represented as

Mathematical Problems in Engineering
In fact, according to Lemma 2, there exist parameters ε 11 > 0 and ε 12 > 0, such that Define Lyapunov function as follows: Based on (16) and (34), we obtain Hence, σ 12 is designed as en,

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 disturbances δ 1 and δ 2 .

Disturbance Observer Designed for Attitude Control.
In order to estimate the disturbance δ 2 , we design the following disturbance observers: where σ 2 is an auxiliary variable, σ 02 is an auxiliary function to be designed, and ρ 2 (v) is a nonlinear function to be designed. Define estimate error as δ 2 � δ 2 − δ 2 . en, the dynamic of the estimate error can be shown as e disturbance observer gain is selected as Choose Lyapunov function as follows: Similarly with V 01 , for V 02 , there exist ε 03 > 0 and ε 04 > 0, such that where Design the auxiliary variable σ 02 as follows: e error dynamics of disturbance (42) can be rewritten as (47)

Attitude Loop Controller
Designed. e predefined tracking signals are position and yaw angle, which are denoted by p r and ψ r . After designing the position loop control, the reasonable attitude tracking signals are obtained, which are given as R 3c and ψ r . Moreover, defining c r � R T 3 ψ and c � θ ϕ ψ , we can confirm that ‖zc r /zc‖ � cos θ ≠ 0, for |θ| < (π/2). Hence, the mapping that from c to c r is a local topological homeomorphism, which means that the dynamic of c r can represent the dynamic of c for |θ| < (π/2). e error dynamic of c r is given by where For (48), dR 3c can be written as For du 1 , we have From (10), dδ 1 is written as follows: where From (50), where For Mathematical Problems in Engineering en, we confirm that us, (48) can be rewritten as e infinitesimal generator of V 21 along with (60) is shown as From Lemma 2, there exist parameters ε 21 > 0, ε 22 > 0, ε 231 > 0, ε 232 > 0, ε 24 > 0, and ε 25 > 0, such that 10 Mathematical Problems in Engineering Furthermore, for T m � m‖u 1 ‖, there exist parameters ε 26 > 0 and ε 27 > 0, such that en, LV 21 (t) satisfies Define V 2 (t) � V 1 + V 21 . Based on the above discussion and according to (39) and (64), we have where H 2 � H 1 + H 21 . Hence, we design the virtual control law w 1c as where k 5 > 0. en, we obtain From (61), the dynamic of R 3e can be rewritten as For (37), we have where E 11 � 0 sin ϕ/cos θ and r c is the virtual control law to be designed. Choose Lyapunov function V 31 as e derivative of V 31 (t) along with (71) is derived as follows: en, the virtual control law r c is designed as At this point, the dynamic of e ψ can be written as where E 1 � 0 (sin ϕ/cos θ) (cos ϕ/cos θ) and e w � (p, q, r) T − (p c , q c , r c ) T . us, based on Lemma 2, there exists ε 28 > 0, such that Define V 3 (t) � V 2 (t) + V 31 (t); combining (68) and (76) yields In what follows, consider the dynamic of angular velocity: (78)

Mathematical Problems in Engineering
Based on (86) and (91), (81) can be written as By using (40), (41), (22), (27), (68), (75), and (93), we get 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.
e 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.
e structure block diagram of the antidisturbance control scheme is shown in Figure 1. e 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. en, the stochastic control theory is applied to analyze the stability of the closed-loop systems. And the stochastic antidisturbance flight controller is constructed.
en, we confirm that which satisfies (8) in Lemma 1 with p � 2. From the discussion in Sections 3 and 4, we have  where which implies that V(t) 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.

Conclusion
is paper studies the tracking control issue for nonlinear helicopter systems under stochastic disturbances and timevarying disturbances. ese 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. e 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. e 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.