Composite block backstepping trajectory tracking control for disturbed unmanned helicopters

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Composite block backstepping trajectory tracking control for disturbed unmanned helicopters ✩

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Xiangyu Wang

a,b,∗

, Xin Yu

a, b

, Shihua Li

a, b

, Jiyu Liu

c

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a

School of Automation, Southeast University, Nanjing 210096, PR China b Key Laboratory of Measurement and Control of Complex Systems of Engineering, Ministry of Education, PR China c Aviation Key Laboratory of Science and Technology on Aircraft Control, Facri, Xian 710065, PR China

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Article history: Received 12 October 2018 Received in revised form 29 November 2018 Accepted 14 December 2018 Available online xxxx Keywords: Unmanned helicopters Disturbances Trajectory tracking control Feedforward-feedback composite control Blocking backstepping control Generalized proportional integral observer

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a b s t r a c t

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In this paper, the position and yaw angle trajectory tracking control problem is studied for unmanned helicopters subject to both matched and mismatched disturbances. To achieve the trajectory tracking goal, a feedforward-feedback composite control scheme is proposed based on the combination of the generalized proportional integral observer and the block backstepping control techniques. The controller design process mainly consists of two stages. In the first stage, some generalized proportional integral observers are developed for the helicopter system to estimate the mismatched, matched disturbances and their (higher-order) derivatives. In the second stage, the composite controller is designed by integrating the block backstepping control method and the disturbance estimates together. The proposed composite scheme guarantees asymptotic tracking performances for the position and yaw angle of the helicopter to the desired trajectories even in the presence of fast time-varying disturbances. Numerical simulations demonstrate the effectiveness of the proposed composite control scheme. © 2018 Elsevier Masson SAS. All rights reserved.

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1. Introduction

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Among various types of unmanned aerial vehicles, unmanned helicopters have significant and remarkable capabilities of vertical taking-off and landing, hovering, as well as flying in narrow, complex and unknown environments at low altitudes and speeds [1,2]. On one hand, because of these prominent advantages, unmanned helicopters have wide application prospects in military and civil fields and their flight control problems have attracted considerable attention in recent years [1–3]. On the other hand, unmanned helicopters are known as multi-input multi-output underactuated nonlinear systems with strong dynamic couplings among different channels [3,4]. Moreover, the flight control of helicopters is severely affected by disturbances, including both internal disturbances (such as model parameter uncertainties and unmodeled dynamics [5]) and external disturbances (such as unknown wind gusts and ground effects [6]). Hence, it is significant and also chal-

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✩

This work was supported by the Aeronautical Science Foundation of China under Grant 20170769004, the National Natural Science Foundation of China under Grants 61873060, 61503078, 61473080, 61633003, the Natural Science Foundation of Jiangsu Province under Grant BK20150626, and the Priority Academic Program Development of Jiangsu Higher Education Institutions. Corresponding author at: School of Automation, Southeast University, Nanjing 210096, PR China. E-mail address: [email protected] (X. Wang).

*

https://doi.org/10.1016/j.ast.2018.12.019 1270-9638/© 2018 Elsevier Masson SAS. All rights reserved.

lenging to study reliable and high-performance flight control for unmanned helicopters. In recent years, many kinds of control methods have been proposed for flight control of unmanned helicopters [7–20]. In general, these control methods can be divided into two categories, namely, linear control methods and nonlinear control methods. Linear control methods, such as proportional-integral-derivative (PID) control [7,8] and linear-quadratic regulator [8–10], are designed based on the approximate linearized models of the helicopters in certain flight envelopes. Since the helicopter systems are inherently nonlinear systems with uncertainties and external disturbances, high flight control performance is usually difficult to be achieved by using linear control methods. In order to achieve better control performance, several nonlinear control methods have been presented for flight control of unmanned helicopters currently, such as robust H ∞ control [11,12], backstepping control [13–16], adaptive control [17], sliding-mode control [18,19] and model-based predictive control [20]. As a typical nonlinear control method, backstepping control is popular in the field on control of unmanned helicopters [13–16], due to its flexibility, systematic design process, and good extensibility [21–23]. The backstepping control design process is a recursive procedure [21]. Specifically, in conventional backstepping control design, for a nth-order system, the system is divided into n first-order subsystems. At each intermediate design step i with

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respect to the ith-order subsystem (composed of the first-order subsystems from the first to the ith), a virtual controller is developed to make this subsystem stable. After such a recursive process from the top to the bottom, the actual controller is designed at the last step n to make the entire closed-loop system stable. When designing a virtual controller, the (higher-order) derivatives of the formerly designed virtual controllers are required. Thus, the complexity of the controller increases with the increase of the system order and complexity of the system model. In practice, many control systems are high-order systems with complex dynamics, such as unmanned helicopter systems. Hence, some more concise control methods are desired. A feasible way is to use the block backstepping control method [24–26]. In contrast to conventional backstepping control, the recursive design of block backstepping control is conducted among the system blocks (where the original system is divided into several blocks, which may cover several first-order subsystems of the original system) rather than among all the first-order subsystems. Based on such block recursive control design process, the design steps and complexity are reduced and a simpler controller is derived finally. Therefore, both the reliability and practicability of the controller are enhanced accordingly. It is worth pointing out that neither backstepping control nor other aforementioned control methods proposed for unmanned helicopters belong to feedback control. Generally speaking, feedback control realizes disturbance rejection through feedback regulation on the tracking errors between the measured outputs and the setpoints. It is not direct and fast enough to handle the effects of the disturbances and generally uses a robust way to reject disturbances with sacrifice of the nominal control performances. To enhance the anti-disturbance ability of the closed-loop systems, an effective way is to use feedforward-feedback composite control with disturbance compensations [27–35]. In practice, the disturbances are often difficult to be directly measured or even if they can be measured, the sensors are expensive. Considering this fact, a feasible manner is to use soft measurements on the disturbances, namely, to construct some estimators to estimate the disturbances. Representative disturbance estimators include disturbance observers [27–32], extended-state observers [33,34] and some other types of estimators [29,35]. Based on disturbance observers and extended-state observers, disturbance observer-based control (DOBC) [27–32] and active disturbance rejection control (ADRC) [33,34] methods are formed (with their combinations with the baseline feedback controllers), respectively. Compared with the pure feedback control, composite control based on disturbance soft measurements is more direct, fast and effective to handle disturbances and often retains the nominal control performances [27–35]. Due to the aforementioned nice features, some composite control methods have been utilized for control of unmanned helicopters [36–42]. For example, [36] proposed a disturbance observer and command filtered backstepping based composite control strategy to make the helicopter track the desired horizontal position asymptotically in the presence of parameter uncertainties and wind disturbances. In these literature, some disturbance observers were presented to estimate the disturbances of unmanned helicopters. Nevertheless, owing to the essential structures of these disturbance observers, in most cases, only constant or slowly timevarying (of which the time derivatives converge to zero as time goes to infinity) disturbances can be accurately estimated. As for relatively fast time-varying disturbances (of which the time derivatives are bounded), only estimates with bounded estimation errors can be obtained. Turning to more fast time-varying disturbances (e.g., parabolic or higher-order disturbances), these disturbance observers are not effective anymore and the estimates even diverge. To this end, a kind of disturbance estimators that can provide accurate estimates for more general types of disturbances is desired.

The generalized proportional integral observer (GPIO) technique is a suitable tool to complete this mission [35,43–45]. A GPIO has the ability to estimate fast time-varying disturbances, as well as their (higher-order) time derivatives. Specifically, the GPIO technique is able to precisely estimate many kinds of disturbances (such as constant, ramp, parabolic, and higher-order disturbances in polynomial forms of time) and provides estimates for more common disturbances which can be (at least locally) modeled by Taylor polynomials plus residual terms (such as sinusoidal disturbances and some other differentiable disturbances) to a considerable accuracy [35,43–45]. In this paper, the position and yaw angle trajectory tracking control problem is studied for unmanned helicopters with disturbances (including both mismatched and matched disturbances, where the matched disturbances are in the same channels as the control inputs while the mismatched disturbances are in different channels from the control inputs). Considering the advantages of block backstepping control in feedback control design and GPIO technique in disturbance estimation, these two techniques are organically integrated together to develop a feedforward-feedback composite tracking control scheme for the disturbed unmanned helicopter system such that the trajectory tracking control objective is asymptotically achieved. First of all, to conduct control design, the unmanned helicopter system is divided into three disturbed subsystems with relatively slight couplings, namely, a second-order altitude subsystem, a second-order yaw subsystem and a forth-order horizontal subsystem. Then the tracking controller design is composed of two stages. In the first stage, for the three subsystems, some generalized proportional integral observers (GPIOs) are designed to precisely estimate the mismatched and matched disturbances and their (higher-order) derivatives, respectively. In the second stage, based on the combination of the block backstepping control method and the disturbance estimates (especially for the forth-order horizontal subsystem), a feedforwardfeedback composite tracking control scheme is proposed. Under the designed composite controller, the position and yaw angle of the helicopter track the desired position and yaw angle trajectories asymptotically. The main contributions of this paper are twofold. Firstly, in this paper, the position and yaw angle trajectory tracking control problem of disturbed unmanned helicopters is investigated and solved under a feedforward-feedback composite control scheme by combining the GPIO technique and block backstepping control method together. Compared with the controllers designed in [13, 14] based on the conventional backstepping control design philosophy (namely, design step by step along the first-order subsystems of the original system), the proposed controller in this paper has a simpler structure and provides stronger anti-disturbance ability for the closed-loop system due to the involvement of the disturbance compensation terms. Secondly, with the help of the GPIO technique, accurate estimates can be offered for many kinds of disturbances by the GPIOs, which are not limited to constant or slowly time-varying disturbances anymore. Even in the presence of fast time time-varying disturbances (such as parabolic and higherorder disturbances), in this paper, it is proved that the proposed composite control scheme still guarantees precise tracking performances of the helicopter system, based on the fact that the disturbances are precisely estimated by the GPIOs. This overcomes the common problem of the disturbance observers designed in [37–41] that only constant or slowly time-varying disturbances can be precisely estimated and only bounded trajectory tracking error results are obtained in the cases with (relatively) fast time-varying disturbances. The rest of this paper is organized as follows. The six-degreeof-freedom (6-DOF) rigid-body model of the unmanned helicopter and the problem formulation are described in Section 2. In Sec-

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tion 3, the main result of the paper is presented. Section 4 shows the numerical simulations. Finally, conclusions are drawn in Section 5.

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2. Preliminaries and problem formulation

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2.1. Mathematical notations

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The abbreviations S t , C t and T t with t ∈ R represent the trigonometric functions sin(t ), cos(t ) and tan(t ), respectively. The operand  · 2 represents the Euclidean norm. The matrix I n denotes n × n identity matrix. The notation “×” denotes the cross product of two vectors.

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2.2. Lemmas and definitions

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Consider the system

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x˙ = f (t , x, u ),

(1)

where f : [0, ∞) × R × R → R is piecewise continuous in t and locally Lipschitz in x and u. The input u (t ) is a piecewise continuous and bounded function of t, ∀t ≥ 0. n

m

n

Definition 1. [21] System (1) is input-to-state stable (ISS) if there exist a class KL function β and a class K function γ such that for any initial state x(t 0 ) and bounded input u (t ), the solution x(t) exists for all t ≥ t 0 and satisfies x(t )2 ≤ β(x(t 0 )2 , t − t 0 ) + γ supt0 ≤τ ≤t u (τ )2 . Lemma 1. [21] For system (1), if the unforced system x˙ = f (t , x, 0) has globally exponentially stable equilibrium point at the origin x = 0, then the system (1) is ISS.

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Lemma 2. [21] If the system (1) is ISS and limt →∞ u (t ) = 0, then the system state will asymptotically converge to zero, that is, limt →∞ x(t ) = 0.

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Lemma 3. [21] For system (1), let V : [0, ∞) × Rn → R be a continuously differentiable function such that α1 (x2 ) ≤ V (t , x) ≤ α1 (x2 ), ∂V ∂V ∂ t + ∂ x f (t , x, u ) ≤ − W 3 (x), and ∀x2 ≥ ρ (u 2 ) > 0 for all (t , x, u ) ∈ [0, ∞) × Rn × Rm , where α1 and α2 are K∞ functions, ρ is a K function, and W 3 (x) is a continuous positive-definite function on Rn . Then the system (1) is ISS.

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2.3. Modeling of a 6-DOF rigid helicopter

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Unmanned helicopters are highly nonlinear systems with significant dynamic couplings. A practical way to deal with modeling of a helicopter is to establish a simplified 6-DOF rigid-body model by treating the other trivial factors affecting the helicopter system as uncertainties or disturbances [46,47]. In order to establish the model of the helicopter, two reference frames are defined. The first one is the inertial frame defined as I = { O e , X e , Y e , Z e }, and the second is the body-fixed frame defined as B = { O b , X b , Y b , Z b }, where the origin is located at the gravity center of the helicopter. The two frames are shown in Fig. 1. The 6-DOF rigid-body model of the unmanned helicopter is expressed as [37,41]

⎧˙ P = V, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ V˙ = ge + 1 R ()( F + F ), 3 d m ⎪ ⎪ ˙ = H () ,  ⎪ ⎪ ⎪ ⎩ ˙ = − J −1 × J + J −1 ( + d ),

(2)

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Fig. 1. Helicopter frames.

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where P = [x, y , z] T and V = [u , v , w ] T are, respectively, positions and velocities in the inertial frame,  = [φ, θ, ψ] T represents the vector of three Euler angles, which are roll angle, pitch angle, and yaw angle, respectively. = [ p , q, r ] T is the vector of angular velocities in the body-fixed frame, g is the acceleration of gravity, e 3 = [0, 0, 1] T , m represents the mass of the helicopter, J = diag{ J xx , J y y , J zz } is the diagonal inertia matrix with respect to the body-fixed frame. F and represent the external force and torque expressed in the body-fixed frame, which are the main power of the helicopter and are originated from the main rotor thrust and the tail rotor thrust. F d and d represent the lumped force and torque disturbances including both internal disturbances (such as model parameter uncertainties and unmodeled dynamics) and external disturbances (such as wind gusts and ground effects). R () is the rotation matrix from the body-fixed frame to the inertia frame, which is given as

⎡

Cθ Cψ R () = ⎣ C θ S ψ −Sθ

Sφ Sθ Cψ − Cφ Sψ Sφ Sθ Sψ + Cφ Cψ Sφ Cθ

⎤

Cφ Sθ Cψ + Sφ Sψ Cφ Sθ Sψ − Sφ Cψ ⎦ . Cφ Cθ

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The attitude kinematic matrix H () is defined as

⎡

1 H () = ⎣ 0 0

Sφ Tθ Cφ S φ /C θ

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⎤

103

Cφ T θ −Sφ ⎦ . C φ /C θ

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The force vector F is expressed as

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F = [0, 0, T ] T ,

(3)

where T is the main rotor thrust controlled by the collective pitch of the main rotor δcol and T = m(− g + Z w w + Z col δcol ). The torque vector is described as

⎡

= J⎣

⎤

La a + Lb b ⎦, Ma a + Mb b N r r + N col δcol + N ped δ ped

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(4)

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where δ ped is the collective pitch of the tail rotor, and a and b are the longitudinal and lateral flapping angles of the main rotor. The flapping angles are originally driven by the lateral cyclic δlat and the longitudinal cyclic δlon . In steady state, the relationship between the flapping angles a, b and the control inputs δlat , δlon is approximated by [49]

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a = −τm q + A lon δlon + A lat δlat , b = −τm p + B lon δlon + B lat δlat ,

(5)

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where τm is the main rotor’s dynamics time constant. By combining (4) and (5) together, the torque vector can be further expressed as

= J ( A + BU ),

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(6)

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where A and −τm L b −τm L a −τm M b −τm M a 0

0

B are

0 0 Nr

coefficient

and B =

0 0 N col

matrices Llon M lon 0

Llat M lat 0

as A = 0 0

, and

N ped

U = [δcol , δlon , δlat , δ ped ] T denotes the control input vector, and Llat = L a Alat + L b B lat , Llon = L a Alon + L b B lon , Mlat = M a Alat + M b B lat , Mlon = M a Alon + M b B lon . In reality, the nominal values of the parameters Z w , Z col , τm , L a , L b , M a , M b , N r , N col , N ped , Llon , Llat , Mlon , Mlat in the above expressions can be obtained by system identification. By combining (2), (3) and (6) together, the 6-DOF rigid-body model of the helicopter is described as

⎧ P˙ = V , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ V˙ = ge 3 + R ()e 3 (− g + Z w w + Z col δcol ) + d V , ⎪ ⎨ ˙ = H () ,  ⎪ ⎪ ⎪ ˙ = − J −1 × J + A + BU + d , ⎪ ⎪ ⎪ ⎪  T ⎩ Y = PT,ψ ,

(7)

1 where d V = m R () F d = [du , d v , d w ] T and d = J −1 d = [d p , dq , dr ] T represent the lumped disturbances acting on the unmanned helicopter, and Y represents the system output. It is worth noticing that since du , d v are in different channels from the control inputs, they are mismatched disturbances. Denote r i , j as the element of ith row and jth column for the rotation matrix R (), i , j = 1, 2, 3. The time derivative of R () is

R˙ () = R () S ( ),

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⎧ Sφ Cφ ⎪ ⎪ q+ r, ⎨ ψ˙ = Cθ Cθ J y y − J xx

⎪ ⎪ ⎩ r˙ = −

J zz

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(9) pq + N r r + N col δcol + N ped δ ped + dr ,

⎡

0 S ( ) = ⎣ r −q

−r 0 p

q

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(c) Horizontal subsystem:

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⎧ P˙ 1 = V 1 , ⎪ ⎪ ⎪ ⎪ ⎨ V˙ 1 = R 1 (− g + Z w w + Z col δcol ) + d V , 1 ⎪ R˙ 1 = H 1 1 , ⎪ ⎪ ⎪ ⎩˙ 1 = f 1 ( 1 ) + A 1 1 + B 1 U 1 + d 1 ,

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(10)

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where the variables and parameters are P 1 = [x, y ] , V 1 = [u , v ] T , R 1 = [r1,3 , r2,3 ] T , 1 = [ p , q] T , U1 = [δlon , δlat ]T , f 1 ( 1 ) =  T −r1,2 r1,1 J −J − zz J xx y y qr , − J xxJ−y yJ zz pr , H1 = , A1 = −r2,2 r2,1     Llon Llat −τm L b −τm L a , and B 1 = . For simplicity, de−τm M b −τm M a Mlon Mlat note d1 = d w , d2 = dr , d3 = du , d4 = d v , d5 = d p , d6 = dq . Then d V 1 = [d3 , d4 ] T , d 1 = [d5 , d6 ] T . T

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3.1. GPIOs design

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Before proceeding, the following assumptions are made on disturbances of system (7).

⎤

Assumption 1. For the helicopter system (7), the disturbances di (i = 1, · · · , 6) have the following forms

−p ⎦ . 0

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Remark 1. It is generally considered that the roll and pitch angles of the helicopter system (7) satisfy that |θ(t )| < π2 and |φ(t )| < π2 for any t ≥ 0 during the normal flight conditions. In practice, these two angles almost never be close to ± π2 by implementing proper control schemes [5,15]. Under such situations, the kinematic matrix H () is nonsingular. Similar explanations are also made in [13,41,48] and the references therein.

di (t ) =



a j t j + μi (t ),

(11)

j =0

3. Tracking controller design

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where a j are unknown constant coefficients, (li )

unknown residual terms and di (t ) =

gers. At least the first li -time derivatives of





μi (t ) represents the

μ(li ) (t ), li are positive inte(li ) μi (t ) exist and μi (t )

are bounded by μi i (t ) ≤ c i with c i being positive constants, and li ≥ 1 for i = 1, 2, 5, 6 and li ≥ 3 for i = 3, 4. (l )

2.4. Control objective The objective of this paper is to design the tracking controller U = [δcol , δlon , δlat , δ ped ] T with four control inputs for the unmanned helicopter system (7) to make the outputs Y (containing both the position and the yaw angle) track the desired position P d = [xd , yd , zd ] T and yaw angle ψd trajectories asymptotically in the presence of disturbances (especially mismatched disturbances).

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l i −1

Remark 2. A general class of disturbances can be described by (11), because many kinds of disturbances (e.g., constant, ramp, parabolic, higher-order, and sinusoidal disturbances) in reality can be modeled (at least locally) by the Taylor polynomial plus a residual term if the disturbances are differentiable with respect to time. The order li relies on the property of the disturbance di (t ). For example, li = 1, 2, and 3 for the unknown piecewise constants, ramp and parabolic disturbances, respectively.

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where S ( ) is a skew symmetric matrix defined as

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(b) Yaw subsystem:

The controller design includes two parts, namely, GPIOs design and the feedforward-feedback composite tracking controller design. In order to deal with the couplings among different channels and design the four control inputs relatively independently, the unmanned helicopter system (7) is divided into three subsystems with relatively mild couplings, including a second-order altitude subsystem, a second-order yaw subsystem and a forth-order horizontal subsystem. (a) Altitude subsystem:



z˙ = w ,

˙ = g + r3,3 (− g + Z w w + Z col δcol ) + d w , w

(8)

On the foundation of Assumption 1, Assumption 2 is made.

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Assumption 2. The disturbances di (t ) (i = 1, · · · , 6) described (li )

by (11) satisfy limt →∞ μi (t ) = 0. To precisely estimate the disturbances di (i = 1, . . . , 6) for the helicopter system (7), the following GPIOs are designed.

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3.1.1. GPIOs for d1 and d3 , d4 For the disturbances d1 in subsystem (8) and d3 , d4 in subsystem (10), the following GPIOs are designed.

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⎧ ξ˙i ,1 = ξi ,2 + βi ,1 ( X i ,1 − ξi ,1 ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ξ˙i ,2 = hi + ξi ,3 + βi ,2 ( X i ,1 − ξi ,1 ), ⎪ ⎪ ⎪ ⎪ ⎨ ξ˙i ,3 = ξi ,4 + βi ,3 ( X i ,1 − ξi ,1 ),

where i = 1, 3, 4, ξi ,1 , ξi ,2 , ξi ,3 , . . . , ξi ,li +1 , and ξi ,li +2 are the esti(li −2)

(li −2)

X i ,1 , e i ,2 = ξi ,2 − X i ,2 , e i ,3 = ξi ,3 − di , . . . , e i ,li +1 = ξi ,li +1 − di , (li −1) e i ,li +2 = ξi ,li +2 − di . Taking the time derivatives of the estimation errors, from (8), (10) and (12), the estimation error systems are obtained

e˙ i = A i e i + σi ,

(13)

where i = 1, 3, 4, e i = [e i ,1 , e i ,2 , e i ,3 , . . . , e i ,li +1 , e i ,li +2 ] T , ⎡ ⎤ −βi ,1 1 · · · 0



 (li ) T

0, 0, · · ·, 0, −di

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(li −1)

mates of X i ,1 , X i ,2 , di , . . . , di and di , and X 1,1 = z, X 1,2 = w, X 3,1 = x, X 3,2 = u, X 4,1 = y, X 4,2 = v, respectively. h1 = g + r3,3 (− g + Z w w + Z col δcol ) and [h3 , h4 ] T = R 1 (− g + Z w w + Z col δcol ). βi ,1 , βi ,2 , . . . , βi ,li +1 , and βi ,li +2 are observer gains to be determined. For GPIOs (12), the estimation errors are defined as e i ,1 = ξi ,1 −

29 31

(12)

⎪ ··· ⎪ ⎪ ⎪ ⎪ ⎪ ˙ ⎪ ξi ,li +1 = ξi ,li +2 + βi ,li +1 ( X i ,1 − ξi ,1 ), ⎪ ⎪ ⎪ ⎩˙ ξi ,li +2 = βi ,li +2 ( X i ,1 − ξi ,1 ),

⎢ .. .. . . ⎢ . . . , and A i = ⎢ ⎣ −βi ,l +1 0 · · · i −βi ,li +2 0 · · ·

σi =

.. ⎥ .⎥ ⎥. 1⎦ 0

(14)

44

of X i ,1 , di , . . . , di i

45

J y y − J xx J zz

(l −1)

, and X 2,1 = r , X 5,1 = p, X 6,1 =

49

q, h2 = − pq + N r r + N col δcol + N ped δ ped and [h5 , h6 ] T = f 1 ( 1 ) + A 1 1 + B 1 U 1 . βi ,1 , βi ,2 , . . . , βi ,li , and βi ,li +1 are observer gains to be determined. For GPIOs (14), the estimation errors are (l −2) defined as e i ,1 = ξi ,1 − X i ,1 , e i ,2 = ξi ,2 − di , . . . , e i ,li = ξi ,li − di i ,

50

e i ,li +1 = ξi ,li +1 − di i

46 47 48

51 52 53 54 55 56 57 58 59 60 61 62 63

(l −1)

. Similarly, the estimation error systems are

i

= 2, 5, 6,

(15)

= [e i ,1 , e i ,2 , . . . , e i ,li +1 ] T , σi = ⎡ ⎤ −βi ,1 1 · · · 0 ⎢   .. .. . . .⎥ ⎢ (l ) T . .. ⎥ . . 0, 0, · · ·, 0, −di i , and B i = ⎢ ⎥. ⎣ −βi ,l 0 ··· 1 ⎦ i −βi ,li +1 0 · · · 0 For simplicity, denote dˆ i and edi as the estimates and estimation errors of the disturbances di (i = 1, . . . , 6), respectively. Then, dˆ 1 = ξ1,3 , dˆ 2 = ξ2,2 , dˆ 3 = ξ3,3 , dˆ 4 = ξ4,3 , dˆ 5 = ξ5,2 , dˆ 6 = ξ6,2 , ed1 = e 1,3 , e d 2 = e 2, 2 , e d 3 = e 3, 3 , e d 4 = e 4, 3 , e d 5 = e 5, 2 , e d 6 = e 6, 2 . where

ei

64 65 66

Proposition 1. If the disturbances di , i = 1, . . . , 6 of the helicopter system (7) (also the subsystems (8)–(10)) satisfy Assumption 1, and the observer gains βi ,1 , . . . , βi ,li+2 , i = 1, 3, 4 of GPIOs (12) and βi ,1 , . . . , βi ,li+1 , i = 2, 5, 6 of GPIOs (14) are selected to make A i in (13) and B i in (15) Hurwitz matrices, then the estimation errors e i (t ), i = 1, . . . , 6 of GPIOs (12) and (14) are bounded and asymptotically converge to the regions



 4λ3 ( P i )c i2  i = e i e i 22 ≤ max λmin ( P i )

69 70 71 72 73 74 75 76 77 78 79



81 82 83 84 85 86 87 88

(16)

,

89 90 91 92 93 94 95

Proof. If the observer gains βi ,1 , . . . , βi ,li+2 , i = 1, 3, 4 of GPIOs (12) are selected such that A i are Hurwitz matrices, there exist positive-definite matrices P i ∈ R(li +2)×(li +2) such that A iT P i + P i A i = − I li +2 . For a certain i (i = 1, 3, 4), construct the Lyapunov function W i (e i ) = e iT P i e i for the estimation error system (13). ˙ i (e i ) along system (13) satisfies that W

97

˙ i (e i ) = e T ( A T P i + P i A i )e i + 2e T P i σi W

104

i

i

i

Based on the basic inequality 21 2 ≤ verified that 2λmax ( P i )c i e i 2 ≤  it follows from (17) that

˙ i (e i ) ≤ −(1 − i )e i 22 + W

Remark 3. The proposed GPIOs (12) and (14) are able to estimate higher-order disturbances and the disturbances are not limited to

1

2

2

2

+ 22 , ∀1 , 2 ∈ R, it is

λ2max ( P i )c 2i 2 i e i 2 + i

for

i > 0. Then

i

λ2max ( P i )c i2

W i (0) −

102 103

106 107 108 109 110

113

i

115 116 117 118

(19)

.

119 120 121

The solution of (19) satisfies that

W i (e i (t )) ≤ e

101

114

i



100

112

(18)

.

W (e )

1− − λmax (iP ) t i

99

111

λ2max ( P i )c i2

˙ i (e i ) ≤ − 1 − i W i (e i ) + W λmax ( P i )

98

105

(17)

Since λmin ( P i )e i 22 ≤ W i (e i ) ≤ λmax ( P i )e i 22 , then λ i ( Pi ) ≤ max i e i 22 ≤ λW i ((ePi )) . By letting i ≤ 1, it gives that min

e˙ i = B i e i + σi ,

68

80

≤ − e i 22 + 2λmax ( P i )c i e i 2

where i = 2, 5, 6, ξi ,1 , ξi ,2 , . . . , ξi ,li , and ξi ,li +1 are the estimates and di i

67

96

43

(l −2)

be bounded or slow time-varying disturbances anymore. Generally speaking, the estimate accuracy is higher with the increase of the order of a GPIO, but the computation is more complicated at the same time. Hence, in practice, a balance should be taken into account between the estimation accuracy and the computational burden. Since disturbances d3 , d4 are mismatched disturbances, the (higher-order) derivatives of dˆ 3 , dˆ 4 need to be used in the controller design. Moreover, the (higher-order) derivative of dˆ 1 will also be used in the block backstepping controller design process. Thus, GPIOs for d1 , d3 , d4 are constructed based on the secondorder subsystems of systems (8), (10) and GPIOs for d2 , d5 , d6 are constructed based on the first-order subsystems of systems (9), (10), respectively.

where λmax ( P i ) and λmin ( P i ) denote the maximum and minimum eigenvalues of the positive definite matrix P i satisfying that A iT P i + P i A i = − I li +2 , i = 1, 3, 4 and B iT P i + P i B i = − I li +1 , i = 2, 5, 6, respectively.

3.1.2. GPIOs for d2 and d5 , d6 For the disturbances d2 in subsystem (9) and d5 , d6 in subsystem (10), the following GPIOs are designed.

⎧ ˙ ⎪ ⎪ ξi ,1 = hi + ξi ,2 + βi ,1 ( X i ,1 − ξi ,1 ), ⎪ ⎪ ⎪ ⎪ ξ˙i ,2 = ξi ,3 + βi ,2 ( X i ,1 − ξi ,1 ), ⎪ ⎨ ··· ⎪ ⎪ ⎪ ˙ ⎪ ξi ,li = ξi ,li +1 + βi ,li ( X i ,1 − ξi ,1 ), ⎪ ⎪ ⎪ ⎩˙ ξi ,li +1 = βi ,li +1 ( X i ,1 − ξi ,1 ),

5

λ3max ( P i )c i2

i (1 − i )

122

 +

λ3max ( P i )c i2

i (1 − i )

123

.

124 125

(20) In addition,  (11− ) ≥ 4, ∀i ∈ (0, 1]. Thus, the estimation errors i i e i (t ), i = 1, 3, 4 are bounded and asymptotically converge to the regions i described in (16) (by letting i = 12 ). The proof on the boundedness and asymptotic convergence with respect to the regions i of the estimation errors e i (t ) (i = 2, 5, 6) (from GPIOs

126 127 128 129 130 131 132

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6

 1 2 3

(14) and the error systems are (15)) is similar to that of the estimation errors e i (t ) (i = 1, 3, 4) (from GPIOs (12) and the error systems are (13)), so it is omitted here. This completes the proof. 2

4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Proposition 2. If the disturbances di , i = 1, . . . , 6 of the helicopter system (7) (also the subsystems (8)–(10)) satisfy Assumptions 1 and 2, and the observer gains βi ,1 , . . . , βi ,li+2 , i = 1, 3, 4 of GPIOs (12) and βi ,1 , . . . , βi ,li+1 , i = 2, 5, 6 of GPIOs (14) are selected to make A i in (13) and B i in (15) Hurwitz matrices, then the estimation errors of GPIOs (12) and (14) asymptotically converge to zero, namely, limt →∞ e i (t ) = 0, i = 1, . . . , 6. Proof. For a certain i (i = 1, 3, 4), take σi as the input of the estimation error system (13). Notice that the reduced system e˙ i = A i e i of system (13) is globally exponentially stable if A i is a Hurwitz matrix. Then from Lemma 1, the estimation error system (13) is



ISS. According to Assumption 2, the input

σi = 0, 0, · · ·, 0, −μ(i li )

T

asymptotically converges to zero. Then it follows from Lemma 2 that the system state of (13) asymptotically converges to zero, that is, limt →∞ e i (t ) = 0. The proof on the asymptotic convergence of the estimation errors e i (t ) (i = 2, 5, 6) to zero (from GPIOs (14) and the error systems are (15)) is similar to that of the estimation errors e i (t ) (i = 1, 3, 4) (from GPIOs (12) and the error systems are (13)), so it is omitted here. This completes the proof. 2

26 27 28 29 30 31 32 33 34

Remark 4. Assumption 2 is helpful to obtain asymptotic convergence for the estimation error systems of GPIOs (12) and (14). Actually, even if Assumption 2 is not satisfied by the disturbances, the estimation error systems (13) and (15) are still ISS with respect to σi , i = 1, . . . , 6. Under Assumption 1, since σi is bounded, by Proposition 1, the disturbance estimation errors still asymptotically converge to a small region around zero by appropriately tuning the observer gains.

35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54

The composite tracking controller design for the helicopter system (7) is based on the combination of the block backstepping control technique and disturbance compensations (by using the disturbance estimates generated from GPIOs (12) and (14)). In the following subsections, the controller design is conducted for the altitude subsystem (8), the yaw subsystem (9) and the horizontal subsystem (10) successively. 3.2.1. Control design for the altitude subsystem (8) Define the altitude and vertical velocity tracking errors as e z = zd − z and e w = w d − w, where zd and w d represent the desired altitude and vertical velocity, respectively. By invoking subsystem (8), the altitude and vertical velocity tracking error subsystem is

e˙ z = z˙ d − z˙ = e w , (21)

56

59 60 61 62 63 64 65 66

δcol =

z¨ d − g − dˆ 1 + k z e z + k w e w Z col r3,3

+

e˙ ψ = ψ˙ d −

Cθ

Cφ

q−

Cθ

71

J y y − J xx J zz

(r − er ),

(24)

r =

Cθ 

Sφ

Cφ

Cθ

δ ped =

ψ˙ d −

pq − N r r − N col δcol − N ped δ ped − d2 .

(25)

r˙  +

g − Zw w Z col

,

75 77

q + kψ e ψ ,

(26)

pq − N r r − N col δcol − dˆ2 +

Cφ e Cθ ψ

+ kr e r

N ped

79 80 81 82 83 85 86 87 88 89

,

90

(27) where kψ and kr are positive controller gains to be determined.

dˆ 2 is the estimate of d2 generated from the GPIO (14) designed to estimate d2 , namely, dˆ 2 = ξ2,2 . By substituting (26) and (27) into (24) and (25), the closed-loop yaw angle tracking error subsystem are formulated as

⎧ Cφ ⎪ ⎪ er , ⎨ e˙ ψ = −kψ e ψ +

91 92 93 94 95 96 97 98 99

Cθ

(28)

C ⎪ ⎪ ⎩ e˙ r = −kr er − φ e ψ + ed2 ,

100 101 102

where ed2 = dˆ 2 − d2 = e 2,2 is the estimation error of d2 .

103

3.2.3. Control design for the horizontal subsystem (10) In the control design, the horizontal subsystem (10) is treated as a two-dimensional forth-order system with three blocks by taking the first two orders as a block. Based on the block backstepping control method, two virtual controllers R  and  are designed to obtain the control inputs U 1 during the three-step recursive T design. For simplicity, denote T m = m , which represents the normalized main rotor thrust. The detailed design procedure is given in the following three steps. Step 1: Let P 1d and V 1d represent the reference horizontal position and velocity, respectively. The horizontal position and velocity tracking errors are defined as e P 1 = P 1d − P 1 and e V 1 = V 1d − V 1 , respectively. Taking the derivatives of e P 1 and e V 1 along subsystem (10) yields that

105

e˙ P 1 = e V 1 ,

104

e˙ V 1 = P¨ 1d − ( R 1 − e R 1 ) T m − d V 1 ,

R1 = 

106 107 108 109 110 111 112 113 114 115 116 117 118 119 120



(22)

where k z and k w are positive controller gains to be determined. dˆ 1 is the estimate of d1 generated from the GPIO (12) designed to estimate d1 , that is, dˆ 1 = ξ1,3 . By combining (21) and (22), the closed-loop altitude and vertical velocity tracking error subsystem becomes

74

84



J y y − J xx J zz

73

78

Choose the virtual controller r  and the control input δ ped as 

72

76



where er = r  − r and r  is the virtual controller to be designed. By invoking subsystem (9), the derivative of er is

e˙ r = r˙  +

68 70

3.2.2. Control design for the yaw subsystem (9) Define the yaw angle tracking error as e ψ = ψd − ψ , where ψd is the desired yaw angle. Taking the derivative of e ψ along subsystem (9) yields that

Sφ

67 69

where ed1 = dˆ 1 − d1 = e 1,3 is the estimation error of d1 .

(29)

P¨ 1d − dˆ V 1 + k V 1 e V 1 + k P 1 e P 1 Tm

121 122

where e R 1 = R 1 − R 1 and R 1 is the virtual controller to be designed. Design R 1 as

Take the control input δcol as

58

e˙ w = −k z e z − k w e w + ed1 ,



˙d−w ˙ = z¨d − g − r3,3 (− g + Z w w + Z col δcol ) − d1 . e˙ w = w

55 57

(23)

Cθ

3.2. Composite tracking controller design



e˙ z = e w ,

123 124 125 126

,

(30)

where k P 1 = diag{k P 1,1 , k P 2,2 } and k V 1 = diag{k V 1,1 , k V 2,2 } are positive-definite diagonal matrices to be determined. dˆ V 1 is the estimate of d V 1 = [d3 , d4 ] T generated from the GPIOs (12) designed to

estimate d3 , d4 , that is, dˆ V 1 = [ξ3,3 , ξ4,3 ] T .

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By combining (29) and (30), it follows that

1 2 3 4 5 6 7 8 9 10 11 12

e˙ V 1 = −k V 1 e V 1 − k P 1 e P 1 + e R 1 T m + ed V , 1

where ed V = dˆ V 1 − d V 1 = [ed3 , ed4 ] T , and ed3 = e 3,3 and ed4 = e 4,3 1 are the estimation errors of d3 and d4 , respectively. Step 2: The derivative of e R 1 is

e˙ R 1

= R˙ 1 − R˙ 1 = R˙ 1 − H 1 ( 1 − e 1 ),

15 16 17 18 19 20 21 22 23 24 25 26 27 28

(32)

where e 1 = 1 − 1 and 1 is the virtual controller to be designed. Take the virtual controller  as



1

13 14

(31)

−1

1 = H 1 



˙

R 1 + kR1 e R1 +

eV1 5

Tm +

e P1 5



where k R 1 = diag{k R 1,1 , k R 2,2 } is a positive-definite diagonal matrix to be determined. It follows from (32) and (33) that

e˙ R 1 = − k R 1 e R 1 + H 1 e 1 −

eV1 5

Tm −

e P1 5

31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

Proof. The proof consists of two steps. The first step is to prove that the closed-loop trajectory tracking error system composed of subsystems (23), (28), and (37) is ISS. Then the position and yaw angle trajectory tracking errors are proved to be bounded ∀t ∈ [0, +∞) in the second step. Step 1: The closed-loop trajectory tracking error system consisting of subsystems (23), (28), (37) is

⎡

(34)

Tm.

Step 3: The derivative of e 1 is





˙ 1 − ˙1= ˙ 1 − f 1 ( 1 ) + A 1 1 + B 1 U 1 + d 1 . e˙ 1 = Design the control inputs U 1 = [δlon , δlat ] as



(35)

⎤

ew

⎡

⎤

0

⎥ ⎢ e ⎥ ⎢ −k z e z − k w e w ⎥ ⎢ ⎥ ⎢ d1 ⎥ Cφ ⎥ ⎢ ⎥ ⎢ 0 ⎥ −kψ e ψ + C θ er ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ e ⎥ Cφ ⎥ ⎢ ⎥ ⎢ d2 ⎥ − kr e r − C e ψ ⎥. θ ⎥=⎢ ⎥+⎢0 ⎥ ⎢ ⎥ ⎢ 2×1 ⎥ e ⎥ V 1 ⎥ ⎢ ⎥ ⎢e ⎥ ⎢ ⎥ ⎢ dV 1 ⎥ − kV 1 e V 1 − k P 1 e P 1 + e R1 Tm ⎥ ⎥ ⎢ ⎦ ⎣ −k R e R + H 1 e − e V 1 T m − e P 1 T m ⎦ ⎣ 02×1 ⎦ 1 1 1 5 5 e d −k 1 e 1 − H 1 e R 1 1

Choose a Lyapunov function for the closed-loop trajectory tracking error system (38) as

V =



(36) where k 1 = diag{k 1,1 , k 2,2 } is a positive-definite diagonal matrix

to be determined. dˆ 1 is the estimate of d 1 = [d5 , d6 ] T generated from the GPIOs (14) designed to estimate d5 , d6 , namely, dˆ 1 = [ξ5,2 , ξ6,2 ] T . From (29), (31), (34), (35) and (36), the closed-loop horizontal tracking error subsystem is expressed as

⎧ e˙ P 1 = e V 1 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ e˙ V 1 = −k V 1 e V 1 − k P 1 e P 1 + e R 1 T m + ed V 1 , V P ⎪ e˙ R 1 = −k R 1 e R 1 + H 1 e 1 − 1 T m − 1 T m , ⎪ ⎪ 5 5 ⎪ ⎪ ⎩ e˙ 1 = −k 1 e 1 − H 1 e R 1 + ed ,

1 2

e 2z +

+

1 ˙ U1 = B− 1 − f 1 ( 1 ) − A 1 1 − dˆ 1 + k 1 e 1 + H 1 e R 1 , 1

e

⎤

e˙ z ⎢ e˙ w ⎢ ⎢ e˙ ψ ⎢ ⎢ e˙ r ⎢˙ ⎢ e P1 ⎢˙ ⎢ eV1 ⎣ e˙ R1 e˙ 1

T

e

⎡

(38)

29 30

(33)

Tm ,

7

e TP 1 e V 1

+

1 10

1 10

e 2w +

e TV 1 e V 1

1 2

e 2ψ + 1

+

2

1

er2 +

2

e TR 1 e R 1

1

=−

e TP 1 e˙ V 1 5 kz 5

where ed 1 are the estimation errors of d5 and d6 , respectively. Based on the above controller design process, the main results of this paper are ready to be given.

kP1 5

5

+

5 − kz − k w

e 2z +

− e TP 1

= dˆ 1 − d 1 = [ed5 , ed6 ] T , and ed5 = e 5,2 and ed6 = e 6,2

+

5 e P 1 + e TP 1

5 e TV 1 e˙ V 1

(39) 1 2 e 2 z

eze w −

+ e r e d2 +

5

5

5I 2 − k P 1 − k V 1 5

T − e TR 1 k R 1 e R 1 − e k e + 1 1 1

e TP 1 ed V 1

kw − 1

+

e z e d1

e TV 1 ed V 1 5

5

eze w ≤ −

5I 2 − k P 1 − k V 1 5

5 − kz − k w 10

e V 1 ≤ − e TP 1

− e TP 1

e 2z −

10

77 78 79 80 81 82 83 84 85 86 87

91 92 93 94 95 96 97 98 99 100

106

− e TV 1

kV 1 − I2 5

107 108

eV1

109 110

e w e d1

111

5

112 113

T

+ e 1 ed 1 .

e 2z −

(40)

114 115

5 − kz − k w 10

5I 2 − k P 1 − k V 1 10 5I 2 − k P 1 − k V 1 10

−k z + k w + 3

10 k P 1 − k V 1 + 5I 2

76

105

117

e 2w .

(41)

10 e P 1 − e TV 1

118 119 120 121 122 123

e P1

124

(42)

eV1 .

125 126 127

Substituting (41) and (42) into (40) yields

V˙ = −

75

116

− e TV 1 kz − k w + 5

74

104

Similarly, if choosing k P 1 + k V 1 − 5I 2 as a positive-definite matrix, then 5I 2 − k P 1 − k V 1 is a negative-definite matrix and

e TP 1

73

103

By choosing k z + k w > 5, then

5 − kz − k w

72

102

e 2w − kψ e 2ψ − kr er2

eV1

+

5

71

101

T + e TR 1 e˙ R 1 + e e˙ 1 1

5

70

90

e z e˙ w e˙ z e w e w e˙ w V˙ = e z e˙ z + + + + e ψ e˙ ψ + er e˙ r + e TP 1 e˙ P 1 5 e TV 1 e˙ P 1

69

89

2 1 T + e 1 e 1 . 2

which means that V is positive definite with respect to the tracking errors e z , e w , e ψ , er , e P 1 , e V 1 , e R 1 , e 1 . Taking the time derivative of V along system (38) yields that

5

68

88

e TP 1 e P 1

+ 1 1 2 3 2 1 2 1 T 1 T 1 T e e + 10 e w ≥ 8 e z + 50 e w , 2 e P 1 e P 1 + 5 e P 1 e V 1 + 10 e V 1 e V 1 ≥ 5 z w 3 T 1 T e e + 50 e V 1 e V 1 . Then it follows from (39) that V ≥ 38 e 2z + 8 P1 P1 1 2 1 2 1 T T e + 2 e ψ + 12 er2 + 38 e TP 1 e P 1 + 50 e V 1 e V 1 + 12 e TR 1 e R 1 + 12 e e 1 ≥ 0, 50 w 1

1

Theorem 1. Consider the unmanned helicopter system (7) under the controller (22), (27), (36). If the disturbances di , i = 1, . . . , 6 of system (7) satisfy Assumption 1, the observer gains of GPIOs (12), (14) and controller gains are chosen to satisfy the following conditions: (a) the observer gains βi ,1 , . . . , βi ,li+2 , i = 1, 3, 4 of GPIOs (12) and βi ,1 , . . . , βi ,li+1 , i = 2, 5, 6 of GPIOs (14) are selected to make estimation error system matrices A i (in (13)) and B i (in (15)) Hurwitz matrices, respectively. (b) For the designed control input δcol in (22), k z + k w − 5 > 0, k z − k w + 5 > 0, −k z + k w + 3 > 0. (c) For the designed control input δ ped in (27), kr > 0, kψ > 0. (d) For the designed control inputs U 1 = [δlon , δlat ] T in (36), k P 1 + k V 1 − 5I 2 , k P 1 − k V 1 + 5I 2 and −k P 1 + k V 1 + 3I 2 are all positive-definite matrices, and k R 1 and k 1 are also positive-definite matrices. Then the position and yaw angle trajectory tracking errors e x (t ), e y (t ), e z (t ), e ψ (t ) are bounded ∀t ∈ [0, +∞).

5

5

eze w +

Based on the basic inequality, it can be obtained that

+ (37)

1

1

67

128

e 2w − kψ e 2ψ − kr er2

−k P 1 + k V 1 + 3I 2 10

129 130 131

eV1

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8

T − e TR 1 k R 1 e R 1 − e k e + 1 1 1

1 2 3

+

4 5

10 11 12 13 14

+

=

e



0 ed1 0 ed2 01×2 edT

V1

1 T e e 5 P 1 dV 1

23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

67 68 69 70 71

− e TP 1 a P 1 e P 1

72 73

(43)

74

e 2v

+ e 2p

+

+ eq2

01×2 edT

n

 12 

ed2 1

i =1 ai b i

+ ed22

. In (43), the Cauchy–Buniakowsky–

79

2

+ ed23

T

e 1

≤

,

n

and

ed

n 2

+ ed24

+ ed25

+ ed26

 12

≤e 2 ed 2 .

The observer gains and controller gains are chosen to satisfy the following conditions (which are the same as those given in Theorem 1): (a) the observer gains of GPIOs (12) and (14) are selected to make estimation error system matrices A i (in (13)) and B i (in (15)) Hurwitz matrices, respectively. (b) k z and k w satisfy that k z + k w > 5, and a z > 0, a w > 0. (c) kr > 0 and kψ > 0. (d) k P 1 and k V 1 satisfy that k P 1 + k V 1 − 5I 2 is a positive-definite matrix, a P 1 , a V 1 are positive-definite matrices. k R 1 and k 1 are positive-definite matrices. Then, from (43), it can be obtained that

V˙ ≤ − a z (1 − ε

)e 2z

− a w (1 − ε

)e 2w

− kψ (1 − ε )e ψ

T

ε)e R 1 − e 1 k 1 (1 − ε

)e 1 − a z e 2z

ε

− a w εe 2w − kψ εe 2ψ − kr εer2 − e TP 1 a P 1 εe P 1 − e TV 1 a V 1 εe V 1 T − e TR 1 k R 1 εe R 1 − e k ε e 1 + e 2 ed 2 1 1

≤ − a z (1 − ε

)e 2z

− a w (1 − ε

)e 2w

2

− kψ (1 − ε )e ψ − kr (1 − ε

)er2

− e TP 1 a P 1 (1 − ε )e P 1 − e TV 1 a V 1 (1 − ε )e V 1 − e TR 1 k R 1 (1 − ε )e R 1 T

− e 1 k 1 (1 − ε )e 1 − ε

e 22

+ e 2 ed 2 ,

(44)

where 0 < ε < 1 and ε is the minimum of a z ε , a w ε , kψ ε , kr ε , a P 1,1 ε , a P 2,2 ε , a V 1,1 ε , a V 2,2 ε , k R 1,1 ε , k R 2,2 ε , k 1,1 ε and k 2,2 ε . When

e 2 ≥

ed 2

ε

, it follows from (44) that

V˙ ≤ − a z (1 − ε

)e 2z

− a w (1 − ε

)e 2w

2

− kψ (1 − ε )e ψ

T

ε)e R 1 − e 1 k 1 (1 − ε)e 1 .

(45)

Define the function ϒ = a z (1 − ε )e 2z + a w (1 − ε )e 2w + kψ (1 − ε )e 2ψ + kr (1 −

− ε )e P 1 + − ε )e V 1 + − ε)e R 1 + e 1 k 1 (1 − ε)e 1 . Then ϒ is positive definite with respect to the tracking errors e z , e w , e ψ , er , e P 1 , e V 1 , e R 1 , e 1 and V˙ ≤ −ϒ .

ε

)er2 T

+

e TP 1 a P 1 (1

e TV 1 a V 1 (1

80

Fig. 2. The feasible region of the controller gains k z , k w .

81 82

Remark 5. According to the condition (b) given in Theorem 1, Fig. 2 shows the feasible region of the controller gains (k z , k w ) (for the designed control input δcol in (22)), which is just the shadowed region. According to the condition (d) given in Theorem 1, the feasible regions of the controller gains (k P 1,1 , k V 1,1 ) and (k P 2,2 , k V 2,2 ) (for the designed control inputs U 1 = [δlon , δlat ] T in (36)) are the same as (k z , k w ). In fact, if k z , k w , k P 1 , k V 1 do not satisfy the conditions (b) and (d), the closed-loop trajectory tracking error system (38) is still ISS as long as the controller gains are selected to make V˙ < 0 for the Lyapunov function V defined in (39) in the presence of disturbances. It is worth noticing that the above feasible regions of k z , k w , k P 1 , k V 1 are only sufficient conditions rather than necessary conditions for the choices of the controller gains. The merit of the above feasible regions lies in that they are open regions.

83 84 85 86 87 88 89 90 91 92 93 94 95 96 97

Proof. As proved in Theorem 1, by regarding ed as the input, the closed-loop trajectory tracking error system (38) is ISS if the observer gains and controller gains satisfy condition (a)–(d). If the disturbances di , i = 1, . . . , 6 satisfy Assumptions 1 and 2, then from Proposition 2, the input ed asymptotically converges to zero. By combining this with Lemma 2 together, it follows that the state elements of the tracking error system (38) asymptotically converge to zero, that is, the position and yaw tracking errors satisfy that e x (t ) → 0, e y (t ) → 0, e z (t ) → 0 and e ψ (t ) → 0 as t → ∞. This completes the proof. 2

98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117

− kr (1 − ε )er2 − e TP 1 a P 1 (1 − ε )e P 1 − e TV 1 a V 1 (1 − ε )e V 1 − e TR 1 k R 1 (1 −

78

Theorem 2. Under the controller (22), (27), (36), if the observer gains of the GPIOs (12), (14) and the controller gains satisfy conditions (a)–(d) stated in Theorem 1, and the disturbances di , i = 1, . . . , 6 of system (7) satisfy Assumptions 1 and 2, then the position and yaw angle track the desired trajectories asymptotically, that is, the position and yaw angle trajectory tracking errors e x , e y , e z , and e ψ asymptotically converge to zero.

2

− kr (1 − ε )er2 − e TP 1 a P 1 (1 − ε )e P 1 − e TV 1 a V 1 (1 − ε )e V 1 − e TR 1 k R 1 (1 −

76 77

e TR 1 T

1

T

75

=

e TV 1

2 i =1 ai i =1 b i , ∀ai , b i ∈ R has 1 1 to obtain the inequality 5 e z ed1 + 5 e w ed1 + er ed2 +  1 T T e e + e e ≤ e 2z + e 2w + er2 + e 2x + e 2y + e 2u + d 5 V 1 dV 1 1 1

17

22

− kψ e ψ − kr er2

e z e w e ψ er e TP 1

been used

21

+ e r e d2

5

+ e 1 ed 1 2



16

20

e w e d1

where a z = (k z − k w + 5)/10, a w = (−k z + k w + 3)/10, a P 1 = (k P 1 − k V 1 + 5I 2 )/10, a V 1 = (−k P 1 + k V 1 + 3I 2 )/10,

Schwarz inequality

19

+

T

1

5

− a w e 2w

15

18

e TV 1 ed V

5

T − e TV 1 a V 1 e V 1 − e TR 1 k R 1 e R 1 − e k e + e 2 ed 2 , 1 1 1

7 9

1

5

≤ − a z e 2z

6 8

e TP 1 ed V

e z e d1

e TR 1 k R 1 (1

Then by Lemma 3, the closed-loop trajectory tracking error system (38) is ISS by taking ed as the input of this system. Step 2: If the disturbances di , i = 1, . . . , 6 satisfy Assumption 1, it follows from Proposition 1 that ed is bounded in t ∈ [0, +∞). According to Definition 1, the state elements of system (38) are bounded. Then e x (t ), e y (t ), e z (t ), e ψ (t ) are bounded in t ∈ [0, +∞). This completes the proof. 2

Remark 6. In the controller design process, block backstepping control method was employed in the design of control inputs U 1 = [δlon , δlat ] T on the forth-order horizontal subsystem (10). Compared with [13,14] where recursive control design schemes based on conventional backstepping control were used to design the control inputs U 1 = [δlon , δlat ] T for the horizontal subsystem, since the first two orders of the horizontal subsystem are taken as one block in this paper, the recursive control is simplified and the design steps are reduced from four to three and then a simpler controller was derived finally.

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Remark 7. In the literature [37–42] on flight control of disturbed unmanned helicopters, disturbance observers (DOs) were presented to estimate the disturbances. However, owing to the essential structures of these disturbance observers, only constant or

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slowly time-varying (of which the time derivatives converge to zero with the time) disturbances can be accurately estimated. As for relatively fast time-varying disturbances (of which the time derivatives are bounded), only estimates with bounded estimation errors can be obtained. While for faster time-varying disturbances (e.g., parabolic or higher-order disturbances), the disturbances estimates even diverge. Specifically, the basic form of a disturbance observer (DO) given in these literature is (like the DO (37) in [38])



dˆ = z + p (x), z˙ = −l(x) gd (x) z − l(x) [gd (x) p (x) + f (x) + g u (x)u] ,

(46)

for the system x˙ = f (x) + g u (x)u + gd (x)d, where z is the internal state of the disturbance observer, p (x) is a nonlinear function to be ∂ p (x) designed, and l(x) is the observer gain satisfying that l(x) = ∂ x . The dynamics of the disturbance estimation error ed = dˆ − d are

˙

e˙ d = dˆ − d˙ = −l(x) gd (x)ed − d˙ ,

(47)

˙

with dˆ = −l(x) gd (x)(dˆ − d) = −l(x) gd (x)ed . By Lemma 1, the estimation error system (47) is ISS, if −l(x) gd (x) is a Hurwitz matrix by choosing an appropriate gain matrix l(x). When the disturbance d is constant (i.e., d˙ (t ) = 0) or slowly time-varying (i.e., limt →+∞ d˙ (t ) = 0), then ed asymptotically converges to zero. As for a disturbance d whose derivative is bounded, ed is bounded, while for a higher-order disturbance, ed diverges. On one hand, in reality, the disturbances of unmanned helicopters including both internal and external disturbances, are not limited to constant, slowly time-varying disturbances or disturbances with bounded time derivatives. Thus, it is necessary to develop some disturbance estimators providing accurate estimates for more general disturbances. On the other hand, owing to the presence of mismatched disturbances (namely, d3 , d4 ) in the helicopter system, the (higherorder) derivatives of the mismatched disturbance estimates are essentially needed for the composite controller design. Since the information of the disturbance d and its derivative d˙ appear in

˙ ¨ ˙ ¨ dˆ and dˆ with dˆ generated from DO (46), dˆ and dˆ cannot be directly used in the composite controller. In addition, for constant

˙ ¨ or slowly time-varying disturbances, dˆ and dˆ can be replaced by zero vectors (since the disturbance estimation error ed asymptotically converges to zero), but for faster time-varying disturbances, some additive errors are involved into the closed-loop system if

˙

¨

still using zero vectors to replace dˆ and dˆ (since the disturbance estimation error ed does not converge to zero). These additive errors bring adverse effects to the performances of the closed-loop system and even make it unstable. To improve the aforementioned problems, disturbance estimators which accurately estimate both disturbances and their (higher-order) derivatives are desired. GPIOs are just a kind of disturbance estimators satisfying such demands.

51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

Remark 8. The proposed controller (22), (27), (36) is a feedforwardfeedback composite controller. With the help of the GPIOs (12) and (14), the effects of the disturbances are coped with directly and efficiently. Moreover, the GPIOs are “patches” of the whole controller, namely, they work when there are disturbances and stop working when the disturbance vanish or there are no disturbances. In detail, for the GPIOs (12), by choosing the initial state values as ξi ,1 (0) = X i ,1 , ξi ,2 (0) = X i ,2 for i = 1, 3, 4 and all the disturbances’ and their derivatives’ estimates as ξi ,3 (0) = 0, . . . , ξi ,li +2 (0) = 0, then it can be verified from the estimation error system (13) that e i (t ) ≡ 0, ∀t ∈ [0, +∞). The similar analysis can be made for the GPIOs (14). Therefore, the controller (22), (27), (36) reduce into its baseline feedback controller automatically. This implies that the nominal performance of the baseline feedback controller is retained by the proposed composite control scheme.

9

4. Numerical simulations

67 68

In this section, numerical simulations are performed to verify the effectiveness and anti-disturbance performance of the proposed block backstepping and GPIOs (BBS+GPIOs) composite control scheme. Moreover, another composite control scheme based on the combination of the block backstepping technique and the nonlinear disturbance observers as (46) (BBS+DOs) is also implemented for comparisons. The model parameters of the helicopter are [41]: g = 9.81 m · s−2 , m = 8.2 kg, Z w = −0.7615 s−1 , Zcol = −131.4125 m/(rad  · s2 ), J = diag{0.18, 0.34, 0.28} kg · m2 , A =



and B =

0 0 1689.5 0 0 0 0 894.5 −0.3705 0 0 135.8



−48.1757 0 0

0

−25.5048 0

0 0 −0.9808

s−1,

s−2 .

xd (t ) =

 yd (t ) =

70 71 72 73 74 75 76 77 78 79 80 81 82

The desired trajectories are composed of two stages, where the trajectory of the first stage is a vertical take-off curve and the trajectory of the second stage is an ‘8’ shaped curve where the height continuous to lift. The detailed expressions are: zd (t ) = 6(1 − e −0.2t ) m for t ≥ 0 s, ψd (t ) = 0 rad for t ≥ 0 s,



69

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0 m, t < 7 s, π (t − 7) m, t ≥ 7 s, 10 1 − cos 36

90 91 92

0 m, t < 7 s,  π (t − 7) m, t ≥ 7 s. 6sin 18

93 94 95

All the model parameters are set with random uncertainties whose proportion are within ±10%, and the external disturbances are imposed on the helicopter system suddenly at t = 20 s.



The external disturbances

[ d1 ,ex d2 ,ex d3 ,ex ] T

T

F dT,ex dT,ex

T



= [ F d1 ,ex F d2 ,ex F d3 ,ex ] T

are described as follows: F d1 ,ex , F d2 ,ex and

 π

d3 ,ex are sinusoidal disturbances as F d1 ,ex = 5 sin 20 (t − 20) kg ·  π − 2 − 2 m · s , F d2 ,ex = −2 sin 15 (t − 20) kg · m · s , d3 ,ex =  π sin 10 (t − 20) kg · m2 · rad · s−2 , respectively. F d3 ,ex is a periodic parabolic disturbance as

96 97 98 99 100 101 102 103 104 105 106 107

! ⎧  2  1 ⎪ 0 . 02 t − 20 + ( + k ) T + 4 . 5 kg · m · s−2 , − ⎪ 1 ⎪ ⎪    4 ⎪ 1 ⎨ t ∈ 20 + kT 1 , 20 + k + T 1 , 2  "   2 F d3 ,ex = 3 ⎪ − 4.5 kg · m · s−2 , ⎪ 0.02 t − 20 + ( 4 + k) T 1 ⎪ ⎪ ⎪     ⎩ t ∈ 20 + k + 12 T 1 , 20 + (k + 1) T 1 , k = 0, 1, 2 . . . , T 1 = 60 s. d1 ,ex is a periodic trapezoid disturbance as

108 109 110 111 112 113 114 115 116 117

⎧ 0.1 [t − (20 + kT 2 )] kg · m2 · rad · s−2 , ⎪     ⎪ ⎪ ⎪ t ∈ 20 + kT 2 , 20 + k + 16 T 2 , ⎪ ⎪ ⎪ 1 kg · m2 · rad · s−2 , ⎪ ⎪ ⎪       ⎪ ⎪ t ∈ 20 + k + 16 T 2 , 20 + k + 13 T 2 , ⎪ ⎪ #  $   ⎪ ⎪ ⎨ −0.1 t − 20 + k + 12 T 2 kg · m2 ·rad · s−2 ,

d1 ,ex = t ∈ 20 + k + 1  T , 20 + k + 2 T , 2 2 ⎪ 3 3 ⎪ ⎪ 2 −2 , ⎪ ⎪ 1 kg · m · rad · s − ⎪    ⎪    ⎪ ⎪ ⎪ t ∈ 20 + k + 23 T 2 , 20 + k + 56 T 2 , ⎪ ⎪ ⎪ ⎪ ⎪ 0.1 {t − [20 + (1 + k) T 2 ]} kg · m2 · rad · s−2 , ⎪     ⎩ t ∈ 20 + k + 56 T 2 , 20 + (1 + k) T 2 ,

118

k = 0, 1, 2 . . . , T 2 = 60 s. d2 ,ex is a periodic ramp disturbance as

132

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k = 0, 1, 2 . . . , T 3 = 30 s. For the three subsystems (8), (9), and (10) of the helicopter system (7), define χ1 = [ z, w ] T , χ2 = [ψ, r ] T , χ3 = [x, u ] T , χ4 = [ y , v ] T , χ5 = [r1,3 , p ] T , χ6 = [r2,3 , q] T . Denote Dˆ i (i = 1, . . . , 6) as, respectively, the estimates of the disturbances di (i = 1, . . . , 6), which are generated from the following disturbance observers like (46) (i = 1, . . . , 6)



ˆ i = ζi + p i (χi ), D





= [0, λ2 ],

f 2 (χ2 ) =

−

J y y − J xx J zz

Sφ q Cθ

+

Cφ Cθ

r



45



49

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pq + N r r + N col δcol

3 = 0, p 3 (χ3 ) = λ3 u, l3 (χ3 ) =

∂ p 3 (χ3 ) ∂ χ3

102 103 104 105

,

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= [0, λ3 ], f 3 (χ3 ) =

109

u , g u ,3 = 0, gd,3 = [0, 1] T (for the DO to estimate d3 , i.e., r 1, 3 T m

110

du );

48

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97

dr );

50

95

1

43

47

94

98

g u ,2 = [0, N ped ] T , gd,2 = [0, 1] T (for the DO to estimate d2 , i.e.,

46

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w , g u ,1 = [0, r3,3 Z col ] T , gd,1 = [0, 1] T (for g + r3,3 (− g + Z w w ) the DO to estimate d1 , i.e., d w ); 2 = δ ped , p 2 (χ2 ) = λ2 r, l2 (χ2 ) = ∂ p 2 (χ2 ) ∂ χ2

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where ζi are the internal states of the disturbance observers; 1 = δcol , p 1 (χ1 ) = λ1 w , l1 (χ1 ) = ∂ p∂1χ(χ1 ) = [0, λ1 ], f 1 (χ1 ) =



89 91

(48)

42 44

88

Fig. 4. Response curves of the position and yaw angle. (a) Lateral position x. (b) Longitudinal position y. (c) Vertical position z. (d) Yaw angle ψ .

96





ζ˙i = −li (χi ) gd,i ζi − li (χi ) gd,i p i (χi ) + f i (χi ) + g u ,i i ,

32 34

80

⎧ 2 −2 ⎪ ⎪ 0.2 [t − (20 + kT 3 )] kg · m · rad  ·s , ⎪ ⎪ ⎨ t ∈ 20 + kT 3 , 20 + k + 2 T 3 , 3

d2 ,ex = 0 , ⎪ ⎪     ⎪ ⎪ ⎩ t ∈ 20 + k + 2 T 3 , 20 + (1 + k) T 3 ,

31 33

79

Fig. 3. Three-dimensional trajectory tracking response curves. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)



4 = 0, p 4 (χ4 ) = λ4 v, l4 (χ4 ) = 

∂ p 4 (χ4 ) ∂ χ4



−

5 = δlon , p 5 (χ5 ) = λ5 p, l5 (χ5 ) = J zz − J y y qr J xx T

∂ p 5 (χ5 )

∂ χ5

112

= [0, λ4 ], f 4 (χ4 ) =

v , g u ,4 = 0, gd,4 = [0, 1] T (for the DO to estimate d4 , i.e., r 2, 3 T m

d v );

111

= [0, λ5 ], f 5 (χ5 ) =

−r1,2 p + r1,1 q , g u ,5 = [0, Llon ] T , gd,5 = − τm L b p − τm L a q + Llat δlat

[0, 1] (for the DO to estimate d5 , i.e., d p ); 6 = δlat , ∂ p 6 (χ6 ) p 6 (χ6 ) = λ6 q, l6 (χ6 ) = = [0, λ6 ], f 6 (χ6 ) = ∂ χ6

−r2,2 p + r2,1 q , g u ,6 = [0, Mlat ] T , − J xxJ−y yJ zz pr − τm M b p − τm M a q + Mlon δlon gd,6 = [0, 1] T (for the DO to estimate d6 , i.e., dq ); In the composite BBS+DOs control scheme, the disturbances d1 , . . . , d6 are estimated by DOs design as (48) and the composite controller has expressions similar to (22), (27) and (36). The controller gains are chosen according to conditions (b)–(d) of Theorem 1 and the observer gains λi (i = 1, . . . , 6) are selected to make −li (χi ) gd,i Hurwitz matrices, namely, they are all positive constants.

113 114

Fig. 5. Response curves of the position and yaw angle tracking errors. (a) Tracking error of x. (b) Tracking error of y. (c) Tracking error of z. (d) Tracking error of ψ .

115 116 117

The initial position and yaw angle of the helicopter system (7) π rad. The initial values are set as P (0) = [0, 0, 0] T m and ψ(0) = 12 of other state elements of system (7) are set to be zero. The initial values of all the state elements in GPIOs (12), (14) and DOs (48) are set to be zero. The simulation time length is 80 s and the time step is set as 0.02 s. To have fair comparisons between the BBS+GPIOs and BBS+DOs composite controllers, the control inputs amplitudes should be at the same levels. Based on this principle, according to the conditions given in Theorem 1, the gains of the proposed BBS+GPIOs composite controller (22), (27), (36) are selected as k z = 2.5, k w = 3, kψ = 2, kr = 2, k P 1 = diag{3.8, 3.9}, k V 1 = diag{1.3, 1.2}, k R 1 = diag{10, 9.5}, and k 1 = diag{13, 14}. To balance the estimation accuracy and computational burden, the orders of GPIOs (12) and (14) for d1 , . . . , d6 are chosen as 6, 6, 6, 6, 3, 3, respec-

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Fig. 6. Time histories of four control inputs. (a) δcol . (b) δlon . (c) δlat . (d) δ ped .

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Fig. 8. Disturbance estimate curves of DOs (48). (a) d1 (i.e., d w ). (b) d2 (i.e., dr ). (c) d3 (i.e., du ). (d) d4 (i.e., d v ). (e) d5 (i.e., d p ). (f) d6 (i.e., dq ).

35 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 60

Fig. 7. Disturbance estimate curves of GPIOs (12) and (14). (a) d1 (i.e., d w ). (b) d2 (i.e., dr ). (c) d3 (i.e., du ). (d) d4 (i.e., d v ). (e) d5 (i.e., d p ). (f) d6 (i.e., dq ).

61 62 63 64 65 66

101 102

36

59

100

tively. According to Proposition 1, the observer gains are chosen as β1,1 = 10, β1,2 = 49, β1,3 = 118, β1,4 = 159, β1,5 = 114, β1,6 = 34 (for the GPIO to estimate d1 ); β2,1 = 10, β2,2 = 48, β2,3 = 115, β2,4 = 154, β2,5 = 110, β2,6 = 32 (for the GPIO to estimate d2 ); β3,1 = 10, β3,2 = 47, β3,3 = 111, β3,4 = 145, β3,5 = 100, β3,6 = 28

(for the GPIO to estimate d3 ); β4,1 = 11, β4,2 = 53, β4,3 = 134, β4,4 = 188, β4,5 = 140, β4,6 = 43 (for the GPIO to estimate d4 ); β5,1 = 11, β5,2 = 40, and β5,3 = 48 (for the GPIO to estimate d5 ); β6,1 = 10, β6,2 = 33, and β6,3 = 36 (for the GPIO to estimate d6 ). The controller gains of the BBS+DOs composite controller are set as k z = 2.6, k w = 3.1, kψ = 3, kr = 3, k P 1 = diag{3.9, 4}, k V 1 = diag{1.4, 1.3}, k R 1 = diag{10.1, 9.6}, and k 1 = diag{13.2, 14.2}. The observer gains of the DOs (48) are λ1 = 2.1, λ2 = 2, λ3 = 2, λ4 = 2.5, λ5 = 4, λ6 = 3. The simulation results are shown in Figs. 3–9. Fig. 3 shows the three-dimensional trajectory curves of the helicopter under the BBS+GPIOs and BBS+DOs composite controllers. Fig. 4 gives the position and yaw angle response curves under the two composite controllers. Fig. 5 depicts the position and yaw angle trajectories tracking error curves. From Figs. 3–5, it is shown that the position and yaw angle trajectories tracking errors e x , e y , e z , e ψ under the proposed BBS+GPIOs composite controller are much smaller than those under the BBS+DOs composite controller, especially after t = 20 s when the external disturbances are imposed onto the helicopter system. This fully demonstrates the better anti-disturbance control performance of the proposed BBS+GPIOs composite controller. Fig. 6 shows the time histories of four control inputs. The waves of the trajectories tracking error curves in Fig. 5 and the control input curves in Fig. 6 around t = 7 s and t = 20 s (and the time after that) are due to the switch between the two stages of desired trajectories and the sudden changes of disturbances, respectively. Figs. 7 and 8 show the disturbances and their estimates under the GPIOs and DOs, respectively. Fig. 9 depicts the estimation error curves of the GPIOs and DOs, respectively. It can be seen

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the position and yaw angle of the helicopter track their desired trajectories asymptotically. Simulations have also validated the proposed composite control scheme.

1 2 3

Conflict of interest statement

5 7

None declared.

9

References

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Fig. 9. Disturbance estimation error curves of the GPIOs (12) and (14), and the DOs (48). (a) Estimation error of d1 (i.e., d w ). (b) Estimation error of d2 (i.e., dr ). (c) Estimation error of d3 (i.e., du ). (d) Estimation error of d4 (i.e., d v ). (e) Estimation error of d5 (i.e., d p ). (f) Estimation error of d6 (i.e., dq ).

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Fig. 10. Response curves of the roll and pitch angles. (a) Roll angle φ . (b) Pitch angle θ .

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from Figs. 7–9 that the estimation errors of the lumped disturbances di (i = 1, . . . , 6) under the proposed GPIOs (12) and (14) are much smaller than those under the DOs (48). Fig. 10 demonstrates the responses curves of the roll and pitch angles under the two composite controllers, which are within the reasonable ranges.

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5. Conclusions

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This paper has provided a novel solution to the trajectory tracking control problem of disturbed unmanned helicopters. By properly embedding the estimates of mismatched and matched disturbances into the block backstepping recursive design procedure, a feedforward-feedback composite tracking control scheme has been derived. Under the proposed composite control scheme,

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