High-order nonlinear differentiator and application to aircraft control

Mechanical Systems and Signal Processing 46 (2014) 227–252

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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp

High-order nonlinear differentiator and application to aircraft control Xinhua Wang n, Bijan Shirinzadeh Robotics and Mechatronics Research Laboratory, Department of Mechanical and Aerospace Engineering, Monash University, Clayton, VIC 3800, Australia

a r t i c l e i n f o

abstract

Article history: Received 5 February 2013 Received in revised form 19 August 2013 Accepted 2 February 2014 Available online 26 February 2014

In this paper, a high-order continuous nonlinear differentiator with lead compensation is presented based on finite-time stability. Not only the proposed high-order nonlinear differentiator can obtain the high-order derivatives of a signal, but also the chattering phenomenon can be reduced sufficiently. The parameters regulation is only required to be satisfied with Routh–Hurwitz Stability Criterion. The presented differentiator is a generalization of sliding mode differentiator and linear high-gain differentiator. The merits of the continuous differentiator include its simplicity, selecting parameters easily, restraining noises sufficiently, decreasing the phase shift and avoiding the chattering phenomenon. The theoretical results are confirmed by computer simulations and an experiment on a quadrotor aircraft: (i) the estimation of flying velocity and acceleration from the position measurement; (ii) a control law is designed based on the presented nonlinear differentiator to track a reference trajectory. & 2014 Elsevier Ltd. All rights reserved.

Keywords: High-order continuous Differentiator Lead compensation Finite-time stability Chattering phenomenon Quadrotor aircraft

1. Introduction Obtaining the velocities and accelerations of tracked targets is crucial for several kinds of systems with correct and timely performances, such as the missile-interception systems [1] and underwater vehicle systems [2], for which disturbances must be restrained. Therefore, the design of high-order differentiators for the estimation of velocity and acceleration should be taken into consideration. Differentiation of signals is an old and well-known problem [3–5] and has attracted more attention in recent years [6–19]. The popular linear high-gain differentiators [6–8] provide the estimates of derivatives of the output up to order n
1. Because the convergent velocities of the state variables are slow in the nonlinear region of the system dynamics, the timelagging phenomenon is inevitable. In [9–11], a differentiator via second-order (high-order) sliding modes algorithm was proposed. The information one needs to know about the signal is an upper bound for Lipschitz constant of nth derivative of the signal. Although high-order sliding mode is introduced and no chattering phenomenon exists in the outputs of derivative estimates up to n
1 orders, nth derivative estimate still contains a chattering phenomenon. If noises exist in the signal, this chattering will magnify these noises near the equilibrium. Moreover, it is difficult to regulate the parameters of a high-order sliding mode differentiator. The parameters are selected through computer simulations.

n

Corresponding author. E-mail address: [email protected] (X. Wang).

http://dx.doi.org/10.1016/j.ymssp.2014.02.003 0888-3270 & 2014 Elsevier Ltd. All rights reserved.

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X. Wang, B. Shirinzadeh / Mechanical Systems and Signal Processing 46 (2014) 227–252

In [12], an integral-chain differentiator based on finite-time stability [20–23] and singular perturbation technique [24,25] has been presented. However, the differentiator is complicated and difficult to compute. In [13], a hybrid second-order differentiator with high-speed convergence was designed to estimate the first-order derivative of a signal. It also has an integral-chain structure, and information of the upper bounds of the first-order derivative and second-order derivatives of the signal is required. Though this rapid-convergent differentiator has excellent dynamical performances, the selection of parameters become complex. In fact, for all integral-chain differentiators, increasing the filter order can improve the noise rejection, but it increases the phase shift. Therefore the estimate error brings out due to both modulus and phase distortion. Moreover, this type of differentiator needs the information of the upper bounds of derivatives of the signal up to nth order. The differentiators in [14–19] only consider the first-order derivative estimate of a signal, because it is very difficult to construct a Lyapunov function for a high-order differentiator. In this paper, a high-order continuous nonlinear differentiator is presented based on finite-time stability. For this differentiator, lead compensation exists in the high-order differentiator, and it decreases the phase shift due to increasing system orders. The continuous differentiator can reduce the chattering phenomenon sufficiently more than the sliding mode differentiator. Moreover, this continuous differentiator is a generalization of high-order sliding mode differentiator and linear high-gain differentiator, and the parameters regulation is only required to be satisfied with the Routh–Hurwitz Stability Criterion. Especially, with respect to the differentiator in [13], the presented nonlinear high-order differentiator with lead compensation can estimate the high-order derivatives of a signal, and the selection of parameters is easier. Moreover, for this differentiator, the information one only needs to know about the signal is an upper bound for Lipschitz constant of nth derivative of the signal. However, for the differentiator in [13], the information of the upper bounds of derivatives of the signal up to second order is required. On the other hand, lead compensations exist in the presented differentiator, the phase shift due to increasing system orders is decreased sufficiently. Moreover, the estimate precision of the presented differentiator is better than that of differentiator in [13]. Finally, the presented nonlinear high-order differentiator is applied to a quadrotor aircraft, and an experiment is presented to observe the performances of the proposed differentiator. Usually controlling a quadrotor aircraft to track a reference trajectory needs the information of the position and attitude. Inertial measurement units (IMUs) can measure the attitude information including the attitude angle and angular rate. For the position information, there are several measurement tools: GPS, indoor motion capture system, laser rangefinder and vision system, etc. Some inertial sensors may be used and supplemented by other sensors which provide usually some position-related information. Representative designs are ultrasonic rangers [34]; GPS module when outdoors and infrared rangers when indoors [35]; carrier phase differential GPS [36]; laser rangefinder [37]; vision system [38–40]; indoor motion capture system [41,42]; and laser rangefinder and vision system [43]. However, these strategies are dependent on the accurate model, and all the states are required to be known. For the system of quadrotor aircraft, the flying velocity is difficult to be measured directly. Airspeed tube can measure the velocity of a fixed-wing aircraft. It is difficult to measure the velocity of a quadrotor aircraft, because the quadrotor aircraft is usually in the flexible and slow-speed flight. Moreover, quadrotor aircrafts are underactuated mechanical systems, which exhibit high nonlinear behaviors. Meanwhile, the influences of aerodynamic disturbances, unmodeled dynamics and parametric uncertainties are not avoidable in modeling. These uncertainties render great challenges in the design of flight control system. In [27,28], the uncertainties including complex dynamic behavior are analyzed by the differential transformation method. The proposed method provides an effective means of gaining insights into the nonlinear dynamics. However, all the states are required to be known. The uncertainties can be estimated, but the velocity cannot be obtained synchronously. Inertial Navigation System (INS) is a self-contained navigation technique in which measurements provided by accelerometers and gyroscopes are used to track the position and orientation of an object relative to a known starting point, orientation and velocity. In INS, inertial measurement units (IMUs) typically contain three orthogonal rate-gyroscopes and three orthogonal accelerometers, measuring angular velocity and linear acceleration, respectively. To calculate the position of the device, the signals from the accelerometers are double integrated. However, the drift phenomenon of position calculation is mainly brought out by the usual integral algorithms [29-33]. They cannot restrain the effect of stochastic noise, especially non-white noise. Such noise leads to the accumulation of additional drift in the integrated signal. On the other hand, the accelerometers in IMU cannot respond to the flexible movement efficiently, and the intensive highfrequency noise exists in the acceleration measurements due to the trembling effects (for instance, motors trembling). Therefore, INS is not suitable to flexible aircrafts, for instance, quadrotor aircrafts, helicopters, and the flying velocity cannot be measured directly. However, INS is more suitable to the navigation of fixed-wing aircrafts. The large inertial movements of fixed-wing aircrafts or missiles are more fit for the use of accelerometer measurements, and no obvious high-frequency trembling exists. By an indoor motion capture system, this paper provides an output-feedback tracking control method based on the presented nonlinear differentiator for an underactuated velocity-sensorless quadrotor aircraft. The unknown velocity and uncertainties are reconstructed by the presented high-order differentiator. Furthermore, two controllers are designed to stabilize the attitude and to force the quadrotor aircraft to track a given reference trajectory, respectively. This paper is organized in the following format. In Section 2, problem statement is given, and preliminaries are introduced in Section 3. In Section 4, our main results of high-order nonlinear differentiator are presented. In Section 5, the main results are proved in detail. In Section 6, the robustness analysis of the high-order continuous nonlinear differentiator with lead compensation is described. In Section 7, the equivalence of continuous nonlinear differentiator and sliding mode

X. Wang, B. Shirinzadeh / Mechanical Systems and Signal Processing 46 (2014) 227–252

229

differentiator is analyzed. In Section 8, a quadrotor aircraft control law based on the presented differentiator is presented. In Section 9, the computational analysis and simulations are given, and the experiments on a quadrotor aircraft are presented in Section 10. Our conclusions are made in Section 11. 2. Problem statement When a controller is designed for a system, state feedback is used to place the poles. In order to use the system states to design a feedback controller, sensors must be adopted to measure the states. However, not all states can be measured directly. Therefore, state observer should be presented to estimate the system states. These observers are designed on condition that the system model must be known. For the system x_ 1 ¼ x2 ;

x_ 2 ¼ u þ Δ;

y ¼ x1

ð1Þ

where Δ is the uncertainty. It is required that the output y tracks the desired trajectory yd(t). Let e1 ¼ x1
yd ðtÞ and e2 ¼ x2
y_ d ðtÞ, and the system error may be written as follows: e_ 1 ¼ e2 ;

e_ 2 ¼ u þΔ
y€ d ðtÞ

ð2Þ

Before designing the controller u, how can one obtain the information of x2 and Δ? The difference method with the firstorder filter can be used. However, the outputs are usually distorted and delayed. For a signal, a reduced-order observer can be designed, for instance, several first-order-derivative differentiators can be used in series. However, such structure is often noise-sensitive, and the serious delay exists. The system may even become unstable. Here, we need to design a full-order differentiator to estimate the high-order derivatives of the signal. Thus, we are interested in designing a high-order continuous differentiator to estimate the derivatives of a signal. Moreover, the chattering phenomenon can be restrained sufficiently without phase distortion. Further, the parameters selection should be easy. 3. Preliminaries The related concepts are presented here. Definition 1 (Bhat and Bemstein [22]). Let us consider a time-invariant system in the form of U

x ¼ f ðxÞ;

f ð0Þ ¼ 0;

x A Rn ;

ð3Þ

n

n

where f : D-R is continuous on open neighborhood D D R of the origin. The origin is said to be a finite-timestable equilibrium of the above system if there exists an open neighborhood N D D of the origin and a function T f : N\f0g-ð0; 1Þ, called the settling-time function, such that the following statements hold: (i) Finite-time-convergence: For every x A N\f0g, ψx is the flow starting from x and defined on ½0; T f ðxÞÞ, ψ x ðtÞ A N\f0g for all t A ½0; T f ðxÞÞ, and limt-T f ðxÞ ψ x ðtÞ ¼ 0. (ii) Lyapunov stability: For every open neighborhood U ε of 0 there exists an open subset U δ of N containing 0 such that, for every x A U δ \f0g, ψ x ðtÞ A U ε for all t A ½0; T f ðxÞÞ. The origin is said to be a globally finite-time-stable equilibrium if it is a finite-time-stable equilibrium with D ¼ N ¼ Rn . Then the system is said to be finite-time-convergent with respect to the origin. Assumption 1. For a system depicted by Eq. (3), there exists ρi A ð0; 1; i ¼ 1; …; n, and a nonnegative constant a such that n

jf j ðz~ 1 ; z~ 2 ; …; z~ n Þ
f j ðz 1 ; z 2 ; …; z n Þj ra ∑ jz~ i
z i jρi

ð4Þ

i¼1

where z~ i ; z i A R; i ¼ 1; …; n, j ¼ 1; …; n. Remark 1. There are a number of nonlinear functions capable of satisfying this assumption. For example, one such function is xρi since jxρi
x ρi j r21
ρi jx
xjρi ; ρi A ð0; 1. Moreover, there are smooth functions also satisfying this property. In fact, it is easy to verify that j sin x
sin xjr 2jx
xjρi for any ρi A ð0; 1. Definition 2 (Haimo [20] and Hong et al. [21] (Homogeneous)). A family of dilations δrρ is a mapping that assigns to every real ρ 40 a diffeomorphism δrρ ðx1 ; …; xn Þ ¼ ðρr1 x1 ; …; ρrn xn Þ

ð5Þ

where x1 ; …; xn are suitable coordinates on Rn and r ¼ ðr 1 ; …; r n Þ with the dilation coefficients r 1 ; …; r n positive real numbers. A vector field f ðxÞ ¼ ðf 1 ðxÞ; …; f n ðxÞÞT is homogeneous of degree m A R with respect to the family of dilations if f i ðρr1 x1 ; …; ρrn xn Þ ¼ ρri þ m f i ðx1 ; …; xn Þ;

i ¼ 1; …; n

The system (3) is called homogeneous if its vector field f is homogeneous.

ð6Þ

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X. Wang, B. Shirinzadeh / Mechanical Systems and Signal Processing 46 (2014) 227–252

Theorem 4.2 (Bhat and Bemstein [22]). Suppose there exists a continuous function V : Rn -R such that the following conditions holds: (i) V is positive definite. (ii) There exist real numbers c 4 0 and β A ð0; 1Þ such that V_ ðxÞ þ cðVðxÞÞβ r0

ð7Þ

Then (3) is globally finite-time stable. Moreover, if N is as in Definition1 and Tf is the setting time function then 1 VðxÞ1
β ð8Þ T f ðxÞ r cð1
βÞ

Proposition 8.1 (Bhat and Bernstein [23]). Let k1 ; …; kn 40 be such that sn þkn sn
1 þ ⋯ þ k2 s þk1 is Hurwitz, and consider the system x_ i ¼ xi þ 1 ;

i ¼ 1; …; n
1;

n

x_ n ¼
∑ ki jxi jαi signðxi Þ

ð9Þ

i¼1

there exists ξ A ð0; 1Þ such that, for every αA ð1
ξ; 1Þ, the origin is globally finite-time-stable equilibrium for Eq. (9) where α1 ; …; αn satisfy αi αi þ 1 ; i ¼ 2; …; n ð10Þ αi
1 ¼ 2αi þ 1
αi with αn þ 1 ¼ 1 and αn ¼ α. Theorem 5.2 (Bhat and Bemstein [22]). Consider the perturbed system of (3) following: U

x ¼ f ðxÞ þ gðt; xðtÞÞ;

xð0Þ ¼ x0

ð11Þ

Suppose there exists a function V : D-R such that V is positive definite and Lipschitz continuous on D, and satisfies (7),
 where ν D D is an open neighborhood of the origin, c 4 0 and β A 0; 12 . Then there exist δ0 4 0, L 4 0, Γ 40, and an open neighborhood U of origin such that, for every continuous function g : R þ  D-Rn with δ ¼ sup J gðt; xðtÞÞ J rδ0

ð12Þ

R þ D

every right maximally defined solution x of (11) with xð0Þ A U is defined on R þ and satisfies xðtÞ A U for all t A R þ and J xðtÞ J r Lδγ ;

t ZΓ

ð13Þ

where γ ¼ ð1
βÞ=β 4 1. 4. Design of high-order nonlinear differentiator In this section, we will design a high-order nonlinear differentiator with lead compensation. The proposed high-order differentiator not only can obtain the high-order derivatives of the signal, but also the phase shift due to increasing system orders can be decreased sufficiently. 4.1. Design of high-order integral-chain nonlinear differentiator In order to present the main results, firstly, we use Proposition 8.1 [23] and singular perturbation technique [24,25] to design a high-order integral-chain continuous nonlinear differentiator, and Theorem 1 is obtained as follows. Theorem 1. If system (9) is satisfied with Proposition8.1 in [23], signal vðtÞ is continuous and n-order derivable, and supt A ½0;1Þ jvðiÞ ðtÞj rhi o 1, i ¼ 1; …; n, then for system x_ i ¼ xi þ 1 ;

i ¼ 1; …; n
1 n

ε x n ¼
k1 jx1
vðtÞjα1 signðx1
vðtÞÞ
∑ ki jεi
1 xi jαi signðxi Þ n_

ð14Þ

i¼2

there exist γ 4 1 and Γ 40, such that, for t ZεΓðΞðεÞeð0ÞÞ jxi
vði
1Þ ðtÞj rLεα2 γ
i þ 1 ;

i ¼ 1; …; n

ð15Þ ð0Þ

where ε A ð0; 1Þ is the perturbation parameter; v ðtÞ ¼ vðtÞ; L is some positive constant; γ ¼ ð1
βÞ=β, β A ð0; α2 =ðα2 þ nÞÞ, n Z 2; ei ¼ xi
vði
1Þ ðtÞ, i ¼ 1; …; n, e ¼ ½e1 ⋯ en T ; and ΞðεÞ ¼ diagf1; ε; …; εn
1 g.

X. Wang, B. Shirinzadeh / Mechanical Systems and Signal Processing 46 (2014) 227–252

231

Remark 2. The design of the integral-chain differentiator (14) needs the information of the boundedness of signal derivatives up to nth orders, i.e., supt A ½0;1Þ jvðiÞ ðtÞj rhi o1, i ¼ 1; …; n. Further, the structure of the high-order integral-chain differentiator increases the filter order, and it can improve the noise rejection, but it increases the phase shift. Thus the estimate errors bring out due to both modulus and phase distortion. Therefore, the design of a differentiator with lead compensation should be considered. 4.2. Design of high-order nonlinear differentiator with lead compensation In the following, we will design a high-order continuous nonlinear differentiator with lead compensation. Theorem 2. For system x_ i ¼ xi þ 1
x_ n ¼


kn
i þ 1 jx1
vðt Þjαn
i þ 1 signðx1
vðt ÞÞ; εi

i ¼ 1; …; n
1

k1 jx1
vðt Þjα1 signðx1
vðt ÞÞ εn

ð16Þ

if signal vðtÞ is continuous and nth-order derivable, and supt A ½0;1Þ jvðnÞ ðtÞj rhn o 1, then there exist γ 4 1 and Γ 4 0, such that, for t ZεΓðΞðεÞeð0ÞÞ jxi
vði
1Þ ðtÞj r Lεnγ
i þ 1 ;

i ¼ 1; …; n

ð17Þ n
1

where ε A ð0; 1Þ is the perturbation parameter; let k1 ; …; kn 40 be such that s þ kn s satisfy n

αn
i þ 1 ¼

iα1 þn
i ; n

þ⋯ þk2 s þ k1 is Hurwitz, and α1 ; …; αn

i ¼ 1; …; n

ð18Þ

with α1 A ð0; 1Þ; vð0Þ ðtÞ ¼ vðtÞ; L is some positive ei ¼ xi
vði
1Þ ðtÞ; i ¼ 1; …; n, n Z 2; and e ¼ ½e1 ⋯ en T .

constant;

γ ¼ ð1
βÞ=β,

β A ð0; 1=2Þ;

ΞðεÞ ¼ diagf1; ε; …; ε

n
1

g;

Remark 3. In differentiator (16), only the condition supt A ½0;1Þ jvðnÞ ðtÞj r hn o1 is required for signal v(t). However, supt A ½0;1Þ jvðiÞ ðtÞj r hi o 1, i ¼ 1; …; n should be satisfied for integral-chain differentiator (14). On the other hand, lead compensations exist in differentiator (16), the phase shift due to increasing system orders is decreased sufficiently. Importantly, for the same γ in (15) and (17), we obtain that nγ
i þ 1 4 α2 γ
iþ 1, i ¼ 1; …; n. Therefore, it implies that for ε A ð0; 1Þ, the ultimate bound (17) on the estimation error is of more higher order than that of (15). Thus, the estimation precision of differentiator (16) is better than that of differentiator (14). 5. Proof of Theorems 1 and 2 Firstly, we present a lemma on finite-time stability of a high-order continuous nonlinear system. Lemma 1. Let k1 ; …; kn 4 0 be such that sn þkn sn
1 þ ⋯ þ k2 s þk1 is Hurwitz, and consider the system x_ i ¼ xi þ 1
kn
i þ 1 jx1 jαn
i þ 1 signðx1 Þ; x_ n ¼
k1 jx1 jα1 signðx1 Þ

i ¼ 1; …; n
1 ð19Þ

there exists ξ A ð0; 1Þ such that, for every αA ð1
ξ; 1Þ, the origin is globally finite-time-stable equilibrium for Eq. (19) where α1 ; …; αn satisfy αn
i þ 1 ¼

iα1 þn
i ; n

i ¼ 1; …; n

ð20Þ

with α1 ¼ α. Proof. From Definition 2 and Eq. (19), let ρri þ 1 xi þ 1
kn
i þ 1 jρr1 x1 jαn
i þ 1 signðx1 Þ ¼ ρri þ m ½xi þ 1
kn
i þ 1 jx1 jαn
i þ 1 signðx1 Þ;
k1 jρr1 x1 jα1 signðxi Þ ¼ ρrn þ m ½
k1 jx1 jα1 signðx1 Þ

i ¼ 1; …; n
1 ð21Þ

Therefore, from (21), we have r i þ 1 ¼ αn
i þ 1 r 1 ¼ r i þ m; i ¼ 1; …; n
1;

α1 r 1 ¼ r n þ m

ð22Þ

Selecting r 1 ¼ 1, thus, the following may be written: m ¼ αn
1;

αn
i þ 1 ¼

iα1 þ n
i ; n

rn þ 1
i ¼

ðn
iÞα1 þ i ; n

i ¼ 1; …; n

ð23Þ

Let k1 ; …; kn 40 be chosen Hurwitz and, for each α 4 0, let f α denote the closed-loop vector field obtained in (19). For each α 40, the vector field f α is continuous. It is also easy to verify that, for each α 4 0, the vector field f α is homogeneous of degree αn
1 ¼ ðα1
1Þ=n, where α1 ¼ α and α1 ; …; αn satisfy (20). Moreover, the vector field f1 is linear with the Hurwitz

232

X. Wang, B. Shirinzadeh / Mechanical Systems and Signal Processing 46 (2014) 227–252

characteristic polynomial sn þkn sn
1 þ ⋯ þ k2 s þk1 . Therefore, by Theorem 6.2 in [23], there exists a positive-definite, radially unbounded, Lyapunov function V : Rn -R such that Lf 1 V is continuous and negative definite. Let A ¼ V
1 ð½0; 1Þ and S ¼ bdA ¼ V
1 ðf1gÞ. Then A and S are compact since V is proper and 0 A S since V is positive definite. Define φ : ð0; 1  S-R by φðα; zÞ ¼ Lf α VðzÞ. Then, φ is continuous and satisfies φð1; zÞ o 0 for all z A S, that is, φðf1g  SÞ  ð
1; 0Þ. Since S is compact, it follows from Lemma 5.8 in [26, p. 169] that there exists ξ 40 such that φðð1
ξ; 1  SÞ  ð
1; 0Þ. It follows that for α A ð1
ξ; 1, Lf α V takes negative values on S. Thus, A is strictly positively invariant under f α for every α A ð1
ξ; 1. By Theorem 6.1 in [23], the origin is a globally asymptotically stable equilibrium under f α for every α A ð1
ξ; 1. The result now follows from Theorems 7.1 and 7.3 in [23] by noting that, for every α A ð1
ξ; 1Þ, the degree of homogeneity of f α is negative. □ 5.1. Proof of Theorem 1 The system error may be obtained for Eq. (14) and the derivatives of v(t) as follows: e_ i ¼ ei þ 1 ;

i ¼ 1; …; n
1 α1

n_

ε e n ¼
k1 je1 j

  ! i
1 αi i
1 n  v d v  i
1 i
1 d nd v signðe1 Þ
∑ ki ε ei þ ε sign e þ
ε  i n i
1  i
1 dt dt dt i¼2  n

ð24Þ

Eq. (24) can be rewritten as dεi
1 ei ¼ εi ei þ 1 ; dt=ε

i ¼ 1; …; n
1

  ! i
1 αi i
1 n  n dεn
1 en d v d v d v  ¼
k1 je1 jα1 signðe1 Þ
∑ ki εi
1 ei þ εi
1 i
1  sign ei þ i
1
εn n  dt=ε dt dt dt i¼2 

ð25Þ

Let z1 ðτÞ ¼ e1 ðt Þ; z2 ðτÞ ¼ εe2 ðt Þ; …; zn ðτÞ ¼ εn
1 en ðt Þ;

τ ¼ t=ε;

z ¼ ½z1 ⋯ zn T ;

i

v i ð τ Þ ¼ εi

d vðtÞ dt

i

;

i ¼ 1; …; n

ð26Þ

therefore, we have z ¼ ΞðεÞe. Eq. (25) can be written as dzi ¼ zi þ 1 ; i ¼ 1; …; n
1 dτ n dzn ¼
k1 jz1 jα1 signðz1 Þ
∑ ki jzi þv i
1 ðτÞjαi signðzi þv i
1 ðτÞÞ
v n ðτÞ dτ i¼2

ð27Þ

Furthermore, Eq. (27) can be rewritten as dzi ¼ zi þ 1 ; i ¼ 1; …; n
1 dτ n n dzn ¼
∑ ki jzi jαi signðzi Þ
∑ fki jzi þv i
1 ðτÞjαi signðzi þv i
1 ðτÞÞ
ki jzi jαi signðzi Þg
v n ðτÞ dτ i¼1 i¼2

ð28Þ

Let n

g n ðτ; zðτÞÞ ¼
∑ fki jzi þ v i
1 ðτÞjαi signðzi þ v i
1 ðτÞÞ
ki jzi jαi signðzi Þg
v n ðτÞ

ð29Þ

i¼2

Therefore, from Assumption 1 and Remark 1, we obtain n

δ¼

sup nþ1

ðτ;zÞ A R

α

J g n ðτ; zðτÞÞ J r ∑ 21
αi ki εαi ði
1Þ hi
i 1 þ εn hn r ερ δ0

ð30Þ

i¼2

α

where supt A ½0;1Þ jvðiÞ ðtÞj rhi o 1, i ¼ 1; …; n, δ0 ¼ ∑ni¼ 2 21
αi ki hi
i 1 þ hn , and ρ ¼ min f minfn; ði
1Þαi gg ¼ α2 i A f2;…;ng

ð31Þ

In fact, it is checked that the recursive form of Eq. (10) may be rewritten in the non-recursive form αi ¼

αn ; ðn
i þ 1Þ
ðn
iÞαn

i ¼ 2; …; n

ð32Þ

We will calculate the minimum value of the following expression: ði
1Þαi ¼

ði
1Þαn ; ðn
iþ 1Þ
ðn
iÞαn

i ¼ 2; …; n

ð33Þ

X. Wang, B. Shirinzadeh / Mechanical Systems and Signal Processing 46 (2014) 227–252

233

Defining the following function Ψ ðsÞ ¼

ðs
1Þαn ; ðn
s þ 1Þ
ðn
sÞαn

s A ð1; n þ 1Þ

ð34Þ

and taking derivative of Ψ ðsÞ with respect to variable s, we obtain dΨ ðsÞ αn ½ðn
s þ1Þ
ðn
sÞαn 
ðs
1Þαn ½
1 þ αn  ¼ ds ½ðn
sþ 1Þ
ðn
sÞαn 2 ¼

αn ½n
ðn
1Þαn  ½ðn
s þ 1Þ
ðn
sÞαn 2

40

ð35Þ

Because αn A ð0; 1Þ, function Ψ ðsÞ is monotone increasing. Moreover, sequence f2; …; ng is monotone increasing in ð1; n þ 1Þ. Therefore min fði
1Þαi g ¼ α2

ð36Þ

i A f2;…;ng

And because n Z2, we obtain ρ ¼ min fminfn; ði
1Þαi gg ¼ α2

ð37Þ

i A f2;…;ng

Furthermore, because ε A ð0; 1Þ and αi A ð0; 1Þ, i ¼ 2; …; n, we obtain max fεði
1Þαi g ¼ ερ ¼ εα2

ð38Þ

i A f2;…;ng

From Proposition 8.1 in [23], Theorem 5.2 in [22] and Eq. (30), for system (28), there exist positive constants μ and Γðzð0ÞÞ, such that J zðτÞ J rμδγ r μðεα2 δ0 Þγ ;

8 τ A ½Γðzð0ÞÞ; 1Þ

ð39Þ

Therefore, from coordinate transformation (26), we obtain J ½e1 ⋯ εn
1 en  J rμðεα2 δ0 Þγ ;

8 t A ½εΓðΞðεÞeð0ÞÞ; 1Þ

ð40Þ

Thus, the following inequality holds: jei j r μðεα2
ði
1Þ=γ δ0 Þγ ¼ Lεα2 γ
i þ 1 ; where L

¼ μδγ0 .

8 t A ½εΓðΞðεÞeð0ÞÞ; 1Þ

ð41Þ

To make α2 γ
iþ 1 4 1, i ¼ 1; …; n, from Theorem 5.2 in [22], we let

β A ð0; minfα2 =ðα2 þ nÞ; 1=2gÞ ¼ ð0; α2 =ðα2 þ nÞÞ

ð42Þ

In fact, from Theorem 4.3 in [22], β can be chosen to be arbitrarily small. Hence, the requirement that β lies on β A ð0; α2 =ðα2 þ nÞÞ is not restrictive. Accordingly, we can obtain α2 ½ð1
βÞ=β
n þ 1 4 1. Therefore, α2 γ
i þ 1 41 for i ¼ 1; …; n. The choice of β leads to α2 γ
iþ 1 4 1 in (41) which implies that for 0 oε o1, the ultimate bound (41) on the estimate error is of higher order than the perturbation. This concludes the proof. □ 5.2. Proof of Theorem 2 The system error for Eq. (16) and the derivatives of v(t) is obtained as follows: kn
i þ 1 je1 jαn
i þ 1 signðe1 Þ; εi n k1 d vðtÞ e_ n ¼
n je1 jα1 signðe1 Þ
n ε dt

e_ i ¼ ei þ 1


i ¼ 1; …; n
1 ð43Þ

Thus, Eq. (43) can be rewritten as dεi
1 ei ¼ εi ei þ 1
kn
i þ 1 je1 jαn
i þ 1 signðe1 Þ; dt=ε

i ¼ 1; …; n
1

n

dεn
1 en d vðtÞ ¼
k1 je1 jα1 signðe1 Þ
εn n dt=ε dt

ð44Þ

Let a coordinate transform be described as follows: τ ¼ t=ε;

zi ðτÞ ¼ εi
1 ei ðt Þ; i ¼ 1; …; n;

n

z ¼ ½z1 ⋯ zn T ;

Therefore, we obtain z ¼ ΞðεÞe, and Eq. (44) can be written as dzi ¼ zi þ 1
kn
i þ 1 jz1 jαn
i þ 1 signðz1 Þ; dτ

i ¼ 1; …; n
1

v n ð τ Þ ¼ εn

d vðtÞ n dt

ð45Þ

234

X. Wang, B. Shirinzadeh / Mechanical Systems and Signal Processing 46 (2014) 227–252

dzn ¼
k1 jz1 jα1 signðz1 Þ
v n ðτÞ dτ

ð46Þ

Therefore, from Assumption 1 and Remark 1, we obtain δ ¼ sup jv n ðτÞj r εn hn

ð47Þ

τ A ½0;1Þ

From Lemma 1, Theorem 5.2 in [22] and Eq. (47), for system (46), there exist positive constants μ and Γðzð0ÞÞ, such that J zðτÞ J r μδγ ¼ μðεn hn Þγ ;

8 τ A ½Γðzð0ÞÞ; 1Þ

ð48Þ

Therefore, from coordinate transform (45), we obtain J ½e1 εe2 ⋯ εn
1 en T J r μðεn hn Þγ ;

8 τ A ½Γðzð0ÞÞ; 1Þ

ð49Þ

Thus, the following inequality holds: jei jr μðεn hn Þγ ε
i þ 1 ¼ Lεnγ
i þ 1 ; where L

γ ¼ μhn .

i ¼ 1; …; n;

8 τ A ½Γðzð0ÞÞ; 1Þ

ð50Þ

To make nγ
i þ 1 4 1, i ¼ 1; …; n, from Theorem 5.2 in [22], we let

β A ð0; 1=2Þ

ð51Þ

In fact, from Theorem 4.3 in [22], β can be chosen to be arbitrarily small. Hence, the requirement that β lies in β A ð0; 1=2Þ is not restrictive. Accordingly, we can obtain γ ¼ ð1
βÞ=β 41. Thus, nγ
n þ 1 4 1 holds. Therefore, nγ
iþ 1 4 1 for i ¼ 1; …; n. The choice of β leads to nγ
iþ 1 4 1 in (50) which implies that for 0 o ε o 1, the ultimate bound (50) on the estimate error is of higher order than the perturbation. This concludes the proof. □ 6. Robustness analysis of high-order nonlinear differentiator with lead compensation In a realistic problem, signal v(t) in differentiator (16) might represent an ideal signal without any disturbance, while stochastic disturbances exist in almost all signals. The following theorem concerns the robustness behavior of the presented high-order nonlinear differentiator under bounded perturbations. Theorem 3. For nonlinear differentiator (16), if the disturbance exists in signal vðtÞ, i.e., vðtÞ ¼ v0 ðtÞ þ dðtÞ, where v0 ðtÞ is the desired signal, dðtÞ is the bounded stochastic disturbance, and supt A ½0;1Þ jdðtÞj r Ld o 1, then there exist γ 4 1 and Γ 4 0, such that, for t ZεΓðΞðεÞeð0ÞÞ jxi
vði0
1Þ ðtÞj rLðδdi Þγ ;

i ¼ 1; …; n

ð52Þ

α 1Þ=γ þ ðLdp =hn Þ∑ni¼ 1 21
αn
i þ 1 kn
i þ 1 ε
ði
1Þ=γ , and where L is some positive constant; δdi ¼ εn
ði
1=α p α 1
αn
i þ 1 n n and Ld o ðð1
ε Þ=∑i ¼ 1 2 kn
i þ 1 Þhn ; supt A ½0;1Þ jv0ðnÞ ðtÞj rhn o 1; Ldp ¼ Lαd1 when

Ld 41; γ ¼ ð1
βÞ=β, and 91 8 0 > > > > > > B C > > > > B C > > > > > > B C > > < B 1 1=C B C ; β∈B0; min C; > B C nlog ε 2> > > > B C ! þ1 > > > α p > > B C > > L > > @ A 1−αn−i þ 1 > > n þ d ∑n > > 2 k log ε n−i þ 1 ; : i¼1 h

δdi ∈ð0; 1Þ, i ¼ 1; …; n; ε A ð0; 1Þ, α 0 oLd ≤1, and Ldp ¼ Lαdn when

n≥2;

n

ΞðεÞ ¼ diagf1; ε; …; ε

n
1

g, and ei ¼ xi
v0ði
1Þ ðtÞ, i ¼ 1; …; n, e ¼ ½e1 ⋯ en T .

Proof. The system error for differentiator (16) and the derivatives of v0 ðtÞ is given by kn
i þ 1 je1 jαn
i þ 1 signðe1
dðt ÞÞ; εi n k1 d v0 ðtÞ e_ n ¼
n je1 jα1 signðe1
dðt ÞÞ
n ε dt e_ i ¼ ei þ 1


i ¼ 1; …; n
1 ð53Þ

Eq. (53) can be rewritten as dεi
1 ei ¼ εi ei þ 1
kn
i þ 1 je1
dðt Þjαn
i þ 1 signðe1
dðt ÞÞ; dt=ε n dεn
1 en d v0 ðtÞ ¼
k1 je1
dðt Þjα1 signðe1
dðt ÞÞ
εn n dt=ε dt

i ¼ 1; …; n
1 ð54Þ

Let τ ¼ t=ε;

zi ðτÞ ¼ εi
1 ei ðt Þ; i ¼ 1; …; n;

n

z ¼ ½z1 ⋯ zn T ;

d ðτÞ ¼ dðt Þ;

v n ð τ Þ ¼ εn

d v0 ðtÞ n dt

ð55Þ

X. Wang, B. Shirinzadeh / Mechanical Systems and Signal Processing 46 (2014) 227–252

235

therefore, we have z ¼ ΞðεÞe, and Eq. (54) can be written as   dzi ¼ zi þ 1
kn
i þ 1 jz1
d ðτÞjαn
i þ 1 sign z1
d ðτÞ ; dτ   dzn ¼
k1 jz1
d ðτÞjα1 sign z1
d ðτÞ
v n ðτÞ dτ

i ¼ 1; …; n
1 ð56Þ

Furthermore, Eq. (56) can be rewritten as   dzi ¼ zi þ 1
kn
i þ 1 jz1 jαn
i þ 1 signðz1 Þ
kn
i þ 1 fjz1
d ðτÞjαn
i þ 1 sign z1
d ðτÞ
jz1 jαn
i þ 1 signðz1 Þg dτ i ¼ 1; …; n
1   dzn ¼
k1 jz1 jα1 signðz1 Þ
k1 fjz1
d ðτÞjα1 sign z1
d ðτÞ
jz1 jα1 signðz1 Þg
v n ðτÞ dτ

ð57Þ

Let gðτ; zðτÞÞ ¼ ½g 1 ðτ; zðτÞÞ ⋯ g n ðτ; zðτÞÞT

ð58Þ

where g i ðτ; zðτÞÞ ¼
kn
i þ 1 fjz1
dðτÞjαn
i þ 1 signðz1
dðτÞÞ
jz1 jαn
i þ 1 signðz1 Þg; i ¼ 1; …; n
1; g n ðτ; zðτÞÞ ¼
k1 fjz1
dðτÞjα1 signðz1
dðτÞÞ
jz1 jα1 signðz1 Þg
v n ðτÞ

ð59Þ

Therefore, from Assumption 1 and Remark 1, we obtain δ¼

n

sup ðτ;zÞ A R

nþ1

α

J gðτ; zðτÞÞ J r ∑ 21
αn
i þ 1 kn
i þ 1 Ldn
i þ 1 þ εn hn i¼1

α

n

r εn hn þ Ldp ∑ 21
αn
i þ 1 kn
i þ 1

ð60Þ

i¼1

α

α

where Ldp ¼ Lαd1 when 0 oLd ≤1, and Ldp ¼ Lαdn when Ld 4 1. In fact, we can calculate, respectively, the minimum and maximal values of the following expression (defined in Eq. (18)): αn
i þ 1 ¼

iα1 þn
i ; n

i ¼ 1; …; n

ð61Þ

Defining the following function: Ψ ðsÞ ¼

sα1 þn
s ; n

s A ð0; þ 1Þ

ð62Þ

and taking derivative of Ψ ðsÞ with respect to variable s, we obtain dΨ ðsÞ α1
1 ¼ o0; ds n

s A ð0; þ 1Þ

ð63Þ

Because α1 A ð0; 1Þ, function Ψ ðsÞ is monotone decreasing with respect to s. Moreover, the sequence f1; …; ng is monotone increasing in ð0; þ 1Þ. Therefore min fαn
i þ 1 g ¼ α1

ð64Þ

max fαn
i þ 1 g ¼ αn

ð65Þ

i A f1;…;ng

and i A f1;…;ng

α

α

α

α

Therefore, Ldp ¼ maxi A f1;…;ng fLdn
i þ 1 g ¼ Lαd1 when 0 o Ld ≤1; and Ldp ¼ maxi A f1;…;ng fLdn
i þ 1 g ¼ Lαdn when Ld 41. From Lemma 1, Theorem 5.2 in [22] and Eq. (60), for system (57), there exist positive constants μ and Γðzð0ÞÞ, such that  γ α ð66Þ J zðτÞ J rμδγ r μ εn hn þ Ldp ∑ni¼ 1 21
αn
i þ 1 kn
i þ 1 ; 8 τ A ½Γðzð0ÞÞ; 1Þ From coordinate transformation (55), we obtain  γ α J ½e1 ⋯ εn
1 en  J r μ εn hn þ Ldp ∑ni¼ 1 21
αn
i þ 1 kn
i þ 1 ;

8 t A ½εΓðΞðεÞeð0ÞÞ; 1Þ

ð67Þ

Thus, the following inequality holds: jei j r Lðδdi Þγ ;

i ¼ 1; …; n;

8 t A ½εΓðΞðεÞeð0ÞÞ; 1Þ

ð68Þ

236

X. Wang, B. Shirinzadeh / Mechanical Systems and Signal Processing 46 (2014) 227–252 α

p where L ¼ μLγv , δdi ¼ εn
ði
1Þ=γ  þðL =hn Þ∑ni¼ 1 21
αn
i þ 1 kn
i þ 1 ε
ði
1Þ=γ ,  1=αd p 1
αn
i þ 1 n n Ld o ðð1
ε Þ=∑i ¼ 1 2 kn
i þ 1 Þhn , then

i ¼ 1; …; n.

If

ε A ð0; 1Þ

and

α

0 o εn þ

Ldp n ∑ 21−αn−i þ 1 kn−i þ 1 o1 hn i ¼ 1

ð69Þ

Furthermore, from Theorem 4.3 in [22], β can be chosen to be arbitrarily small. Hence, the requirement that β lies on 91 8 0 > > > > > > B C > > > > B C > > > > > > B C > > = < B 1 1 C B C ð70Þ ; β∈B0; min C > B C nlog ε 2> > > > > B ! þ 1 >C > α > > B C > > Ldp n > > @ A 1−αn−i þ 1 > > n > > kn−i þ 1 ; :log ε þ h ∑i ¼ 1 2 n

is not restrictive. Accordingly, we can obtain 9 8 > > > > > > > > = < n log ε ! γ ¼ ð1−βÞ=β 4 max ; 1 α p > > Ld n 1−αn−i þ 1 > > > > > > kn−i þ 1 ∑ 2 ; :log εn þ hn i ¼ 1 Therefore γ log εn þ

ð71Þ

! α Ldp n ∑i ¼ 1 21−αn−i þ 1 kn−i þ 1 onlog ε hn

ð72Þ

i.e., α

εn þ

Ldp n ∑ 21−αn−i þ 1 kn−i þ 1 o εn=γ hn i ¼ 1

ð73Þ

Therefore, from ε A ð0; 1Þ and γ 4 n, we can obtain εn=γ oεi−1=γ ;

i ¼ 1; …; n

ð74Þ

Then, δdi ¼ εn
ði
1Þ=γ þ

α

Ldp n 1
αn
i þ 1 ∑ 2 kn
i þ 1 ε
ði
1Þ=γ o1; hn i ¼ 1

i ¼ 1; …; n

ð75Þ

The choice of β leads to γ 4 1 in (68) which implies that for δdi ∈ð0; 1Þ, the ultimate bound (68) on the estimate error is of higher order than the perturbation. Consequently, the presented high-order nonlinear differentiator (16) leads us to perform rejection of low-level persistent disturbances. This concludes the proof. □ 7. Equivalence of continuous nonlinear differentiator and sliding mode differentiator Interestingly, we find that, when α1 -0 þ , the presented nonlinear differentiator (16) has the same structure with sliding mode differentiator in [10], and the suitable parameters for sliding mode differentiator can be selected easily. For sliding mode differentiator [10] x_ 1 ¼ v1 ;

v1 ¼ x2
λ1 C 1=n jx1
vðtÞjðn
1Þ=n signðx1
vðtÞÞ

x_ 2 ¼ v2 ;

v2 ¼ x3
λ2 C 1=ðn
1Þ jx2
v1 jðn
2Þ=ðn
1Þ signðx2
v1 Þ

⋯ x_ n
1 ¼ vn
1 ;

vn
1 ¼ xn
λn
1 C 1=2 jxn
1
vn
2 j1=2 signðxn
1
vn
2 Þ

x_ n ¼
λn C signðxn
vn
1 Þ

ð76Þ

the parameters are defined from Proposition 4 in [10], where C is the Lipschitz constant of signal vðn
1Þ ðtÞ, and λ1 ; …; λn are obtained through computer simulations. In the following, we give a method of parameters selection for sliding mode differentiator (76). Theorem 4. If the parameters in (76) are selected as follows:  n 1 k
iþ1 C¼ ; λ1 ¼ kn ; λi ¼ ðn
niÞ=ðn ; i ¼ 2; …; n
1;
i þ 1Þ ε kn
i þ 2

λn ¼ k1

ð77Þ

X. Wang, B. Shirinzadeh / Mechanical Systems and Signal Processing 46 (2014) 227–252

237

then the following equalities are true after a finite time of a transient process x1 ¼ vðtÞ;

xi ¼ vði
1Þ ðtÞ; i ¼ 2; …; n ðn
1Þ

where C is Lipschitz constant of v

ð78Þ

ðtÞ, k1 ; …; kn and ε are defined in Lemma1 or Theorem2.

Proof. It is easy to check that recursive form differentiator (76) may be rewritten in the non-recursive form x_ 1 ¼ x2
λ1 C 1=n jx1
vðtÞjðn
1Þ=n signðx1
vðtÞÞ ðn
2Þ=ðn
1Þ λ2 C 2=n jx1
vðtÞjðn
2Þ=n signðx1
vðtÞÞ x_ 2 ¼ x3
λ1

⋯ 1=ðn
1Þ 1=ðn
2Þ 1=2 x_ n
1 ¼ xn
ðλ1 λ2 ⋯λn
2 λn
1 ÞC ðn
1Þ=n jx1
vðtÞj1=n signðx1
vðtÞÞ x_ n ¼
λn C signðx1
vðtÞÞ

ð79Þ

For the presented continuous nonlinear differentiator (16), α1 A ð0; 1Þ is arbitrary and can be sufficiently small. From the continuity of α1 selection, when α1 -0 þ , differentiator (16) is still stable. Therefore, differentiator (16) with αn
i þ 1 ¼ ðn
iÞ=n; i ¼ 1; …; n, has the same structure as sliding mode differentiator (79). Then, the only task that remains is to match their parameters for the two differentiators. For Lipschitz constant C of vðn
1Þ ðtÞ, we can select it as large as possible. Let C 1=n ¼

1 ; ε

λ1 ¼ kn ;

ðn
2Þ=ðn
1Þ

λ1

1=ðn
1Þ 1=ðn
2Þ 1=2 λ2 ⋯λn
2 λn
1

λ2 ¼ kn
1 ; …; λ1

¼ k2 ;

λn ¼ k1

ð80Þ

therefore, when α1 ¼ 0, Eqs. (16) and (76) (i.e., Eq. (79)) are equivalent. Moreover, from (80), we can carry out λ1 ; …; λn as follows: λ1 ¼ kn ; λi ¼

kn
i þ 1 ðn
iÞ=ðn
i þ 1Þ

kn
i þ 2

This concludes the proof.

; i ¼ 2; …; n
1;

λn ¼ k1

ð81Þ

□

Remark 4. Because switching function exists in the last differential equation of sliding mode differentiator (76), although the output x1 ; …; xn
1 are smooth, the output xn is non-smooth, and this is generally referred to as the chattering phenomenon. Further, if noises exist in the signal, this chattering phenomenon will magnify noises near the equilibrium. In some velocity–acceleration feedback systems, this chattering in xn can make motors trembling. Therefore, the chattering must be removed sufficiently in the output xn of a differentiator. High-order continuous nonlinear differentiator (16) with lead compensation is designed to remove the chattering phenomenon, and the parameters can be selected easily. Further, when α1 -0 þ , differentiator (16) (i.e., differentiator (76)) is still stable, the term
λn C signðxn
vn
1 Þ in (76) can counteract n n n n the effect of d vðtÞ=dt . Therefore, C in (76) can be selected to be smaller than 1=εn , and only C Zjd vðtÞ=dt j is required. Further, differentiator (76) is still stable. Remark 5. For the presented continuous nonlinear differentiator (16), if α1 ¼ 1 is selected, then the linear high-gain differentiator is obtained [7]. 8. Application to quadrotor aircraft In this section, the high-order nonlinear differentiator application to a quadrotor aircraft is presented in order to observe the performance of the proposed differentiator. We consider the following: trajectory tracking for the quadrotor aircraft based on the differentiator estimations. The presented high-order nonlinear differentiator estimates the flying velocity, acceleration and the uncertainties from the position measurement by a Vicon system (Motion Capture System). 8.1. Modeling of quadrotor aircraft The quadrotor aircraft is shown in Fig. 1, and the forces and torques of the quadrotor aircraft are denoted in Fig. 2. Let Ξ g ¼ ðEx ; Ey ; Ez Þ denote the right handed inertial frame, and Ξ b ¼ ðEbx ; Eby ; Ebz Þ denote the frame attached to the aircraft's fuselage whose origin is located at its center of gravity. (ψ; θ; ϕ) describes the aircraft orientation expressed in the classical yaw, pitch and roll angles (Euler angles). m is the mass of the aircraft. We use cθ for cos θ and sθ for sin θ. Rbg is the transformation matrix representing the orientation of the rotorcraft from frame Ξb to Ξg, i.e., 2 3 cψ cθ cψ sθ sϕ
sψ cϕ cψ sθ cϕ þ sψ sϕ 6 7 ð82Þ Rbg ¼ 4 sψ cθ sψ sθ sϕ þ cψ cϕ sψ sθ cϕ
cψ sϕ 5
sθ cθ sϕ cθ cϕ The reactive torque generated, in free air, by the rotor i due to rotor drag is given by Q i ¼ kω2i

ð83Þ

238

X. Wang, B. Shirinzadeh / Mechanical Systems and Signal Processing 46 (2014) 227–252

Fig. 1. Quadrotor aircraft.

Fig. 2. Forces and torques of quadrotor aircraft.

and the total thrust generated by the four rotors is given by 4

4

i¼1

i¼1

F ¼ ∑ F i ¼ b ∑ ω2i

ð84Þ

where F i ¼ bω2i is the lift generated by the rotor i in free air, and k; b4 0 are two parameters depending on the density of air, the size, shape, and pitch angle of the blades, as well as other factors. Considering the bounded uncertainties (Δx ; Δy ; Δz ) and (Δψ ; Δθ ; Δϕ ) in the aircraft, the equations of motion written in terms of the center of mass C in the fixed axes of coordinate (x; y; z) are then mx€ ¼ ðcψ sθ cϕ þ sψ sϕ ÞF þ Δx my€ ¼ ðsψ sθ cϕ
cψ sϕ ÞF þΔy mz€ ¼ cθ cϕ F
mg þ Δz

ð85Þ

4

J z ψ€ ¼ ∑ ð
1Þi þ 1 Q i þ Δψ i¼1

J y θ€ ¼ ðF 1
F 3 Þl þΔθ J x ϕ€ ¼ ð
F 2 þF 4 Þl þ Δϕ

ð86Þ

where Jx, Jy and Jz are the three-axis moments of inertias; m is the mass of the aircraft; g is the gravity acceleration; l is the distance between each rotor and the center of gravity. The measurement outputs are yop1 ¼ x; yop51 ¼ θ;

yop2 ¼ y;

yop3 ¼ z;

_ yop52 ¼ θ;

yop42 ¼ ψ_ ;

yop41 ¼ ψ;

yop61 ¼ ϕ;

yop62 ¼ ϕ_

ð87Þ

Here, the aircraft is required to move from ð0; 0; hz Þ to ðh0 ; 0; hz Þ, and the desired trajectory is arranged as follows: xd ¼ h0 ð1
e
0:5km at Þ; 2

yd ¼ 0;

y_ d ¼ 0;

x_ d ¼ h0 km ate
0:5km at ;

y€ d ¼ 0;

2

zd ¼ hz ;

z_ d ¼ 0;

x€ d ¼ h0 km að1
km at 2 Þe
0:5km at ; 2

z€ d ¼ 0

where hz, h0, a and km are positive constants. This space motion trajectory is easy to be implemented.

ð88Þ

X. Wang, B. Shirinzadeh / Mechanical Systems and Signal Processing 46 (2014) 227–252

239

8.2. Differentiator design for quadrotor aircraft Corollary 1. (i) For position dynamics (85) of quadrotor aircraft, the third-order differentiators are designed as follows:   ki;3 jx
yopi jα3 sign xi;1
yopi x_ i;1 ¼ xi;2
εi i;1   k jxi;1
yopi jα2 sign xi;1
yopi x_ i;2 ¼ xi;3
i;2 2 εi   ki;1 x_ i;3 ¼
3 jxi;1
yopi jα1 sign xi;1
yopi εi

ð89Þ

_ y, _ z_ , Δx, Δy and Δz can be estimated, i.e., for t Z t s where yop1 ¼ x, yop2 ¼ y, yop3 ¼ z, i ¼1, 2, 3. Then x, _ r Lε3γ
1 ; jx1;2
xj _ rLε3γ
1 ; jx2;2
yj jx3;2
z_ j r Lε

3γ
1

jΔx
ðmx1;3
ðcψ sθ cϕ þsψ sϕ ÞFÞj r Lε3γ
2 ; jΔy
ðmx2;3
ðsψ sθ cϕ
cψ sϕ ÞFÞj rLε3γ
2 jΔz
ðmx3;3
cθ cϕ F þ mgÞj rLε3γ
2

;

ð90Þ

(ii) For attitude dynamics (86) of quadrotor aircraft, the second-order differentiators are designed as follows: α2 α1     k  k    x_ i;2 ¼ xi;3 − i;2 xi;2 −yopi2  sign xi;2 −yopi2 x_ i;3 ¼ − i;1 xi;2 −yopi2  sign xi;2 −yopi2 2 εi εi

ð91Þ

_ yop62 ¼ ϕ, _ i¼4, 5, 6. Then Δψ , Δθ and Δϕ can be estimated, i.e., for t Zt s where yop42 ¼ ψ_ , yop52 ¼ θ,  !   4    Δψ
J z x4;3
∑ ð
1Þi þ 1 Q i  r Lε2γ
1 ; Δθ
ðJ y x5;3
ðF 1
F 3 ÞlÞj r Lε2γ
1 ;   i¼1 jΔϕ
ðJ x x6;3
ð
F 2 þ F 4 ÞlÞj r Lε2γ
1

ð92Þ

8.3. System errors for quadrotor aircraft For reference trajectory (xd ; yd ; zd ), let e1 ¼ x
xd , e2 ¼ x_
x_ d , e3 ¼ y
yd , e4 ¼ y_
y_ d , e5 ¼ z
zd , e6 ¼ z_
z_ d . Then, for the position dynamics (85), the system error is me€ 1 ¼ ðcψ sθ cϕ þ sψ sϕ ÞF
mx€ d þ Δx me€ 3 ¼ ðsψ sθ cϕ
cψ sϕ ÞF
my€ d þΔy me€ 5 ¼ cθ cϕ F
mg
mz€ d þ Δz Let

2

3 e1 6e 7 ep ¼ 4 3 5; e5

ð93Þ

3 3 2 upx ðcψ sθ cϕ þsψ sϕ ÞF 6 u 7 6 ðs s c
c s ÞF 7 ψ ϕ up ¼ 4 py 5 ¼ 4 ψ θ ϕ 5; upz cθ cϕ F

2

2

6 Ξp ¼ 4

3
mx€ d
my€ d 7 5;
mz€ d
mg

2

3 Δx 6Δ 7 Δp ¼ 4 y 5; Δz

ð94Þ

the system error (93) for the position dynamics can be rewritten as e€ p ¼ m
1 ðup þΞ p þ Δp Þ

ð95Þ

For the reference attitude angle (ψ d ; θd ; ϕd ), let e7 ¼ ψ
ψ d , e8 ¼ ψ_
ψ_ d , e9 ¼ θ
θd , e10 ¼ θ_
θ_ d , e11 ¼ ϕ
ϕd , e12 ¼ ϕ_
ϕ_ d . Then, for the attitude dynamics (86), the system error is 4

J z e€ 7 ¼ ∑ ð
1Þi þ 1 Q i þΔψ
J z ψ€ i¼1

J y e€ 9 ¼ ðF 1
F 3 Þl þΔθ
J y θ€ d J x e€ 11 ¼ ð
F 2 þ F 4 Þl þ Δϕ
J x ϕ€ Let 2

3

2

3

ð96Þ 2

4

3

iþ1 6 ∑ ð
1Þ Q i 7 7 i¼1 6e 7 6u 7 6 7 ea ¼ 4 9 5; ua ¼ 4 ay 5 ¼ 6 6 ðF 1
F 3 Þl 7; 4 5 e11 uaz ð
F 2 þ F 4 Þl 2 3 2 3
J z ψ€ Δψ 6Δ 7 6
J θ€ d 7 y Ξa ¼ 4 5; Δa ¼ 4 θ 5; J ¼ diagfJ z ; J y ; J x g; Δϕ
J x ϕ€

e7

uax

ð97Þ

240

X. Wang, B. Shirinzadeh / Mechanical Systems and Signal Processing 46 (2014) 227–252

the system error (96) for the attitude dynamics can be rewritten as e€ a ¼ J
1 ðua þΞ a þ Δa Þ

ð98Þ

For quadrotor aircraft (85) and (86), the flying velocity cannot be measured directly, and the uncertainties exist. The proposed control law and its performance based on high-order nonlinear differentiator are presented in the following theorems. 8.4. Controller design for position dynamics

Theorem 5. For position dynamics (85), to track reference trajectory (xd ; yd ; zd ), if the controller is designed as b p
mðkp1 ep þkp2 eb_ p Þ up ¼
Ξ p
Δ

ð99Þ

by Δ b z , kp1 ; kp2 4 0, and b p ¼ ½Δ bx Δ where e1 ¼ x−xd , e^ 2 ¼ x1;2 −x_ d , e3 ¼ y−yd , e^ 4 ¼ x2;2 −y_ d , e5 ¼ z−zd , e^ 6 ¼ x3;2 −z_ d , Δ b x ¼ mx1;3
ðcψ sθ cϕ þ sψ sϕ ÞF Δ b y ¼ mx2;3
ðsψ sθ cϕ
cψ sϕ ÞF Δ b z ¼ mx3;3
cθ cϕ F þmg Δ

ð100Þ

then position error dynamic system (95) rendered by controller (99) will converge asymptotically to the origin, i.e., the tracking errors ep -0 and e_ p -0 as t-1. b_
p_ J r Lε3γ
1 , J Δ b p
Δp J r Lε3γ
2 , where Proof. In the light of Corollary 1, for t Z ts, the observation signals J p p p 2 3 2 3 x_ xb_ 6 7 6 7 b_ ¼ 6 yb_ 7 p_ p ¼ 4 y_ 5; p p 4 5 z_ zb_

ð101Þ

Considering controller (99), the closed-loop error system for the position dynamics is b p
Δp Þ e€ p ¼
kp1 ep
kp2 eb_ p
m
1 ðΔ b_
p_ Þ
m
1 ðΔ b p
Δp Þ ¼
kp1 ep
kp2 e_ p
kp2 ðp p p For t Z t s and sufficiently small ε, selecting the Lyapunov function be e_ p -0 as t-1. This concludes the proof. □

ð102Þ V p ¼ kp1 eTp ep þ 12 e_ Tp e_ p ,

we can obtain that ep -0 and

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi From the definition of up in (94), we know that F ¼ J up J 2 , where J up J 2 ¼ u2px þ u2py þ u2pz . Therefore, it follows that b p
mðkp1 ep þ kp2 eb_ p Þ J 2 . F ¼ J
Ξ p
Δ Especially, the reference trajectory is required along the x-direction. Therefore, the reference angles ϕd ¼ ψ d ¼ 0 are required. Moreover, from the flying characteristic of quadrotor aircraft, F a0. Accordingly, from upz ¼ cθ cϕ F in (94), we obtain the reference pitch angle as follows: θd ¼ arccosðupz =FÞ

ð103Þ

8.5. Controller design for attitude dynamics

Theorem 6. For attitude dynamics (86), to track reference attitude (ψ d ; θd ; ϕd ), if the controller is designed as b a
J½ka1 ea þka2 e_ a  ua ¼
Ξ a
Δ

ð104Þ

bθ Δ b ϕ , and b a ¼ ½Δ bψ Δ where ka1 ; ka2 40, Δ 4

b ψ ¼ J z x4;3
∑ ð
1Þi þ 1 Q i Δ i¼1

b θ ¼ J y x5;3
ðF 1
F 3 Þl Δ b ϕ ¼ J x x6;3
ð
F 2 þ F 4 Þl Δ

ð105Þ

then attitude error dynamic system (98) rendered by controller (104) will converge asymptotically to the origin, i.e., the tracking errors ea -0 and e_ a -0 as t-1.

X. Wang, B. Shirinzadeh / Mechanical Systems and Signal Processing 46 (2014) 227–252

241

b a
Δa J r Lε2γ
1 . Considering controller (104), the Proof. In the light of Corollary 1, for t Z ts, the observation signals J Δ closed-loop error system for attitude dynamics is b a
Δa Þ e€ a ¼
ka1 ea
ka2 e_ a
J
1 ðΔ For t Zt s and sufficiently small ε, selecting the Lyapunov function be e_ a -0 as t-1. This concludes the proof. □

ð106Þ V a ¼ ka1 eTa ea þ 12 e_ Ta e_ a ,

we can obtain that ea -0 and

9. Computational analysis and simulations 9.1. Comparison between the third-order sliding mode differentiator and the third-order continuous nonlinear differentiator In the following, function 10 sin ðtÞ is selected as signal vðtÞ. (1) Third-order sliding mode differentiator: x_ 1 ¼ v1 ; v1 ¼ x2
λ1 C 1=3 jx1
vðtÞj2=3 signðx1
vðtÞÞ x_ 2 ¼ v2 ; v2 ¼ x3
λ2 C 1=2 jx2
v1 j1=2 signðx2
v1 Þ x_ 3 ¼
λ3 C signðx3
v2 Þ (2) Third-order continuous nonlinear differentiator: k3 jx1
vðt Þjðα þ 2Þ=3 signðx1
vðt ÞÞ ε k2 x_ 2 ¼ x3
2 jx1
vðt Þjð2α þ 1Þ=3 signðx1
vðt ÞÞ ε k1 x_ 3 ¼
3 jx1
vðt Þjα signðx1
vðt ÞÞ ε x_ 1 ¼ x2


Parameters: α ¼ 0:5, ε ¼ 0:1, k3 ¼ 6, k2 ¼ 11, k1 ¼ 6, λ1 ¼ k3 , λ2 ¼ k2 =ðk3 Þ1=2 , λ3 ¼ k1 , C¼ 10. Fig. 3(a)–(c) shows the signal tracking and derivatives estimation, respectively, by the third-order sliding mode differentiator. Though signal tracking x1 and velocity estimation x2 are smooth (see Fig. 3(a) and (b)), acceleration estimation x3 is continuous but non-smooth (see Fig. 3(d)). The intensive chattering phenomena happen due to discontinuous differentiator structure. Fig. 4(a)–(c) shows the signal tracking and derivatives estimation, respectively, by the third-order continuous nonlinear differentiator. From Fig. 4(a)–(d), signal tracking output x1, velocity estimation x2 and acceleration estimation x3 are all smooth. The chattering phenomenon is avoided due to continuous differentiator structure. Moreover, the selection of parameters is very easy. 9.2. Discretization of nonlinear differentiator In the following, the discretization performances of the presented differentiators will be investigated. The popular 4thorder Runge–Kutta method and the Backward Euler Discretization approach will be used, respectively, for the third-order nonlinear differentiator. (1) The discrete form of the third-order nonlinear differentiator using the 4th-order Runge–Kutta Method is m11 ¼ T ðx2 ðk
1Þ
Rk3 jx1 ðk
1Þ
vðkÞjα3 signðx1 ðk
1Þ
vðkÞÞÞ   m21 ¼ T x3 ðk
1Þ
R2 k2 jx1 ðk
1Þ
vðkÞjα2 signðx1 ðk
1Þ
vðkÞÞ   m31 ¼
T R3 k1 jx1 ðk
1Þ
vðkÞjα1 signðx1 ðk
1Þ
vðkÞÞ  α3    m21 m11 m11  
Rk3 x1 ðk
1Þ þ
vðkÞ sign x1 ðk
1Þ þ
vðkÞ m12 ¼ T x2 ðk
1Þ þ 2 2 2  α2    m31 m11 m11  
R2 k2 x1 ðk
1Þ þ
vðkÞ sign x1 ðk
1Þ þ
vðkÞ m22 ¼ T x3 ðk
1Þ þ 2 2 2  α1    m11 m11  
vðkÞ sign x1 ðk
1Þ þ
vðkÞ m32 ¼
T R3 k1 x1 ðk
1Þ þ 2 2  α3    m22 m12 m12  
Rk3 x1 ðk
1Þ þ
vðkÞ sign x1 ðk
1Þ þ
vðkÞ m13 ¼ T x2 ðk
1Þ þ 2 2 2  α2    m32 m12 m12  
R2 k2 x1 ðk
1Þ þ
vðkÞ sign x1 ðk
1Þ þ
vðkÞ m23 ¼ T x3 ðk
1Þ þ 2 2 2  α1    m12 m12  
vðkÞ sign x1 ðk
1Þ þ
vðkÞ m33 ¼
T R3 k1 x1 ðk
1Þ þ 2 2  α3    m23 m13 m13  
Rk3 x1 ðk
1Þ þ
vðkÞ sign x1 ðk
1Þ þ
vðkÞ m14 ¼ T x2 ðk
1Þ þ 2 2 2  α2    m33 m13 m13   m24 ¼ T x3 ðk
1Þ þ
R2 k2 x1 ðk
1Þ þ
vðkÞ sign x1 ðk
1Þ þ
vðkÞ 2 2 2

242

X. Wang, B. Shirinzadeh / Mechanical Systems and Signal Processing 46 (2014) 227–252

Fig. 3. Sliding mode differentiator. (a) Signal tracking. (b) Velocity estimation. (c) Acceleration estimation. (d) The magnified figure of (c).

 α1    m13 m13  
vðkÞ sign x1 ðk
1Þ þ
vðkÞ m34 ¼
T R3 k1 x1 ðk
1Þ þ 2 2 x1 ðkÞ ¼ x1 ðk
1Þ þ 16 ðm11 þ2m12 þ 2m13 þ m14 Þ

X. Wang, B. Shirinzadeh / Mechanical Systems and Signal Processing 46 (2014) 227–252

243

Fig. 4. Continuous nonlinear differentiator. (a) Signal tracking. (b) Velocity estimation. (c) Acceleration estimation. (d) The magnified figure of (c).

244

X. Wang, B. Shirinzadeh / Mechanical Systems and Signal Processing 46 (2014) 227–252

Fig. 5. Differentiator discretizations without noise. (a) Signal tracking (T ¼ 0:001 s). (b) Velocity and acceleration estimation (T ¼ 0:001 s). (c) Signal tracking (T ¼ 0:05 s). (d) Velocity and acceleration estimation (T ¼ 0:05 s).

X. Wang, B. Shirinzadeh / Mechanical Systems and Signal Processing 46 (2014) 227–252

245

Fig. 6. Differentiator discretizations with noise. (a) Signal tracking (T ¼ 0:001 s). (b) Velocity and acceleration estimation (T ¼ 0:001 s). (c) Signal tracking (T ¼ 0:05 s). (d) Velocity and acceleration estimation (T ¼ 0:05 s).

246

X. Wang, B. Shirinzadeh / Mechanical Systems and Signal Processing 46 (2014) 227–252

x2 ðkÞ ¼ x2 ðk
1Þ þ 16 ðm21 þ2m22 þ 2m23 þ m24 Þ x3 ðkÞ ¼ x3 ðk
1Þ þ 16 ðm31 þ2m32 þ 2m33 þ m34 Þ (2) The discrete form of the third-order nonlinear differentiator using the Backward Euler Discretization Approach is z1 ðkÞ ¼ z1 ðk
1Þ þ Tz2 ðk
1Þ
TRk3 jz1 ðk
1Þ
vðkÞjα3 signðz1 ðk
1Þ
vðkÞÞ z2 ðkÞ ¼ z2 ðk
1Þ þ Tz3 ðk
1Þ
TR2 k2 jz1 ðkÞ
vðkÞjα2 signðz1 ðkÞ
vðkÞÞ z3 ðkÞ ¼ z3 ðk
1Þ
TR3 k1 jz1 ðkÞ
vðkÞjα1 signðz1 ðkÞ
vðkÞÞ where vðkÞ ¼ sin ðtÞ, t¼kT, k ¼ 1; …; N. 9.2.1. Case 1 (without noise) Parameters: α1 ¼ 0:5, R ¼ 1=ε ¼ 4, k3 ¼ 6, k2 ¼ 11, k1 ¼ 6. Initial values: x1 ð0Þ ¼ 0, x2 ð0Þ ¼ 0, x3 ð0Þ ¼ 2; z1 ð0Þ ¼ 0, z2 ð0Þ ¼ 0, z3 ð0Þ ¼ 0. The sampling time is selected as T ¼0.001 and T¼ 0.05, respectively. Fig. 5(a) and (b) describes the signal tracking and derivatives estimation for signal v(k) without noise on condition that T ¼0.001. It is shown that the estimate results by the 4th-order Runge–Kutta Method are little better than those by the Backward Euler Discretization Approach. Fig. 5(c) and (d) describes the signal tracking and derivatives estimation for signal v(k) without noise under condition that T ¼0.05. It is shown that the estimate results by the 4th-order Runge–Kutta Method still have the satisfying performances. However, the estimates are unstable by the Backward Euler Discretization Approach. 9.2.2. Case 2 (with noise) Parameters: α1 ¼ 0:15, R ¼ 1=ε ¼ 3, k3 ¼ 6, k2 ¼ 11, k1 ¼ 6. Initial values: x1 ð0Þ ¼ 0, x2 ð0Þ ¼ 0, x3 ð0Þ ¼ 2; z1 ð0Þ ¼ 0, z2 ð0Þ ¼ 0, z3 ð0Þ ¼ 0. The sampling time is selected as T ¼0.001 and T¼ 0.05, respectively. Fig. 6(a) and (b) describes the signal tracking and derivatives estimation for signal v(k) with noise on condition that T ¼0.001. It is shown that the estimate results by the 4th-order Runge–Kutta Method are little better than those by the Backward Euler Discretization Approach. Fig. 6(c) and (d) describes the signal tracking and derivatives estimation for signal v(k) with noise on condition that T ¼0.05. It is also shown that the estimate results by the 4th-order Runge–Kutta Method have the satisfying performances. However, the estimations are unstable by the Backward Euler Discretization Approach. Therefore, we can find that when sampling time is sufficiently small, for the nonlinear differentiator, the discretization estimates by the 4th-order Runge–Kutta Method and the Backward Euler Discretization Approach, respectively, almost have the same precision. However, when sampling time is relatively large, obvious different estimate performances exist for the two discretization methods. The Backward Euler Discretization Approach becomes unstable. While the 4th-order Runge– Kutta Method still has the satisfying estimate performance.

10. Experiment In this section, we present a real-time experimental result obtained when applying the nonlinear high-order differentiator and controller proposed in the previous sections to a quadrotor aircraft. The quadrotor aircraft prototype has been designed and shown in Fig. 1. In the quadrotor aircraft, four Electric Speed Controllers (ESC, RCEBL35X) based on pulse width modulation (PWM) are adopted to regulate the speeds of the four Brushless Direct Current (BLDC) motors, respectively. The peaking current of each ESC is 45 A, the constant current is 35 A, and the input voltage is 5.5–16.8 V. The shortest regulating period of ESC is equal to 2.5 ms by tests. A high-precision IMU (XsensMTI AHRS) is selected as the _ ϕ) _ can be measured directly. A DSP (TMS320LF2812) is taken as the driven board, which has attitude sensor, and (ψ; θ; ϕ; ψ_ ; θ; multiple PWM output channels. The control period is 8 ms. The voltage range of the BLDC motors is 6–18 V, the weight of each motor is 78 g, the max power is 340 W, and the maximum thrust is 1400 g. The four rotors are all plastic airscrew. Vicon (i.e., Motion Capture System) is a positioning system with a sub-millimetre precision. The real-time position of quadrotor aircraft is measured by a Vicon system. In the Vicon system, the cameras are connected to the motion capture processing unit, which in turn is connected to the application PC. The application PC is also connected to a receiver that receives RC transmitter signals. Further, the application PC is connected to a transmitter. The basic procedure for positioning the aircraft in real time is as follows: the cameras capture the positions of markers fixed on the aircraft and sent them to the motion capture processing unit. In the processing PC, the data are dealt with for computation. Then the position information is sent to the processing PC. In the application PC, the velocity and uncertainties are estimated by the nonlinear differentiator. Then through the transmitter, the information of position and estimates is sent to the processor in the aircraft for feedback control.

X. Wang, B. Shirinzadeh / Mechanical Systems and Signal Processing 46 (2014) 227–252

247

10.1. Measurement outputs Here, the quadrotor tracks a given trajectory (xd ; yd ; zd ). For aircraft (85) and (86), position (x; y; zÞ can be obtained by a _ ϕ) _ is measured by a high-precision IMU Vicon system (i.e., Motion Capture System). The attitude information (ψ; θ; ϕ; ψ_ ; θ; _ y; _ z_ ) and acceleration ðx; € y; € z€ Þ are unknown. Moreover, (Δx ; Δy ; Δz ) and (Δψ ; Δθ ; Δϕ ) are bounded (XsensMTI AHRS). Velocity ðx; uncertainties. 10.2. Discrete nonlinear differentiators 10.2.1. Discrete differentiator for the position dynamics (85) _ y; _ z_ ) The following discretization form of the third-order nonlinear differentiator is designed to estimate the velocity (x; € y; € z€ ) of the quadrotor aircraft from the position measurement (px ; py ; pz ) by the Vicon system: and acceleration (x;

 mi11 ¼ T xi2 ðk−1Þ−Ri ki3 jxi1 ðk−1Þ−pi ðkÞjα3 sign xi1 ðk−1Þ−pi ðkÞ 
 mi21 ¼ T xi3 ðk−1Þ−R2i ki2 jxi1 ðk−1Þ−pi ðkÞjα2 sign xi1 ðk−1Þ−pi ðkÞ 
 mi31 ¼ −T R3i ki1 jxi1 ðk−1Þ−pi ðkÞjα1 sign xi1 ðk−1Þ−pi ðkÞ      m m m   mi12 ¼ T xi2 ðk−1Þ þ i21 −Ri ki3 xi1 ðk−1Þ þ i11 −pi ðkÞα3 sign xi1 ðk−1Þ þ i11 −pi ðkÞ 2 2 2      m m m   mi22 ¼ T xi3 ðk−1Þ þ i31 −R2i ki2 xi1 ðk−1Þ þ i11 −pi ðkÞα2 sign xi1 ðk−1Þ þ i11 −pi ðkÞ 2 2 2      m m   mi32 ¼ −T R3i ki1 xi1 ðk−1Þ þ i11 −pi ðkÞα1 sign xi1 ðk−1Þ þ i11 −pi ðkÞ 2 2     m m m   mi13 ¼ T xi2 ðk−1Þ þ i22 −Ri ki3 xi1 ðk−1Þ þ i12 −pi ðkÞα3 sign xi1 ðk−1Þ þ i12 −pi ðkÞ 2 2 2      m m m   mi23 ¼ T xi3 ðk−1Þ þ i32 −R2i ki2 xi1 ðk−1Þ þ i12 −pi ðkÞα2 sign xi1 ðk−1Þ þ i12 −pi ðkÞ 2 2 2      m m   mi33 ¼ −T R3i ki1 xi1 ðk−1Þ þ i12 −pi ðkÞα1 sign xi1 ðk−1Þ þ i12 −pi ðkÞ 2 2

 mi14 ¼ T xi2 ðk−1Þ þ mi23 −Ri ki3 jxi1 ðk−1Þ þmi13 −pi ðkÞjα3 sign xi1 ðk−1Þ þmi13 −pi ðkÞ 
 mi24 ¼ T xi3 ðk−1Þ þ mi33 −R2i ki2 jxi1 ðk−1Þ þ mi13 −pi ðkÞjα2 sign xi1 ðk−1Þ þmi13 −pi ðkÞ 
 mi34 ¼ −T R3i ki1 jxi1 ðk−1Þ þ mi13 −pi ðkÞjα1 sign xi1 ðk−1Þ þ mi13 −pi ðkÞ xi1 ðkÞ ¼ xi1 ðk−1Þ þ 16 ðmi11 þ 2mi12 þ 2mi13 þmi14 Þ xi2 ðkÞ ¼ xi2 ðk−1Þ þ 16 ðmi21 þ 2mi22 þ 2mi23 þmi24 Þ xi3 ðkÞ ¼ xi3 ðk−1Þ þ 16 ðmi31 þ 2mi32 þ 2mi33 þmi34 Þ where (px ðkÞ; py ðkÞ; pz ðkÞ) is the position vector in the right handed inertial frame Ξg, and it is measured by the Vicon system. xx1 ðkÞ, xy1 ðkÞ and xz1 ðkÞ track the positions in x, y and z directions, respectively; xx2 ðkÞ, xy2 ðkÞ and xz2 ðkÞ estimate the velocities in x, y and z directions, respectively; xx3 ðkÞ, xy3 ðkÞ and xz3 ðkÞ estimate the accelerations in x, y and z directions, respectively. Moreover, from Eq. (90), we obtain the estimate of (Δx , Δy , Δz ). 10.2.2. Discrete differentiator for attitude dynamics (86) The following discretization form of the second-order nonlinear differentiator is designed to estimate the uncertainties _ ϕ) _ by an IMU (XsensMTI AHRS): (Δψ , Δθ , Δϕ ) in the attitude dynamics from the measurement of (ωψ ; ωθ ; ωϕ )¼(ψ_ ; θ; ni11 ¼ T ðxi2 ðk−1Þ−Ri k2 jxi1 ðk−1Þ−ωi ðkÞjα2 signðxi1 ðk−1Þ−ωi ðkÞÞÞ   ni21 ¼ −T R2i k1 jxi1 ðk−1Þ−ωi ðkÞjα1 signðxi1 ðk−1Þ−ωi ðkÞÞ      n n n   ni12 ¼ T xi2 ðk−1Þ þ i21 −Ri k2 xi1 ðk−1Þ þ i11 −ωi ðkÞα2 sign xi1 ðk−1Þ þ i11 −ωi ðkÞ 2 2 2      n n   ni22 ¼ −T R2i k1 xi1 ðk−1Þ þ i11 −ωi ðkÞα1 sign xi1 ðk−1Þ þ i11 −ωi ðkÞ 2 2     n n n   ni13 ¼ T xi2 ðk−1Þ þ i22 −Ri k2 xi1 ðk−1Þ þ i12 −ωi ðkÞα2 sign xi1 ðk−1Þ þ i12 −ωi ðkÞ 2 2 2      n n   ni23 ¼ −T R2i k1 xi1 ðk−1Þ þ i12 −ωi ðkÞα1 sign xi1 ðk−1Þ þ i12 −ωi ðkÞ 2 2 ni14 ¼ T ðxi2 ðk−1Þ þ ni23 −Ri k2 jxi1 ðk−1Þ þni13 −ωi ðkÞjα2 signðxi1 ðk−1Þ þ ni13 −ωi ðkÞÞÞ   ni24 ¼ −T R2i k1 jxi1 ðk−1Þ þ ni13 −ωi ðkÞjα1 signðxi1 ðk−1Þ þni13 −ωi ðkÞÞ xi1 ðkÞ ¼ xi1 ðk−1Þ þ 16 ðni11 þ 2ni12 þ 2ni13 þ ni14 Þ xi2 ðkÞ ¼ xi2 ðk−1Þ þ 16 ðni21 þ 2ni22 þ 2ni23 þ ni24 Þ

248

X. Wang, B. Shirinzadeh / Mechanical Systems and Signal Processing 46 (2014) 227–252

Fig. 7. Velocity and acceleration estimations in x-direction. (a) Estimation of x. (b) Estimation of vx . (c) Estimation of ax . (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

where (ωψ ðkÞ; ωθ ðkÞ; ωϕ ðkÞ) is the angular rate vector in the frame Ξb attached to the aircraft's fuselage, and it is measured by the IMU (XsensMTI AHRS). xψ1 ðkÞ, xθ1 ðkÞ and xϕ1 ðkÞ track the angular rates ψ_ , θ_ and ϕ_ in yaw, pitch and roll directions, respectively; xψ2 ðkÞ, xθ2 ðkÞ and xϕ2 ðkÞ estimate the angular accelerations ψ€ , θ€ and ϕ€ in yaw, pitch and roll directions, respectively. Moreover, from Eq. (92), we obtain the estimate of (Δψ , Δθ , Δϕ ). 10.3. Parameters of quadrotor control system The parameters for the aircraft control system are given as follows: Quadrotor aircraft (85) and (86): m ¼2.33 kg, g ¼ 9:8 m=s2 , l ¼ 0:4 m, J x ¼ 0:16 N m, J y ¼ 0:16 N m, J z ¼ 0:32 N m; b¼0.1, k ¼ 5  10
3 . Differentiators (89) for position dynamics (85): Rx ¼ 1=εx ¼ 2, kx1 ¼ 0:425, kx2 ¼ 0:75, kx3 ¼ 1:5, αx ¼ 0:5; Ry ¼ 1=εy ¼ 1:2, ky1 ¼ 0:425, ky2 ¼ 0:75, ky3 ¼ 1:5, αy ¼ 0:8; Rz ¼ 1=εz ¼ 1:2, kz1 ¼ 0:425, kz2 ¼ 0:75, kz3 ¼ 1:5, αz ¼ 0:5, T ¼ 0:008 s. Differentiators (91) for attitude dynamics (86): Ri ¼ 1=εi ¼ 2, ki1 ¼ 2:45, ki2 ¼ 1:25 (i ¼ ψ, θ, ϕ), α1 ¼ 0:5. Reference trajectory (88): h0 ¼ 10 m, a ¼ 5 m=s2 , km ¼ 0:01, hz ¼ 2 m.

X. Wang, B. Shirinzadeh / Mechanical Systems and Signal Processing 46 (2014) 227–252

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Fig. 8. Velocity and acceleration estimations in y-direction. (a) Estimation of y. (b) Estimation of vy . (c) Estimation of ay . (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

Controllers (99) and (104): kp1 ¼ 2:5, kp2 ¼ 1:25; ka1 ¼ 0:5, ka2 ¼ 1. 10.4. Comparison between the estimation by the differentiator and the calculation by IMU In the following, a navigation algorithm is used to calculate the position and velocity in the inertial frame Ξg using the measurements in IMU. Furthermore, the estimates by the presented nonlinear differentiator are compared with the results of the navigation algorithm. In the IMU (XsensMTI AHRS), three orthogonal accelerometers and three orthogonal rate-gyroscopes are contained. Therefore, the triaxial acceleration vector ab ¼ ½axb ayb azb T in body frame Ξb can be measured. The triaxial acceleration vector ag ¼ ½ax ay az T in the inertial frame Ξg can be written as ag ¼ Rbg ab þ ½0 0 gT . In this experiment, the quadrotor aircraft tracks reference trajectory (xd ; yd ; zd ) without the information of position and _ ϕ) _ is measured by the high-precision _ y; _ z_ ) is unknown. The attitude information of ab and (ψ; θ; ϕ; ψ_ ; θ; velocity, i.e., (x; y; z; x; IMU (XsensMTI AHRS). The integral algorithm by the trapezoidal rule [29] is used to calculate the position (x; y; z) and velocity (vx ; vy ; vz ) of the quadrotor aircraft from the acceleration outputs ag ¼ ½ax ay az T . We will compare the calculations of position and velocity by

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Fig. 9. Velocity and acceleration estimations in z-direction. (a) Estimation of z. (b) Estimation of vz . (c) Estimation of az . (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

the acceleration measurements with the real position by Vicon and the velocity estimations by the nonlinear differentiators, respectively. 10.5. Experiment results In this experiment, the quadrotor aircraft tracks reference trajectory (xd ; yd ; zd ) without the information of velocity and uncertainties. The nonlinear differentiators (89) and (91) are designed to estimate the velocity (vx ; vy ; vz ), acceleration (ax ; ay ; az ) and the uncertainty (Δx, Δy, Δz, Δψ , Δθ , Δϕ ) of the quadrotor aircraft from the measurements of position (x; y; z) and _ ϕ). _ The controllers (99) and (104) are designed to stabilize the flight dynamics. the attitude angular rate (ψ_ ; θ; Fig. 7 describes the position tracking, velocity and acceleration estimations in the x-direction; Fig. 8 describes the position tracking, velocity and acceleration estimations in the y-direction; Fig. 9 describes the position tracking, velocity and acceleration estimations in the z-direction. In Figs. 7–9, green lines denote the calculations of position and velocity from the acceleration measurements in IMU. In the experiment above, though high-frequency stochastic noises exist in the measurement signals, the velocity and acceleration estimations by the presented nonlinear differentiators and the tracking results by the designed controller have

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satisfying qualities. The stochastic noises are restrained sufficiently by the nonlinear differentiators. In the tracking outputs, not only the dynamical performances are fast, but also the tracking precisions are accurate. The acceleration information provided by the three orthogonal accelerometers in IMU is usually accurate. From Figs. 7(c), 8(c) and 9(c), except for much noise in the acceleration measurements in IMU, there are no obvious errors between the acceleration estimations by the nonlinear differentiator and the acceleration measurement by IMU. However, from the green lines in Figs. 7(a), 8(a) and 9(a), the obvious position drifts exist in the outputs of the navigation algorithm. Moreover, the intensive high-frequency noise exists in the acceleration measurements due to trembling effects (for instance, motors trembling). The drift phenomenon of position calculation is mainly brought out by the usual integral algorithms. The integral algorithms cannot restrain the effect of stochastic noise (especially non-white noise). Such noise leads to the accumulation of additional drift in the integrated signal. On the other hand, our research plant is a quadrotor aircraft. The aircraft is in flexible and low-speed flight. The accelerometers in IMU cannot response such flexible movement efficiently, and the flying velocity cannot be measured directly. This navigation algorithm by the acceleration measurements in IMU is more suitable to a fixed-wing aircraft, and the flying velocity can be measured by airspeed tube. The large inertial movement of fixed-wing aircraft is more suitable to the use of accelerometers, and no obvious high-frequency trembling exists.

11. Conclusions In this paper, a high-order continuous nonlinear differentiator with lead compensation is presented based on finite-time stability. Because of its continuous structure, the chattering phenomenon can be reduced sufficiently than that of the sliding mode differentiator. Moreover, this continuous differentiator may be considered as the generalization of high-order sliding mode differentiator and linear high-gain differentiator, and the parameters selection become more convenient and readily easy. Our future work is to apply the presented differentiator to a fixed-wing aircraft system control.

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