Global smooth sliding mode controller for flexible air-breathing hypersonic vehicle with actuator faults

Aerospace Science and Technology 92 (2019) 563–578

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Aerospace Science and Technology www.elsevier.com/locate/aescte

Global smooth sliding mode controller for flexible air-breathing hypersonic vehicle with actuator faults Yibo Ding, Xiaogang Wang ∗ , Yuliang Bai, Naigang Cui School of Astronautics, Harbin Institute of Technology, Harbin 150001, China

a r t i c l e

i n f o

Article history: Received 22 November 2018 Received in revised form 27 May 2019 Accepted 20 June 2019 Available online 21 June 2019 Keywords: Flexible air-breathing hypersonic vehicle Fast finite-time high-order regulator Integral sliding surface Global sliding mode Generalized smooth second-order sliding mode reaching law Smooth fixed-time observer

a b s t r a c t A global smooth sliding mode controller (GSSMC) is proposed for flexible air-breathing hypersonic vehicle (FAHV) under actuator faults and parametric uncertainties, consisting of global fast finitetime integral sliding surface (GFFIS), generalized smooth second-order sliding mode reaching law (GSRL) and smooth fixed-time observer. Firstly, nonlinear control-oriented model of FAHV is processed using input/output feedback linearization with flexible effects and actuator faults modeling as lumped matched disturbances. Secondly, a GFFIS is established to ensure finite-time convergence of states without singularity based on a newly proposed fast finite-time high-order regulator (FFR). The FFR is improved from standard finite-time high-order regulator via dilation rescaling, which can accelerate response speed avoiding complicated parameters selection. Meanwhile, GFFIS can eliminate initial reaching phase to enhance robustness of system due to characteristic of global convergence. Thirdly, a GSRL is presented to ensure finite-time convergence of sliding mode vector and its derivative without chattering based on a generalized smooth second-order sliding mode control algorithm, the stability and finite convergence time of which is analyzed via Lyapunov criteria in detail. Then, a smooth fixed-time observer is applied to estimate lumped disturbances in fixed time and avoid effects of parametric uncertainties. With the three components, GSSMC can drive FAHV subject to actuator faults and parametric uncertainties to follow desired values in finite time with smooth control signals. Ultimately, three sets of simulations are performed to verify the effectiveness of the methods proposed. © 2019 Elsevier Masson SAS. All rights reserved.

1. Introduction Air-breathing hypersonic vehicle has attracted great attention in recent years on account of its large flight envelope, quickresponse and maneuverability. However, strong couplings between propulsion and aerodynamics, static instability and flexible effects aroused by slender geometry all raise difficulties in control system design [1,2]. In addition, considering air-breathing hypersonic vehicle often operates under extreme flight conditions, it is susceptible to the effects of actuator faults and parametric uncertainties, which will lead to poor control performance and even catastrophic accidents [3,4]. Therefore, it is necessary to investigate control algorithm with strong robustness, quick response and high precision for flexible air-breathing hypersonic vehicle (FAHV).

*

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

https://doi.org/10.1016/j.ast.2019.06.032 1270-9638/© 2019 Elsevier Masson SAS. All rights reserved.

The longitudinal dynamic model of FAHV has been developed during the past decades. Bolender [5] presents a truth model considering major characteristic of FAHV: static instability, nonminimum phase and strong couplings between propulsion and aerodynamic systems. Meanwhile, forebody and afterbody of FAHV are assumed as cantilever beams fixed at barycenter, in order to describe the flexible effects accurately [6]. However, considering the complexity of true model, a curve-fitted model is introduced in [2] to approximate the expressions of force and moment as simplified forms. To facilitate the design of controller, Parker [7] proposes a control-oriented model via neglecting flexible states and weak elevator couplings. In this paper, FAHV is modeled as a highly nonlinear curve-fitted model with control algorithm developed based on control-oriented model. In addition, actuators of FAHV are assumed to suffer from gain fault and deviation fault at the same time induced by the leakage of hydraulic fluid or sensor fault in an actuator system [8–10]. By means of considering flexible effects and actuator faults as lumped matched disturbances, input/output feedback linearization method is applied to

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Y. Ding et al. / Aerospace Science and Technology 92 (2019) 563–578

transform nonlinear control-oriented model into an affine nonlinear form. FAHV GSSMC GFFIS FFR GSRL GSSOSM SSOSM

flexible air-breathing hypersonic vehicle global smooth sliding mode controller global fast finite-time integral sliding surface fast finite-time high-order regulator generalized smooth second-order sliding mode reaching law generalized smooth second-order sliding mode control algorithm smooth second-order sliding mode control algorithm

Considering FAHV is always required to perform critical tasks and the cost of it is expensive, it is essential to develop effective fault-tolerant control algorithm (FTC) to keep FAHV reliable even under actuator faults. The existing FTC methods commonly fall into two main categories [11,12]: active FTC and passive FTC. Compared with active FTC, passive FTC does not need reconfiguration of controller which improves reliability of control system and avoids time delay induced by reconfiguration mechanism. There are various control theories developed for passive FTC of hypersonic flight vehicle such as back-stepping control [13,14], adaptive control [15–17], neural networks control [6,18], fuzzy control [19, 20], sliding mode control [8,9,21] and so on. Among them, sliding mode control is recommended owing to its characteristics of finitetime convergence and robustness relative to matched disturbances [22]. In this paper, in order to achieve high-precision finite-time control for FAHV under actuator faults and parametric uncertainties with smooth control signal, a global smooth sliding mode controller (GSSMC) is proposed with three components: global fast finite-time integral sliding surface (GFFIS), generalized smooth second-order sliding mode reaching law (GSRL) and smooth fixedtime observer. A variety of sliding surfaces have been developed in recent years such as: linear sliding surface, terminal sliding surface, nonsingular terminal sliding surface, integral sliding surface and so on. Linear sliding surface could only drive states to converge exponentially. Terminal sliding surface is able to realize finite-time convergence of states, but singularity phenomenon is inevitable. Nowadays, many researchers [9,23] utilize integral sliding surface to avoid singularity based on arbitrary order continuous finite-time regulator, which is firstly presented by Bhat [24]. However, the adjustment for coefficients of finite-time high-order regulator is complicated, which makes it hard to achieve arbitrarily fast convergence speed. In order to improve it, a fast finite-time high-order regulator (FFR) is proposed in this paper with a gain incorporating into standard finite-time high-order regulator via dilation rescaling. The FFR presented is able to accelerate response speed of system by increasing value of gain simply without complicated adjustment process of coefficients. In addition, stability of FFR is proved strictly via Lyapunov criteria and finite convergence time is estimated. Based on the novel FFR, a GFFIS is developed in this paper to make tracking errors of FAHV converge in finite time without singularity. In addition, GFFIS eliminates initial reaching phase, which ensures that system is initially on sliding surface. Considering invariance property of sliding mode control only exists in sliding phase, GFFIS could enhance robustness of system and improve response speed, compared with conventional sliding mode control. Traditional reaching laws mainly include: constant rate reaching law, exponential reaching law and power rate reaching law, etc. Considering conventional reaching laws could only ensure convergence of sliding mode variable, a GSRL is presented based on generalized smooth second-order sliding mode control algorithm (GSSOSM) to make sliding mode variable and its derivative converge to zero at the same time. GSSOSM proposed in this paper is improved from smooth second-order sliding mode control algorithm (SSOSM) presented in [25]. The stability of GSSOSM is

proved via Lyapunov criteria in detail and finite convergence time is provided, which is shown to be smaller than SSOSM. Compared with generalized super-twisting algorithm in [26] and [27], control signal produced by GSRL is not only continuous but also smooth, which avoids chattering phenomena in conventional sliding mode control. In order to improve the robustness respect to actuator faults and parametric uncertainties, a smooth fixed-time observer [28] is introduced. Considering FAHV is statically unstable, conventional observers cannot ensure the convergence of estimated error before states of FAHV escape to infinity under abrupt actuator faults. Therefore, smooth fixed-time observer is applied in this paper to estimate lumped disturbances and derivatives of error vector accurately in fixed time. In addition, the output of smooth fixed-time observer is smooth, which avoids chattering phenomena occurring in conventional sliding mode observer [29,30]. Finally, compensation of lumped disturbances enhances the fault-tolerant performance, while estimation for derivatives of error vector enhances the robustness with respect to parametric uncertainties. By means of combining the three components GFFIS, GSRL and smooth fixedtime observer, the novel GSSMC can drive FAHV subject to actuator faults and parametric uncertainties to follow desired values in finite time with smooth control signal. Compared with existing disturbance observer based sliding mode controllers, the GSSMC proposed shows great advantages at each of the three components. The method in [21] combines quasi-continuous high-order sliding mode controller and homogeneous high-order sliding mode observer. The disadvantage of quasi-continuous high-order sliding mode controller is that chattering is inevitable when tracking error approaches zero. The disadvantage of homogeneous high-order sliding mode observer is that estimated error can only converge in finite time. Compared with it, the smooth fixed-time observer applied in this paper can estimate disturbances exactly in fixed time, which surely has a faster response speed. In addition, the control signal produced by GSSMC is smooth which avoids chattering phenomena. The method in [31] combines exponential convergent sliding surface, power rate reaching law and homogeneous high-order sliding mode observer. Compared with exponential convergent sliding surface, the GFFIS can make state error converge in finite time, which can achieve higher convergence precision and faster response speed. Compared with power rate reaching law, GSRL can make sliding mode variable achieve higher second-order convergence accuracy with smooth signal. The method in [32] combines finite-time integral sliding surface, fast super-twisting reaching law and fast super-twisting disturbance observer. The finite-time integral sliding surface of it is established based on standard finite-time high-order regulator. Compared with it, the GFFIS in this paper can accelerate response speed of system by increasing value of gain simply. Compared with fast super-twisting reaching law, the output of GSRL is not only continuous but also smooth. In addition, smooth fixedtime observer can achieve faster response speed compared with fast super-twisting disturbance observer. In summary, the contributions of this paper are given as follows: 1. A novel FFR is proposed and the stability is proved strictly. Compared with standard finite-time high-order regulator, FFR is able to accelerate response speed of system simply without complicated adjustment process of coefficients. 2. A GFFIS is developed based on FFR to realize finite-time convergence of tracking errors without singularity. Meanwhile, initial reaching phase is eliminated, which can enhance robustness of system and improve response speed.

Y. Ding et al. / Aerospace Science and Technology 92 (2019) 563–578

3. A GSSOSM is developed to achieve faster response speed compared with nominal SSOSM. Stability of it is proved via Lyapunov criteria. 4. GSSMC proposed can guarantee finite-time convergence for FAHV under actuator faults and parametric uncertainties with smooth control signal, which avoids chattering in conventional sliding mode control. The organization of the rest of paper is structured below: In Section 2, a curve-fitted model of FAHV is introduced. Section 3 presents control-oriented model, actuator faults model and input/output feedback linearization technique. Section 4 proposes novel GSSMC including three components. Stability and finite convergence time of GFFIS and GSRL are analyzed via Lyapunov criteria in detail. In Section 5, three sets of simulation are performed to verify the effectiveness and superiority of the FFR, GSSOSM and GSSMC proposed. 2. Flexible hypersonic vehicle model In this section, a longitudinal model of FAHV is presented which is developed by Bolender [2] named curve-fitted model. In order to describe the flexible effects more accurately, fuselage is modeled as a pair of cantilever beams clamped at barycenter of fuselage. The equations of longitudinal dynamics are given below

(1)

where V , h, γ , α , Q , η1 , η2 are vehicle velocity, altitude, flight path angle, angle of attack, pitch rate, generalized modal coordinates of forebody and afterbody respectively. Variables m, g , I y y denote mass, gravitational acceleration and moment of inertia. Notations ζi , ωi , ψ˜ i are damping ratio, natural frequency and inertial coupling parameter. Variables ki satisfy ki = 1 + ψ˜ i2 / I y y . Notations T , D , L , M , N i represent thrust, drag, lift, pitching moment and generalized forces which are expressed using curve-fitted approximations as follows: 2

α 3 + C Tα α 2 + C Tα α + C T0 δ2

α 2 + C αD α + C De δe2 + C δDe δe + C 0D δ







2

α α2 + C α α + C 0 M = z T T + q¯ S c¯ C M M ,α + c e δe ,α M ,α

N 1 = N 1α

2

N 2 = N 2α

2



α 2 + N 2α α + N 2δe δe + N 20 

ρ = ρ0 exp −(h − h0 )/h s

3

C Tα = β1 (h, q¯ )Φ + β2 (h, q¯ ) C Tα = β5 (h, q¯ )Φ + β6 (h, q¯ )

¯ )/m − g sin γ V˙ = ( T cos α − D

(4)

α 2 + C αD α + C 0D 

2

(2)

in which δe , Φ are elevator angular deflection and fuel-to-air ratio. Readers are recommended to refer to Tables A1–A7 in Appendix of [7] for detailed definitions and values of model coefficients.

2



dc = (ψ˜ 1 η¨ 1 + ψ˜ 2 η¨ 2 )/ I y y

(5)

In order to possess full relative vector degree, a dynamic extension is required for Φ . Selecting command value Φc as a new input to replace Φ , a second-order actuator model is appended as below:

Φ¨ = −2ζ ωΦ˙ − ω2 Φ + ω2 Φc

(6)

where damping ratio and natural frequency of actuator dynamics are chosen as: ζ = 0.7, ω = 20 [7]. The appended second-order engine dynamics (6) incorporates the lag associated with Φ in a real model of FAHV, which increases the fidelity of model [31]. Control input and output vector are selected as: u = [δe Φc ] T and y = [ V γ ] T respectively. The desired values V d and hd are smoothed from step commands V c and hc by two pre-filters:

×

C Tα = β3 (h, q¯ )Φ + β4 (h, q¯ )



g Q = q¯ S c¯ ce

(s2

hd = h c



C T0 = β7 (h, q¯ )Φ + β8 (h, q¯ )

2

α α2 + C α α + C 0 f Q = z T T + q¯ S c¯ C M M ,α ,α M ,α

Vd = Vc ×

α 2 + N 1α α + N 10

q¯ = ρ V /2 2

A feedback linearization technique is performed based on control-oriented model in this section and then the input-output error dynamics in case of actuator faults is derived. Considering rigid-flexible couplings as disturbances and neglecting weak elevator couplings, equation (1) turns to a controloriented model.



k2 η¨ 2 = −2ζ2 ω2 η˙ 2 − ω22 η2 + N 2 − ψ˜ 2 M / I y y − ψ˜ 2 ψ˜ 1 η¨ 1 / I y y



3. Control-oriented model and input/output feedback linearization

¯ = q¯ S C αD D

k1 η¨ 1 = −2ζ1 ω1 η˙ 1 − ω12 η1 + N 1 − ψ˜ 1 M / I y y − ψ˜ 1 ψ˜ 2 η¨ 2 / I y y

L = q¯ S C Lα α + C Le δe + C L0

The nominal flight is considered to be at a trimmed hypersonic cruising regime. Therefore, system states and control inputs are required to be maintained within admissible ranges given in Table 1 referenced from [34], which determines the flight envelope and value ranges of control inputs.

in which

Q˙ = ( M + ψ˜ 1 η¨ 1 + ψ˜ 2 η¨ 2 )/ I y y

2

(3)

Q˙ = f Q + g Q δe + dc

α˙ = Q − γ˙





α˙ = Q − γ˙

γ˙ = ( L + T sin α )/(mV ) − g / V cos γ

D = q¯ S C α D



L = q¯ S C Lα α + C L0

γ˙ = ( L + T sin α )/(mV ) − g / V cos γ

h˙ = V sin γ

3

Considering the deflection of elevator produces a significant amount of lift and results in a non-minimum phase behavior, an additional canard is added to compensate undesirable contribution of elevator to lift [33]. The deflection of canard δc is ganged with elevator deflection using a negative gain to ensure δ δ C Le δe + C Lc δc = 0. Thus, the lift L is expressed as

h˙ = V sin γ

V˙ = ( T cos α − D )/m − g sin γ

T = C Tα

565

2 ωd1 × ωd2 2 + 2 × ζd1 × ωd1 s + ωd1 )(s + ωd2 )

(7)

2 2 ωd1 × ωd2 2 2 (s2 + 2 × ζd1 × ωd1 s + ωd1 )(s2 + 2 × ζd2 × ωd2 s + ωd2 )

(8) Defining tracking error of altitude as eh = h − hd , the desired value of γ is designed as [35]:



γd = arcsin (h˙ d − k P eh )/ V



(9)

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Table 1 Admission ranges for states and control inputs. Parameter

Lower bound

Upper bound

Parameter

Lower bound

Upper bound

h V

21336 m 2133.6 m/s −5 deg −3 deg

41148 m 3352.8 m/s 10 deg 3 deg

Q

−10 deg/s 0.05 −30 deg

10 deg/s 1.5 30 deg

α γ

Φ δe

Fig. 1. The architecture of the proposed controller scheme for FAHV.

where k P > 0. Gain fault and deviation fault are the faults commonly appearing on FAHV actuators. With assumption that the two faults are encountered at the same time, control signal under faults denoted by u f is modeled as:

u f = F gu + Fd

(10)

in which F g is gain fault matrix while F d is derivative fault vector. They are expressed below:

F g = diag( F g1 F d = [ F d1

F g2 )

F d2 ] T

(11)

where 0 < F g1 , F g2 ≤ 1. If F g = E 2 and F d = [0 0] , the actuators are working in normal mode. The error vector of output is defined as e = y − y d = [e 1 e 2 ] T = [ V − V d γ − γd ] T . Therefore, an affine nonlinear form of error dynamics under actuator faults is derived from control-oriented model via input/output feedback linearization [36]. T

... ... ... e = y − yd

...

= F + G u f + G d dc − y d

...

= F + G ( F g u + F d ) + G d dc − y d   ... = F + G u − y d + G ( F g − E 2 ) u + F d + G d dc

(12)

The expressions of F , G and G d are given in Appendix A in detail. The lumped disturbances of system under actuator faults are defined as

D f = [D f 1





D f 2 ] T = G ( F g − E 2 ) u + F d + G d dc

(13)

˙ h are assumed Remark 1. Only the rigid-body states V , α , γ , Φ, Φ, to be measurable such that they could be used in controller design. Considering the flexible states η1 , η2 , η˙ 1 , η˙ 2 and faults information F g and F d are unmeasurable terms, they cannot be compensated accurately by feedback. Therefore, the actuator error and flexible effects must be considered as unknown lumped disturbances, which require to be dealt with via robustness of controller.

Considering FAHV is statically unstable and prone to suffer from strong disturbances, control object is to design a robust smooth controller to make output y track the desired value y d in finite time for FAHV under actuator faults and parameter uncertainties. Meanwhile, the values of states and control inputs fall into the ranges in Table 1 all the time during the process of maneuver. 4. Global smooth sliding mode controller design A global smooth sliding mode controller (GSSMC) is presented for FAHV subject to actuator faults and parametric uncertainties in this section, composed of global fast finite-time integral sliding surface (GFFIS), generalized smooth second-order sliding mode reaching law (GSRL) and smooth fixed-time observer. The architecture of the controller proposed is given below in Fig. 1. 4.1. Global fast finite-time integral sliding surface design In order to achieve finite-time convergence of the tracking error vector e without singularity, a GFFIS is developed based on a fast finite-time high-order regulator (FFR) proposed in this subsection. The FFR presented is able to improve convergence speed via increasing value of gain simply without complicated adjustment process of coefficients. In addition, GFFIS can eliminate initial reaching phase, so as to enhance robustness of system and improve response speed. Theorem 1. Consider a general third-order chain of integrator

x˙ 1 = x2 x˙ 2 = x3 x˙ 3 = u

(14)

with a fast finite-time high-order regulator (FFR) given as 1 −α 1

u = −k F 1 L F

1 −α 2

sigα1 (x1 ) − k F 2 L F

− k F 3 L 1F−α3 sigα3 (x3 )

sigα2 (x2 ) (15)

Y. Ding et al. / Aerospace Science and Technology 92 (2019) 563–578

in which L F is a gain used to improve standard finite-time high-order regulator via dilation rescaling and it is satisfied with L F ≥ 1. Parameters k F 1 , k F 2 , k F 3 are selected to ensure that polynomial s3 + k F 3 s2 + k F 2 s + k F 1 is Hurwitz. The notation sigr (x) represents |x|r sgn(x). The origin of system (14) s finite-time stable with αi satisfying

α i −1 =

α i α i +1

i = 2, 3

2α i +1 − α i

(16)

where α4 = 1, α3 = αε ∈ (1 − ε , 1) for sufficiently small positive number ε . The upper bound of convergence time for (14) decreases along with the increase of L F . Proof. Substituting (15) into (14) results in the following continuous system:

x˙ 1 = x2 x˙ 2 = x3 1 −α 1

x˙ 3 = −k F 1 L F

1 −α 2

sigα1 (x1 ) − k F 2 L F

sigα2 (x2 )

1 −α − k F 3 L F 3 sigα3 (x3 )

(17)

AF = ⎣

0 0

−k F 1 L 1F−α1

1 0

−k F 2 L 1F−α2

0 1

−k F 3 L 1F−α3

⎤ (18)

1−α3 2

1−α

1−α2

s +kF 2 L F

s+

k F 1 L F 1 . According to Routh stability criterion, matrix A F is Hurwitz if the following conditions are satisfied 1 −α 3

>0

1 −α kF 1 L F 1 > 0 1 −α 1 −α 1 −α kF 3 L F 3 · kF 2 L F 2 − kF 1 L F 1 1 −α kF 3 L F 3

1 −α 3

>0

(19)

1 −α 3

2(1−αε )

kF 3

z1 = x1 / L F

2αε (1−αε ) 3−2αε

LF

(20)

αε ∈ (1 − ε, 1), variables α3 = αε , α2 = αε /(2 − αε ) and α1 = αε /(3 − 2αε ) belong to (0, 1). The following inequality holds: 2 − αε

−

= (1 − αε )

χ (24)

z2 = x2 / L F

z3 = x3 / L F

(25)

z˙ 2 = z3 z˙ 3 = −k F 1 sigα1 ( z1 ) − k F 2 sigα2 ( z2 ) − k F 3 sigα3 ( z3 )

2αε (1 − αε ) 3 − 2αε 2αε2 − 8αε + 6

(2 − αε )(3 − 2αε )

>0

(21)

(26)

System (17) and system (26) have the same convergence time. Considering s3 + k F 3 s2 + k F 2 s + k F 1 is Hurwitz, there exists a symmetric positive definite matrix P F 1 satisfied with a Lyapunov equation:

in which Q

⎡

(27)

F1

is symmetric positive definite and A F 1 is defined as

F1

1 0 −k F 2

⎤

0 1 ⎦ −k F 3

1/r z2 2

1/r z3 3 ] T

(28)

ϕ 1T (ζ ) P F 1 ϕ 1 (ζ ),

where ϕ 1 = and ζ = [ z1 z2 z3 ] . The proof process of finite-time convergence for system (26) is similar to system (17). Functions V F 1 (ϕ 1 ) and V˙ F 1 (ϕ 1 ) are homogeneous in vector ζ of degree 2 and 1 + αε respectively with respect to the same weights r i . Thus, the following inequality holds [39]:

When

2(1 − αε )



According to the conclusion acquired in [37], the derivative of Lyapunov function V F (ϕ ) is also negative definite with αε ∈ (1 − ε , 1) for a sufficiently small ε > 0. Thus, system (17) is asymptotically stable. Let the homogeneity degree of x1 , x2 , x3 be r1 , r2 and r3 respectively. Then system (17) is homogeneous of degree αε − 1 with respect to vector χ according to the definition of homogeneous vector fields in [38]. On the basis of Theorem 7.1 in [24], system (17) could converge to origin in finite time. A dilation rescaling of states in system (17) is taken as

1/r [ z1 1

k F 1 α 3 −α 1 L kF 3 F kF 1

= −χ Q F χ < 0

Define a Lyapunov function as V F 1 (ϕ 1 ) =

kF 3 L F

= k F 2 L F 2−αε −

T

0 AF1 = ⎣ 0 −k F 1

· k F 2 L 1F−α2 − k F 1 L 1F−α1

= k F 2 L 1F−α2 −



V˙ F (χ ) = χ˙ T P F χ + χ T P F χ˙ = χ T A TF P F + P F A F

P F 1 A F 1 + A TF 1 P F 1 = − Q

in which

kF 3 L F

1/r

z˙ 1 = z2

⎦

The characteristic polynomial of A F is s3 + k F 3 L F

kF 3 L F

1/r

in which ϕ = [x1 1 x2 2 x3 3 ] T , χ = [x1 x2 x3 ] T , r1 = 3 − 2αε , r2 = 2 − αε , and r3 = 1. Considering A F is Hurwitz, system χ˙ = A F χ is asymptotically stable. Selecting V F (χ ) = χ T P F χ as a Lyapunov function for system χ˙ = A F χ , the derivative of V F (χ ) is satisfied with

System (17) in the new coordinates is expressed by

Define matrix A F as

⎡

1/r

567

T

1+αε

V˙ F 1 (ϕ 1 ) ≤ −c 1 V F 12 (ϕ 1 )

(29)

where c 1 is a positive constant. Considering system ζ˙ = A F 1 ζ is asymptotically stable, it can be inferred from Rayleigh’s inequalities that V F 1 (ζ ) ≤ λmax ( P F 1 )ζ 2 and V˙ F 1 (ζ ) ≤ −λmin ( Q F 1 )ζ 2 . Therefore,

λmin ( Q F 1 ) V F 1 (ζ ) λmax ( P F 1 ) [λmin ( Q F 1 ) − δ] V F 1 (ζ ) <− λmax ( P F 1 )

Considering s3 + k F 3 s2 + k F 2 s + k F 1 is Hurwitz, it is known that k F 2 > k F 1 /k F 3 . Therefore, when L F ≥ 1, the conditions in (19) are always satisfied, which means A F is Hurwitz. Thus, there exists a symmetric positive definite matrix P F satisfied with

V˙ F 1 (ζ ) ≤ −

P F A F + A TF P F = − Q

in which δ is an arbitrarily small positive number. In consideration of continuity of the right-hand side of (26) with respect to αε , the following inequality is true [28]:

F

(22)

where Q F is an arbitrary positive definite and symmetric matrix. A scalar positive Lyapunov function for system (17) is defined as

V F (ϕ ) = ϕ T (χ ) P F ϕ (χ )

(23)

V˙ F 1 (ϕ 1 ) < −

[λmin ( Q F 1 ) − δ] 1+2αε V F 1 (ϕ 1 ) λmax ( P F 1 )

(30)

(31)

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Y. Ding et al. / Aerospace Science and Technology 92 (2019) 563–578

In view of δ is arbitrarily small, (31) can be written as

V˙ F 1 (ϕ 1 ) ≤ −

1+αε 2

λmin ( Q F 1 ) V λmax ( P F 1 ) F 1

(ϕ 1 )

4.2. Generalized smooth second-order sliding mode reaching law scheme

(32)

Therefore, the finite convergence time of system (26) is satisfied with

 λmax ( P F 1 )(1 − αε ) 1−2αε  V F1 ϕ 1 (t 0 ) 2λmin ( Q F 1 )

1 −α λmax ( P F 1 )(1 − αε ) 1−2αε (33) ≤ λmax ( P F 1 ) ϕ 1 (t 0 ) ε 2λmin ( Q F 1 ) in which ϕ 1 (t 0 ) is initial condition of system (26). Considering L F ≥ 1, 0 < αε < 1 and 1 = r3 < r2 < r1 , the following inequality TF ≤

holds:

1−αε  3−αε (1 − αε ) ϕ (t 0 ) 2 TF ≤ ( P F 1) λmax 1/r 2λmin ( Q F 1 ) L 1

(34)

In order to ensure sliding mode vector S and its derivative S˙ converge at the same time, a GSRL is developed in this subsection based on a novel GSSOSM. The stability of GSSOSM is proved via Lyapunov criteria and finite convergence time is estimated, which is shown to be smaller than conventional smooth second-order sliding mode control algorithm (SSOSM). Theorem 2. Consider a nominal first-order system

x˙ = u + ξ

(39)

in the presence of disturbance ξ , which has a Lipschitz constant L ξ . A generalized smooth second-order sliding mode control algorithm (GSSOSM) is proposed as follows

F

where ϕ (t 0 ) is initial condition of system (17). For a given initial value ϕ (t 0 ), it is obvious that upper bound of convergence time decreases with increasing gain L F , which enhances response speed of system avoiding complicated parameters selection for k F i . 2 Remark 2. The original finite-time regulator presented by Bhat in [24] requires the recalculation of k F i to improve response speed of system, which complicates the process of gain adjustment. In this paper, for any set of parameters k F i satisfying Hurwitz condition, convergence speed of system (14) can be accelerated via increasing value of L F simply, without complicated parameters selection for k F i . Define a novel GFFIS for error dynamics (12) as follows:

t S = e¨ − e¨ (0) +

v F dτ

(35)

0

where e¨ (0) is initial value of error vector e¨ and v F = [ v F 1 v F 2 ] T is expressed as 1−α1i

v F i = k F 1i L F i

1−α3i

+ k F 3i L F i

1−α2i

sigα1i (e i ) + k F 2i L F i sigα3i (¨e i )

α3i = αεi ,

(36) 2

α2i = αεi /(2 − αεi ),

α1i = αεi /(3 − 2αεi )

(40)

in which

φ S1 = μ1 sig(m1 −1)/m1 (x) + μ2 x m1 − 1 2 (m1 −2)/m1 φ S2 = μ1 sig (x) m1 2m1 − 1

+ where

m1

μ1 μ2 sig(m1 −1)/m1 (x) + μ22 x

(41)

γ1 , γ2 are chosen to ensure s2 + γ1 s + γ2 be Hurwitz and

μ1 , μ2 , L S are positive design coefficients. Parameter m1 is satisfied with m1 > 2. If there is no disturbance ξ , both state x and its derivative x˙ in (39) could converge to origin in finite time under control algorithm (40). If disturbance ξ exists in system (39), the system is guaranteed to be uniformly ultimately bounded in finite time. The size of convergence domain is proportional to L ξ and inversely proportional to L S .

1/m1

x˙ = −γ1 L S

˙ = −γ w

(37)

(38)

According to Theorem 1, error vector e can converge to origin in finite time. Remark 3. The GFFIS proposed is capable of avoiding singularity phenomenon which always exists when conventional terminal sliding surface is applied. Remark 4. The global sliding surface in (35) can initially eliminate reaching phase of sliding mode control, which ensures states locate on sliding surface initially. Considering the system is invariant with respect to matching disturbances on sliding surface, the GFFIS proposed can improve response speed and enhance robustness of system.

1/m diag( L S 1

of

χS

φ S1 + w

2/m1 φ S2 2LS

in which

with αε i ∈ (1 − εi , 1) for sufficiently small positive numbers εi . When derivative of sliding mode vector S is equal to zero, it follows that

... e = −v F

φ S1 + x2

2/m x˙ 2 = −γ2 L S 1 φ S2

Proof. The resulting closed-loop system (39) with control input (40) and (41) takes the form:

sigα2i (˙e i )

in which L F i ∈ [1, ∞) and s + k F 3i s + k F 2i s + k F 1i is Hurwitz for i = 1, 2. Parameters α1i , α2i , α3i , (i = 1, 2) are defined as 3

1/m1

u = −γ1 L S

+

(42)

 represents ξ˙ . Define χ S = L 0−1 [φ S1 w ]T with L 0 = 2/m

∂φ

L S 1 ). In consideration of φ S2 = ∂ xS1 φ S1 , the derivative along system (42) is expressed as

1 χ˙ S = L − 0

∂φ S1 ∂x

−1 ∂φ S1

= L0

∂x 





x˙ ∂φ



˙ / ∂ xS1 w 1/m1

−γ1 L S

2/m1

−γ2 L S

φ S1 + w

φ S1 +  / ∂φ∂ xS1  −1/m

−γ1 φ S1 + L S

1

=

∂φ S1 ∂x

=

 ∂φ S1  1/m1 1 ˜ LS A S χ S + L− 0  ∂x

−2/m1

−γ2 φ S1 + L S 

in which A S =

−γ1 1 −γ2 0





and

w

 / ∂φ∂ xS1 (43)



˜ = 0  / ∂φ∂ xS1 

T

. Considering

s2 + γ1 s + γ2 is Hurwitz, there exists a symmetric positive definite matrix P S satisfying

A TS P S + P S A S = − Q

S

(44)

Y. Ding et al. / Aerospace Science and Technology 92 (2019) 563–578

where Q S is an arbitrary positive definite and symmetric matrix. Selecting a quadratic Lyapunov function as V S = χ TS P S χ S and considering Rayleigh’s inequality λmin ( Q S )χ S 2 ≤ χ TS Q S χ S , the derivative of V S is satisfied with

V˙ S = χ˙ TS P S χ S + χ TS P S χ˙ S

1/m1

When L S that

(m1 −2)/(m21 −m1 )

m1

Based on inequality |x|1/m1 ≤ |φ S1 /μ1 |1/(m1 −1) ≤ (χ S  L S 1 / μ1 )1/(m1 −1) and V S ≤ λmin ( P S )χ S 2 , the following result is obtained.

m /(m1 −1)

λmin ( Q S )

(m1 − 1)μ1 1

m1 /(m1 −1)

(m1 −2)/(m21 −m1 ) λmin ( Q S )(m1 − 1)μ1 (2m1 −3)/(2m1 −2)

= −L S

( P S )m1

λmax

(2m1 −3)/(2m1 −2)

× VS

(51) 2 P S  L ξ L

m1

 × χ S −1/(m1 −1) + μ2 χ S 2 m 1 −2 m2 −m1 1

(2m −3)/(2m1 −2)

=

2m1 λmax1

(m1 −2)/(m21 −m1 )

LS

(P S)

= −L S

λmin ( Q S )(m1 − 1)μ (2m1 −3)/(2m1 −2)

λmax 1/m1 λmin ( Q S )μ2 − LS VS λmax ( P S )

( P S )m1

2m1 −3 2m1 −2

λmin ( Q S )μ

1/m1

(47)

1/(m21 −m1 )

LS

1/(2m −2) λmax 1 ( P S )(m1



×

m1 μ2 m1 /(m1 −1) 1

− 1)μ

1/(2m1 −2) VS (t 0 )

(48)

1

∂φ S1 −2/m1 ˜

χ S  P S  L − χ S  P S | | 0  = 2L S ∂x

(49)

Similarly to the analysis in Case 1, derivative of V S is satisfied with 1/m1

 1  −1 μ1 (m1 − 1) μ1 m1 −1 λmin ( Q S ) χ S  m1 −1 1/m m1



LS

1

−2

m

+ μ2 χ S 2 + 2L S 1 χ S  P S | | m 1 −2 m2 −m1 1

= −L S −

m1 m −1

λmin ( Q S )

1/m L S 1 λmin ( Q S )

m1

(52)

χ S (m1 −2)/(m1 −1) ≥

 P S | |, it follows that 1/m1

λmin ( Q S )μ2 χ S 2 = − L S

λmin ( Q S )μ2 V S (53) λmax ( P S )

(4−3m1 )/(m2 −m1 ) 1

2m1  P S  L ξ L S

m /(m1 −1)

(m1 −1)λmin ( Q S )μ1 1

T S ≤ T S2 =

 ×

λmax ( P S )

](m1 −1)/(m1 −2)

1/m1 2LS

λmin ( Q S )μ



V S (t 0 ) λmax ( P S ) m1 /(m1 −1) 2(m1 −1)/(m1 −2)  ln

(m1 − 1)λmin ( Q S )μ1

(4−3m1 )/(m21 −m1 )

(54)

2m1  P S  L ξ L S

(m1 − 1)μ1 1 m1

2m1 −3

χ S  m1 −1 −2

μ2 χ S 2 + 2L Sm1 χ S  P S | |

 −3/m1 2 P S  L ξ L S χ S  ≤ min , λmin ( Q S )μ2  (4−3m1 )/(m21 −m1 ) (m1 −1)/(m1 −2)  2m1  P S  L ξ L S m /(m1 −1)

Case 2: If disturbance ξ exists in system (42) and satisfies |ξ˙ | = | | ≤ L ξ , it follows that

V˙ S ≤ − L S

(m1 −1)μ1 1

(m1 − 1)λmin ( Q S )μ1 1

in which V S (t 0 ) is the initial value of Lyapunov function V S .

2

m /(m1 −1)

λmin ( Q S )

(t 0 )

Synthesizing the above two conditions, system (42) can converge into the following region in finite time:

μ2 λmin ( Q S )

 × ln 1 +

VS

can be reached in finite time. Convergence time is satisfied with

VS

2(m1 − 1)λmax ( P S ) LS

−2/m1

2L S

1/m1

According to Theorem 1 in [40], system (42) is finite-time convergent and convergence time satisfies

TS ≤

(m1 −2)/(m21 −m1 )

When L S

1/(2m1 −2)

m1 /(m1 −1) 1

Therefore, the region χ S  ≤ [

m1 /(m1 −1) 1

can be reached in finite

T S ≤ T S1

V˙ S ≤ − L S

1

LS

−3/m1

Therefore, the region χ S  ≤ λ ( Q S)μ 2 min S time. Convergence time is satisfied with

1/m

  μ1 (m1 − 1) μ1 1/(m1 −1) 1/m V˙ S ≤ − L S 1 λmin ( Q S ) 1/m

m1

× χ S 

(45)

∂φ S1 1/m1 T χ S Q Sχ S L ∂x S   μ1 (m1 − 1) −1/m1 1/m ≤ − L S 1 λmin ( Q S ) |x| + μ2 χ S 2 (46)

χ S  P S | |, it follows

(2m1 −3)/(m1 −1)

 ∂φ S1  1/m1 T 1 ˜ − L S χ S Q S χ S + 2χ TS P S L − 0  ∂x

1  ∂φ S1  1/m1 ˜

− L S λmin ( Q S )χ S 2 + 2χ S  P S  L − ≤ 0  ∂x

V˙ S = −

−2/m1

λmin ( Q S )μ2 χ S 2 ≥ 2L S

V˙ S ≤ − L S

=

Case 1: If there is no disturbance ξ in system (42), Equation (45) changes to

569

(50)

(55)

Convergence time is satisfied with T S ≤ max( T S1 , T S2 ). It is observed from (55) that convergence domain is sensitive to Lipschitz constant of disturbance L ξ . However, convergence domain can be reduced by increasing control gain L S . 2 Traditional reaching laws of sliding mode control have serious shortcomings, such as: chattering and lower convergence accuracy. In order to improve them, super-twisting algorithm can be utilized as reaching law, which can make sliding mode variable and its derivative converge to zero in finite time at the same time. Thus, the sliding mode variable can have two-order convergence precision. Meanwhile, super-twisting reaching law is continuous, which suppresses chattering due to hiding discontinuous sign function in an integral term. In order to improve convergence speed of supertwisting algorithm further, a generalized super-twisting algorithm is developed, which retains the advantages of super-twisting algorithm. However, output signals of super-twisting algorithm and generalized super-twisting algorithm are only continuous.

570

Y. Ding et al. / Aerospace Science and Technology 92 (2019) 563–578

Remark 5. Compared with super-twisting algorithm and generalized super-twisting algorithm in [26] and [27], the control signal provided by GSSOSM is not only continuous but also smooth, which avoids chattering and is more suitable for controller design of FAHV. Remark 6. If parameter μ2 = 0, the GSSOSM changes to nominal SSOSM. With regard to system (39) without disturbance ξ , the stability of SSOSM is proved in [25] and [41] via LaSalle theorem and homogeneity, but the convergence time cannot be estimated. In this paper, convergence time of SSOSM can be derived from (47) easily. In addition, the convergence time of GSSOSM with regard to system (39) without disturbance ξ is provided in (48), which is shown to be smaller than nominal SSOSM. Remark 7. With regard to system (39) with disturbance ξ , the stability of SSOSM is proved in [42] with the condition that disturbance is eliminated. However, the condition cannot be satisfied in many practical situations. In this paper, system (39) is proved to be uniformly ultimately bounded without need for the overly strict condition. Remark 8. The main focus of GSSOSM is theoretical innovation. Therefore, under the condition of existing engineering computing power, its computational complexity is relatively high and it is not easy to be applied to practical engineering. Despite this, the existing control design attempts can be used as a theoretical basis when engineering computing power is improved in the future. In order to realize high-precision finite-time convergence of sliding mode vector and its derivative, the GSSOSM in Theorem 2 is served as a reaching law. By means of combining GFFIS in (35) and generalized smooth second-order sliding mode reaching law (GSRL), a sliding mode controller for FAHV is proposed as follows:



... 1/m u = G −1 − F + y d − v F − γ1 L S 1 Φ S1

−γ

2/m1 2LS



t Φ S2 dτ

(56)

0

4.3. Smooth fixed-time observer design Although sliding mode variable S and its derivative S˙ can be uniformly ultimately bounded using the controller in (56), the size of convergence domain is proportional to Lipschitz constant of disturbance. When strong disturbances occur in FAHV, such as actuator faults, the controller in (56) cannot achieve satisfactory performance. In order to achieve higher control precision and enhance robustness for actuator faults and parametric uncertainties, a smooth fixed-time observer [28] is introduced to estimate lumped disturbances and derivatives of error vector rapidly.

e˙ˆ 1 = −κ B1 L B

sigα B1 (ˆe 1 − e ) − k B1 sigβ B1 (ˆe 1 − e ) + eˆ 2

e˙ˆ 2 = −κ B2 L B

sigα B2 (ˆe 1 − e ) − k B2 sigβ B2 (ˆe 1 − e ) + eˆ 3

(1−α B1 ) (1−α B2 )

e˙ˆ 3 = −κ B3 L B

(1−α B3 )

sigα B3 (ˆe 1 − e ) − k B3 sigβ B3 (ˆe 1 − e ) ... + eˆ 4 + ( Fˆ + Gˆ u − yd )

e˙ˆ 4 = −κ B4 L B

sigα B4 (ˆe 1 − e ) − k B4 sigβ B4 (ˆe 1 − e ) + eˆ 5

e˙ˆ 5 = −κ B5 L B

sigα B5 (ˆe 1 − e ) − k B5 sigβ B5 (ˆe 1 − e )

(1−α B4 ) (1−α B5 )

where L B = diag( L A1 L A2 ) > 0. Parameters κ Bi = diag(κ Bi1 κ Bi2 ) and k Bi = diag(k Bi1 k Bi2 ) for i = 1, . . . , 5 are selected to make polynomials s5 + κ B1 j s4 + κ B2 j s3 + κ B3 j s2 + κ B4 j s + κ B5 j and s5 + k B1 j s4 + k B2 j s3 + k B3 j s2 + k B4 j s + k B5 j ( j = 1, 2) be Hurwitz. Coefficient α Bi and β Bi satisfy the recurrent relations α Bi = i α B ε − (i − 1) and β Bi = i β B ε − (i − 1) for i = 1, . . . , 5, where α B ε and β B ε belong to interval (1 − ε B , 1) and (1, 1 + ε B1 ) respectively with sufficiently small positive numbers ε B and ε B1 . Variables Fˆ and Gˆ are calculated via (A.1) using the estimated values of observer eˆ 2 and eˆ 3 to replace e˙ and e¨ . According to Theorem 2 in [28], with gain L B designed based on the bounds of disturbances, the estimated values of observer eˆ 1 , eˆ 2 , eˆ 3 and eˆ 4 can exactly converge to e, e˙ , e¨ and D f + ( F − Fˆ ) + ( G − Gˆ )u respectively in fixed time T o . The composite controller with strong fault-tolerant performance is established via compensating eˆ 4 into (56). Meanwhile, derivatives of error vector e˙ and e¨ used in controller are replaced by outputs of smooth fixed-time observer eˆ 2 and eˆ 3 , in order to compensate unmatched disturbances induced by flexible effects, actuator faults, parametric uncertainties of aerodynamic and structural coefficients. Therefore, the novel GSSMC proposed in this paper is given as follows:

where

Φ S1 = μ1 sig Φ S2 =

(m1 −1)/m1

m1 − 1 m1

+

u = Gˆ

( S ) + μ2 S

μ21 sig(m1 −2)/m1 ( S )

2m1 − 1 m1

μ1 μ2 sig

(m1 −1)/m1

−1

−γ 2 2S

(S) + μ



...

1/m − Fˆ + yd − eˆ 4 − vˆ F − γ1 L S 1 Φˆ S1

2/m1 2LS

with μ1 , μ2 > 0, m1 > 2. Polynomial s + γ1 s + γ2 is Hurwitz and L S ∈ R2×2 is a positive definite diagonal matrix. With regard to a vector p = [ p 1 , . . . , pn ], the notation sigr ( p ) denotes [sigr ( p 1 ), . . . , sigr ( pn )] T . In (56), parameter L S is used to adjust the convergence domain and response speed of sliding mode variable S , while coefficients L F i of S in (36) are used to adjust the convergence speed of error vector e. Remark 9. In [43], the invertibility of G during the flight process over admission ranges in Table 1 is verified by nonlinear optimization method.

t

 Φˆ S2 dτ

(59)

0

(57)

2

(58)

in which Gˆ , Fˆ , vˆ F , Φˆ S1 and Φˆ S2 are calculated using eˆ 2 and eˆ 3 . The errors between estimated values and actual values are defined as below:

e˜ 2 = eˆ 2 − e˙ ,



e˜ 3 = eˆ 3 − e¨ ,

˜ f = eˆ 4 − D f + ( F − Fˆ ) + ( G − Gˆ )u D G˜ = Gˆ − G ,

F˜ = Fˆ − F ,



v˜ F = vˆ F − v F

(60)

The estimated value of sliding mode vector Sˆ is calculated as: t Sˆ = eˆ 3 − eˆ 3 (0) + 0 vˆ F dτ . Therefore, combining (12), (35) and (59),

the dynamics of Sˆ is satisfied with:

Y. Ding et al. / Aerospace Science and Technology 92 (2019) 563–578

  |ˆe 31 | | D L1 | + | vˆ F 1 |   α11  1 + |e 1 | ≤ |ˆe 31 | | D L1 | + k F 11 L 1F− 1   1 −α  α21  1 + |ˆe 21 | + k F 31 L F 1 31 1 + |ˆe 31 | + k F 21 L 1F− 1

... ... ˙ Sˆ = e˙ˆ 3 + vˆ F = e + e˙˜ 3 + vˆ F = F + G u + D f − y d + e˙˜ 3 + vˆ F ... = F + Gˆ u + ( G − Gˆ )u + D f − yd + e˙˜ 3 + vˆ F ... 1/m = F − Fˆ + yd − eˆ 4 − vˆ F − γ1 L S 1 Φˆ S1

−γ

2/m1 2LS

t

1−α31 1−α21 1−α11 2 2 ≤ eˆ 31 + (k F 11 L F 1 |e1 | + k F 21 L F 1 |ˆe21 | + k F 31 L F 1 |ˆe31 |)

... Φˆ S2 dτ + ( G − Gˆ )u + D f − yd + e˙˜ 3 + vˆ F

2

= −γ

−γ

2/m1 2LS

t

+ Φˆ S2 dτ − D˜ f + e˙˜ 3

(61)

≤

e˙ = eˆ 2 − e˜ 2

2

˙ e˙ˆ 3 = Sˆ − vˆ F

(62) (1−α B2 )

where σ e2 = κ B2 L B sigα B2 (ˆe 1 − e ) + k B2 sigβ B2 (ˆe 1 − e ). Next, the stability of whole system will be proved. Firstly, the boundedness of Sˆ , e, eˆ 2 and eˆ 3 during the convergence of smooth fixed-time observer when t ∈ (0, T o ) will be verified. Lyapunov functions of velocity and flight path angle subsystems are chosen as follows [44]:

2

| Sˆ 1 |dτ

0

Vγ =

1

 t

2

+

+ 2

| Sˆ 2 |dτ

1

Sˆ 12 +

2

1 2

e 21 +

1 2

eˆ 221 +

1 2

eˆ 231

1 2

Sˆ 22 +

1 2

e 22 +

1 2

eˆ 222 +

1 2

eˆ 232

S1 1/m1 1 L S1

0

μ1 + γ1 L 1S1/m1 (μ1 + μ2 )| Sˆ 1 |

 

t     2/m1 φˆ S21dτ  γ2 L S1  

(66)

0

t 

2/m1

≤ γ2 L S1 +

m1 − 1 m1

0

≤γ (63)

(65)

      γ1 L 1/m1 φˆ S11  ≤ γ1 L 1/m1 μ1 1 + | Sˆ 1 | + μ2 | Sˆ 1 |

2m1 − 1 m1

m1 − 1 m1

+γ

2/m1 2 L S1



  2 ˆ ˆ μ1 μ2 1 + | S 1 | + μ2 | S 1 | dτ

t 

2/m1 2 L S1



μ21 1 + | Sˆ 1 |



0

+

α31 2 α21 + k F 21 L 1F− + k F 31 L 1F− ) 1 1

2

≤γ

2

α31 2 2 α21 2 2 + (k F 21 L 1F− ) eˆ 21 + (k F 31 L 1F− ) eˆ 31 ] 1 1

eˆ 231 + (| D L1 | + k F 11 L F 1

S1

e˙ˆ 2 = eˆ 3 − σ e2

VV =

2 1−α11 2 2 ) e1

1−α11

˜ f + e˙˜ 3 . The dynamics of e, eˆ 2 Define the symbol D L as: D L = − D and eˆ 3 can be obtained as:

1

eˆ 231 + (| D L1 | + k F 11 L F 1

eˆ 231 + 3[(k F 11 L F 1

0

 t

α31 2 α21 + k F 21 L 1F− + k F 31 L 1F− ) 1 1

1−α11

0 1/m1 ˆ Φ S1 1LS

571

t 

μ21 +

m1 − 1 m1

0



2m1 − 1

μ1 μ2 dτ

m1

μ21 +

2m1 − 1 m1



μ1 μ2 + μ22 | Sˆ 1 |dτ

Taking velocity subsystem as an example, the derivative of V V can be calculated according to (61) and (62) as:

V˙ V = | Sˆ 1 |

t

Substituting (65), (66) and (67) into (64), it can be obtained that:

| Sˆ 1 |dτ + Sˆ 1 S˙ˆ 1 + e 1 e˙ 1 + eˆ 21 e˙ˆ 21 + eˆ 31 e˙ˆ 31

0

= | Sˆ 1 |

t

V˙ V ≤



2/m1

t

− γ2 L S1

φˆ S21dτ + D L1

| Sˆ 1 |dτ )2

2 eˆ 221 + eˆ 231

+

2

Sˆ 12 + D 2L1

+

2

eˆ 221 + σe22 1

+

e 21 + eˆ 221 2

+

e 21 + e˜ 221 2

   1/m + |ˆe 31 | γ1 L 1 φˆ S11  S1

2

0

≤

+ e 1 (ˆe 21 − e˜ 21 ) + eˆ 21 (ˆe 31 − σe2 1 ) + eˆ 31 ( S˙ˆ 1 − vˆ F 1 )

1 + [γ

 t

  | Sˆ 1 |dτ + | Sˆ 1 || D L1 | + |e 1 | |ˆe 21 | + |˜e 21 |

×

2/m1 m1 −1 2 L S1 ( m1

2

2 | Sˆ 1 |dτ

μ21 + 2mm11−1 μ1 μ2 + μ22 )]2 1/m1

+

2 + [γ1 L S1

(μ1 + μ2 )]2

2

Sˆ 12

0

0

     1/m + |ˆe 21 | |ˆe 31 | + |σe2 1 | + |ˆe 31 | −γ1 L S1 1 φˆ S11   

t   2/m1 ˆ − γ2 L S1 φ S21dτ  + | D L1 | + | vˆ F 1 | 

0

  

t       2/m1 ˆ + γ2 L S1 φ S21dτ  + | D L1 | + | vˆ F 1 |  



0

≤ | Sˆ 1 |

t

Sˆ 12 + (

+

1/m | Sˆ 1 |dτ + Sˆ 1 −γ1 L S1 1 φˆ S11

0

t

(67)

+

1−α11 2

2 + 3(k F 11 L F 1

)

2 1−α31 2

+ (64)

0

According to inequalities: |ab| ≤ (a2 + b2 )/2, |x| p < 1 + |x|, (a + b + c )2 ≤ 3(a2 + b2 + c 2 ) for ∀a, b, c , x ∈ R, p ∈ (0, 1), it can be derived that:

+ +

7 + 3(k F 31 L F 1

)

2

[γ

2/m1 2 L S1

t

( mm1 −1 1 0

e 21 +

1−α21 2

3 + 3(k F 21 L F 1 2

)

eˆ 221

1/m

eˆ 231 +

μ + 2 1

(γ1 L S1 1 μ1 )2 2

2m1 −1 m1

μ1 μ2 )dτ ]2

2 1−α11

D 2L1 + e˜ 221 + σe22 1 + (| D L1 | + k F 11 L F 1

≤ Ψ × VV + Ω

α31 2 α21 + k F 21 L 1F− + k F 3i L 1F− ) 1 1

2

(68)

572

Y. Ding et al. / Aerospace Science and Technology 92 (2019) 563–578

in which

Ψ = max{Ξ V 1 , Ξ V 2 , Ξ V 3 , Ξ V 4 , Ξ V 5 } 2/m1  t m1 −1 2 1/m ( m1 μ1 + (γ1 L S1 1 μ1 )2 [γ2 L S1 0 + Ω= 2

+

2m1 −1 m1

μ1 μ2 )dτ ]2

2

1−α11

D 2L1 + e˜ 221 + σe22 1 + (| D L1 | + k F 11 L F 1

1−α31 2

α21 + k F 21 L 1F− + k F 3i L F 1 1

)

2

(69) where

ΞV 1 =

1 2

+

1



γ

2

2/m1 2 L S1

1/m1

ΞV 2 = ΞV 4 =

2 + [γ1 L S1



m1 − 1 m1

μ +

(μ1 + μ2 )]2

2 1−α21 2

3 + 3(k F 21 L F 1

)

2

,

2 1

2m1 − 1 m1

2 2 2

μ1 μ2 + μ

,

1−α11 2

ΞV 3 =

ΞV 5 =

2 + 3(k F 11 L F 1 2

)

,

1−α31 2

7 + 3(k F 31 L F 1 2

)

(70)

Considering D L1 , e˜ 21 , σe2 1 are all bounded during t ∈ (0, T o ), t Ω is bounded. Therefore, V V , 0 | Sˆ 1 |dτ , Sˆ 1 , e 1 , eˆ 21 and eˆ 31 are all

bounded, which means that they cannot escape to infinity before the convergence of smooth fixed-time observer. Similarly to velocity subsystem, the variables of flight path angle subsystem V γ ,

t

| Sˆ 2 |dτ , Sˆ 2 , e 2 , eˆ 22 and eˆ 32 are all bounded during the conver0 gence of smooth fixed-time observer when t ∈ (0, T o ). Since the smooth fixed-time observer can achieve exact estima˜ f , G˜ , F˜ , v˜ F and tion in fixed time T o , estimated errors e˜ 2 , e˜ 3 , D

σ e2 are all equal to zero after T o . Thus, D L = 02×1

and S = Sˆ .

Then, sliding mode variable S and its derivative S˙ can converge to origin in finite time according to Theorem 2. Therefore, equation (38) is satisfied which drives error vector to converge in finite time along sliding surface. Remark 10. In order to avoid adverse effects on tracking performance of system when initial observation error is too large, before the outputs of smooth fixed-time observer converge to desired values, the derivatives of e can be calculated approximately according to (A.2) of Appendix A as below:



e˙ = y˙ − y˙ d = V˙

T

− y˙ d   ˙ π1w ˙ T − y¨ d e¨ = y¨ − y˙ d = ω1 w

γ˙

(71)

in which y˙ d and y¨ d are acquired accurately from pre-filters in (7) and (8). Remark 11. Lumped disturbances D f + ( F − Fˆ ) + ( G − Gˆ )u are composed of three parts: flexible effects, actuator error, uncertainties of aerodynamic and structural coefficients. By means of analyzing system characteristics of the above three parts, bounds of lumped disturbances can be obtained initially. Then, combined with several iterative simulations, theoretical bounds of lumped disturbances can be explored roughly. In order to enhance robustness, gain L B of smooth fixed-time observer can be selected conservatively by means of enlarging bounds of disturbances slightly. 5. Simulation results In this section, three subsections of simulation are carried out to verify the effectiveness of the method proposed in detail. The first one simulates the proposed FFR in a third-order continuous system to demonstrate characteristic of it. The second one applies the GSSOSM in a first-order system. Two cases are included to illustrate the performance and superiority more sufficiently.

Fig. 2. System response of different control gains.

The third one performs simulations on longitudinal model of FAHV subject to actuator faults and parametric uncertainties. The GSSMC presented is simulated compared with conventional LQR controller with integral augmentation and the method in [32] to verify the advantages of the method proposed. 5.1. Simulations for fast finite-time high-order regulator Considering a third-order continuous system as (14), parameters of FFR in (15) are chosen as: k F 1 = 1.1, k F 2 = 2.12, k F 3 = 2, α3 = 0.9, α2 = 9/11, α1 = 0.75. Initial conditions are set as: x1 (0) = −5, x2 (0) = 10 and x3 (0) = 5. Simulations are performed in three cases with different gain L F :

Case A.1:

LF = 1

Case A.2:

LF = 5

Case A.3:

L F = 25

(72)

The results in Fig. 2 demonstrate that FFR is able to drive states to converge in finite time. Due to the effect of sampling step, states can only converge to a neighborhood near zero. Convergence domains for Case A.1, Case A.2 and Case A.3 are |x1 | < 1.5 × 10−27 , |x1 | < 7 × 10−27 and |x1 | < 4 × 10−26 respectively. Finite convergence times for Case A.1, Case A.2 and Case A.3 are 29.2 s, 25.1 s and 22 s respectively. It is observed that convergence time of system decreases along with the increase of control gain L F . Therefore, convergence speed of system can be accelerated via increasing value of L F simply, avoiding complicated parameters selection for k F i when using conventional finite-time regulator. 5.2. Simulations for generalized smooth second-order sliding mode control algorithm Case A: Considering the first-order system in (39) without disturbance ξ , GSSOSM proposed is simulated compared with conventional generalized super-twisting algorithm (GSTA) [26] and nominal smooth second-order sliding mode control algorithm (SSOSM) [25]. Initial value is set as: x(0) = −5. Design parameters are selected as: γ1 = 1.5, γ2 = 1.1, μ1 = μ2 = 1, m1 = 3 and L S = 1. Case B: Considering the first-order system in (39) with disturbance ξ , simulations of GSSOSM are performed to verify characteristic of uniformly bounded convergence. Initial value is set as: x(0) = −5. Different simulation conditions are applied as follows:

Y. Ding et al. / Aerospace Science and Technology 92 (2019) 563–578

573

Table 2 Initial conditions. Parameter

Value

Parameter

Value

Parameter

Value

h V

25908 m 2347.60 m/s 1.5153 deg 0 deg

Q

0 deg/s 0.2514 0 11.4635 deg

η1 η˙ 1 η2 η˙ 2

1.8710 0 0.3565 0

α γ

Φ Φ˙ δe

spectively. Finite convergence times for Case B.1, Case B.2 and Case B.3 are 1.08 s, 1.11 s and 0.92 s respectively. The simulation results of Case B.1 and Case B.2 indicate the size of convergence domain is proportional to Lipschitz constant of ξ . Then, the results of Case B.2 and Case B.3 indicate the size of convergence domain is inversely proportional to control gain L S . 5.3. Simulations of FAHV In this subsection, three different controllers are applied to a high-fidelity nonlinear curve-fitted model of FAHV stated in Equation (1) under the same flight maneuver. Case A uses conventional LQR controller with integral augmentation proposed in [7] as a contrast simulation, while Case B compares the GSSMC and the existing method in [32]. The nominal parameters of FAHV are selected according to [2] and [7] as: m = 4378.17 kg, I y y = 6.779 × 105 kg m2 . The elastic parameters are calculated as: ψ˜ 1 = −423.195, ψ˜ 2 = 382.799, ω1 = 16.020, ω2 = 19.579, ζ1 = ζ2 = 0.02. FAHV is assumed to suffer from actuator faults including gain faults and deviation faults 40 seconds after the simulation begins. Fault factors are given as: F g = diag(0.6 0.7), F d = [0.15 0.02] T . Define P r as values of real structural coefficients: mass m, moment of inertia I y y and reference area S, while P 0 are the nominal values of them. Notation C r represents values of real aerodynamic coefficients and C 0 are nominal values. Parametric uncertainties are defined as follows [18]:

Fig. 3. System response of different control methods in Case A.









P r = P 0 1 + 0.2 sin(0.1π × t )

C r = C 0 1 + 0.1 cos(0.1π × t )

The step commands V c and hc are set as 2407.92 m/s and 26212.8 m respectively, while parameters in (7) and (8) are designed as: ωd1 = 0.03, ωd2 = 0.1, ζd1 = 0.95, ζd2 = 0.5. The parameter in Equation (9) is designed as: k P = 0.3. The initial conditions of vehicle dynamics are listed in Table 2:

Fig. 4. System response of different simulation conditions in Case B.

Case B.1:

L S = 10 ξ = 5 + 2 sin(3t ) + cos(1.5t )

Case B.2:

L S = 10 ξ = 6 + 3 sin(4t ) + cos(1.5t )

Case B.3:

L S = 50 ξ = 6 + 3 sin(4t ) + cos(1.5t )

(74)

(73)

The remaining design parameters are selected as the same as those given in Case A. In Fig. 3, blue solid line represents the result of GSSOSM. Red dashed line and black dotted line represent the results of SSOSM in [25] and GSTA in [26] respectively. System can be finite-time convergent under the three methods. It can be seen that GSTA can attenuate chattering phenomena but cannot eliminate them completely. Compared with it, control signal of GSSOSM is not only continuous but also smooth. Therefore, chattering can be avoided, which confirms the descriptions in Remark 5. Convergence domain of GSSOSM is |x| < 5 × 10−8 and finite convergence time is 3.5 s. Compared with nominal SSOSM, the convergence speed of GSSOSM is faster obviously, which confirms the descriptions in Remark 6. In Fig. 4, Lipschitz constants of ξ in Case B.1, Case B.2 and Case B.3 are equal to 7.5, 13.5 and 13.5 respectively. System can be uniformly ultimately bounded in finite time and convergence domains of the three cases are |x| < 0.043, |x| < 0.121 and |x| < 0.007 re-

Case A: An LQR controller with integral augmentation proposed in [7] is applied as a contrast simulation. The design parameters are selected as: Q V = diag(100 100 100 100), R V = 0.1, Q γ = diag(10 10 10 10), R γ = 0.5. It can be seen from Fig. 5, Fig. 6, Fig. 7 and Fig. 8 that velocity, height and flight path angle cannot track the desired values accurately due to the effects of parameter uncertainties and actuator faults. Poor control performance may lead to mission failure or even catastrophic accidents, which needs to be improved by a controller with stronger robustness. Case B: In order to realize high convergence precision, the GSSMC proposed is applied in the second case. Meanwhile, conventional disturbance observer based sliding mode control in [32] is utilized as comparative simulation. Parameters of GFFIS are chosen as: k F 11 = k F 12 = 1.1, k F 21 = k F 22 = 2.12, k F 31 = k F 32 = 2, α31 = α32 = 0.9, α21 = α22 = 9/11, α11 = α12 = 0.75, L F 1 = 2, L F 2 = 5. Parameters of GSRL are chosen as: μ1 = μ2 = 1, γ1 = 1.5, γ2 = 1.1, m1 = 3, L S = diag(30 5 × 10−7 ). Parameters in smooth fixed-time observer are given as: k B1 = κ B1 = 5E 2 , k B2 = κ B2 = 10.03E 2 , k B3 = κ B3 = 9.3E 2 , k B4 =

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Y. Ding et al. / Aerospace Science and Technology 92 (2019) 563–578

Fig. 5. Velocity tracking performance.

Fig. 8. Fuel-to-air ratio and elevator angular deflection.

κ B4 = 4.57E 2 , k B5 = κ B5 = 1.1E 2 , α B ε = 0.83, β B ε = 1.1, L B = diag(5000 0.02). The comparative simulation in [32] is composed of finite-time integral sliding surface, fast super-twisting reaching law and fast super-twisting disturbance observer. The finite-time integral sliding surface is established based on finite-time high-order regulator, while the FFR proposed in this paper is improved from it to accelerate response speed avoiding complicated parameters selection. The fast super-twisting reaching law is only continuous, while GSRL in this paper is not only continuous but also smooth. The smooth fixed-time observer applied in this paper can achieve fast tracking speed and less chattering compared with the fast supertwisting disturbance observer applied in [32]. The expression of controller in [32] is given as follow:

u D = Gˆ

−1



... − Fˆ + yd − eˆ T 2 − vˆ D − Φˆ D1 −

t

 Φˆ D2 dτ

(75)

0

in which vˆ D = [ vˆ D1 vˆ D2 ] is expressed as: T

Fig. 6. Height tracking performance.

vˆ Di = k D1i sigα1i (ˆe 1i ) + k D2i sigα2i (ˆe 2i ) + k D3i sigα3i (ˆe 3i )

(76)

where s + k D3i s + k D2i s + k D1i is Hurwitz for i = 1, 2. Notations Φˆ D1 and Φˆ D2 are defined as: 3

2

1/2 Φˆ D1 = μ D1 L D sig1/2 ( Sˆ D ) + μ D2 L D Sˆ D Φˆ D2 = μ D3 L D sgn( Sˆ D ) + μ D4 L 2D Sˆ D

Sˆ D = eˆ 3 +

t vˆ D dτ

(77)

0

Output of fast super-twisting disturbance observer eˆ T 2 is calculated as: 1/2 e˙ˆ T 1 = −κ T 1 L T sig1/2 (ˆe T 1 − Sˆ D ) − k T 1 L T (ˆe T 1 − Sˆ D ) ... + eˆ T 2 + ( Fˆ + Gˆ u − yd + vˆ D )

e˙ˆ T 2 = −κ T 2 L T sgn(ˆe T 1 − Sˆ D ) − k T 2 L 2T (ˆe T 1 − Sˆ D )

Fig. 7. Flight path angle tracking performance.

(78)

In order to make a fair comparison with GSSMC, comparative simulation also uses smooth fixed-time observer to estimate derivatives of state error. Therefore, outputs of smooth fixed-time observer eˆ 1 , eˆ 2 , eˆ 3 can exactly converge to e, e˙ , e¨ respectively in fixed time.

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575

Fig. 9. Velocity tracking performance.

Fig. 11. Flight path angle tracking performance.

Fig. 10. Height tracking performance.

Fig. 12. Angle of attack and pitch rate.

Design parameters of finite-time integral sliding surface for method in [32] are selected as: k D11 = k D12 = 1.1, k D21 = k D22 = 2.12, k D31 = k D32 = 2. Parameters of fast super-twisting reaching law are selected as: μ D1 = 1.5, μ D3 = 1.1, μ D2 = μ D4 = 1, L D = diag(30 5 × 10−7 ). Parameters of fast super-twisting disturbance observer are chosen as: κ T 1 = 1.5E 2 , k T 1 = E 2 , κ T 2 = 1.1E 2 , k T 2 = E 2 , L T = diag(5 5 × 10−5 ). Parameters of smooth fixed-time observer are designed as the same as those in GSSMC. The simulation results in Fig. 9, Fig. 10 and Fig. 11 illustrate the effectiveness of GSSMC in realizing control of FAHV subject to parameter uncertainties and actuator faults. The velocity and flight path angle tracking errors are able to realize higher-precision convergence in finite time, while height error achieves exponential convergence. After actuator faults occur at 40 s, convergence domains for velocity tracking error e 1 , height tracking error eh , flight path angle tracking error e 2 using GSSMC are |e 1 | < 0.0013 m/s, |eh | < 0.04033 m and |e 2 | < 0.002039 deg respectively. The corresponding convergence times are 56.93 s, 86.97 s and 60.02 s respectively. In contrast, convergence domains of e 1 , eh and e 2 using conventional method in [32] are |e 1 | < 0.012 m/s, |eh | < 0.461 m and |e 2 | < 0.02198 deg respectively. The corresponding convergence times are 60.83 s, 103 s and 67.1 s respectively. By means of comparing the results in Fig. 9, Fig. 10 and Fig. 11, it is obvious

that e 1 , eh and e 2 using GSSMC can achieve faster response speed and higher convergence precision. Fig. 12 and Fig. 14 demonstrate that states and control inputs are always kept within admission ranges depicted by Table 1 throughout the process of maneuver. It can be observed that chattering of control signals using the method in [32] is significantly severe compared with that using GSSMC. On account of it, the flexible states depicted in Fig. 13 using GSSMC are not excited during the whole flight maneuver, which indicates the effectiveness of GSSMC in restraining the effect of flexible vibration. Fig. 15 and Fig. 16 are tracking performance of smooth fixed-time observer. The results show that eˆ 1 and eˆ 4 can realize accurate tracking for e and D f + ( F − Fˆ ) + ( G − Gˆ )u respectively in fixed time. By means of compensating observer output eˆ 4 into controller to establish GSSMC, the ability of coping with actuator faults is enhanced remarkably. In addition, comparing Fig. 5, Fig. 6, Fig. 7 with Fig. 9, Fig. 10, Fig. 11, it can be seen the effects induced by parametric uncertainties are eliminated efficiently owing to estimating e˙ and e¨ accurately using eˆ 2 and eˆ 3 . Fig. 17 gives the curves of sliding mode variables for velocity channel and flight path angle channel. It can be seen that initial sliding mode variables are equal to zero, which means initial states of system are located on sliding surface. Such global sliding surface eliminates

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Y. Ding et al. / Aerospace Science and Technology 92 (2019) 563–578

Fig. 15. Observer estimation performance for velocity channel. Fig. 13. Generalized modal coordinates.

Fig. 14. Fuel-to-air ratio and elevator angular deflection.

Fig. 16. Observer estimation performance for flight path angle channel.

the process of reaching phase, so as to improve response speed and enhance robustness. 6. Conclusion In this paper, a GSSMC is presented for FAHV subject to actuator faults and parametric uncertainties. The novel controller consists of GFFIS, GSRL and smooth fixed-time observer. Firstly, a novel FFR is proposed to achieve finite-time convergence, the response speed of which can be accelerated via tuning value of a gain simply without complicated parameter adjustment. GFFIS is designed based on FFR to avoid singularity and eliminate initial reaching phase, so as to improve response speed and enhance robustness. Secondly, GSRL is presented to ensure high-precision finite-time convergence of sliding mode vector and its derivative without chattering. Thirdly, a smooth fixed-time observer is utilized to estimate lumped disturbances and derivatives of states in fixed time. By means of combining the three components, GSSMC could drive tracking errors of FAHV to converge in finite time with smooth control signal. Meanwhile, actuator faults and parametric uncertainties can be solved efficiently.

Fig. 17. Sliding mode variables curves.

Y. Ding et al. / Aerospace Science and Technology 92 (2019) 563–578

  Π 2 = π 21 π 22 π 23 π 24 π 25

Declaration of Competing Interest

π 21

The authors declare that there is no conflict of interest.

⎡

⎢ ⎢ =⎢ ⎢ ⎣

Acknowledgements This work was supported by the National Natural Science Foundation of China [grant number 61703125]. Appendix A. Definitions of F , G , G d

F=

 

G=

fV

fγ

gV1 g γ1



Gd = gV3

T

gV2 g γ2 g γ3

577



⎤ ∂ 2 L /∂ V 2 /(mV ) − 2∂ L /∂ V /(mV 2 ) + 2( L + T sin α )/(mV 3 ) − 2g cos γ / V 3 ⎥ ∂ 2 L /∂ V ∂ α /(mV ) − (∂ L /∂ α + ∂ T /∂ α sin α + T cos α )/(mV 2 ) ⎥ ⎥ − g sin γ / V 2 ⎥ 2 ⎦ −∂ T /∂Φ sin α /(mV ) 2 2 2 ∂ L /(∂ V ∂ h)/(mV ) − ∂ L /∂ h/(mV ) + ∂ g /∂ h cos γ / V

π 22

⎤ ∂ 2 L /(∂ V ∂ α )/(mV ) − (∂ L /∂ α + ∂ T /∂ α sin α + T cos α )/(mV 2 ) ⎥ ⎢ [∂ 2 L /∂ α 2 + (∂ 2 T /∂ α 2 − T ) sin α + 2∂ T /∂ α cos α ]/(mV ) ⎥ ⎢ ⎥ 0 =⎢ ⎥ ⎢ 2 ⎦ ⎣ [∂ T /(∂ α ∂Φ) sin α + ∂ T /∂Φ cos α ]/(mV ) ∂ 2 L /(∂ α ∂ h)/(mV ) ⎡



π 23 = − g sin γ / V 2 0 g cos γ / V 0 ∂ g /∂ h sin γ / V T

T

π 24 =

¨¯ + w ˙ T Ω2 w ˙ f V = ω1 w



−∂ T /∂Φ sin α /(mV 2 ) [∂ 2 T /(∂ α ∂Φ) sin α + ∂ T /∂Φ cos α ]/(mV ) 0 0 0

T

π 25

¯ /∂ α )/m g V 1 = g Q (∂ T /∂ α cos α − T sin α − ∂ D

⎡

⎤ ∂ 2 L /(∂ V ∂ h)/(mV ) − ∂ L /∂ h/(mV 2 ) + ∂ g /∂ h cos γ / V 2 ⎢ ⎥ ∂ 2 L /(∂ α ∂ h)/(mV ) ⎢ ⎥ ⎢ ⎥ =⎢ ∂ g /∂ h sin γ / V ⎥ ⎣ ⎦ 0 2 2 2 2 ∂ L /∂ h /(mV ) − ∂ g /∂ h cos γ / V

2

g V 2 = ω (∂ T /∂Φ cos α )/m

¯ /∂ α )/m g V 3 = (∂ T /∂ α cos α − T sin α − ∂ D ¨¯ + w ˙ T Π2 w ˙ fγ = π 1 w g γ1 = g Q (∂ T /∂ α sin α + T cos α + ∂ L /∂ α )/(mV )

(A.4)

2

g γ2 = ω (∂ T /∂Φ sin α )/(mV ) g γ3 = (∂ T /∂ α sin α + T cos α + ∂ L /∂ α )/(mV )

References

(A.1)

in which

T



α˙ γ˙ Φ˙ h˙ ⎡ ˙ ω1 w ⎢ ˙ − π x + fQ 1 ⎢ ¨¯ = ⎢ ˙ π1w w ⎢ ⎣ −2ζ ωΦ˙ − ω2 Φ V˙ sin γ + V γ˙ cos γ ˙ = V˙ w

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

(A.2)

⎡

⎤T −∂ D¯ /∂ V ⎢ ∂ T /∂ α cos α − T sin α − ∂ D¯ /∂ α ⎥ ⎥ 1 ⎢ ⎥ ω1 = ⎢ −mg cos γ ⎢ ⎥ m⎣ ⎦ ∂ T /∂Φ cos α −∂ D¯ /∂ h − ∂ g /∂ hm sin γ   Ω 2 = ω21 ω22 ω23 ω24 ω25 /m T  ω21 = −∂ 2 D¯ /∂ V 2 −∂ 2 D¯ /(∂ α ∂ V ) 0 0 −∂ 2 D¯ /(∂ V ∂ h) ⎡ ⎤ −∂ 2 D¯ /(∂ α ∂ V ) ⎢ (∂ 2 T /∂ α 2 − T ) cos α − 2∂ T /∂ α sin α − ∂ 2 D¯ /∂ α 2 ⎥ ⎢ ⎥ ⎥ 0 ω22 = ⎢ ⎢ ⎥ ⎣ ⎦ ∂ 2 T /(∂ α ∂Φ) cos α − ∂ T /∂Φ sin α −∂ 2 D¯ /(∂ α ∂ V ) T  ω23 = 0 0 mg sin γ 0 −m cos γ ∂ g /∂ h T  ω24 = 0 ∂ 2 T /(∂ α ∂Φ) cos α − ∂ T /∂Φ sin α 0 0 0 T  ω25 = −∂ 2 D¯ /(∂ V ∂ h) −∂ 2 D¯ /(∂ α ∂ V ) −m cos γ ∂ g /∂ h 0 0 (A.3)

π1

⎤T ⎡ (∂ L /∂ V + ∂ T /∂ V sin α )/(mV ) − ( L + T sin α )/(mV 2 ) + g cos γ / V 2 ⎥ ⎢ (∂ L /∂ α + ∂ T /∂ α sin α + T cos α )/(mV ) ⎥ ⎢ ⎥ =⎢ g sin γ / V ⎥ ⎢ ⎦ ⎣ ∂ T /∂Φ sin α /(mV ) (∂ L /∂ h − m cos γ ∂ g /∂ h)/(mV )

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