Adaptive control of underactuated flight vehicles with moving mass

Aerospace Science and Technology 85 (2019) 75–84

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Adaptive control of underactuated flight vehicles with moving mass Jianqing Li a , Sai Chen a , Chaoyong Li a,∗ , Changsheng Gao b , Wuxing Jing b a b

College of Electrical Engineering, Zhejiang University, 310027 Hangzhou, China Department of Aerospace Engineering, Harbin Institute of Technology, 150001 Harbin, China

a r t i c l e

i n f o

Article history: Received 19 September 2018 Received in revised form 6 November 2018 Accepted 5 December 2018 Available online 6 December 2018 Keywords: Moving mass control Adaptive control Underactuated control Immersion and invariance Flight control system

a b s t r a c t The configuration of internal moving masses is a key challenge for applying moving mass control technology to flight vehicle control. A novel configuration with a large mass ratio moving mass and reaction jets is proposed for bank-to-turn control. The control system of the proposed configuration consists of the attitude dynamics and the moving mass dynamics, which are coupled by the additional inertia moment of moving mass. To deal with the coupling, the integrated control of an attitude-servo system and a lateral underactuated control based on immersion and invariance theory is presented. To overcome the uncertainties in the flight vehicle model, immersion and invariance theory is employed to design an estimator for the unknown aerodynamic parameters. The estimator has an additional nonlinear term which adjusts the performance of the estimation error. The simulation results show that the proposed attitude-servo controller for the longitudinal subsystem can enhance the response of the system and the underactuated controller of the lateral subsystem can reduce fuel consumption. © 2018 Elsevier Masson SAS. All rights reserved.

1. Introduction Moving mass control technology has been applied to many fields as a control methodology [1], especially in the attitude control of reentry vehicles. Over the past few decades, the research on the reentry vehicles with moving mass has mainly concerned the moving mass configurations, control mechanisms, and design of the control systems. There are many different control mechanisms being developed with different configurations. For example, a configuration with two or three moving masses has been applied in kinetic warheads to offer the capacity of longitudinal and lateral maneuvering [2,3]. Although the configuration with multiple moving masses shows outstanding and various maneuvers, it is difficult to implement for engineering because more actuators limit the utilization of the internal space. Thus, the single movingmass control system is proposed to provide roll control ability [4], or generate a trim angle of attack (AOA) for maneuvering [5]. In order to improve the maneuver of the single moving mass flight vehicle, some novel configurations have been developed. Gao [6] developed a moving mass rail which can rotate with respect to the roll axis of the body so that the vehicle can track pitch and yaw channel commands. He designed different controllers based

*

Corresponding author. E-mail addresses: [email protected] (J. Li), [email protected] (S. Chen), [email protected] (C. Li), [email protected] (C. Gao), [email protected] (W. Jing). https://doi.org/10.1016/j.ast.2018.12.003 1270-9638/© 2018 Elsevier Masson SAS. All rights reserved.

on the sliding mode theory and feedback linearization theory for the attitude loop and servo loop. Wei [7] proposed a combination of bank-to-turn control with the single moving mass and reaction jets to control the pitch and roll channel. To implement the moving mass control technology in engineering further, we presented a novel configuration with a large mass ratio moving mass [8]. Due to the increase of the mass ratio, this configuration can increase the maneuver performance and improve availability of the internal space. However, there exist two issues worth investigating in the proposed configuration. The first issue is that the proposed moving mass control scheme is an underactuated system and the attitude motion is not affected directly by the control inputs. The proposed moving mass configuration with large mass ratio results in serious dynamical coupling which makes the dynamic behavior complex. However, the coupling offers controllability between the passive channels (uncontrolled by actuators) and the active channels (controlled by actuators). Underactuated systems are used for reducing weight, cost or energy consumption, while still keeping an adequate degree without reducing the reachable configuration space. However, some undesired properties can increase the controller design difficulty, such as dynamic coupling, nonlinearity, and nonholonomic constraints. Partial feedback linearization [9] and energy based control [10] are developed for underactuated control. However, they cannot guarantee global stability. A model reduction by transforming the underactuated system to cascade normal forms is presented [11], which makes the application of the backstepping and sliding-mode control possible [12]. In addition, model predic-

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J. Li et al. / Aerospace Science and Technology 85 (2019) 75–84

tive control [13], hybrid switching control [14], and inverse optimal control [15] are studied in different areas. The second issue is that there are various uncertainties in the aerodynamic parameters and nonlinear nature of dynamical effects with the increase of the mass ratio of the moving mass to the vehicle body mass. To overcome this shortcoming, some nonlinear adaptive control methods are applied to the flight control system such as classical adaptive control [16], adaptive feedback linearization control [17], and adaptive iterative learning control [18]. These approaches give a certainty-equivalent adaptive law to overcome uncertainties in the nonlinear dynamics of the reentry vehicle, but its estimation shows a poor performance, which may cause the undesired response of the closed-loop system [19]. Recently, a new method for adaptive control of uncertain nonlinear systems based on immersion and invariance (I&I) theory was presented [20,21]. This method is different from most of the nonlinear stabilization methods which require knowledge of a Lyapunov function. Any trajectory of the controlled system is the image through the mapping of the target system which is asymptotically stable. Then, the designed control law renders the manifold attractive and invariant, which can ensure the stability of closed-loop system. In addition, an outstanding feature of adaptive control based on I&I is to introduce an extra term in the parameter update law. This new term shapes the manifold into which the adaptive system is immersed. Due to its significant nonlinear adaptive performance, most investigators designed parameter estimators based on I&I theory to estimate aerodynamic parameters or unknown disturbances of the flight vehicles [22–24]. There is also an observer which can estimate the unmeasurable states [25,26]. In view of these problems, the purpose of this paper is to develop a new controller to the underactuated moving mass system by application of the I&I method. The coupling index is investigated to reveal the coupling characteristics of underactuated moving mass system and evaluate the coupling effects on unactuated channels. To take advantage of the coupling and overcome the uncertainties of parameters, the I&I method is employed to design the underactuated controller and estimator. In particular, the longitudinal subsystem is an integrated attitude-servo control system and the lateral subsystem is only controlled by the roll torque. The inner and outer manifolds established by the target system of I&I method are used to design the asymptotically stable controller. Thus, the advantage of the proposed underactuated control scheme is its efficient response and low fuel consumption. Simulation results show that the proposed control scheme can ensure good performance efficiently in the novel configuration with moving mass. 2. Problem definition A sketch of the moving mass flight vehicle is shown in Fig. 1. It consists of three major components, namely, a main vehicle body, an internal single moving mass that can move on an internal rail, and a reaction control system. The moving mass is fixed with the vehicle body at O , and another side c can translate on the rail under the actuator. The mass center of the moving mass and vehicle body are denoted by p and b, and the mass center of the system is denoted by s. The mass center offset caused by the loose moving mass can generate a trim AOA resulting from aerodynamic drag. To achieve the desired orientation, the roll channel is controlled by the reaction control system (RCS). It means that the RCS jets only generates the roll moment in the proposed control scheme. The body frame is fixed on the vehicle body at O . The xb -axis is aligned with the longitudinal axis of the projectile. The zb -axis is in the symmetric plane and perpendicular to the xb -axis. The y b -axis completes a right-handed coordinate system. The moving mass reference frame can be obtained from the vehicle body frame

Fig. 1. A moving mass flight vehicle.

via a z-axis rotation by the deflection angle δ . The two reference frames are linked by the following transformation matrix:

⎡

cos δ C PB = ⎣ − sin δ 0

sin δ cos δ 0

⎤

0 0 ⎦. 1

(1)

m P and m B are the masses of the moving mass and the body, respectively. The mass of system is m S = m P + m B . μ P = m P /m S , μ B = m B /m S are the mass ratios of the moving mass and body mass relative to the system, respectively. L P and L B are the distance of center of mass of the moving mass and center of mass of the body from O . r o is the distance vector of O with respect to an inertial reference frame. r op and r ob are the distance vectors from O to the center of mass of moving mass and center of mass of body, respectively. r sp and r sb are the distance vectors from the center of mass of system to the center of mass of moving mass and center of mass of body. Summing the angular momentum of the moving mass P and the body B with respect to the mass center of system yields the angular momentum of the system:

HS = HP + HB = I P · (ω B / I + ω P / B ) + I B · ω B / I + m P r sp × r˙ op + m B r sb × r˙ ob ,

(2)

where ω B / I is the inertial angular velocity vector of the body. ω P / B is the relative angular velocity of the moving mass with respect to the body B. According to the momentum theorem, the derivative of angular momentum of the system in the inertial reference frame is given by: I

dH S dt

= =

I

d( I P · ω P / I + I B · ω B / I )



dt

+ m P r sp × r¨ op + m B r sb × r¨ ob

MS.

(3)



M S is the total external moments acting on the flight vehicle which can be written as



M S = M RCS + r sq × F aero

= M RCS + M B + r sb × F aero ,

(4)

where r sq is the distance vector from the system center of mass to the center of pressure. r sb × F aero is the additional aerodynamic moment caused by the deflection of the moving mass, which is the main control moment. M RCS is the control moment generated by RCS. By the relative differential principle, r¨ op and r¨ ob are given in the body reference frame by

J. Li et al. / Aerospace Science and Technology 85 (2019) 75–84

r¨ op = ωP / B × r op + ωB / I × r op + ω P / B × (ω P / B × r op )

+ 2ω B / I × (ω P / B × r op ) + ω B / I × (ω B / I × r op ),

M p2 = −r op × m P μ B (5)

r¨ ob = ωB / I × r ob + ω B / I × (ω B / I × r ob ), where (•) denotes the time derivative in the body reference frame. The components of the rotational dynamics equation expressed in the body reference frame are obtained by substituting Eq. (4) and Eq. (6) into Eq. (3)



× ×T I B + J P + μ B m P r bp r bp







× ×T ωB / I + J P + μ B m P r bp r op ωP / B

= M B + M RCS + M p1 + M b1 + M j1 + r sb × F a − ω B / I 

× ( I B + J P )ω B / I + J P ω P / B ,

(6)

where

M p1 = −r sp × m p



+ 2ω B / I × (ω P / B × r op ) + ω B / I × (ω B / I  M b1 = −r sb × m B ω B / I × (ω B / I × r ob ) ,

J P = ω P / B × J P − J P × ω P / B .

(•)× denotes the matrix resulted from the cross-product operator acting on (•). In view of the dynamic behavior of the moving mass, the dynamic equation of the internal moving mass must be established. According to the momentum theorem, the derivative of angular momentum of moving mass P with respect to the mass center in the inertial reference frame is given I

dH P

=

dt

I

d( I P · ω P / I ) dt

= r pc × F C + r po × F O ,

FC mP

+

FO mP

(7)

+ g,

r¨ s = r¨ o + r¨ os = r¨ o + μ P r¨ op + μ B r¨ ob =

(8) ms g + F a ms

.

(9)

Substituting Eq. (8) into Eq. (7): I

d( I P · ω P / I ) dt



= r pc × F C + r po × m P (¨r o + r¨ op ) − F C − m P g .

Then, substituting Eq. (9) into Eq. (10):

d( I P · ω P / I ) dt



= r oc × F C − r op × m P μ B (¨r op − r¨ ob ) + μ P F a .

The components of the rotational dynamics equation of moving mass expressed in the body reference frame are obtained: × ×T

J P + m P μ B r op r op



ω P /B

where

⎤

M xRCS M RCS = ⎣ 0 ⎦ , 0

β

Cz β

⎤

⎡

⎤

⎡

⎢ α ⎥ F a = ⎣ Y ⎦ = ⎣ C y α ⎦ qS ,

ω mx x L x V ωy ω my L my x L + V y + V ωz mα + mzV L z z

ω

ω

α

⎤ ⎥ ⎥ qS L , ωx ⎥ ⎦

ω

where q = ρ V 2 /2. α and β are the angle of attack and angle of β sideslip, respectively. C x , C αy , and C z are aerodynamic coefficients. β

ω

ωy

ω

ω

z x x mα are the aerodynamic moment z , m z , m x , m y , m y , and m z coefficients. ρ is the atmospheric density, S is the cross-sectional area, L is the reference length, and V is the magnitude of flight vehicle velocity. The distance vectors and the angular velocity are described in the body frame

r ob = [− L B , 0, 0] T

˙T ω P / B = [0, 0, δ]

ω B / I = [ωx , ω y , ωz ]T ,

Eq. (6) and Eq. (12) constitute the complete attitude dynamic model of moving mass flight vehicle. It can be seen that the attitude and the moving mass dynamics are coupled by the moment of inertia of the moving mass and the coupling increases with an increasing mass of the moving mass. To analyze dynamic coupling and design the controller, the dynamic model should be simplified. The products of the angular velocity are assumed to be small quantities and be negligible. Then, the components of the dynamics are given by

˙ x + I y ω˙ y = M y + μ P Z ( L B − L P cos δ), I yx ω

˙ z + I zδ δ¨ = M z + μ P Y ( L P cos δ − L B ) + μ P L P X sin δ, Izω

(13)

˙ z + I δ δ¨ = μ P L P ( X sin δ + Y ) + M C , Iδz ω

I x = I B1 + I P 1 cos2 δ + I P 2 sin2 δ + μ B m P L 2P sin2 δ, I y = I B2 + I P 1 sin2 δ + I P 2 cos2 δ + μ B m P ( L B − L P cos δ)2 ,





I z = I B3 + I P 3 + m P μ B L 2P + L 2B − 2L P L B cos δ ,





I xy = ( I P 1 − I P 2 ) cos δ sin δ + m P μ B L P L B sin δ − L 2P sin δ cos δ ,





I δ = I P 3 + m P μ B L 2P .

= M p2 + M b2 + M j2 + r oc × F C − r op × μ P F a − ω B/I × J P ω P /I ,

−C x

I zδ = I P 3 + m P μ B L 2P − L P L B cos δ ,



 ×T × ×T + J P − m P μB r× op r ob + m P μ B r op r op ω B / I 

⎡

where

(11)



⎤

˙ x + I xy ω˙ y = μ P L P Z sin δ + M x + M xRC S , I xω

(10)

I

−X

r op = [− L P cos δ, − L P sin δ, 0] T ,

where F O is the force acting on O . F C is servo force exerted on the moving mass. By Newton’s second law, the translational dynamics equations of the moving mass and the center mass of system, respectively, are given by Eq. (8) and Eq. (9):

r¨ P = (¨r o + r¨ op ) =

⎡

Mx ⎢ ⎢ M B = ⎣ M y ⎦ = ⎢ mβ β ⎣ y Mz

M j1 = M j2 = − J P (ω B / I + ω P / B ), J P = C TP B I P C P B ,

ω P / B × (ω P / B × r op )

I B = diag{ I B1 , I B2 , I B3 } and I P = diag{ I P 1 , I P 2 , I P 3 } are the 3 × 3 inertia tensors of the body and moving mass about their center of mass, respectively. F a is the vector of aerodynamic force at the center of pressure. M B is the vector of aerodynamic moment about the center of mass of the vehicle body. M C = r oc × F C is servo torque. The aerodynamic force and torque are described in the body frame:

⎡

× r op ) ,



 + 2ω B / I × (ω P / B × r op ) + ω B / I × (ω B / I × r op ) ,

 M b2 = r op × m P μ B ω B / I × (ω B / I × r ob ) .

Z

ω P / B × (ω P / B × r op )

77

(12)

Equation (13) describes an underactuated system, which has four degrees of freedom and two control inputs. The actual control inputs are the rolling moment M xRCS generated by the RCS jets and servo torque M C . The underactuated system is composed of two subsystems: one is the longitudinal underactuated subsystem and

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J. Li et al. / Aerospace Science and Technology 85 (2019) 75–84

Fig. 2. Coupling index for the moving mass system.

another one is the lateral underactuated subsystem. It should be noted that the longitudinal underactuated subsystem is constituted by the attitude system and the servo system, which generates a trim AOA by the deflection angle [10]. Thus, the controller design for the longitudinal subsystem is an integrated attitude-servo controller. In general, there are various uncertainties in the reentry vehicle dynamics. In order to verify the adaptive control law, the aerodynamic parameters are assumed to be uncertain and defined as follows

β ωy ωz α β T . x θ = m y , mαz , mω x , m y , mz , C x , C y , C z 3. Dynamic coupling of underactuated system

γ as the roll angle and let qu = [α , β]T , qa = [γ , δ]T , ˙ T , the dynamics can be rearranged p u = [ω y , ωz ] T , and pa = [ωx , δ] Define

as follows:



I uu I au

I ua I aa

where

I uu = I aa =







Iy 0

0 Iz

Ix 0

0 Iδ

F u (q, p ) = F a (q, p ) =

 

p˙ u p˙ a



 =

F u (q, p ) F a (q, p )



 ,



I ua =

I yx 0

I au =

I xy 0



,



 +

0 I zδ 0 Iδz

0 u

 (14)

,

 ,  ,

M y + μ P Z ( L B − L P cos δ) M z + μ P Y ( L P cos δ − L B ) + μ P L P X sin δ



μ P L P Z sin δ + M x , μ P L P ( X sin δ + Y )



u=

M RCS MC

 ,



.

Note that the moments of inertia are related to the mass ratio of moving mass and the deflection angle. The passive channels cannot be controlled because these channels are not equipped with actuators. The passive channels, however, are dynamically coupled to the active channels because of the presence of non-zero offdiagonal elements in the inertia matrix. ˙ y the second line of For the lateral subsystem, factoring ω Eq. (13) we obtain





ω˙ y = I −y 1 M y + μ P Z ( L B − L P cos δ) − I yx ω˙ x .

(15)

We focus on the angular acceleration relationship between the roll channel and the yaw channel, and rewrite Eq. (15) as

ω˙¯ y = − I −y 1 I yx ω˙ x = G xy ω˙ x ,

(16)

1 ¯˙ y = ω˙ y + I − where ω y [ M y + M x + μ P Z ( L B − L P cos δ)]. The accel-

˙¯ y can be viewed as a virtual acceleration of the passive eration ω channel, generated by the acceleration of the active one. The gain matrix G xy shows the coupling between the active and the passive channels. To quantify this coupling, we define the acceleration coupling index as

ρxy (δ) =

n  i =1

   I xy  , I 

σi (G xy ) = −

(17)

y

where σ is a singular value function. This index indicates the intensity that the actuation inputs control the passive channel. The bigger the value is, the easier the control of passive channel is. The coupling index between the AOA and the deflection angle of moving mass also is given by similar method as follows:

ρz (δ) =

n  i =1

   I zδ   σi (G zδ ) = − . Iz

(18)

This index indicates the ability that the moving mass control the AOA. The characteristics of these coupling indexes can be illustrated in Fig. 2 to demonstrate the coupling effect between the active channel and the passive channel. Fig. 2a illustrates the yaw and roll coupling index with respect to the deflection angle for different mass ratios. It can be seen that the coupling increases with increasing mass ratio of the moving mass. ρxy has the maximum value and the minimum value in some particular point, which corresponds to the strongest and the weakest coupling. In the weakest coupling point like δ = 0, the motion of the moving mass has no effect on the yaw channel. Fig. 2b illustrates the coupling index between the deflection angle and the pitch channel for different mass ratios. Note that more mass ratio leads to the curves changing from concave to convex. An interesting aspect of this result is that when the mass ratio is close to 0.7, the coupling index curve becomes an approximate straight line. The coupling index is meaningful, since it can be used for designing the moving mass parameters to control attitude efficiently. 4. Flight control system design 4.1. Cascade normal forms for the moving mass flight vehicle We introduce cascade normal forms for underactuated mechanical systems that are convenient for control design. Consider the

J. Li et al. / Aerospace Science and Technology 85 (2019) 75–84

underactuated system in Eq. (14) and the kinematic differential equations in [27], the dynamic equations can be transformed to

4.2. Controller design

q˙ u = h1 (qu ) p u + h2 (qu ) pa + h3 (qu )qu , q˙ a = pa ,

(19)

p˙ u = J aa F u (q, p ) + J ua F a (q, p ) − J ua u , p˙ a = J ua F u (q, p ) + J uu F a (q, p ) + J uu u , where



h1 (qu ) = h3 (qu ) =

− tan β sin α 1 cos α



qS

0

−C αy



2 J aa = I aa I uu − I au



J uu = I aa I uu −

− 1

 h2 (qu ) =

,

− tan β cos α 0 sin α



0



2 J ua = I aa I uu − I au

I aa ,

2 −1 I au I uu .



0

,

β

C z cos β

0

mV cos β



− 1

I ua ,

It can be seen that the control input u has a direct effect on the dynamics of both variables p u and pa . The deflection angle, the AOA, and the sideslip angle are small. Then, the following global change of coordinates:

⎧ ⎪ ⎪ z 1 = q u , −1 ⎨ z 2 = p u + I uu I au pa , ⎪ z 3 = qa , ⎪ ⎩ z 4 = pa ,

transforms the dynamics of (19) into a cascade nonlinear system in strict feedback form





1 z˙ 1 = h1 z 2 − I − uu I au z 4 + h 2 z 4 + h 3 z 1 ,







1 z˙ 2 = J 1 c 1 z 1 + c 2 z 3 + c 3 z 2 − I − uu I ua z 4 ,

(21)

z˙ 3 = z 4 ,

v = ( J ua c 1 + J uu c 4 ) z 1 + ( J ua c 2 + J uu c 5 ) z 3

 1 + J ua c 3 z 2 − I − uu I au z 4 + ( J ua c xy + J uu c 6 ) z 4 + J uu c n ,   β β 0 m y qS L + μ P C z qS ( L B − L P ) , c1 = α α m z qS L + μ P C y qS ( L P − L B )

 c4 =

 c6 =

 cn =

0 μP L P X

0 μ P L P qSC αy ω mx x qsL 2 V

0

μP L P Z δ 0

ξ˙ 2 = J 1 (c 1 ξ 1 + c 3 ξ 2 ).

(22)

The coefficient matrix of the above equation is



h3 J 1c1

h1 J 1c3



.

(23)

It is noted that A is a constant matrix. After some simple calculations, the eigenvalues of A all have negative real parts. Therefore, the target system (22) has a globally asymptomatically stable equilibrium at the origin (0, 0). To accomplish the control law design we need to prove that conditions of I&I theorem hold. We fix π1 (ξ ) = ξ 1 and π2 (ξ ) = ξ 2 . According to the immersion condition, the mapping is clearly given by

e = z 4 + λ z 3 = z˙ 3 + λ z 3 ,

1 J 1 = J aa − I − uu I au J ua ,

c2 =

ξ˙ 1 = h1 ξ 2 + h3 ξ 1 ,

(24)

Then, according to the above mappings, we define the off-themanifold variable as

where

0 0



x = π (ξ ) = [ξ 1 , ξ 2 , 0, 0] T .

z˙ 4 = v + J uu u ,



,

The method of I&I stabilization of nonlinear systems includes four conditions for the target system, immersion condition, implicit manifold, and manifold attractivity and trajectory boundedness [20]. An I&I-stabilizable system is implemented by two primary steps. Firstly, I&I requires the selection of a target dynamical system which is locally asymptotically stable and of dimension smaller than the dimension of the original system. Then, we design a control law so that the closed-loop trajectories remain bounded and asymptotically converge to the attractive manifold. In the controller design, two assumptions are considered: 1) the AOA, sideslip angle, and deflection angle are small; 2) frozen operation design points are chosen, which means the aerodynamic coefficients, dynamic pressure, and velocity are considered as constants [28,29]. We first start with selecting the target system. When z 3 = z 4 = 0, the target dynamical system (ξ 1 , ξ 2 ) has been chosen as the image of the subsystem (z 1 , z 2 ). The target dynamics can be written as

A= (20)

79

0 0



0



 c3 =

,

m y qsL 2 V





0 , 0

,

 c5 =

 c xy =



ωy

0 0 0

ω m z z qsL 2 V

0 μ P L P qSC x

ω m y x qsL 2 V

0

0

,  ,



0 0

,

.

It is noted that the control input u acts directly on the state variable z 4 , and acts implicitly on other state variables via z 4 . The controller aims to make the attitude follow a reference command, then the target behavior of the moving mass system corresponds to [ p , q] T = 0, which is equivalent to z = 0.

(25)

where λ = diag{λ1 , λ2 } is a positive coefficient matrix. This makes the condition of implicit manifold hold. Notice that if e → 0, z 3 converges exponentially to zero with the rate of convergence λ. Then, z 4 also converges to zero. Hence, a control input should be designed to make the trajectories of the closed-loop system bounded and e converges to zero. To this end, let

e˙ = − K e ,

(26)

where K = diag{k1 , k2 } represents the rate of exponential convergence of e to zero. Substituting z˙ 3 and z˙ 4 of Eq. (21) in e˙ gives

e˙ = v + J uu u + λ z 4 .

(27)

The control input is of the form 1 u = − J− uu ( K e + v + λ z 4 ).

(28)

The outer manifold M 2 is reached when the off-the-manifold variable e converges exponentially toward the manifold e = 0 at a speed determined by K . Then, the inner manifold M 1 is reached from outside when the off-the-manifold z 3 converges exponentially to zero with the rate of convergence λ. Finally, the manifold M 1 with the target system has a locally asymptotically stable equilibrium at the origin. Note that, K and λ must be selected to

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J. Li et al. / Aerospace Science and Technology 85 (2019) 75–84

ensure the attractivity of both manifolds, that render the closeloop system globally asymptotically stable. According to the last condition of manifold attractivity and trajectory boundedness, it is also necessary to prove boundedness of the trajectories of the closed-loop system with control law and the off-the-manifold coordinate e. First, to prove boundedness of ξ 1 and ξ 2 in the manifold M 1 , let V 1 (ξ 1 , ξ 2 ) be a positive-definite function as follows

V 1 (ξ 1 , ξ 2 ) =

1 2

1

ξ 1T ξ 1 + ξ 2T ξ 2 .

(29)

2

Note that when z 3 = z 4 = 0, the state variables ξ 1 and ξ 2 correspond to the variable z 1 and z 2 respectively. Differentiating V 1 and using Eq. (22) gives

V˙ 1 = ξ 1T h3 ξ 1 + ξ 2T J 1 c 3 ξ 2 + ξ 1T h1 ξ 2 + ξ 2T J 1 c 1 ξ 1 .

(30)

Because V 1 is a positive definite function and V˙ 1 ≤ 0, one finds that ξ 1 and ξ 2 are bounded. Now, we define a positive definite function V 2 ( z ) in the manifold M 2 which corresponds to e = z 4 + λ z 3 = 0.

V2 =

1 2

z 1T z 1 +

1 2

z 2T z 2 +

1 2

z 3T z 3 .

(31)

σ1  2

2 σ1 z 3T λ T λ z 3

+

z 3T z 3



−

4.3. Parameter estimator In I&I adaptive design, the manifold is defined as the difference between the parameters and estimations. Therefore, to render the manifold invariant and attractive can guarantee the estimation errors are bounded and convergent. The first line and the second line of Eq. (13) can be rearranged as follows:

p˙ = W ( p u , pa )θ + G u ,

(39)

where

G = [− J uu , J ua ] T , W ( p u , pa ) = −1

β L2

0

0

ω y L2 /V

0

ωx L 2

0

α L2

0

0

ωz L 2 V

0

μ P L P sin δ μ P α ( L P cos δ − L B )

0 0

0 0

0 0

0 0

0 0

0

0

μ P L P sin δ

μP α L P

ωx L 2 V

0

V

0

μ P β( L B − L P cos δ)

0



λ2i + 1 − λi < 0,

(32)

< 0,

(33)

i = 1, 2,

(34)

yield boundedness of trajectories in the manifold M 2 . Thus, this subsystem corresponding to e = 0 is globally asymptotically stable. The last step is to complete the claim. Consider now the positive definite and proper function for the closed loop system

V 3 (z, e) =

1 2

z 1T z 1 +

1 2

z 2T z 2 +

1 2

z 3T z 3 +

1 2

e2 ,

(35)

.

μ P L P β sin δ 0

˙ W

f

= −aW f + W ,

u˙ f = −au f + u ,

1 V˙ 3 ( z , e ) = V˙ 2 + z 1T h2 e + z 2T J 1 c 3 I − uu I ua e + z 3 e + e e˙ 1 2 || z T h2 + z 2T J 1 c 3 I − uu I ua + z 3 || ≤ V˙ 2 + 1 2 σ2   σ2 + − K e T e,

2

< 0,

(41)

It should be pointed out that the additive exponentially decaying signal caused by the filtering has been ignored [24]. Now let the parameter error be

s = θˆ + β − θ ,

(42)

where θˆ is partial estimate of the unknown parameter θ and the nonlinear function β is added to θˆ to obtain the full estimate. Unlike the traditional parameter estimation method (the classical certainty-equivalence adaptive control method), the nonlinear function changes simple integral action of parameter update law to proportional-integral action, which enhances the flexibility of estimator design. The nonlinear function is selected as

β = η W Tf p f ,

˙ ˙ Tf p f + η W T ( W f θ + G u f ) s˙ = θˆ + η W f

˙ = θˆ + η(−aW f + W )T p f + η W Tf ( W f θ + G u f ).

(36)

(43)

(37)

(44)

In view of Eq. (44), the update law to cancel known function is chosen as

 ˙ θˆ = −η(−aW f + W )T p f − η W Tf W f (θˆ + β) + G u f .

σ2 > 0. Similarly, setting σ2 such that

1 2 || z 1T h2 + z 2T J 1 c 3 I − uu I ua + z 3 ||

p˙ f = W f θ + G u f .

where η > 0. The dynamics of the estimation error can obtained by differentiating s. Its derivative is

and note that z 4 = z˙ 3 = e − λ z 3 . Then,

2 σ2



0

where a > 0. Then, after filtering both sides of Eq. (39), one obtains

z 3T λ z 3 ,

2 σ1

σ1 

(38)

(40)

and selecting λ such that

V˙ 2 +

i = 1, 2,

1 T T 2 2 || z 2T J 1 I − uu I ua − z 1 h 2 || + || z 2 J 1 c 2 ||

1 T T 2 2 || z 2T J 1 I − uu I ua − z 1 h 2 || + || z 2 J 1 c 2 ||

for any

1 2 || z 1T h2 + z 2T J 1 c 3 I − uu I ua + z 3 || , 4 V˙ 2

p˙ f = −a p f + p ,

for any σ1 > 0. From the last inequality above we conclude that, setting σ1 such that

2

2

>−

To avoid solving the partial differential equation, Let us introduce filtered signals given by

= z 1T (h1 z 2 + h2 z 4 + h3 z 1 )

  1 + z 3T z˙ 3 + z 2T J 1 c 1 z 1 + c 2 z 3 + c 3 z 2 − I − uu I ua z 4 

T 1 T T T = V˙ 1 + z 2 J 1 I − uu I ua − z 1 h 2 λ z 3 + z 2 J 1 c 2 z 3 − z 3 λ z 3

V˙ 1 +

σ2

yield the desired result of attractivity of the manifold M 2 . We conclude that, when the controller gains are satisfied for the conditions (34) and (38), the closed loop system has a globally asymptotically stable equilibrium at zero with I&I controller. 2

qS I

V˙ 2 ( z , e = 0)

+

Ki >



Using Eq. (21), the dynamics of V 2 is given by

≤ V˙ 1 +

and setting K such that

(45)

Substituting the update law in Eq. (44), one obtains the parameter error dynamics of the form

J. Li et al. / Aerospace Science and Technology 85 (2019) 75–84

81

Fig. 3. Attitude response of different controller.

s˙ = −η W Tf W f s.

(46)

To examine the stability of the closed-loop system, consider a Lyapunov function

V = V3 + Vs = V3 +

1 2η

s T s.

(47)

Differentiating the Lyapunov function and we obtain

2

αc = 1 deg, βc = 0 deg, γc = 10 deg, 0 ≤ t < 1.5 s, αc = 2 deg, βc = 0 deg, γc = 0 deg, 1.5 ≤ t ≤ 3 s.

5.1. Comparison of different nominal controller

(48)

Since V is a positive-definite function and V˙ ≤ 0, indicating boundedness for all states. Then the closed loop system has a globally stable equilibrium at zero. Moreover, the trajectories of the closed-loop system asymptotically converge to the attractive manifold

  Ω = ( W f , s) : W f s = 0 .



The controller gains are chosen as K = diag{15, 50} and λ = diag{5, 10}, which satisfy the condition (34) and (38) of the stability proof by simple calculation.

V˙ = V˙ 3 + V˙ s 1 2 || z T h2 + z 2T J 1 c 3 I − uu I ua + z 3 || ≤ V˙ 2 + 1 2 σ2   σ2 + − K e T e −  W f s2 .

The purposes of a BTT control are to develop the desired acceleration and keep sideslip angle small. The initial conditions are set to α (0) = γ (0) = 0 deg, and β(0) = 1 deg. The command angles are set to

(49)

The estimation θˆ +β can be applied to overcome the parametric uncertainties and guarantee the stability of the system. However, it should be noted that convergence of the function W f s to zero does not imply convergence of the parameter error e to zero. That is, the parameter estimates cannot converge to their corresponding true values. This is because the reference trajectory is not persistently exciting (PE) so that it cannot guarantee the convergence of the estimated parameters [27]. 5. Simulation In this section, numerical simulation results of the proposed controller applied to the underactuated moving mass system are presented. The parameters of the flight vehicle are chosen as m S = 1000 kg, μ P = 0.6, S = 3.8 m2 , L = 4.11 m, I P 1 = 38 kg m2 , I P 2 = I P 3 = 380 kg m2 , and I B1 = 104 kg m2 , I B2 = I B3 = 440 kg m2 . The flight height and velocity are 20 km and 3500 m/s, respectively.

To show the effectiveness of the proposed underactuated controller, a controller based on dynamic inversion (DI) is carried out [28]. In the DI controller, the longitudinal underactuated subsystem is divided into the attitude control and the servo-loop control. The lateral subsystem is fully actuated, which means the RCS jets are used to produce the rolling moment M xRCS and the yaw moment M yRCS for the yaw and roll channel control. The linear programming technique is applied to solve the RCS control allocation. In this test, the aerodynamic parameters are known and certain. The tracking performance of different control scheme are shown in Fig. 3. As it can be seen, the proposed underactuated controller shows good dynamic performance and steady-state performance, ensuring a rapid convergence to the commanded angles accurately. Unlike the separate design of the attitude and servo system, the control parameters are easy to be chosen because the proposed control simplifies design process. The proposed controller can achieve better performance than the DI controller. The control inputs with the servo force and RCS torques are shown in Fig. 4. In the RCS control, an objective is to minimize the number of jets that are used in an effort to minimize propellant usage. Thus, the sum of working time 4 of all thrusters can be defined as a performance index J = i =1 t i . We calculate that the indexes of the proposed controller and the DI controller are 0.85 s and 1.29 s from the simulation results, respectively. It can be seen that the underactuated control reduces the fuel consumption by around 34%. This is because the underactuated lateral subsystem is only controlled by the rolling moment which can be generated by

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Fig. 4. Control inputs of different controller.

Fig. 5. Attitude response of different configurations.

two thrusters (thruster 1, 3 or thruster 2, 4). Apparently, the proposed control scheme can achieve less energy consumption and prolong working life more than the fully actuated control. 5.2. Comparison of different configurations We now consider the case for the comparison of the point moving mass configuration and proposed configuration. The mass ratio of the point moving mass configuration is set as 0.1 [9]. The simulation results are shown in Fig. 5 and Fig. 6. As can be seen from the simulation results, the AOA tracking with large mass ratio shows faster convergence and less overshoot than the case with small mass ratio. Due to the inertia coupling between the roll and yaw channel increases with increasing large mass ratio, the sideslip angle response of large mass ratio converges faster than that of small mass ratio. The mass ratio also determines the amplitude of deflection angle, which indicates that the large mass ratio contributes to a decrease in the deflection angle. In Fig. 6, we recognize that the larger mass ratio requires more servo force. Obviously, the moving mass with large mass ratio can provide greater control authority but need more power to

Fig. 6. Control inputs of different configurations.

J. Li et al. / Aerospace Science and Technology 85 (2019) 75–84

83

Fig. 7. Adaptive attitude tracking with uncertainties.

Fig. 8. Control inputs and estimation.

drive. The fuel consumption index of small mass ratio is 0.88 s and the roll control process is mostly unchanged. Clearly, the proposed configuration can enhance the attitude dynamic behavior and control authority. 5.3. Adaptive controller To examine the performance of the proposed adaptive controller, we now consider the case that the aerodynamic parameters are unknown. Moreover, the controller is tested in two scenarios with 30% and −30% of uncertainty of deviation from the nominal parameters’ value. The gain of the update law is η = 5. As seen in Fig. 7, the performance of the proposed adaptive controller remains the same effects that compared with the ideal case. The control in-

puts and the parameter estimation error norms are shown in Fig. 8. Obviously, the magnitude of the servo force decreases with the decrease of the value of the aerodynamic parameters. It should be noted that the norm of estimate error ||s|| fails to converge to zero. However, the norm of W s converges to zero. This is because of the non-PE nature of the reference command. All simulation results illustrate that the proposed control scheme can achieve a good tracking performance of attitude and overcome uncertain aerodynamic parameters. 6. Conclusion The proposed configuration of the moving mass with large mass ratio results in two couplings: 1) between the AOA and the actua-

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tor mass dynamics, and 2) the yaw and the roll dynamics. The coupling index indicates the amount of coupling available for the purpose of controlling the passive channel through the active channel. To design the controller based on the immersion and invariance theory, an adaptive controller for the underactuated moving mass system is presented. The proposed controller renders the outer and inner manifolds attractive and ensures stability of the closed-loop system. In the longitudinal subsystem, the controller integrate the moving mass dynamics into the attitude dynamics. This integrated design of attitude-servo system can deliver fast-responding control systems by exploiting attitude and moving mass dynamics coupling. In the lateral subsystem, the underactuated control through inertia coupling can reduce the fuel consumption and increase the payload of vehicles. Then, the parameter estimator based on attractive manifolds is applied to deal with uncertain aerodynamic parameters. The dynamics of the estimation error can be specified by an extra function to guarantee good estimation performance. The simulation results demonstrate the features of the proposed underactuated control with uncertain aerodynamic parameters.

[8] J.Q. Li, C.S. Gao, W.X. Jing, P.X. Wei, Dynamic analysis and control of novel moving mass flight vehicle, Acta Astronaut. 131 (2017) 36–44. [9] M.W. Spong, The swing up control problem for the Acrobot, IEEE Control Syst. 15 (1) (1995) 49–55. [10] R. Lozano, I. Fantoni, J.B. Dan, Stabilization of the inverted pendulum around its homoclinic orbit, Syst. Control Lett. 40 (3) (2000) 197–204. [11] R. Olfati-Saber, Normal forms for underactuated mechanical systems with symmetry, IEEE Trans. Autom. Control 47 (2) (2002) 305–308. [12] R. Xu, Ümit Özgüner, Sliding mode control of a class of underactuated systems, Automatica 44 (1) (2008) 233–241. [13] J.M. Peng, C.Y. Li, X. Ye, Cooperative control of high-order nonlinear systems with unknown control directions, Syst. Control Lett. 113 (2018) 101–108. [14] S. Kang, J.N. Wang, C.Y. Li, J.Y. Shan, Nonlinear optimal control with disturbance rejection for asteroid landing, J. Franklin Inst. 355 (16) (2018) 8027–8048. [15] N.M. Horri, P. Palmer, Practical implementation of attitude-control algorithms for an underactuated satellite, J. Guid. Control Dyn. 35 (1) (2012) 40–45. [16] R. Ortega, T. Yu, Robustness of adaptive controllers – a survey, Automatica 25 (5) (1989) 651–677. [17] Q.L. Hu, Y. Meng, C. Wang, Y. Zhang, Adaptive backstepping control for airbreathing hypersonic vehicles with input nonlinearities, Aerosp. Sci. Technol. 73 (2018) 289–299. [18] M. Yu, C.Y. Li, Robust adaptive iterative learning control for discrete-time nonlinear systems with time-iteration-varying parameters, IEEE Trans. Syst. Man Cybern. Syst. 47 (7) (2017) 1737–1745.

Conflict of interest statement

[19] E.R. van Oort, L. Sonneveldt, Q.P. Chu, J.A. Mulder, A comparison of adaptive control designs for an over-actuator fighter aircraft model, in: AIAA Guidance, Navigation, and Control Conference and Exhibit, AIAA, 2008. [20] A. Astolfi, R. Ortega, Immersion and invariance: a new tool for stabilization and adaptive control of nonlinear systems, IEEE Trans. Autom. Control 48 (4) (2003) 590–606. [21] A. Astolfi, D. Karagiannis, R. Ortega, Nonlinear and Adaptive Control with Applications, Springer-Verlag, London, 2008, pp. 276–309. [22] K.W. Lee, S.N. Singh, Noncertainty-equivalent adaptive missile control via immersion and invariance, J. Guid. Control Dyn. 33 (3) (2010) 655–665. [23] B. Zhao, B. Xian, Y. Zhang, X. Zhang, Nonlinear robust adaptive tracking control of a quadrotor UAV via immersion and invariance methodology, IEEE Trans. Ind. Electron. 62 (5) (2015) 2891–2902. [24] J. Zhang, Q. Li, N. Cheng, B. Liang, Adaptive dynamic surface control for unmanned aerial vehicles based on attractive manifolds, J. Guid. Control Dyn. 36 (36) (2013) 1776–1783. [25] Z. Liu, X. Tan, R. Yuan, G. Fan, Immersion and invariance-based output feedback control of air-breathing hypersonic vehicles, IEEE Trans. Autom. Sci. Eng. 13 (1) (2015) 1–9. [26] J. Hu, H. Zhang, Bounded output feedback of rigid-body attitude via angular velocity observers, J. Guid. Control Dyn. 36 (4) (2015) 1240–1248. [27] C.Y. Li, W.X. Jing, C.S. Gao, Adaptive backstepping-based flight control system using integral filters, Aerosp. Sci. Technol. 13 (2009) 105–113. [28] B. Xu, X. Huang, D. Wang, F. Sun, Dynamic surface control of constrained hypersonic flight models with parameter estimation and actuator compensation, Asian J. Control 16 (1) (2014) 162–174. [29] O.U. Rehman, B. Fidan, I.R. Petersen, Uncertainty modeling and robust minimax LQR control of multivariable nonlinear systems with application to hypersonic flight, Asian J. Control 14 (5) (2012) 1180–1193.

The authors declared that they have no conflicts of interest to this work. Acknowledgements This work is supported by National Natural Science Foundation of China (Grant Nos. 11572097, 11802268, 61503333, 91748128). References [1] J.Q. Li, C.S. Gao, C.Y. Li, W.X. Jing, A survey on moving mass control technology, Aerosp. Sci. Technol. 82–83 (2018) 594–606. [2] Z. Zhang, Y.K. Wang, J.Q. Mao, Moving-mass control of hypersonic vehicles based on fuzzy tree inverse method, Sci. China, Technol. Sci. 42 (11) (2012) 1379–1390. [3] P.K. Menon, G.D. Sweriduk, E.J. Ohlmeyer, D.S. Malyevac, Integrated guidance and control of moving-mass actuated kinetic warheads, J. Guid. Control Dyn. 27 (1) (2004) 118–126. [4] T. Petsopoulos, F.J. Regan, J. Barlow, Moving-mass roll control system for fixedtrim re-entry vehicle, J. Spacecr. Rockets 33 (1) (1996) 54–60. [5] R.D. Robinett, B.R. Sturgis, S.A. Kerr, Moving mass trim control for aerospace vehicles, J. Guid. Control Dyn. 19 (5) (1996) 1064–1070. [6] C.S. Gao, W.X. Jing, P.X. Wei, Research on application of single moving mass in the reentry warhead maneuver, Aerosp. Sci. Technol. 30 (1) (2013) 108–118. [7] P.X. Wei, C.S. Gao, W.X. Jing, Roll control problem for the long-range maneuverable warhead, Aircr. Eng. Aerosp. Technol. 86 (5) (2014) 440–446.