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PDE model-based boundary control of a spacecraft with double flexible appendages under prescribed performance
Journal Pre-proofs PDE Model-Based Boundary Control of a Spacecraft with Double Flexible Appendages under Prescribed Performance Junteng Ma, Hao Wen, Dongping Jin PII: DOI: Reference:
S0273-1177(19)30728-8 https://doi.org/10.1016/j.asr.2019.09.050 JASR 14475
To appear in:
Advances in Space Research
Received Date: Revised Date: Accepted Date:
29 March 2019 16 September 2019 29 September 2019
Please cite this article as: Ma, J., Wen, H., Jin, D., PDE Model-Based Boundary Control of a Spacecraft with Double Flexible Appendages under Prescribed Performance, Advances in Space Research (2019), doi: https:// doi.org/10.1016/j.asr.2019.09.050
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PDE Model-Based Boundary Control of a Spacecraft with Double Flexible Appendages under Prescribed Performance Junteng Ma, Hao Wen+, Dongping Jin State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, 210016 Nanjing, P. R. China
Revised Manuscript for Advances in Space Research
Running head:
Model-based boundary control with flexible appendages
Number of pages:
27
Number of tables:
1
Number of figures: 12 Postal address:
As stated above
E-mail:
[email protected] (for Junteng Ma) [email protected] (for Hao Wen) [email protected] (for Dongping Jin)
+Corresponding
author
Abstract
This paper investigates a boundary control scheme of a spacecraft with double flexible appendages under prescribed performance. The flexible spacecraft system comprises a rigid central hub and two flexible appendages regarded as continuum models, so that the motion of the system can be portrayed by using partial differential equations (PDEs). In this paper, only one control torque and two control forces are applied to guarantee the desired attitude angle of the spacecraft and simultaneously suppress the vibration of the two flexible appendages. Moreover, the angle tracking error of the spacecraft can be restricted in a small residual set under a minimum convergence rate by adopting the prescribed performance technique (PPT). The stability of the boundary control is analyzed by employing LaSalle’s invariance principle. Finally, the feasibility of the proposed controller is verified through numerical results. Keywords: Boundary control; Flexible spacecraft; Prescribed performance; Partial differential equation
2
1. Introduction Spacecraft has become an important symbol of the most challenging high-tech fields in space technology (Kane, 1980; Wertz and Larson, 2005). To enhance the communication performance and reduce energy expenditures, many spacecraft are equipped with long and lightweight flexible appendages such as antennas and solar arrays. Due to the sophisticated surroundings and the requirement of high accuracy, the rigid-flexible coupling dynamics of the spacecraft with elastic bodies have drawn the attention of many researchers (Chen and Wen, 2018). Commonly, the flexible spacecraft system is discretized into ordinary differential equations (ODEs) based on the lumped parameter method (LPM) (Chilan et al., 2017; Herber et al., 2014), the finite element method (FEM) (Jonker, 1990) or the assumed modes method (AMM) (Chen et al., 2018; Wen et al., 2017). Various studies for the control issues of the flexible spacecraft system have been extensively explored in recent years. For example, Nagashio et al. (2014) devised a robust controller to stabilize the Engineering Test Satellite VIII (ETS-VIII) spacecraft and performed successfully ground experiments verified by flight tests. Xiao et al. (2017) proposed a control scheme with nonlinear estimator to stabilize the attitude of flexible satellites with highaccuracy, and the estimator was applied to compensate for unknown flexible vibrations. Tayyebtaher and Esmaeilzadeh (2017) studied a model predictive control of a flexible satellite, in which the performance of the controller was optimized by the Genetic Algorithm. Due to the restriction of the physical characteristic and safety factors in practice, it is necessary to consider the common constraint condition of the control systems. In other words, serious instability may be caused by violations of the output constraints in physical instruments and control systems (Goodwin et al., 2006; Glattfelder and Schaufelberger, 2012). In automatic control, a faster convergence rate may require less time to stabilize the system. Accordingly, a new control scheme with prescribed performance was invented to
3
simultaneously bound the tracking errors and guarantee a minimum convergence rate (Bechlioulis and Rovithakis, 2008, 2009, 2011). Shahvali and Shojaei (2017) proposed a distributed neural adaptive control for networked uncertain Euler-Lagrange systems by using prescribed performance technique. Wei et al. (2018) investigated an adaptive model-free constrained prescribed performance control strategy for the flexible spacecraft to capture an unknown object. Hu et al. (2018) performed an adaptive fault-tolerant controller for a spacecraft based on the prescribed performance in the situation of uncertain parameters, unknown disturbances and input saturations. Wang et al. (2018) investigated an adaptive neural tracking control for an uncertain nonlinear system, and the prescribed performance function was utilized to ensure the transient and steady-state bounds. Nevertheless, the aforementioned studies on devising the controller were entirely formulated based on ODEs. However, the dynamic model based on ODEs is an approximate one with modal truncated errors, which may produce spillover instability. In recent years, significant interest has been associated with the original dynamic model-based PDEs, which is an infinite dimensional continuum (Ma et al., 2018). With the progress for the control scheme of the flexible spacecraft, it is more difficult to devise a composite PDE controller. He and Ge (2015) established the PDE dynamic model for a flexible satellite and designed a single-point control force to eliminate the vibration of two appendages. Rad et al. (2018a) addressed a PDE boundary control problem of a flexible satellite in planar motion and achieved the control goal by employing only one control torque in the absence of a damping effect. To track the objectives and eliminate the vibrations, Rad et al. (2018b) introduced the PDE boundary control for a flexible satellite on it two forces and one torque acting while ignoring damping. Considering the system output constraints, Meng et al. (2016) performed a boundary control of a flexible satellite based on PDEs, and a barrier Lyapunov function was introduced to restrict the output signals. Moreover, Liu and Liu (2017, 2018) creatively
4
introduced the prescribed performance technique into the boundary control for a flexible manipulator on the basis of the original PDEs. Cao and Liu (2018) studied the PDE boundary control issue for a rigid-flexible manipulator with prescribed performance. From the previous investigations based on PDE models, it can be seen that output constraints were considered in the PDE controller of the flexible spacecraft system, while the convergence rate was usually ignored. This paper focuses on the PDE boundary control of a flexible spacecraft system with a prescribed performance, where only one control torque and two boundary forces are employed to achieve attitude tracking and flexible vibration elimination. In this study, a smooth hyperbolic function is utilized to transform the system output errors into equivalent unlimited errors. With the prescribed performance method, the attitude tracking error can converge to a specified bounded value under a minimum convergence rate. Then, the asymptotic convergence of the boundary control is demonstrated using LaSalle’s invariance principle. Finally, the feasibility of the proposed boundary controller is illustrated by comparison with a PID controller in the numerical simulations. The remainder of this paper is organized as follows: In Section 2, the flexible spacecraft model is constructed based on the original PDEs, and the prescribed performance function is detailed. In Section 3, the controller of the flexible system is devised with prescribed performance, and the asymptotic convergence is verified. In Section 4, numerical cases demonstrate the advantage of the proposed controller. In Section 5, a summary of this paper is presented.
5
2. Problem Formulation 2.1 PDE modeling of the spacecraft with double flexible appendages As illustrated in Fig. 1, the spacecraft with double flexible appendages consists of the following parts: a rigid spacecraft and two Euler-Bernoulli beams. The inertial frame is denoted as XOY, and the central body is fixed at the origin O . The local frame xoy can describe the behavior of the flexible system. In addition, the elastic deflection of the flexible beam at position x for time t is denoted as w( x, t ) , and the rotation angle of the rigid body for time t is denoted as (t ) . The central control torque (t ) and boundary control force f (t ) are utilized to regulate the desired attitude angle and suppress the flexible vibration simultaneously when the spacecraft rotates slowly in a plane (Chen et al., 2019). In practice, the control force f (t ) can be provided by fixed thrusters at the endpoint of the flexible appendage (Rad et al., 2018b).
f (t )
r
o
O
(t )
y
X
(t )
w( x, t ) x
Y
f (t )
Fig. 1 Configuration of the spacecraft with double flexible appendages. Remark 1. For convenience, the symbol (t) will be omitted in the following discussion, such as (t ) and (t ) . Moreover, the symbols are introduced as follows:
() x () / x , () xx 2 () / x 2 , () xxx 3 () / x 3 , () xxxx 4 () / x 4 , ( ) () / t , ( ) 2 () / t 2 , ()(3) 3 () / t 3 , ()( n ) n () / t n , and n 1, , . 6
Assumption 1. The axial deformation and damping effect of the flexible appendages are not considered in this paper. Assumption 2. If the total kinetic energy of the system is bounded for t [0, ) , then and w ( x, t ) are bounded for ( x, t ) [0, l ] [0, ) (Queiroz et al., 2001). Lemma 1. Let ( x, t ) R with ( x, t ) [0, l ] [0, ) satisfy the condition (Hardy et al., 1959)
(0, t ) 0 then the following inequalities hold: l
2 ( x, t ) l x2 ( x, t )dx 0
If the boundary function x (0, t ) 0 holds true, then the following inequalities hold: l
x2 ( x, t ) l xx2 ( x, t )dx 0
Lemma 2. Let 1 ( x, t ) R and 2 ( x, t ) R be defined on ( x, t ) [0, l ] [0, ) , then the following inequalities hold (Rahn, 2001):
12 ( x , t ) 22 ( x , t ) 1 ( x , t ) 2 ( x , t )
12 ( x, t ) 22 ( x, t ) 1 ( x, t ) 2 ( x, t ) where is a positive constant. To facilitate the analysis, define the auxiliary variable z ( x, t ) ( x r ) w( x, t ) , where r is the radius of the central rigid body. According to the boundary condition at the origin o , the deflection w(0, t ) and the rotation angle wx (0, t ) are zero at an arbitrary time t. Then, subsequent conclusions can be derived, such as z (0, t ) r w(0, t ) r , z x (0, t ) , z x (0, t ) , and n z / x n n w / x n (n 2) .
7
In this paper, the PDE dynamic model of the spacecraft with double flexible appendages is established by adopting Hamilton’s principle
t2
t1
( E P W )dt 0
(1)
where () is the variation of () , and E , P and W are the variation in the kinetic energy, potential energy, and virtual work done by the nonconservative force, respectively (Liberzon, 2012). The total kinetic energy of the dynamic system is presented as
E
1 2 l 2 I h z ( x, t )dx 0 2
(2)
where I h is the rotational inertia of the central rigid body, is the mass per unit length of the flexible beams, and l is the length of each beam. Based on Assumption 1, the axial deformation of the flexible beam is neglected such that the potential energy of the beam can be obtained as l
P EIwxx2 ( x, t )dx 0
(3)
where EI denotes the flexural rigidity of the beams. Considering Assumption 1, the damping effect of the flexible beam is neglected and the virtual work is given by
W (t ) (t ) 2 f (t ) z (l , t )
(4)
Accordingly, the PDE dynamic model of the spacecraft with double flexible appendages has the following forms
z ( x, t ) EIz xxxx ( x, t ) 0
(5)
I h 2 EIz xx (0, t ) 2rEIz xxx (0, t ) (t ) 0
(6)
z xx (l , t ) 0 , f (t ) EIz xxx (l , t )
(7)
w(0, t ) wx (0, t ) 0
(8)
8
or ( x , t )] EIwxxxx ( x , t ) 0 [( x r ) w
(9)
I h 2 EIwxx (0, t ) 2 rEIwxxx (0, t ) (t ) 0
(10)
wxx (l , t ) 0 , f (t ) EIwxxx (l , t )
(11)
w(0, t ) wx (0, t ) 0
(12)
2.2 Statement of the prescribed performance technique To introduce the prescribed performance technique, a decay function (t ) is defined as an exponential form for R R . Moreover, the positive (t ) can converge to the ultimate limit
( 0) for t . In this section, a smooth hyperbolic function is proposed to transform the system tracking error into the equivalent unrestricted error so as to ensure transient and steady-state tracking error (Bechlioulis and Rovithakis, 2008, 2009, 2011). To limit the tracking error, the inequality is established as follows
(t ) e(t ) (t ) , t 0
(13)
where e(t ) denotes the output tracking error. In the above mathematical expression, (t ) is the decaying performance function and can be expressed as
(t ) 0 e t , t 0
(14)
where 0 0 , 0 , 0 (0) and lim (t ) for all t 0 . t
As shown in Fig. 2, the absolute value |e(t )| at an arbitrary time t can be regulated in the range between the upper bound (t ) and the lower bound (t ) , where (t ) is the prespecified performance index. In addition, the convergence rate of transient and steady-state tracking error e(t ) can be ensured to exceed the convergence rate of (t ) .
9
Decay function
0
e(0)
(t)
e(t)
(t) 0
Time (s) Fig. 2 Diagram of the prescribed performance technique. To transform the original system into an unrestricted one, an error transformation should be introduced as
e(t ) (t )T ( ) , t 0
(15)
where T ( ) tanh( ) and is the equivalent unconstrained tracking error. According to the property of the hyperbolic tangent function, one obtains
1 T ( ) 1 , t 0
(16)
Substituting Eq. (16) into Eq. (15) yields
(t ) e(t ) (t ) , t 0
(17)
Obviously, it is consistent with inequality (13). By observing Eq. (15), the inverse solution of T ( ) is
e(t ) 1 (t ) e(t ) , t 0 ln (t ) 2 (t ) e(t )
T 1
(18)
If can be guaranteed bounded for t 0 , Equation (13) will hold true. Differentiating Eq. (18) with respect to time gives
1 e e e e , t 0 2 e e 2 e2
10
(19)
3. Boundary control design and stability analysis In this section, the control scheme is devised to guarantee the desired attitude angle of the rigid body and simultaneously eliminate the vibration of flexible beams, that is, d ,
0 , w( x, t ) 0 and w ( x, t ) 0 . The prescribed performance technique will be introduced to ensure that the tracking error converges to a specified small range. In the boundary controller, the angle tracking error can be denoted as
e(t ) d , t 0
(20)
where d represents the desired attitude angle of the central rigid body, e(t ) , and
e(t ) . To analyze the stability of the dynamic model, a Lyapunov candidate function is constructed in the following form
V V1 V2 V3
(21)
where V1 denotes the sum of the kinetic energy and potential energy of the dynamic system,
V2 the tracking error index, and V3 the cross term, namely l
l
l
l
0
0
0
0
V1 z 2 ( x, t )dx EI wxx 2 ( x, t )dx z 2 ( x, t )dx EI z xx 2 ( x, t )dx V2
1 2 1 1 I h 1e 2 2 2 2 2 2
l
l
0
0
V3 I h ee 2 xz ( x, t )[e wx ( x, t )]dx 2 rz ( x, t )[e wx ( x, t )]dx where 1 0 , 2 0 and 0 . After applying Lemma 1 and 2, an upper bound can be obtained as follows
11
(22) (23) (24)
l 1 1 V3 I h e 2 I h e 2 2 l z 2 ( x, t )dx l 2 e 2 0 2 2 l
l
l
0
0
0
l 3 wxx2 ( x, t )dx 2 r z 2 ( x, t )dx rle 2 rl 2 wxx2 ( x, t )dx l 1 1 ( I h l 2 rl )e 2 I h e 2 2 (l r ) z 2 ( x, t )dx 0 2 2
(25)
l
( l 3 rl 2 ) wxx2 ( x, t )dx 0
I 2 l 2 2 rl l 3 rl 2 , , 2 (l r ), max h 1, EI 1
Letting
the
above
inequality can be rewritten as
V3 (V1 V2 )
(26)
As a result, V 0 . Then, the derivative of Eq. (21) with respect to time reads V V1 V2 V3
(27)
where l l V1 2 z ( x, t ) z ( x, t )dx 2 EI z xx ( x, t )zxx ( x, t )dx 0
0
l
l
0
0
2 EI z ( x, t )z xxxx ( x, t )dx 2 EI z xx ( x, t )zxx ( x, t )dx l l 2 EI z ( x, t )z xxxx ( x, t )dx 2 EI z xx ( x, t ) zx ( x, t ) 0 z xxx ( x, t )zx ( x, t )dx 0 0 l
2 EI z ( x, t )z xxxx ( x, t )dx 2 EI z xx ( x, t ) zx ( x, t ) 0 z xxx ( x, t ) z ( x, t ) 0 z xxxx ( x, t )z ( x, t )dx 0 0 l l 2 EI z xx ( x, t ) zx ( x, t ) 0 z xxx ( x, t ) z ( x, t ) 0 2 EIwxx (0, t ) (t ) 2 EIwxxx (l , t ) z (l , t ) 2rEIwxxx (0, t )(t ) l
l
ee V2 I h 1 2 I h ee 1ee 2
l
(28)
l
(29)
e[ 2 EIwxx (0, t ) 2rEIwxxx (0, t )] 1ee 2 V3 A1 A2 A3 A4 A5
(30)
A1 I h e 2 I h ee
(31)
where
12
l
A2 2 xz( x, t )[e wx ( x, t )]dx
(32)
0 l
A3 2 xz ( x, t )[e w x ( x, t )]dx
(33)
0
l
A4 2 rz( x, t )[e wx ( x, t )]dx
(34)
0 l
A5 2 rz ( x, t )[e w x ( x, t )]dx
(35)
0
First, Eq. (32) can be expressed as l
A2 2 EI xz xxxx ( x, t )[e wx ( x, t )]dx 0
l
l
2 EI xz xxxx ( x, t )edx 2 EI xz xxxx ( x, t ) wx ( x, t )dx 0
0
2 EIe xz xxx ( x, t ) 0 z xxx ( x, t )dx 0 l
l
l l l 2 EI xwx ( x, t ) wxxx ( x, t ) 0 wx ( x, t ) wxxx ( x, t )dx xwxx ( x, t ) wxxx ( x, t )dx 0 0 3 l 2 EIe lwxxx (l , t ) wxx (0, t ) 2 EI lwx (l , t ) wxxx (l , t ) wxx2 ( x, t )dx 2 0
(36)
l
2 EIlewxxx (l , t ) 2 EIewxx (0, t ) 2 EIlwx (l , t ) wxxx (l , t ) 3 EI wxx2 ( x, t )dx 0
Second, Eq. (33) can be derived as l
A3 2 xz ( x, t ) zx ( x, t )dx 0
l l l 2 xz 2 ( x, t ) z 2 ( x, t )dx xz ( x, t ) zx ( x, t )dx 0 0 0 l
l
2 lz (l , t ) 2 z ( x, t )dx 2 xz ( x, t ) zx ( x, t )dx 2
2
0
(37)
0
l
lz 2 (l , t ) z 2 ( x, t )dx 0
Third, Eq. (34) can be obtained as l
A4 2 EI rz xxxx ( x, t )[e wx ( x, t )]dx 0
l
l
0
0
2 EI rz xxxx ( x, t )edx 2 EI rz xxxx ( x, t ) wx ( x, t )dx l l l 2 EIre z xxx ( x, t ) 0 2 EIr wx ( x, t ) wxxx ( x, t ) 0 wxx ( x, t ) wxxx ( x, t )dx (38) 0 l 1 2 EIre wxxx (l , t ) wxxx (0, t ) 2 EIr wx (l , t ) wxxx (l , t ) wxx2 ( x, t ) 0 2 2 2 EIrewxxx (l , t ) 2 EIrewxxx (0, t ) 2 EIrwx (l , t ) wxxx (l , t ) EIrwxx (0, t )
13
Fourth, Eq. (35) can be expressed as l
l
0
0
A5 2 rz ( x, t ) zx ( x, t )dx r z 2 ( x, t ) rz 2 (l , t ) rz 2 (0, t )
(39)
According to Eq. (10), one has
V 2 EIwxx (0, t )(t ) 2 EIwxxx (l , t ) z (l , t ) 2rEIwxxx (0, t )(t )
e[ 2 EIwxx (0, t ) 2rEIwxxx (0, t )] 1ee 2 I h e 2
e[ 2 EIwxx (0, t ) 2rEIwxxx (0, t )] 2 EI (l r )ewxxx (l , t )
(40)
l
2 EIewxx (0, t ) 2 EI (l r ) wx (l , t ) wxxx (l , t ) 3 EI wxx2 ( x, t )dx 0
l
(l r ) z 2 (l , t ) z 2 ( x, t )dx 2 EIrewxxx (0, t ) EIrwxx2 (0, t ) rz 2 (0, t ) 0
Lemma 3. Given two constants 3 0 and 4 0 , the control strategy in this paper is defined as follows
(t ) 1e 2
e2 2
3e
f (t ) 4 z (l , t )
(41) (42)
Under the initial condition 0 |e(0)| (0) , the tracking error can be constrained within a prespecified range that exponentially decays with time. Additionally, the boundary controller with the prescribed performance is asymptotically stable: when t , one has d ,
0 , w( x, t ) 0 and w ( x, t ) 0 . Proof: With control law (41) and (42), the derivative of Lyapunov candidate function can be rewritten as l e V ( I h 3 )e 2 ( ) 2 2 2 1 e 2 3 ee 3 EI wxx2 ( x, t )dx 0 e l
z 2 ( x, t )dx (l r ) z 2 (l , t ) 2 4 z 2 (l , t ) 2 4 (l r ) wx (l , t ) z (l , t ) 0
(43)
2 4 (l r )ez (l , t ) rEIwxx2 (0, t ) rz 2 (0, t )
Based on subsequent inequalities 3 ee
3e 2 31e 2 1 14
(44)
2 4 (l r ) wx (l , t ) z (l , t ) 2 4 (l r ) 2 wx2 (l , t ) 2 4 (l r ) 2 4 (l r )ez (l , t ) 2 4 (l r ) 3e 2 2 4 (l r )
z 2 (l , t )
2
z 2 (l , t )
3
l wx2 (l , t ) 3 EI w ( x, t )dx EI 2 EI wxx2 ( x, t )dx 0 0 l l
2 xx
(45)
(46)
(47)
where 1 , 2 and 3 are positive constants. Hence, the following conclusion can be obtained
e V ( I h 3 31 )e 2 ( ) 2 2 2 [ 3 1 2 4 (l r ) 3 ]e 2 e 1 l
l
2 EI wxx2 ( x, t )dx z 2 ( x, t )dx 0
[2 4
0
2 4 (l r )
2
[2 4 (l r ) 2
EI l
2 4 (l r )
3
(l r )]z 2 (l , t )
(48)
]wx2 (l , t ) rEIwxx2 (0, t ) rz 2 (0, t )
By choosing parameters , 1 , 3 , 4 , 1 , 2 and 3 to satisfy the following conditions
1 I h 3 31 0 2 0
3 4 2 4
3 1 2 4 (l r ) 3 0 1
2 4 (l r )
2
2 4 (l r )
3
5 2 4 (l r ) 2
EI l
(l r ) 0
0
And noting that e 0 , one has
e V 1e 2 2 2 2 2 3e 2 4 z 2 (l , t ) 0 e
15
(49)
LaSalle’s invariance principle is adopted to analyze the stability of the boundary control. When V 0 , it can be concluded through Eq. (49) that
e e z (l , t ) 0
(50)
z (l , t ) 0 e
(51)
z xxxx (l , t ) wxxxx (l , t ) 0
(52)
Then
With Eqs. (51) and (5), one has
Substituting Eq. (51) into Eq. (9) yields
( x, t ) EIwxxxx ( x, t ) 0 w
(53)
Using the variable separation method, the solution of Eq. (53) can be derived as follows
w( x, t ) W ( x)T (t ) ( x, t ) EIwxxxx ( x, t ) 0 wxxxx ( x, t ) w
(54)
EI
( x, t ) w
( x, t ) W ( x) T(t ) wxxxx ( x, t ) Wxxxx ( x) T (t ), w W ( x) T(t ) xxxx EI T (t ) W ( x) Wxxxx ( x) W ( x) 0
(55)
(56)
By setting s 4 , the solution of Eq. (56) arrives at
W ( x) c1 cosh sx c2 sinh sx c3 cos sx c4 sin sx
(57)
By combining Eq. (52) with Eqs. (11) and (12), the following results can be obtained
W (0) Wx (0) Wxx (l ) Wxxxx (l ) 0
(58)
Then, a set of equations is established as
c1 c3 0 c c 0 2 4 c1 cosh sl c2 sinh sl c3 cos sl c4 sin sl 0 c1 cosh sl c2 sinh sl c3 cos sl c4 sin sl 0 After a series of operations, Equation (59) can be simplified as 16
(59)
c4 (sinh sl cos sl sin sl cosh sl ) 0
(60)
with the solutions being ci 0 (i 1, 2,3, 4) for all s. Therefore, one has W ( x) 0 , w( x, t ) 0 , and w ( x, t ) 0 . Accordingly, the PDE boundary control with a prescribed performance in this paper is asymptotically stable by utilizing the extended LaSalle’s invariance principle; that is, when t , one has d , 0 , w( x, t ) 0 and w ( x, t ) 0 . Since V 0 , it is clear that V
will be bounded. Based on Assumption 2 and Eqs. (21)-(24), it can be found that all the other terms in V are bounded except the term containing 2 . As a consequence, is bounded for
t 0 and then the system tracking error can be constrained within a prespecified range, which is (t ) e(t ) (t ) for t 0 .
4. Case Studies In this section, some case studies are illustrated to verify the feasibility of the proposed boundary control with prescribed performance. The structural parameters of the spacecraft with double flexible appendages are listed in Table 1. The numerical results under the following
PID
control
strategy
are
also
included
for
comparison
purposes:
t
(t ) k1e k2 e k3 edt ( t 0 ), where k1 , k2 and k3 denote the positive gain factors. 0 Furthermore, the proposed control is evaluated under the effect of parameter uncertainties, sensor noise and time delay (UND). In UND cases, the controller design is still based on the structural parameters in Table 1 but dynamic simulation uses the following ‘true’ parameters: the length of each beam is 6 m, the flexural stiffness of the beam is 144 Nm2 and the rotational inertia of the central rigid body is 320 kgm2. In addition, the sensor noise and the time delay in the torque input are supposed to be 0.001sin(0.1t ) rad and 0.1 s, where the delayed control torque is not applied until t 0.1 s . 17
Table 1 Structural parameters of the spacecraft. Parameters
Values
Length of each beam, l
5m
Radius of the rigid body, r
0.5m
Flexural stiffness of the beam, EI
120Nm2
Mass per unit length,
20kg/m
Inertia of the central rigid body, Ih
400kgm2
The aim of all the simulations is to regulate the rotation attitude angle of the rigid body to the specified value, d 0.5 rad , with no any residual vibration of the elastic appendages. To constrain the tracking error of the attitude angle, the absolute value |e(t )| of no more than
(t ) and the minimum convergence rate are permitted when 0.05 in Eq. (14). Once the parameters are properly adjusted in the prescribed performance function, the transient and steady-state output error can be guaranteed bounded. Finally, the attitude angle error and transverse vibration of the beams can converge to zero with the proposed controller. The case studies are investigated in below. Case 1: The central rigid body is actuated by a control torque under the PID control scheme with k1 25 , k2 115 and k3 0.5 . The parameters in the prescribed performance function were set at 0 0.6 and 0.02 . The coefficients in the controller were selected as 1 20 , 2 0.1 , 3 250 and 4 1 . Figures 3 and 4 indicate the rotation attitude angle and angular velocity responses of the spacecraft. One can see that under the proposed control scheme, the rotation angle reaches the desired value and the angular velocity decreases to zero within 160 seconds, whereas the angular responses under the PID controller is still significant at the end of simulation. Besides, the vibration at the tip of the beam ( x l ) with the proposed control reduces more rapidly than that with the PID controller as indicated in Fig. 5. Figure 6 presents the time histories of the rotation attitude angle error of the rigid body by employing the proposed control with and 18
without UND. Nevertheless, it can be seen that the tracking error falls within the predefined range and asymptotically converges to zero. Thus, the conclusion |e(t )| (t ) holds true. Further, the deformation of each beam gradually reduces as shown in Fig. 7. Comparatively, as shown in Fig. 6, it is worth noting that the angle tracking error of the rigid body with the PID control has a larger amplitude exceeding outside the prespecified bounds (red dash lines shown in the figure). Although the PID controller can eliminate the steady state error with the help of integral, it may increase the system response time and make a larger error amplitude. 1.0
(Proposed) d
0.8
Angle tracking (rad)
Angle tracking (rad)
1.0
(PID)
0.6 0.4 0.2 0.0
0
20
40
60
80
(PID)
0.6 0.4 0.2 0.0
100 120 140 160 180 200
(Proposed) d
0.8
0
20
40
60
80
100 120 140 160 180 200
Time (s)
Time (s)
(a)
(b)
0.06
(Proposed) (PID)
0.04 0.02 0.00 -0.02 -0.04
0
20
40
60
80
100 120 140 160 180 200
Angular velocity response (rad/s)
Angular velocity response (rad/s)
Fig. 3 Angle tracking under (a) the PID and proposed controllers without UND and (b) the PID and proposed controllers with UND.
Time (s)
0.06
(Proposed) (PID)
0.04 0.02 0.00 -0.02 -0.04
0
20
40
60
80
100 120 140 160 180 200
Time (s)
(a)
(b)
Fig. 4 Angular velocity response under (a) the PID and proposed controllers without UND and (b) the PID and proposed controllers with UND.
19
0.4
w(l,t) (Proposed) w(l,t) (PID)
Deflection at the tip (m)
Deflection at the tip (m)
0.2
0.0
-0.2
-0.4
0.0 -0.2 -0.4 -0.6
20
0
40
60
w(l,t) (Proposed) w(l,t) (PID)
0.2
80 100 120 140 160 180 200
0
20
40
60
Time (s)
80 100 120 140 160 180 200
Time (s)
(a)
(b)
Fig. 5 Deflection at the tip ( x l ) under (a) the PID and proposed controllers without UND and (b) the PID and proposed controllers with UND. 0.6
e(t) (Proposed) (t) e(t) (PID)
0.4 0.2
Angle tracking error (rad)
Angle tracking error (rad)
0.6
0.0 -0.2 -0.4 -0.6
0
20
40
60
80
100 120 140 160 180 200
e(t) (Proposed) (t) e(t) (PID)
0.4 0.2 0.0 -0.2 -0.4 -0.6
0
20
40
60
80 100 120 140 160 180 200
Time (s)
Time (s)
(a)
(b)
Fig. 6 Angle tracking error under (a) the PID and proposed controllers without UND and (b) the PID and proposed controllers with UND.
(a)
20
(b) Fig. 7 Deformation of the flexible appendages with (a) the PID control and (b) the proposed control. Case 2: Assume that the parameters in the prescribed performance function are identical to those in Case 1, the gain coefficients of PID and the proposed controller were set at k1 25 ,
k2 750 , k3 1 , 1 25 , 2 0.1 , 3 400 and 4 1 . As shown in Figs. 8 and 9, the rotation attitude angle achieves the desired value, and the angular velocity smoothly converges to zero under the proposed control scheme, which indicates a better performance with larger control gains. The deflection at the tip of the beam ( x l ) is shown in Fig. 10. One can see from Fig. 10 that the deflection gradually decreases to zero. However, the tracking error under PID control goes beyond the prescribed bounds [red dash lines in Fig. 11] even when a higher control gain is applied. In contrast, the scenarios do not occur under the proposed control with and without UND as shown in Fig. 11. Similarly, one can see that the vibration at the tip of each beam gradually decreases to zero, while the deflection of which is eliminated after a short time, as shown in Fig. 12.
21
0.8
(Proposed) d
0.6
Angle tracking (rad)
Angle tracking (rad)
0.8
(PID)
0.4 0.2 0.0
0
20
40
60
80
100 120 140 160 180 200
(Proposed) d
0.6
(PID)
0.4 0.2 0.0
0
20
40
60
80 100 120 140 160 180 200
Time (s)
Time (s)
(a)
(b)
0.03
(Proposed) (PID)
0.02 0.01 0.00 -0.01
0
20
40
60
80 100 120 140 160 180 200
Angular velocity response (rad/s)
Angular velocity response (rad/s)
Fig. 8 Angle tracking under (a) the PID and proposed controllers without UND and (b) the PID and proposed controllers with UND. 0.03
(Proposed) (PID)
0.02 0.01 0.00 -0.01
0
20
40
60
80 100 120 140 160 180 200
Time (s)
Time (s)
(a)
(b)
Fig. 9 Angular velocity response under (a) the PID and proposed controllers without UND and (b) the PID and proposed controllers with UND. 0.2 w(l,t) (Proposed) w(l,t) (PID)
Deflection at the tip (m)
Deflection at the tip (m)
0.2
0.0
-0.2
-0.4
w(l,t) (Proposed) w(l,t) (PID) 0.0
-0.2
-0.4
0
20
40
60
80 100 120 140 160 180 200
Time (s)
0
20
40
60
80
100 120 140 160 180 200
Time (s)
(a)
(b)
Fig. 10 Deflection at the tip ( x l ) under (a) the PID and proposed controllers without UND and (b) the PID and proposed controllers with UND.
22
0.6 Angle tracking error (rad)
Angle tracking error (rad)
0.6
e(t) (Proposed) (t) e(t) (PID)
0.4 0.2 0.0 -0.2 -0.4 -0.6
0
20
40
60
80 100 120 140 160 180 200
e(t) (Proposed)
0.4
(t)
e(t) (PID)
0.2 0.0 -0.2 -0.4 -0.6
0
20
40
60
80
100 120 140 160 180 200
Time (s)
Time (s)
(a)
(b)
Fig. 11 Angle tracking error under (a) the PID and proposed controllers without UND and (b) the PID and proposed controllers with UND.
(a)
(b) Fig. 12 Deformation of the flexible appendages with (a) the PID control and (b) the proposed control.
23
5. Conclusions The motion of a spacecraft with one central rigid body and two flexible appendages is described by PDEs. Only one control torque and two boundary forces can simultaneously regulate the rotational attitude angle and eliminate the vibration of the flexible appendages. By utilizing the prescribed performance technique, the output error of the dynamic system can be restricted to a predefined range, and the convergence rate can be ensured to be greater than a prespecified value. The stability of the boundary control is demonstrated using the extended LaSalle’s invariance principle, with the performance of the proposed boundary controller being verified through numerical simulations. Competing Interests The authors declare that there is no conflict of interest regarding the publication of this paper. Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant no. 11832005) and the Research Fund of State Key Laboratory of Mechanics and Control of Mechanical Structures (Grant no. NUAA MCMS-0118G01).
References Bechlioulis, C.P., Rovithakis, G.A., 2008. Robust adaptive control of feedback linearizable MIMO nonlinear systems with prescribed performance. IEEE Trans. Autom. Control 53(9), 2090-2099. Bechlioulis, C.P., Rovithakis, G.A., 2009. Adaptive control with guaranteed transient and steady state tracking error bounds for strict feedback systems. Automatica 45(2), 532-538.
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Bechlioulis, C.P., Rovithakis, G.A., 2011. Robust partial-state feedback prescribed performance control of cascade systems with unknown nonlinearities. IEEE Trans. Autom. Control 56(9), 2224-2230. Chilan, C.M., Herber, D.R., Nakka, Y.K., Chung, S.J., Allison, J.T., Aldrich, J.B., AlvarezSalazar, O.S., 2017. Co-design of strain-actuated solar arrays for spacecraft precision pointing and jitter reduction. AIAA J. 55(9), 3180-3195. Cao, F.F., Liu J.K., 2018. Boundary control for a constrained two-link rigid-flexible manipulator with prescribed performance. Int. J. Control 91(5), 1091-1103. Chen, T., Shan, J., Ramkumar, G., 2018. Distributed adaptive control for multiple underactuated Lagrangian systems under fixed or switching topology. Nonlinear Dyn. 93(3), 1705-1718. Chen, T., Wen, H., 2018. Autonomous assembly with collision avoidance of a fleet of flexible spacecraft based on disturbance observer. Acta Astronaut. 147, 86-96. Chen, T., Wen, H., Wei, Z.T., 2019. Distributed attitude tracking for multiple flexible spacecraft described by partial differential equations. Acta Astronaut. 159, 637-645. Goodwin, G., Seron, M.M., De, Doná, J.A., 2006. Constrained Control and Estimation: An Optimisation Approach. Springer Science & Business Media, USA. Glattfelder, A.H., Schaufelberger, W., 2012. Control Systems with Input and Output Constraints. Springer Science & Business Media, London. Hardy, G.H., Littlewood, J.E., Polya, G., 1959. Inequalities. Cambridge University Press, Cambridge, Massachusetts. Herber, D.R., McDonald, J.W., Alvarez-Salazar, O.S., Krishnan, G., Allison, J.T., 2014. Reducing spacecraft jitter during satellite reorientation maneuvers via solar array dynamics. The 15th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Atlanta, GA, USA.
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He, W., Ge, S.S., 2015. Dynamic modeling and vibration control of a flexible satellite. IEEE Trans. Aerosp. Electron. Syst. 51(2), 1422-1431. Hu, Q., Shao, X., Guo, L., 2018. Adaptive fault-tolerant attitude tracking control of spacecraft with prescribed performance. IEEE-ASME Trans. Mechatron. 23(1), 331-341. Jonker, B., 1990. A finite element dynamic analysis of flexible manipulators. Int. J. Robot. Res. 9(4), 59-74. Kane, T.R., Levinson, D.A., 1980. Formulation of equations of motion for complex spacecraft. J. Guid. Control Dyn. 3(2), 99-112. Liberzon, D., 2012. Calculus of Variations and Optimal Control Theory: A Concise Introduction. Princeton University Press, New Jersey, USA. Liu, Z.J., Liu, J.K., 2017. Boundary control of a flexible robotic manipulator with output constraints. Asian J. Control 19(1), 332-345. Liu, Z.J., Liu, J.K., 2018. Adaptive iterative learning boundary control of a flexible manipulator with guaranteed transient performance. Asian J. Control 20(3), 1027-1038. Meng, T.T., He, W., Yang, H., Liu, J.K., You, W., 2016. Vibration control for a flexible satellite system with output constraints. Nonlinear Dyn. 85(4), 2673-2686. Ma, J.T., Jin, D.P., Wei, Z.T., Chen, T., Wen, H., 2018. Boundary control of a flexible manipulator based on a high order disturbance observer with input saturation. Shock Vib. Volume 2018, Article ID 2086424, 10 pages. Nagashio, T., Kida, T., Hamada, Y., Ohtani, T., 2014. Robust two-degrees-of-freedom attitude controller design and flight test result for engineering test satellite-VIII spacecraft. IEEE Trans. Control Syst. Technol. 22(1), 157-168. Queiroz, M.S.D., Dawson, D.M., Nagarkatti, S.P., Zhang, F.M., Bentsman, J., 2001. Lyapunov-based control of mechanical systems. Appl. Mech. Rev. 54(5), B81.
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Rahn, C.D., 2001. Mechatronic Control of Distributed Noise and Vibration: A Lyapunov Approach. Springer Verlag, Berlin. Rad, H.K., Salarieh, H., Alasty, A., Vatankhah, R., 2018a. Boundary control of antisymmetric vibration of satellite with flexible appendages in planar motion with exponential stability. Acta Astronaut. 147, 219-230. Rad, H.K., Salarieh, H., Alasty, A., Vatankhah, R., 2018b. Boundary control of flexible satellite vibration in planar motion. J. Sound Vibr. 432, 549-568. Shahvali, M., Shojaei, K., 2017. Distributed control of networked uncertain Euler–Lagrange systems in the presence of stochastic disturbances: a prescribed performance approach. Nonlinear Dyn. 90(1), 697-715. Tayyebtaher, M., Esmaeilzadeh, S.M., 2017. Model predictive control of attitude maneuver of geostationary flexible satellite based on genetic algorithm. Adv. Space Res. 60(1), 57-64. Wertz, J.R., Larson, W.J., 2005. Space Mission Analysis and Design (Space Technology Library). Microcosm Press and Kluwer Academic Publishers, El Segundo, CA, USA. Wen, H., Chen, T., Jin, D.P., Hu, H.Y., 2017. Passivity-based control with collision avoidance for a hub-beam spacecraft. Adv. Space Res. 59(1), 425-433. Wang, X., Yin, X., Shen, F., 2018. Disturbance observer based adaptive neural prescribed performance control for a class of uncertain nonlinear systems with unknown backlashlike hysteresis. Neurocomputing 299, 10-19. Wei, C.S., Luo, J.J., Dai, H.H., Yuan, J.P., 2018. Adaptive model-free constrained control of postcapture flexible spacecraft: a Euler–Lagrange approach. J. Vib. Control 24(20), 48854903. Xiao, B., Yin, S., Kaynak, O., 2017. Attitude stabilization control of flexible satellites with high accuracy: an estimator-based approach. IEEE-ASME Trans. Mechatron. 22(1), 349358.
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S0273-1177(19)30728-8 https://doi.org/10.1016/j.asr.2019.09.050 JASR 14475
To appear in:
Advances in Space Research
Received Date: Revised Date: Accepted Date:
29 March 2019 16 September 2019 29 September 2019
Please cite this article as: Ma, J., Wen, H., Jin, D., PDE Model-Based Boundary Control of a Spacecraft with Double Flexible Appendages under Prescribed Performance, Advances in Space Research (2019), doi: https:// doi.org/10.1016/j.asr.2019.09.050
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PDE Model-Based Boundary Control of a Spacecraft with Double Flexible Appendages under Prescribed Performance Junteng Ma, Hao Wen+, Dongping Jin State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, 210016 Nanjing, P. R. China
Revised Manuscript for Advances in Space Research
Running head:
Model-based boundary control with flexible appendages
Number of pages:
27
Number of tables:
1
Number of figures: 12 Postal address:
As stated above
E-mail:
[email protected] (for Junteng Ma) [email protected] (for Hao Wen) [email protected] (for Dongping Jin)
+Corresponding
author
Abstract
This paper investigates a boundary control scheme of a spacecraft with double flexible appendages under prescribed performance. The flexible spacecraft system comprises a rigid central hub and two flexible appendages regarded as continuum models, so that the motion of the system can be portrayed by using partial differential equations (PDEs). In this paper, only one control torque and two control forces are applied to guarantee the desired attitude angle of the spacecraft and simultaneously suppress the vibration of the two flexible appendages. Moreover, the angle tracking error of the spacecraft can be restricted in a small residual set under a minimum convergence rate by adopting the prescribed performance technique (PPT). The stability of the boundary control is analyzed by employing LaSalle’s invariance principle. Finally, the feasibility of the proposed controller is verified through numerical results. Keywords: Boundary control; Flexible spacecraft; Prescribed performance; Partial differential equation
2
1. Introduction Spacecraft has become an important symbol of the most challenging high-tech fields in space technology (Kane, 1980; Wertz and Larson, 2005). To enhance the communication performance and reduce energy expenditures, many spacecraft are equipped with long and lightweight flexible appendages such as antennas and solar arrays. Due to the sophisticated surroundings and the requirement of high accuracy, the rigid-flexible coupling dynamics of the spacecraft with elastic bodies have drawn the attention of many researchers (Chen and Wen, 2018). Commonly, the flexible spacecraft system is discretized into ordinary differential equations (ODEs) based on the lumped parameter method (LPM) (Chilan et al., 2017; Herber et al., 2014), the finite element method (FEM) (Jonker, 1990) or the assumed modes method (AMM) (Chen et al., 2018; Wen et al., 2017). Various studies for the control issues of the flexible spacecraft system have been extensively explored in recent years. For example, Nagashio et al. (2014) devised a robust controller to stabilize the Engineering Test Satellite VIII (ETS-VIII) spacecraft and performed successfully ground experiments verified by flight tests. Xiao et al. (2017) proposed a control scheme with nonlinear estimator to stabilize the attitude of flexible satellites with highaccuracy, and the estimator was applied to compensate for unknown flexible vibrations. Tayyebtaher and Esmaeilzadeh (2017) studied a model predictive control of a flexible satellite, in which the performance of the controller was optimized by the Genetic Algorithm. Due to the restriction of the physical characteristic and safety factors in practice, it is necessary to consider the common constraint condition of the control systems. In other words, serious instability may be caused by violations of the output constraints in physical instruments and control systems (Goodwin et al., 2006; Glattfelder and Schaufelberger, 2012). In automatic control, a faster convergence rate may require less time to stabilize the system. Accordingly, a new control scheme with prescribed performance was invented to
3
simultaneously bound the tracking errors and guarantee a minimum convergence rate (Bechlioulis and Rovithakis, 2008, 2009, 2011). Shahvali and Shojaei (2017) proposed a distributed neural adaptive control for networked uncertain Euler-Lagrange systems by using prescribed performance technique. Wei et al. (2018) investigated an adaptive model-free constrained prescribed performance control strategy for the flexible spacecraft to capture an unknown object. Hu et al. (2018) performed an adaptive fault-tolerant controller for a spacecraft based on the prescribed performance in the situation of uncertain parameters, unknown disturbances and input saturations. Wang et al. (2018) investigated an adaptive neural tracking control for an uncertain nonlinear system, and the prescribed performance function was utilized to ensure the transient and steady-state bounds. Nevertheless, the aforementioned studies on devising the controller were entirely formulated based on ODEs. However, the dynamic model based on ODEs is an approximate one with modal truncated errors, which may produce spillover instability. In recent years, significant interest has been associated with the original dynamic model-based PDEs, which is an infinite dimensional continuum (Ma et al., 2018). With the progress for the control scheme of the flexible spacecraft, it is more difficult to devise a composite PDE controller. He and Ge (2015) established the PDE dynamic model for a flexible satellite and designed a single-point control force to eliminate the vibration of two appendages. Rad et al. (2018a) addressed a PDE boundary control problem of a flexible satellite in planar motion and achieved the control goal by employing only one control torque in the absence of a damping effect. To track the objectives and eliminate the vibrations, Rad et al. (2018b) introduced the PDE boundary control for a flexible satellite on it two forces and one torque acting while ignoring damping. Considering the system output constraints, Meng et al. (2016) performed a boundary control of a flexible satellite based on PDEs, and a barrier Lyapunov function was introduced to restrict the output signals. Moreover, Liu and Liu (2017, 2018) creatively
4
introduced the prescribed performance technique into the boundary control for a flexible manipulator on the basis of the original PDEs. Cao and Liu (2018) studied the PDE boundary control issue for a rigid-flexible manipulator with prescribed performance. From the previous investigations based on PDE models, it can be seen that output constraints were considered in the PDE controller of the flexible spacecraft system, while the convergence rate was usually ignored. This paper focuses on the PDE boundary control of a flexible spacecraft system with a prescribed performance, where only one control torque and two boundary forces are employed to achieve attitude tracking and flexible vibration elimination. In this study, a smooth hyperbolic function is utilized to transform the system output errors into equivalent unlimited errors. With the prescribed performance method, the attitude tracking error can converge to a specified bounded value under a minimum convergence rate. Then, the asymptotic convergence of the boundary control is demonstrated using LaSalle’s invariance principle. Finally, the feasibility of the proposed boundary controller is illustrated by comparison with a PID controller in the numerical simulations. The remainder of this paper is organized as follows: In Section 2, the flexible spacecraft model is constructed based on the original PDEs, and the prescribed performance function is detailed. In Section 3, the controller of the flexible system is devised with prescribed performance, and the asymptotic convergence is verified. In Section 4, numerical cases demonstrate the advantage of the proposed controller. In Section 5, a summary of this paper is presented.
5
2. Problem Formulation 2.1 PDE modeling of the spacecraft with double flexible appendages As illustrated in Fig. 1, the spacecraft with double flexible appendages consists of the following parts: a rigid spacecraft and two Euler-Bernoulli beams. The inertial frame is denoted as XOY, and the central body is fixed at the origin O . The local frame xoy can describe the behavior of the flexible system. In addition, the elastic deflection of the flexible beam at position x for time t is denoted as w( x, t ) , and the rotation angle of the rigid body for time t is denoted as (t ) . The central control torque (t ) and boundary control force f (t ) are utilized to regulate the desired attitude angle and suppress the flexible vibration simultaneously when the spacecraft rotates slowly in a plane (Chen et al., 2019). In practice, the control force f (t ) can be provided by fixed thrusters at the endpoint of the flexible appendage (Rad et al., 2018b).
f (t )
r
o
O
(t )
y
X
(t )
w( x, t ) x
Y
f (t )
Fig. 1 Configuration of the spacecraft with double flexible appendages. Remark 1. For convenience, the symbol (t) will be omitted in the following discussion, such as (t ) and (t ) . Moreover, the symbols are introduced as follows:
() x () / x , () xx 2 () / x 2 , () xxx 3 () / x 3 , () xxxx 4 () / x 4 , ( ) () / t , ( ) 2 () / t 2 , ()(3) 3 () / t 3 , ()( n ) n () / t n , and n 1, , . 6
Assumption 1. The axial deformation and damping effect of the flexible appendages are not considered in this paper. Assumption 2. If the total kinetic energy of the system is bounded for t [0, ) , then and w ( x, t ) are bounded for ( x, t ) [0, l ] [0, ) (Queiroz et al., 2001). Lemma 1. Let ( x, t ) R with ( x, t ) [0, l ] [0, ) satisfy the condition (Hardy et al., 1959)
(0, t ) 0 then the following inequalities hold: l
2 ( x, t ) l x2 ( x, t )dx 0
If the boundary function x (0, t ) 0 holds true, then the following inequalities hold: l
x2 ( x, t ) l xx2 ( x, t )dx 0
Lemma 2. Let 1 ( x, t ) R and 2 ( x, t ) R be defined on ( x, t ) [0, l ] [0, ) , then the following inequalities hold (Rahn, 2001):
12 ( x , t ) 22 ( x , t ) 1 ( x , t ) 2 ( x , t )
12 ( x, t ) 22 ( x, t ) 1 ( x, t ) 2 ( x, t ) where is a positive constant. To facilitate the analysis, define the auxiliary variable z ( x, t ) ( x r ) w( x, t ) , where r is the radius of the central rigid body. According to the boundary condition at the origin o , the deflection w(0, t ) and the rotation angle wx (0, t ) are zero at an arbitrary time t. Then, subsequent conclusions can be derived, such as z (0, t ) r w(0, t ) r , z x (0, t ) , z x (0, t ) , and n z / x n n w / x n (n 2) .
7
In this paper, the PDE dynamic model of the spacecraft with double flexible appendages is established by adopting Hamilton’s principle
t2
t1
( E P W )dt 0
(1)
where () is the variation of () , and E , P and W are the variation in the kinetic energy, potential energy, and virtual work done by the nonconservative force, respectively (Liberzon, 2012). The total kinetic energy of the dynamic system is presented as
E
1 2 l 2 I h z ( x, t )dx 0 2
(2)
where I h is the rotational inertia of the central rigid body, is the mass per unit length of the flexible beams, and l is the length of each beam. Based on Assumption 1, the axial deformation of the flexible beam is neglected such that the potential energy of the beam can be obtained as l
P EIwxx2 ( x, t )dx 0
(3)
where EI denotes the flexural rigidity of the beams. Considering Assumption 1, the damping effect of the flexible beam is neglected and the virtual work is given by
W (t ) (t ) 2 f (t ) z (l , t )
(4)
Accordingly, the PDE dynamic model of the spacecraft with double flexible appendages has the following forms
z ( x, t ) EIz xxxx ( x, t ) 0
(5)
I h 2 EIz xx (0, t ) 2rEIz xxx (0, t ) (t ) 0
(6)
z xx (l , t ) 0 , f (t ) EIz xxx (l , t )
(7)
w(0, t ) wx (0, t ) 0
(8)
8
or ( x , t )] EIwxxxx ( x , t ) 0 [( x r ) w
(9)
I h 2 EIwxx (0, t ) 2 rEIwxxx (0, t ) (t ) 0
(10)
wxx (l , t ) 0 , f (t ) EIwxxx (l , t )
(11)
w(0, t ) wx (0, t ) 0
(12)
2.2 Statement of the prescribed performance technique To introduce the prescribed performance technique, a decay function (t ) is defined as an exponential form for R R . Moreover, the positive (t ) can converge to the ultimate limit
( 0) for t . In this section, a smooth hyperbolic function is proposed to transform the system tracking error into the equivalent unrestricted error so as to ensure transient and steady-state tracking error (Bechlioulis and Rovithakis, 2008, 2009, 2011). To limit the tracking error, the inequality is established as follows
(t ) e(t ) (t ) , t 0
(13)
where e(t ) denotes the output tracking error. In the above mathematical expression, (t ) is the decaying performance function and can be expressed as
(t ) 0 e t , t 0
(14)
where 0 0 , 0 , 0 (0) and lim (t ) for all t 0 . t
As shown in Fig. 2, the absolute value |e(t )| at an arbitrary time t can be regulated in the range between the upper bound (t ) and the lower bound (t ) , where (t ) is the prespecified performance index. In addition, the convergence rate of transient and steady-state tracking error e(t ) can be ensured to exceed the convergence rate of (t ) .
9
Decay function
0
e(0)
(t)
e(t)
(t) 0
Time (s) Fig. 2 Diagram of the prescribed performance technique. To transform the original system into an unrestricted one, an error transformation should be introduced as
e(t ) (t )T ( ) , t 0
(15)
where T ( ) tanh( ) and is the equivalent unconstrained tracking error. According to the property of the hyperbolic tangent function, one obtains
1 T ( ) 1 , t 0
(16)
Substituting Eq. (16) into Eq. (15) yields
(t ) e(t ) (t ) , t 0
(17)
Obviously, it is consistent with inequality (13). By observing Eq. (15), the inverse solution of T ( ) is
e(t ) 1 (t ) e(t ) , t 0 ln (t ) 2 (t ) e(t )
T 1
(18)
If can be guaranteed bounded for t 0 , Equation (13) will hold true. Differentiating Eq. (18) with respect to time gives
1 e e e e , t 0 2 e e 2 e2
10
(19)
3. Boundary control design and stability analysis In this section, the control scheme is devised to guarantee the desired attitude angle of the rigid body and simultaneously eliminate the vibration of flexible beams, that is, d ,
0 , w( x, t ) 0 and w ( x, t ) 0 . The prescribed performance technique will be introduced to ensure that the tracking error converges to a specified small range. In the boundary controller, the angle tracking error can be denoted as
e(t ) d , t 0
(20)
where d represents the desired attitude angle of the central rigid body, e(t ) , and
e(t ) . To analyze the stability of the dynamic model, a Lyapunov candidate function is constructed in the following form
V V1 V2 V3
(21)
where V1 denotes the sum of the kinetic energy and potential energy of the dynamic system,
V2 the tracking error index, and V3 the cross term, namely l
l
l
l
0
0
0
0
V1 z 2 ( x, t )dx EI wxx 2 ( x, t )dx z 2 ( x, t )dx EI z xx 2 ( x, t )dx V2
1 2 1 1 I h 1e 2 2 2 2 2 2
l
l
0
0
V3 I h ee 2 xz ( x, t )[e wx ( x, t )]dx 2 rz ( x, t )[e wx ( x, t )]dx where 1 0 , 2 0 and 0 . After applying Lemma 1 and 2, an upper bound can be obtained as follows
11
(22) (23) (24)
l 1 1 V3 I h e 2 I h e 2 2 l z 2 ( x, t )dx l 2 e 2 0 2 2 l
l
l
0
0
0
l 3 wxx2 ( x, t )dx 2 r z 2 ( x, t )dx rle 2 rl 2 wxx2 ( x, t )dx l 1 1 ( I h l 2 rl )e 2 I h e 2 2 (l r ) z 2 ( x, t )dx 0 2 2
(25)
l
( l 3 rl 2 ) wxx2 ( x, t )dx 0
I 2 l 2 2 rl l 3 rl 2 , , 2 (l r ), max h 1, EI 1
Letting
the
above
inequality can be rewritten as
V3 (V1 V2 )
(26)
As a result, V 0 . Then, the derivative of Eq. (21) with respect to time reads V V1 V2 V3
(27)
where l l V1 2 z ( x, t ) z ( x, t )dx 2 EI z xx ( x, t )zxx ( x, t )dx 0
0
l
l
0
0
2 EI z ( x, t )z xxxx ( x, t )dx 2 EI z xx ( x, t )zxx ( x, t )dx l l 2 EI z ( x, t )z xxxx ( x, t )dx 2 EI z xx ( x, t ) zx ( x, t ) 0 z xxx ( x, t )zx ( x, t )dx 0 0 l
2 EI z ( x, t )z xxxx ( x, t )dx 2 EI z xx ( x, t ) zx ( x, t ) 0 z xxx ( x, t ) z ( x, t ) 0 z xxxx ( x, t )z ( x, t )dx 0 0 l l 2 EI z xx ( x, t ) zx ( x, t ) 0 z xxx ( x, t ) z ( x, t ) 0 2 EIwxx (0, t ) (t ) 2 EIwxxx (l , t ) z (l , t ) 2rEIwxxx (0, t )(t ) l
l
ee V2 I h 1 2 I h ee 1ee 2
l
(28)
l
(29)
e[ 2 EIwxx (0, t ) 2rEIwxxx (0, t )] 1ee 2 V3 A1 A2 A3 A4 A5
(30)
A1 I h e 2 I h ee
(31)
where
12
l
A2 2 xz( x, t )[e wx ( x, t )]dx
(32)
0 l
A3 2 xz ( x, t )[e w x ( x, t )]dx
(33)
0
l
A4 2 rz( x, t )[e wx ( x, t )]dx
(34)
0 l
A5 2 rz ( x, t )[e w x ( x, t )]dx
(35)
0
First, Eq. (32) can be expressed as l
A2 2 EI xz xxxx ( x, t )[e wx ( x, t )]dx 0
l
l
2 EI xz xxxx ( x, t )edx 2 EI xz xxxx ( x, t ) wx ( x, t )dx 0
0
2 EIe xz xxx ( x, t ) 0 z xxx ( x, t )dx 0 l
l
l l l 2 EI xwx ( x, t ) wxxx ( x, t ) 0 wx ( x, t ) wxxx ( x, t )dx xwxx ( x, t ) wxxx ( x, t )dx 0 0 3 l 2 EIe lwxxx (l , t ) wxx (0, t ) 2 EI lwx (l , t ) wxxx (l , t ) wxx2 ( x, t )dx 2 0
(36)
l
2 EIlewxxx (l , t ) 2 EIewxx (0, t ) 2 EIlwx (l , t ) wxxx (l , t ) 3 EI wxx2 ( x, t )dx 0
Second, Eq. (33) can be derived as l
A3 2 xz ( x, t ) zx ( x, t )dx 0
l l l 2 xz 2 ( x, t ) z 2 ( x, t )dx xz ( x, t ) zx ( x, t )dx 0 0 0 l
l
2 lz (l , t ) 2 z ( x, t )dx 2 xz ( x, t ) zx ( x, t )dx 2
2
0
(37)
0
l
lz 2 (l , t ) z 2 ( x, t )dx 0
Third, Eq. (34) can be obtained as l
A4 2 EI rz xxxx ( x, t )[e wx ( x, t )]dx 0
l
l
0
0
2 EI rz xxxx ( x, t )edx 2 EI rz xxxx ( x, t ) wx ( x, t )dx l l l 2 EIre z xxx ( x, t ) 0 2 EIr wx ( x, t ) wxxx ( x, t ) 0 wxx ( x, t ) wxxx ( x, t )dx (38) 0 l 1 2 EIre wxxx (l , t ) wxxx (0, t ) 2 EIr wx (l , t ) wxxx (l , t ) wxx2 ( x, t ) 0 2 2 2 EIrewxxx (l , t ) 2 EIrewxxx (0, t ) 2 EIrwx (l , t ) wxxx (l , t ) EIrwxx (0, t )
13
Fourth, Eq. (35) can be expressed as l
l
0
0
A5 2 rz ( x, t ) zx ( x, t )dx r z 2 ( x, t ) rz 2 (l , t ) rz 2 (0, t )
(39)
According to Eq. (10), one has
V 2 EIwxx (0, t )(t ) 2 EIwxxx (l , t ) z (l , t ) 2rEIwxxx (0, t )(t )
e[ 2 EIwxx (0, t ) 2rEIwxxx (0, t )] 1ee 2 I h e 2
e[ 2 EIwxx (0, t ) 2rEIwxxx (0, t )] 2 EI (l r )ewxxx (l , t )
(40)
l
2 EIewxx (0, t ) 2 EI (l r ) wx (l , t ) wxxx (l , t ) 3 EI wxx2 ( x, t )dx 0
l
(l r ) z 2 (l , t ) z 2 ( x, t )dx 2 EIrewxxx (0, t ) EIrwxx2 (0, t ) rz 2 (0, t ) 0
Lemma 3. Given two constants 3 0 and 4 0 , the control strategy in this paper is defined as follows
(t ) 1e 2
e2 2
3e
f (t ) 4 z (l , t )
(41) (42)
Under the initial condition 0 |e(0)| (0) , the tracking error can be constrained within a prespecified range that exponentially decays with time. Additionally, the boundary controller with the prescribed performance is asymptotically stable: when t , one has d ,
0 , w( x, t ) 0 and w ( x, t ) 0 . Proof: With control law (41) and (42), the derivative of Lyapunov candidate function can be rewritten as l e V ( I h 3 )e 2 ( ) 2 2 2 1 e 2 3 ee 3 EI wxx2 ( x, t )dx 0 e l
z 2 ( x, t )dx (l r ) z 2 (l , t ) 2 4 z 2 (l , t ) 2 4 (l r ) wx (l , t ) z (l , t ) 0
(43)
2 4 (l r )ez (l , t ) rEIwxx2 (0, t ) rz 2 (0, t )
Based on subsequent inequalities 3 ee
3e 2 31e 2 1 14
(44)
2 4 (l r ) wx (l , t ) z (l , t ) 2 4 (l r ) 2 wx2 (l , t ) 2 4 (l r ) 2 4 (l r )ez (l , t ) 2 4 (l r ) 3e 2 2 4 (l r )
z 2 (l , t )
2
z 2 (l , t )
3
l wx2 (l , t ) 3 EI w ( x, t )dx EI 2 EI wxx2 ( x, t )dx 0 0 l l
2 xx
(45)
(46)
(47)
where 1 , 2 and 3 are positive constants. Hence, the following conclusion can be obtained
e V ( I h 3 31 )e 2 ( ) 2 2 2 [ 3 1 2 4 (l r ) 3 ]e 2 e 1 l
l
2 EI wxx2 ( x, t )dx z 2 ( x, t )dx 0
[2 4
0
2 4 (l r )
2
[2 4 (l r ) 2
EI l
2 4 (l r )
3
(l r )]z 2 (l , t )
(48)
]wx2 (l , t ) rEIwxx2 (0, t ) rz 2 (0, t )
By choosing parameters , 1 , 3 , 4 , 1 , 2 and 3 to satisfy the following conditions
1 I h 3 31 0 2 0
3 4 2 4
3 1 2 4 (l r ) 3 0 1
2 4 (l r )
2
2 4 (l r )
3
5 2 4 (l r ) 2
EI l
(l r ) 0
0
And noting that e 0 , one has
e V 1e 2 2 2 2 2 3e 2 4 z 2 (l , t ) 0 e
15
(49)
LaSalle’s invariance principle is adopted to analyze the stability of the boundary control. When V 0 , it can be concluded through Eq. (49) that
e e z (l , t ) 0
(50)
z (l , t ) 0 e
(51)
z xxxx (l , t ) wxxxx (l , t ) 0
(52)
Then
With Eqs. (51) and (5), one has
Substituting Eq. (51) into Eq. (9) yields
( x, t ) EIwxxxx ( x, t ) 0 w
(53)
Using the variable separation method, the solution of Eq. (53) can be derived as follows
w( x, t ) W ( x)T (t ) ( x, t ) EIwxxxx ( x, t ) 0 wxxxx ( x, t ) w
(54)
EI
( x, t ) w
( x, t ) W ( x) T(t ) wxxxx ( x, t ) Wxxxx ( x) T (t ), w W ( x) T(t ) xxxx EI T (t ) W ( x) Wxxxx ( x) W ( x) 0
(55)
(56)
By setting s 4 , the solution of Eq. (56) arrives at
W ( x) c1 cosh sx c2 sinh sx c3 cos sx c4 sin sx
(57)
By combining Eq. (52) with Eqs. (11) and (12), the following results can be obtained
W (0) Wx (0) Wxx (l ) Wxxxx (l ) 0
(58)
Then, a set of equations is established as
c1 c3 0 c c 0 2 4 c1 cosh sl c2 sinh sl c3 cos sl c4 sin sl 0 c1 cosh sl c2 sinh sl c3 cos sl c4 sin sl 0 After a series of operations, Equation (59) can be simplified as 16
(59)
c4 (sinh sl cos sl sin sl cosh sl ) 0
(60)
with the solutions being ci 0 (i 1, 2,3, 4) for all s. Therefore, one has W ( x) 0 , w( x, t ) 0 , and w ( x, t ) 0 . Accordingly, the PDE boundary control with a prescribed performance in this paper is asymptotically stable by utilizing the extended LaSalle’s invariance principle; that is, when t , one has d , 0 , w( x, t ) 0 and w ( x, t ) 0 . Since V 0 , it is clear that V
will be bounded. Based on Assumption 2 and Eqs. (21)-(24), it can be found that all the other terms in V are bounded except the term containing 2 . As a consequence, is bounded for
t 0 and then the system tracking error can be constrained within a prespecified range, which is (t ) e(t ) (t ) for t 0 .
4. Case Studies In this section, some case studies are illustrated to verify the feasibility of the proposed boundary control with prescribed performance. The structural parameters of the spacecraft with double flexible appendages are listed in Table 1. The numerical results under the following
PID
control
strategy
are
also
included
for
comparison
purposes:
t
(t ) k1e k2 e k3 edt ( t 0 ), where k1 , k2 and k3 denote the positive gain factors. 0 Furthermore, the proposed control is evaluated under the effect of parameter uncertainties, sensor noise and time delay (UND). In UND cases, the controller design is still based on the structural parameters in Table 1 but dynamic simulation uses the following ‘true’ parameters: the length of each beam is 6 m, the flexural stiffness of the beam is 144 Nm2 and the rotational inertia of the central rigid body is 320 kgm2. In addition, the sensor noise and the time delay in the torque input are supposed to be 0.001sin(0.1t ) rad and 0.1 s, where the delayed control torque is not applied until t 0.1 s . 17
Table 1 Structural parameters of the spacecraft. Parameters
Values
Length of each beam, l
5m
Radius of the rigid body, r
0.5m
Flexural stiffness of the beam, EI
120Nm2
Mass per unit length,
20kg/m
Inertia of the central rigid body, Ih
400kgm2
The aim of all the simulations is to regulate the rotation attitude angle of the rigid body to the specified value, d 0.5 rad , with no any residual vibration of the elastic appendages. To constrain the tracking error of the attitude angle, the absolute value |e(t )| of no more than
(t ) and the minimum convergence rate are permitted when 0.05 in Eq. (14). Once the parameters are properly adjusted in the prescribed performance function, the transient and steady-state output error can be guaranteed bounded. Finally, the attitude angle error and transverse vibration of the beams can converge to zero with the proposed controller. The case studies are investigated in below. Case 1: The central rigid body is actuated by a control torque under the PID control scheme with k1 25 , k2 115 and k3 0.5 . The parameters in the prescribed performance function were set at 0 0.6 and 0.02 . The coefficients in the controller were selected as 1 20 , 2 0.1 , 3 250 and 4 1 . Figures 3 and 4 indicate the rotation attitude angle and angular velocity responses of the spacecraft. One can see that under the proposed control scheme, the rotation angle reaches the desired value and the angular velocity decreases to zero within 160 seconds, whereas the angular responses under the PID controller is still significant at the end of simulation. Besides, the vibration at the tip of the beam ( x l ) with the proposed control reduces more rapidly than that with the PID controller as indicated in Fig. 5. Figure 6 presents the time histories of the rotation attitude angle error of the rigid body by employing the proposed control with and 18
without UND. Nevertheless, it can be seen that the tracking error falls within the predefined range and asymptotically converges to zero. Thus, the conclusion |e(t )| (t ) holds true. Further, the deformation of each beam gradually reduces as shown in Fig. 7. Comparatively, as shown in Fig. 6, it is worth noting that the angle tracking error of the rigid body with the PID control has a larger amplitude exceeding outside the prespecified bounds (red dash lines shown in the figure). Although the PID controller can eliminate the steady state error with the help of integral, it may increase the system response time and make a larger error amplitude. 1.0
(Proposed) d
0.8
Angle tracking (rad)
Angle tracking (rad)
1.0
(PID)
0.6 0.4 0.2 0.0
0
20
40
60
80
(PID)
0.6 0.4 0.2 0.0
100 120 140 160 180 200
(Proposed) d
0.8
0
20
40
60
80
100 120 140 160 180 200
Time (s)
Time (s)
(a)
(b)
0.06
(Proposed) (PID)
0.04 0.02 0.00 -0.02 -0.04
0
20
40
60
80
100 120 140 160 180 200
Angular velocity response (rad/s)
Angular velocity response (rad/s)
Fig. 3 Angle tracking under (a) the PID and proposed controllers without UND and (b) the PID and proposed controllers with UND.
Time (s)
0.06
(Proposed) (PID)
0.04 0.02 0.00 -0.02 -0.04
0
20
40
60
80
100 120 140 160 180 200
Time (s)
(a)
(b)
Fig. 4 Angular velocity response under (a) the PID and proposed controllers without UND and (b) the PID and proposed controllers with UND.
19
0.4
w(l,t) (Proposed) w(l,t) (PID)
Deflection at the tip (m)
Deflection at the tip (m)
0.2
0.0
-0.2
-0.4
0.0 -0.2 -0.4 -0.6
20
0
40
60
w(l,t) (Proposed) w(l,t) (PID)
0.2
80 100 120 140 160 180 200
0
20
40
60
Time (s)
80 100 120 140 160 180 200
Time (s)
(a)
(b)
Fig. 5 Deflection at the tip ( x l ) under (a) the PID and proposed controllers without UND and (b) the PID and proposed controllers with UND. 0.6
e(t) (Proposed) (t) e(t) (PID)
0.4 0.2
Angle tracking error (rad)
Angle tracking error (rad)
0.6
0.0 -0.2 -0.4 -0.6
0
20
40
60
80
100 120 140 160 180 200
e(t) (Proposed) (t) e(t) (PID)
0.4 0.2 0.0 -0.2 -0.4 -0.6
0
20
40
60
80 100 120 140 160 180 200
Time (s)
Time (s)
(a)
(b)
Fig. 6 Angle tracking error under (a) the PID and proposed controllers without UND and (b) the PID and proposed controllers with UND.
(a)
20
(b) Fig. 7 Deformation of the flexible appendages with (a) the PID control and (b) the proposed control. Case 2: Assume that the parameters in the prescribed performance function are identical to those in Case 1, the gain coefficients of PID and the proposed controller were set at k1 25 ,
k2 750 , k3 1 , 1 25 , 2 0.1 , 3 400 and 4 1 . As shown in Figs. 8 and 9, the rotation attitude angle achieves the desired value, and the angular velocity smoothly converges to zero under the proposed control scheme, which indicates a better performance with larger control gains. The deflection at the tip of the beam ( x l ) is shown in Fig. 10. One can see from Fig. 10 that the deflection gradually decreases to zero. However, the tracking error under PID control goes beyond the prescribed bounds [red dash lines in Fig. 11] even when a higher control gain is applied. In contrast, the scenarios do not occur under the proposed control with and without UND as shown in Fig. 11. Similarly, one can see that the vibration at the tip of each beam gradually decreases to zero, while the deflection of which is eliminated after a short time, as shown in Fig. 12.
21
0.8
(Proposed) d
0.6
Angle tracking (rad)
Angle tracking (rad)
0.8
(PID)
0.4 0.2 0.0
0
20
40
60
80
100 120 140 160 180 200
(Proposed) d
0.6
(PID)
0.4 0.2 0.0
0
20
40
60
80 100 120 140 160 180 200
Time (s)
Time (s)
(a)
(b)
0.03
(Proposed) (PID)
0.02 0.01 0.00 -0.01
0
20
40
60
80 100 120 140 160 180 200
Angular velocity response (rad/s)
Angular velocity response (rad/s)
Fig. 8 Angle tracking under (a) the PID and proposed controllers without UND and (b) the PID and proposed controllers with UND. 0.03
(Proposed) (PID)
0.02 0.01 0.00 -0.01
0
20
40
60
80 100 120 140 160 180 200
Time (s)
Time (s)
(a)
(b)
Fig. 9 Angular velocity response under (a) the PID and proposed controllers without UND and (b) the PID and proposed controllers with UND. 0.2 w(l,t) (Proposed) w(l,t) (PID)
Deflection at the tip (m)
Deflection at the tip (m)
0.2
0.0
-0.2
-0.4
w(l,t) (Proposed) w(l,t) (PID) 0.0
-0.2
-0.4
0
20
40
60
80 100 120 140 160 180 200
Time (s)
0
20
40
60
80
100 120 140 160 180 200
Time (s)
(a)
(b)
Fig. 10 Deflection at the tip ( x l ) under (a) the PID and proposed controllers without UND and (b) the PID and proposed controllers with UND.
22
0.6 Angle tracking error (rad)
Angle tracking error (rad)
0.6
e(t) (Proposed) (t) e(t) (PID)
0.4 0.2 0.0 -0.2 -0.4 -0.6
0
20
40
60
80 100 120 140 160 180 200
e(t) (Proposed)
0.4
(t)
e(t) (PID)
0.2 0.0 -0.2 -0.4 -0.6
0
20
40
60
80
100 120 140 160 180 200
Time (s)
Time (s)
(a)
(b)
Fig. 11 Angle tracking error under (a) the PID and proposed controllers without UND and (b) the PID and proposed controllers with UND.
(a)
(b) Fig. 12 Deformation of the flexible appendages with (a) the PID control and (b) the proposed control.
23
5. Conclusions The motion of a spacecraft with one central rigid body and two flexible appendages is described by PDEs. Only one control torque and two boundary forces can simultaneously regulate the rotational attitude angle and eliminate the vibration of the flexible appendages. By utilizing the prescribed performance technique, the output error of the dynamic system can be restricted to a predefined range, and the convergence rate can be ensured to be greater than a prespecified value. The stability of the boundary control is demonstrated using the extended LaSalle’s invariance principle, with the performance of the proposed boundary controller being verified through numerical simulations. Competing Interests The authors declare that there is no conflict of interest regarding the publication of this paper. Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant no. 11832005) and the Research Fund of State Key Laboratory of Mechanics and Control of Mechanical Structures (Grant no. NUAA MCMS-0118G01).
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