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Attitude control without angular velocity measurement for flexible satellites
CJA 1030 29 March 2018 Chinese Journal of Aeronautics, (2018), xxx(xx): xxx–xxx
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Chinese Society of Aeronautics and Astronautics & Beihang University
Chinese Journal of Aeronautics [email protected] www.sciencedirect.com
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Attitude control without angular velocity measurement for flexible satellites
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Qinghua ZHU a,b,d, Guangfu MA a, Xiaoting WANG c, Aiguo WU c,*
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a
Department of Control Science and Control Engineering, Harbin Institute of Technology, Harbin 150001, China Shanghai Aerospace Control Technology Institute, Shanghai 201109, China c Harbin Institute of Technology Shenzhen Graduate School, Shenzhen 518055, China d Shanghai Key Laboratory of Aerospace Intelligent Control Technology, Shanghai 201109, China b
Received 16 May 2017; revised 21 June 2017; accepted 12 October 2017
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KEYWORDS
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Attitude control; Flexible satellites; Modal variables; Quaternion models; Passivity
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Abstract In this paper, by using quaternion models, the problem of attitude control is investigated for a class of flexible satellites. Two control laws are presented for the considered flexible satellite models to guarantee convergence of the closed-loop systems without using angular velocity measurement. One is in the form of a partial state feedback for the case where the modal variable is available, and the other is in the form of an observer-based partial state feedback for the case where the modal variable cannot be measured. Finally, an example is employed to illustrate the effectiveness of the proposed control laws. Ó 2018 Production and hosting by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and Astronautics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).
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1. Introduction
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The attitude control of spacecrafts and satellites has important applications for some space missions such as pointing and formation flying. This topic has attracted much attention from a considerable number of researchers.1–4 For attitude control, much investigation is based on the unit quaternion representation.5–8 In Ref. 5, some attitude controllers with the structure of a Proportional-Derivative (PD) feedback plus feed-forward
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* Corresponding author. E-mail address: [email protected] (A. WU). Peer review under responsibility of Editorial Committee of CJA.
Production and hosting by Elsevier
were designed for a rigid body. In Ref. 6, a sliding mode control law was designed and applied to spacecraft attitude tracking maneuvers when the inertia of a spacecraft was not exactly known. In Ref. 7, attitude control was considered for a rigid spacecraft with the control signal constrained by a common maximum magnitude in the presence of bounded unknown disturbances. A sliding mode controller was designed for such a type of spacecrafts to achieve global stability. The designed controller was in the form of a proportional feedback plus a smooth switch-like feedback with an auxiliary time-varying attitude gain function. High-order sliding mode controllers were designed in Ref. 8 for attitude control of a rigid spacecraft. A merit of the designed controller is that the phenomena of chattering can be eliminated. In Ref. 9, two Fault-Tolerant Control (FTC) schemes were derived for spacecraft attitude stabilization with external disturbances. In Ref. 10, the quaternion model of a rigid spacecraft was firstly transformed into
https://doi.org/10.1016/j.cja.2018.03.019 1000-9361 Ó 2018 Production and hosting by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and Astronautics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: ZHU Q et al. Attitude control without angular velocity measurement for flexible satellites, Chin J Aeronaut (2018), https://doi.org/ 10.1016/j.cja.2018.03.019
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the Lagrange-like, and then two robust sliding mode controllers were proposed to solve attitude tracking problems in the absence of both model uncertainties and external disturbances as well as in their presence. In the aforementioned attitude control laws, the angular velocity of a rigid spacecraft was used in the construction of an attitude control law. However, in some circumstances, it is not easy to measure the angular velocity. Therefore, it is necessary to design an attitude control law without angular velocity measurement. Such a controller was designed in Ref. 11 by using a nonlinear filter of the quaternion to replace the angular velocity feedback. In Ref. 12, a velocity-free attitude stabilization scheme was proposed for a rigid spacecraft. In this control scheme, an angular velocity observer-like system was explicitly designed to construct the stabilizing feedback. In Ref. 13, two simple Saturated Proportional-Derivative (SPD) controllers were proposed for asymptotic stabilization of a rigid spacecraft with actuator constraints and without velocity measurement. In Ref. 14, a continuous angular velocity observer with fractional power functions was proposed to estimate the angular velocity via quaternion attitude information. For flexible spacecraft, the effect of the motion of the elastic appendages must be taken into consideration, and thus the attitude control problem is more complicated. In Ref. 15, a dynamic controller was proposed for the attitude control of a flexible spacecraft under the assumption that the modal variables describing flexible elements were not available. In Ref. 16, an adaptive sliding mode control law with a hybrid sliding surface was proposed for a flexible spacecraft to minimize the effect of uncertainties and disturbances. In Ref. 17, an adaptive control law was proposed to solve the attitude tracking problem for flexible spacecrafts subject to a gravity-gradient disturbance under inertia matrix uncertainty. In Ref. 18, a nonlinear observer-based state feedback control law was designed to ensure the control objectives for attitude tracking. In Refs. 15–18, attitude control laws for flexible spacecraft were designed based on the unit quaternion representation. In Ref. 19, the three-axis attitude tracking control problem was investigated in presence of parameter uncertainties and disturbances based on the modified Rodrigues parameterization. An attitude control law was presented in the form of a nonlinear PD term plus a switching function about a sliding variable. In this paper, we consider the problem of attitude control for flexible satellites based on the unit quaternion representation. It is assumed that the modal variables describing flexible elements are not measurable. For such a class of flexible satellites, a dynamic controller is given to achieve stability for the closed-loop system. The designed controller has two features. One is that it is in the form of an observer-based state feedback. The other is that the angular velocity feedback is not used. 2. Motion equations of a flexible satellite and problem formulation In this section, the mathematical model of a flexible satellite is given. We adopt the unit quaternion to describe the attitude of a satellite. The associated quaternion is given by q0 ð1Þ q¼ qv
Q. ZHU et al. with
2
q0 ¼ cosðU=2Þ;
105 106
3
q1 6 7 qv ¼ 4 q2 5 ¼ sinðU=2Þ q3
ð2Þ 108
where is the unit Euler axis, and U is the rotation angel about the Euler axis. The quaternion components are not independent on each other, and they satisfy a single constraint as q20 þ qTv qv ¼ 1
Nðq0 ; qv Þ ¼ qv ; q0 I where
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For the quaternion q in Eq. (1), define the following matrix N:
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q v
116 117
ð3Þ
q v
is the cross-product matrix defined by 3 0 q3 q2 6 7 q 0 q1 5 v ¼ 4 q3 q2 q1 0
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119 120 121
2
ð4Þ 123
With the preceding notation, the quaternion kinematics equation is given as 6 q_ 0 1 ¼ NT ðq0 ; qv Þx ð5Þ q_ ¼ 2 q_ v
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where x is the satellite angular velocity. Due to the property of the matrix N, there holds q_ 0 ð6Þ x ¼ 2Nðq0 ; qv Þ q_ v
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Under the hypothesis of small deformations, by using the Euler theorem, the dynamic equations of a flexible satellite can be given by
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_ þu Jx_ þ dT € g ¼ x ðJx þ dT gÞ
ð7Þ
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€ g þ Cg_ þ Kg ¼ dx_
ð8Þ
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where J is the total inertia matrix which is symmetric, u is the external torque acting on the main body of the satellite, and g is the modal coordinate vector. In the modal Eq. (8), C and K are respectively the damping matrix and the stiffness matrix, which are in the following forms:
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C ¼ diagf2f1 xn1 ; 2f2 xn2 ; . . . ; 2fN xnN g K ¼ diagfx2n1 ; x2n2 ; . . . ; x2nN g
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d is the coupling matrix between flexible and rigid dynamics. In this paper, N elastic modes are considered. The corresponding natural frequencies are xni, i = 1, 2,. . ., N, and the associated dampings are fi, i = 1, 2,. . ., N. From Eqs. (7) and (8), the following dynamic equations of the flexible satellite can be obtained15: 8 T T 1 > < x_ ¼ Jmb ½x ðJmb x þ d wÞ þ d ðCw þ Kg CdxÞ þ u g_ ¼ w dx > :_ w ¼ ðCw þ Kg CdxÞ
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ð9Þ
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In dynamic Eq. (9),
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Jmb ¼ J d d
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is the main body inertia matrix and
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T
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w ¼ g_ þ dx is the total velocity of the flexible appendages. By summarizing derivation, a satellite with flexible appendages can be described by the mathematical models in Eqs. (5) and (9). Compared with attitude sensors, angular velocity sensors are more expensive. Therefore, it is practically important to design a control law without angular velocity measurement. In this paper, we aim to design such a control law so that limt!1 qv ðtÞ ¼ 0;
lim gðtÞ ¼ 0;
lim wðtÞ ¼ 0;
t!1
t!1
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2
2
w ð17Þ
Denote
ð11Þ
where Cv ðsÞ is a 4 4 linear time-invariant, strictly positive real, strictly proper transfer matrix. With the aid of this filter, we design the following partial state feedback controller for the flexible satellite system described by Eqs. (5) and (9): 2 3 qv 6 7 u ¼ F4 g 5 2Nðq0 ; qv Þz ð12Þ w "
T # K I P C C
ð13Þ
where k is the control parameter, in the control gain of Eq. (13), the matrix P will be determined later. The main result of this section is given in the following theorem. Theorem 1. Consider the satellite system described by Eqs. (5) and (9). Let Cv(s) be a 4 4 linear time-invariant, strictly positive real, strictly proper transfer matrix, and the symmetric matrix P satisfies the following Lyapunov matrix equation P
0 K
I 0 þ C K
I C
T P ¼ 2Q
ð14Þ
for a prescribed positive definite matrix Q. Then, under the control law in Eqs. (11)–(13), the closed-loop system achieves the performance in Eq. (10). Proof. Consider an arbitrary minimal realization of Cv ðsÞ as follows:
x_ ¼ A1 x þ B1 q_ z ¼ C1 x
ð15Þ
Since Cv ðsÞ is linear time-invariant, strictly positive real, strictly proper, then according to Kalman-YakubovichPopv’s lemma, there exist positive definite matrices P1 and Q1 such that AT1 P1
þ P1 A1 ¼ Q1 ;
P1 B1 ¼
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V1 ¼ k½ðq0 1Þ2 þ qTv qv
ð18Þ
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1 V2 ¼ xT Jmb x 2
ð19Þ
V3 ¼ 2xT P1 x
ð20Þ
Thus, V = V1 + V2 + V3 + V4. Next, we give the time derivatives for the four functions V1, V2, V3, and V4 along the solution of the closed-loop system. For the function V1, from Eq. (5), we have
with
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g
1 1 þ qTv qv þ xT Jmb x þ 2xT P1 x þ ½gT ; wT P
z ¼ Cv ðsÞq_
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V ¼ k½ðq0 1Þ
2
g 1 T T V4 ¼ ½g ; w P 2 w
F ¼ kI; d
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For the closed-loop system under the control law in Eqs. (11)–(13), we consider the following Lyapunov function:
In this section, it is assumed that the modal variables g and w are available. Since the angular velocity is not available, we introduce the following filter:
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t!1
3. Partial state feedback control
T
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lim xðtÞ ¼ 0 ð10Þ
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3
CT1
ð16Þ
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ð21Þ
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V_ 1 ¼ 2k½q0 1; qTv q_ 1 ¼ 2k½q0 1; qTv NT ðq0 ; qv Þx 2 T ¼ k½q0 1; qTv ½qv q0 I q v x T ¼ kðq0 1ÞqTv x þ kqTv ðq0 I qT v Þx ¼ kqv x
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ð22Þ
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For the function V2, by using Eqs. (5), (9), and (12), we have
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T V_ 2 ¼ xT Jmb x_ ¼ xT Jmb J1 mb ðx ðJmb x þ d wÞ þdT ðCw þ Kg CdxÞ þ uÞ
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¼ xT ½dT ðCw þ Kg CdxÞ þ u 2 0
3 1 qv 6 7 C B ¼ xT @dT ðCw þ Kg CdxÞ F4 g 5 2Nðq0 ; qv ÞzA w ¼ xT dT ðCw þ Kg CdxÞ 2xT Nðq0 ; qv Þz 2 3 " T # qv K I 6 7 P xT kI; dT 4g5 C C w
¼ xT dT ðCw þ Kg CdxÞ 4q_ T z T ! K g I T g þ xT kqv dT þ dT P C w w C I dx 4zT q_ ¼ xT kqv xT dT Cdx þ ½gT ; wT P C ð23Þ
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For the function V3, it follows Eqs. (15), (16), and (9) that
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_ V_ 3 ¼ 4xT P1 x_ ¼ 4xT P1 ðA1 x þ B1 qÞ ¼ 4xT P1 A1 x þ 4xT P1 B1 q_ ¼ 4xT P1 A1 x þ 4xT C1 q_ ¼ 2xT Q1 x þ 4zT q_
ð24Þ
For the function V4, it can be obtained from Eq. (9) that
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g_ V_ 4 ¼ ½gT ; wT P _ w 0 I g I dx ¼ ½gT ; wT P K C w C 0 I I g ¼ ½gT ; wT P dx ½gT ; wT P K C w C
With the previous time derivatives for the functions Vi, i = 1,2,3,4, it can be obtained that V_ ¼ V_ 1 þ V_ 2 þ V_ 3 þ V_ 4
I ¼ x kqv x d Cdx þ ½g ; w P dx 4zT q_ C 0 I g 2xT Q1 x þ 4zT q_ þ ½gT ; wT P K C w I dx ½gT ; wT P C g ¼ xT dT Cdx 2xT Q1 x ½gT ; wT Q 60 w ð26Þ kqTv x
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ð25Þ
T
T T
T
T
Since V is a continuously differentiable, radially unbounded, and positive definite function, and V_ 6 0 over the entire state, then the global asymptotic stability can be sta as the larted by using the LaSalle invariant theorem. Define X gest invariant set, which is contained in X ¼ fðq; x; x; g; wÞ : V_ ¼ 0g ¼ fðq; x; x; g; wÞ : x ¼ 0; x ¼ 0; g ¼ 0; w ¼ 0g
ð27Þ
Then we obtain Jmb x_ ¼ 0 on X. With this, it follows from the first expression in Eq. (9) that
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their estimates. Define the estimates of the modal variables g ^ respectively. We construct the following and w as ^ g and w, observer to obtain the estimates of g and w: " # ^ ^ g_ 0 I g _ ð28Þ ¼ _^ ^ 2TdNðq0 ; qv Þq K C w w
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where K I I P2 T¼ P1 3 C C C
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and the positive definite matrices P2 and P3 are determined later. With the aid of the observer in Eq. (28), now we construct the following dynamic controller for the satellite system in Eqs. (5) and (9): 2 3 qv 6 7 u ¼ F4 ^ ð29Þ g 5 2Nðq0 ; qv Þz ^ w where F is given by Eq. (13), and z is constructed by Eq. (11).
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Remark 2. When the observer in Eq. (28) for estimating the flexible modals g and w is constructed, the basic structure of the sub-equation on g w in Eq. (9) is utilized. With the ^ of g and w, the overall controller is then estimations ^ g and w obtained by replacing g and w in the control law Eq. (12) with ^ ^ g and w.
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Remark 3. Compared with some existing observers in flexible spacecraft, for example in Ref. 15, an important feature of the observer in Eq. (28) is that angular velocity measurement is not used. Thus, the overall dynamic output feedback controller in Eqs. (28), (29) can be implemented without angular velocity measurement.
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Theorem 2. Consider the satellite system described by Eqs. (5) and (9). Let Cv(s) be a 4 4 linear time-invariant, strictly positive real, strictly proper transfer matrix, and the symmetric matrices P2 and P3 satisfy the following Lyapunov matrix equation:
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x ðJmb x þ dT wÞ þ dT ðCw þ Kg CdxÞ þ u ¼ 0 which implies that on the set O, there holds u = 0. By using Eqs. (12), (13), and (15), it can be obtained that u ¼ kqv dT ðK PÞg dT ðI þ PÞCw 2Nðq0 ; qv ÞC1 x ¼ 0 From the preceding expression, it can be derived that ¼ fðq; x; x; g; wÞ : x ¼ 0; qv ¼ 0 on the set X. Therefore, X x ¼ 0; qv ¼ 0; g ¼ 0; w ¼ 0g; and the global asymptotic stability is proven. h Remark 1. In the proposed stabilizing control law of Eqs. (12) and (13), the need for direct angular velocity measurement is removed since the filter Cv(s), or equivalently the filter in Eq. (15), is used. 4. Observer-based feedback controllers In the designed control law in Eqs. (11)–(13), modal measurements are needed. However, in many cases, it is difficult to measure vibration modals. Thus, it is necessary to construct a control law when modal measurements are not available. For this end, we adopt the idea of an observer-based control law. For these vibration modals, we use an observer to give their estimation, and then the overall control law can be obtained by replacing the modals in the controller Eqs. (11)–(13) with
0 I 0 þ Pi K C K
I C
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T Pi ¼ 2Qi ; i ¼ 2; 3
ð30Þ 336
for two prescribed positive definite matrices Q2 and Q3. Then under the control law Eqs. (29), (28), and (11), qv ; x; x; g, and w tend to zero asymptotically for any initial condition.
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Proof. Consider an arbitrary minimal realization of Cv(s) as in Eqs. (15) and (16). For the closed-loop system under the control law Eqs. (29) and (28), introduce the errors
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^ eg ¼ g ^ g; ew ¼ w w and denote
g 1 V4 ¼ ½gT ; wT P2 2 w
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ð31Þ
345 346 347
eg 1 V5 ¼ ½eTg ; eTw P3 2 ew
338
ð32Þ
349 350
ð33Þ
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In addition, we also define functions V1, V2, and V3 as in Eqs. (18), (19), and (20), respectively. For the closed-loop system, we consider the following Lyapunov function:
For the function V5, from Eqs. (9), (31), (28), and (6), it can be derived that
V ¼ V1 þ V2 þ V3 þ V4 þ V5
V_ 5 ¼ ½eTg ; eTw P3
ð34Þ
By calculation, it can be found that the time derivatives of V1 and V3 are the same as those in the proof of Theorem 1. Next, we give the time derivatives of functions V2, V4, and V5 along the solution of the closed-loop system. For the function V2, by using Eqs. (5), (9), (29), and (31), we have T V_ 2 ¼ xT Jmb x_ ¼ xT Jmb J1 mb ðx ðJmb x þ d wÞ þdT ðCw þ Kg CdxÞ þ uÞ 0 1 2 3 qv B C 6 7 ¼ xT @dT ðCw þ Kg CdxÞ F4 ^g 5 2Nðq0 ; qv ÞzA ^ w
¼ xT dT ðCw þ Kg CdxÞ 4zT q_ xT kqv T eg K I g P2 xT dT ew C C w T I g P2 ¼ xT dT Cdx xT kqv þ xT dT C w T e K I g 4zT q_ P2 þ xT dT ew C C I T T T T T dx ¼ x d Cdx x kqv þ ½g ; w P2 C I K 4zT q_ þ ½eTg ; eTw dx P2 C C 367 368
370
0
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I
#" # g
K C w " #" # 0 I g K C
w
"
I
C " # I C
# dx
" ## ^ g_ ^_ w
dx
With the previous preliminaries, it can be obtained from Eq. (14) that
g I ½gT ; wT P2 dx w C
Fig. 1 Quaternion behavior for case of partial state feedback in Eqs. (11)–(13).
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V_ ¼ V_ 1 þ V_ 2 þ V_ 3 þ V_ 4 þ V_ 5
For function V4, it can be obtained from Eq. (9) that
I C
¼ ½eTg ; eTw P3
e_ w ""
#
ð36Þ
ð35Þ g_ V_ 4 ¼ ½gT ; wT P2 _ w 0 ¼ ½gT ; wT P2 K
¼ ½eTg ; eTw P3
e_ g
½eTg ; eTw P3 2TdNðq0 ; qv Þq_ "" #" # " # # 0 I g I T T ¼ ½eg ; ew P3 dx K C w C #" # " "" # # ^ 0 I g I T T ½eg ; ew P3 dx ^ K C w C "" # " ## K I 1 T T P2 dx ½eg ; ew P3 P3 C C " #" # "" # " ## eg 0 I I K T T T T dx P2 ½eg ; ew ¼ ½eg ; ew P3 K C ew C C
¼ xT dT ðCw þ Kg CdxÞ 2xT Nðq0 ; qv Þz 0 2 31 " T # qv K I B 6 ^ 7C P2 þ xT @ kI; dT 4 g 5A C C ^ w
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"
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K I ¼ kqTv x xT dT Cdx xT kqv ½eTg ; eTw P2 dx C C eg I 0 I þ ½gT ; wT P2 dx þ ½eTg ; eTw P3 C K C ew 0 I g I þ ½gT ; wT P2 ½gT ; wT P2 dx K C w C I K dx 4zT q_ 2xT Q1 x þ 4zT q_ P2 þ ½eTg ; eTw C C eg g 60 ½eTg ; eTw Q3 ¼ xT dT Cdx 2xT Q1 x ½gT ; wT Q2 ew w ð37Þ
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Next, along a similar line to that in the proof of Theorem 1, the conclusion of this theorem can be obtained according to the LaSalle invariant theorem. h
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5. Simulation results
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In this section, we apply the control law Eqs. (12), (29), and (28) to a satellite model with four modes to demonstrate the previous theoretical results. The main parameters of the flexible satellite taken from Ref. 20 are as follows. The main body inertia matrix (in kg m2 ) is 2 3 350 3 4 6 7 Jmb ¼ 4 3 270 10 5 4 10 190
385
and the coupling matrix between flexible and rigid dynamics (in kg1=2 m) is
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6:45637 6 1:25619 6 d¼6 4 1:11687 397 398 399 400
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1:23637
3
1:27814 0:91756 2:48901
2:15629 1:67264 7 7 7 0:83674 5
2:6581
1:12503
For the four natural modes of the flexible satellite, the natural frequencies are as follows: xn1 ¼ 0:7681 rad=s; xn2 ¼ 1:1038 rad=s; xn3 ¼ 1:8733 rad=s;xn4 ¼ 2:5496 rad=s and the dampings are given as f1 ¼ 0:005607;
f2 ¼ 0:00862;
f3 ¼ 0:01283;
f4 ¼ 0:02516
Fig. 3 Quaternion behavior for case of observer-based state feedback in Eqs. (28) and (29).
The maneuver of the considered flexible satellite is a rotation of 160° with an Euler axis, i.e., ¼ ½0:267261; 0:801784; 0:534523T Thus, the initial attitude described by the quaternion is 2 3 2 3 q1 ð0Þ 0:263201 6 7 6 7 q0 ð0Þ ¼ 0:173648; qv ð0Þ ¼ 4 q2 ð0Þ 5 ¼ 4 0:789603 5 q3 ð0Þ 0:526402 and the initial angular velocity of the spacecraft is xð0Þ ¼ ½0; 0; 0T In addition, the initial values of the four modals of the flexible appendages are gi ¼ 0:001;
wi ¼ 0:001;
i ¼ 1; 2; 3; 4
Firstly, we apply the partial state feedback controller Eqs. (11)–(13) to the flexible satellite. The parameters of the controller are as follows: k ¼ 150; B1 ¼ 6I44 ; A1 ¼ 13I44 ; Q1 ¼ 2600I88 ; Q ¼ 0:15I88 In addition, the parameter P is determined by solving the Lyapunov matrix Eq. (14). The behavior of the quaternion q ¼ q0 þ q1 i þ q2 j þ q3 k is shown in Fig. 1 where i; j and k are the fundamental quaternion units. The modal displace-
Fig. 4 Modal displacements for case of partial state feedback in Eqs. (28) and (29).
ments are shown in Fig. 2. From Fig. 1, it can be seen that the quaternion q of the closed-loop system is asymptotically convergent. Secondly, the observer-based feedback controller in Eqs. (28), (29) is also used for the flexible satellite, and the parameters are chosen to be as follows: k ¼ 300; B1 ¼ 6I44 ; A1 ¼ 15I44 ; Q2 ¼ 0:034I88 ; Q3 ¼ 10I88
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Q1 ¼ 3500I88 ;
In addition, the matrices P2 and P3 are determined by Eq. (30). In this case, the behavior of the quaternion q ¼ q0 þ q1 i þ q2 j þ q3 k is shown in Fig. 3, and the modal displacements are shown in Fig. 4. From Fig. 3, it can be seen that the quaternion q is also asymptotically convergent when the observer-based control law is used. In addition, it can be seen from Figs. 2 and 4 that the vibration amplitude of the flexible modals under the observer-based control law is bigger than that under the state feedback control law.
Fig. 2 Modal displacements for case of partial state feedback in Eqs. (11)–(13).
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6. Conclusions
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In this paper, attitude control is considered without angular velocity measurement for satellites with flexible appendages
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based on quaternion models. Firstly, a partial state feedback control law is proposed for this kind of satellites to guarantee convergence of the quaternion behaviors of the considered satellites. Secondly, when the modal variables describing flexible elements are not measurable, an observer-based partial state feedback control law is also presented. By applying this dynamic controller to the considered satellites, the convergence of the closed-loop system can also be guaranteed. A common feature of the two control laws proposed in this paper is that angular velocity measurement is not used. In future, we will use the idea in this paper to the problem of attitude tracking for flexible satellites. Acknowledgements
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This work was co-supported by the Major Program of National Natural Science Foundation of China (Nos. 61690210, 61690212), Shenzhen Municipal Basic Research Project for Discipline Layout (No. JCYJ20170413112722597), and Shenzhen Municipal Project for Basic Research (Nos. JCYJ20170307150952660, JCYJ20170307150227897).
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References
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1. Liu H, Guo L, Zhang Y. An anti-disturbance PD control scheme for attitude control and stabilization of flexible spacecrafts. Nonlinear Dyn 2011;67(3):2081–8. 2. Xiao B, Hu QL, Zhang Y. Adaptive sliding mode fault yolerant attitude tracking control for flexible spacecraft under actuator saturation. IEEE Trans Control Syst Technol 2012;20(6):1605–12. 3. Guo Y, Song S. Adaptive finite-time backstepping control for attitude tracking of spacecraft based on rotation matrix. Chin J Aeronaut 2014;27(2):375–82. 4. Sun H, Li S. Composite control method for stabilizing spacecraft attitude in terms of Rodrigues parameters. Chin J Aeronaut 2013;26(3):687–96. 5. Wen TY, Kreutz-Delgado K. The attitude control problem. IEEE Trans Autom Control 1991;36(10):1148–62.
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7 6. Lo SC, Chen YP. Smooth sliding-mode control for spacecraft attitude tracking maneuvers. J Guid Control Dyn 1995;18 (6):1345–9. 7. Wallsgrove RJ, Akella MR. Globally stabilizing saturated attitude control in the presence of bounded unknown disturbances. J f Guid Control Dyn 2005;28(5):957–63. 8. Tiwari PM, Janardhanan S, Nabi MU. Attitude control using higher order sliding mode. Aerosp Sci Technol 2016;54:108–13. 9. Shen Q, Wang D, Zhu S, Poh EK. Integral-type sliding mode fault-tolerant control for attitude stabilization of spacecraft. IEEE Trans Control Syst Technol 2015;23(3):1131–8. 10. Wu S, Radice G, Gao Y, Sun Z. Quaternion-based finite time control for spacecraft attitude tracking. Acta Astronaut 2011;69:48–58. 11. Lizarralde F, Wen JT. Attitude control without angular velocity measurement: a passivity approach. IEEE Trans Autom Control 1996;41(3):468–72. 12. Benziane L, Benallegue A, Chitour Y, Tayebi A. Velocity-free attitude stabilization with inertial vector measurements. Int J Robust Nonlinear Control 2016;26:2478–93. 13. Su Y, Zheng C. Velocity-free saturated PD controller for asymptotic stabilization of spacecraft. Aerosp Sci Technol 2014;39:6–12. 14. Gui H, Vukovich G. Finite-time angular velocity observers for rigid-body attitude tracking with bounded inputs. Int J Robust Nonlinear Control 2017;27:15–38. 15. Gennaro SD. Passive attitude control of flexible spacecraft from quaternion measurements. J Optim Theory Appl 2003;116 (1):41–60. 16. Shahravi M, Kabganian M, Alasty AM. Adaptive robust attitude control of a flexible spacecraft. Int J Robust Nonlinear Control 2006;16(6):287–302. 17. Gennaro SD. Adaptive robust tracking of flexible spacecraft in presence of disturbances. J Optim Theory Appl 1998;98(3):545–68. 18. Gennaro SD. Output attitude tracking for flexible spacecraft. Automatica 2002;38(10):1719–26. 19. Jin E, Sun Z. Robust attitude tracking control of flexible spacecraft for achieving globally asymptotic stability. Int J Robust Nonlinear Control 2009;19:1201–23. 20. Hu QL. Robust adaptive backstepping attitude and vibration control with L2-gain performance for flexible spacecraft under angular velocity constraint. J Sound Vib 2009;327(3–5):285–98.
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Chinese Society of Aeronautics and Astronautics & Beihang University
Chinese Journal of Aeronautics [email protected] www.sciencedirect.com
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Attitude control without angular velocity measurement for flexible satellites
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Qinghua ZHU a,b,d, Guangfu MA a, Xiaoting WANG c, Aiguo WU c,*
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6 7 8 9
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a
Department of Control Science and Control Engineering, Harbin Institute of Technology, Harbin 150001, China Shanghai Aerospace Control Technology Institute, Shanghai 201109, China c Harbin Institute of Technology Shenzhen Graduate School, Shenzhen 518055, China d Shanghai Key Laboratory of Aerospace Intelligent Control Technology, Shanghai 201109, China b
Received 16 May 2017; revised 21 June 2017; accepted 12 October 2017
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KEYWORDS
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Attitude control; Flexible satellites; Modal variables; Quaternion models; Passivity
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Abstract In this paper, by using quaternion models, the problem of attitude control is investigated for a class of flexible satellites. Two control laws are presented for the considered flexible satellite models to guarantee convergence of the closed-loop systems without using angular velocity measurement. One is in the form of a partial state feedback for the case where the modal variable is available, and the other is in the form of an observer-based partial state feedback for the case where the modal variable cannot be measured. Finally, an example is employed to illustrate the effectiveness of the proposed control laws. Ó 2018 Production and hosting by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and Astronautics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).
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1. Introduction
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The attitude control of spacecrafts and satellites has important applications for some space missions such as pointing and formation flying. This topic has attracted much attention from a considerable number of researchers.1–4 For attitude control, much investigation is based on the unit quaternion representation.5–8 In Ref. 5, some attitude controllers with the structure of a Proportional-Derivative (PD) feedback plus feed-forward
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* Corresponding author. E-mail address: [email protected] (A. WU). Peer review under responsibility of Editorial Committee of CJA.
Production and hosting by Elsevier
were designed for a rigid body. In Ref. 6, a sliding mode control law was designed and applied to spacecraft attitude tracking maneuvers when the inertia of a spacecraft was not exactly known. In Ref. 7, attitude control was considered for a rigid spacecraft with the control signal constrained by a common maximum magnitude in the presence of bounded unknown disturbances. A sliding mode controller was designed for such a type of spacecrafts to achieve global stability. The designed controller was in the form of a proportional feedback plus a smooth switch-like feedback with an auxiliary time-varying attitude gain function. High-order sliding mode controllers were designed in Ref. 8 for attitude control of a rigid spacecraft. A merit of the designed controller is that the phenomena of chattering can be eliminated. In Ref. 9, two Fault-Tolerant Control (FTC) schemes were derived for spacecraft attitude stabilization with external disturbances. In Ref. 10, the quaternion model of a rigid spacecraft was firstly transformed into
https://doi.org/10.1016/j.cja.2018.03.019 1000-9361 Ó 2018 Production and hosting by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and Astronautics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: ZHU Q et al. Attitude control without angular velocity measurement for flexible satellites, Chin J Aeronaut (2018), https://doi.org/ 10.1016/j.cja.2018.03.019
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the Lagrange-like, and then two robust sliding mode controllers were proposed to solve attitude tracking problems in the absence of both model uncertainties and external disturbances as well as in their presence. In the aforementioned attitude control laws, the angular velocity of a rigid spacecraft was used in the construction of an attitude control law. However, in some circumstances, it is not easy to measure the angular velocity. Therefore, it is necessary to design an attitude control law without angular velocity measurement. Such a controller was designed in Ref. 11 by using a nonlinear filter of the quaternion to replace the angular velocity feedback. In Ref. 12, a velocity-free attitude stabilization scheme was proposed for a rigid spacecraft. In this control scheme, an angular velocity observer-like system was explicitly designed to construct the stabilizing feedback. In Ref. 13, two simple Saturated Proportional-Derivative (SPD) controllers were proposed for asymptotic stabilization of a rigid spacecraft with actuator constraints and without velocity measurement. In Ref. 14, a continuous angular velocity observer with fractional power functions was proposed to estimate the angular velocity via quaternion attitude information. For flexible spacecraft, the effect of the motion of the elastic appendages must be taken into consideration, and thus the attitude control problem is more complicated. In Ref. 15, a dynamic controller was proposed for the attitude control of a flexible spacecraft under the assumption that the modal variables describing flexible elements were not available. In Ref. 16, an adaptive sliding mode control law with a hybrid sliding surface was proposed for a flexible spacecraft to minimize the effect of uncertainties and disturbances. In Ref. 17, an adaptive control law was proposed to solve the attitude tracking problem for flexible spacecrafts subject to a gravity-gradient disturbance under inertia matrix uncertainty. In Ref. 18, a nonlinear observer-based state feedback control law was designed to ensure the control objectives for attitude tracking. In Refs. 15–18, attitude control laws for flexible spacecraft were designed based on the unit quaternion representation. In Ref. 19, the three-axis attitude tracking control problem was investigated in presence of parameter uncertainties and disturbances based on the modified Rodrigues parameterization. An attitude control law was presented in the form of a nonlinear PD term plus a switching function about a sliding variable. In this paper, we consider the problem of attitude control for flexible satellites based on the unit quaternion representation. It is assumed that the modal variables describing flexible elements are not measurable. For such a class of flexible satellites, a dynamic controller is given to achieve stability for the closed-loop system. The designed controller has two features. One is that it is in the form of an observer-based state feedback. The other is that the angular velocity feedback is not used. 2. Motion equations of a flexible satellite and problem formulation In this section, the mathematical model of a flexible satellite is given. We adopt the unit quaternion to describe the attitude of a satellite. The associated quaternion is given by q0 ð1Þ q¼ qv
Q. ZHU et al. with
2
q0 ¼ cosðU=2Þ;
105 106
3
q1 6 7 qv ¼ 4 q2 5 ¼ sinðU=2Þ q3
ð2Þ 108
where is the unit Euler axis, and U is the rotation angel about the Euler axis. The quaternion components are not independent on each other, and they satisfy a single constraint as q20 þ qTv qv ¼ 1
Nðq0 ; qv Þ ¼ qv ; q0 I where
110 111 112 114
For the quaternion q in Eq. (1), define the following matrix N:
109
q v
116 117
ð3Þ
q v
is the cross-product matrix defined by 3 0 q3 q2 6 7 q 0 q1 5 v ¼ 4 q3 q2 q1 0
115
119 120 121
2
ð4Þ 123
With the preceding notation, the quaternion kinematics equation is given as 6 q_ 0 1 ¼ NT ðq0 ; qv Þx ð5Þ q_ ¼ 2 q_ v
124
where x is the satellite angular velocity. Due to the property of the matrix N, there holds q_ 0 ð6Þ x ¼ 2Nðq0 ; qv Þ q_ v
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Under the hypothesis of small deformations, by using the Euler theorem, the dynamic equations of a flexible satellite can be given by
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133
135 136 137
_ þu Jx_ þ dT € g ¼ x ðJx þ dT gÞ
ð7Þ
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€ g þ Cg_ þ Kg ¼ dx_
ð8Þ
140 142
where J is the total inertia matrix which is symmetric, u is the external torque acting on the main body of the satellite, and g is the modal coordinate vector. In the modal Eq. (8), C and K are respectively the damping matrix and the stiffness matrix, which are in the following forms:
143 144 145 146 147 148
C ¼ diagf2f1 xn1 ; 2f2 xn2 ; . . . ; 2fN xnN g K ¼ diagfx2n1 ; x2n2 ; . . . ; x2nN g
150
d is the coupling matrix between flexible and rigid dynamics. In this paper, N elastic modes are considered. The corresponding natural frequencies are xni, i = 1, 2,. . ., N, and the associated dampings are fi, i = 1, 2,. . ., N. From Eqs. (7) and (8), the following dynamic equations of the flexible satellite can be obtained15: 8 T T 1 > < x_ ¼ Jmb ½x ðJmb x þ d wÞ þ d ðCw þ Kg CdxÞ þ u g_ ¼ w dx > :_ w ¼ ðCw þ Kg CdxÞ
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ð9Þ
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In dynamic Eq. (9),
152 153 154 155 156 157
160 161
Jmb ¼ J d d
163
is the main body inertia matrix and
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T
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Attitude control without angular velocity 165 167 168 169 170 171 172 173 174 175
w ¼ g_ þ dx is the total velocity of the flexible appendages. By summarizing derivation, a satellite with flexible appendages can be described by the mathematical models in Eqs. (5) and (9). Compared with attitude sensors, angular velocity sensors are more expensive. Therefore, it is practically important to design a control law without angular velocity measurement. In this paper, we aim to design such a control law so that limt!1 qv ðtÞ ¼ 0;
lim gðtÞ ¼ 0;
lim wðtÞ ¼ 0;
t!1
t!1
179 180 181 182 184 185 186 187 188 189
191 192 193
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2
2
w ð17Þ
Denote
ð11Þ
where Cv ðsÞ is a 4 4 linear time-invariant, strictly positive real, strictly proper transfer matrix. With the aid of this filter, we design the following partial state feedback controller for the flexible satellite system described by Eqs. (5) and (9): 2 3 qv 6 7 u ¼ F4 g 5 2Nðq0 ; qv Þz ð12Þ w "
T # K I P C C
ð13Þ
where k is the control parameter, in the control gain of Eq. (13), the matrix P will be determined later. The main result of this section is given in the following theorem. Theorem 1. Consider the satellite system described by Eqs. (5) and (9). Let Cv(s) be a 4 4 linear time-invariant, strictly positive real, strictly proper transfer matrix, and the symmetric matrix P satisfies the following Lyapunov matrix equation P
0 K
I 0 þ C K
I C
T P ¼ 2Q
ð14Þ
for a prescribed positive definite matrix Q. Then, under the control law in Eqs. (11)–(13), the closed-loop system achieves the performance in Eq. (10). Proof. Consider an arbitrary minimal realization of Cv ðsÞ as follows:
x_ ¼ A1 x þ B1 q_ z ¼ C1 x
ð15Þ
Since Cv ðsÞ is linear time-invariant, strictly positive real, strictly proper, then according to Kalman-YakubovichPopv’s lemma, there exist positive definite matrices P1 and Q1 such that AT1 P1
þ P1 A1 ¼ Q1 ;
P1 B1 ¼
223
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V1 ¼ k½ðq0 1Þ2 þ qTv qv
ð18Þ
229 230
1 V2 ¼ xT Jmb x 2
ð19Þ
V3 ¼ 2xT P1 x
ð20Þ
Thus, V = V1 + V2 + V3 + V4. Next, we give the time derivatives for the four functions V1, V2, V3, and V4 along the solution of the closed-loop system. For the function V1, from Eq. (5), we have
with
221 222
232 233
218 220
g
1 1 þ qTv qv þ xT Jmb x þ 2xT P1 x þ ½gT ; wT P
z ¼ Cv ðsÞq_
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V ¼ k½ðq0 1Þ
2
g 1 T T V4 ¼ ½g ; w P 2 w
F ¼ kI; d
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For the closed-loop system under the control law in Eqs. (11)–(13), we consider the following Lyapunov function:
In this section, it is assumed that the modal variables g and w are available. Since the angular velocity is not available, we introduce the following filter:
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t!1
3. Partial state feedback control
T
196
lim xðtÞ ¼ 0 ð10Þ
177
178
3
CT1
ð16Þ
236
ð21Þ
238 239 240 241 242 243
V_ 1 ¼ 2k½q0 1; qTv q_ 1 ¼ 2k½q0 1; qTv NT ðq0 ; qv Þx 2 T ¼ k½q0 1; qTv ½qv q0 I q v x T ¼ kðq0 1ÞqTv x þ kqTv ðq0 I qT v Þx ¼ kqv x
235
ð22Þ
245
For the function V2, by using Eqs. (5), (9), and (12), we have
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T V_ 2 ¼ xT Jmb x_ ¼ xT Jmb J1 mb ðx ðJmb x þ d wÞ þdT ðCw þ Kg CdxÞ þ uÞ
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247
¼ xT ½dT ðCw þ Kg CdxÞ þ u 2 0
3 1 qv 6 7 C B ¼ xT @dT ðCw þ Kg CdxÞ F4 g 5 2Nðq0 ; qv ÞzA w ¼ xT dT ðCw þ Kg CdxÞ 2xT Nðq0 ; qv Þz 2 3 " T # qv K I 6 7 P xT kI; dT 4g5 C C w
¼ xT dT ðCw þ Kg CdxÞ 4q_ T z T ! K g I T g þ xT kqv dT þ dT P C w w C I dx 4zT q_ ¼ xT kqv xT dT Cdx þ ½gT ; wT P C ð23Þ
250
For the function V3, it follows Eqs. (15), (16), and (9) that
251 252
_ V_ 3 ¼ 4xT P1 x_ ¼ 4xT P1 ðA1 x þ B1 qÞ ¼ 4xT P1 A1 x þ 4xT P1 B1 q_ ¼ 4xT P1 A1 x þ 4xT C1 q_ ¼ 2xT Q1 x þ 4zT q_
ð24Þ
For the function V4, it can be obtained from Eq. (9) that
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Q. ZHU et al.
g_ V_ 4 ¼ ½gT ; wT P _ w 0 I g I dx ¼ ½gT ; wT P K C w C 0 I I g ¼ ½gT ; wT P dx ½gT ; wT P K C w C
With the previous time derivatives for the functions Vi, i = 1,2,3,4, it can be obtained that V_ ¼ V_ 1 þ V_ 2 þ V_ 3 þ V_ 4
I ¼ x kqv x d Cdx þ ½g ; w P dx 4zT q_ C 0 I g 2xT Q1 x þ 4zT q_ þ ½gT ; wT P K C w I dx ½gT ; wT P C g ¼ xT dT Cdx 2xT Q1 x ½gT ; wT Q 60 w ð26Þ kqTv x
263 264 265 266 267 268 269
271 272 273
ð25Þ
T
T T
T
T
Since V is a continuously differentiable, radially unbounded, and positive definite function, and V_ 6 0 over the entire state, then the global asymptotic stability can be sta as the larted by using the LaSalle invariant theorem. Define X gest invariant set, which is contained in X ¼ fðq; x; x; g; wÞ : V_ ¼ 0g ¼ fðq; x; x; g; wÞ : x ¼ 0; x ¼ 0; g ¼ 0; w ¼ 0g
ð27Þ
Then we obtain Jmb x_ ¼ 0 on X. With this, it follows from the first expression in Eq. (9) that
274 276 277 278 279 281 282 283 284 285
286 287 288 289
290
291 292 293 294 295 296 297 298
their estimates. Define the estimates of the modal variables g ^ respectively. We construct the following and w as ^ g and w, observer to obtain the estimates of g and w: " # ^ ^ g_ 0 I g _ ð28Þ ¼ _^ ^ 2TdNðq0 ; qv Þq K C w w
299
where K I I P2 T¼ P1 3 C C C
305 306
300 301 302
304
308
and the positive definite matrices P2 and P3 are determined later. With the aid of the observer in Eq. (28), now we construct the following dynamic controller for the satellite system in Eqs. (5) and (9): 2 3 qv 6 7 u ¼ F4 ^ ð29Þ g 5 2Nðq0 ; qv Þz ^ w where F is given by Eq. (13), and z is constructed by Eq. (11).
309 310 311 312 313
315 316
Remark 2. When the observer in Eq. (28) for estimating the flexible modals g and w is constructed, the basic structure of the sub-equation on g w in Eq. (9) is utilized. With the ^ of g and w, the overall controller is then estimations ^ g and w obtained by replacing g and w in the control law Eq. (12) with ^ ^ g and w.
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Remark 3. Compared with some existing observers in flexible spacecraft, for example in Ref. 15, an important feature of the observer in Eq. (28) is that angular velocity measurement is not used. Thus, the overall dynamic output feedback controller in Eqs. (28), (29) can be implemented without angular velocity measurement.
323
Theorem 2. Consider the satellite system described by Eqs. (5) and (9). Let Cv(s) be a 4 4 linear time-invariant, strictly positive real, strictly proper transfer matrix, and the symmetric matrices P2 and P3 satisfy the following Lyapunov matrix equation:
329
318 319 320 321 322
324 325 326 327 328
x ðJmb x þ dT wÞ þ dT ðCw þ Kg CdxÞ þ u ¼ 0 which implies that on the set O, there holds u = 0. By using Eqs. (12), (13), and (15), it can be obtained that u ¼ kqv dT ðK PÞg dT ðI þ PÞCw 2Nðq0 ; qv ÞC1 x ¼ 0 From the preceding expression, it can be derived that ¼ fðq; x; x; g; wÞ : x ¼ 0; qv ¼ 0 on the set X. Therefore, X x ¼ 0; qv ¼ 0; g ¼ 0; w ¼ 0g; and the global asymptotic stability is proven. h Remark 1. In the proposed stabilizing control law of Eqs. (12) and (13), the need for direct angular velocity measurement is removed since the filter Cv(s), or equivalently the filter in Eq. (15), is used. 4. Observer-based feedback controllers In the designed control law in Eqs. (11)–(13), modal measurements are needed. However, in many cases, it is difficult to measure vibration modals. Thus, it is necessary to construct a control law when modal measurements are not available. For this end, we adopt the idea of an observer-based control law. For these vibration modals, we use an observer to give their estimation, and then the overall control law can be obtained by replacing the modals in the controller Eqs. (11)–(13) with
0 I 0 þ Pi K C K
I C
330 331 332 333 334
T Pi ¼ 2Qi ; i ¼ 2; 3
ð30Þ 336
for two prescribed positive definite matrices Q2 and Q3. Then under the control law Eqs. (29), (28), and (11), qv ; x; x; g, and w tend to zero asymptotically for any initial condition.
337
Proof. Consider an arbitrary minimal realization of Cv(s) as in Eqs. (15) and (16). For the closed-loop system under the control law Eqs. (29) and (28), introduce the errors
340
^ eg ¼ g ^ g; ew ¼ w w and denote
g 1 V4 ¼ ½gT ; wT P2 2 w
339
341 342 343
ð31Þ
345 346 347
eg 1 V5 ¼ ½eTg ; eTw P3 2 ew
338
ð32Þ
349 350
ð33Þ
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Attitude control without angular velocity 353 354 355 356 358 359 360 361 362 363 364
5
In addition, we also define functions V1, V2, and V3 as in Eqs. (18), (19), and (20), respectively. For the closed-loop system, we consider the following Lyapunov function:
For the function V5, from Eqs. (9), (31), (28), and (6), it can be derived that
V ¼ V1 þ V2 þ V3 þ V4 þ V5
V_ 5 ¼ ½eTg ; eTw P3
ð34Þ
By calculation, it can be found that the time derivatives of V1 and V3 are the same as those in the proof of Theorem 1. Next, we give the time derivatives of functions V2, V4, and V5 along the solution of the closed-loop system. For the function V2, by using Eqs. (5), (9), (29), and (31), we have T V_ 2 ¼ xT Jmb x_ ¼ xT Jmb J1 mb ðx ðJmb x þ d wÞ þdT ðCw þ Kg CdxÞ þ uÞ 0 1 2 3 qv B C 6 7 ¼ xT @dT ðCw þ Kg CdxÞ F4 ^g 5 2Nðq0 ; qv ÞzA ^ w
¼ xT dT ðCw þ Kg CdxÞ 4zT q_ xT kqv T eg K I g P2 xT dT ew C C w T I g P2 ¼ xT dT Cdx xT kqv þ xT dT C w T e K I g 4zT q_ P2 þ xT dT ew C C I T T T T T dx ¼ x d Cdx x kqv þ ½g ; w P2 C I K 4zT q_ þ ½eTg ; eTw dx P2 C C 367 368
370
0
373
I
#" # g
K C w " #" # 0 I g K C
w
"
I
C " # I C
# dx
" ## ^ g_ ^_ w
dx
With the previous preliminaries, it can be obtained from Eq. (14) that
g I ½gT ; wT P2 dx w C
Fig. 1 Quaternion behavior for case of partial state feedback in Eqs. (11)–(13).
375 376 377 378
V_ ¼ V_ 1 þ V_ 2 þ V_ 3 þ V_ 4 þ V_ 5
For function V4, it can be obtained from Eq. (9) that
I C
¼ ½eTg ; eTw P3
e_ w ""
#
ð36Þ
ð35Þ g_ V_ 4 ¼ ½gT ; wT P2 _ w 0 ¼ ½gT ; wT P2 K
¼ ½eTg ; eTw P3
e_ g
½eTg ; eTw P3 2TdNðq0 ; qv Þq_ "" #" # " # # 0 I g I T T ¼ ½eg ; ew P3 dx K C w C #" # " "" # # ^ 0 I g I T T ½eg ; ew P3 dx ^ K C w C "" # " ## K I 1 T T P2 dx ½eg ; ew P3 P3 C C " #" # "" # " ## eg 0 I I K T T T T dx P2 ½eg ; ew ¼ ½eg ; ew P3 K C ew C C
¼ xT dT ðCw þ Kg CdxÞ 2xT Nðq0 ; qv Þz 0 2 31 " T # qv K I B 6 ^ 7C P2 þ xT @ kI; dT 4 g 5A C C ^ w
366
"
371 372
K I ¼ kqTv x xT dT Cdx xT kqv ½eTg ; eTw P2 dx C C eg I 0 I þ ½gT ; wT P2 dx þ ½eTg ; eTw P3 C K C ew 0 I g I þ ½gT ; wT P2 ½gT ; wT P2 dx K C w C I K dx 4zT q_ 2xT Q1 x þ 4zT q_ P2 þ ½eTg ; eTw C C eg g 60 ½eTg ; eTw Q3 ¼ xT dT Cdx 2xT Q1 x ½gT ; wT Q2 ew w ð37Þ
380
Next, along a similar line to that in the proof of Theorem 1, the conclusion of this theorem can be obtained according to the LaSalle invariant theorem. h
381
5. Simulation results
384
In this section, we apply the control law Eqs. (12), (29), and (28) to a satellite model with four modes to demonstrate the previous theoretical results. The main parameters of the flexible satellite taken from Ref. 20 are as follows. The main body inertia matrix (in kg m2 ) is 2 3 350 3 4 6 7 Jmb ¼ 4 3 270 10 5 4 10 190
385
and the coupling matrix between flexible and rigid dynamics (in kg1=2 m) is
393
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1:23637
3
1:27814 0:91756 2:48901
2:15629 1:67264 7 7 7 0:83674 5
2:6581
1:12503
For the four natural modes of the flexible satellite, the natural frequencies are as follows: xn1 ¼ 0:7681 rad=s; xn2 ¼ 1:1038 rad=s; xn3 ¼ 1:8733 rad=s;xn4 ¼ 2:5496 rad=s and the dampings are given as f1 ¼ 0:005607;
f2 ¼ 0:00862;
f3 ¼ 0:01283;
f4 ¼ 0:02516
Fig. 3 Quaternion behavior for case of observer-based state feedback in Eqs. (28) and (29).
The maneuver of the considered flexible satellite is a rotation of 160° with an Euler axis, i.e., ¼ ½0:267261; 0:801784; 0:534523T Thus, the initial attitude described by the quaternion is 2 3 2 3 q1 ð0Þ 0:263201 6 7 6 7 q0 ð0Þ ¼ 0:173648; qv ð0Þ ¼ 4 q2 ð0Þ 5 ¼ 4 0:789603 5 q3 ð0Þ 0:526402 and the initial angular velocity of the spacecraft is xð0Þ ¼ ½0; 0; 0T In addition, the initial values of the four modals of the flexible appendages are gi ¼ 0:001;
wi ¼ 0:001;
i ¼ 1; 2; 3; 4
Firstly, we apply the partial state feedback controller Eqs. (11)–(13) to the flexible satellite. The parameters of the controller are as follows: k ¼ 150; B1 ¼ 6I44 ; A1 ¼ 13I44 ; Q1 ¼ 2600I88 ; Q ¼ 0:15I88 In addition, the parameter P is determined by solving the Lyapunov matrix Eq. (14). The behavior of the quaternion q ¼ q0 þ q1 i þ q2 j þ q3 k is shown in Fig. 1 where i; j and k are the fundamental quaternion units. The modal displace-
Fig. 4 Modal displacements for case of partial state feedback in Eqs. (28) and (29).
ments are shown in Fig. 2. From Fig. 1, it can be seen that the quaternion q of the closed-loop system is asymptotically convergent. Secondly, the observer-based feedback controller in Eqs. (28), (29) is also used for the flexible satellite, and the parameters are chosen to be as follows: k ¼ 300; B1 ¼ 6I44 ; A1 ¼ 15I44 ; Q2 ¼ 0:034I88 ; Q3 ¼ 10I88
436 437 438 439 440 441
Q1 ¼ 3500I88 ;
In addition, the matrices P2 and P3 are determined by Eq. (30). In this case, the behavior of the quaternion q ¼ q0 þ q1 i þ q2 j þ q3 k is shown in Fig. 3, and the modal displacements are shown in Fig. 4. From Fig. 3, it can be seen that the quaternion q is also asymptotically convergent when the observer-based control law is used. In addition, it can be seen from Figs. 2 and 4 that the vibration amplitude of the flexible modals under the observer-based control law is bigger than that under the state feedback control law.
Fig. 2 Modal displacements for case of partial state feedback in Eqs. (11)–(13).
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6. Conclusions
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In this paper, attitude control is considered without angular velocity measurement for satellites with flexible appendages
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Please cite this article in press as: ZHU Q et al. Attitude control without angular velocity measurement for flexible satellites, Chin J Aeronaut (2018), https://doi.org/ 10.1016/j.cja.2018.03.019
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Attitude control without angular velocity 456 457 458 459 460 461 462 463 464 465 466 467
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based on quaternion models. Firstly, a partial state feedback control law is proposed for this kind of satellites to guarantee convergence of the quaternion behaviors of the considered satellites. Secondly, when the modal variables describing flexible elements are not measurable, an observer-based partial state feedback control law is also presented. By applying this dynamic controller to the considered satellites, the convergence of the closed-loop system can also be guaranteed. A common feature of the two control laws proposed in this paper is that angular velocity measurement is not used. In future, we will use the idea in this paper to the problem of attitude tracking for flexible satellites. Acknowledgements
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This work was co-supported by the Major Program of National Natural Science Foundation of China (Nos. 61690210, 61690212), Shenzhen Municipal Basic Research Project for Discipline Layout (No. JCYJ20170413112722597), and Shenzhen Municipal Project for Basic Research (Nos. JCYJ20170307150952660, JCYJ20170307150227897).
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Please cite this article in press as: ZHU Q et al. Attitude control without angular velocity measurement for flexible satellites, Chin J Aeronaut (2018), https://doi.org/ 10.1016/j.cja.2018.03.019