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Station-keeping control for a stratospheric airship platform via fuzzy adaptive backstepping approach
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Advances in Space Research xxx (2012) xxx–xxx www.elsevier.com/locate/asr
Station-keeping control for a stratospheric airship platform via fuzzy adaptive backstepping approach Yueneng Yang, Jie Wu, Wei Zheng ⇑ Staff Room of Flight Dynamics and Control, College of Aerospace Science and Engineering, National University of Defense Technology, No. 47 Yanwachi Street, Kaifu District, Changsha 410073, China Received 1 August 2012; received in revised form 10 October 2012; accepted 18 October 2012
Abstract This paper presents a novel approach for station-keeping control of a stratospheric airship platform in the presence of parametric uncertainty and external disturbance. First, conceptual design of the stratospheric airship platform is introduced, including the target mission, configuration, energy sources, propeller and payload. Second, the dynamics model of the airship platform is presented, and the mathematical model of its horizontal motion is derived. Third, a fuzzy adaptive backstepping control approach is proposed to develop the station-keeping control system for the simplified horizontal motion. The backstepping controller is designed assuming that the airship model is accurately known, and a fuzzy adaptive algorithm is used to approximate the uncertainty of the airship model. The stability of the closed-loop control system is proven via the Lyapunov theorem. Finally, simulation results illustrate the effectiveness and robustness of the proposed control approach. Ó 2012 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Stratospheric airship platform; Conceptual design; Backstepping control; Fuzzy logic system; Adaptation law; Station keeping
1. Introduction Near space is quantitatively defined as the range of earth altitudes from 20 km to 100 km, below which aerial aircraft can produce sufficient lift for steady flight, and above which the atmosphere is rarefied enough for satellites to orbit with meaningful lifetimes (Young et al., 2009). There is a growing worldwide interest in a new concept consisting of using autonomous fight vehicles as platforms operating for extended periods of time at an altitudes around 20 km to accomplish various missions. As a typical lighterthan-air vehicle, the stratosphere airship has an enormous, yet untapped potential as low-speed, or even steady platforms for various applications, such as telecommunication, broadcasting relays, region navigation, environmental ⇑ Corresponding author. Tel./fax: +86 731 84573139.
E-mail addresses: [email protected] (Y.N. Yang), wujie_nudt@ 163.com (J. Wu), [email protected] (W. Zheng).
monitoring, scientific exploration, and so on (Chu et al., 2007). Especially, the stratosphere airship provides a unique and promising platform for disaster perception, which has several advantages over satellites and scouts: compared with satellites, the platform is highly costeffective, very mobile, fast to deploy and convenient to retrieve; compared with scouts, the platform has long duration, wide coverage and great survivability (Smith et al., 2011). Station-keeping flight is a unique feature of the stratosphere airship, in contrast to a fixed-wing aircraft. A key technical challenge for the airship platform is autonomous station keeping, or the ability to remain fixed over a specified geo-location in the presence of external disturbances. The problem of airship station keeping has received special attention in the literatures. Nagabhushan & Tomlinson (1982) presented a feedback control method for a quadrotor heavy lift airship with a sling load, and investigated the dynamics and control characteristics of the airship
0273-1177/$36.00 Ó 2012 COSPAR. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.asr.2012.10.014
Please cite this article in press as: Yang, Y.N., et al. Station-keeping control for a stratospheric airship platform via fuzzy adaptive backstepping approach. J. Adv. Space Res. (2012), http://dx.doi.org/10.1016/j.asr.2012.10.014
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Y.N. Yang et al. / Advances in Space Research xxx (2012) xxx–xxx
while it is hovering. Zwaan et al. (2000) proposed an image-based control strategy for an airship, and developed a PID controller for the station-keeping problem. However, PID controller cannot assure closed-loop performance specifications for different operating conditions. Schmidt (2007) developed a control law followed a conventional loop-shaping approach compatible with the characteristic dynamics of a large high altitude airship, and focused on the analysis of the station-keeping performance of the airship. These control methods abovementioned have limitations because it was developed based on the linear model, neglecting dynamic nonlinearity and coupling effects between longitudinal and lateral motion. Benjovengo (2009) adopted the sliding mode control approach to develop the control system for an autonomous airship, covering the full flight envelope from hovering to aerodynamic flight. Azinheira & Moutinho (2006) introduced a backstepping control approach for hover stabilization problem of an airship, which is robust against unmatched dynamics. These works presented several referenced control approaches for unmanned airships. However, researches on station-keeping control for the stratosphere airship platform are still rarely documented in published works. Motivated by the above studies, the current paper addresses the station-keeping problem of an airship platform in the presence of parametric uncertainty and external disturbance. We proposed a fuzzy adaptive backstepping control (FABC) scheme to tackle the planar stationkeeping problem of the airship platform. The backstepping control is a systematic and recursive design methodology for nonlinear systems. The idea of backstepping design is to select some appropriate functions of state variables as pseudo-control inputs for lower dimension subsystems of the overall system (Chen et al., 2012). It is known as a nonlinear recursive design method based on a control Lyapunov function, and provides the flexibility to treat nonlinear terms and guarantee the stability of a closed-loop system. The fuzzy logic system (FLS) has been widely applied to many control problems because they do not need an accurate mathematical model of the control system and they can cooperate with human expert knowledge (Yu et al., 2010). The FLS is a promising way to deal with the control problems of nonlinear systems containing highly uncertain nonlinear functions. It has been shown that FLS can be used to approximate any nonlinear function over a convex compact region (Wang et al., 2011). Based on this property, we employ the FLS to approximate the uncertainty in the dynamics model of the airship platform. In addition, an adaptive law is used to update the parameters of the FLS (Tong & Li, 2010). Combining all these terms, the proposed control approach has the merits of asymptotically steady and robustness against model uncertainty. Moreover, the stability of the closed-loop control system is proven via the Lyapunov theorem, and effectiveness and robustness of the proposed control scheme is demonstrated via simulation studies.
This paper is organized as follows. In Section 2, the conceptual design of the stratospheric airship platform is presented. In Section 3, the dynamics model and the simplified planar motion of the platform is formulated. Section 4 proposes the control scheme for designing the station-keeping control system. In Section 5, simulation studies test the performance of the proposed control system. Finally, conclusions are given in Section 6. 2. Conceptual design of the stratospheric airship platform A schematic representation of the stratospheric airship platform is illustrated in Fig. 1. The near space disaster perception system comprises stratospheric airship platforms that may cover kilometers of earth area. The platform has a task of maintaining a payload in a geostationary position at an altitude of approximately 20 km for several days or weeks, and could be used for disaster perception (Chu et al., 2007; Yang et al., 2012d). The stratospheric airship platform is designed to be capable of station keeping at an altitude of approximately 20 km, and to have long-term airborne presence of several days or weeks. The conceptual design configuration of the airship platform, which is consistent with the stationkeeping mission, is summarized in Table 1 (Mueller et al., 2004; Smith et al., 2011). The airship platform consists of an axis-symmetric, teardrop-shaped hull with solar power cells, propellers, tail fins, gondola and payload (Smith et al., 2011; Yang et al., 2012a). The hull is composed of an airpressurized envelope to maintain its shape, and internal divided bags filled with helium as a buoyant gas. Two air ballonets are installed inside the hull, which are controlled by a pneumatic system of pipes and valves. These ballonets can be blown up with air and deflated, respectively during the descent and climb operations, in order to handle altitude variations without losing helium from the hull and avoiding any significant change in the hull shape. The electric power is available from solar cells distributed to the propulsive motors, the flight control system and the payload mission. Propulsive propellers are mounted on the larboard, starboard and stern of the platform, and four tail fins of cross shape are installed on the rear end of the hull. The payload including optical cameras and other imaging equipments can accomplish missions such as disaster perception (Chu et al., 2007; Yang et al., 2012b). 3. Modeling of the stratospheric airship platform 3.1. Airship platform kinematics and dynamics The motion of an airship platform is usually represented by a set of kinematics and dynamics equations that describe its evolution in a six degrees-of-freedom (6DOF) space. The coordinate system is depicted in Fig. 2. OeXeYeZe is an earth-fixed inertial frame (E frame), with the origin on the surface of the earth, the X-axis points north, the Y-axis points east, and the Z-axis points down. obxbybzb
Please cite this article in press as: Yang, Y.N., et al. Station-keeping control for a stratospheric airship platform via fuzzy adaptive backstepping approach. J. Adv. Space Res. (2012), http://dx.doi.org/10.1016/j.asr.2012.10.014
Y.N. Yang et al. / Advances in Space Research xxx (2012) xxx–xxx
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Fig. 1. Disaster perception using stratospheric airship platforms.
Table 1 Conceptual design parameters of the platform. Design parameters
Value
Mission altitude h/km Maximum airspeed v/(m s–1) Length of the airship L/m Maximal diameter D/m Volume of the airship V/m3 Payloads mass m/kg
20 25 200 50 0.62 106 0.5 103
The generalized velocities of an airship are expressed by V = [v, x]T, where v = [u, v, w]T defines the linear velocities in the B frame, namely, the forward, lateral and vertical velocities; and x = [p, q, r]T denotes the angular velocities about each axis of the B frame (Yang et al., 2012c). Considering these motion variables, the 6-DOF kinematics equations of an airship can be written as (Yang et al., 2012d): g_ ¼ JðgÞV;
ð1Þ
where is the body-fixed frame (B frame), with the origin on the center of volume, the x-axis points forward, the y-axis points right, and z-axis points downward (Li et al., 2009). Under the established coordinate frames, the generalized coordinates of an airship are expressed by g = [P, X]T, where P = [x, y, z]T denotes the relative position with respect to the E frame, and X = [h, w, u]T defines the attitude angles, respectively, the pitch, yaw, and roll angles.
2
cos w cos h 6 J 1 ¼ 4 sin w cos h sin h
cos w sin h sin / sin w sin h sin / þ cos w cos / cos h sin /
JðgÞ ¼
J1
033
033
J2
;
ð2Þ
with J1 and J2 respectively being the rotation matrix from the B frame to the E frame and the transform matrix from angular velocities to attitude angle rates. The corresponding expressions of the two matrices can be expressed as follows:
3 cos w sin h cos / þ sin w sin / 7 sin w sin h cos / cos w sin / 5; cos h cos /
Please cite this article in press as: Yang, Y.N., et al. Station-keeping control for a stratospheric airship platform via fuzzy adaptive backstepping approach. J. Adv. Space Res. (2012), http://dx.doi.org/10.1016/j.asr.2012.10.014
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Fig. 2. Coordinate systems of an airship platform.
2
0
6 J2 ¼ 4 0 1
cos / sec h sin / tan h sin /
sin /
3
Under these assumptions, a simplified three degreesof-freedom model of planar motion is obtained as follows:
7 sec h cos / 5: tan h cos /
The 6-DOF dynamics equations of an airship can be expressed as follows (Lee et al., 2007; Cai et al., 2007): M V_ þ CðVÞV þ DðVÞV þ GðgÞ ¼ s;
M V_ þ CðVÞV þ DðVÞV ¼ s;
ð4Þ
g_ ¼ JðgÞV;
ð5Þ
where
ð3Þ
2
where M is the inertia matrix, C(V) is the centrifugal and Coriolis matrix, D(V) is the damping matrix, G(g) is the vector of gravitational and buoyant forces and moments, and s denotes the control forces and moments (Cai et al., 2007).
6 M¼4
3.2. Model of the station-keeping airship platform Considering the station-keeping control problem, we can restrict the six-dimensional dynamics to the horizontal plane by making the following assumptions: Assumption 1: The dynamics associated with the pitch and roll motions are negligible (Zhang et al., 2008). When the airship is cruising at a constant altitude, as shown in Fig. 3, pitch and roll variables are very small, and therefore, their effect on the motion in the horizontal plane can be neglected. Assumption 2: We consider an airship with neutral buoyancy, that is, the gravitation is equal to buoyancy. Therefore, the resultant forces of gravitation and buoyancy have no effect on the dynamics in the horizontal motion (Zhang et al., 2008).
m X u_
0
0 0
m Y v_ 0
2 6 CðVÞ ¼ 4 2
0
3
7 0 5; I 33 N r_
0 0
0 0
ðm Y v_ Þv
ðm X u_ Þu 3 0 7 0 5;
3 ðm Y v_ Þv 7 ðm X u_ Þu 5; 0
X u 0 6 DðVÞ ¼ 4 0 Y v 0 0 N r 2 3 cos w sin w 0 6 7 JðXÞ ¼ 4 sin w cos w 0 5; 0 0 1
where m is the gross mass; X u_ , Y v_ , N r_ , Xu, Yv, and Nr are the added inertial parameters (Cai, 2006); g = [x, y, w]T denotes the position and orientation in the E frame; V = [u, v, r]T denotes the forward, lateral, and yaw angular velocities; and s = [su, sv, sr]T denotes the forward force, lateral force, and yaw moment (Yang et al., 2012d). Eq. (4) is evidently only an explicit function of V, but the problem formulation should be an explicit function of g.
Please cite this article in press as: Yang, Y.N., et al. Station-keeping control for a stratospheric airship platform via fuzzy adaptive backstepping approach. J. Adv. Space Res. (2012), http://dx.doi.org/10.1016/j.asr.2012.10.014
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4. Station-keeping control system design The station-keeping problem is the design of a control law that asymptotically stabilizes both the position and orientation of the airship platform, to remain fixed over a specified geo-location in the presence of external disturbances. In the present section, we adopt the fuzzy adaptive backstepping approach to design the control system that actualizes station keeping of the airship platform. 4.1. Idea backstepping control Assume that the model parameters of the system described in Eq. (13) are accurately known, the design of backstepping control for the airship paltform is described step-by-step as follows: (Chen et al., 2012; Tong et al., 2010). Step 1: Define the following error: e1 ¼ g gd ¼ x1 yd ;
Fig. 3. Airship motion in station-keeping mode.
Therefore, we should transform Eq. (4) into an explicit function of g (Bagheri & Moghaddam, 2009). From Eq. (5), we obtain 1
_ V ¼ J ðgÞg:
ð6Þ
Differentiating Eq. (6), we derive
ð7Þ
ð15Þ
Define the following virtual control a1 ¼ k 1 e_ 1 þ y_ d ;
ð16Þ
ð17Þ
Substituting Eqs. (16) and (17) into Eq. (15), it is obtained that e_ 1 ¼ e2 þ a1 y_ d ¼ e2 k 1 e1 :
1 _ g MJ 1 ðgÞJðgÞJ ðgÞg_ MJ 1 ðgÞ€
ð18Þ
A candidate Lyapunov function for e1 is defined as
þ CðVÞJ 1 ðgÞg_ þ DðVÞJ 1 ðgÞg_
1 V 1 ¼ eT1 e1 : 2
1 _ ¼ MJ 1 ðgÞ€ g þ ½CðVÞ MJ 1 ðgÞJðgÞJ ðgÞg_
þ DðVÞJ 1 ðgÞg_ ¼ s:
ð19Þ
Differentiating Eq. (19) and using Eq. (18), it is obtained that
Define ð9Þ
1 _ C g ðVÞ ¼ ½CðVÞ MJ 1 ðgÞJðgÞJ ðgÞ;
ð10Þ
Dg ðVÞ ¼ DðVÞJ 1 ðgÞ:
ð11Þ
Eq. (4) can then be rewritten as g þ C g ðgÞg_ þ Dg ðgÞg_ ¼ s: M g ðgÞ€
e_ 1 ¼ x_ 1 y_ d ¼ x2 y_ d :
e2 ¼ x2 a1 :
Substituting Eq. (7) into Eq. (4), we obtain
M g ¼ MJ 1 ðgÞ;
where yd = gd is the command position and orientation. The derivative of e1 is:
where k1 is a positive constant. Define
1
@J ðgÞ g_ þ J 1 ðgÞ€ V_ ¼ g @t 1 _ ðgÞg_ þ J 1 ðgÞ€ g: ¼ J 1 ðgÞJðgÞJ
ð14Þ
ð12Þ
_ then Eq. We select the state variables as x1 = g, x2 ¼ g, (12) can be represented as: 8 > < x_ 1 ¼ x2 1 1 ð13Þ x_ 2 ¼ M 1 g s M C g x2 M g Dg x2 > : y ¼ x1 :
V_ 1 ¼ eT1 e_ 1 ¼ eT1 ðe2 k 1 e1 Þ ¼ k 1 eT1 e1 þ eT1 e2 :
ð20Þ
As shown by Eq. (20), if e2 = 0 then V_ 1 < 0. Therefore, we can conclude the system described in Eq. (13) is stable under the assumption e2 equals to zero. However, e2 is not always equals to zero, and we shall further design the control system to assure the error e2 equals to zero (Lee et al., 2007; Liu, 2008). Step 2: Differentiating Eq. (17), it is obtained that e_ 2 ¼ x_ 2 a_ 1 :
ð21Þ
Substituting Eq. (13) into Eq. (21), it is obtained that 1 1 _ 1: e_ 2 ¼ M 1 g s M g C g x2 M g Dg x2 a
ð22Þ
Please cite this article in press as: Yang, Y.N., et al. Station-keeping control for a stratospheric airship platform via fuzzy adaptive backstepping approach. J. Adv. Space Res. (2012), http://dx.doi.org/10.1016/j.asr.2012.10.014
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A candidate Lyapunov function is defined as 1 V 2 ¼ V 1 þ eT2 M g e2 : 2
ð23Þ
Differentiating Eq. (23), it is obtained that 1 1 1 _ V_ 2 ¼ V_ 1 þ eT2 M g e_ 2 þ e_ T2 M g e2 þ eT2 M g e2 2 2 2 1 _ ¼ k 1 eT1 e1 þ eT1 e2 þ eT2 M g e_ 2 þ eT2 M g e2 2 1 _ ¼ k 1 eT1 e1 þ eT1 e2 þ eT2 M g ðx_ 2 a_ 1 Þ þ eT2 M g e2 : 2
the airship model is accurately known. Then, the FLS is used to approximate the uncertainty of the airship model (Lin, 2010). In addition, an adaptive law is adopted to update the control parameters of the FLS (Tong & Li, 2010; Montaseri and Yazdanpanah, 2012). Fig. 4 depicts the block diagram of the control system. (1) FLS approximator.
ð24Þ
Based on the ideal backstepping controller in Eq. (26), we define T
Substituting Eq. (13) into Eq. (24), it is obtained that V_ 2 ¼ k1 eT1 e1 þ eT1 e2 þ eT2 ðM g a_ 1 C g a1 Dg a1 þ sÞ:
ð25Þ
According to Eq. (25), we can design the control law as follows: s ¼ M g a_ 1 þ C g a1 þ Dg a1 e1 k 2 e2 ;
ð26Þ
where k2 is a positive constant. Substituting Eq. (26) into Eq. (25), it is obtained that V_ 2 ¼ k 1 eT1 e1 þ eT1 e2 þ eT2 M g a_ 1 C g a1 Dg a1 þ M g a_ 1 þ C g a1 þ Dg a1 e1 k 2 e2 ¼ k 1 eT1 e1 k 2 eT2 e2 : ð27Þ
Since k1 and k2 are both positive definite, it is concluded that V_ 2 is less than zero from Eq. (27). Therefore, the stability of the control system is proven via Eq. (27). 4.2. Fuzzy adaptive backstepping control Since the model parameters of the airship platform may be unknown or perturbed in practical application, the ideal backstepping controller in Eq. (26) cannot be precisely obtained. Thus, a FABC system is proposed to develop the control system (Yu et al., 2010; Tong et al., 2009). The backstepping controller is first designed assuming that
f ðxÞ ¼ ½f1 ðxÞ; f2 ðxÞ; f3 ðxÞ ¼ M g a_ 1 þ C g a1 þ Dg a1 :
ð28Þ
Since Mg, Cg and Dg are uncertain in practice, we use the FLS to approximate f(x). Usually, a FLS consists of four parts: the knowledge base, the fuzzifier, the fuzzy inference engine working on fuzzy rules, and the defuzzifier. Suppose the FLS is constructed from the following IF-THEN rules (Wang et al., 2011): R(i): If x1 is F i1 and . . . and xn is F in , then yi is Bi, where x = [x1, x2, . . .xn]T and yi are the FLS input and output, respectively. F ij and Bi are the fuzzy sets. Construct the fuzzy basic function as (Chang et al., 2011): ni ðxÞ ¼ P
Pnj¼1 lF ij ðxj Þ h i; m n i¼1 Pj¼1 lF ij ðxj Þ
ð29Þ
where lF ij ðxj Þ is the membership function of xj. Specially, using the product inference engine, singleton fuzzifier and center average defuzzifier (Wang et al., 2011; Tong & Li, 2010), we have Pm n i¼1 ki ðPj¼1 lF ij ðxj ÞÞ i P : ð30Þ y ¼ m n i¼1 ðPj¼1 lF ij ðxj ÞÞ T
Denoting k ¼ ½k1 ; k2 ; . . . kn and n(x) = [n1(x), n2(x), . . . nm(x)]T, then Eq. (30) can be expressed as y i ¼ kT nðxÞ:
ð31Þ
Fig. 4. The control system block diagram.
Please cite this article in press as: Yang, Y.N., et al. Station-keeping control for a stratospheric airship platform via fuzzy adaptive backstepping approach. J. Adv. Space Res. (2012), http://dx.doi.org/10.1016/j.asr.2012.10.014
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Lemma 1 Tong & Li, 2010. Let g(x) be a continuous function defined on a compact set X. Then for any constant e > 0, there exists a FLS described in Eq. (32) such as supjgðxÞ kT nðxÞj < e:
ð32Þ
x2X
By Lemma 1, FLS are universal approximators, i.e., they can approximate any smooth function on a compact space. Due to this approximation capability, we apply the above-mentioned FLS to approximate the nonlinear function f(x). The corresponding FLS f^ðxÞ ¼ ½f^ 1 ðxÞ; T f^ 2 ðxÞ; f^ 3 ðxÞ for the nonlinear function is described as follows: Pm
i¼1 f^ 1 ðxÞ ¼ Pm
Pm
n i¼1 ðPj¼1 lF ij ðxj ÞÞ
i¼1 f^ 2 ðxÞ ¼ Pm
Pm f^ 3 ðxÞ ¼
k1i ðPnj¼1 lF ij ðxj ÞÞ k2i ðPnj¼1 lF ij ðxj ÞÞ
n i¼1 ðPj¼1 lF ij ðxj ÞÞ
n i¼1 k3i ðPj¼1 lF ij ðxj ÞÞ Pm n i¼1 ðPj¼1 lF i ðxj ÞÞ
¼ kT1 n1 ;
ð33Þ
¼ kT2 n2 ;
ð34Þ
¼ kT3 n3 :
ð35Þ
j
Denoting U ¼ ½k1 ; k2 ; . . . kn T and f(x) = [n1(x), n2(x), . . ., nm(x)]T, then Eqs. (33)–(35) can be expressed as: f^ðxÞ ¼ UT fðxÞ
ð36Þ
7
Differentiating Eq. (40), it is obtained that 1 ~T _ U V_ ¼ V_ 2 U c 1 T ~ U _ ¼ k 1 eT1 e1 k 2 eT2 e2 þ eT2 f UT fðxÞ U c h i T ¼ k 1 eT1 e1 k 2 eT2 e2 þ eT2 f U fðxÞ h T i 1 _ ~TU þ eT2 U fðxÞ UT fðxÞ U c T
6 k 1 eT1 e1 k 2 eT2 e2 þ keT2 k kf U fðxÞk 1 ~T _ þ eT2 ðUT fðxÞÞ U U c 1 1 2 6 k 1 eT1 e1 k 2 eT2 e2 þ keT2 k þ e2 2 2 T T 1 T _ : ~ þU e2 fðxÞ U c
ð41Þ
Substituting Eq. (38) into Eq. (41), it is obtained that 1 1 2 V_ 6 k 1 eT1 e1 k 2 eT2 e2 þ keT2 k þ e2 2 2 1 ~ T ðeT fðxÞÞT cðeT fT ðxÞÞT 2qU þU 2 2 c 1 1 2q ~ T ¼ k 1 eT1 e1 k 2 eT2 e2 þ eT2 e2 þ e2 þ U U 2 2 c
1 q 1 T ¼ k 1 eT1 e1 k 2 eT1 e2 þ ð2U U 2UT UÞ þ e2 : 2 c 2 ð42Þ
(3) Adaptive law. The optimal parameter vector is defined as (Tong et al., 2009) U ¼ arg min supjf^ðxÞ f ðxÞj : ð37Þ x2Rn
The adaptive law is defined as (Tong & Li, 2010): _ ¼ cðeT fT ðxÞÞT 2qU; U 2
ð38Þ
where c > 0 and q > 0 are both the optional parameters. The error between U* and U is defined as: ~ ¼ U U: U
ð39Þ
According to ðU UÞT ðU UÞ P 0, it is obtained that T
T
2U U 2UT U 6 UT U þ U U :
ð43Þ
Substituting Eq. (43) into Eq. (42), it is obtained that
_V 6 k 1 eT e1 k 2 1 eT e2 þ q UT U UT U þ 1 e2 1 2 2 c 2
1 q T U U UT U ¼ k 1 eT1 e1 k 2 eT2 e2 þ 2 c 2q T 1 þ U U þ e2 : ð44Þ c 2 According to the following equation ~ ¼ ðUT UT ÞðU UÞ ~TU U
(3) Stability analysis (Liu, 2008; Montaseri & Yazdanpanah, 2012). A candidate Lyapunov function is defined as 1 1 1 ~T ~ 1 ~T ~ V ¼ eT1 e1 þ eT2 M g e2 þ U U¼V2þ U U: 2 2 2c 2c
T
T
¼ U U þ UT U U U UT U T
6 2U U þ 2UT U;
ð45Þ
it is obtained that ð40Þ
1 ~T ~ T UT U U U 6 U U: 2
ð46Þ
Please cite this article in press as: Yang, Y.N., et al. Station-keeping control for a stratospheric airship platform via fuzzy adaptive backstepping approach. J. Adv. Space Res. (2012), http://dx.doi.org/10.1016/j.asr.2012.10.014
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Y.N. Yang et al. / Advances in Space Research xxx (2012) xxx–xxx
Substituting Eq. (46) into Eq. (44), it is obtained that
~TU ~ _V 6 k 1 eT e1 k 2 1 eT e2 q 1 U 1 2 2 c 2 2q T 1 þ U U þ e2 c 2
1 T 1 q ~T ~ T ¼ k 1 e1 e1 k 2 e2 M g M g e2 ðU UÞ 2 2c 2q T 1 þ U U þ e2 : ð47Þ c 2 It is supposed that kM g k 6 r, where r is positive definite (Liu, 2008). Through this assumption, we obtain M 1 g 6 r1 I, where I is the 3 3 identify matrix. Define a0 ¼ min k 1 ; r1 k 2 12 ; 12 q (Liu, 2008; Montaseri & Yazdanpanah, 2012), we derive the following equation from Eq. (47):
q ~T ~ 2q T 1 V_ 6 a0 eT1 e1 þ eT2 M g e2 þ U U þ U U þ e2 c c 2 2q T 1 2 ¼ a1 V 2 þ U U þ e ¼ a1 V 2 þ C; ð48Þ c 2
Fig. 5. Station-keeping control with accurate parameters and without disturbances.
T
where a1 = 2a0, and C ¼ 2qc U U þ 12 e2 . We can obtain the following equation deriving from Eq. (48) V ðtÞ 6 V ð0Þea1 t þ
C C ð1 ea1 t Þ 6 V ð0Þ þ ; a1 a1
ð49Þ
where V(0) is the initial value. Eq. (49) shows that the candidate Lyapunov function V is bounded. Consequently, boundedness of all signals is verified. It is proven that the candidate Lyapunov function V is bounded, and the closed-loop system controlled is bounded via (49). 5. Simulation results To evaluate the designed control system, repetitive simulation tests were performed via numerical simulation. The control system was simulated using the variable step Runge–Kutta integrator in MATLAB. The model parameters of the airship platform are available in the related literatures (Cai, 2006; Smith et al., 2011). The command state in station-keeping mode is gd = [xd, yd, wd]T = [500 m, 500 m, 0 rad]T, and the initial state of the airship platform is set to be g0 = [x0, y0, w0]T = [10 m, 10 m, 0 rad]T. Simulation results were obtained for two cases: (1) when the model parameters are accurately known, and (2) when there are parametric uncertainties and external disturbances. In all of the following simulations we applied the same control parameters for the station-keeping control. Case 1: The following simulations concern the stationkeeping control design based on the accurate model parameters. Simulation results of this case are shown in Figs. 5–8.
Fig. 6. Control errors with accurate parameters and without disturbances.
Fig. 5 shows the simulation results of station-keeping control with accurate parameters and without external disturbances. In Fig. 5 it can be observed that the airship platform trends to the target point precisely, which demonstrates that the proposed approach succeeds in stationkeeping control for the airship platform. Fig. 6 shows that the position and orientation errors of the airship platform with accurate parameters and without external disturbances. The position error in x-direct and ydirect both converge to zero within 300 s. The orientation error is varying with a maximum magnitude of 0.4 rad at the first 300 s, and then converges to zero. The abovementioned results demonstrate that the station-keeping control is accomplished with precision using the designed control system.
Please cite this article in press as: Yang, Y.N., et al. Station-keeping control for a stratospheric airship platform via fuzzy adaptive backstepping approach. J. Adv. Space Res. (2012), http://dx.doi.org/10.1016/j.asr.2012.10.014
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Fig. 7. Generalized velocities with accurate parameters and without disturbances.
Fig. 9. Station-keeping disturbances.
control
9
with
inaccurate
parameters
and
Fig. 10. Control errors with inaccurate parameters and disturbances. Fig. 8. Control disturbances.
inputs
with
accurate
parameters
and
without
Fig. 7 displays the airship platform velocities, namely, the forward speed, lateral speed, and yaw angular velocity. As shown in Fig. 7, the forward speed decreases from 10 to 0 m/s within 300 s, the lateral speed decreases from 6 to 0 m/s within 200 s, and the yaw angular velocity slowly varies at the first 300 s, and then converges to 0 rad/s. Fig. 8 displays the control inputs, namely, the control force in forward direction, the control force in lateral direction, and the control moment in yaw direction. As shown by the transition curves of the control inputs in Fig. 8, the changes in su, sv and sr are both asymptotically conver-
gent, and meet the controlling force and moment requirements for the station-keeping control perfectly. Case 2: We concern the robustness properties of the designed control system to parametric uncertainties and external disturbances. We conducted simulations in which errors of the order of 5% on all parameters were assumed. In practice, the external disturbances mainly may be the wind disturbances. We assume that the wind disturbances in lateral direction are dw = 10cos(t) m/s, where 10 m/s is the wind velocity, cos(t) is the cosine function, that is, wind disturbances vary in form of a cosine function with a magnitude of 10 m/s. Simulation results concerning the inaccurate model parameters and wind disturbances are shown in Figs. 9–12.
Please cite this article in press as: Yang, Y.N., et al. Station-keeping control for a stratospheric airship platform via fuzzy adaptive backstepping approach. J. Adv. Space Res. (2012), http://dx.doi.org/10.1016/j.asr.2012.10.014
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Y.N. Yang et al. / Advances in Space Research xxx (2012) xxx–xxx
imum magnitude of 0.4 rad at the first 300 s, and then converges to zero. The abovementioned simulation results demonstrate that the proposed approach accomplishes station-keeping control precisely despite parametric uncertainties and external disturbances. The general velocities are depicted in Fig. 11. The variation of these velocities in case 2 are a little different from those in case 1, due to the parametric uncertainties and external disturbances, As shown by the transition curves of the control inputs in Fig. 12, the control inputs actualize the station-keeping control in the presence of the parametric uncertainties and external disturbances. In contrast to case 1, the control input su converges to a neighborhood of zero, and slowly oscillates within. In contrast, the control inputs are different from the case with accurate parameters and without external disturbances. Fig. 11. Generalized disturbances.
velocities
with
inaccurate
parameters
and
6. Conclusions This paper proposes a novel conceptual design for stratospheric airship platform and a novel approach for station-keeping control of the stratospheric airship platform. The main results of the work are outlined as (1) introduction of a novel conceptual design of the stratospheric airship platform; (2) proposition of an original formulation of station-keeping control problem with an appropriate change of variable allowing the application of backstepping approach; (3) design of a FABC system that actualizes the station-keeping control of the airship platform against the parametric uncertainty and external disturbance; (4) and finally, demonstration the effectiveness and robustness of the proposed control approach. The proposed conceptual design and control scheme provide a promising approach for disaster perception using the stratospheric station-keeping airship platforms. Acknowledgments
Fig. 12. Control inputs with inaccurate parameters and disturbances.
Fig. 9 presents the simulation results of station-keeping control with inaccurate parameters and wind disturbances. From Fig. 9, it is clear that the proposed control approach succeeds in station keeping for the airship platform accurately despite parametric uncertainties and external disturbances. The trace in case 2 is more flexuous than that in case 1 due to the wind disturbances in lateral direction, as show in the magnified image. The control errors in position and orientation are shown in Fig. 10. In contrast to case 1, the position error in xdirection converges to a very small neighborhood of zero with 300 s. The position error in y-direction gradually converges to zero. The orientation error is varying with a max-
This research is supported by the Hunan Provincial Innovation Foundation for Postgraduates (CX2011B005) and the National University of Defense Technology Innovation Foundation for Postgraduates (B110105). References Azinheira, J.R., Moutinho, A. Airship hover stabilization using a backstepping control approach. J. Guid. Contr. Dynam. 29 (4), 903– 914, 2006. Bagheri, A., Moghaddam, J.J. Simulation and tracking control based on neural-network strategy and sliding-mode control for underwater remotely operated vehicle. Neurocomputing 72, 1934–1950, 2009. Benjovengo, F.P., Sliding mode control approaches for an autonomous unmanned airship. In: Proc. 18th AIAA Lighter-Than-Air Syst. Technol. Conf., 1–12, 2009. Cai, Z.L., Research on dynamical modeling and nonlinear control of a stratospheric airship. Dr Dissertation, Shanghai JIAO TONG University, Shanghai, China. 2006.
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Advances in Space Research xxx (2012) xxx–xxx www.elsevier.com/locate/asr
Station-keeping control for a stratospheric airship platform via fuzzy adaptive backstepping approach Yueneng Yang, Jie Wu, Wei Zheng ⇑ Staff Room of Flight Dynamics and Control, College of Aerospace Science and Engineering, National University of Defense Technology, No. 47 Yanwachi Street, Kaifu District, Changsha 410073, China Received 1 August 2012; received in revised form 10 October 2012; accepted 18 October 2012
Abstract This paper presents a novel approach for station-keeping control of a stratospheric airship platform in the presence of parametric uncertainty and external disturbance. First, conceptual design of the stratospheric airship platform is introduced, including the target mission, configuration, energy sources, propeller and payload. Second, the dynamics model of the airship platform is presented, and the mathematical model of its horizontal motion is derived. Third, a fuzzy adaptive backstepping control approach is proposed to develop the station-keeping control system for the simplified horizontal motion. The backstepping controller is designed assuming that the airship model is accurately known, and a fuzzy adaptive algorithm is used to approximate the uncertainty of the airship model. The stability of the closed-loop control system is proven via the Lyapunov theorem. Finally, simulation results illustrate the effectiveness and robustness of the proposed control approach. Ó 2012 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Stratospheric airship platform; Conceptual design; Backstepping control; Fuzzy logic system; Adaptation law; Station keeping
1. Introduction Near space is quantitatively defined as the range of earth altitudes from 20 km to 100 km, below which aerial aircraft can produce sufficient lift for steady flight, and above which the atmosphere is rarefied enough for satellites to orbit with meaningful lifetimes (Young et al., 2009). There is a growing worldwide interest in a new concept consisting of using autonomous fight vehicles as platforms operating for extended periods of time at an altitudes around 20 km to accomplish various missions. As a typical lighterthan-air vehicle, the stratosphere airship has an enormous, yet untapped potential as low-speed, or even steady platforms for various applications, such as telecommunication, broadcasting relays, region navigation, environmental ⇑ Corresponding author. Tel./fax: +86 731 84573139.
E-mail addresses: [email protected] (Y.N. Yang), wujie_nudt@ 163.com (J. Wu), [email protected] (W. Zheng).
monitoring, scientific exploration, and so on (Chu et al., 2007). Especially, the stratosphere airship provides a unique and promising platform for disaster perception, which has several advantages over satellites and scouts: compared with satellites, the platform is highly costeffective, very mobile, fast to deploy and convenient to retrieve; compared with scouts, the platform has long duration, wide coverage and great survivability (Smith et al., 2011). Station-keeping flight is a unique feature of the stratosphere airship, in contrast to a fixed-wing aircraft. A key technical challenge for the airship platform is autonomous station keeping, or the ability to remain fixed over a specified geo-location in the presence of external disturbances. The problem of airship station keeping has received special attention in the literatures. Nagabhushan & Tomlinson (1982) presented a feedback control method for a quadrotor heavy lift airship with a sling load, and investigated the dynamics and control characteristics of the airship
0273-1177/$36.00 Ó 2012 COSPAR. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.asr.2012.10.014
Please cite this article in press as: Yang, Y.N., et al. Station-keeping control for a stratospheric airship platform via fuzzy adaptive backstepping approach. J. Adv. Space Res. (2012), http://dx.doi.org/10.1016/j.asr.2012.10.014
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Y.N. Yang et al. / Advances in Space Research xxx (2012) xxx–xxx
while it is hovering. Zwaan et al. (2000) proposed an image-based control strategy for an airship, and developed a PID controller for the station-keeping problem. However, PID controller cannot assure closed-loop performance specifications for different operating conditions. Schmidt (2007) developed a control law followed a conventional loop-shaping approach compatible with the characteristic dynamics of a large high altitude airship, and focused on the analysis of the station-keeping performance of the airship. These control methods abovementioned have limitations because it was developed based on the linear model, neglecting dynamic nonlinearity and coupling effects between longitudinal and lateral motion. Benjovengo (2009) adopted the sliding mode control approach to develop the control system for an autonomous airship, covering the full flight envelope from hovering to aerodynamic flight. Azinheira & Moutinho (2006) introduced a backstepping control approach for hover stabilization problem of an airship, which is robust against unmatched dynamics. These works presented several referenced control approaches for unmanned airships. However, researches on station-keeping control for the stratosphere airship platform are still rarely documented in published works. Motivated by the above studies, the current paper addresses the station-keeping problem of an airship platform in the presence of parametric uncertainty and external disturbance. We proposed a fuzzy adaptive backstepping control (FABC) scheme to tackle the planar stationkeeping problem of the airship platform. The backstepping control is a systematic and recursive design methodology for nonlinear systems. The idea of backstepping design is to select some appropriate functions of state variables as pseudo-control inputs for lower dimension subsystems of the overall system (Chen et al., 2012). It is known as a nonlinear recursive design method based on a control Lyapunov function, and provides the flexibility to treat nonlinear terms and guarantee the stability of a closed-loop system. The fuzzy logic system (FLS) has been widely applied to many control problems because they do not need an accurate mathematical model of the control system and they can cooperate with human expert knowledge (Yu et al., 2010). The FLS is a promising way to deal with the control problems of nonlinear systems containing highly uncertain nonlinear functions. It has been shown that FLS can be used to approximate any nonlinear function over a convex compact region (Wang et al., 2011). Based on this property, we employ the FLS to approximate the uncertainty in the dynamics model of the airship platform. In addition, an adaptive law is used to update the parameters of the FLS (Tong & Li, 2010). Combining all these terms, the proposed control approach has the merits of asymptotically steady and robustness against model uncertainty. Moreover, the stability of the closed-loop control system is proven via the Lyapunov theorem, and effectiveness and robustness of the proposed control scheme is demonstrated via simulation studies.
This paper is organized as follows. In Section 2, the conceptual design of the stratospheric airship platform is presented. In Section 3, the dynamics model and the simplified planar motion of the platform is formulated. Section 4 proposes the control scheme for designing the station-keeping control system. In Section 5, simulation studies test the performance of the proposed control system. Finally, conclusions are given in Section 6. 2. Conceptual design of the stratospheric airship platform A schematic representation of the stratospheric airship platform is illustrated in Fig. 1. The near space disaster perception system comprises stratospheric airship platforms that may cover kilometers of earth area. The platform has a task of maintaining a payload in a geostationary position at an altitude of approximately 20 km for several days or weeks, and could be used for disaster perception (Chu et al., 2007; Yang et al., 2012d). The stratospheric airship platform is designed to be capable of station keeping at an altitude of approximately 20 km, and to have long-term airborne presence of several days or weeks. The conceptual design configuration of the airship platform, which is consistent with the stationkeeping mission, is summarized in Table 1 (Mueller et al., 2004; Smith et al., 2011). The airship platform consists of an axis-symmetric, teardrop-shaped hull with solar power cells, propellers, tail fins, gondola and payload (Smith et al., 2011; Yang et al., 2012a). The hull is composed of an airpressurized envelope to maintain its shape, and internal divided bags filled with helium as a buoyant gas. Two air ballonets are installed inside the hull, which are controlled by a pneumatic system of pipes and valves. These ballonets can be blown up with air and deflated, respectively during the descent and climb operations, in order to handle altitude variations without losing helium from the hull and avoiding any significant change in the hull shape. The electric power is available from solar cells distributed to the propulsive motors, the flight control system and the payload mission. Propulsive propellers are mounted on the larboard, starboard and stern of the platform, and four tail fins of cross shape are installed on the rear end of the hull. The payload including optical cameras and other imaging equipments can accomplish missions such as disaster perception (Chu et al., 2007; Yang et al., 2012b). 3. Modeling of the stratospheric airship platform 3.1. Airship platform kinematics and dynamics The motion of an airship platform is usually represented by a set of kinematics and dynamics equations that describe its evolution in a six degrees-of-freedom (6DOF) space. The coordinate system is depicted in Fig. 2. OeXeYeZe is an earth-fixed inertial frame (E frame), with the origin on the surface of the earth, the X-axis points north, the Y-axis points east, and the Z-axis points down. obxbybzb
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Fig. 1. Disaster perception using stratospheric airship platforms.
Table 1 Conceptual design parameters of the platform. Design parameters
Value
Mission altitude h/km Maximum airspeed v/(m s–1) Length of the airship L/m Maximal diameter D/m Volume of the airship V/m3 Payloads mass m/kg
20 25 200 50 0.62 106 0.5 103
The generalized velocities of an airship are expressed by V = [v, x]T, where v = [u, v, w]T defines the linear velocities in the B frame, namely, the forward, lateral and vertical velocities; and x = [p, q, r]T denotes the angular velocities about each axis of the B frame (Yang et al., 2012c). Considering these motion variables, the 6-DOF kinematics equations of an airship can be written as (Yang et al., 2012d): g_ ¼ JðgÞV;
ð1Þ
where is the body-fixed frame (B frame), with the origin on the center of volume, the x-axis points forward, the y-axis points right, and z-axis points downward (Li et al., 2009). Under the established coordinate frames, the generalized coordinates of an airship are expressed by g = [P, X]T, where P = [x, y, z]T denotes the relative position with respect to the E frame, and X = [h, w, u]T defines the attitude angles, respectively, the pitch, yaw, and roll angles.
2
cos w cos h 6 J 1 ¼ 4 sin w cos h sin h
cos w sin h sin / sin w sin h sin / þ cos w cos / cos h sin /
JðgÞ ¼
J1
033
033
J2
;
ð2Þ
with J1 and J2 respectively being the rotation matrix from the B frame to the E frame and the transform matrix from angular velocities to attitude angle rates. The corresponding expressions of the two matrices can be expressed as follows:
3 cos w sin h cos / þ sin w sin / 7 sin w sin h cos / cos w sin / 5; cos h cos /
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Fig. 2. Coordinate systems of an airship platform.
2
0
6 J2 ¼ 4 0 1
cos / sec h sin / tan h sin /
sin /
3
Under these assumptions, a simplified three degreesof-freedom model of planar motion is obtained as follows:
7 sec h cos / 5: tan h cos /
The 6-DOF dynamics equations of an airship can be expressed as follows (Lee et al., 2007; Cai et al., 2007): M V_ þ CðVÞV þ DðVÞV þ GðgÞ ¼ s;
M V_ þ CðVÞV þ DðVÞV ¼ s;
ð4Þ
g_ ¼ JðgÞV;
ð5Þ
where
ð3Þ
2
where M is the inertia matrix, C(V) is the centrifugal and Coriolis matrix, D(V) is the damping matrix, G(g) is the vector of gravitational and buoyant forces and moments, and s denotes the control forces and moments (Cai et al., 2007).
6 M¼4
3.2. Model of the station-keeping airship platform Considering the station-keeping control problem, we can restrict the six-dimensional dynamics to the horizontal plane by making the following assumptions: Assumption 1: The dynamics associated with the pitch and roll motions are negligible (Zhang et al., 2008). When the airship is cruising at a constant altitude, as shown in Fig. 3, pitch and roll variables are very small, and therefore, their effect on the motion in the horizontal plane can be neglected. Assumption 2: We consider an airship with neutral buoyancy, that is, the gravitation is equal to buoyancy. Therefore, the resultant forces of gravitation and buoyancy have no effect on the dynamics in the horizontal motion (Zhang et al., 2008).
m X u_
0
0 0
m Y v_ 0
2 6 CðVÞ ¼ 4 2
0
3
7 0 5; I 33 N r_
0 0
0 0
ðm Y v_ Þv
ðm X u_ Þu 3 0 7 0 5;
3 ðm Y v_ Þv 7 ðm X u_ Þu 5; 0
X u 0 6 DðVÞ ¼ 4 0 Y v 0 0 N r 2 3 cos w sin w 0 6 7 JðXÞ ¼ 4 sin w cos w 0 5; 0 0 1
where m is the gross mass; X u_ , Y v_ , N r_ , Xu, Yv, and Nr are the added inertial parameters (Cai, 2006); g = [x, y, w]T denotes the position and orientation in the E frame; V = [u, v, r]T denotes the forward, lateral, and yaw angular velocities; and s = [su, sv, sr]T denotes the forward force, lateral force, and yaw moment (Yang et al., 2012d). Eq. (4) is evidently only an explicit function of V, but the problem formulation should be an explicit function of g.
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4. Station-keeping control system design The station-keeping problem is the design of a control law that asymptotically stabilizes both the position and orientation of the airship platform, to remain fixed over a specified geo-location in the presence of external disturbances. In the present section, we adopt the fuzzy adaptive backstepping approach to design the control system that actualizes station keeping of the airship platform. 4.1. Idea backstepping control Assume that the model parameters of the system described in Eq. (13) are accurately known, the design of backstepping control for the airship paltform is described step-by-step as follows: (Chen et al., 2012; Tong et al., 2010). Step 1: Define the following error: e1 ¼ g gd ¼ x1 yd ;
Fig. 3. Airship motion in station-keeping mode.
Therefore, we should transform Eq. (4) into an explicit function of g (Bagheri & Moghaddam, 2009). From Eq. (5), we obtain 1
_ V ¼ J ðgÞg:
ð6Þ
Differentiating Eq. (6), we derive
ð7Þ
ð15Þ
Define the following virtual control a1 ¼ k 1 e_ 1 þ y_ d ;
ð16Þ
ð17Þ
Substituting Eqs. (16) and (17) into Eq. (15), it is obtained that e_ 1 ¼ e2 þ a1 y_ d ¼ e2 k 1 e1 :
1 _ g MJ 1 ðgÞJðgÞJ ðgÞg_ MJ 1 ðgÞ€
ð18Þ
A candidate Lyapunov function for e1 is defined as
þ CðVÞJ 1 ðgÞg_ þ DðVÞJ 1 ðgÞg_
1 V 1 ¼ eT1 e1 : 2
1 _ ¼ MJ 1 ðgÞ€ g þ ½CðVÞ MJ 1 ðgÞJðgÞJ ðgÞg_
þ DðVÞJ 1 ðgÞg_ ¼ s:
ð19Þ
Differentiating Eq. (19) and using Eq. (18), it is obtained that
Define ð9Þ
1 _ C g ðVÞ ¼ ½CðVÞ MJ 1 ðgÞJðgÞJ ðgÞ;
ð10Þ
Dg ðVÞ ¼ DðVÞJ 1 ðgÞ:
ð11Þ
Eq. (4) can then be rewritten as g þ C g ðgÞg_ þ Dg ðgÞg_ ¼ s: M g ðgÞ€
e_ 1 ¼ x_ 1 y_ d ¼ x2 y_ d :
e2 ¼ x2 a1 :
Substituting Eq. (7) into Eq. (4), we obtain
M g ¼ MJ 1 ðgÞ;
where yd = gd is the command position and orientation. The derivative of e1 is:
where k1 is a positive constant. Define
1
@J ðgÞ g_ þ J 1 ðgÞ€ V_ ¼ g @t 1 _ ðgÞg_ þ J 1 ðgÞ€ g: ¼ J 1 ðgÞJðgÞJ
ð14Þ
ð12Þ
_ then Eq. We select the state variables as x1 = g, x2 ¼ g, (12) can be represented as: 8 > < x_ 1 ¼ x2 1 1 ð13Þ x_ 2 ¼ M 1 g s M C g x2 M g Dg x2 > : y ¼ x1 :
V_ 1 ¼ eT1 e_ 1 ¼ eT1 ðe2 k 1 e1 Þ ¼ k 1 eT1 e1 þ eT1 e2 :
ð20Þ
As shown by Eq. (20), if e2 = 0 then V_ 1 < 0. Therefore, we can conclude the system described in Eq. (13) is stable under the assumption e2 equals to zero. However, e2 is not always equals to zero, and we shall further design the control system to assure the error e2 equals to zero (Lee et al., 2007; Liu, 2008). Step 2: Differentiating Eq. (17), it is obtained that e_ 2 ¼ x_ 2 a_ 1 :
ð21Þ
Substituting Eq. (13) into Eq. (21), it is obtained that 1 1 _ 1: e_ 2 ¼ M 1 g s M g C g x2 M g Dg x2 a
ð22Þ
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A candidate Lyapunov function is defined as 1 V 2 ¼ V 1 þ eT2 M g e2 : 2
ð23Þ
Differentiating Eq. (23), it is obtained that 1 1 1 _ V_ 2 ¼ V_ 1 þ eT2 M g e_ 2 þ e_ T2 M g e2 þ eT2 M g e2 2 2 2 1 _ ¼ k 1 eT1 e1 þ eT1 e2 þ eT2 M g e_ 2 þ eT2 M g e2 2 1 _ ¼ k 1 eT1 e1 þ eT1 e2 þ eT2 M g ðx_ 2 a_ 1 Þ þ eT2 M g e2 : 2
the airship model is accurately known. Then, the FLS is used to approximate the uncertainty of the airship model (Lin, 2010). In addition, an adaptive law is adopted to update the control parameters of the FLS (Tong & Li, 2010; Montaseri and Yazdanpanah, 2012). Fig. 4 depicts the block diagram of the control system. (1) FLS approximator.
ð24Þ
Based on the ideal backstepping controller in Eq. (26), we define T
Substituting Eq. (13) into Eq. (24), it is obtained that V_ 2 ¼ k1 eT1 e1 þ eT1 e2 þ eT2 ðM g a_ 1 C g a1 Dg a1 þ sÞ:
ð25Þ
According to Eq. (25), we can design the control law as follows: s ¼ M g a_ 1 þ C g a1 þ Dg a1 e1 k 2 e2 ;
ð26Þ
where k2 is a positive constant. Substituting Eq. (26) into Eq. (25), it is obtained that V_ 2 ¼ k 1 eT1 e1 þ eT1 e2 þ eT2 M g a_ 1 C g a1 Dg a1 þ M g a_ 1 þ C g a1 þ Dg a1 e1 k 2 e2 ¼ k 1 eT1 e1 k 2 eT2 e2 : ð27Þ
Since k1 and k2 are both positive definite, it is concluded that V_ 2 is less than zero from Eq. (27). Therefore, the stability of the control system is proven via Eq. (27). 4.2. Fuzzy adaptive backstepping control Since the model parameters of the airship platform may be unknown or perturbed in practical application, the ideal backstepping controller in Eq. (26) cannot be precisely obtained. Thus, a FABC system is proposed to develop the control system (Yu et al., 2010; Tong et al., 2009). The backstepping controller is first designed assuming that
f ðxÞ ¼ ½f1 ðxÞ; f2 ðxÞ; f3 ðxÞ ¼ M g a_ 1 þ C g a1 þ Dg a1 :
ð28Þ
Since Mg, Cg and Dg are uncertain in practice, we use the FLS to approximate f(x). Usually, a FLS consists of four parts: the knowledge base, the fuzzifier, the fuzzy inference engine working on fuzzy rules, and the defuzzifier. Suppose the FLS is constructed from the following IF-THEN rules (Wang et al., 2011): R(i): If x1 is F i1 and . . . and xn is F in , then yi is Bi, where x = [x1, x2, . . .xn]T and yi are the FLS input and output, respectively. F ij and Bi are the fuzzy sets. Construct the fuzzy basic function as (Chang et al., 2011): ni ðxÞ ¼ P
Pnj¼1 lF ij ðxj Þ h i; m n i¼1 Pj¼1 lF ij ðxj Þ
ð29Þ
where lF ij ðxj Þ is the membership function of xj. Specially, using the product inference engine, singleton fuzzifier and center average defuzzifier (Wang et al., 2011; Tong & Li, 2010), we have Pm n i¼1 ki ðPj¼1 lF ij ðxj ÞÞ i P : ð30Þ y ¼ m n i¼1 ðPj¼1 lF ij ðxj ÞÞ T
Denoting k ¼ ½k1 ; k2 ; . . . kn and n(x) = [n1(x), n2(x), . . . nm(x)]T, then Eq. (30) can be expressed as y i ¼ kT nðxÞ:
ð31Þ
Fig. 4. The control system block diagram.
Please cite this article in press as: Yang, Y.N., et al. Station-keeping control for a stratospheric airship platform via fuzzy adaptive backstepping approach. J. Adv. Space Res. (2012), http://dx.doi.org/10.1016/j.asr.2012.10.014
Y.N. Yang et al. / Advances in Space Research xxx (2012) xxx–xxx
Lemma 1 Tong & Li, 2010. Let g(x) be a continuous function defined on a compact set X. Then for any constant e > 0, there exists a FLS described in Eq. (32) such as supjgðxÞ kT nðxÞj < e:
ð32Þ
x2X
By Lemma 1, FLS are universal approximators, i.e., they can approximate any smooth function on a compact space. Due to this approximation capability, we apply the above-mentioned FLS to approximate the nonlinear function f(x). The corresponding FLS f^ðxÞ ¼ ½f^ 1 ðxÞ; T f^ 2 ðxÞ; f^ 3 ðxÞ for the nonlinear function is described as follows: Pm
i¼1 f^ 1 ðxÞ ¼ Pm
Pm
n i¼1 ðPj¼1 lF ij ðxj ÞÞ
i¼1 f^ 2 ðxÞ ¼ Pm
Pm f^ 3 ðxÞ ¼
k1i ðPnj¼1 lF ij ðxj ÞÞ k2i ðPnj¼1 lF ij ðxj ÞÞ
n i¼1 ðPj¼1 lF ij ðxj ÞÞ
n i¼1 k3i ðPj¼1 lF ij ðxj ÞÞ Pm n i¼1 ðPj¼1 lF i ðxj ÞÞ
¼ kT1 n1 ;
ð33Þ
¼ kT2 n2 ;
ð34Þ
¼ kT3 n3 :
ð35Þ
j
Denoting U ¼ ½k1 ; k2 ; . . . kn T and f(x) = [n1(x), n2(x), . . ., nm(x)]T, then Eqs. (33)–(35) can be expressed as: f^ðxÞ ¼ UT fðxÞ
ð36Þ
7
Differentiating Eq. (40), it is obtained that 1 ~T _ U V_ ¼ V_ 2 U c 1 T ~ U _ ¼ k 1 eT1 e1 k 2 eT2 e2 þ eT2 f UT fðxÞ U c h i T ¼ k 1 eT1 e1 k 2 eT2 e2 þ eT2 f U fðxÞ h T i 1 _ ~TU þ eT2 U fðxÞ UT fðxÞ U c T
6 k 1 eT1 e1 k 2 eT2 e2 þ keT2 k kf U fðxÞk 1 ~T _ þ eT2 ðUT fðxÞÞ U U c 1 1 2 6 k 1 eT1 e1 k 2 eT2 e2 þ keT2 k þ e2 2 2 T T 1 T _ : ~ þU e2 fðxÞ U c
ð41Þ
Substituting Eq. (38) into Eq. (41), it is obtained that 1 1 2 V_ 6 k 1 eT1 e1 k 2 eT2 e2 þ keT2 k þ e2 2 2 1 ~ T ðeT fðxÞÞT cðeT fT ðxÞÞT 2qU þU 2 2 c 1 1 2q ~ T ¼ k 1 eT1 e1 k 2 eT2 e2 þ eT2 e2 þ e2 þ U U 2 2 c
1 q 1 T ¼ k 1 eT1 e1 k 2 eT1 e2 þ ð2U U 2UT UÞ þ e2 : 2 c 2 ð42Þ
(3) Adaptive law. The optimal parameter vector is defined as (Tong et al., 2009) U ¼ arg min supjf^ðxÞ f ðxÞj : ð37Þ x2Rn
The adaptive law is defined as (Tong & Li, 2010): _ ¼ cðeT fT ðxÞÞT 2qU; U 2
ð38Þ
where c > 0 and q > 0 are both the optional parameters. The error between U* and U is defined as: ~ ¼ U U: U
ð39Þ
According to ðU UÞT ðU UÞ P 0, it is obtained that T
T
2U U 2UT U 6 UT U þ U U :
ð43Þ
Substituting Eq. (43) into Eq. (42), it is obtained that
_V 6 k 1 eT e1 k 2 1 eT e2 þ q UT U UT U þ 1 e2 1 2 2 c 2
1 q T U U UT U ¼ k 1 eT1 e1 k 2 eT2 e2 þ 2 c 2q T 1 þ U U þ e2 : ð44Þ c 2 According to the following equation ~ ¼ ðUT UT ÞðU UÞ ~TU U
(3) Stability analysis (Liu, 2008; Montaseri & Yazdanpanah, 2012). A candidate Lyapunov function is defined as 1 1 1 ~T ~ 1 ~T ~ V ¼ eT1 e1 þ eT2 M g e2 þ U U¼V2þ U U: 2 2 2c 2c
T
T
¼ U U þ UT U U U UT U T
6 2U U þ 2UT U;
ð45Þ
it is obtained that ð40Þ
1 ~T ~ T UT U U U 6 U U: 2
ð46Þ
Please cite this article in press as: Yang, Y.N., et al. Station-keeping control for a stratospheric airship platform via fuzzy adaptive backstepping approach. J. Adv. Space Res. (2012), http://dx.doi.org/10.1016/j.asr.2012.10.014
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Y.N. Yang et al. / Advances in Space Research xxx (2012) xxx–xxx
Substituting Eq. (46) into Eq. (44), it is obtained that
~TU ~ _V 6 k 1 eT e1 k 2 1 eT e2 q 1 U 1 2 2 c 2 2q T 1 þ U U þ e2 c 2
1 T 1 q ~T ~ T ¼ k 1 e1 e1 k 2 e2 M g M g e2 ðU UÞ 2 2c 2q T 1 þ U U þ e2 : ð47Þ c 2 It is supposed that kM g k 6 r, where r is positive definite (Liu, 2008). Through this assumption, we obtain M 1 g 6 r1 I, where I is the 3 3 identify matrix. Define a0 ¼ min k 1 ; r1 k 2 12 ; 12 q (Liu, 2008; Montaseri & Yazdanpanah, 2012), we derive the following equation from Eq. (47):
q ~T ~ 2q T 1 V_ 6 a0 eT1 e1 þ eT2 M g e2 þ U U þ U U þ e2 c c 2 2q T 1 2 ¼ a1 V 2 þ U U þ e ¼ a1 V 2 þ C; ð48Þ c 2
Fig. 5. Station-keeping control with accurate parameters and without disturbances.
T
where a1 = 2a0, and C ¼ 2qc U U þ 12 e2 . We can obtain the following equation deriving from Eq. (48) V ðtÞ 6 V ð0Þea1 t þ
C C ð1 ea1 t Þ 6 V ð0Þ þ ; a1 a1
ð49Þ
where V(0) is the initial value. Eq. (49) shows that the candidate Lyapunov function V is bounded. Consequently, boundedness of all signals is verified. It is proven that the candidate Lyapunov function V is bounded, and the closed-loop system controlled is bounded via (49). 5. Simulation results To evaluate the designed control system, repetitive simulation tests were performed via numerical simulation. The control system was simulated using the variable step Runge–Kutta integrator in MATLAB. The model parameters of the airship platform are available in the related literatures (Cai, 2006; Smith et al., 2011). The command state in station-keeping mode is gd = [xd, yd, wd]T = [500 m, 500 m, 0 rad]T, and the initial state of the airship platform is set to be g0 = [x0, y0, w0]T = [10 m, 10 m, 0 rad]T. Simulation results were obtained for two cases: (1) when the model parameters are accurately known, and (2) when there are parametric uncertainties and external disturbances. In all of the following simulations we applied the same control parameters for the station-keeping control. Case 1: The following simulations concern the stationkeeping control design based on the accurate model parameters. Simulation results of this case are shown in Figs. 5–8.
Fig. 6. Control errors with accurate parameters and without disturbances.
Fig. 5 shows the simulation results of station-keeping control with accurate parameters and without external disturbances. In Fig. 5 it can be observed that the airship platform trends to the target point precisely, which demonstrates that the proposed approach succeeds in stationkeeping control for the airship platform. Fig. 6 shows that the position and orientation errors of the airship platform with accurate parameters and without external disturbances. The position error in x-direct and ydirect both converge to zero within 300 s. The orientation error is varying with a maximum magnitude of 0.4 rad at the first 300 s, and then converges to zero. The abovementioned results demonstrate that the station-keeping control is accomplished with precision using the designed control system.
Please cite this article in press as: Yang, Y.N., et al. Station-keeping control for a stratospheric airship platform via fuzzy adaptive backstepping approach. J. Adv. Space Res. (2012), http://dx.doi.org/10.1016/j.asr.2012.10.014
Y.N. Yang et al. / Advances in Space Research xxx (2012) xxx–xxx
Fig. 7. Generalized velocities with accurate parameters and without disturbances.
Fig. 9. Station-keeping disturbances.
control
9
with
inaccurate
parameters
and
Fig. 10. Control errors with inaccurate parameters and disturbances. Fig. 8. Control disturbances.
inputs
with
accurate
parameters
and
without
Fig. 7 displays the airship platform velocities, namely, the forward speed, lateral speed, and yaw angular velocity. As shown in Fig. 7, the forward speed decreases from 10 to 0 m/s within 300 s, the lateral speed decreases from 6 to 0 m/s within 200 s, and the yaw angular velocity slowly varies at the first 300 s, and then converges to 0 rad/s. Fig. 8 displays the control inputs, namely, the control force in forward direction, the control force in lateral direction, and the control moment in yaw direction. As shown by the transition curves of the control inputs in Fig. 8, the changes in su, sv and sr are both asymptotically conver-
gent, and meet the controlling force and moment requirements for the station-keeping control perfectly. Case 2: We concern the robustness properties of the designed control system to parametric uncertainties and external disturbances. We conducted simulations in which errors of the order of 5% on all parameters were assumed. In practice, the external disturbances mainly may be the wind disturbances. We assume that the wind disturbances in lateral direction are dw = 10cos(t) m/s, where 10 m/s is the wind velocity, cos(t) is the cosine function, that is, wind disturbances vary in form of a cosine function with a magnitude of 10 m/s. Simulation results concerning the inaccurate model parameters and wind disturbances are shown in Figs. 9–12.
Please cite this article in press as: Yang, Y.N., et al. Station-keeping control for a stratospheric airship platform via fuzzy adaptive backstepping approach. J. Adv. Space Res. (2012), http://dx.doi.org/10.1016/j.asr.2012.10.014
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Y.N. Yang et al. / Advances in Space Research xxx (2012) xxx–xxx
imum magnitude of 0.4 rad at the first 300 s, and then converges to zero. The abovementioned simulation results demonstrate that the proposed approach accomplishes station-keeping control precisely despite parametric uncertainties and external disturbances. The general velocities are depicted in Fig. 11. The variation of these velocities in case 2 are a little different from those in case 1, due to the parametric uncertainties and external disturbances, As shown by the transition curves of the control inputs in Fig. 12, the control inputs actualize the station-keeping control in the presence of the parametric uncertainties and external disturbances. In contrast to case 1, the control input su converges to a neighborhood of zero, and slowly oscillates within. In contrast, the control inputs are different from the case with accurate parameters and without external disturbances. Fig. 11. Generalized disturbances.
velocities
with
inaccurate
parameters
and
6. Conclusions This paper proposes a novel conceptual design for stratospheric airship platform and a novel approach for station-keeping control of the stratospheric airship platform. The main results of the work are outlined as (1) introduction of a novel conceptual design of the stratospheric airship platform; (2) proposition of an original formulation of station-keeping control problem with an appropriate change of variable allowing the application of backstepping approach; (3) design of a FABC system that actualizes the station-keeping control of the airship platform against the parametric uncertainty and external disturbance; (4) and finally, demonstration the effectiveness and robustness of the proposed control approach. The proposed conceptual design and control scheme provide a promising approach for disaster perception using the stratospheric station-keeping airship platforms. Acknowledgments
Fig. 12. Control inputs with inaccurate parameters and disturbances.
Fig. 9 presents the simulation results of station-keeping control with inaccurate parameters and wind disturbances. From Fig. 9, it is clear that the proposed control approach succeeds in station keeping for the airship platform accurately despite parametric uncertainties and external disturbances. The trace in case 2 is more flexuous than that in case 1 due to the wind disturbances in lateral direction, as show in the magnified image. The control errors in position and orientation are shown in Fig. 10. In contrast to case 1, the position error in xdirection converges to a very small neighborhood of zero with 300 s. The position error in y-direction gradually converges to zero. The orientation error is varying with a max-
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Please cite this article in press as: Yang, Y.N., et al. Station-keeping control for a stratospheric airship platform via fuzzy adaptive backstepping approach. J. Adv. Space Res. (2012), http://dx.doi.org/10.1016/j.asr.2012.10.014