Discrete-time pure-tension sliding mode predictive control for the deployment of space tethered satellite with input saturation

Journal Pre-proof Discrete-time Pure-tension Sliding Mode Predictive Control for the Deployment of Space Tethered Satellite with Input Saturation Xiaolei Li, Guanghui Sun, Xiangyu Shao PII:

S0094-5765(20)30071-0

DOI:

https://doi.org/10.1016/j.actaastro.2020.02.009

Reference:

AA 7875

To appear in:

Acta Astronautica

Received Date: 7 September 2019 Revised Date:

3 February 2020

Accepted Date: 4 February 2020

Please cite this article as: X. Li, G. Sun, X. Shao, Discrete-time Pure-tension Sliding Mode Predictive Control for the Deployment of Space Tethered Satellite with Input Saturation, Acta Astronautica, https:// doi.org/10.1016/j.actaastro.2020.02.009. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier Ltd on behalf of IAA.

Discrete-time Pure-tension Sliding Mode Predictive Control for the Deployment of Space Tethered Satellite with Input Saturation Xiaolei Li, Guanghui Sun∗, Xiangyu Shao Department of Control Science and Engineering, Harbin Institute of Technology, Harbin 150001, China

Abstract A novel discrete-time pure-tension sliding mode predictive control scheme is presented to realize the deployment of space tethered satellite system in consideration of saturated input, which inherits the property of explicitly handling constraints from model predictive control, and the remarkable robustness from sliding mode control. Considering the digital control characteristics of the practical engineering, a discrete-time nonlinear model for space tethered satellite system is derived based on the discrete-time Euler-Lagrange theorem. Then, a discrete-time underactuated sliding manifold and a sliding mode predictive equation are raised to guarantee a more stable and faster tension control. Meanwhile, given the input saturation, an auxiliary controller is involved in the proposed controller to compensate saturated tension. By using Lyapunov stability theory, the accessibility of the discrete-time sliding manifold and the asymptotic stability of the deployment process are certificated. Finally, groups of numerical simulations for the deployment are demonstrated to verify the effectiveness of the proposed control scheme. Keywords: Space tethered satellite, Sliding mode predictive control, Discrete-time Euler-Lagrange, Underactuated system, Input saturation

∗ Corresponding

author Email addresses: [email protected] (Xiaolei Li), [email protected] (Guanghui Sun), [email protected] (Xiangyu Shao)

Preprint submitted to Journal of LATEX Templates

February 6, 2020

1. Introduction Space tethered satellite (STS) system is an advanced artificial multi-satellites technology [1], which generally has two-body structure like a dumbbell comprised of mother satellite, subsatellite and a connecting space tether. With the 5

help of space tether [2], STS has a significant application prospect in the field of space missions, such as satellite capture [3], space tug [4], space shuttle [5], tethered robot [6], artificial gravity [7], and so on. Since the first STS system structured by Mario Grossi and Guiseppe Colombo in early 1970s [8], up to now, STS has become a critical research topic for the space missions. Orbital

10

experiments of STS have been implemented in several space missions [1], for example, Gemini, CHARGE, SEDS, TSS-1R [9], YES2 [10], and so forth. Due to the flexibility of space tether [11], only tension but not support force can be provided during the deployment, which leads to underactuated features and input saturation. Accordingly, the stable and fast deployment of

15

space tether [12] is one of the significant and challenging issues of STS subjects, on which many effective control schemes and devices have been focusing. For example, given the underactuated dynamics caused by space tether, an extended time-delay autosynchronization (ETDAS) was put forward in [13] to deal with the underactuated dynamics of tethered satellite system (TSS), by stabilizing

20

the chaotic motion to a periodic motion. An anti-windup hierarchical sliding sliding mode control (SMC) law for the constrained and underactuated transfer thrusts of tethered space robots (TSR) was proposed in [14], to realize the towing removal of debris swing in orbit. Considering the saturated tension of space tether, an adaptive update rate was introduced in the hierarchical SMC

25

to eliminate the effect of the saturated input for the multi-satellite tethered system (MSTS) [15]. A fractional-order fuzzy SMC was developed to realize deployment of STS with a view to input saturation [16]. To the best of our knowledge, most of the researchers focused little attention on the discrete-time nonlinear dynamics of actual STS system, and relatively

30

few research have been reported, compared with the continuous-time linear dy-

2

namics. Nevertheless, the latter is based on the traditional analog processor and usually needs the local linearization near the equilibrium point, which makes it insufficient to completely describe the system dynamics of practical STS, and may bring about inferior control performance or failure of deployment. As more 35

and more digital processors are implemented in STS system, the discrete-time nonlinear dynamics are playing a more and more critical role in the tension control for deployment. Consequently, for the discrete-time nonlinear dynamics of STS, it is worthwhile to research a better discrete-time tension control scheme for the deployment.

40

Among the various control schemes, SMC [17] has incomparable superiority in disposing of system uncertainties and external perturbations, which was extensively used to obtain the stable deployment performance. For instance, Zhao designed a super-twisting adaptive SMC scheme for TSR, to eliminate the vibration of the tether to assure a successful capture in station-keeping phase

45

[18]. Kang et al. defined two sub-sliding manifolds for the actuated and unactuated states separately [19], which could generate a fractional-order SMC scheme to make all states move toward to the desired states for deployment of TSS. As we all know, the switching control law consists of the reaching phase and the sliding phase. Between them, the latter ensures excellent stability of

50

SMC and the former determines dynamic performance of SMC. In present research, the reaching phase of SMC still needs to be improved and optimized, which originates high-order sliding control or hybrid control methods to obtain better dynamic performance. As far as SMC is concerned, model predictive control (MPC) can optimize

55

its dynamic performance by accelerating the reaching phase, and even can handle the constraints of input and state explicitly. There are some research have been accomplished for combining SMC with MPC in some cases, for example, Xu et al. presented an enhanced model predictive discrete-time sliding mode control (MPDSMC) scheme for the nanopositioning system [20], whose distinct

60

advantage is only a second-order model of system required, without the hysteresis model. Huang et al. raised a sliding mode predictive tracking control 3

(SMPTC) strategy for a steer-by-wire (SbW) system with uncertain dynamics to control the wheels to follow the driver’s command closely [21]. McNinch and Ashrafıuon studied an optimizing sliding mode cascade control structure for 65

underactuated unmanned surface vessel systems (USVS) [22], whose optimal sliding surface parameters is determined by discrete-time nonlinear MPC. However, as far as we know, there are relatively few reports corresponding to the hybrid control of SMC and MPC utilized for the deployment of STS, hence it is worthy and significant to research the proposed control scheme in this paper.

70

It is worth noting that the out-of-plane angle is higher-order coupled with the in-plane angle, which implies the former has little influence on the pitch motion in the tether deployment [19]. Therefore, we pay more attention to the in-plane dynamics of STS, without considering the out-of-plane motion. Furthermore, given the obvious shortcomings and limitations of thruster-aid scheme [23], the

75

pure-tension control is more suitable and effective for profound study of STS system. In this paper, we will study the nonlinear pure-tension deployment of STS system via discrete-time sliding mode predictive control (DSMPC) scheme, and further taking input saturation into account to obtain better deployment performance of STS. The main works and contributions of this paper can be

80

summarized as follows: (1) A novel discrete-time Euler-Lagrange model is designed for STS, which is more appropriative to evaluate deployment performance and stability by sample interval. Moreover, the discrete-time model is only comprised of position information, and leading to the deployment of STS with only length

85

and angle measurements, which aids deployment of space tether in practical missions. (2) A DSMPC scheme is firstly applied for the deployment of STS as we know, which can deal with underactuated dynamics and discrete-time nonlinearity concurrently. By means of the discrete-time underactuated sliding manifold

90

and the sliding mode predictive equation presented in this paper, a faster and more stable deployment of STS can be achieved in comparison with the

4

classical controller. (3) An auxiliary controller is put forward to compensate the influence of input saturation in this paper, by which we can verify the asymptotic stability of 95

proposed control scheme and obtain preferable deployment performance of STS. The section arrangement of this paper is organized as follows. The discretetime Euler-Lagrange model of STS is given in Section 2. Section 3 presents the main design process of DSMPC controller. In Section 4, the accessibility

100

of underactuated sliding manifold and the asymptotic stability of DSMPC in consideration of input saturation are proposed. Section 5 demonstrates the contrastive simulations of various control schemes for tether deployment, and Section 6 concludes this paper in short.

2. System model 105

Given that researching the control scheme of the discrete-time nonlinear STS is the main purpose of this paper, and the dumbbell model has little difference with basic motion state of STS system in practical missions, thus we choose dumbbell model for the modelling and controlling of tether deployment below. Based on the two-body structure of dumbbell model, a typical and simplified

110

rigid model of STS is shown in Fig.1, whose main components include: subsatellite, mother satellite, and an insulated space tether connecting the two satellites. For the sake of generality, three relevant coordinate systems are given as follows, which is contributed to illustrate the equivalent model and physical significance: The first is the inertial coordinate system O1 XYZ, whose origin is the Earth’s

115

center O1 . O1 X locates on the Earth’s equator and points to the vernal equinox, O1 Z coincides with the Earth’s axis of rotation, and O1 Y is the product from the right-hand rule of O1 XYZ. The next is the orbital coordinate system O2 xyz with the origin at the centroid of STS, where O2 x denotes the orbital direction of STS, O2 z points toward the Earth’s center O1 from the centroid O2 , similarly,

120

O2 y assists O2 xyz in accomplishing the right-hand rule. Specially, the pitch 5

motion for the deployment of STS is indicated by the in-plane angle θ below. The last is the STS’s body coordinate system O2 x′ y ′ z ′ , due to the in-plane angle θ, O2 x′ y ′ z ′ can be generated by revolving O2 xyz around O2 y(O2 y ′ ) with θ degrees by assuming O2 y and O2 y ′ coincided.

Figure 1: Geometric representation of STS system

125

Thanks to the reasonable definitions and assumptions for the deployment of STS proposed by Fujii [11] and Kang [24], we can establish the following assumptions to simplify the modeling process. Assumption 1. The space tether is insulated, massless and springless. Moreover, the tether can remain rigid due to its saturated tension, which satisfies

130

the basic assumptions of the two-body dumbbell structure. Assumption 2. The length of space tether is much greater than the size of satellites during the deployment, which indicates that the satellites can be regarded as particles, without considering variation of attitude in the modeling process.

135

Assumption 3. The Earth is a perfect sphere with its centroid coincides with its geometric center, and the orbit of STS is circular with only in-plane motion considered. 6

Assumption 4. If the mass ratio of mother satellite to subsatellite is greater than 100, subsatellite will not affect the running states of mother satellite during 140

the deployment of STS. 2.1. Continuous-time Lagrange model In the light of the above coordinate systems and assumptions, the kinetic energy K and the potential energy V for the deployment of STS can be presented as follows [25]: 1 1 2 ˙ 1 ˙2 mΩ2 R02 + ml ¯ (θ + Ω)2 + m ¯l 2 2 2 1 V = −mΩ2 R02 + mΩ ¯ 2 l2 [1 − 3 cos2 θ] 2

K=

(1)

m ¯ = (m1 m2 )/m, m = m1 + m2 where m ¯ and m separately express the equivalent mass and the total mass of satellites, inside m1 and m2 denote the mass of mother satellite and subsatellite. R0 is the length of O1 O2 , while θ, l and Ω indicate the in-plane angle, the length 145

of space tether and the orbital angular velocity of STS system, respectively. Accordingly, based on the Lagrangian mechanics theory, the continuous-time Lagrangian function of STS can be expressed by L = K − V : L=

3 1 ˙2 1 2 ˙ mΩ2 R02 + ml ¯ (θ + Ω)2 + m ¯l 2 2 2 1 ¯ 2 l2 (1 − 3 cos2 θ) − mΩ 2

(2)

Lemma 1. In order to attain a smooth deployment of STS, the generalized coordinate vector q = (θ, l) must satisfy the Euler-Lagrange equation:   d ∂L ∂L − =Q dt ∂ q˙ ∂q

(3)

where Q denotes the generalized force of system, and it can be decomposed into (Qθ , Ql ) corresponding to the generalized vector q. Particularly, in the proposed pure-tension control scheme, we don’t take the auxiliary angle thrusters into 150

account, i.e. Qθ = 0 and Ql = T , where T is the tether tension.

7

Subsequently, the following Lagrangian dynamic equations of STS can be set up from Eq. (3) [26]: ˙ ˙ + 3Ω2 cos θ sin θ = 0 θ¨ + 2(l/l)(Ω + θ) h i ¨l − l (Ω + θ) ˙ 2 + (3 cos2 θ − 1)Ω2 = −T /m ¯

(4)

Due to a space tether mission usually takes a long time to complete the deployment, it is more convenient and clear to work with non-dimensional equations [27]. Therefore, we measure system time in units of the true anomaly τ , and there exists the following dimensionless conversions: τ = Ωt Tˆ =

T mΩ ¯ 2L

(5)

λ = l/Lm − 1 where τ indicates the angle of position vector between orbital perigee and STS at a certain time, Tˆ, λ and Lm denote the dimensionless tether tension, the dimensionless tether length and the total length of tether, respectively. Therefore, we can get the following dimensionless Lagrangian dynamics: λ˙ ˙ + 3 cos θ sin θ = 0 θ¨ + 2( )(1 + θ) λ+1 ¨ − (λ + 1)[(1 + θ) ˙ 2 − 1 + 3 cos2 θ] = −Tˆ λ

(6)

Until now, the continuous-time Lagrange model of STS is presented as above, which can be employed in the derivation process of underactuated sliding manifold for the tether deployment below. 2.2. Discrete-time Euler-Lagrange model By introducing the discrete-time state vector qk = (θk , lk ), the discrete-time Lagrangian function of STS is given based on the forward-Euler discretization

8

(FED) method: ∆qk = qk+1 − qk = (θk+1 − θk , lk+1 − lk ) Ld (qk , qk+1 ) = L (qk , ∆qk /h)  2 1 2 θk+1 − θk 3 ¯ k +Ω = mΩ2 R02 + ml 2 2 h  2 1 1 lk+1 − lk + m ¯ − mΩ ¯ 2 lk2 (1 − 3 cos2 θk ) 2 h 2

(7)

Invoking the discrete-time Euler-Lagrange theorem can raise capability of anti-jamming and anti-delay for the model of STS, which will prominently enhance the deployment performance. Therefore, we have the following discretetime Euler-Lagrange equations established. D1 Ld (qk , qk+1 ) + D3 Ld (qk−1 , qk ) = 0

(8)

D2 Ld (qk , qk+1 ) + D4 Ld (qk−1 , qk ) = −T (k) where ∂Ld (qk , qk+1 ) ∂θk ∂Ld (qk , qk+1 ) D2 Ld (qk , qk+1 ) = ∂lk ∂Ld (qk−1 , qk ) D3 Ld (qk−1 , qk ) = ∂θk ∂Ld (qk−1 , qk ) D4 Ld (qk−1 , qk ) = ∂lk D1 Ld (qk , qk+1 ) =

(9)

Consequently, the discrete-time dynamic equation for the deployment of STS can be derived from Eq. (8). θk+1 = (θk − hΩ) +

lk+1

2 lk−1 (θk − θk−1 + hΩ) lk2

− 3h2 Ω2 cos θk sin θk  2 θk+1 − θk = l k h2 + Ω + lk + (lk − lk−1 ) h h2 − h2 Ω2 lk (1 − 3 cos2 θk ) − T (k) m ¯

155

9

(10)

Remark 1. For the dynamics of deployment mission Eq. (10), the following inequality should always be satisfied: 0 < l0 ≤ lmin ≤ lk ≤ lmax

(11)

During the practical deployment mission, lk = 0 only appears in the preparatory stage of STS, which means that space tether has not been deployed at all, i.e. subsatellite is still in the cabin of mother satellite. But it is worth noting that the tether can not exert tension in this case, which is contrary to Assump160

tion 1 and leads into model singularity. In order to deploy the tether effectively, a certain distance should be deployed artificially in advance [10]. Therefore, introducing the lmin not only ensures the physical meaning, but also avoids model singularity of STS system. According to Eq. (5), a discrete-time dimensionless length of space tether λk = lk /Lm − 1 is introduced to make the system states have the same equilibrium point, hence the Eq. (11) can be modified as: Ξλ = {λk | −1 < λmin ≤ λk ≤ λmax ≤ 0}

(12)

2

h And for the sake of convenience, we introduce uk = − mL ¯ m T (k) to denote the

discrete-time tension control. Therefore, the discrete-time dynamics Eq. (10) can be converted to the following iterations: θk+1 = (θk − hΩ) + 2

(λk−1 + 1)2 (θk − θk−1 + hΩ) (λk + 1)2

(13)

2

− 3h Ω cos θk sin θk

λk+1 = (λk + 1)h2

 2

θk+1 − θk +Ω h 2

2

+ 2λk

(14)

2

− λk−1 − h Ω (λk + 1)(1 − 3 cos θk ) + uk So far, the discrete-time Euler-Lagrange model of STS is raised by the afore165

said equations, which is the considerable model for implementing simulation experiments of the deployment control. 10

3. Control design In this section, a novel discrete-time pure-tension sliding mode predictive controller is proposed for the deployment, whose sliding manifold is devised 170

upon underactuated property of STS [22]. Moreover, based on the optimal control theory, the local optimal solution of DSMPC is derived on the output of discrete-time MPC within the predictive horizon, which can be equivalent to the switching control term of discrete-time SMC, making the sliding mode converge to the equilibrium state quickly.

Figure 2: Block diagram of DSMPC scheme

175

As shown in Fig. 2, DSMPC scheme for the state variables of STS is presented on the whole, which consists of the modules of DSMPC controller, input compensation and STS model. The DSMPC controller combines robust MPC scheme with discrete SMC scheme, whose output can be expressed as usmpc = ueq + umpc , via establishing and calculating of discrete-time underac-

180

tuated sliding manifold and the sliding mode predictive equation. Furthermore, to compensate the influence of input saturation in STS system, the module of input compensation is presented in Fig. 2, which is composed by a auxiliary system and a input saturation function. Therefore, the output of DSMPC with compensation is denoted as uk = ucom + usmpc , so that the de-

185

ployment output of STS system can be obtained with proposed control scheme.

11

3.1. Discrete-time underactuated sliding manifold With the help of the continuous-time Lagrangian model of STS Eq. (6), the first-order differential cascade model for the deployment can be generated by   ˙ λ, λ˙ [23]: defining the continuous-time state vector as q = θ, θ, q˙1 = q2 2q4 (1 + q2 ) − 3 cos(q1 ) sin(q1 ) 1 + q3

q˙2 = −

(15)

q˙3 = q4   q˙4 = (1 + q3 ) (1 + q2 )2 − 1 + 3 cos2 (q1 ) − Tˆ

According to [28], a classical SMC strategy for a class of underactuated systems has been proposed by Xu, whose relevant parameters are defined as follows: 2q4 (1 + q2 ) − 3 cos(q1 ) sin(q1 ) 1 + q3   f2 = (1 + q3 ) (1 + q2 )2 − 1 + 3 cos2 (q1 )

f1 = −

e1 = q1

(16)

e2 = q2 e3 = f1 (q1 , q2 , q3 , q4 ) Then, the sliding manifold of STS can be constructed in the the following form: s = ce = c1 e1 + c2 e2 + e3

(17)

Substituting Eq. (16) into Eq. (17), the discrete-time sliding mode equation can be acquired by employing the sampling interval H = hΩ via the FED method: sk = c1 (θk − θk−1 )hΩ + c2 θk−1 h2 Ω2 −

2(λk − λk−1 )(hΩ + θk − θk−1 ) λk−1 + 1

− h2 Ω2 (3 cos θk−1 sin θk−1 + αλk−1 )

12

(18)

Considering the reaching condition of the discrete-time sliding mode: (sk+1 − sk )sk < 0, the variation of sliding mode ∆sk = sk+1 − sk should be deduced firstly: ∆s = sk+1 − sk = c1 [(θk+1 − 2θk + θk−1 )] hΩ + c2 (θk − θk−1 )h2 Ω2 − 3h2 Ω2 (cos θk sin θk − cos θk−1 sin θk−1 )

(19)

− F (λk+1 , λk ) + F (λk , λk−1 ) − αh2 Ω2 (λk − λk−1 ) where F (λk , λk−1 ) =

2(λk − λk−1 )(hΩ + θk − θk−1 ) λk−1 + 1

(20)

Substituting dimensionless tether length Eq. (14) into Eq. (19), the equivalent control equation can be yielded by means of solving ∆sk = sk+1 − sk = 0, and further the output of DSMPC controller can be derived as follows, by replacing the switching control term with umpc : usmpc (k) = ueq (k) + usw (k)   A(k) ueq (k) = − + C(k) B(k)

(21)

usw (k) = umpc (k) where A(k) = c1 [(θk+1 − 2θk + θk−1 )] hΩ + c2 (θk − θk−1 )h2 Ω2 − F (λk+1 , λk ) + F (λk , λk−1 ) − αh2 Ω2 (λk − λk−1 ) − 3h2 Ω2 (cos θk sin θk − cos θk−1 sin θk−1 ) 2(hΩ + θk+1 − θk ) B(k) = − λk + 1 θk+1 − θk ) C(k) = (λk + 1)h2 ( + Ω)2 + 2λk − λk−1 h − h2 Ω2 (λk + 1)(1 − 3 cos2 θk ) + uk

13

(22)

Under the proposed control algorithm Eq. (21), the sliding mode dynamics of STS can be expressed as: sk+1 = sk + A(k) + B(k) [C(k) + usmpc (k)]

(23)

= sk + B(k)umpc (k) 3.2. Sliding mode predictive equation In the light of the iterative form of Eq. (23), we can acquire the Np -step sliding mode predictive Equation:  sk+Np =sk + B(k)umpc (k) + B(k + 1)umpc (k + 1)  + · · · + B(k + Np − 1)umpc (k + Np − 1)

(24)

where Np denotes the predictive horizon of the DSMPC.

Let Γ indicates the Np -dimensional column vector with elements are all 1, the remaining algebraic matrices are defined as follows: h iT S(k) = sk+1 , sk+2 , · · · , sk+Np h iT U (k) = umpc (k), umpc (k + 1), · · · , umpc (k + Np − 1)   B(k) 0 ··· 0     0  B(k) B(k + 1) · · ·   Φ(k) =  . .  . .. ..   .. 0   B(k)

B(k + 1) · · ·

(25)

B(k + Np − 1)

Consequently, the sliding mode predictive equation Eq. (24) can be rewritten

as the matrix form: S(k) = Γsk + Φ(k)U (k)

190

(26)

Remark 2. In the Eq. (25), Φ(k) has the lower triangular matrix form, which demonstrates the time causality of discrete-time MPC scheme that the control input and predicted output are independent of each other. Such that the disturbances caused by the variation of control input can be effectively reduced, which will enhance the robustness of the DSMPC scheme. 14

195

3.3. Local optimization problem The control objective of DSMPC is to attract the sliding mode sk converge to sliding manifold asymptotically, which must guarantee the tension of tether remain strictly positive during the deployment of STS. Therefore, the control effect of DSMPC should be paid more attention instead of its variation of amplitude. To achieve this goal, we adopt the following form of cost function to study the effectiveness and stability of DSMPC controller: J(k) =

Np X

kΥs (sk+i|k − rk )k2 +

i=1

NX c −1

kΥu umpc (k + j|k)k2

(27)

j=0

where Np and Nc separately indicate the predictive horizon and the control horizon, and Np ≥ Nc in general. rk is the reference input at the time instant k. Υs and Υu denote the weight coefficient of the predictive output error and the control input respectively. 200

Remark 3. The control performance of DSMPC controller is determined by Υs and Υu . Theoretically, the larger weight coefficient signifies the better control performance we can achieve. It is a remarkable fact that if the weight coefficient Υs and Υu are invariant during the whole predictive horizon, the sliding mode predictive gain of DSMPC can be calculated off-line and checked on-line. Suppose that Np = Nc for the sake of simplification and clarity, hence the cost function Eq. (27) can be simplified as the linear quadratic form in accordance with Eq. (26): J(k) = S(k)T QS(k) + U (k)T RU (k) = (Γsk + Φ(k)U (k))T Q(Γsk + Φ(k)U (k))

(28)

+ U (k)T RU (k) Calculating the locally optimal control sequence to minimize cost function Eq. (28) is one of the most essential issues of MPC. To acquire the extreme solution within the predictive horizon, the optimization problem should satisfy

15

the following condition based on the optimal control theory:   ∂J(k) = 2 Φ(k)T QΓsk + Φ(k)T QΦ(k)U (k) + RU (k) U (k)

(29)

T

= Φ(k) Q(Γsk + ΦU (k)) + RU (k) = 0

205

Remark 4. (Receding-Horizon) The basic feature of MPC is receding-horizon (rolling-horizon), whose performance index at each time instant is only related to the limited horizon, namely the predictive horizon, in contrast with global optimization. Furthermore, the 210

predictive horizon will move forward as sampling time goes on, and only the first element of the optimal control sequence can be employed for DSMPC scheme Eq. (21). It illustrates the predictive property of discrete-time MPC, such that various of distortions and error problems can be corrected effectively and timely during the practical control. Accordingly, we present the optimal control solution of DSMPC within the predictive horizon: U (k) = −(Φ(k)T QΦ(k) + R)−1 Φ(k)T QΓsk

(30)

umpc (k) = [1, 0, · · · , 0]1×m U (k) = Kmpc (k)sk 215

where Kmpc denotes predictive control gain of sliding mode sk . Based on the convergence condition of the discrete-time sliding mode, a positive constant η is introduced to consummate the constraint of sliding mode. Besides, given the proposed DSMPC is the pure-tension control scheme, the variation of tether length should be taken into consideration to avert loosening or breaking the space tether. Thus, the following state constraint of sliding mode should be satisfied: Ξs = {sk | ksk+1 k ≤ ηksk k}

(31)

where η is a positive constant less than 1. According to the principle of quasi sliding mode, a vicinity ǫs is designed for the sliding manifold to stabilize the terminal sliding mode. Similarly, to 16

stabilize terminal tether deployment, a vicinity ǫ∆ of terminal length variation is raised, which can be expressed in the following constraint set:    ksk+Np k ≤ ǫs ,  Ξt = (sk , ∆λk )  k∆λk+Np k ≤ ǫ∆ 

(32)

For the convenience of calculation and explanation, we suppose that the

control effect is applied in the whole prediction horizon, i.e. Np = Nc . Subsequently, based on the Eq. (21), Eq. (23), Eq. (31) and Eq. (32), the optimization problem for the deployment of STS can be defined as: min

u(k),...,u(k+Np −1)

s.t.

J(k) =

Np X

F (sk+a|k , uk+a−1|k )

a=1

sk+1 = sk + B(k)umpc (k), uk = ueq (k) + umpc (k),

(33)

sk+a|k ∈ Ξs , λk+a|k ∈ Ξλ ,
sk+Np , ∆λk+Np ∈ Ξt 4. Stability analysis According to the formulas above, the DSMPC input can be expressed by substituting Eq. (30) into Eq. (21):   A(k) usmpc (k) = − + C(k) + Kmpc (k)sk B(k)

(34)

Under the proposed input, the sliding mode dynamics Eq. (23) can be reformulated as: sk+1 = sk + B(k)Kmpc (k)sk = [1 + B(k)Kmpc (k)] sk = s0

k Y

(35)

[1 + B(i)Kmpc (i)]

i=0

Remark 5. Accordingly, we can conclude that the sliding mode is relevant to 220

the initial mode s0 and B(k)Kmpc (k). Moreover, given the upper bound of 17

B(k)Kmpc (k) represented as σ, if σ ∈ (−2, 0), then the sliding mode sk will asymptotically converge to the sliding manifold. In the practical space missions, the space tether is deployed and tightened through the tension generated by the reel type deployer in mother satellite. So 225

it is worthwhile to emphasize that space tether does not have the compressive force similar to rigid rod. Precisely, space tether can only be stretched, but not be compressed, which indicates the direction of tether tension must be nonnegative. In order to assure the asymptotic stability for the deployment of STS with

230

input constraint, firstly, a saturation function should be put forward to make the system states strictly bounded.    Umin , u ≤ Umin    Sat(u) = u, Umin ≤ u ≤ Umax     U max , u ≥ Umax

(36)

2

h where Umin < Umax < 0, due to uk = − mL ¯ m T (k).

For the sake of compensating the influence of above input saturation, an auxiliary control variable is introduced for the proposed control scheme according 235

to [29]: ξ˙ =

where

  

h  − βξ +

0,

ξ<δ

i ξ ˙ λ) , Θ(ξ, ∆u, θ, 2 kξk

(37)

ξ≥δ

#2 ˙ 2(1 + θ) ˙ λ) = (ρξ + ∆u) Θ(ξ, ∆u, θ, λ+1 "

(38)

∆u = Sat(u) − u, β ∈ (0, 1), ξ(0)&ρ&δ > 0

˙ λ) ≥ 0 and ξ > 0 are valid, which Remark 6. It is noticeable that Θ(ξ, ∆u, θ, leads to ξ˙ ≤ 0 and makes the auxiliary control variable monotonically decrease to terminal region Ξξt = {ξ | δ > ξ > 0, sk ∈ R}. 18

240

Theorem 1. Quoting the FED method, the discrete-time auxiliary control variable can be established as:    ξk+1 =  (1 − β)ξk −

ξk ,

ξk < δ

ξk kξk k2 Θ(ξk , ∆uk ),

(39)

ξk ≥ δ

where

2

Θ(ξk , ∆uk ) = [B(k)(ρξ + ∆uk )] ≥ 0

(40)

by which we can define the following input compensation for DSMPC controller, and the new control scheme can be derivated, namely discrete-time input compensation-based SMPC (DICSMPC): ucom (k) = ρ · ξk

(41)

uk = usmpc (k) + ucom (k) By implementing the above controller, the sliding mode can be driven onto the designed sliding manifold Eq. (17), to make the system states for the deployment of STS asymptotically stable. Proof. Based on the sliding manifold Eq. (17), one can get that sk = 0 is the globally asymptotical stable equilibrium surface of sliding mode, thus, we can define the following Lyapunov function candidate for sliding manifold: V1 (k) = s2k

(42)

then the forward difference of V1 (k) is given as: ∆V1 (k) = s2k+1 − s2k , sk 6= 0

(43)

Under the DICSMPC scheme Eq. (41), the reaching condition of sliding

19

manifold can be deduced as follows: sk+1 − sk = B(k)Kmpc (k)sk + B(k)(ρξ + ∆u) sk+1 + sk = [2 + B(k)Kmpc (k)] sk + B(k)(ρξ + ∆u) s2k+1 − s2k = B(k)Kmpc (k) [2 + B(k)Kmpc (k)] s2k

(44)

+ [2 + 2B(k)Kmpc (k)] (ρξ + ∆u) · B(k)sk + [B(k)(ρξ + ∆u)]

2

⇔ ∆V1 (k) ≤ σ(2 + σ)s2k + (2 + 2σ)(ρξ + ∆u) · B(k)sk + [B(k)(ρξ + ∆u)]2 Invoking the Remark 5, the control parameter σ should subject to (−2, 0) to guarantee the asymptotic convergence of sliding mode, such that we can choose σ = −1 to eliminate the uncertain influence of sk and minimize the gain coefficient of the s2k , simplifying the reaching condition as: ∆V1 (k)|σ=−1 ≤ −s2k + [B(k)(ρξ + ∆u)]

2

(45)

2

Due to Θ(ξk , ∆uk ) = [B(k)(ρξ + ∆u)] , thus ∆V1 (k) ≤ −s2k + Θ(ξk , ∆uk ) can be satisfied. Considering the auxiliary control variable ξk , we construct a novel Lyapunov function candidate as follows: V2 (k) = V1 (k) + ξk2 = s2k + ξk2

(46)

according to the Remark 6, the forward difference of V2 (k) can be delivered as: ∆V2 (k) = V2 (k + 1) − V2 (k) 2 = V1 (k + 1) − V1 (k) + ξk+1 − ξk2

(47)

≤ ∆V1 (k) + ξk+1 ξk − ξk2 245

In the light of the auxiliary control variable Eq. (39), the following two cases of ∆V2 (k) can be taken into consideration:

20

Case 1. ξ < δ ⇒ ξk+1 = ξk ∆V2 (k) = V2 (k + 1) − V2 (k) ≤ ∆V1 (k) + ξk+1 ξk − ξk2 ≤ −s2k + Θ(ξk , ∆uk ) + ξk (ξk+1 − ξk )

(48)

≤ −s2k + Θ(ξk , ∆uk ) h ih i p p ≤ − sk + Θ(ξk , ∆uk ) sk − Θ(ξk , ∆uk )

Introducing set ΞΘ to indicate the terminal attraction region of sliding mode, n o p we can draw a conclusion that if lim sk ∈ ΞΘ = sk | |sk | ≤ Θ(ξk , ∆uk ) , k→∞

then ∆V2 (k) ≤ 0, proving the asymptotic stability of DICSMPC scheme. Case 2. ξ ≥ δ ⇒ ξk+1 = (1 − β)ξk −

ξk kξk k2 Θ(ξk , ∆uk )

∆V2 (k) = V2 (k + 1) − V2 (k) ≤ ∆V1 (k) + ξk+1 ξk − ξk2 ≤

−s2k

+ Θ(ξk , ∆uk ) − ξk



 βξk2 + Θ(ξk , ∆uk ) ) ξk

(49)

≤ −s2k + Θ(ξk , ∆uk ) − βξk2 − Θ(ξk , ∆uk ) ≤ −(s2k + βξk2 ) ≤ 0 which illustrates that under the input compensation, sliding mode will converge to the sliding manifold asymptotically. Furthermore, to analyze the stability condition, the following inequality derivation can be constructed from Eq. (49): V2 (k + 1) − V2 (k) ≤ −(s2k + βξk2 ) 2 − s2k − ξk2 ≤ −(s2k + βξk2 ) s2k+1 + ξk+1 2 s2k+1 + ξk+1 ≤ (1 − β)ξk2 2 s2k+1 + ξk+1 ≤ (1 − β)(s2k + ξk2 ) 2 s2k+1 + ξk+1 ≤ (1 − β)k (s20 + ξ02 ) 250

Consequently, lim V2 (k) = s2k + ξk2 → 0 , if β ∈ (0, 1). k→∞

This completes the proof. 21

(50)

Remark 7. Considering the input saturation of the tether tension, we can conclude that V2 (k) = s2k + ξk2 will asymptotically converge to the origin when k → ∞. Furthermore, the sliding mode sk will be uniform bounded and stable, 255

if there exists sup (Θ(ξk , ∆uk )) to make the proposed control input uk bounded.

5. Numerical simulations In this section, the numerical simulations for the deployment of STS is implemented to verify the effectiveness of the proposed control scheme. The simulation parameters of practical STS system are presented in the Table 1, refer to the YES2 phase 1 deployment [10]. Table 1: Parameters of STS system

Properties

symbol

value

dimension

Mother satellite mass

m1

6530

kg

Subsatellite mass

m2

12

kg

Space tether mass

m3

0

kg

Sample interval

h

1

s

Orbital altitude

R0

260

km

Space tether length

Lm

3.5

km

Pre-deploy tether length

l0

0.175

Orbital angular velocity

Ω

1.17 × 10

km −3

rad/s

260

In accordance with [12], the tether deployment task can be depicted by regulating the initial state (θ0 , λ0 ) = (0.1, −0.95) approach to the terminal state (θt , λt ) = (0, 0), inside defining (θ1 , λ1 ) = (0.1, −0.95 + 0.3Ω) for the sake of generality. It can also be equivalent to draw the sliding mode converge to the 265

sliding manifold sk = 0, i.e. the reference value of Eq. (27) can be defined as rk = 0. Furthermore, we assume that STS is located in the initial true anomaly with τ0 = 0 rad when the subsatellite begins to deploy. To satisfy the accessibility and stability of the sliding manifold, we determine the sliding manifold with c1 = 1.20, c2 = 5.12, α = 3.50 via the trial and 22

error method. In consideration of sliding mode predictive optimization problem Eq. (33), the following constraints should be satisfied to obtain the local optimal control within the predictive horizon: Ξs = {sk | ksk+1 k ≤ 0.7ksk k} Ξλ = {λk | −0.8 ≤ λk ≤ 0}  Ξu = uk | −6.86 × 10−6 ≤ uk < 0    ksk+Np k ≤ 1 × 10−3 ,  Ξt = (sk , ∆λk )  k∆λ k ≤ 1 × 10−5 

(51)

k+Np

where Ξs , Ξλ , Ξu , Ξt denote the constraint sets of sliding mode, tether length, discrete-time tension control, terminal states respectively. Subsequently the parameters of cost function and auxiliary controller are given as follow: Np = Nc = 5, Q = 20, R = 30 ξ0 = 5, β = 1 × 10−3 , ρ = 7 × 10−5 , δ = 1 × 10−5

(52)

To illustrate the effectiveness of proposed scheme, four different schemes are employed to implement comparative experiments of STS, which contains the 270

discrete-time input compensation-based sliding mode predictive control (DICSMPC), discrete-time sliding mode predictive control (DSMPC), discrete-time sliding mode control (DSMC) and continuous-time tension control (CTC) [12]. In this paper, we are capable of providing numerical simulations for the deployment of STS revealed as Fig. 3 to Fig. 8, namely the trajectories of

275

in-plane angle, tether length, sliding mode, discrete-time tension, variation of tether length and in-plane angle, respectively. Fig. 3 illustrates the trajectories of in-plane angle for DICSMPC, DSMPC, DSMC and CTC, respectively. It is clear that the in-plane angle regulated by CTC has obvious lag effects in contrast to the discrete-time methods. Under

280

the same initial conditions, CTC need about true anomaly 10 rad to reach the desired state, which indicates the obvious advantages of the proposed discretetime methods. From the magnified area on the right, we can get that DSMPC and DICSMPC have faster convergence rate than DSMC, which owes to the 23

Figure 3: In-plane angle of the deployment

dynamic optimization of MPC for the reaching phase. Particularly, DICSMPC 285

can compete the planar deployment of about true anomaly 4.5 rad. In addition, from the magnified area on the left, we can find that the trajectory of DICSMPC possesses slight overshoot of about 0.06 rad compared with DSMPC in the early stage, which is due to the influence of input saturation. But with input compensation, the trajectory of DICSMPC is smoother and faster than DSMPC, which

290

implies the better deployment performance and application value of DICSMPC in the practical space missions. Fig.4 depicts dynamic property of tether length for the deployment. From the magnified area on the right, we can get that tether length of DSMC converges to the origin after about true anomaly 7.5 rad, whereas DICSMPC and DSMPC

295

only need about true anomaly 4.5 rad and 5.5 rad respectively. At the magnified area on the left, it is clear that the trajectory of DICSMPC has faster liberation rate than DSMPC, while the trajectory of DSMC and CTC needs much longer deployment time. Moreover, the entire deployment of DICSMPC is fast and flat, which is more suitable for the stable and fast deployment of space tether.

300

Combining the results of Fig. 3 and Fig. 4, we can conclude that the deployment mission of the proposed discrete-time methods have remarkable deploy-

24

Figure 4: Tether length of the deployment

ment performances contrasted to CTC, regarding to rise-time and settle-time. Furthermore, DICSMPC and DSMPC separately accomplishes the deployment in true anomaly 1.6 rad and 0.6 rad earlier than DSMC, which indicates the 305

dynamic optimization capability of MPC. And DICSMPC has a smoother and faster deployment performance than DSMPC, such that the effect of input compensation is also emphasized by simulation results. Fig. 5 plots the trajectories of sliding mode controlled by DICSMPC, DSMPC and DSMC, which are all on the basis of the same discrete-time sliding manifold

310

Eq. (18) for comparison purposes. As shown in Fig. 5, one can find that the trajectory of DSMC has obvious overshoots of positive and negative direction in the early stage, which are similar to impulse signals and may deteriorate or ruin the actual deployment performance. From the magnified area on the left, we can get that the trajectory of DICSMPC and DSMPC is flatter and more coherent

315

than DSMC, such that the former are more beneficial to practical engineering application. In addition, as illustrated in the magnified area on the right, it is clear that DICSMPC has small overshoots of about −1 × 10−6 compared with DSMPC, which fully demonstrates the effect of the input compensation. Fig. 6 gives out the simulation results of the discrete-time tether tension

25

4

× 10-6 DICSMPC DSMPC DSMC

2

0

×10-6

sk

-2

×10-7

2 -4

2

0

-6

-8

-2

0

-4

-2

-6

-4

-8 0

0.5

1

-6 2.5

3

3.5

4

4.5

5

6

7

8

9

-10 0

1

2

3

4

10

True anomaly (rad)

Figure 5: Sliding mode of the deployment

9 DICSMPC DSMPC DSMC

8 7

0.1

T(k) (N)

6

1

5

0.05

0.5

4 3

0

0 0.7

0.75

0.8

6

6.5

2 1 0 0

1

2

3

4

5

6

7

8

True anomaly (rad)

Figure 6: Tether tension of the deployment

26

9

10

320

T (k). On the whole, it is clear that the control cost of DICSMPC is relatively smaller than DSMPC and DSMC, because of its trajectory of tether tension is smooth and flat with no obvious oscillation. From Fig. 6, we can find that the trajectory of DSMC have large oscillations with maximum 8.82 N at the beginning of deployment. It is not in line with the actual situation of initial free

325

deployment, moreover the instantaneous tension and relaxation of space tether will affect the stability of the deployment process and may cause the wobble of the subsatellite. And it is clear that the curve of DSMPC is smoothed by dynamic optimization of MPC, compared with DSMC. However, it still has relatively large oscillations with maximums 2.52 N and 1.24 N, respectively in the

330

range of (0.43, 0.46) rad and (5.90, 6.55) rad of true anomaly. Whereas DICSMPC can effectively compensates the oscillations of DSMPC with the input compensation aided, as shown in magnified areas. A better curve of tension con¯ m trol is obtained, with its maximum is less than T (k)max = − mL h2 Umin = 0.2875

N, which satisfies the constraint set of the discrete-time tension control. 6

× 10-4

×10-4 5

1

4

2

0

∆ λk

1

DICSMPC DSMPC DSMC

×10-5

0

1.4

3

-2 6

7

8

9

2

1

0 0

1

2

3

4

5

6

7

8

9

10

True anomaly (rad)

Figure 7: Variation of tether length for the deployment

335

Fig. 7 presents the variation of tether length with respect to true anomaly. Given the total length of tether is 3.5 km, the measurement precision of length will require reaching 10−3 m. As shown in Fig. 7, the variation of tether length 27

belongs to the magnitude of 10−4 , which fully satisfies the measurement precision of the industrial length sensors. From the magnified area on the left, it is 340

clear that the trajectory of DICSMPC have smaller valley of about 4.75 × 10−5. Furthermore, the variation of tether length for DICSMPC possesses better convergence rate, which can converge to the origin after about true anomaly 5.2 rad. While DSMPC and DSMC require about true anomaly 9.5 rad to reach the steady-state, from the magnified area on the right. In general, the convergence

345

rate of DSMPC is faster than DSMC, however without the aids of input compensation, the trajectory of DSMPC generates a chattering and a long-period fluctuation around the steady-state and affect stability of the deployment. 5

× 10-4 DICSMPC DSMPC DSMC

∆ θ k (rad)

0

×10-5 5 -5

0 -5 5

6

7

8

9

-10 0

1

2

3

4

5

6

7

8

9

10

True anomaly (rad)

Figure 8: Variation of in-plane angle for the deployment

Fig. 8 shows the variation of in-plane angle for the deployment. From the results, the variation of in-plane angle locates in the magnitude of 10−4 , which 350

is easy to meet the measurement precision of the industrial angle sensors. From the magnified area, we we can see that the trajectories of DSMPC and DSMC possess a long-period fluctuation around the steady-state, which will evidently extend the deployment time of the space missions. While it is obvious that the trajectory of DICSMPC has a little volatility on the whole deployment and a

355

shorter settle-time of about true anomaly 5 rad, which is more conducive to the 28

stable and fast deployment of space tether.

6. Conclusion This paper develops a novel discrete-time pure-tension sliding mode predictive control for the deployment of STS, with taking saturated input into 360

account. In the light of the discrete-time Euler-Lagrange model, it has been proved by the multigroup simulation results, that the proposed control scheme without velocity sensor and estimator is more considerable and applicable for the tension control of STS in reality. In particular, to handle the input saturation of STS, a discrete-time auxiliary controller is involved in DSMPC scheme

365

to achieve a faster and more stable deployment performance. In a word, from the perspective of theoretical research and practical application, it is the first time that DSMPC has been successfully employed in the tether deployment of STS with preferable control performance than classical SMC, which will make a significant reference value for the space missions.

370

7. Acknowledgments This work was supported by the General Program of National Natural Science Foundation of China under Grant 61673009.

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Highlights: 1. A discrete-time sliding mode predictive control scheme is proposed to realize the deployment of space tethered satellite system. 2. A discrete-time Euler-Lagrange model is developed for space tethered satellite system. 3. The discrete-time sliding manifold is typical underactuated and nonlinear coupled, subject to input and states constraints. 4. Given the input saturation, an auxiliary controller is put forward to guarantee the asymptotic stability analysis of control. 5. Considering the limitations of auxiliary thrusters, the discrete-time sliding mode predictive control scheme is pure-tension control with only length and angle measurements.

Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: