Relative Position Model Predictive Control of Double Cube Test-Masses Drag-Free Satellite with Extended Sliding Mode Observer

'e drag-free satellites, being space-borne ultrahigh precise measurement platforms, have played irreplaceable roles in a great number of space science missions such as navigation, earth science, fundamental physics, and astrophysics. Most of these missions have to be performed based on the satellites placed with double cube test-masses, which makes the satellite layout and control strategy bemore complex.'is paper investigates the orbit keeping control problem of a class of low Earth orbit drag-free satellites with double cube test masses. A disturbance observer-based composite control method is proposed, which consists of an extended sliding mode observer and the tube-based robust model predictive control approach. In this design, the observer is proposed to estimate the relative position and velocity of the satellite and the external space disturbance force. A tube-based robust model predictive control scheme is then developed to stabilize the satellite orbit control systems in the presence of actuator saturation, state constraints, and additive stochastic noises. Finally, a simulation example is presented to demonstrate the efficacy and superiority of the proposed orbit control method.


Introduction
In recent years, the drag-free satellites [1], being space-borne ultrahigh precise measurement platforms, have played irreplaceable roles in many space science missions, such as the test of equivalence principle [2], the measurement of the Earth gravity field [3], and the detection of gravitational waves [4]. e drag-free satellites possess many advantages; for example, they can provide autonomous precision orbit determination, map the static and time-varying components of the Earth's mass distribution more accurately, deepen the understanding of the fundamental force of gravity, eventually open up a new window to the universe through the detection and observation of gravitational waves, and so forth.
e key technology of the drag-free satellite is the gravitational reference sensor (GRS), which insulates an internal free-floating test mass (TM, also called proof mass) from both external disturbances and disturbances caused by the spacecraft itself [5]. e drag-free satellites can be divided into two types [6]. e first one is the "accelerometer" drag-free mode, where an electrostatic accelerometer is used as the primary sensor and an electrostatic suspension actuator is paired to maintain the TM to be centred in its cage; therefore it can counter the disturbance forces acting on the spacecraft [7][8][9][10][11]. e second one is free-falling TM mode, in which the satellite provides indirect drag-free behaviour by tracking the movement of the free-falling TM in the cage [12][13][14][15]. In particular, the structure of a satellite with two cube TMs is always regarded as the primary layout in these missions, which makes the GRS be more complex and the control system design work be more challenging. For example, as shown in Figure 1, the drag-free satellite containing two cube TMs will enter a low Earth orbit (LEO), which decays more rapidly due to the decelerating effects of the Earth's atmosphere.
is paper investigates the orbit control problem for a class of LEO drag-free satellites with double cube TMs based on the relative position dynamics with state constraints, actuator saturation, and the additive stochastic disturbances. In this design, an extended sliding mode observer method is developed to estimate the system states. en, a tube-based robust model predictive control (TRMPC) approach is presented [52] to stabilize the resulting orbit control systems. In particular, the TRMPC design approach is divided into the following two steps: (i) an offline evaluation of the constraints to ensure the uncertain future trajectories to lie in sequence of sets, which is called tubes, and (ii) an online MPC scheme designed for the nominal trajectories. As a result, the orbit control issue for the drag-free satellite is solved. e remaining of this paper is organized as follows. Section 2 formulates the design problem under investigation; the design of disturbance observation-based composite control strategy is presented in Section 3. A demonstrated simulation example is provided in Section 4, and the paper is concluded in Section 5.

Problem Formulation
According to [22,53], for a LEO satellite containing two TMs, as shown in Figure 2, the drag-free control strategy is defined as follows: TM1 is chosen as the gravitational reference, which flies freely in a pure gravitational orbit, and the satellite and TM2 are controlled to follow TM1; then the linearized relative position dynamics between the TMs and the satellite are given as follows: where i � 0 means the satellite and i � 2 means the TM2; ρ 1i (t), m i , and F c,1i (t) represent the relative position variables, the mass of the satellite or the TM2, and the control forces; F d,1i ′ (t) is the sum of all kinds of disturbances, which for the satellite, the air drag, the solar radiation pressure, and the thruster quantization error and other stochastic noises is considered primarily, and, for the TM2, it mainly consists of electrostatic interference signals, actuator quantization error, and other stochastic noises [9,40].
where ω 0 is orbit angular velocity of the TM1 and K t and D t are the damping factors between the TM and its condenser cage [54]. e desired positions and velocities of the satellite and TM2 are denoted as ρ d 10 (t), _ ρ d 10 (t), ρ d 12 (t), and _ ρ d 12 (t), respectively. e desired positions ρ d 10 (t) and ρ d 12 (t) are some constants dependent on the layout of the satellite. e relative velocities are defined as follows: _ ρ d 10 (t) � 0 and _ ρ d 12 (t) � 0. Define the following error variables for case of presentation: By substituting (3) into (1), one can obtain Due to the limit volume of the capacitor cage, the relative error variables should satisfy ρ 1i (t) ≤ ρ max and _ ρ 1i (t) ≤ _ ρ max , which are the state constraints. For convenience, formations (4) should be translated into a state space model. By 2 Mathematical Problems in Engineering where ω i (t) and ξ i (t) (i � 1, 2) denote the unknown stochastic noises and unknown estimable disturbances, respectively. en, the state space model is given as follows: where Before designing the disturbance observer-based composite control strategy, the following assumptions are given about ω i (t) and ξ i (t) (i � 1, 2).

Assumption 2.
e unknown stochastic noise ω i (t) (i � 1, 2) is assumed to be discontinuous but bounded subject to ‖ω i (t)‖ ≤ ϖ i and belongs to a bounded and convex subset W ⊂ R n containing the origin in its interior.

Design of Disturbance Observer-Based
Composite Control e control system composition of a LEO satellite with two cube TMs is shown in Figure 3. To achieve the "drag-free" goal and eliminate the effects of external disturbances (the environment disturbances may contain the atmosphere and the solar radiation pressure, the thruster quantization error, the electrostatic noises, the actuator quantization error, etc.), a disturbance observer-based composite control approach is proposed, which consists of an extended sliding mode observer and a tube-based robust model predictive control. e detailed design process is as follows. First, for system (5), the observer is proposed to estimate states x i for state feedback control and ξ i (t) for active disturbance rejection control u ffc,i . Second, a tube-based robust model predictive control u i � u fbc,i + u ffc,i is adopted, which can cope with state constraints and actuator saturation and attenuate the effects of additive stochastic noises. For convenience, in the following discussion, the subscript i in (5) is omitted.

Design of Extended Sliding Mode Observers.
Motivated by the augmented strategy method in [34,36,55], we define the following extended vectors and matrices: For system (7), consider the following continuous-time extended SMO [55]: where and v 2 (t) is the discontinuous term designed as follows: , and λ max (F T F) and λ min (F T F) are the maximal and minimal nonzero eigenvalues of matrix F T F, respectively.

Remark 1.
It is worth mentioning that the main function of the proposed extended SMO is to estimate the state and disturbances vectors simultaneously, which is a design basis of the subsequent composite control law. Define ; then the error system is derived as follows: Lemma 1 (see [56]). System (5) has the relative degree n with respect to the unknown input ξ(t) (i.e., the system is strongly observable).

Lemma 2. If the matrix pair (A, C) is detectable and the condition
holds, then the pair (A, C) is detectable; that is, there exists an observer gain matrix L such that A − LC is Hurwitz.
Proof. It can be derived that When λ ≠ 0, for the matrix in right side of equation (12), we have at means en, the following rank condition holds: Case 2. When λ � 0, if rank condition (11) holds, we have Finally, for both cases, we can always imply To attenuate the influence of ω(t) for error system (10), we define the prescribed H ∞ performance index as follows: where ψx(t) is the weighted estimation error and ψ is the weight matrix. e following main theorem provides the existence condition of the proposed observer (8).

Theorem 1.
e solution of observer error system (10) is asymptotically stable with the prescribed H ∞ performance index if there exist positive and definite matrix Q > 0 and matrix F with appropriate dimensions, in minimizing c 1 > 0, such that the following matrix constraints hold: Moreover, the observer gain is designed as Proof. For system (10), select the Lyapunov function as
Case 4. ‖y‖ ≤ (ε/η); in this case, it can be derived that Based on Cases 3 and 4, we can conclude that To minimize the effect of the disturbance on the estimation error in the sense of L 2 norm, we consider the following constraint: In light of Schur complement, it is easy to derive that Accordingly, it can be seen that W(t) ≤ 0 is ensured provided that Q < 0 holds. erefore, the solution of error system (10) is asymptotically stable with the prescribed H ∞ performance index as t ⟶ ∞.

Remark 2.
It should be pointed out that, in eorem 1, the equality constraint QD � C T F T can be rewritten as Hence, we can introduce the following condition: where c 2 > 0 is a parameter to be designed. en the design problem of observer gains L can be converted into the following minimization problem: subject to (20) and (28).

Design of Tube-Based Robust Model Predictive Control.
For equations (5), ξ i (t) is estimated and eliminated through the extended sliding mode observer; thus we have to compensate it by disturbance compensation feed-forward control. Translate formulations (5) without ξ i (t) into the discrete model of the form in (30) as follows: As mentioned before, the controller is designed as u k � u k,ffc + u k,fbc , where is the feed-forward control part to compensate ξ k and u k,fbc is the model predictive control law to be designed. Define the estimation error as 6 Mathematical Problems in Engineering en, system (30) becomes As proved in the observer design results in eorem 1, we have ‖ξ k ‖ ⟶ 0 with k ⟶ ∞; thus ‖ω k − ξ k ‖ ≤ ‖ω k ‖ + ‖ξ k ‖ � (1 + β)‖ω k ‖ holds, where β > 0 is a small constant and β ⟶ 0 when k ⟶ ∞.
Define d k � ω k − ξ k ; it satisfies ‖d k ‖ ≤ (1 + β)ϖ, which belongs to the bounded and convex subset W. Hence, system (30) can be written in the following form: e investigated system (30) is subject to hard constraints on both state and input vectors with the following form: x ∈ X, where X and U are polytopes.
To solve this control problem, a robust MPC algorithm is considered [57] by repeatedly solving an optimal control problem, where the finite horizon quadratic cost J N (x, u) to be minimized at the current time k is In (36), N ∈ R + is the MPC prediction horizon, Q ∈ R n×n , Q > 0, R ∈ R m×m , R > 0 and P is the solution of the algebraic Riccati equation [57].
Due to the presence of the unknown disturbance d k , we rewrite the state vector x i|k of the system as the sum of a nominal part z i|k and an error part e i|k in the following form: where e i|k denotes the deviation of the real state e i|k with respect to the nominal one. Design the following feedback policy (39) for system (30): where v i|k denotes the nominal input vector and the gain matrix K should be selected such that A K � A d + B d K is Schur-stable; then the corresponding nominal and error dynamics can be described, respectively, as follows: Hence, the finite horizon optimal quadratic cost (36) can be redefined in terms of nominal state z k and control input v k as (42) and the finite horizon optimal control problem can be reformulated as follows.
Definition 1. Given the nominal system dynamics (40), cost (42), and nominal constraints set Z, V, Z f , the nominal robust MPC finite horizon optimization problem can be described as e solution of (43) is the optimal nominal control sequence v * 0|k � [v * 0|k (0; z k ), . . . , v * T− 1|k (T − 1; z k )] and the first control action, that is, κ N (z k ): � v * 0|k (0; z k ), represents the optimal control v i|k to be applied to system (40). e proposed control law applied on the uncertain system (40), according to the control policy adopted, is e composite closed-loop system then satisfies For the TRMPC approach, the matrix K in the control policy (39) is designed to stabilize system (30). Consider the following closed-loop system: Hence, the satisfaction of the following condition aims to define the feedback gain K that stabilizes the system.

Mathematical Problems in Engineering
In order to robustly satisfy the mission constrains, they are tightened to allow the trajectories of the uncertain system, affected by disturbance, to lie in a tube centred on the nominal one, where each trajectory is related to a particular realization of the uncertainty at each time step k. e derivations of the nominal state, input, and terminal constrains set Z, V, Z f are described according to the approach proposed in [57], such that the constraints in (39) of system (30) are satisfied for every realization of the disturbance sequence ω by suitable design of the tube.
We now define S K (∞): where ⊕ is the Minkowski sum and A j K W: � A j K d k |d k ∈ W} is the set multiplication; the uncertain set of the error e i|k is the minimal robust positive invariant set for en the state and control input vector constraints in (35) are satisfied provided that where ⊝ denotes the Pontryagin set difference. It is obvious that the terminal constraint for system (30) at time instant N is ensured if the normal system (40) satisfies the tighter constraint Moreover, these assertions only make sense if the disturbance set W is sufficiently small to satisfy the following Assumption 3, as defined in [57]. Assumption 3. (Restricted disturbances for constraints satisfaction) S K ⊂ X and K × S K ⊂ U. e next step is to define a robust positively invariant set S K for [19] to obtain the tighter constraints acting on the nominal system. en the constraints are considered for the TRMPC problem. Once the uncertainty set W is evaluated, an inner approximation of the nominal constraint set can be constructed. In this design, we adopt the following strategy presented in [57].

Algorithm 1. Computation of Z and V
(1) Define the linear state constraint as: (2) Construct the nominal state constraint inequality Choose a suitable α ∈ (0, 1) and N such that Hence, the observer-based model predictive control strategy could be formally described by the following algorithm.

Algorithm 2.
Disturbance observer-based model predictive control strategy.
(1) Initialization: at time k � 0, set x k � z k � x(0) where x(0) denotes the current state. (2) At time k, considering the current state (x k , z k ), based on the disturbance estimation ξ k from the observer (8), solve the nominal optimal control problem (43) to obtain the nominal control vector v k : � v * 0|k (0; z k ) and the control input vector (3) If the nominal optimal control problem (43) is infeasible, adopt safety/recovery procedure. (4) Apply the control u k to the system (45) and (46); (5) Calculate the estimation x k+1 from the observer (8) as successor state x k+1 of the system (30), and calculate the successor state z k+1 of the nominal system (40). (6) Set (x k , z k ) � (x k+1 , z k+1 ), set k � k + 1 , and go to (2).

Simulation Results
In this section, a numerical simulation is carried out to verify the effectiveness of the extended sliding mode-based TRMPC approach. Suppose that the satellite is flying at a little eccentric low Earth orbit with altitude of 300 km. e parameters in (1) are given in Table 1 [54].
For the relative position motion between the TM1 and the satellite, define F d,10 � F drag + F thruster , where e resolution and the maximum value of the thruster are 10 − 6 N and 0.4 N. Besides, a zero mean white noise with mean squared error being 10 − 8 N is added as the stochastic disturbance.

Conclusions
is paper has considered the relative position control of the drag-free satellite with double cube test-masses in the presence of external disturbance, additive stochastic disturbances, actuator quantization error, actuator saturation, and state constraints. An extended sliding mode observer method is adopted to estimate the state vector and external disturbance, based on which a tube-based robust model predictive control scheme is developed. e designed control method can not only cope with the constraints of control and state but also attenuate the effect of additive stochastic noises. Future work will be focused on the consideration of relative attitude dynamics between the test-masses and the satellite.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.