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

Authors: Enyou Wang; Jinxiu Zhang; Huayi Li; Ming Liu
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

The 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 be more complex. This 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.


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## Body

## 1. 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]. The 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.The 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]. The drag-free satellites can be divided into two types [6]. The 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 [7–11]. The 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 [12–15]. 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.Figure 1
A kind of drag-free satellite structure with double test-masses.In the research area of control for drag-free satellites, a trade-off should be considered among system complexity, fuel conservation, cost of components, operations, performance, and so forth [16]. In fact, it is always difficult for the drag-free satellite to obtain control satisfactory performance due to the existence of external time-varying environmental disturbances and the unmodelled internal uncertainties. To solve this problem, various types of robust control algorithms have been proposed, such as model predictive control (MPC) [17], H∞ control [18, 19], μ-synthesis control [20], embedded model control (EMC) technique [21], and quantitative feedback theory [22]. However, these design results can only realize disturbance attenuation but not disturbance rejection. In order to improve closed-loop control performance, the disturbance observer-based control [23–31] and active disturbance rejection control [32–39] have been developed for drag-free satellite to pursue more ideal control system performance [25, 40–50]. However, how to design a suitable observer to estimate the disturbances more precisely is still an open and attractive problem [51].This 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. Then, 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.The remaining of this paper is organized as follows. Section2 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.


## 2. 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:(1)ρ¨1it+Mtρ˙1it+Ntρ1it=1miFc,1it+Fd,1i′t,i=0or2,where i=0 means the satellite and i=2 means the TM2; ρ1it, mi, and Fc,1it represent the relative position variables, the mass of the satellite or the TM2, and the control forces; Fd,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].(2)Mt=0−2ω002ω000000−Kt,Nt=−3ω00000000ω02−Dt,where ω0 is orbit angular velocity of the TM1 and Kt and Dt are the damping factors between the TM and its condenser cage [54].Figure 2
The relative position dynamics between the TMs and the satellite, with respect to TM1 Hill reference system.The desired positions and velocities of the satellite and TM2 are denoted asρ10dt, ρ˙10dt, ρ12dt, and ρ˙12dt, respectively. The desired positions ρ10dt and ρ12dt are some constants dependent on the layout of the satellite. The relative velocities are defined as follows: ρ˙10dt=0 and ρ˙12dt=0. Define the following error variables for case of presentation:(3)ρ˜1it=ρ1it−ρ1idt,ρ˜˙1it=ρ˙1it−ρ˙1idt,i=0or2.By substituting (3) into (1), one can obtain(4)ρ˜¨1it+Mtρ˜˙1it+Ntρ˜1it=1miFc,1it+Fd,1it,i=0or2.Due to the limit volume of the capacitor cage, the relative error variables should satisfyρ˜1it≤ρ˜max and ρ˜˙1it≤ρ˜˙max, which are the state constraints. For convenience, formations (4) should be translated into a state space model. By choosing x1t=ρ˜10Tt,ρ˜˙10TtT and x2t=ρ˜12Tt,ρ˜˙12TtT, u1t=Fc,10t, u2t=Fc,12t, Fd,10t=ω1t+ξ1t, and Fd,12t=ω2t+ξ2t, where ωit and ξiti=1,2 denote the unknown stochastic noises and unknown estimable disturbances, respectively. Then, the state space model is given as follows:(5)x˙it=Aixit+Biuit+Bω,iωit+ξit,yit=Cixit,where(6)Ai=03I3NtMt,B1=Bω,1=031m0I3,B2=Bω,2=031m2I3,Ci=I303,i=1,2.Before designing the disturbance observer-based composite control strategy, the following assumptions are given aboutωi(t) and ξiti=1,2.Assumption 1.
The unknown external disturbanceξiti=1,2 satisfies the following: ξit<ϱi and ξ˙it<ςi, where ϱi and ςi are known real constants.Assumption 2.
The unknown stochastic noiseωiti=1,2 is assumed to be discontinuous but bounded subject to ωit≤ϖi and belongs to a bounded and convex subset W⊂ℝn containing the origin in its interior.


## 3. Design of Disturbance Observer-Based Composite Control

The control system composition of a LEO satellite with two cube TMs is shown in Figure3. 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. The detailed design process is as follows. First, for system (5), the observer is proposed to estimate states xi for state feedback control and ξit for active disturbance rejection control uffc,i. Second, a tube-based robust model predictive control ui=ufbc,i+uffc,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.Figure 3
The control system schematic diagram of a LEO satellite with two cube TMs; the satellite and TM2 are controlled, respectively.


### 3.1. Design of Extended Sliding Mode Observers

Motivated by the augmented strategy method in [34, 36, 55], we define the following extended vectors and matrices:(7)x¯t=xTt,ξTtT,A¯=ABω00,B¯=B0,D¯=0Ip,B¯w=Bw0,C¯=C0.For system (7), consider the following continuous-time extended SMO [55]:(8)x¯^˙t=A¯x¯^t+B¯ut+Ly˜t+D¯Fv2t,where y˜t=yt−C¯x¯^t and v2t is the discontinuous term designed as follows:(9)v2t=ηy˜ty˜t,ify˜t>εη,η2y˜tε,ify˜t≤εη,where η>λ¯F/λ¯F2 and λ¯F=λmaxFTF, λ¯F=λminFTF, and λmaxFTF and λminFTF are the maximal and minimal nonzero eigenvalues of matrix FTF, 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.
Definex¯˜t=x¯t−x¯^t; then the error system is derived as follows:(10)x¯˜˙t=A¯−LC¯x¯˜t+D¯ξ˙t−Fv2t+B¯wωt.Lemma 1 (see [56]).
System (5) has the relative degree n with respect to the unknown input ξt (i.e., the system is strongly observable).Lemma 2.
If the matrix pairA,C is detectable and the condition(11)rankABωC0=n+p,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(12)rankA¯−λIn+pC¯=rankA−λInBw0p×n−λIpC0q×p.Case 1.
Whenλ≠0, for the matrix in right side of equation (12), we have(13)Inλ−1Bw00Ip000IqA−λInBw0p×n−λIpC0q×p=A−λIn0p×n0p×n−λIpC0q×p,which means that(14)rankA−λInBw0p×n−λIpC0q×p=rankInλ−1Bw00Ip000IqA−λInBw0p×n−λIpC0q×p=rankA−λIn0n×p0p×n−λIpC0q×p=rankA−λInC¯+p.
That means(15)rankA¯−λIn+pC¯=rankA−λInC¯+p.
Hence, if the pairA,C is detectable, we have(16)rankA−λInC¯=n.
Then, the following rank condition holds:(17)rankA−λIn+pC¯=n+p.Case 2.
Whenλ=0, if rank condition (11) holds, we have(18)rankA¯C¯=rankABwC0=n+p.
Finally, for both cases, we can always implyrankA−λIn+pC¯=n+p, λ∈ℂ, Reλ≥0, which implies that the pair A¯,C¯ is detectable.
To attenuate the influence ofωt for error system (10), we define the prescribed H∞ performance index as follows:(19)H∞=supωtL2≠0ψx¯˜tL2ωtL2≤γ1,where ψx¯˜t is the weighted estimation error and ψ is the weight matrix. The following main theorem provides the existence condition of the proposed observer (8).Theorem 1.
The 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 γ1>0, such that the following matrix constraints hold:(20)Q¯=A¯TQ+QA¯−C¯TC¯+ψTψQB¯wB¯wTQ−γ1I<0,QD¯=C¯TFT.
Moreover, the observer gain is designed asL=1/2Q−1C¯T.Proof.
For system (10), select the Lyapunov function as Vt=x¯˜TtQx¯˜t with Q>0 being designed; it is easy to obtain that(21)V˙t=x¯˜˙TtQx¯˜t+x¯˜TtQx¯˜˙t=2x¯˜TtQA¯−LC¯x¯˜t+D¯ξ˙t−Fv2t+B¯wωt=x¯˜TtA¯TQ+QA¯−C¯TC¯x¯˜t+2x¯˜tC¯TFTξ˙t−Fv2t+x¯˜TtQB¯wωt+ωTtB¯wTQx¯˜t=x¯˜TtA¯TQ+QA¯−C¯TC¯x¯˜t+2y˜FTξ˙t−Fv2t+2x¯˜TtQB¯wωt.
In the following analysis, letYt=2y˜FTξ˙t−Fv2t; we consider two cases separately.Case 3.
y˜>ε/η; in this case, we have(22)Yt=2y˜FTξ˙t−Fηy˜ty˜t≤2y˜Fρ−F2η≤2y˜λ¯Fρ−λ¯F2η≤0,where η can be chosen such that η>λ¯F/λ¯F2ρ.Case 4.
y˜≤ε/η; in this case, it can be derived that(23)Yt=2y˜FTξ˙t−Fη2y˜tε≤2ρλ¯Fy˜−2η2λ¯F2y˜2ε≤2ηλ¯F2y˜−2η2λ¯F2y˜2ε≤−2λ¯F2ηy˜ε−ε22+λ¯F2ε2≤2λ¯F2ηε/ηε−ε22+λ¯F2ε2≤0.
Based on Cases3 and 4, we can conclude that(24)V˙t=Yt+x¯˜TtA¯TQ+QA¯−C¯TC¯x¯˜t+2x¯˜TtQB¯wωt≤x¯˜TtA¯TQ+QA¯−C¯TC¯x¯˜t+2x¯˜TtQB¯wωt.
To minimize the effect of the disturbance on the estimation error in the sense ofL2 norm, we consider the following constraint:(25)Wt=V˙t+x¯˜TtψTψx¯˜t−γ1ωTtωt≤0.
In light of Schur complement, it is easy to derive that(26)Wt≤x¯˜TtA¯TQ+QA¯−C¯TC¯x¯˜t+x¯˜TtQB¯wωt+ωTtB¯wTQx¯˜t+x¯˜TtψTψx¯˜Tt−γ1ωTtωt≤x¯˜TtωTtQ¯x¯˜tωt<0.
Accordingly, it can be seen thatWt≤0 is ensured provided that Q¯<0 holds. Therefore, 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 Theorem1, the equality constraint QD¯=C¯TFT can be rewritten as(27)TraceQD¯−C¯TFTTQD¯−C¯TFT=0.
Hence, we can introduce the following condition:(28)QD¯−C¯TFTTQD¯−C¯TFT<γ2I,where γ2>0 is a parameter to be designed. Then the design problem of observer gains L can be converted into the following minimization problem:(29)minγ1,γ2subject to20and28.


### 3.2. Design of Tube-Based Robust Model Predictive Control

For equations (5), ξit 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 ξit into the discrete model of the form in (30) as follows:(30)xk+1=Adxk+Bduk+Bωdξk+ωk.As mentioned before, the controller is designed asuk=uk,ffc+uk,fbc, where(31)uk,ffc=−ξ^k,is the feed-forward control part to compensate ξk and uk,fbc is the model predictive control law to be designed. Define the estimation error as(32)ξ˜k=ξ^k−ξk.Then, system (30) becomes(33)xk+1=Adxk+Bduk,fbc+Bωdωk−ξ˜k.As proved in the observer design results in Theorem1, we have ξ˜k⟶0 with k⟶∞; thus ωk−ξ˜k≤ωk+ξ˜k=1+βωk holds, where β>0 is a small constant and β⟶0 when k⟶∞.Definedk=ωk−ξ˜k; it satisfies dk≤1+βϖ, which belongs to the bounded and convex subset W. Hence, system (30) can be written in the following form:(34)xk+1=Adxk+Bduk,fbc+Bωddk.The investigated system (30) is subject to hard constraints on both state and input vectors with the following form:(35)x∈X,u∈U,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 JNx,u to be minimized at the current time k is(36)JNxk,uk=∑i=0N−1xi|kTQxi|k+ui|kTℛui|k+xN|kTPxN|k.In (36), N∈ℝ+ is the MPC prediction horizon, Q∈ℝn×n, Q>0, ℛ∈ℝm×m, R>0 and P is the solution of the algebraic Riccati equation [57].(37)Ad+BdKTPAd+BdK+Q+KTℛK=P.Due to the presence of the unknown disturbancedk, we rewrite the state vector xi|k of the system as the sum of a nominal part zi|k and an error part ei|k in the following form:(38)xi|k=zi|k+ei|k,where ei|k denotes the deviation of the real state ei|k with respect to the nominal one.Design the following feedback policy (39) for system (30):(39)ui|k=vi|k+Kxi|k−zi|k,where vi|k denotes the nominal input vector and the gain matrix K should be selected such that AK=Ad+BdK is Schur-stable; then the corresponding nominal and error dynamics can be described, respectively, as follows:(40)zi+1|k=Adzi|k+Bdvi|k,z0|k=x0|k,(41)ei+1|k=AKei|k+Bωddi|k,e0|k=0.Hence, the finite horizon optimal quadratic cost (36) can be redefined in terms of nominal state zk and control input vk as(42)JNzk,vk=∑i=0N−1zi|kTQzi|k+vi|kTℛvi|k+zN|kTPzN|k,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,ℤf, the nominal robust MPC finite horizon optimization problem can be described as(43)minvJNzk,vks.t.zi+1|k=Adzi|k+Bdvi|k,z0|k=xkzi|k∈ℤ,i∈1,Nvi|k∈V,i∈0,N−1zi|k∈ℤf.
The solution of (43) is the optimal nominal control sequence v0|k∗=v0|k∗0;zk,…,vT−1|k∗T−1;zk and the first control action, that is, κ˜Nzk:=v0|k∗0;zk, represents the optimal control vi|k to be applied to system (40).
The proposed control law applied on the uncertain system (40), according to the control policy adopted, is(44)ui|k=vi|k+Kxi|k−zi|k=κNxk,zk=κ˜Nzk+Kxk−zk.
The composite closed-loop system then satisfies(45)xi+1|k=Adxi|k+BdκNi,xk,zk+Bωddi|k,(46)zi+1|k=Adzi|k+Bdκ˜Ni,zk.
For the TRMPC approach, the matrixK in the control policy (39) is designed to stabilize system (30). Consider the following closed-loop system:(47)xi+1|k=Ad+BdKxi|k+Bdvi|k+Bωddi|k.
Hence, the satisfaction of the following condition aims to define the feedback gainK that stabilizes the system.(48)Ad+BdKTP^Ad+BdK−P^<0,P^>0.
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 stepk. The derivations of the nominal state, input, and terminal constrains set ℤ,V,ℤf 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 defineSK∞:=AK0W⊕AK1W⊕⋯=∑j=0∞AKjW, where ⊕ is the Minkowski sum and AKjW:=AKjdk|dk∈W is the set multiplication; the uncertain set of the error ei|k is the minimal robust positive invariant set for(49)xi+1|k=Adxi|k+Bωddi|k,dk∈W.
Then the state and control input vector constraints in (35) are satisfied provided that(50)zi|k∈X⊝SK∞,vi|k∈U⊝KSK∞,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(51)zN∈ℤ⊆X−SK∞,ℤf⊆ℤ.
Moreover, these assertions only make sense if the disturbance setW is sufficiently small to satisfy the following Assumption 3, as defined in [57].Assumption 3.
(Restricted disturbances for constraints satisfaction)SK⊂X and K×SK⊂U.
The next step is to define a robust positively invariant setSK for [19] to obtain the tighter constraints acting on the nominal system. Then 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 ofℤ and V(1)
Define the linear state constraint as:X=xi|k∈ℝn|axi|k<b;(2)
Construct the nominal state constraint inequalityazi|k≤b−maxaei|k|ei|k∈SK∞=b−Φ∞;(3)
Approximate the upper value ofΦ∞ as ΦN=maxa∑i=0N−1AKidi|k|di|k∈W;(4)
Choose a suitableα∈0,1 and N such that AKNdi|k∈αW, compute Φ∞≤1−α−1ΦN;(5)
Compute the nominal state constraint setℤ:=zi|k∈ℝn|azi|k≤b−1−α−1ΦN;(6)
Compute the nominal control constraint setV:=vi|k∈ℝm|a′vi|k≤b′−K1−α−1ΦN, where U:=ui|k∈ℝm|a′ui|k≤b′.
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 timek=0, set xk=zk=x0 where x0 denotes the current state.(2)
At timek, considering the current state (xk,zk), based on the disturbance estimation ξ^k from the observer (8), solve the nominal optimal control problem (43) to obtain the nominal control vector vk:=v0|k∗0;zk and the control input vector uk=vk+Kxk−zk−ξ^k.(3)
If the nominal optimal control problem (43) is infeasible, adopt safety/recovery procedure.(4)
Apply the controluk to the system (45) and (46);(5)
Calculate the estimationx^k+1 from the observer (8) as successor state xk+1 of the system (30), and calculate the successor state zk+1 of the nominal system (40).(6)
Set (xk,zk) = (xk+1,zk+1), set k=k+1 , and go to (2).


## 3.1. Design of Extended Sliding Mode Observers

Motivated by the augmented strategy method in [34, 36, 55], we define the following extended vectors and matrices:(7)x¯t=xTt,ξTtT,A¯=ABω00,B¯=B0,D¯=0Ip,B¯w=Bw0,C¯=C0.For system (7), consider the following continuous-time extended SMO [55]:(8)x¯^˙t=A¯x¯^t+B¯ut+Ly˜t+D¯Fv2t,where y˜t=yt−C¯x¯^t and v2t is the discontinuous term designed as follows:(9)v2t=ηy˜ty˜t,ify˜t>εη,η2y˜tε,ify˜t≤εη,where η>λ¯F/λ¯F2 and λ¯F=λmaxFTF, λ¯F=λminFTF, and λmaxFTF and λminFTF are the maximal and minimal nonzero eigenvalues of matrix FTF, 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.
Definex¯˜t=x¯t−x¯^t; then the error system is derived as follows:(10)x¯˜˙t=A¯−LC¯x¯˜t+D¯ξ˙t−Fv2t+B¯wωt.Lemma 1 (see [56]).
System (5) has the relative degree n with respect to the unknown input ξt (i.e., the system is strongly observable).Lemma 2.
If the matrix pairA,C is detectable and the condition(11)rankABωC0=n+p,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(12)rankA¯−λIn+pC¯=rankA−λInBw0p×n−λIpC0q×p.Case 1.
Whenλ≠0, for the matrix in right side of equation (12), we have(13)Inλ−1Bw00Ip000IqA−λInBw0p×n−λIpC0q×p=A−λIn0p×n0p×n−λIpC0q×p,which means that(14)rankA−λInBw0p×n−λIpC0q×p=rankInλ−1Bw00Ip000IqA−λInBw0p×n−λIpC0q×p=rankA−λIn0n×p0p×n−λIpC0q×p=rankA−λInC¯+p.
That means(15)rankA¯−λIn+pC¯=rankA−λInC¯+p.
Hence, if the pairA,C is detectable, we have(16)rankA−λInC¯=n.
Then, the following rank condition holds:(17)rankA−λIn+pC¯=n+p.Case 2.
Whenλ=0, if rank condition (11) holds, we have(18)rankA¯C¯=rankABwC0=n+p.
Finally, for both cases, we can always implyrankA−λIn+pC¯=n+p, λ∈ℂ, Reλ≥0, which implies that the pair A¯,C¯ is detectable.
To attenuate the influence ofωt for error system (10), we define the prescribed H∞ performance index as follows:(19)H∞=supωtL2≠0ψx¯˜tL2ωtL2≤γ1,where ψx¯˜t is the weighted estimation error and ψ is the weight matrix. The following main theorem provides the existence condition of the proposed observer (8).Theorem 1.
The 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 γ1>0, such that the following matrix constraints hold:(20)Q¯=A¯TQ+QA¯−C¯TC¯+ψTψQB¯wB¯wTQ−γ1I<0,QD¯=C¯TFT.
Moreover, the observer gain is designed asL=1/2Q−1C¯T.Proof.
For system (10), select the Lyapunov function as Vt=x¯˜TtQx¯˜t with Q>0 being designed; it is easy to obtain that(21)V˙t=x¯˜˙TtQx¯˜t+x¯˜TtQx¯˜˙t=2x¯˜TtQA¯−LC¯x¯˜t+D¯ξ˙t−Fv2t+B¯wωt=x¯˜TtA¯TQ+QA¯−C¯TC¯x¯˜t+2x¯˜tC¯TFTξ˙t−Fv2t+x¯˜TtQB¯wωt+ωTtB¯wTQx¯˜t=x¯˜TtA¯TQ+QA¯−C¯TC¯x¯˜t+2y˜FTξ˙t−Fv2t+2x¯˜TtQB¯wωt.
In the following analysis, letYt=2y˜FTξ˙t−Fv2t; we consider two cases separately.Case 3.
y˜>ε/η; in this case, we have(22)Yt=2y˜FTξ˙t−Fηy˜ty˜t≤2y˜Fρ−F2η≤2y˜λ¯Fρ−λ¯F2η≤0,where η can be chosen such that η>λ¯F/λ¯F2ρ.Case 4.
y˜≤ε/η; in this case, it can be derived that(23)Yt=2y˜FTξ˙t−Fη2y˜tε≤2ρλ¯Fy˜−2η2λ¯F2y˜2ε≤2ηλ¯F2y˜−2η2λ¯F2y˜2ε≤−2λ¯F2ηy˜ε−ε22+λ¯F2ε2≤2λ¯F2ηε/ηε−ε22+λ¯F2ε2≤0.
Based on Cases3 and 4, we can conclude that(24)V˙t=Yt+x¯˜TtA¯TQ+QA¯−C¯TC¯x¯˜t+2x¯˜TtQB¯wωt≤x¯˜TtA¯TQ+QA¯−C¯TC¯x¯˜t+2x¯˜TtQB¯wωt.
To minimize the effect of the disturbance on the estimation error in the sense ofL2 norm, we consider the following constraint:(25)Wt=V˙t+x¯˜TtψTψx¯˜t−γ1ωTtωt≤0.
In light of Schur complement, it is easy to derive that(26)Wt≤x¯˜TtA¯TQ+QA¯−C¯TC¯x¯˜t+x¯˜TtQB¯wωt+ωTtB¯wTQx¯˜t+x¯˜TtψTψx¯˜Tt−γ1ωTtωt≤x¯˜TtωTtQ¯x¯˜tωt<0.
Accordingly, it can be seen thatWt≤0 is ensured provided that Q¯<0 holds. Therefore, 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 Theorem1, the equality constraint QD¯=C¯TFT can be rewritten as(27)TraceQD¯−C¯TFTTQD¯−C¯TFT=0.
Hence, we can introduce the following condition:(28)QD¯−C¯TFTTQD¯−C¯TFT<γ2I,where γ2>0 is a parameter to be designed. Then the design problem of observer gains L can be converted into the following minimization problem:(29)minγ1,γ2subject to20and28.


## 3.2. Design of Tube-Based Robust Model Predictive Control

For equations (5), ξit 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 ξit into the discrete model of the form in (30) as follows:(30)xk+1=Adxk+Bduk+Bωdξk+ωk.As mentioned before, the controller is designed asuk=uk,ffc+uk,fbc, where(31)uk,ffc=−ξ^k,is the feed-forward control part to compensate ξk and uk,fbc is the model predictive control law to be designed. Define the estimation error as(32)ξ˜k=ξ^k−ξk.Then, system (30) becomes(33)xk+1=Adxk+Bduk,fbc+Bωdωk−ξ˜k.As proved in the observer design results in Theorem1, we have ξ˜k⟶0 with k⟶∞; thus ωk−ξ˜k≤ωk+ξ˜k=1+βωk holds, where β>0 is a small constant and β⟶0 when k⟶∞.Definedk=ωk−ξ˜k; it satisfies dk≤1+βϖ, which belongs to the bounded and convex subset W. Hence, system (30) can be written in the following form:(34)xk+1=Adxk+Bduk,fbc+Bωddk.The investigated system (30) is subject to hard constraints on both state and input vectors with the following form:(35)x∈X,u∈U,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 JNx,u to be minimized at the current time k is(36)JNxk,uk=∑i=0N−1xi|kTQxi|k+ui|kTℛui|k+xN|kTPxN|k.In (36), N∈ℝ+ is the MPC prediction horizon, Q∈ℝn×n, Q>0, ℛ∈ℝm×m, R>0 and P is the solution of the algebraic Riccati equation [57].(37)Ad+BdKTPAd+BdK+Q+KTℛK=P.Due to the presence of the unknown disturbancedk, we rewrite the state vector xi|k of the system as the sum of a nominal part zi|k and an error part ei|k in the following form:(38)xi|k=zi|k+ei|k,where ei|k denotes the deviation of the real state ei|k with respect to the nominal one.Design the following feedback policy (39) for system (30):(39)ui|k=vi|k+Kxi|k−zi|k,where vi|k denotes the nominal input vector and the gain matrix K should be selected such that AK=Ad+BdK is Schur-stable; then the corresponding nominal and error dynamics can be described, respectively, as follows:(40)zi+1|k=Adzi|k+Bdvi|k,z0|k=x0|k,(41)ei+1|k=AKei|k+Bωddi|k,e0|k=0.Hence, the finite horizon optimal quadratic cost (36) can be redefined in terms of nominal state zk and control input vk as(42)JNzk,vk=∑i=0N−1zi|kTQzi|k+vi|kTℛvi|k+zN|kTPzN|k,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,ℤf, the nominal robust MPC finite horizon optimization problem can be described as(43)minvJNzk,vks.t.zi+1|k=Adzi|k+Bdvi|k,z0|k=xkzi|k∈ℤ,i∈1,Nvi|k∈V,i∈0,N−1zi|k∈ℤf.
The solution of (43) is the optimal nominal control sequence v0|k∗=v0|k∗0;zk,…,vT−1|k∗T−1;zk and the first control action, that is, κ˜Nzk:=v0|k∗0;zk, represents the optimal control vi|k to be applied to system (40).
The proposed control law applied on the uncertain system (40), according to the control policy adopted, is(44)ui|k=vi|k+Kxi|k−zi|k=κNxk,zk=κ˜Nzk+Kxk−zk.
The composite closed-loop system then satisfies(45)xi+1|k=Adxi|k+BdκNi,xk,zk+Bωddi|k,(46)zi+1|k=Adzi|k+Bdκ˜Ni,zk.
For the TRMPC approach, the matrixK in the control policy (39) is designed to stabilize system (30). Consider the following closed-loop system:(47)xi+1|k=Ad+BdKxi|k+Bdvi|k+Bωddi|k.
Hence, the satisfaction of the following condition aims to define the feedback gainK that stabilizes the system.(48)Ad+BdKTP^Ad+BdK−P^<0,P^>0.
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 stepk. The derivations of the nominal state, input, and terminal constrains set ℤ,V,ℤf 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 defineSK∞:=AK0W⊕AK1W⊕⋯=∑j=0∞AKjW, where ⊕ is the Minkowski sum and AKjW:=AKjdk|dk∈W is the set multiplication; the uncertain set of the error ei|k is the minimal robust positive invariant set for(49)xi+1|k=Adxi|k+Bωddi|k,dk∈W.
Then the state and control input vector constraints in (35) are satisfied provided that(50)zi|k∈X⊝SK∞,vi|k∈U⊝KSK∞,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(51)zN∈ℤ⊆X−SK∞,ℤf⊆ℤ.
Moreover, these assertions only make sense if the disturbance setW is sufficiently small to satisfy the following Assumption 3, as defined in [57].Assumption 3.
(Restricted disturbances for constraints satisfaction)SK⊂X and K×SK⊂U.
The next step is to define a robust positively invariant setSK for [19] to obtain the tighter constraints acting on the nominal system. Then 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 ofℤ and V(1)
Define the linear state constraint as:X=xi|k∈ℝn|axi|k<b;(2)
Construct the nominal state constraint inequalityazi|k≤b−maxaei|k|ei|k∈SK∞=b−Φ∞;(3)
Approximate the upper value ofΦ∞ as ΦN=maxa∑i=0N−1AKidi|k|di|k∈W;(4)
Choose a suitableα∈0,1 and N such that AKNdi|k∈αW, compute Φ∞≤1−α−1ΦN;(5)
Compute the nominal state constraint setℤ:=zi|k∈ℝn|azi|k≤b−1−α−1ΦN;(6)
Compute the nominal control constraint setV:=vi|k∈ℝm|a′vi|k≤b′−K1−α−1ΦN, where U:=ui|k∈ℝm|a′ui|k≤b′.
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 timek=0, set xk=zk=x0 where x0 denotes the current state.(2)
At timek, considering the current state (xk,zk), based on the disturbance estimation ξ^k from the observer (8), solve the nominal optimal control problem (43) to obtain the nominal control vector vk:=v0|k∗0;zk and the control input vector uk=vk+Kxk−zk−ξ^k.(3)
If the nominal optimal control problem (43) is infeasible, adopt safety/recovery procedure.(4)
Apply the controluk to the system (45) and (46);(5)
Calculate the estimationx^k+1 from the observer (8) as successor state xk+1 of the system (30), and calculate the successor state zk+1 of the nominal system (40).(6)
Set (xk,zk) = (xk+1,zk+1), set k=k+1 , and go to (2).


## 4. 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. The parameters in (1) are given in Table 1 [54].Table 1
Parameters of satellite used in the simulation.
SymbolsParametersm0145 kgm11 kgm21 kgω00.0011569 rad/sρ˜max0.1 mρ˜˙max0.1 m/sKt10.0390.0390.03910.0390.0390.0391×10−6 N/mDt1.4×10−11I3 N/(m/s)For the relative position motion between the TM1 and the satellite, defineFd,10=Fdrag+Fthruster, where(52)Fdrag=m0×2.15×10−7sin0.00114t0.15×10−4sin0.00114t−2.35×10−40.20×10−5sin0.00114t−2.05×10−5N.The resolution and the maximum value of the thruster are10−6 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.For the relative position motion between TM1 and TM2, defineFd,12=Felec+Factuator, where(53)Felec=m1×1×10−6sin0.04t+π31×10−6sin0.05t1×10−6sin0.05t+π2N.The resolution and the maximum value of the thruster are10−7 N and 60×10−6 N. Similarly, a zero mean white noise with mean squared error being 10−8 N is added as the stochastic disturbance.The simulation results are shown in Figures4–10 as follows. Define the estimated error ξ˜it=ξ^it−ξit (i = 1, 2), as shown in Figures 4 and 5, and ξit can be estimated precisely by the ESMO. From Figures 6–11, the relative motion variables ρ˜10t, ρ˜˙10t, ρ˜12t, and ρ˜˙12t could be estimated well by the ESMO. In addition, the stable control accuracy of the composite control approach has achieved 10−7 or 10−8, which shows that the developed extended sliding mode observer method and tube-based model predictive control law are effective.Figure 4
Estimation of the disturbancesξ1t by the ESMO.
(a)(b)(c)(d)Figure 5
Estimation of the disturbancesξ2t by the ESMO.
(a)(b)(c)(d)Figure 6
Simulated relative positionsρ˜10,xt,ρ˜˙10,xt and their estimations between TM1 and the satellite.
(a)(b)Figure 7
Simulated relative positionsρ˜10,yt,ρ˜˙10,yt and their estimations between TM1 and the satellite.
(a)(b)Figure 8
Simulated relative positionsρ˜10,zt,ρ˜˙10,zt and their estimations between TM1 and the satellite.
(a)(b)Figure 9
Simulated relative positionsρ˜12,xt,ρ˜˙12,xt and their estimations between TM1 and TM2.
(a)(b)Figure 10
Simulated relative positionsρ˜12,yt,ρ˜˙12,yt and their estimations between TM1 and TM2.
(a)(b)Figure 11
Simulated relative positionsρ˜12,zt,ρ˜˙12,zt and their estimations between TM1 and TM2.
(a)(b)


## 5. Conclusions

This 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. The 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.



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