Attitude Synchronization of Heterogenous Flexible Spacecrafts by Measurement-Based Feedback With Disturbance Suppression

In this paper, we investigate multiple flexible spacecrafts attitude synchronization with parametric uncertainties and external disturbances. A large number of results on this problem rely on accurate measurements, ignoring measurement noises in attitude and angular velocity feedback. However, the existence of inherent noises and accumulated measurement errors are invertible, such that the performance of synchronization may be damaged. By developing a decentralized measurement-based attitude controller, we achieve attitude synchronization having input-to-state stability (ISS) with respect to measurement noises as external input. Moreover, thanks to the ISS-based robust design procedure, each of flexible spacecraft attitude control subsystem gains modeled external disturbance rejection, as well as unmodeled actuator disturbances attenuation and robustness with regarding to uncertain parameters.


I. INTRODUCTION
Attitude synchronization, as one of the most important and fundamental problems in the field of multiple spacecrafts control, has been studied intensively nowadays, see [1]- [6] and references thereof for background materials.
Such a critical problem implemented in the past literatures relies on either accurate sensor data [7], [8] or specific structure, such as well-designed observers [9], [10] and filters [11]. On the one hand, when sensor data are used in feedback, measurement errors caused by different kinds of sensors, are often ignored due to the fact that the sensors can be chosen with high precision. Whereas in practice, high precision sensors are often much expensive. Besides, no matter what sensor is chosen, the sensor data used for control law calculation is always biased such that the control performance may be damaged, even destroyed [12]. On the other hand, The associate editor coordinating the review of this manuscript and approving it for publication was Haibin Sun . some specific filters are designed to make the sensor data more reliable, see [13]- [15] for examples. However, there are lack of studies on robust attitude synchronization with sensor error quantified in the performance analysis.
Moreover, uncertain parameters handling and external disturbances suppression are most attractive problems in attitude synchronization [16]- [19]. There are a large number of results have been presented in the past two decades. On the one hand, uncertainty due to the uncertain systematic parameters, is an inescapable obstacle to compensate the nontrivial time-varying uncertain torques due to the nonlinearity of attitude dynamics. In most existing work, adaptive control is used to tackle the uncertain parameters, see [20]- [23]. On the other hand, disturbance suppression is studied as disturbance rejection or disturbance attenuation problems in different aspects. In literature, disturbance rejection filters and disturbance observer based controllers, are proposed to compensate the external disturbance torques by estimation or reconstruction of the disturbances, see [24], [25]. Another useful tool to achieve disturbance rejection is the so-called internal model approach, see [26]. In [20], [27], modeled disturbances are rejected by the internal model based controller design for a rigid spacecraft, while the results are extended to flexible spacecraft attitude regulation in [28]- [30].
However, these results limit real implementations of robust attitude synchronization due to the fact that they fail to obtain an external stability. In engineering scenarios, there exists unmodeled disturbances, such as external winds, actuator noises, see [31], such that the external stability is crucial to the stability of the controlled heterogenous flexible spacecraft attitude closed-loop system. This problem is investigated as a disturbance attenuation problem, see [32]- [34]. These results are based on classical robust control which is lack of precision and fails to modeled disturbance rejection.
In this study, we focus on robust multiple attitude synchronization for uncertain flexible spacecrafts via measurement-based feedback. Based on Sontag's input-tostate stability(ISS) [35], [36], which characterizes the property from input to state with giving a quantitative relationship, we design a measurement-based input-to-state stabilizing controller from measurement error to state, achieving not only robust attitude synchronization, but also disturbance suppression.

A. CONTRIBUTION
In the present study, our main contribution is three-fold.
• First, a decentralized dynamic output feedback controller is proposed for a specific modified Rodriguez parameter-based decentralized attitude error system with unknown parameters. In the past literature, uncertain parameters in heterogenous spacecrafts attitude synchronization is mostly tackled by adaption as aforementioned. That hinders attitude synchronization with input-to-state stability characterization [37]. The controller proposed in this study gives rise to robustness property for the resultant closed-loop system.
• Second, with respect to measurement error as external input, input-to-state stability of the decentralized attitude error closed-loop system is characterized. Thus, the effect of the sensor error is quantized. This consequence is quite novel such that the each flexible spacecraft has the quantified measurement noise tolerance capability.
• Last but not least, based on the internal model based regulation theory, we achieve external disturbances suppression in the sense of actuator-side. Specifically speaking, sinusoidal disturbances, also called modeled disturbances, are rejected by internal model based controller design manner, while unmodeled disturbances are attenuated by ISS-based design procedure.

B. ORGANIZATION
Section II gives the mathematical model of heterogenous flexible spacecrafts with some normative assumptions. Section III states the preliminaries and formulates the problem under investigation. Section IV presents the main results of this study. Section V shows the numerical simulation results for illustration. Section VI closes the paper.

C. NOTATION
· is the Euclidean norm of a vector or the induced Euclidean norm of a matrix. I n is the n-dimensional identity matrix.
is of class K and for each fixed s > 0, β(s, t) decreases to zero as t → ∞. L n ∞ denotes the set of all measurable essentially bounded functions from R ≥0 to R n . For a function u(·) ∈ L n ∞ , u(·) ∞ = sup t≥0 u(t) . γ 1 • γ 2 (·) denotes the composite function γ 1 (γ 2 (·)) of compatible dimensions functions γ 1 , γ 2 . For any column vectors Manipulations × and · are standard cross-product and dot-product respectively, i.e., for O = V\{0} = {1, 2, · · · , N } is the flexible spacecraft index set. A B represents the correspondence relationship from A to B.

II. MATHEMATICAL MODEL
In this section, we first present the attitude dynamics mathematical model of flexible spacecrafts. Then, some universal assumptions are given for formulating the problem this paper mainly concerned in the next section.
Consider the standing attitude dynamics model of a group of N heterogeneous flexible spacecrafts in terms of modified Rodriguez parameters (see [38,Equation (338)] and [39,Equations (2)and(3)]), described bẏ where, (2a), (2b) and (2c) are kinematics subsystem, dynamics subsystem and flexible appendages vibration equation, respectively. Furthermore, for each i ∈ O, • σ i ∈ R 3 is the modified Rodriguez vector defined as VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
where q v ∈ R 3 and q 4 ∈ R are vector and scalar part of a quaternion respectively, see [38, pp. 475], and each component σ ij ∈ R, j = 1, 2, 3 is called a modified Rodriguez parameter.
is the angular velocity and the skew symmetric matrix of ω i is which is an asymmetric and non-singular matrix satisfying, • η i ∈ R n η i is the modal coordinate vector of the i-th flexible spacecraft appendages while n η i is the flexible modes of η i .
• d i ∈ R 3 is the time-varying external disturbance torque effects on the i-th flexible spacecraft.
• J i ∈ R 3×3 is the inertia matrix of the whole structure of the i-th flexible spacecraft, that to be constant, symmetric, positive definite and bounded.
• δ i ∈ R n η i ×3 is the coupling matrix between the elastic and the rigid structure of the i-th flexible spacecraft.
• C i ∈ R n η i ×n η i and K i ∈ R n η i ×n η i are symmetric and positive definite matrices, determined by with the natural frequency matrix i ∈ R n η i ×n η i and the corresponding damping ratio matrix ς i ∈ R n η i ×n η i .
• u i ∈ R 3 is the control torque of the i-th flexible spacecraft. In the present study, to achieve attitude synchronization of heterogenous flexible spacecrafts, besides the spacecrafts of (2), we set a virtual leader specified by a constant trajectory σ 0 ∈ R 3 with angular velocity free, i.e., When attitude of all spacecrafts track virtual leader (6) simultaneously, attitude synchronization of (2) is achieved. Relevant to the above leader-following system composed of (2) and (6), we specify the following assumptions.
Assumption 1: For each i ∈ O, J i , δ i , C i , K i are uncertain but bounded matrices, satisfying with some known Assumption 2: For the disturbance d i in (2b) with each i ∈ O, it assumed to be a combination of a constant and multiple sinusoids, i.e., with n p ∈ Z + and D ij0 , A ijp , ijp , ψ ijp ∈ R. Assumption 3: The attitude and angular velocity of each flexible spacecraft can be measured by its sensors with unknown bounded measurement noises, i.e., each spacecraft can get measurements of σ and ω as where ω i and σ i are the angular velocity measurement by sensors equipped on itself, and ( iσ , iω ) are piecewise continuous bounded measurement noises. Remark 4: All above assumptions are reasonable. Specifically speaking, Assumption 1 and Assumption 3 are the facts due to limitations of the physical system with finite structure and unavoidable measurement noises. Assumption 2 is reasonable due to the disturbance modelling technique, see [27] for an example. Moreover, note that, for the disturbance (8), there exists an autonomous systeṁ where v i (t) ∈ R n v i is an exogenous signal, generating the external disturbance. All the eigenvalues of S i ∈ R n v i ×n v i are distinct lying on the imaginary axis referring to the frequencies of the external disturbance, (S i , L i ) is observable, and the initial condition v i (0) starts from a specified compact set V i ∈ R n v i .

III. PRELIMINARIES & PROBLEM FORMULATION
In this section, we show the outstanding concept of inputto-state stability in the sense of Sontag ([36]) and standard measurement-based feedback control problem formulation ( [40]). Based on those priori knowledge, we state the complete description of heterogenous flexible spacecrafts attitude synchronization problem.

A. INPUT-TO-STATE STABILITY NOTION
Consider a general nonlinear system of the following forṁ with state x and input u, and the vector field f is Lipschitz continuous in its arguments. Suppose system (11) is forward complete for each initial condition x(0) and each control input The following definition is the so-called inputto-state stability, used as an effective tool in the analysis and design of nonlinear control systems, see [36] for more details.
Definition 5 ([36, pp. 8]): The system (11) is said to be ISS with state x and input u if there exist functions β ∈ KL and γ ∈ K such that each solution x(t) of (11) satisfies Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
The above specific function γ is referred to as a gain function of (11).
Note that there is an equivalent way to characterize the ISS property of (11) as follows.

Remark 7:
As pointed out in [36, Theorem 5], a system is ISS if and only if it admits a smooth ISS-Lyapunov function.

B. MEASUREMENT-BASED FEEDBACK CONTROL PROBLEM
Here we consider system (11) having the following specific formẋ where x i ∈ R n i , i = 1, 2 is the state but unavailable for design of the control input u ∈ R m . ∈ L n 2 ∞ is the measurement disturbance and χ ∈ R n 2 is the measured output used for controller design. The vector fields F i , G, W , i = 1, 2 are not completely known but are sufficiently smooth. For the uncertain system (13), we list the following technical assumptions.
Assumption 8: The systemẋ 2 = F 2 (x 2 ) + G(x 2 )u is globally asymptotically stabilizable, i.e., there exist a Lipschitz continuous control law φ(x 2 ) and a Lyapunov function V (x 2 ) such that, for all x 2 ∈ R n 2 , where α x , α x , α x ∈ K ∞ . Assumption 9: Let G := (G 1 , · · · , G m ). There exist a constant d v ≥ 0 and a K ∞ -function v such that, for each 1 ≤ i ≤ m, the signs of L G i V (x 2 ) and L G i V (x 2 + τ ) are the same as long as Assumption 10: The x 1 -subsystem with W (x 1 , x 2 , u) as output and (x 2 , u) as input satisfies following property Then we state the following lemma which is crucial to the solvability verification of the forthcoming problem in this study. Lemma 11 ( [40,Theorem 5.1]): If the Assumptions 8-10 are satisfied, then there exists a measurement-based feedback controller of the following form such that the closed-loop system composed of (13) and (16) is ISS with state (x 1 , x 2 ) and input µ.

C. PROBLEM STATEMENT
At this moment, we concentrate our attention on attitude synchronization problem of the leader-following flexible spacecrafts composed of the virtual leader (6) and the followers (2). We define an attitude error vector to describe the deviation between the current attitude σ i of each spacecraft and the leader attitude σ d of (6) by satisfying, by (1), (6) and [41,Equation (27)], Then, we further define the following coordinate as transformed angular velocity, in which k 1 is a positive constant to be designed later. Moreover, followed by [28,Equation (23)], we define another coordinate leading to a cascade auxiliary systeṁ which meets the structure of (2c) with x i2 = η i . Associated with equations (17), (19), (21) and (2), we establish a decentralized attitude error system based on the new coordinate (x i , e i , z i ) written bẏ where the matrix and vector fields are specified by Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
From (9), we know that the measurement of z i is the case thatẑ where the unknown constructive noise iz ∈ R 3 is collected by By Assumption 3, iz is also unknown but piecewise continuous bounded, and here it is called translated gyro noise.
Here, we are ready to formulate the main problem of multiple attitude error stabilization of system (22) by a measurementbased decentralized output feedback controller of the follow- Problem 12: For the decentralized attitude error system (22), the problem of heterogeneous attitude synchronization undertaken in this paper is to seek a measurement-based output feedback controller in the form of such that for each i ∈ O, the following condition hold. 1) The system trajectory of (e i , z i , x i ) always exists and bounded.
2) The external harmonic disturbance d i of (8) can be rejected by designed controller eventually, i.e., 3) When measurement noises are ignored, which means that, 4) When measurement noises can not be ignored, i.e., Remark 13: From the formulation of Problem 12, we make an attempt to design a measurement-based feedback controller to gain robust attitude consensus of heterogenous flexible spacecrafts. As a sharp contrast to past literature in multiple spacecrafts attitude control, we achieve not only robust attitude synchronization by measurement feedback but also modeled disturbance rejection property.

A. SOLVABILITY OF PROBLEM 1
To solve Problem 12, it is necessary for us to show the solvability of stabilizing (22) by a decentralized measurementbased feedback control law. In other words, we shall confirm the probability of finding such a controller (26).
Note that, for each i ∈ O, system (22) accommodates the specific form of (13) with Therefore, we shall to verify Assumptions 8-10 for system (22). After confirmation of those assumptions, the solvability of Problem 12 can be guaranteed according to Lemma 11. In what follows, we list the details of verification of each assumption respectively.
(i) In line with Assumption 8, for the examined systeṁ Hence, Assumption 8 is satisfied.
are the same. That verifies Assumption 9.
(iii) Consider system composed of (36a) and (36b) with vector fields given by (23a) and (23b). In line with Assumption 10, we need to show that the composed system has the IOS property (see [36, pp. 30] for explanation) characterized by (15) with h i (x i , e i , z i ) as output and z i as input.
We first claim that the composed system is ISS with state x i , e i and input z i as the following proposition states.
Proposition 14: Consider system composed of (36a) and (36b). We state that, for each i ∈ O, there exists a number k 1 > 0 of (19) and a positive definite matrix P i , such that is an ISS-Lyapunov function for that system. Moreover, it satisfies, along the trajectories of the composed system, we havė for functions γ 1i ∈ K ∞ , and in particular, where ρ i is a positive constant.
Proof: Now we rewrite the composed system aṡ Then, we construct a candidate ISS-Lyapunov function of the form (33) with a positive definite matrix P i ∈ R 2n η i ×2n η i satisfying the following Lyapunov equation Therefore, by (23a), there exists K ∞ functions Calculating the time derivative of V i1 (x i , e i ) of (33), along the trajectories of (36), we havė In (39), using (4) giveṡ For (40), by using the Young's inequality: for nonnegative real numbers a and b and positive numbers p and q such that 1/p + 1/q = 1, ab ≤ a p p + b q q , we have, for each i ∈ O and any ε i > 0, Moreover, from (7), (23b) and (37), we obtain for some constants c i1 and c i2 . Substituting the above in (40) givesV Hence, we can choose k 1 > 0 and ε i > 0 such that (34) and (35). The proof of Proposition 14 is complete. Remark 15: Proposition 14 propose an ISS characterization of the inverse dynamics of the decentralized attitude error system (22). By Remark 7, there exist functions β i(x,e) ∈ KL and γ iz ∈ K such that for each input z i ∈ L 3 ∞ and each initial condition (x i (0), e i (0)), the solution exists for each t > 0 and satisfies Then, recalling the function h i (x i , e i , z i ) of (23d) which is considered as the output of the composed system, we state the following lemma.
Lemma 16: where α ih (s) = s + s 2 + s 3 is a K ∞ function. Proof: From (23d), we have the following inequality VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
Note that, for any vector s := [s 1 , s 2 , s 3 ] T ∈ R 3 , we have Then, it can be calculated that the biggest eigenvalue of S is s 2 1 + s 2 2 + s 2 3 , leading to the following fact s × = s Together with (4) gives Now, by considering each part of (44) separately, we obtain Substituting the above in (44) gives where parameters are specified by leading to (43). The proof is complete. Associated with (42), (43) can be rewritten by It can be straightforwardly known that β ih ∈ KL, γ ih ∈ K. Thus, Assumption 10 is confirmed.
So far, we have noted that Assumption 8 to 10 are verified for the decentralized attitude error system (22). Hence, the solvability of Problem 12 can be straightforwardly obtained by Lemma 11, i.e., there exists a measurement-based feedback controller of the form (26), such that, when the disturbance is rejected, the closed-loop system composed of (22) and (16) is ISS with state (x i , e i , z i ) and input iz . By the concept of ISS aforementioned, conditions 1 to 4 in Problem 12 is satisfied.

B. PROBLEM CONVERSION
Let us focus on the external harmonic disturbances of (8). Harmonic disturbances in the form of (8) can be rejected by internal model based regulation theory, which is a famous framework to solve controller synthesis problems for nonlinear systems with uncertainties. Thus, the well controlled decentralized attitude error system can be achieved asymptotically stable by accurate feedback, see [26], [42] for more background materials.
Followed by design procedure in nonlinear output regulation framework shown in [26, Chapter 6], we first solve the relevant regulator equations to obtain the zero-error constrain input (see [26, pp. 83]). Consider the attitude error system (22), we have the following fact Together with (10), it gives the zero-error constrain input as Hence, the zero-error constraint (53) leads to a zeroing polynomial with some positive constantn i = 3 j=1n ij > 0 as order, dependent on the index i. Then by the internal model construction method ( [26, Chapter 6]), a canonical internal model is designed aṡ for any controllable pair Here, for each i ∈ O, there is a smooth function θ i (v), which is called steady-state generator ( [26, Chapter 6]), and a matrix i such that Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.

Now let the new coordinate and input bē
Associated with (22), it leads to the following translated attitude error systeṁ where the vector fields are specified bȳ From (56), it is obviously seen that the disturbance effects on attitude error system (22) is rejected by internal model based output regulation theory. Thus the problem we concerned can be transformed into the following form.
Problem 17: For the translated attitude error system (56), the problem of robust attitude maneuver design with unknown mesurement noise undertaken in this paper is to seek a measurement-based output feedback controller of the form where d io is the actuator noise, such that the trajectories of the closed-loop system comprising of (56) and (58) always exists and bounded; moreover, when ( iσ , iω , d io ) = (0, 0, 0), the trajectories converges to (x i , e i , z i ) = (0, 0, 0). Remark 18: As described in Problem 17, we make an attempt to design a measurement-based feedback controller for solving the problem of attitude error synchronization with uncertainties and measurement noises. As a sharp contrast to past literature in attitude control as aforementioned, the problem we concerned refers to not only robust attitude consensus design with uncertain parameters but also attenuate the effect of unknown gyro noise. By the definition of ISS stated in [36], the controller of (58) is well designed implies that asymptotic attitude consensus of a group of uncertain spacecrafts is achieved when actuator and measurement noises can be neglected, i.e., when ( iσ , iω , d io ) = (0, 0, 0), the trajectories of closed-loop system converges to (x i , e i , z i ) = (0, 0, 0).

C. INPUT-TO-STATE STABILIZATION BY MEASUREMENT-BASED FEEDBACK DESIGN
From (9), (24) and (25), we know that the controller to be designed of (58) can only utilize the sensor data ofẑ. Here, we state that the controller can be designed by the following lemma.
Lemma 19: Consider the subsystem (56d) of translated attitude error system (56) with measurement (24). There is an output feedback controller of the form such that the closed-loop system has an ISS-Lyapunov function with state z i and input (x i , e i ,ξ i , iσ , iω , d io ), i.e., the following two conditions hold.
• The ISS-Lyapunov function V i2 (z i ) satisfies, along the trajectories of (60), Proof: Firstly, there are two K ∞ functions defined in (32) such that (62) is verified. Next, we rewrite the closedloop z-subsystem composed of (60) and (24) by Then, the time derivative of the candidate smooth ISS-Lyapunov function V i2 (z i ) of (61) satisfies, along the trajectories of (64), Followed by [43] and [26, lemma 7.8], there are constants b i1 , b i2 , b i3 , b i4 > 0 such that, using (52) Moreover, we specify the designed function κ(·) with a constant k 2 > 1, as 84460 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.

In (65), it holds that
In above, by the Young's inequality, we obtain, for any ε > 0, Summarized from (67) to (68), we have Finally, substituting (66) and (69) in (65) giveṡ For the ctranslated gyro noise iz , using (25), we note that Substituting above to (70), we havė In (71), we can choose positive constants k 2 and ε such that which confirms (63) with K ∞ functions specified by The proof is complete. Then, Problem 17 can be solved by the decentralized measurement-based attitude controller (59) is shown rigorously by the following theorem.
Remark 22: According to Theorem 21, the problem of heterogeneous uncertain flexible spacecraft attitude consensus with modeled disturbance by measurement-based feedback control is well solved. Compared with past literature, the result in this manuscript is a synthesis of decentralized attitude stabilization and disturbance rejection and attenuation. Moreover, Theorem 21 gives a general framework of multiple flexible spacecraft attitude consensus by measurementbased feedback control achieving sensor and actuator noises tolerance.

V. AN ILLUSTRATIVE EXAMPLES
In this section, we give two examples to show the effectiveness of the proposed algorithm. from different perspective Example 23: Consider the attitude consensus problem for four homogenous flexible spacecrafts described by (2) whose inertia moments are specified by (18,12,10) kg · m 2 The the coupling matrix is set as The controller in Theorem 21 is specified by, for i = 1, 2, 3, 4, j = 1, 2, 3, The responses of the attitude and angular velocity are shown from Figures 1 to Figures 6, while the flexible mode is shown in Figure 7. The following consequences can be obtained.
• By comparing attitude control performance of (σ 1 , ω 1 ) and (σ 2 , ω 2 ), it can be seen that measurement noises can be attenuated by ISS-based controller procedure we proposed.
• By comparing attitude control performance of (σ 1 , ω 1 ) and (σ 3 , ω 3 ), it can be seen that harmonic disturbances constructed by sinusoidal signals can be rejected by internal model based attitude controller we proposed.
• Focus on attitude control performance from (σ 1 , ω 1 ) to (σ 4 , ω 4 ), we have known that the attitude controller we proposed achieves disturbances suppression, including external modeled disturbances rejection, sensor and actuator noises attenuation. Example 24: (2) Furthermore, we consider a general attitude consensus scenario. Four flexible spacecraft numbered by i = 1, 2, 3, 4. In this scenario four spacecraft are different structure such that they are a group of heterogenous flexible spacecrafts. Then, different modeled and unmodeled disturbances effect on each spacecraft.
Consider inertia moments of each spacecraft are specified by (10,10,8) kg · m 2 J 2 = diag(5, 1, 2) kg · m 2   The the coupling matrix is set as The reference (6)     The controller in Theorem 21 is specified by, for i = 1, 2, 3, 4, j = 1, 2, 3,     14 show the decentralized attitude controller we proposed has the capacity of heterogenous attitude synchronization with disturbance suppression.

VI. CONCLUSION
In this paper, we develop measurement-based attitude control strategy such that measurement-based robust attitude regulation for uncertain heterogenous rigid spacecrafts with external disturbances is achieved. For a translated attitude control system, we achieve input-to-state stabilization in presence of measurement error by an output feedback controller. The approach implements an internal model based attitude controller and accomplish attitude consensus with disturbance suppression design.