Fault-tolerant Sliding Mode Controller and Active Vibration Control Design for Attitude Stabilization of a Flexible Spacecraft in the Presence of Bounded Disturbances

This paper concerns vibration control and attitude stabilization of a flexible spacecraft with faulty actuators. The PID-based sliding mode fault-tolerant scheme is developed to preserve the system against bounded external disturbances, rigid-flexible body interactions, and partial actuator failures. The proposed control law, which combines the advantages of the PID and SMC, is proposed to enhance the robustness and reduce the steady state errors while reducing complexity and the computational burden and preserving the great properties of the SMC controller. It has been shown that the SMC controller is effective in accommodating different actuator fault scenarios and behaves healthily. Additionally, an active vibration control (AVC) law utilizing a strain rate feedback (SRF) algorithm and piezoelectric (PZT) sensors/actuators is activated during the maneuver to compensate for residual vibrations resulting from attitude dynamics and actuator failures. Numerical simulations demonstrate the proposed schemes' superiority in fault tolerance and robustness compared to conventional approaches .


1.Introduction
In recent years, with the improvement of space technologies, the attitude control system has a higher role in a modern space mission, especially combined with actuator faults and external disturbances. The control system for spacecraft, regardless of their missions and attitude maneuvers, needs to be accurate and faulttolerant [1][2][3].
There are passive and active approaches to fault tolerant control (FTC). Implementing an online fault detection scenario for active fault tolerant control algorithms is necessary compared to passive ones [4,5]. Moreover, with a single fixed controller, a passive FTC algorithm can simultaneously handle many possible actuator faults [6]. So in this approach, in order to reduce the computational burden, the control algorithms are modified according to the detected fault signals.
On the other hand, many large and flexible appendages in aerospace systems can easily reduce spacecraft performance and their attitude control systems. Faults caused by the actuators, which can occur at unknown moments with unknown values and patterns, can excite high-frequency modes of flexible appendages and lead to system instability [7][8][9][10].
Many control approaches have been developed over the past few decades to minimize the effects of disturbances and actuator faults. The following are some effective FTC approaches: model predictive control [11], neural network-based control [12], adaptive fuzzy logicbased control [13], back-stepping control [14], and sliding mode control (SMC) [15].
Several studies have investigated the robustness and ease of implementation of a classical sliding mode faulttolerant controller [15][16][17]. Compared with other control approaches, the sliding mode is able to resolve disturbances and/or different systems uncertainties [18]. Moreover, the conventional SMC's weakness is the chattering phenomenon associated with its high switching frequency, singularity, and inability to meet the finite time convergence. Despite the SMC's remarkable properties, it should be improved to stabilize systems quickly when faced with fault effects. In order to address these concerns, several types of SMCs are proposed, such as terminal SMC (TSMC), Nonsingular TSMC, boundary layer method, high-order SMC, etc. The complexity of these algorithms, however, will significantly affect the system's complexity, real-time implementation, and the required processing burden. In order to simultaneously increase the robustness and performance of the SMC, PID-based SMCs have been developed. Such an approach aims to improve the robustness of SMC by incorporating the advantages of PID controllers into its design procedure. A PID controller using a sliding mode control approach offers significantly better performance than a classical PID controller in view of the fact that the PID SMC does not rely on uncertainty [19]. Moreover, flexibility has not been considered in most of the previous studies dealing with attitude SMC fault tolerant control of systems with nonlinear fully coupled rigid-flexible bodies dynamic.
We propose a more reliable mechanism that utilizes PID-based SMC in order to overcome this weakness. Hu proposed a sliding mode fault-tolerant controller for the stability of flexible spacecraft using a redundant actuator [16].
The problem of residual vibration control has received great attention from spacecraft designers. Using PZT materials as actuators or sensors is an effective method for actively suppressing vibrations. The main characteristics of PZT materials are their high-frequency response, low power consumption, lightweight, and high stiffness. Numerous studies have been conducted on actively using PZT material to control the vibrations of flexible spacecraft structures. A flexible spacecraft's attitude maneuvers can be suppressed using PZT patches and the component synthesis vibration suppression approach [20]. Song developed positive position feedback (PPF) control in order to dampen the vibration of the flexible structure of spacecraft [21]. In a single-axis maneuver, the SRF and SMC techniques are utilized to suppress vibrations [22].
Among the presented methods, the SRF law has a wider active damping region and the ability to stabilize more than one vibration mode. In addition, it is easy to implement. In this method, the structural velocity coordinate (strain rate) is multiplied by a negative gain and feedback to the structure. This paper is focused on a passive fault tolerant PIDbased SMC approach to stabilize the system with fully coupled rigid-flexible body dynamics interactions equipped with three faulty momentum generation actuators. The SRF control approach is also contributed to suppressing the residual vibrations during and after the attitude maneuver. A comparative study with healthy systems validates the proposed model and algorithm, and the overall system stability is proven by applying the Lyapunov theorem. Following is a summary of the main contributions: The proposed method has demonstrated superior performance and robustness compared to conventional methods [23][24][25]. By utilizing AVC to compensate for extra vibrations, the proposed method is superior to FTC approaches without AVC [26,27]. Following is the rest of the paper's organization. The dynamics of the flexible spacecraft and actuator faults are described in Section 2. SMC-based PID fault tolerant control and SRF-based vibration suppression for flexible panels are discussed in Section 3. Section 4 presents the simulation results of the FTC system with and without AVC in the presence of external disturbances, and the last section concludes the paper.

Mathematical modelling of flexible spacecraft and actuator faults
The flexible spacecraft dynamic modelling is considered a rigid-flexible body dynamic, consisting of a main rigid hub and two PZT mounted flexible appendages attached symmetrically. Elastic deformation of panels during multi-axis attitude maneuvers has been modelled using the Euler-Bernoulli beam theory. A fixed reference frame is used to separate the attitude motions of the spacecraft from its translational motions. Figure 1 shows the flexible spacecraft consisting of a rigid hub and two flexible panels equipped with piezoelectric sensor/actuator patches. The equations of the motion of the flexible spacecraft with PZT patches are given as [22]: and denote control torque, external disturbances on the hub, external disturbances on flexible panels, the k th modal components for appendages, PZT sensor/actuator gain amplifier, respectively. The M, C, and K are the mass, damping, and stiffness matrices, A, N, and P are the parameters related to PZT patches that can be decomposed in sensor and actuator parts corresponding to the sensor/actuator voltages A and A , subscripts R, F, and RF represent the rigid, flexible and rigid-flexible dynamics, and superscripts a and s denote as PZT sensor and actuator, respectively.
The unit quaternions (which are also called Euler's symmetric parameters), as state parameters, describes the body frame's attitude orientation of flexible spacecraft q = [ q : ] ∈ R × , are defined as: (2 ) where 0 ≤ Φ( ) ≤ 2 refers to a rigid body's rotation around the Euler axis e( ). The quaternions are related to the vector of angular velocities ω = [ ] by the following relation: The dynamic modelling of actuator faults is as follows: It should be noted that if f ( ) is equal to zero, it means that the actuator works normally. If f ( ) is between zero and one, the actuator loses its effectiveness partially and has not yet completely failed. Therefore, by rewriting the dynamics according to Eq. (4), we have:

Controller design
This section presents two control approaches; the faulttolerant PID-based SMC and the SRF for AVC. Initially, we discussed a simple SMC with actuator faults. Next, the fault-tolerant property is added to the sliding mode structure, followed by the AVC algorithm.

Assumption 4:
The displacement of flexible parts ‖ ‖ and their derivative ‖ ‖ are considered to be bounded [28].

Conventional sliding mode control
A sliding surface can be defined by taking angular velocity and quaternion vectors [29]: where k is a positive constant. Convergence of state variables requires satisfaction of the sliding surface equation (S = 0). The convergence property is given in the following lemma.
Lemma 1: If a proper sliding mode controller can be designed to persuade the sliding mode condition S S < 0, then desired maneuver can be realized. As a result, the spacecraft state parameters and will converge to zero.
Proof: By satisfying the sliding conditions from sliding mode theory, the system is lastly forced to be in sliding mode: S = ω + q : = 0 (7) A candidate Lyapunov function is as follows: Using Eq. (3), the time derivative of the Lyapunov function becomes: (9) = −2 = ω q : Substituting the sliding surface Eq. (7) in Eq. (9), it follows that: (10) = −q : q : It can be shown = 0 only if q : = 0. Subsequently, is a Lyapunov function such that the q → 0. Therefore, we can easily derive ω → 0 from Eq. (7) and can guarantee the system's stability by introducing the sliding vector Eq. (6). Then, this completes the proof of lemma 1.
The time derivative of the sliding surface along the attitude dynamics model Eq. (5) leads to: The sliding surface's time derivative should be zero to derive the equivalent control, which gives:

Fault-tolerant SMC
In order to compensate for actuator faults as well as external disturbances, the SMC law is designed in such a way that the sliding surface can always be reached. The proposed fault-tolerant SMC is considered to be [31]: (17) This indicates that even with external disturbances and partial loss of actuator effectiveness, the sliding motion can be maintained; this concludes the proof.

Active vibration control
In order to create high-precision maneuvers, in this section, an active vibration control algorithm has been designed using PZT patches. The output current of the piezoelectric sensor is converted to the voltage of the sensor using a signal regulator with a gain of and applied to the piezoelectric actuators with the proportional gain factor of the controller. The following equation can represent the output voltage of piezoelectric sensors [32]: where ψ ( ) is the element shape function, ( ) is the circuit current, , , ℎ and represent the PZT charge/voltage constant, width, length and thickness respectively. The input voltage to the actuator is given by: is the controller gain matrix. The relative control force applied to the patches is as follows: where and is the young's modulus and strain constant of PZT layer, respectively.

Numerical Simulation and Results
This section presents numerical simulations for flexible spacecraft systems to illustrate the performance of the proposed fault-tolerant sliding mode and AVC law and compare the dynamics without the control laws under the actuator faults and external disturbances. All the simulations have been carried out by using the Newmark-Beta numerical integration method on the MATLAB/Simulink software.
The parameters for the main body and panels of flexible spacecraft are considered to be: density  As can be seen from Figure 1, the control effort required for the PID-based SMC algorithm along with SRF for the case with and without FTC against the actuator fault scenarios. As can be seen, the fault-tolerant SMC can attain its desired objective. Also, the controller without FTC fails against the second fault scenario at = 30 .
As shown in Figs. 2 and 3, the fault-tolerant SMC law (17) achieved a good attitude stabilization performance despite partial loss of actuator effectiveness and external disturbances approximately in 40 seconds.
Control performance in the missions with pointing accuracy and system agility requirements is also greatly influenced by considering elastic vibrations during the control design procedure. Figure 4 illustrates the first three flexible modes for systems with and without the FTC. As can be seen, actuator failures lead to extra vibration on flexible parts, which can lead to the fracture of large flexible structures. However, the proposed PIDbased SMC considering Eq. (18), can attenuate the extra vibrations.  Using SRF simultaneously with attitude control has also significantly reduced the vibration caused by flexible body dynamics. Figures 5 and 6 illustrate the time history of control effort and vibration modes with and without AVC for the fault-tolerant SMC algorithm, respectively. As can be seen, AVC has led to smooth attitude control commands, resulting in more accurate maneuvers (the oscillations settle within 25 seconds). It is noteworthy that several factors contribute to the excitation of high-frequency modes in the flexible parts, including structural couplings, external disturbances, and actuator faults and failures. It has been shown that active vibration control algorithms significantly reduce control effort as well as residual vibrations, which can interact with one another.

Conclusion
This paper develops fault-tolerant PID-based sliding mode and SRF control schemes for the attitude stabilization problem of a flexible spacecraft in the presence of partial loss of actuator effectiveness and external disturbances simultaneously. Introducing PID into SMC increases robustness and transient response while preserving the great features of the SMC. Using SRF in conjunction with attitude control can help eliminate residual vibrations both during and after the maneuver and reduce vibration-induced effects on rigid body dynamics. Initially, a PID-based SMC law is designed for asymptotic attitude stabilization. Next, the proposed FTC scheme fully compensates for the actuator faults' effects from the maneuver's beginning. In the proposed control design approach, no system identification procedure was required to identify faults, nor was a fault detection and isolation procedure required. Furthermore, a lower bound is not required for the actuator's effectiveness. The Lyapunov criterion is used to ensure the stability of the entire hybrid system, and numerical simulations are given to confirm the proposed FTC in the presence of actuator fault scenarios and external disturbances.