Complexity

Volume 2019, Article ID 1645042, 16 pages

https://doi.org/10.1155/2019/1645042

## Adaptive Neural Network-Based Satellite Attitude Control by Using the Dynamic Inversion Technique and a VSCMG Pyramidal Cluster

University of Craiova, Faculty of Electrical Engineering, Craiova, Romania

Correspondence should be addressed to Romulus Lungu; moc.oohay@ugnul_sulumor

Received 7 June 2018; Revised 10 November 2018; Accepted 19 December 2018; Published 6 January 2019

Academic Editor: Avimanyu Sahoo

Copyright © 2019 Mihai Lungu and Romulus Lungu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The paper presents an adaptive system for the control of small satellites’ attitude by using a pyramidal cluster of four variable-speed control moment gyros as actuators. Starting from the dynamic model of the pyramidal cluster, an adaptive control law is designed by means of the dynamic inversion method and a feed-forward neural network-based nonlinear subsystem; the control law has a proportional-integrator component (for the control of the reduced-order linear subsystem) and an adaptive component (for the compensation of the approximation error associated with the function describing the dynamics of the nonlinear system). The software implementation and validation of the new control architecture are achieved by using the Matlab/Simulink environment.

#### 1. Introduction

Small satellites (mass < 500 kg) are becoming popular due to their low cost of development and shorter realization time; as a result, there has been a lot of effort to push satellite technology to smaller sizes/mass which would enable the small satellites to accomplish different missions as larger satellites; therefore, the modern control architectures’ design for small satellites is a continuing challenge. The two major components of a satellite’s attitude control system (ACS) are the* actuator* and the* control algorithm*.

Various types of actuators include the reaction wheels, magnetic rods, torque coils, thrusters, and Control Moment Gyroscopes (CMGs). A CMG acts as a torque amplifier, being suitable for three-axis slew maneuvering by providing the necessary torques via gambling a spinning flywheel [1]. Their main components are the* flywheel* (the spinning rotor) and the* gimbal* (the pivot about which the flywheel assembly is rotated); the produced torque’s magnitude is directly proportional to the inertia of the flywheel, the angular speed of the flywheel, and the rate of gimbal’s rotation. The three main advantages of CMGs are the large torque amplification, the momentum storage capacity, and the light weight; these advantages make the control moment gyros better for agility maneuvers than the other widely used actuators, the reaction and the momentum wheels. Depending on the gimbal axes, a CMG can be distinguished to a Single Gimbal CMG (SGCMG), Variable-Speed CMG (VSCMG), and Double Gimbal CMG (DGCMG). SGCMG is a CMG with a constant speed momentum wheel, gimbaled in one axis only; from the torque point of view, the most powerful CMGs are the VSCMGs since these can generate significant control couples (up to 3000 Nm); this kind of actuators can generate much greater torque with less energy when compared to ordinary reaction wheels, normally used on small satellites.

A VSCMG is a hybrid actuator combining a reaction wheel with a SGCMG; in contrast to a classical CMG, the wheel speed of a VSCMG is allowed to vary continuously. A reaction wheel or a conventional SGCMG can only generate a torque along a single direction, while a VSCMG can generate a torque that lies anywhere on a plane perpendicular to the gimbal axis [2, 3]. Thus, the second advantage of VSCMGs with respect to classical CMGs is an additional degree of freedom due to the available rotor torque; this is a useful for continuous CMG singularity avoidance and VSCMG cluster reorientation. DGCMGs offer two control axes compared to SGCMGs but they are mechanically more complex and expensive and have not the same capability in one axis because the total torque output on one axis is dependent on two gimbal axes for the DGCMG while for the SGCMG it is only dependent on one [2]. Thirdly, research has shown that the practical benefits of VSCMGs are readily available using conventional CMGs with alterations to CMG cluster steering and CMG rotor motor control laws. Another advantage of the VSCMGs is their possibility of storing the energy in the same time with the control of the satellite’s attitude. Moreover, the extra degree of freedom of a VSCMG can be used for additional purposes, for instance, for combined attitude and power tracking control and/or singularity avoidance.

A minimum of three VSCMGs is needed to practically demonstrate the 3-axis control for a small satellite (S), but, in many cases, a performing ACS utilizes four of these actuators (positioned in the form of a pyramidal cluster [2, 3]) to avoid performance inefficiencies known as internal singularities (no torque can be produced for certain sets of gimbal angles) [1]; if a pyramidal cluster of four CMGs is not used, to solve the problem of internal singularities, some avoidance techniques (null motion, singular value decomposition, differential geometry, topology, etc.) have been developed [2, 4, 5].

In most works, only one pyramidal configuration [6] is used, each pyramid face being inclined at an angle of *β*=54.73 deg from the horizontal; a first innovative element of this paper is related to the fact that the pyramidal cluster presented in the next section has been never used to small satellites’ attitude control. For the pyramidal cluster used here, the following will be achieved: (1) the determination of the rotations for the CMGs’ gyroscopic frames, directions of the kinetic moments, angular rates, and gyroscopic torques (vector diagram); (2) the determination of the projection for CMGs’ gyroscopic torques on the axes of the satellite tied frame, as well as the obtaining of the expressions for the equivalent kinetic moment and resultant gyroscopic torque of the cluster; (3) the determination of the expression of cluster’s cosine matrix for the angles made by the CMGs’ spin axes, transversal axes, and gyroscopic frames’ axes with the minisatellite tied frame’s axes.

The tasks of an ACS are to make the spacecraft (satellite) achieve fine pointing, rapid maneuvering, accurate tracking, and other desired performances. The control problem of small satellites’ attitude in the presence of uncertainties has been investigated in many research papers [7–14]. One possible solution to the problem of attitude control with limited communication is to utilize signal quantization with recently proposed quantizer [7]. Using this idea, a new control architecture for spacecraft attitude stabilization with control torque was quantized by a logarithmic quantizer, and a guideline to choose the quantizer parameters has been proposed in [15]. In [8], an output feedback structured model reference adaptive controller is developed for spacecraft rendezvous and docking problems, but no frictional effects were assumed to be present in the actuator model. The aim of the paper [9] was the design of an attitude tracking controller for CMG-actuated satellites, which is shown to achieve accurate attitude tracking in the presence of unmodeled external disturbance torques, parametric uncertainty, and nonlinear CMG disturbances. To handle the disturbances and the nonlinearities that do not obey the linear-in-the-parameters assumption, fuzzy control or neural network-based control methods have been used [16, 17]; in [16], an approximation-based adaptive fuzzy control architecture is designed for the compensation of nonlinear strict-feedback systems’ unmodeled dynamics; here, the fuzzy logic technique is employed to approximate the unknown nonlinearities, while the compensation of the time-delays is achieved via a Lyapunov–Krasovskii functional. A steering law is designed in [6] to perform the multifunction of satellites’ attitude control and energy storage at the same time, the singularity avoidance being achieved by using the method of the inner product. Reference [18] proposes an effective method to determine the desired spacecraft attitude command through the proper choice of the reference frame for the small satellites, while the papers [11–13] are concentrated on the attitude tracking problem in the Earth staring work mode by means of iterative learning control [11], adaptive fault tolerant control [12], and robust adaptive control with unknown actuator nonlinearity [13]. A model-error control synthesis approach was used in [19] to cancel the effects of modeling errors and external disturbances on the system, but the designed control law requires a model-error term to cancel the effects of a time delay. An adaptive control law is designed in [20], but the controller assumes no dynamic uncertainty in the control torque. Thus, it can be concluded that the development of robust attitude controllers, using different pyramidal clusters of VSCMGs, is still an open issue. Bearing in mind this, and, taking into account the fact that, till now, there is no report on the design of an adaptive ACS using a pyramidal cluster of variable-speed control moment gyros, a feed-forward neural network (NN), a reference model, a linear dynamic compensator, and the dynamic inversion approach, another key goal and innovative element of this paper is the design of such an adaptive ACS, with good tracking performances and robust stability with respect to disturbances. The adaptive control law of the new ACS will consist of four components: (1) a signal provided by the reference model; (2) a signal provided by the linear dynamic compensator (useful for stabilization); (3) an adaptive command (modeled by a feed-forward neural network) which should compensate the inversion error (approximation error of the dynamic model’s nonlinear functions); (4) a signal depending on the NN’s weights, on the Frobenius norm of the neural network’s ideal matrix, and on the observer’s estimation error.

Concluding, the two main goals of our work are as follows: (i) the use for satellites’ stabilization/control of a new pyramidal configuration of four VSCMGs, configuration that has been never used till now in any satellite’s ACS; (ii) the design of a neural network-based ACS using the dynamic inversion control technique. This work has the following innovative elements: (a) the usage for the first time of the pyramidal cluster from Figure 1(b); (b) for the chosen VSCMG cluster, the matrix expressing the dependence between the variation velocity of the cluster’s resultant kinetic moment and the vector of the gyroscopic frames’ angular rates, as well as the satellite’s matrix of the inertia moments have been calculated; (c) the obtaining of the control law consisting of the vector of the gyroscopic frames’ angular rates and the vector of gyros’ angular accelerations, as well as the structure of the adaptive architecture which uses a pyramidal cluster of VSCMGs, a feed-forward neural network, a reference model, a dynamic compensator, and the dynamic inversion approach.