This study aims to analyze the nonlinear dynamics of a satellite attitude control system equipped with reaction wheels and a PD controller. Based on the angular momentum conservation theorem for a closed mechanical system, the nonlinear equations of the attitude control system dynamics are presented as a linear system of differential equations with time-varying parameters. The asymptotic properties of the angular momentum of a mechanical system including a satellite and reaction wheels as rigid bodies are investigated. A relation has been established between the dynamic parameters of the attitude control system and the initial value of the angular momentum of the satellite. The issue of asymptotic stability for differential equations with time-varying parameters is simplified to the asymptotic stability problem for the ultimate homogeneous system of linear differential equations with constant elements. The dependencies of the dynamic parameters of the attitude control system on the constant parameters of this ultimate system of linear differential equations, as well as the initial values of the satellite’s angular momentum, enable us to apply proven and effective engineering methods. These methods are used not only for analyzing the stability of the control system but also for synthesizing the parameters of the control law based on the quality requirements of transient processes such as the stability margin, responsiveness, oscillation, transient time, and overshoot. In this case, the calculation of the control law parameters will be grounded in exact equations, not on approximate equations of the control system dynamics obtained by linearization.

The problem of optimal synthesis of a nonlinear system’s control law parameters of satellite attitude control under constraints is considered. The maximum stability degree of a system is considered as an optimality criterion. The reaction wheels’ control moments and angular velocities achievable within the drives’ physical characteristics are considered as constraints. A linear model of the system converted from the original nonlinear model was used to solve the problem. The decomposition method, based on the transition from real time to relative time, is used to separate the tasks of optimality criterion maximization and constraints consideration. A method of synthesizing the optimal form of system transient process according to the maximum stability degree criterion is developed. An analytical method for determining the optimal values of control law parameters is proposed, based on the structural features of the linear differential equations system. A method that takes into account constraints on control moments and angular velocities of reaction wheels, based on transition scale from relative time to real time and its calculation algorithm, ensuring that conditions of all constraints are met, is developed. An example of solving the problem of optimal system synthesis is considered. The results demonstrate the effectiveness and simplicity of the proposed methods and algorithm for targeted selection of a system’s technical characteristics, taking into account constraints on reaction wheels’ maximum angular velocities and control moments values.

This paper proposes a coefficient-based method for designing a rocket angular stabilization system that yields an explicit, direct solution for determining the parameters of a PID control law from specified stability and performance indices. Linearized equations of motion in the yaw channel with a PID law are formulated, and the system transfer function and its characteristic polynomial are obtained. Quantitative stability and performance indices are introduced, defined via the coefficients of the characteristic polynomial, and expressed directly through the control law parameters. Based on sufficient conditions for stability and performance, resolving inequalities are derived for the control parameters that meet the prescribed stability and performance requirements. It is shown that the solution set of these inequalities is non-empty; i.e., the problem of finding the required control law parameters always has a solution. In the control-parameter space, the boundaries of the system’s stability and performance regions are explicitly described. A decomposition method and its extension are proposed for the design problem by the required transient process shape and transient time, splitting it into two sequential subproblems: obtaining the required shape and obtaining the required speed. A simple and effective adaptive algorithm is proposed to maintain constant stability and performance indices under variable engine thrust. The results of applying the proposed design methods to the development of a rocket angular stabilization system test stand are presented. Numerical experiments convincingly demonstrate the high effectiveness of the proposed coefficient-based design method—combining sufficient stability and performance conditions with the decomposition approach—for designing the test stand system.

The goal is the optimal synthesis of the parameters of the control law for a nonlinear satellite attitude control system under constraints related to the saturation effect of reaction wheels. The maximum stability degree of the control system is considered as a criterion for the optimality of the parameters of the control law, and the absolute values of the control torques and angular velocities of the reaction wheels, achievable within the limits of the physical characteristics of the drives, are considered as constraints. The solution to the optimal synthesis problem is based on the transformation of the nonlinear control system model into a linear model, within the framework of the problem of global asymptotic stability of the control system is also guaranteed. The problem is decomposed into a subproblems of obtaining the optimal transient process shape in the control system and considering the constraints. A method of synthesizing the optimal transient process shape based on the criterion of maximum stability degree had been developed. An analytical method of determining the optimal values of the control law parameters, based on decomposing the normalized characteristic polynomial in the linear model into three identical factors had been proposed. A method with regards to the constraints of the control torque and angular speed of the reaction wheels, based on transitioning from relative to real time is developed. An algorithm for calculation of the transition scale from relative to real time, ensuring the fulfillment of all constraints in the optimal synthesis problem, is proposed. An example demonstrating the efficiency and simplicity of the proposed methods and algorithm for targeted selection of technical characteristics of the control system, considering the maximum absolute values of angular speeds and control torques of the reaction wheels, is provided.