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 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.

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.

The concept of an extended PID controller is introduced for the first time. This controller combines the properties of two well-known variants of the classical PID controller. The extended PID controller includes two additional parameters in addition to the three parameters of the classical PID controller. Its properties are examined using the yaw channel of a rocket angular stabilization system equipped with an extended PID controller. Linearized equations of motion for the yaw channel of the rocket angular stabilization system with the extended PID controller are formulated. The transfer function of the rocket angular stabilization system and its characteristic polynomial are obtained. Stability and performance indices of the rocket angular stabilization system are introduced. These indices are determined from the coefficients of the characteristic polynomial and are expressed directly in terms of the parameters of the extended PID controller of the stabilization system. Based on sufficient conditions for stability and performance of the stabilization system, systems of algebraic inequalities are derived with respect to the required values of the control-law parameters that satisfy the stability and performance requirements of the stabilization system. It is shown that the set of their solutions is nonempty. It is demonstrated that the introduction of additional extension parameters into the classical PID controller makes it possible to vary the zeros of the transfer function of the stabilization system. This enables the stability and performance requirements of the rocket angular stabilization system to be satisfied independently of one another. At the same time, by varying the additional parameters, the zeros of the transfer function of the stabilization system can be made equal to its poles. This changes the structure of the transfer function by reducing its order by two. Numerical experimental studies of the dynamics of the rocket angular stabilization system are carried out using the technical characteristics of the developed test bench for the rocket angular stabilization system. The results confirm the high effectiveness of the extended PID controller: 1) the transient response retains an aperiodic character, which follows directly from the form of the transfer function of the extended PID controller; 2) the settling time is reduced by a factor of 7.53 compared with the classical PID controller.

Широко применяемый вариант классического PID-регулятора имеет три параметра закона управления, выражающие коэффициенты передачи отклонения управляемой переменной от программного значения, интеграла и производной этого отклонения. Менее распространенный вариант использует коэффициенты передачи текущего значения управляемой переменной и ее производной по пропорциональному и дифференциальному каналам. Динамические характеристики систем автоматического управления с одним и тем же объектом для этих вариантов различаются, поскольку передаточная функция замкнутой системы в первом варианте имеет нули, а во втором не имеет. В статье вводится понятие расширенного PID-регулятора, сочетающего свойства обоих вариантов классического PID-регулятора. К трем базовым параметрам добавляются два безразмерных коэффициента, задающих веса управляющих сигналов. В результате управляющий сигнал формируется как комбинация пяти выходных сигналов вместо трех. Приведены результаты численных исследований влияния коэффициентов расширения на устойчивость и качество системы угловой стабилизации тестовой установки ракеты, которые показывают, что применение при проектировании СУС расширенного PID-регулятора позволило уменьшить время переходного процесса в 5,4 раза, сохраняя при этом устойчивость СУС и апериодичность ее переходного процесса.

The concept of an extended PID controller is introduced. This controller combines the properties of two well-known variants of the classical PID controller. The extended PID controller includes two additional parameters in addition to the three parameters of the classical PID controller. Its properties are examined using the yaw channel of a rocket angular stabilization system equipped with an extended PID controller. Linearized equations of motion for the yaw channel of the rocket angular stabilization system with the extended PID controller are formulated. The transfer function of the rocket angular stabilization system and its characteristic polynomial are obtained. Stability and performance indices of the rocket angular stabilization system are introduced. These indices are determined from the coefficients of the characteristic polynomial and are expressed directly in terms of the parameters of the extended PID controller of the stabilization system. Based on sufficient conditions for stability and performance of the stabilization system, systems of algebraic inequalities are derived with respect to the required values of the control-law parameters that satisfy the stability and performance requirements of the stabilization system. It is shown that the set of their solutions is nonempty. It is demonstrated that the introduction of additional extension parameters into the classical PID controller makes it possible to vary the zeros of the transfer function of the stabilization system. This enables the stability and performance requirements of the rocket angular stabilization system to be satisfied independently of one another. At the same time, by varying the additional parameters, the zeros of the transfer function of the stabilization system can be made equal to its poles. This changes the structure of the transfer function by reducing its order by two. Numerical experimental studies of the dynamics of the rocket angular stabilization system are carried out using the technical characteristics of the developed test bench for the rocket angular stabilization system. The results confirm the high effectiveness of the extended PID controller: (1) the transient response retains an aperiodic character, which follows directly from the form of the transfer function of the extended PID controller, and (2) the settling time is reduced by a factor of 7.53 compared with the classical PID controller.