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.

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.