Asymptotic Expansion of the Terminal Control Law of the Output of Two-Tame-Scale Systems

Abstract

Research objective. The research objective is development of asymptotic methods for synthesis of the terminal control laws for the two-tame-scale plants. A two-tame-scale property is typical for he moving objects - aircraft, ships, etc. All of them are characterized by fast rotational and slow translational motions. In power electric drives the mechanical motions of the armature are slow, and the electric transients in circuits are fast. This circumstance hinders formation of the control laws because of the necessity for a frequent time discretization of the transients and performance operations on the ill-conditioned matrices. It is also known under the name of a system rigidity. In case of a terminal control it manifests itself most dramatically due to the non-stationary nature of the process of bringing an object to a predetermined position within a finite period of time. The paper proposes an asymptotic approach to the development of the terminal control law for the two-tame-scale plants. It allows us to decompose the control process into the fast and slow components, which are calculated separately in two time scales, followed by their arrangement in a holistic solution. This simplifies a dynamic representation of the controller and accelerates the numerical procedure for initiation of the control actions. Formulation of the problem. In the article the problem of an asymptotic decomposition for a controlled system is solved with fast and slow motions described by means of the linear differential equations with a small parameter at some derivatives. Terminal control is achieved by minimizing the corresponding quadratic quality functional defined on a finite time interval. Such a statement leads to a singularly perturbed optimal control problem which can be solved in the form of synthesis, and an asymptotic behavior of the control by the method of a boundary layer can be achieved. Research results. As a result of the research, the asymptotic behavior of the terminal control law is built within small singular parameters, uniform on the interval of the control time. The problem is solved for the general case of the output control, which distinguishes the obtained results from the well-known works, devoted exclusively to the state control problems. For this purpose, a special multiplicative decomposition of the control feedback loop matrix into three more simple matrices is used. Improved estimates of the asymptotic expansion remainder, more accurate than the existing estimates, are proved. Conclusion. Application of the developed asymptotic behavior based on its precision estimates allows us to build effective laws for a terminal control of the two-tame-scale plants. The proposed approach solves the problem of the rigidity, simplifies the terminal control laws, and makes their accuracy close to the optimal laws of the terminal control.