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• THE MATHEMATICAL PROBLEM OF OPTIMAL CONTROL OF THE STRING

  G. V. Kupovykh, A. G. Klovo , I. A. Lyapunova

  2020-11-22

  Abstract ▼

  It is generally accepted that optimal control problems or system design problems determine
  for a given object or system of control objects a law or a certain control sequence of actions that
  provide a maximum or minimum of a given set of system quality criteria. In this case, the speed
  problem can be considered, i.e. the problem of bringing the system to a given state in the shortest
  time. We also study the problems of minimizing a given functional for a fixed time of system management.
  Optimal control is closely related to the choice of the most rational modes for managing
  complex objects. A lot of works has been devoted to the problem of control, in addition, wellknown
  mathematical schools are currently engaged in such research. In problems with concentrated
  parameters, the systems under study are described by ordinary differential equations or
  their systems. In this case, the Pontryagin maximum principle plays an important role in this
  study. For partial differential equations, we talk about systems with distributed parameters. In thispaper, we investigate the possibility of synthesizing optimal control of a single system with distributed
  parameters. A model of string oscillation under the influence of control functions under
  boundary conditions is considered. The role of the choice of the functional to be minimized in creating
  opportunities for the synthesis of optimal control. In this case, the control action is searched
  for at each point of the time interval, which leads to the possibility of constructing it explicitly.
  The conditions for the existence of optimal control everywhere in the corresponding functional
  spaces are formulated. In a specific statement of the problem, everywhere optimal control is explicitly
  constructed.

• SYNTHESIS OF A ROBUST ACS WITH A DYNAMIC COMPENSATOR AND A STATE OBSERVER WITH CORRECTIVE FEEDBACK BASED ON SIGMOID FUNCTIONS

  N.О. Luzhevsky , V. F. Lubentsov , Е.V. Lubentsova

  2026-02-27

  Abstract ▼

  A methodology for analyzing a robust automatic control system for linear dynamic control objects with a delay in the uncertainty of information about the parameters of the mathematical model and interference in the measured signals is described. The system implements a widely used PID controller and a dynamic compensator (DC) for the inertial part of the control object, implemented using estimates of state variables obtained on the basis of a state observer (SO). It is noted that any stability criterion can be used for the asymptotic stability of the controller and observer, but to ensure the maximum degree of stability and the required quality indicators of the transient process, it is convenient to use the maximum degree of stability criterion. In this paper, instead of derivatives obtained by differentiation, it is proposed to use in the dynamic compensator of the inertia of the object estimates of the state variables of the object obtained with the help of a state observer. Another difference from the known ones is the use of sigmoid functions in the corrective feedback of the state observer and the implementation of an additional effect to the main one based on the estimation error, compensating for external disturbance at the input of the object. The parametric synthesis of a state observer was used to study the impact of disturbances and noise on the performance of an automated control system (ACS) for an industrial interval-defined plant with various parameters. A robust typical PID controller with optimal parameters for maximum stability was calculated, taking into account the compensation of the inertial part of the object. For this purpose, a dynamic compensator (DC) was implemented using the parameters of a nominal (calculated) model of the object with the worst combination of parameters obtained on the basis of the interval model of the control object with a delay. The conducted research has established that the structure of the ACS with a typical PID controller with a sequential dynamic compensator for the inertia of the object and a state observer with corrective feedback based on sigma functions ensures the simplicity of the ACS synthesis methodology that is robust to changes in the parameters of the object and the action of unmeasured disturbances and uncontrolled interference.