Chetverikov

pp. 13–40

pp. 105-127

The paper proposes a new method to solve the terminal control problem for dynamical systems. The method is based on the complement of initial system by equations for the derivatives of the control and on the reformulation of the terminal control problem into two associated Cauchy problems. It is shown that this method is applied to the flat systems and generalizes the previously used approach. Here, the solution of the terminal control problem lies among the solutions of an arbitrary determined system of ordinary differential equations of appropriate order. This feature of the new method can be used for control design in the case of flat systems taking into consideration the constraints. In addition, an example is given to demonstrate the possibility to apply this method to the non-flat systems.

For four-dimensional model describing the helicopter motion along a given horizontal line the problem of automated synthesis of programmed motion is solved. This solution provides a helicopter motion from a given state of rest in the given state of rest. Time to perform the maneuver is not set. The considered helicopter model is a control dynamic system, which is not a flat system. For such systems general approaches to the terminal problem solution are presently unknown. To solve the terminal problem two approaches are applied. The first approach is based on the use of finite symmetry, which converts the initial conditions of the problem in the final conditions. The use of such symmetry can reduce the number of final conditions. The second approach is based on the use of covers and consists in the construction of a special mapping, which for two given dynamic systems surjectively maps the set of solutions of the first system in the set of solutions of the second system. The programmed motion in this case can be found as the solution of two related specifically posed Cauchy problems for these dynamic systems. The resulting programmed control is a piecewise continuous function of time.

This paper deals with systems that are reduced to linear control systems by composition of a dynamic feedback transformation with the change of variables. Such systems form the widest class of systems for which control algorithms were developed. An invariant description of algebra of vector fields which define linearizing dynamic feedback was obtained by methods of infinite-dimensional differential geometry. This result can be used for checking dynamic linearizability of particular control systems and in theoretical studies for creating examples of dynamically linearizable systems with certain properties or describing the whole class of such systems with a given dimension.

 The authors consider an aircraft with four propellers (four-propeller rotorcraft). The propellers are attached to two bars rigidly fastened in the middle. The propellers on different bars are rotated in opposite directions. Changes of tractive forces of the propellers allows to control motion of the rotorcraft. The mathematical model of such an aircraft represents a dynamical system with a 12-dimensional state and 4-dimensional control. Dynamic feedback linearizing this system is constructed in the article. The feedback is used to track given reference trajectories with stability at the stages of take-off and landing. Acceptability of the found control is verified. Results of numerical simulation demonstrate efficiency of the proposed approach.

 Control systems that have a linearizing output are called flat. Solutions of such system are expressed through functions of this output and a finite set of their derivatives according to the system. Such output allows to synthesize control algorithm in the form of dynamic feedback. In the article the problem of flatness checking is investigated by methods of infinite-dimensional differential geometry. Earlier this problem was reformulated as the problem of search of an invertible differential operator which transforms a column of given 1-forms to a column of exact 1-forms. In the article an equation for such an operator is derived. The solubility of this equation means flatness of system.