What is a controlled system with an aftereffect and what does it have to do with airplanes?

The control system is a set of means for controlling the controlled object, it can be any dynamic system or its model. The state of an object is characterized by some quantitative values changing in time. When describing most phenomena, it used to be assumed that the system in question is subject to the law of causality: the future state of the system does not depend on the past states and is determined only by the present (most often the system is described by an equation containing state variables and rates of their changes, i.e. we come either to the ordinary differential equation or to the partial differential equation).

Upon closer examination, however, it often becomes apparent that this is only a first approximation to the true situation, and a more realistic model must include some of the antecedent states of the system. If, say, this is some physical or technical problem, then the force acting on a material point depends on the position and velocity of the point not only at a given time, but also at a point in time prior to the given one. In other words, the state of the evolving system at any moment of time affects the character of evolution (velocity, acceleration) not only at the same moment of time, but also at subsequent ones. This effect is called lag or aftereffect. Thus, we are talking about processes in which, in principle, there is a lag, and the presence of such a lag often has a significant impact on the course of the process. In the case of a control system with an after-effect, the state of the object is also affected by its previous configuration.

What does this task have to do with airplanes?

The mathematical model under study describes many real feedback control systems. For example, the stabilization of the altitude of a passenger airplane. The process of gaining altitude is controlled by a number of devices: aircraft engines, rudders and others. But since for the operation of the above devices we need to have information from altitude sensors, and this information is received and processed with some delay, the above dynamic process is described by differential-difference equations containing both values of altitude and its derivatives at a given time t, and values of the above function at the previous time t-Δ, when this information was measured. Such equations are called differential-difference equations.

However, in the theory of control systems with an aftereffect arising due to the presence of feedback, there is a problem about the settling of this system for a finite time. This problem was solved by N.N. Krasovsky for stationary systems described by systems of differential-difference equations of lagging type. In the present study, the differential-difference equations of the neutral type are considered.

“For the system to calm down in the case of ordinary equations, you must bring it to equilibrium, then turn off the control. If it is a linear system, it will remain in the zero state without external influences. However, in the case under consideration, there is a lag. Therefore, the behavior of the system is affected by its prehistory. Even if we bring it to zero, the system must be kept at zero for the time corresponding to the lag. Thus, there are infinitely many solutions to the problem. It is important to find the solution that gives the least amount of energy” — said Amina Adhamova.

In spite of the fact that the solutions of this problem are infinitely many, to find the optimal solution it is necessary to minimize the quadratic functional, then go to the equivalent boundary value problem for the system of second order differential-difference equations and find its classical solution.

The main difference of this research is that the parameters of the real control system depend on time, i.e. the system can work differently at different moments of time, although previously we considered systems with constant coefficients, which display only the model component of the system. Along with this, the system under study is multidimensional, because the real control system has many different parameters.

For reference:

The aim of the Contest is to create conditions for young scientists-postgraduate students to prepare their dissertations for the degree of Candidate of Sciences, to assist in employment and retention of young scientists in Russian scientific organizations. The competition is organized by the Russian Foundation for Basic Research (RFBR).