Cancer Immunoediting and beyond in 2021

Human cancer has been depicted as a non-linear dynamic system that is discontinuous in space and time, but progresses through different sequential states (Figure 1) [...]

Determining the effect of fuzziness in the parameters of a linear dynamic system on its stability

This paper considers a relevant issue related to the influence exerted by the fuzziness in linear dynamic system parameters on its stability. It is known that the properties of automated control systems can change under the influence of parametric disturbances. To describe the change in such properties of the system, the concept of roughness is used. It should be noted that taking into consideration the fuzziness in the parameters of mathematical models could make it possible at the design stage to assess all the risks that may arise as a result of an uncontrolled change in the parameters of dynamic systems during their operation. To prevent negative consequences due to variance in the parameters of mathematical models, automated control systems are designed on the basis of the requirement for ensuring a certain margin of stability of the system in terms of its amplitude and phase. At the same time, it remains an open question whether such a system would satisfy the conditions of roughness. Parameters of the mathematical model of a system are considered as fuzzy quantities that have a triangular membership function. This function is inconvenient for practical use, so it is approximated by the Gaussian function. That has made it possible to obtain formulas for calculating the characteristic polynomial and the transfer function of the open system, taking into consideration the fuzziness of their parameters. When investigating the system according to Mikhailov’s criterion, it was established that the dynamic system retains stability in the case when the parameters of the characteristic equation are considered as fuzzy quantities. It has been determined that the quality of the system significantly deteriorated in terms of its stability that could make it enter a non-steady state. When using the Nyquist criterion, it was established that taking into consideration the fuzziness in the parameters of the transfer function did not affect the stability of the closed system but there was a noticeable decrease in the system stability reserve both in terms of phase and amplitude. The relative decrease in the margin of stability for amplitude was 16 %, and for phase ‒ 17.4 %.

Modeling the trajectory of motion of a linear dynamic system with multi-point conditions

<abstract><p>The motion of the linear dynamic system with given properties is modeled; conditions for system state at various arbitrarily points in time are given. Simulated movement carried out due to the calculated input vector function. The method of undefined coefficients is used to construct the input vector function and the corresponding trajectory. The proposed method consists in the formation of the state vector function, the trajectory of motion and the input vector function in exponential-polynomial form, that is, in the form of linear combinations of the powers of the time parameter with vector coefficients. This linear combination is complemented by a scalar exponential function with an additional parameter in the exponent to change the type of trajectory. To find the introduced coefficients, formulas and a linear algebraic system are formed. To find the introduced coefficients, the formed linear combinations are substituted directly into the equations describing the dynamic system and into the given multipoint conditions for finding the entered coefficients. All this leads to obtaining algebraic formulas and linear algebraic systems. Only the matrices included in the system that describe the dynamics of the model (and similar matrices with higher exponents) are the coefficients for the unknown parameters of the resulting algebraic system. It is proved that the fulfillment of the condition Kalman is sufficient for the solvability of the resulting system. To substantiate the solvability of the system, the properties of finite-dimensional mappings are used: decomposition of spaces into subspaces, projectors on subspaces, semi-inverse operators. But for the practical use of the proposed method, it is sufficient to solve the obtained linear algebraic system and use the obtained linear formulas. The correctness of the obtained model is investigated. Due to the non-uniqueness of the solution to the problem posed, the trajectory of motion can be unstable. It is revealed which components of the desired coefficients are arbitrary. It is showed which ones to choose, to make the movement steady, that is, so that small changes in the given multi-point values, as well as a small change parameters of the dynamic system corresponded to a small change in the trajectory of motion. An example is given of constructing trajectories of a material point in a vertical plane under the action of a reactive force in order to hit a given point with a given speed.</p></abstract>

Pseudochaotic behavior of deterministic automatic control systems of technical devices

А method for generating a pseudo-random response in a linear dynamic system is considered. An example of the system transition to pseudochaotic behavior is given. The calculation of fatigue loads in the toggle plate by the given method is presented. The advantage of this method in calculating fatigue loads is shown.

Optimization of a linear dynamic system with periodic changing parameters

The article deals with the problem of optimal control of a linear dynamic system with periodic parameters. The qualitative theory of such problems is developed very fully if the period of coefficients of the system is not very small. With a small period, there are serious difficulties with integration. Therefore, it is reasonable to supplement the constructive methods of solution with asymptotic ones. The article presents such an approach that the method of averaging is used to construct an auxiliary (basic) problem, estimates of the proximity of solutions to the initial and basic problems are obtained.