One approximation in the optimal stabilization problem for the system of periodic delayed differential equations

2020 ◽  
Author(s):  
R. I. Shevchenko
Keyword(s):  
Differential Equations ◽  
Stabilization Problem ◽  
Optimal Stabilization ◽  
Delayed Differential Equations

DISCRETE PROCEDURE OF OPTIMAL STABILIZATION FOR PERIODIC LINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS

2018 ◽  
pp. 891-906
Author(s):  
Roman Ivanovich Shevchenko ◽  
Yuri Filippovich Dolgii
Keyword(s):  
Differential Equations ◽  
Discrete Time ◽  
Periodic Problem ◽  
Optimal Stabilization ◽  
Systems Of Differential Equations ◽  
Special Procedure ◽  
Periodic Linear ◽  
System States ◽  
Linear Periodic Systems ◽  
Time Periodic

We propose procedure to solve the optimal stabilization problem for linear periodic systems of differential equations. Stabilizing controls, formed as a feedback, are defined by the system states at the fixed instants of time. Equivalent discrete-time linear periodic problem of optimal stabilization is considered. We propose a special procedure for the solution of discrete periodic Riccati equation. We investigate the relation between continuous-time and discrete-time periodic optimal stabilization problems. The proposed method is used for stabilization of mechanical systems.

One-sided variations on interval trees

2003 ◽  
Vol 40 (03) ◽  
pp. 654-670 ◽  
Author(s):  
Yoshiaki Itoh ◽  
Hosam M. Mahmoud
Keyword(s):  
Differential Equations ◽  
Path Length ◽  
Analytic Approach ◽  
Tauberian Theorems ◽  
Random Structure ◽  
Delayed Differential Equations ◽  
Interval Tree ◽  
Binary Case ◽  
Interval Trees ◽  
Parameters Of Interest

The binary interval tree is a random structure that underlies interval division and parking problems. Five incomplete one-sided variants of binary interval trees are considered, providing additional flavors and variations on the main applications. The size of each variant is studied, and a Gaussian tendency is proved in each case via an analytic approach. Differential equations on half scale and delayed differential equations arise and can be solved asymptotically by local expansions and Tauberian theorems. Unlike the binary case, in an incomplete interval tree the size determines most other parameters of interest, such as the height or the internal path length.

scholarly journals STABILIZATION OF HAMILTONIAN SYSTEMS WITH A GUARANTEED ESTIMATE OF QUALITY OF MANAGEMENT

2017 ◽  
Vol 20 (7) ◽  
pp. 32-44
Author(s):  
E.V. Kurkina
Keyword(s):  
Hamiltonian System ◽  
Direct Method ◽  
Zero Solution ◽  
Variable Length ◽  
Constant Sign ◽  
Functional Requirements ◽  
Stabilization Problem ◽  
Optimal Stabilization ◽  
Stabilizing Control

Stabilization problem with guaranteed estimate of quality of management of zero solution of nonautonomous Hamiltonian system was solved. It arises from the problem of optimal stabilization by reducing functional requirements for a estimate: instead of minimizing it is only necessary that it excelled to a pre-assessment. The solution is obtained by means of synthesis of active program control, acting to the system, and stabilizing control of feedback. The problem is solved analytically by the direct method of Lyapunov's stability theory with Lyapunov's function with constant sign derivatives. As examples, problems of synthesis and stabilization of program motions of homogeneous rod of variable length and variable-length pendulum in a rotating plane are solved.

Analysis of Composite Blade/Casing Rub Stability Through Delayed Differential Equations

2019 ◽  
Vol 142 (1) ◽  
Author(s):  
Jiaguangyi Xiao ◽  
Yong Chen ◽  
Jie Tian ◽  
Hua Ouyang ◽  
Anjenq Wang
Keyword(s):  
Differential Equations ◽  
Machine Tool ◽  
Experimental Investigations ◽  
Reduced Order ◽  
Composite Blade ◽  
Characteristic Roots ◽  
Engine Industry ◽  
Delayed Differential Equations ◽  
Inner Surface ◽  
Aero Engine

Abstract To improve aerodynamic efficiencies, the clearances between blades and casings are becoming smaller and smaller in the aero-engine industry, which might lead to the interactions between these components. These unexpected interactions are known as the so-called blade/casing rubs. Abradable materials are implemented on the inner surface of the casings to reduce the potential damages caused by it. However, failures may still arise from blade/casing rubs according to experimental investigations and actual accidents. In this paper, a reduced-order delayed differential equations (DDEs) are used to simplify the rubbing process between composite blade and casing. It is assumed that the removal of the abradable material in blade/casing rubbing process shares a resemblance with machine tool chatters encountered in machining. The DDEs are established with centrifugal stiffness and the impacts of stacking sequences on the blade damping taking into consideration. Semidiscretization method (SDM) is used to study the stabilities of the simplified system, which is verified by cluster treatment of characteristic roots (CTCR) and direct integrations. The results show that the stacking sequences, rub positions, blade damping, and stiffness could have much impact on the relatively dangerous interaction regimes. With the help of this method, one can assist the design processes of the composite blade-casing interface in initial aero-engine structural designs.

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