scholarly journals SYNTHESIS OF CONTROLLERS FOR NONLINEAR CONTROL OBJECTS ON BASIS OF NUMERICAL METHODS FOR SOLVING DIFFERENTIAL EQUATIONS

2018 ◽  
pp. 67-72
Author(s):  
Yu. P. Kornyushin ◽  
D. A. Akimenko ◽  
P. Yu. Kornyushin
Keyword(s):  
Differential Equations ◽  
Nonlinear Control ◽  
Dynamic Properties ◽  
Reference Signal ◽  
Reference Trajectory ◽  
Algebraic Equations ◽  
Entire Study ◽  
Right Hand ◽  
Study Interval ◽  
The Right

The algorithm of parametric synthesis of regulators for nonlinear control objects is presented in the article, in which knowledge of the real trajectory of the system motion is not required in an explicit form. The essence of the algorithm is as follows: the dynamic properties of the control system are always determined by the right-hand side of the system of differential equations written in the normal form of Cauchy. Depending on the values of the regulator parameters that are also included in the right-hand side, the system of equations may have one or another solution. If we substitute the reference trajectory into the scheme of numerical integration of differential equations describing the dynamics of the system, then at each step of integration it can be considered as a system of algebraic equations for the desired parameters of the regulator. For each discrete value of the reference signal, there is a set of desired regulator parameters from which the desired values are determined, for example, as weighted averages over the entire study interval, or, as some limiting values. In the proposed algorithm, in an implicit form, the criterion of optimality is the norm in the space of convergent numerical sequences.

scholarly journals Resolvent of nonautonomous linear delay functional differential equations

2015 ◽  
Vol 2 (1) ◽  
Author(s):  
Joël Blot ◽  
Mamadou I. Koné
Keyword(s):  
Continuous Function ◽  
Differential Equations ◽  
Differentiable Function ◽  
Functional Differential Equations ◽  
Initial Value ◽  
Complete Proof ◽  
Right Hand ◽  
Linear Delay ◽  
Functional Differential ◽  
The Right

AbstractThe aim of this paper is to give a complete proof of the formula for the resolvent of a nonautonomous linear delay functional differential equations given in the book of Hale and Verduyn Lunel [9] under the assumption alone of the continuity of the right-hand side with respect to the time,when the notion of solution is a differentiable function at each point, which satisfies the equation at each point, and when the initial value is a continuous function.

scholarly journals On acyclicity of quadratic system with two non-rough foci

2021 ◽  
pp. 41-46
Author(s):  
Адам Дамирович Ушхо
Keyword(s):  
Differential Equations ◽  
Limit Cycles ◽  
Phase Plane ◽  
Second Order ◽  
Quadratic System ◽  
Equilibrium States ◽  
System Of Differential Equations ◽  
Right Hand ◽  
The Right

Доказывается, что система дифференциальных уравнений, правые части которой представляют собой полиномы второй степени, не имеет предельных циклов, если в ограниченной части фазовой плоскости она имеет только два состояния равновесия и при этом они являются состояниями равновесия второй группы. It is proved that a system of differential equations, the right-hand sides of which are second-order polynomials, has no limit cycles if it has only two equilibrium states in the bounded part of the phase plane, and they are the equilibrium states of the second group.

Almost-conservative second-order differential equations

1975 ◽  
Vol 77 (1) ◽  
pp. 159-169 ◽  
Author(s):  
H. P. F. Swinnerton-Dyer
Keyword(s):  
Differential Equations ◽  
Rate Of Convergence ◽  
Second Order ◽  
Good Function ◽  
Second Order Differential Equations ◽  
Right Hand ◽  
Image Position ◽  
The Right

During the last thirty years an immense amount of research has been done on differential equations of the formwhere ε > 0 is small. It is usually assumed that the perturbing term on the right-hand side is a ‘good’ function of its arguments and that its dependence on t is purely trigonometric; this means that there is an expansion of the formwhere the ωn are constants, and that one may impose any conditions on the rate of convergence of the series which turn out to be convenient. Without loss of generality we can assumeand for convenience we shall sometimes write ω0 = 0. Often f is assumed to be periodic in t, in which case it is implicit that the period is independent of x and ẋ (We can also allow f to depend on ε, provided it does so in a sensible manner.)

scholarly journals II. On the solution of linear differential equations

1848 ◽  
Vol 138 ◽  
pp. 31-54 ◽  
Keyword(s):  
Differential Equations ◽  
Differential Expression ◽  
Linear Differential Equations ◽  
Right Hand ◽  
Independent Variable ◽  
Linear Differential ◽  
The Right

If the operation of differentiation with regard to the independent variable x be denoted by the symbol D, and if ϕ (D) represent any function of D composed of integral powers positive or negative, or both positive and negative, it may easily be shown, that ϕ (D){ψ x. u } = ψ x. ϕ (D) u + ψ' x. ϕ' (D) u + ½ψ" x. ϕ" (D) u + 1/2.3 ψ"' x. ϕ"' (D) u + . . . (1.) and that ϕx .ψ(D) u = ψ(D){ ϕx. u } - ψ'(D){ ϕ'x. u } + ½ψ"(D){ ϕ"x. u } - 1/2.3ψ"'(D){ ϕ"'x. u } + . . (2.) and these general theorems are expressions of the laws under which the operations of differentiation, direct and inverse, combine with those operations which are de­noted by factors, functions of the independent variable. It will be perceived that the right-hand side of each of these equations is a linear differential expression; and whenever an expression assumes or can be made to assume either of these forms, its solution is determined; for the equations ϕ (D){ψ x. u } = P and ϕx . ψ(D) u = P are respectively equivalent to u = (ψ x ) -1 { ϕ (D)} -1 P and u = {ψ(D)} -1 (( ϕx ) -1 P).

scholarly journals L-STABLE (4,2)-METHOD OF THE FOURTH ORDER FOR SOLVING STIFF PROBLEMS

2017 ◽  
Vol 17 (8) ◽  
pp. 59-68
Author(s):  
E.A. Novikov
Keyword(s):  
Differential Equations ◽  
Fourth Order ◽  
Maximal Order ◽  
Stiff Problems ◽  
System Of Differential Equations ◽  
Maximum Order ◽  
Order Of Accuracy ◽  
Right Hand ◽  
The Right

(M,k)-methods for solving stiff problems, in which on each step two times the right-hand side of the system of differential equations is calculated are investigated. It is shown that the maximum order of accuracy of the L-stable (m,2)-method is equal to four. (4,2)-method of maximal order is built.

scholarly journals An estimation from within of the reachable set of nonlinear R. Brockett integrator with small nonlinearity

2013 ◽  
Vol 23 (3) ◽  
pp. 355-365
Author(s):  
Aboubacar Moussa ◽  
Mikhail Nikolskii
Keyword(s):  
Differential Equations ◽  
Reachable Set ◽  
Right Hand ◽  
Similar Theme ◽  
The Right

In this paper, the nonlinear R. Brockett integrator with small nonlinear addition to the right-hand side of the corresponding differential equations is considered. More precisely, investigating the possibility to estimate from within the corresponding reachable set, we have obtained an efficient form of the ellipsoidal estimation from within. We used our previous results on the similar theme.

scholarly journals I. On the analytical theory of the attraction of solids bounded by surfaces of a hypothetical class including the ellipsoid

1860 ◽  
Vol 150 ◽  
pp. 1-11
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Analytical Theory ◽  
Constant Independent ◽  
Analytical Treatment ◽  
Partial Differential ◽  
Closed Surfaces ◽  
Right Hand ◽  
The Right

The following investigation is the result of an attempt to simplify the analytical treatment of the problem of the Attraction of Ellipsoids. The application to this particular case, of certain known propositions relating to closed surfaces in general, showed that the principal theorems could easily be deduced without taking account of any other properties of the ellipsoid than those expressed by two differential equations, of which the truth is evident on inspection. In fact if we take the equation x 2 / a 2 + h + y 2 / b 2 + h + z 2 / c 2 + h = k , we see at once that the expression on the left side, considered as a function of x, y, z, h , satisfies the two partial differential equations d 2 u / dx 2 + d 2 u / dy 2 + d 2 u / dz 2 = 2 (1/ a 2 + h + 1/ b 2 + h + 1/ c 2 + h ) ( du/dx ) 2 + ( du/dy ) 2 + ( du/dz ) 2 + 4 du/dh = 0, and these equations express all that we require to know about the ellipsoid, except the fact that the surface is capable of being extended to infinity in every direction by the variation of h , without ceasing to be closed. But it appeared also that the success of the method depended only on the circumstance that the right-hand member of the first equation, and the coefficient of du/dh in the second, are constants independent of k . It was therefore possible to generalize the process by taking indeterminate functions of h for these two constants. As, however, the coefficient of du/dh could always be reduced to a constant independent of h , by taking a function of h as a parameter instead of h , we may suppose, without loss of generality, that this reduction has been effected.

SET-VALUED DYNAMICS IN PROBLEMS OF MATHEMATICAL THEORY OF CONTROL PROCESSES

2012 ◽  
Vol 26 (25) ◽  
pp. 1246010 ◽  
Author(s):  
TATIANA FILIPPOVA
Keyword(s):  
Control System ◽  
Differential Equations ◽  
Nonlinear Control ◽  
Control Systems ◽  
Mathematical Theory ◽  
Quadratic System ◽  
Nonlinear Control System ◽  
Reachable Sets ◽  
The Right ◽  
Differential Control

The dynamics and properties of set-valued states of differential control systems with uncertainties in initial data are studied. It is assumed that the dynamical system has a special structure, in which nonlinear terms in the right-hand sides of related differential equations are quadratic in state coordinates. We construct external and internal ellipsoidal estimates of reachable sets of nonlinear control system and find differential equations of proposed ellipsoidal estimates of reachable sets of nonlinear control system. The results obtained for quadratic system nonlinearities are extended to other types of control systems under uncertainty.

Sign in / Sign up

Export Citation Format

Share Document