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15.6. Linear and nonlinear operators. Dynamic system operator

Lecture



In presenting the theory of transformation of random functions, we will use the concept of an operator widely used in mathematics and technology.

The concept of an operator is a generalization of the concept of a function. When we establish a functional connection between two variables   15.6.  Linear and nonlinear operators.  Dynamic system operator and   15.6.  Linear and nonlinear operators.  Dynamic system operator and write:

  15.6.  Linear and nonlinear operators.  Dynamic system operator . (15.6.1)

then under the symbol   15.6.  Linear and nonlinear operators.  Dynamic system operator we understand the rule by which the given value   15.6.  Linear and nonlinear operators.  Dynamic system operator quite definite value is matched   15.6.  Linear and nonlinear operators.  Dynamic system operator . Sign   15.6.  Linear and nonlinear operators.  Dynamic system operator there is a symbol of some transformation to which it is necessary to subject the value   15.6.  Linear and nonlinear operators.  Dynamic system operator , To obtain   15.6.  Linear and nonlinear operators.  Dynamic system operator . Accordingly, the type of this transformation functions can be linear and nonlinear, algebraic, transcendental, etc.

Similar concepts and the corresponding symbols are applied in mathematics and in those cases when not values, but functions are subjected to conversion.

Consider some function   15.6.  Linear and nonlinear operators.  Dynamic system operator and set a certain rule   15.6.  Linear and nonlinear operators.  Dynamic system operator according to which function   15.6.  Linear and nonlinear operators.  Dynamic system operator converted to another function   15.6.  Linear and nonlinear operators.  Dynamic system operator . We write this transformation in the following form:

  15.6.  Linear and nonlinear operators.  Dynamic system operator . (15.6.2)

Examples of such transformations can be, for example, differentiation:

  15.6.  Linear and nonlinear operators.  Dynamic system operator , (15.6.3)

integration:

  15.6.  Linear and nonlinear operators.  Dynamic system operator , (15.6.4)

etc.

Rule   15.6.  Linear and nonlinear operators.  Dynamic system operator according to which function   15.6.  Linear and nonlinear operators.  Dynamic system operator converted to function   15.6.  Linear and nonlinear operators.  Dynamic system operator , we will call the operator; for example, we will say: the differentiation operator, the integration operator, the operator of the solution of a differential equation, etc.

Defining the operator, we considered only the transformation function   15.6.  Linear and nonlinear operators.  Dynamic system operator to another function   15.6.  Linear and nonlinear operators.  Dynamic system operator same argument   15.6.  Linear and nonlinear operators.  Dynamic system operator . It should be noted that such a preservation of the argument when defining the operator is not at all mandatory: the operator can convert the function   15.6.  Linear and nonlinear operators.  Dynamic system operator to a function of another argument   15.6.  Linear and nonlinear operators.  Dynamic system operator , eg:

  15.6.  Linear and nonlinear operators.  Dynamic system operator , (15.6.5)

Where   15.6.  Linear and nonlinear operators.  Dynamic system operator - some function dependent, besides argument   15.6.  Linear and nonlinear operators.  Dynamic system operator , also from parameter   15.6.  Linear and nonlinear operators.  Dynamic system operator .

But since when analyzing errors of dynamic systems, the most natural argument is time   15.6.  Linear and nonlinear operators.  Dynamic system operator , we will limit ourselves here to considering operators that transform a single argument function.   15.6.  Linear and nonlinear operators.  Dynamic system operator to another function of the same argument.

If a dynamic system converts a function arriving at its input   15.6.  Linear and nonlinear operators.  Dynamic system operator in function   15.6.  Linear and nonlinear operators.  Dynamic system operator :

  15.6.  Linear and nonlinear operators.  Dynamic system operator ,

that operator   15.6.  Linear and nonlinear operators.  Dynamic system operator is called a dynamic system operator.

In the more general case, not one but several functions arrive at the system input; equally, several functions may appear at the output of the system; in this case, the system operator converts one set of functions into another. However, for the sake of simplicity, we consider here only the most elementary case of the transformation of one function into another.

Transformations or operators applied to functions can be of various types. The most important for practice is the class of so-called linear operators.

Operator   15.6.  Linear and nonlinear operators.  Dynamic system operator is called linear homogeneous if it has the following properties:

1) the operator can be applied by term to the sum of functions:

  15.6.  Linear and nonlinear operators.  Dynamic system operator ; (15.6.6)

2) constant value   15.6.  Linear and nonlinear operators.  Dynamic system operator can be taken out of the operator sign:

  15.6.  Linear and nonlinear operators.  Dynamic system operator . (15.6.7)

From the second property, by the way, it follows that for a linear homogeneous operator the property

  15.6.  Linear and nonlinear operators.  Dynamic system operator , (15.6.8)

that is, at zero input, the response of the system is zero.

Examples of linear homogeneous operators:

1) differentiation operator:

  15.6.  Linear and nonlinear operators.  Dynamic system operator ;

2) integration operator:

  15.6.  Linear and nonlinear operators.  Dynamic system operator ;

3) the operator of multiplication by a certain function   15.6.  Linear and nonlinear operators.  Dynamic system operator :

  15.6.  Linear and nonlinear operators.  Dynamic system operator ,

4) integration operator with a given "weight"   15.6.  Linear and nonlinear operators.  Dynamic system operator :

  15.6.  Linear and nonlinear operators.  Dynamic system operator

etc.

In addition to linear homogeneous operators, there are also linear inhomogeneous operators.

Operator   15.6.  Linear and nonlinear operators.  Dynamic system operator is called linear inhomogeneous if it consists of a linear homogeneous operator with the addition of some well-defined function   15.6.  Linear and nonlinear operators.  Dynamic system operator :

  15.6.  Linear and nonlinear operators.  Dynamic system operator , (15.6.9)

Where   15.6.  Linear and nonlinear operators.  Dynamic system operator - linear homogeneous operator.

Examples of linear inhomogeneous operators:

one)   15.6.  Linear and nonlinear operators.  Dynamic system operator ,

2)   15.6.  Linear and nonlinear operators.  Dynamic system operator ,

3)   15.6.  Linear and nonlinear operators.  Dynamic system operator .

Where   15.6.  Linear and nonlinear operators.  Dynamic system operator ,   15.6.  Linear and nonlinear operators.  Dynamic system operator ,   15.6.  Linear and nonlinear operators.  Dynamic system operator - well-defined functions, and   15.6.  Linear and nonlinear operators.  Dynamic system operator - function convertible by operator.

In mathematics and engineering, the conventional form of recording operators, similar to algebraic symbolism, is widely used. Such symbolism in some cases allows to avoid complex transformations and write formulas in a simple and convenient form.

For example, the operator of differentiation is often denoted by the letter   15.6.  Linear and nonlinear operators.  Dynamic system operator :

  15.6.  Linear and nonlinear operators.  Dynamic system operator ,

placed as a multiplier in front of the expression to be differentiated. With this record

  15.6.  Linear and nonlinear operators.  Dynamic system operator

tantamount to writing

  15.6.  Linear and nonlinear operators.  Dynamic system operator .

Double differentiation is indicated by a multiplier.   15.6.  Linear and nonlinear operators.  Dynamic system operator :

  15.6.  Linear and nonlinear operators.  Dynamic system operator

etc.

Using such symbolism, in particular, it is very convenient to write down differential equations.

Let, for example, the work of a dynamic system   15.6.  Linear and nonlinear operators.  Dynamic system operator described by a linear differential equation with constant coefficients relating the response of the system   15.6.  Linear and nonlinear operators.  Dynamic system operator with impact   15.6.  Linear and nonlinear operators.  Dynamic system operator . In the usual notation, this differential equation has the form:

  15.6.  Linear and nonlinear operators.  Dynamic system operator

  15.6.  Linear and nonlinear operators.  Dynamic system operator . (15.6.10)

In symbolic form, this equation can be written as:

  15.6.  Linear and nonlinear operators.  Dynamic system operator

  15.6.  Linear and nonlinear operators.  Dynamic system operator .

Where   15.6.  Linear and nonlinear operators.  Dynamic system operator - differentiation operator.

Denoting for short polynomials with respect to   15.6.  Linear and nonlinear operators.  Dynamic system operator in right and left parts

  15.6.  Linear and nonlinear operators.  Dynamic system operator ,

  15.6.  Linear and nonlinear operators.  Dynamic system operator ,

we write the equation in an even more compact form:

  15.6.  Linear and nonlinear operators.  Dynamic system operator . (15.6.11)

Finally, formally solving equation (15.6.11) with respect to   15.6.  Linear and nonlinear operators.  Dynamic system operator , you can symbolically write the solution operator of a linear differential equation in “explicit” form:

  15.6.  Linear and nonlinear operators.  Dynamic system operator . (15.6.12)

Using the same symbolism, one can also write in the operator form a linear differential equation with variable coefficients. In its usual form, this equation has the form:

  15.6.  Linear and nonlinear operators.  Dynamic system operator

  15.6.  Linear and nonlinear operators.  Dynamic system operator . (15.6.13)

Denoting polynomials relatively   15.6.  Linear and nonlinear operators.  Dynamic system operator whose coefficients depend on

  15.6.  Linear and nonlinear operators.  Dynamic system operator ,

  15.6.  Linear and nonlinear operators.  Dynamic system operator ,

You can write a differential equation operator in the form:

  15.6.  Linear and nonlinear operators.  Dynamic system operator . (15.6.14)

In the future, as necessary, we will use such a symbolic form of recording operators.

The dynamic systems encountered in engineering are often described by linear differential equations. In this case, as is easily seen, the system operator is linear.

A dynamic system whose operator is linear is called a linear dynamic system.

In contrast to linear operators and systems, non-linear systems and operators are considered. Examples of nonlinear operators can serve

  15.6.  Linear and nonlinear operators.  Dynamic system operator ,   15.6.  Linear and nonlinear operators.  Dynamic system operator ,   15.6.  Linear and nonlinear operators.  Dynamic system operator ,

as well as solving a nonlinear differential equation, at least

  15.6.  Linear and nonlinear operators.  Dynamic system operator .

A dynamic system whose operator is not linear is called a nonlinear system.

In practice, linear systems are very common. In connection with the linearity of these systems, the apparatus of the theory of random functions can be applied with great efficiency to the analysis of their errors. Just as the numerical characteristics of linear functions of ordinary random variables can be obtained from the numerical characteristics of the arguments, the characteristics of a random function at the output of a linear dynamic system can be determined if the operator of the system and the characteristics of a random function at its input are known.

Even more often than linear systems, in practice there are systems that are not strictly linear, but within certain limits that allow linearization. If random perturbations at the system input are small enough, then almost any system can be considered - within these small perturbations - as approximately linear, just as with sufficiently small random changes of the arguments, almost any function can be linearized.

Approximation of approximate linearization of differential equations is widely used in the theory of errors of dynamic systems.

In the future, we will consider only linear (or linearizable) dynamical systems and the corresponding linear operators.


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis