You get a bonus - 1 coin for daily activity. Now you have 1 coin

16.3. Linear transformations of random functions defined by canonical decompositions

Lecture



Let the input of some linear system   16.3.  Linear transformations of random functions defined by canonical decompositions random function arrives   16.3.  Linear transformations of random functions defined by canonical decompositions (fig. 16.3.1).

  16.3.  Linear transformations of random functions defined by canonical decompositions

Fig. 16.3.1.

The system converts the function   16.3.  Linear transformations of random functions defined by canonical decompositions by means of a linear operator and at the output we get a random function

  16.3.  Linear transformations of random functions defined by canonical decompositions . (16.3.1)

Suppose a random function   16.3.  Linear transformations of random functions defined by canonical decompositions given by its canonical decomposition:

  16.3.  Linear transformations of random functions defined by canonical decompositions . (16.3.2)

We determine the response of the system to this effect. Since the system operator is linear,

  16.3.  Linear transformations of random functions defined by canonical decompositions . (16.3.3)

Considering the expression (16.3.3), it is easy to verify that it represents nothing more than a canonical decomposition of a random function   16.3.  Linear transformations of random functions defined by canonical decompositions with mathematical expectation

  16.3.  Linear transformations of random functions defined by canonical decompositions (16.3.4)

and coordinate functions

  16.3.  Linear transformations of random functions defined by canonical decompositions . (16.3.5)

Thus, in the linear transformation of the canonical expansion of a random function   16.3.  Linear transformations of random functions defined by canonical decompositions canonical decomposition of a random function is obtained   16.3.  Linear transformations of random functions defined by canonical decompositions , and the mean and coordinate functions undergo the same linear transformation.

If random function   16.3.  Linear transformations of random functions defined by canonical decompositions at the output of the linear system obtained in the form of canonical decomposition

  16.3.  Linear transformations of random functions defined by canonical decompositions , (16.3.6)

then its correlation function and variance are quite simple:

  16.3.  Linear transformations of random functions defined by canonical decompositions , (16.3.7)

  16.3.  Linear transformations of random functions defined by canonical decompositions . (16.3.8)

This makes it particularly convenient canonical expansions compared to any other decompositions in elementary functions.

Let us consider in more detail the application of the method of canonical expansions to the definition of the reaction of a dynamical system to a random input action, when the operation of the system is described by a linear differential equation, in the general case with variable coefficients. We write this equation in operator form:

  16.3.  Linear transformations of random functions defined by canonical decompositions . (16.3.9)

According to the above rules of linear transformations of random functions, the expectation of exposure and reaction must satisfy the same equation:

  16.3.  Linear transformations of random functions defined by canonical decompositions . (3/16/10)

Similarly, each of the coordinate functions must satisfy the same differential equation:

  16.3.  Linear transformations of random functions defined by canonical decompositions ,   16.3.  Linear transformations of random functions defined by canonical decompositions . (3/16/11)

Thus, the problem of determining the response of a linear dynamic system to a random effect has been reduced to the usual mathematical problem of integration   16.3.  Linear transformations of random functions defined by canonical decompositions ordinary differential equations containing ordinary, non-random functions. Since, when solving the basic problem of analyzing a dynamic system — determining its response to a given effect — is the task of integrating a differential equation that describes the operation of the system, it is solved in one way or another, when solving equations (16.3.10) and (16.3.11) new mathematical difficulties does not occur. In particular, the same integrating systems or modeling devices that are used to analyze the operation of the system without random perturbations can be successfully applied to solve these equations.

It remains to clarify the question of the initial conditions under which equations (16.3.10) and (16.3.11) should be integrated.

First, we consider the simplest case when the initial conditions for a given dynamical system are nonrandom. In this case, when   16.3.  Linear transformations of random functions defined by canonical decompositions conditions must be met:

  16.3.  Linear transformations of random functions defined by canonical decompositions (3/16/12)

Where   16.3.  Linear transformations of random functions defined by canonical decompositions - non-random numbers.

Conditions (16.3.12) can be written more compactly:

  16.3.  Linear transformations of random functions defined by canonical decompositions   16.3.  Linear transformations of random functions defined by canonical decompositions (16.3.13)

understanding the term “derivative of zero order”   16.3.  Linear transformations of random functions defined by canonical decompositions the function itself   16.3.  Linear transformations of random functions defined by canonical decompositions .

Let us find out under what initial conditions the equations (16.3.10) and (16.3.11) should be integrated. For this we find   16.3.  Linear transformations of random functions defined by canonical decompositions derivative of the function   16.3.  Linear transformations of random functions defined by canonical decompositions and put in it   16.3.  Linear transformations of random functions defined by canonical decompositions :

  16.3.  Linear transformations of random functions defined by canonical decompositions .

Considering (16.3.12), we have:

  16.3.  Linear transformations of random functions defined by canonical decompositions . (3/16/14)

Since the value   16.3.  Linear transformations of random functions defined by canonical decompositions not random, the variance of the left side of equality (16.3.14) should be zero:

  16.3.  Linear transformations of random functions defined by canonical decompositions . (3/16/15)

Since all dispersions   16.3.  Linear transformations of random functions defined by canonical decompositions values   16.3.  Linear transformations of random functions defined by canonical decompositions are positive, then equality (16.3.15) can be realized only when

  16.3.  Linear transformations of random functions defined by canonical decompositions (3/16/16)

for all   16.3.  Linear transformations of random functions defined by canonical decompositions .

Substituting   16.3.  Linear transformations of random functions defined by canonical decompositions in the formula (16.3.14), we get:

  16.3.  Linear transformations of random functions defined by canonical decompositions . (16.3.17)

From equality (16.3.17) it follows that the equation (16.3.10) for the expectation must be integrated under the given initial conditions (16.3.12):

  16.3.  Linear transformations of random functions defined by canonical decompositions (3/16/18)

As for equations (16.3.11), they should be integrated with zero initial conditions:

  16.3.  Linear transformations of random functions defined by canonical decompositions

  16.3.  Linear transformations of random functions defined by canonical decompositions . (3/16/19)

Consider a more complicated case when the initial conditions are random:

  16.3.  Linear transformations of random functions defined by canonical decompositions (3/16/20)

Where   16.3.  Linear transformations of random functions defined by canonical decompositions - random variables.

In this case, the response at the output of the system can be found as a sum:

  16.3.  Linear transformations of random functions defined by canonical decompositions , (16.3.21)

Where   16.3.  Linear transformations of random functions defined by canonical decompositions - solution of the differential equation (16.3.9) with zero initial conditions;   16.3.  Linear transformations of random functions defined by canonical decompositions - the solution of the same differential equation, but with a zero right-hand side under given initial conditions (16.3.20). As is known from the theory of differential equations, this solution is a linear combination of initial conditions:

  16.3.  Linear transformations of random functions defined by canonical decompositions , (16.3.22)

Where   16.3.  Linear transformations of random functions defined by canonical decompositions - non-random functions.

Decision   16.3.  Linear transformations of random functions defined by canonical decompositions can be obtained by the above method in the form of canonical decomposition. Correlation function of a random function   16.3.  Linear transformations of random functions defined by canonical decompositions can be found by the usual methods of adding random functions (see   16.3.  Linear transformations of random functions defined by canonical decompositions 15.8).

It should be noted that in practice there are very often cases where, for moments of time sufficiently distant from the beginning of the random process, the initial conditions no longer affect its flow: the transients caused by them have time to die out. Systems with this property are called asymptotically stable. If we are interested in the reaction of an asymptotically stable dynamical system on time intervals sufficiently distant from the beginning, then we can restrict ourselves to studying the solution   16.3.  Linear transformations of random functions defined by canonical decompositions obtained with zero initial conditions. For sufficiently remote from the initial moments of time, this solution will be valid under any initial conditions.


Comments


To leave a comment
If you have any suggestion, idea, thanks or comment, feel free to write. We really value feedback and are glad to hear your opinion.
To reply

Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis