Lecture 2.10

You have to define the properties of stationary and not stationary processes.

(Time-invariant random processes).

№6

6.2 Consider the methods to define the spectral density using the realizations of a random process obtained as a result of experiments.

1^st step.

You have to obtain the required number of random function’s realizations.

2^nd step.

You have to define the values of expectation, variance and correlation function.

3^rd step.

You have to use the following formulas of the Fourier transform (which allows make the transfer from a time function to Fourier representation) (Lecture 2.12):

(25.1)

(25.2)

The power spectral density (PSD) and the correlation function of random processes are connected via the direct and inverse double-sided Fourier transform.

(25.10)

(25.12)

6.3 Define stability indicators of the control system with the following roots of its characteristic equation:

Lecture 2.5:

In terms of linear systems, we recognize that the stability requirement may be defined in terms of the location of the poles of the closed-loop transfer function. The closed-loop system transfer function is written as

(18.2)

where is the characteristic equation whose roots are the poles of the closed-loop system.

This transfer function includes M zeros, N poles at the origin, Q poles on the real axis, and R pairs of complex conjugate poles.

The output response for an impulse function input (when N = 0) is then

(18.3)

where and are constants that depend on , , , K, and . To obtain a bounded response, the poles of the closed-loop system must be in the left-hand portion of the S-plane.

Thus, a necessary and sufficient condition fora feedback system to be stable is that all the poles of the system transfer function have negative real parts.