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.
1st step.
You have to obtain the required number of random function’s realizations.
2nd step.
You have to define the values of expectation, variance and correlation function.
3rd 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.
