The feedback of the proportional control system with the parameters of assignment 7 has the properties of the first grad

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The feedback of the
proportional control system with the parameters of assignment 7 has
the properties of the first grade aperiodic element with time
constant T = 0,m sec. Estimate stability of CS by means of
Routh-Hurwitz criteria. If CS is not stable, then using the Routh
table determine what value of kas will result in the system’
stability. Calculate the stationary error at this value of kas.
Theoretical
fundamentals
The differential equations are often
of higher grade. Thus the solution of such equations is
complicated, therefore the criteria of stability estimation are
applied. These criteria could be both algebraic and with the
application of complex values. One of the mostly applied algebraic
criteria is Routh-Hurwitz criteria, that calculates the artificial
coefficients of the table. For example, the following
characteristic equation
0,07s^3 +0,79s^2
+s +9 =0.
The first two rows of
the table are formed with the coefficients of the equation: the
first row with the highest grade coefficient and each further
second lower coefficient; the second row is formed with the second
after the highest and again further each second after it, etc.
Thus,
0,07
1 0
0,79
9 0.
The first artificial
coefficient is the difference of the cross product of the first and
second rows values divided by the very left coefficient of the
second row:
(0,79.1 - 0,07.9)
/0,79 =0,202 (0,79.0 - 0,07.0) /0,79=0 0
The second artificial
coefficients row is formed in a similar way to the previous, but
with the coefficients of the second real and previous artificial
rows:
(0,202.9 - 0,79.0)
/0,202 =9 (0,202.0 - 0,79.0) /0,202=0 0
0
0
0.
Therefore, finally the
table looks like the following:
0,07
1 0
0,79
9 0
0,202
0 0
9
0 0.
If all the
coefficients of the left column are positive then the system is
stable.
NOTE: DON'T SOLVE
the assignment below, solve the one above. Assignment 7
is just for parameters which was mentioned above:
Assignment 7. A
proportional control system with the transfer function of plant Wob
=3 / [1 + 0,n.s + 0,(1+m).s2], transfer
function of feedback Was = (1
– 0,n) and amplification factor of
proportional controller kp= (20 + m). Estimate stability of the
system. Calculate the roots of the characteristic equation and, if
an oscillation process exists, calculate the period of the
escillations.
For the stable system calculate the stationary value and amplitude
of the oscillations maximum peak, for the input signal X0 =
(5 + n)V.
Calculate kp , necessary for a stationary error of 1%, as well as
calculate the roots for this case, period of oscillations and
amplitude of the oscillations maximum peak.