The transfer function of a typical tape-drive system is given by KG(s) = K(s + 4)/ s(s + 0.5)(s + 1)(s2 = 0.4s + 4)
where time is measured in milliseconds. Using Routh's stability criterion, determine the range of K for which this system is stable when the characteristic equation is 1 + KG(s) = 0.

the range of K can be said to be :  -3.59 < K< 0.35

Explanation:

The transfer function of a typical tape-drive system is given by;

calculating the characteristics equation; we have:

1 + KG(s) = 0

We can compute a Simulation Table for the Routh–Hurwitz stability criterion Table as  follows:

            1                     5.1                          2+ K

           1.9                   6.2                           4K

            1.83                      0

              4K                         0

S                  0                            0

We need to understand that in a given stable system; all the elements in the first column is usually greater than zero

So;

11.34  - 1.9(X) > 0

7.54 +2.1 K > 0

2.1 K > - 7.54

K > - 7.54/2.1

K > - 3.59

Also

4K >0

K > 0/4

K > 0

Similarly;

XY - 7.32 K > 0

0.54(2.1K+7.54)>7.32 K

11.45 K < 4.07

K < 4.07/11.45

K < 0.35

Thus the range of K can be said to be :  -3.59 < K< 0.35

energy can cross the boundaries of a closed system in the form of heat and mass

explanation: