5. 3. Solved Problems

Example 3: Number of Negative Roots Given the characteristic equation below, select the number of roots which will be located to the right of the imaginary axis

(G- 2005-47)

s4+5s3-s2-17s+12=0

(a) One (b) Two (c) Three (d) Zero

Solution:

The characteristic equation is: s4+5s3-s2-17s+12=0 

From the first test of Routh analysis, it is found that the given system is unstable. To find the number of roots to the right of the imaginary axis, the second test of Routh analysis is to be done.

Elements of Routh array are calculated as follows:

Row 1 1 \(-1\) 12
2 5 \(-17\) 0
3 \(\dfrac{5\times(-1)-1\times(-17)}{5}=2.4\) \(\dfrac{5\times12-1\times0}{5}=12 \quad\) 0
4 \(\dfrac{2.4\times(-17)-5\times12}{2.4} = -42 \quad\) 0 0
5 \(\dfrac{(-42)\times(12)-2.4\times0}{(-42)} =12\) 0 0

The elements of the first column of Routh array are: \([1 \quad 5 \quad 2.4 \ -42 \quad 12]\). There are two changes here: one from 2.4 to \(-42\) (i.e., positive to negative), another from \(-42\) to 12 (i.e., negative to positive). Hence, there are two roots to the right of the imaginary axis.

s4 +5s3-s2-17s+12=0

>> p = [1 5 -1 -17 12];

>> roots(p)

ans = -4 -3 1 1

The standard procedure fails if we encounter any of the following situations in the formulation of the array.

  1. A row of all zeros appears.

  2. First element of a row, appearing in first column of the array is zero, but the entire row is not all zeros.