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There are frequently questions about stability that you need to answer. Some of the questions might be.
To apply the Routh criterion, you need to form an array from the polynomial coefficients. Then, after some computations, you get information that will let you determine how many roots of the polynomial are in the LHP. We can understand this best with an example.
s^{3} + s^{2} + 3s + 10
Looking at this polynomial, it looks innocent enough. All of the coefficients are positive integers. You might not expect this to be the denominator of an unstable system. In fact, this polynomial can be factored into this form.
(s + 2)(s^{2}  s + 5)
The quadratic factor has two roots in the RHP.
We will work with this polynomial, and apply the Routh criterion to it
as an example. To begin, we put the coefficients of the original
polynomial into an array (the Routh array)









The last two rows are filled in by doing some computation. Call the "1" colored green the pivot element. The computations start by multiplying the two entries colored green  the lower "1"  the pivot element and the "3". Call that result, R1. Then you multiply the two entries in blue. Call that result, R2. Then you divide by the green "1"  the pivot element  and place the result in the leftmost column. (Since it's a "1" here it won't make any difference, but it won't always be a "1".) That computation is given by:
Result = [R1  R2]/[Pivot element]
The result is shown below  with the result
forming a new entry in the s^{1} row. Since there
are no more elements in the first two rows, that ends the computations
for the s^{1} row.










Finally, we need to get the elements in the
last row. As in the preceding row, there will really be only one
element in this example, but larger systems will require more than that.
The new pivot element will be the first element in the last row modified.
That's the "7". You need to multiply 7 times 10 and divide by 7.
That's the formal algorithm, and  in this case  it just moves the 10
down to the last spot in the first column.










10 
Now, what Routh showed was that the sign changes in the first column tell you how many roots the polynomial has in the RHP. In this case, there are two sign changes in the first column.
s^{4} + 10s^{3} + 35s^{2} + 50s + 124
We form the array in the same way (using
coefficients of every other power in each row), but we now have three entries
in the top row  the coefficients of s^{4},
s^{2},
and s^{0}.

















We proceed the same way, doing the cross
multiplication. However, in this case, we need to examine how we
get 124 in the second entry in the s^{2} row.

















To get the 124 in the second entry in the s^{2} row we use the same pivot element. It will always be the first element in the last row computed (or entered). In this case it will be "10". What we do is multiply the 10 and the 124. Notice how we multiply the first and the third column elements. (Before, we multiplied the first and the second column elements.) We always use the same pivot element. Next, if there had been an element in the box below the 124 entry, we would have multipled that by 1  the very first entry at the top left in the table. The computations are:


















The computations are:
Consider this polynomial.
s^{4} + 4s^{3} + 6s^{2} + 4s + 30
Form the array as above
(using coefficients of every other power in each row), with three entries
in the top row  the coefficients of s^{4},
s^{2},
and s^{0}.

















What is the the value
of the term indicated by "??" ?
Is there another element in the third row  the s^{2} row?

















What is the the value of the term indicated by "??" ?
And, we hope you got that one right also. If not, remember the following.Is there another element in the fourth row  the s^{1} row?


















You should realize  from the preceding examples  that the last element is just "130". It's the element in the s^{2} row  at the end.
And, finally, you get this table:


















And there are two sign changes, so there are two roots of the polynomial in the RHP.
The polynomial in the example above can be related to a closed loop system. Consider the block diagram below.
Assume that the gain, K, is variable, and that G(s) is given by:
G(s) = 1/[(s + 1)^{4}]
Then, the closed loop transfer function is given by:
G_{CL}(s) = K/[(s + 1)^{4} + K]
The polynomial in the example is the closed loop denominator for K = 29.
Now, think about what happens when K is variable. Then the closed loop denominator is given by:
s^{4} + 4s^{3} + 6s^{2} + 4s + 1 + K
We can form the Routh
array for this polynomial.


















The critical element is (16  4K)/5. That element is negative whenever K is larger than 4, and when that element is negative, the closed loop system has poles in the RHP, and the closed loop system is unstable.
The interesting and useful conclusion is that the Routh array can be used to determine gain limits for stability  at least for systems small enough that you can do the computations for the Routh array.
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