Routh-Hurwitz Criterion :
The Routh-Hurwitz criteria will focus on the denominator polynomial D(s).
Therefore, if N is odd, the top row will be all the odd coefficients. If N is even, the top row will be all the even coefficients. We can fill in the remainder of the Routh Array as follows:
Now, we can define all our b, c, and other coefficients, until we reach row s0. To fill them in, we use the following formulae:
And
For each row that we are computing, we call the left-most element in the row directly above it the pivot element. For instance, in row b, the pivot element is aN-1, and in row c, the pivot element is bN-1 and so on and so forth until we reach the bottom of the array.
To obtain any element, we negate the determinant of the following matrix, and divide by the pivot element:
Where:
In terms of k l m n, our equation is:
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Stability Criteria
The Routh-Hurwitz stability criterion provides a simple algorithm to
decide whether or not the zeros of a polynomial are all in the left half
of the complex plane (such a polynomial is called at times "Hurwitz"). A
Hurwitz polynomial is a key requirement for a linear continuous-time
time invariant to be stable (all bounded inputs produce bounded
outputs).
- Necessary stability conditions
- Conditions that must hold for a polynomial to be Hurwitz.
- Sufficient stability conditions
- Conditions that if met imply that the polynomial is stable. However, a polynomial may be stable without implying some or any of them.
Routh-Hurwitz Criteria
The Routh-Hurwitz criteria is comprised of three separate tests that
must be satisfied. If any single test fails, the system is not stable
and further tests need not be performed. For this reason, the tests are
arranged in order from the easiest to determine to the hardest.
The Routh Hurwitz test is performed on the denominator of the transfer function, the characteristic equation. For instance, in a closed-loop transfer function with G(s) in the forward path, and H(s) in the feedback loop, we have:
The Routh Hurwitz test is performed on the denominator of the transfer function, the characteristic equation. For instance, in a closed-loop transfer function with G(s) in the forward path, and H(s) in the feedback loop, we have:
If we simplify this equation, we will have an equation with a numerator N(s), and a denominator D(s):
The Routh-Hurwitz criteria will focus on the denominator polynomial D(s).
Routh-Hurwitz Tests
Here are the three tests of the Routh-Hurwitz Criteria. For convenience,
we will use N as the order of the polynomial (the value of the highest
exponent of s in D(s)). The equation D(s) can be represented generally
as follows:
- Rule 1
- All the coefficients ai must be present (non-zero)
- Rule 2
- All the coefficients ai must be positive (equivalently all of them must be negative, with no sign change)
- Rule 3
- If Rule 1 and Rule 2 are both satisfied, then form a Routh array from the coefficients ai. There is one pole in the right-hand s-plane for every sign change of the members in the first column of the Routh array (any sign changes, therefore, mean the system is unstable).
The Routh Array
The Routh array is formed by taking all the coefficients ai of D(s), and staggering them in array form. The final columns for each row should contain zeros:
Therefore, if N is odd, the top row will be all the odd coefficients. If N is even, the top row will be all the even coefficients. We can fill in the remainder of the Routh Array as follows:
Now, we can define all our b, c, and other coefficients, until we reach row s0. To fill them in, we use the following formulae:
And
For each row that we are computing, we call the left-most element in the row directly above it the pivot element. For instance, in row b, the pivot element is aN-1, and in row c, the pivot element is bN-1 and so on and so forth until we reach the bottom of the array.
To obtain any element, we negate the determinant of the following matrix, and divide by the pivot element:
Where:
- k is the left-most element two rows above the current row.
- l is the pivot element.
- m is the element two rows up, and one column to the right of the current element.
- n is the element one row up, and one column to the right of the current element.
In terms of k l m n, our equation is: