Routh Hurwitz Stability Criterion

For a continuous-time linear time-invariant system, asymptotic stability requires every characteristic root to have strictly negative real part. Any right-half-plane root is unstable. Roots on the imaginary axis require separate multiplicity analysis and do not give asymptotic stability. In network synthesis and control, the Hurwitz Criterion and the Routh test are related algebraic ways to determine root location without explicitly solving the polynomial. Together they are commonly called the Routh Hurwitz (R-H) Stability Criterion.

Hurwitz Criterion

Hurwitz determinants are formed from the real coefficients of the characteristic polynomial. Assume a non-zero leading coefficient and use a consistent sign normalisation. The characteristic equation is:

An n^th-order polynomial has n leading principal Hurwitz determinants.

Here is how to write determinants from the coefficients of the characteristic equation. Follow these steps for a k^th order characteristic equation:
Determinant one : The value of this determinant is given by |a₁| where a₁ is the coefficient of s^n-1 in the characteristic equation.
Determinant two : The value of this determinant is given by

Here number of elements in each row is equal to determinant number and we have determinant number here is two. The first row consists of first two odd coefficients and second row consists of first two even coefficients.
Determinant three : The value of this determinant is given by

Here number of elements in each row is equal to determinant number and we have determinant number here is three. The first row consists of first three odd coefficients, second row consists of first three even coefficients and third row consists of first element as zero and rest of two elements as first two odd coefficients.
Determinant four: The value of this determinant is given by,

Here number of elements in each row is equal to determinant number and we have determinant number here is four. The first row consists of first three four coefficients, second row consists of first four even coefficients, third row consists of first element as zero and rest of three elements as first three odd coefficients the fourth row consists of first element as zero and rest of three elements as first three even coefficients.

The pattern extends to the general Hurwitz matrix shown below:

For a real polynomial with a positive leading coefficient and no degree loss, all leading principal Hurwitz determinants must be positive for every root to lie strictly in the open left half-plane. A zero or negative determinant fails that strict Hurwitz condition and requires further root-location analysis.

Routh Stability Criterion

The Routh criterion arranges the real polynomial coefficients into a table. Under its standard assumptions, first-column sign changes count roots in the open right half-plane. No sign change, together with proper handling of zero cases, establishes that no roots lie there. Consider the characteristic equation:

1) Necessary coefficient check: After normalising the leading coefficient as positive, a strictly Hurwitz real polynomial must satisfy both conditions below:

1. Every coefficient must be real and have the same positive sign.
2. No coefficient may be zero. These conditions are necessary but are not sufficient for order three or higher.

2) Routh-array test: Construct the Routh array and inspect its first column. Use the following coefficient order:

• The first row contains coefficients of powers n, n – 2, n – 4 and so on: a₀ a₂ a₄ a₆ …
• The second row contains coefficients of powers n – 1, n – 3, n – 5 and so on: a₁ a₃ a₅ a₇ …
• The elements of third row can be calculated as:
  (1) First element : Multiply a₀ with the diagonally opposite element of next column (i.e. a₃) then subtract this from the product of a₁ and a₂ (where a₂ is diagonally opposite element of next column) and then finally divide the result so obtain with a₁. Mathematically we write as first element

(2) Second element: Use the same two-row determinant pattern for the next column. With the notation shown, combine a₀ and a₅ with a₁ and a₄, where a₄ is the adjacent even coefficient, then divide by a₁ using the sign convention in the displayed formula:

Continue this pattern across the third row.
The fourth row uses the two rows immediately above it.
(1) First element: With the page’s notation, combine b₁ and a₃ with a₁ and b₂, where b₂ is the adjacent third-row coefficient, then divide by b₁:

(2) Second element: Apply the same pattern to b₁, a₅, a₁ and b₃, with b₃ in the next column. The displayed expression uses a₁ as its denominator, so verify that denominator against the row recurrence before using the formula:

Continue across the fourth row.
Repeat the recurrence until the final row.
The number of sign changes in the first column equals the number of open right-half-plane roots. With a positive leading entry, no sign changes and no imaginary-axis roots give strict continuous-time stability.
The special zero cases require the following procedures.

(1) Zero first-column entry: If the first entry of a row is zero while another entry in that row is non-zero, direct recursion would divide by zero.
Replace the zero by a positive ε, complete the array and inspect first-column signs as ε tends to zero from the positive side. Count sign changes in the limiting sequence. Do not require every finite table entry to be positive; only the first-column sign pattern determines the right-half-plane root count.

(2) Complete row of zeros: A zero row indicates an even polynomial factor whose roots are symmetric about the origin. The roots may include imaginary-axis pairs, mirrored real roots or mirrored complex roots, so the zero row alone does not prove marginal stability. Form the auxiliary polynomial from the row above, differentiate it and use the derivative coefficients to replace the zero row. Continue the Routh array and factor or solve the auxiliary polynomial. The system is marginally stable only if any imaginary-axis roots are simple and all remaining roots lie strictly in the left half-plane; repeated imaginary-axis roots or any right-half-plane roots make it unstable.