Routh–Hurwitz Stability: Root Counts, Array Construction, and Special Cases

All flashcards

What is a Hurwitz polynomial?

A polynomial whose roots all have strictly negative real parts.

How does a characteristic root relate to a continuous-time system mode?

A simple root $p$ corresponds to an exponential component $e^{pt}$; repeated roots can also produce polynomial factors in time.

What does the Routh–Hurwitz criterion test?

Whether a continuous-time characteristic polynomial has all roots in the open left half-plane, using coefficient-based calculations.

How do the Routh and Hurwitz methods differ in form?

Routh uses a recursive array and first-column signs; Hurwitz uses a coefficient matrix and leading principal determinants.

Name two tests used for discrete-time stability.

Examples include the Schur–Cohn criterion and the Jury test.

Quiz questions

1. For a regular Routh array with no exceptional zero cases, what do first-column sign changes count?Answer: Roots in the right half-plane.
2. In a Routh array, a row begins with zero but has other nonzero entries. What standard procedure allows the calculation to continue?Answer: Replace the leading zero temporarily with $\varepsilon>0$, then take the limit as $\varepsilon\to0^+$.
3. When an entire Routh-array row is zero, how is the auxiliary polynomial used to continue the procedure?Answer: Form it from the row above, differentiate it, and use the derivative coefficients.
4. For a third-order polynomial $a_3s^3+a_2s^2+a_1s+a_0$ with positive leading coefficient, which additional inequality is required for strict stability?Answer: $a_2a_1-a_3a_0>0$.
5. How is a generalized Sturm chain generated from the real polynomials $P_0(y)$ and $P_1(y)$?Answer: By successive Euclidean divisions that produce a sequence of remainders.