Routh–Hurwitz stability criterion

In the <a href="/facts/Control_theory/aIuERpn6">control system theory</a>, the Routh–Hurwitz stability criterion is a mathematical test that is a <a href="/facts/Necessary_and_sufficient/G45aDCY9">necessary and sufficient</a> condition for the <a href="/facts/Stable_polynomial/lxJv9NSL">stability</a> of a <a href="/facts/Linear_time-invariant_system/kEmj0QDC">linear time-invariant</a> (LTI) <a href="/facts/Dynamical_system/782gREl5">dynamical system</a> or <a href="/facts/Control_system/8CEjIDlS">control system</a>. A <a href="/facts/Stability_theory/Su5sHnxs">stable system</a> is one whose output signal is bounded; the position, velocity or energy do not increase to infinity as time goes on. The Routh test is an efficient recursive algorithm that English mathematician <a href="/facts/Edward_John_Routh/aEHbbUQK">Edward John Routh</a> proposed in 1876 to determine whether all the <a href="/facts/Root_of_a_function/mlK3dhoT">roots</a> of the <a href="/facts/Characteristic_polynomial/7m8roDqo">characteristic polynomial</a> of a <a href="/facts/Linear_system/uEl20Kc1">linear system</a> have negative real parts. German mathematician <a href="/facts/Adolf_Hurwitz/urNgHh58">Adolf Hurwitz</a> independently proposed in 1895 to arrange the coefficients of the polynomial into a square matrix, called the <a href="/facts/Routh%25E2%2580%2593Hurwitz_matrix/txJkIFF7">Hurwitz matrix</a>, and showed that the polynomial is stable if and only if the sequence of determinants of its principal submatrices are all positive. The two procedures are equivalent, with the Routh test providing a more efficient way to compute the Hurwitz determinants (
 
 
 
 
 Δ
 
 i
 
 
 
 
 {\displaystyle \Delta _{i}}
 
) than computing them directly. A polynomial satisfying the Routh–Hurwitz criterion is called a <a href="/facts/Hurwitz_polynomial/t2Ys4jtI">Hurwitz polynomial</a>.
The importance of the criterion is that the roots p of the characteristic equation of a <a href="/facts/Linear_system/uEl20Kc1">linear system</a> with negative real parts represent solutions ept of the system that are stable (<a href="/facts/BIBO_stability/RxD8h2f5">bounded</a>). Thus the criterion provides a way to determine if the <a href="/facts/Equations_of_motion/25msr1vE">equations of motion</a> of a <a href="/facts/Linear_system/uEl20Kc1">linear system</a> have only stable solutions, without solving the system directly. For discrete systems, the corresponding stability test can be handled by the Schur–Cohn criterion, the <a href="/facts/Jury_stability_criterion/8hh2uGYN">Jury test</a> and the <a href="/facts/Bistritz_stability_criterion/gYYuLf7t">Bistritz test</a>. With the advent of computers, the criterion has become less widely used, as an alternative is to solve the polynomial numerically, obtaining approximations to the roots directly.
The Routh test can <a href="/facts/Derivation_of_the_Routh_array/KgxM6NaE">be derived</a> through the use of the <a href="/facts/Euclid%2527s_algorithm/wApql29c">Euclidean algorithm</a> and <a href="/facts/Sturm%2527s_theorem/fn2MG3wu">Sturm's theorem</a> in evaluating <a href="/facts/Cauchy_index/s48jHJhw">Cauchy indices</a>. Hurwitz derived his conditions differently.

Routh–Hurwitz stability criterion open-in-new

Routh–Hurwitz stability criterion