Q4Control System

The Routh-Hurwitz criterion determines the number of roots of a polynomial that lie in the right-half s-plane purely from its coefficients, without needing to factor or numerically solve the polynomial. It is constructed as a triangular array: the first two rows are formed directly from the characteristic equation's coefficients (alternating terms in each row), and every subsequent row is computed from the two rows immediately above it using a standard cross-multiplication rule, continuing until the array has as many rows as the polynomial has degree, plus one. The key theorem is that the number of sign changes encountered while reading down the first column of the completed array exactly equals the number of characteristic roots in the right-half plane — zero sign changes confirms stability, while any sign change (or a special case such as a zero appearing in the first column) signals instability or requires special handling.

Characteristic equation: s⁵ + s⁴ + 2s³ + 0·s² + 2s + 4 = 0 (note the s² coefficient is 0, already a warning sign per the necessary condition for stability). Constructing the Routh array:

Row s³ coefficients: b1 = (1×2 - 1×0)/1 = 2, b2 = (1×2 - 1×4)/1 = -2. Row s³: [2, -2]. Row s² coefficients: c1 = (2×0 - 1×(-2))/2 = (0+2)/2 = 1, c2 = (2×4 - 1×0)/2 = 4. Row s²: [1, 4]. Row s¹: d1 = (1×(-2) - 2×4)/1 = (-2-8)/1 = -10. Row s¹: [-10]. Row s⁰: e1 = 4 (from cross-multiplication with the s² row's second term and zero, effectively carrying down 4).

The first column is: 1, 1, 2, 1, -10, 4. There are two sign changes in this column (from +1 at the s² row to -10 at the s¹ row, and back from -10 to +4 at the s⁰ row). By the Routh-Hurwitz criterion, the number of sign changes in the first column equals the number of roots of the characteristic equation lying in the right-half s-plane. Since there are 2 sign changes, the system has 2 roots in the right-half plane, and therefore the system is unstable.

Note on the initial zero coefficient. It is worth observing that this instability could actually have been predicted immediately, without constructing the full array, simply from the necessary (but not sufficient) condition for stability: the characteristic equation s⁵+s⁴+2s³+0·s²+2s+4 = 0 is missing its s² term (coefficient is zero), which alone is sufficient to conclude the system cannot be stable, since a necessary condition for all roots to lie in the left-half plane is that every power of s from sⁿ down to s⁰ must be present with a non-zero, same-signed coefficient. The full Routh array construction carried out above is nonetheless the standard rigorous method, since it additionally reveals precisely how many roots (here, 2) lie in the right-half plane, which the quick necessary-condition check alone cannot provide.