Q4Control System

(a) Routh-Hurwitz Stability Analysis

Given characteristic equation (as printed): s⁶+s⁵-2s⁴-3s³-7s³-4s²-4=0. Note that this equation as given contains two separate s³ terms (-3s³ and -7s³), which is not a valid standard polynomial form — combining these (as they must represent a single s³ coefficient, likely split across a line/OCR artifact in the original source) gives the corrected characteristic equation:

Constructing the Routh array from the coefficients (1, 1, -2, -10, -4, 0, -4):

Row-by-row derivation: s⁴ row: b1=(1×(-10)-1×(-2))/1 = -8... wait, using the standard cross-multiplication formula b1 = (s5_col1×s6_col2 - s6_col1×s5_col2)/s5_col1 = (1×(-2)-1×(-10))/1 = (-2+10)/1 = 8; b2 = (1×(-4)-1×0)/1 = -4; b3 = (1×(-4)-1×0)/1 = -4. s³ row: c1 = (8×(-10)-1×(-4))/8 = (-80+4)/8 = -9.5; c2 = (8×0-1×(-4))/8 = 4/8 = 0.5. s² row: d1 = ((-9.5)×(-4)-8×0.5)/(-9.5) = (38-4)/(-9.5) = 34/(-9.5) = -3.579; d2 = ((-9.5)×(-4)-8×0)/(-9.5) = 38/(-9.5) = -4. s¹ row: e1 = ((-3.579)×0.5-(-9.5)×(-4))/(-3.579) = (-1.789-38)/(-3.579) = -39.79/(-3.579) = 11.12. s⁰ row: f1 = -4 (brought down directly, as the last non-zero term).

First column of the Routh array: 1, 1, 8, -9.5, -3.579, 11.12, -4.

Counting sign changes: examining consecutive signs in this first column — (+1, +1): no change; (+1, +8): no change; (+8, -9.5): 1 change; (-9.5, -3.579): no change; (-3.579, +11.12): 2nd change; (+11.12, -4): 3rd change. Total: 3 sign changes.

Conclusion: since there are 3 sign changes in the first column of the Routh array, the characteristic equation has exactly 3 roots located in the right-half of the s-plane (confirmed independently by direct numerical root-finding, which yields roots at approximately s = 2.259, s = 0.263±j0.599 — three roots with positive real parts — along with s = -1.393±j1.484 and s = -1 in the left-half plane). Since one or more roots lie in the right-half plane, the system described by this characteristic equation is unstable.

(b) Stability, Its Importance, and Absolute vs Relative Stability

Stability of a control system, in the Bounded-Input-Bounded-Output (BIBO) sense most commonly used in linear control system analysis, is the property that guarantees the system's output remains bounded for every bounded input, and equivalently (for a linear system) requires that all roots (poles) of the system's characteristic equation lie strictly in the left half of the s-plane — a system with even a single pole on the imaginary axis is marginally (critically) stable (producing sustained, non-decaying oscillations for certain inputs), and a system with any pole in the right-half plane is unstable, producing an output that grows without bound over time.

Importance of stability: stability is the single most fundamental and essential requirement for any practical control system, since an unstable system's output will grow without bound (in practice, until some physical component saturates, overheats, or fails catastrophically), rendering the system completely unusable and potentially dangerous — no other performance specification (accuracy, speed of response, disturbance rejection) has any practical meaning or value if the underlying system is not first stable; stability analysis (via Routh-Hurwitz, root locus, Bode, or Nyquist methods) is therefore always the first and most critical step in any control system design or analysis process, performed before any other performance optimization is even attempted.

Absolute stability vs relative stability: absolute stability is a simple, binary classification — a system is determined to be either stable or unstable, with no indication of how close to the boundary of instability a stable system actually is (a system could be technically stable yet have poles located extremely close to the imaginary axis, giving a very slow, poorly-damped, oscillatory response that is stable in the strict mathematical sense but practically undesirable). Relative stability, by contrast, provides a quantitative measure of how far a stable system's actual operating point is from the boundary of instability — quantified through measures such as gain margin and phase margin (from Bode/Nyquist analysis, indicating how much additional gain or phase lag could be tolerated before the system becomes unstable), the damping ratio ζ (indicating how oscillatory the transient response is), or the specific real-part magnitude of the dominant closed-loop poles (indicating how quickly transients decay) — relative stability measures are essential in practical control system design, since a design goal is typically not merely to achieve bare absolute stability, but to achieve stability with an adequate margin, ensuring robust, well-damped performance despite real-world parameter variations, modeling uncertainties, and external disturbances that a marginally-stable design would not tolerate reliably.

Root Distribution via the Routh-Hurwitz Special Case Method

The result above (3 right-half-plane roots) can also be cross-checked using the Routh-Hurwitz special technique for locating the number of roots in each half-plane without full numerical root-finding: for a 6th-order polynomial, if the first column of the completed Routh array has p sign changes reading top to bottom, then exactly p roots lie in the right-half s-plane, and the remaining (6-p) roots lie in the left-half plane (assuming no row is identically zero, which is the case here since no auxiliary-equation condition arose). With p=3 sign changes found in the first column (1,1,8,-9.5,-3.579,11.12,-4), exactly 3 roots lie in the right half-plane and the remaining 3 lie in the left half-plane — consistent with the independent numerical root-finding cited above (2.259; 0.263±j0.599 in the RHP; -1.393±j1.484; -1 in the LHP), which itself serves as the definitive confirmation of the Routh-array sign-change count.

It is instructive to note what would need to change for this system to be rendered stable: since the instability here arises from an inherently unstable open-loop/characteristic-polynomial structure (rather than merely excessive loop gain in an otherwise well-behaved system), no simple gain adjustment alone could stabilize it — a full compensator redesign (adding lead/lag networks, or relocating poles via full state feedback) would be required to shift the dominant right-half-plane roots into the left-half plane, in contrast to gain-limited instability cases (such as the root-locus problem elsewhere in this paper) where a simple reduction in loop gain K below a critical Kmax is sufficient to restore stability.

Relation to Root Locus and Frequency-Domain Methods

The Routh-Hurwitz criterion used here is one of three principal classical stability-analysis techniques taught in this subject, the other two being the root-locus method and the Nyquist/Bode frequency-response methods; all three are fundamentally equivalent in the sense that they each determine whether any characteristic-equation roots lie in the right-half s-plane, but they offer different practical advantages. Routh-Hurwitz is purely algebraic and gives a fast yes/no stability verdict (plus the exact count of unstable roots) directly from the coefficients, without needing to actually solve for the roots or sweep any parameter, making it ideal for quickly checking a single, fixed characteristic equation such as the one given here. Root locus, by contrast, shows graphically how the closed-loop pole locations move as a single parameter (typically gain K) is varied continuously, making it more suited to design problems where the gain is still a free parameter to be chosen. Frequency-response methods (Bode, Nyquist) work directly from measured or modeled open-loop frequency response and additionally yield quantitative relative-stability measures (gain margin, phase margin) that neither Routh-Hurwitz nor root locus directly provide. For the present problem, since the characteristic equation is completely fixed (no free parameter K to sweep), Routh-Hurwitz is the most direct and efficient method, and this is precisely why it was the technique specified for this question.