Q6Control System

(a) M and N Circles

M-circles and N-circles are families of curves plotted on the complex G(jω) plane (the same plane used for polar/Nyquist plots), used as a graphical tool to directly determine the closed-loop frequency response magnitude and phase of a unity-feedback system from its open-loop frequency response, without needing to explicitly compute the algebraic closed-loop transfer function C(jω)/R(jω) = G(jω)/(1+G(jω)) at every frequency of interest.

M-circles are loci of points in the G(jω)-plane where the closed-loop magnitude |C(jω)/R(jω)| = M is constant, for a specified value of M; each different value of M corresponds to a different circle (except M=1, which is a straight line), with the family of M-circles all centered on the real axis, becoming smaller and shifting further to the left as M increases above 1, and smaller and shifting to the right as M decreases below 1.

N-circles are loci of points where the closed-loop phase angle ∠[C(jω)/R(jω)] = α is constant, for a specified phase value α; each N-circle passes through both the origin and the point (-1,0) in the G-plane, with its center located on the vertical line through (-0.5,0) at a height determined by the specific phase angle α represented.

Physical significance in stability analysis: by overlaying the M and N circle families directly onto the open-loop polar/Nyquist plot G(jω), the closed-loop frequency response (both magnitude M(ω) and phase α(ω) at every frequency ω) can be read off graphically simply by noting which M-circle and N-circle the open-loop polar plot curve passes through at each frequency point, without needing separate algebraic calculation. This provides a powerful graphical technique for control system design: it directly reveals the closed-loop resonant peak Mr (the maximum M-circle value the polar plot touches, indicating the peak closed-loop frequency-response magnitude and its associated resonant frequency ωr), and shows how close the open-loop polar plot approaches the critical point (-1,0) — since M-circles become infinitely large (approaching a straight vertical line) exactly at the critical point (-1,0), a polar plot passing very close to this point indicates a very high resonant peak Mr, corresponding to a lightly-damped, poorly-stable closed-loop system approaching the boundary of instability, giving M and N circles direct value as a graphical relative-stability assessment tool in addition to their primary use for closed-loop frequency-response determination.

(b) Nyquist Stability Investigation for G(s)H(s) = K/[s(1+sT1)(1+sT2)]

Given: open-loop transfer function G(s)H(s) = K/[s(1+sT1)(1+sT2)], a Type-1 system (one pole at the origin) with two additional real poles at s=-1/T1 and s=-1/T2 (assuming T1, T2 > 0, both poles are in the left-half plane).

Step 1 — Open-loop poles on/inside the Nyquist contour: the system has one pole at the origin (on the imaginary axis, requiring the standard small semicircular indentation around the origin in the Nyquist D-contour) and two poles at -1/T1, -1/T2, both in the left-half plane (i.e., not enclosed by the standard Nyquist contour). Therefore, the number of open-loop poles in the right-half plane, P, is P=0.

Step 2 — Nyquist plot behavior: substituting s=jω: G(jω)H(jω) = K/[jω(1+jωT1)(1+jωT2)]. As ω→0⁺, the magnitude →∞ and the phase → -90° (dominated by the pole at the origin); as ω→∞, the magnitude →0 and the phase → -270° (-90° from the origin pole, plus -90° from each of the two first-order lag terms). The polar plot therefore starts (at ω→0⁺) at infinite magnitude along the -90° direction, and spirals inward, crossing the negative real axis at some finite frequency before continuing to spiral into the origin as ω→∞ with phase approaching -270°.

Step 3 — Real-axis crossing point: the real-axis crossing (where the imaginary part of G(jω)H(jω) equals zero, i.e., where the phase angle equals exactly -180°) occurs at the frequency where the sum of the individual phase contributions from the two first-order lag terms equals 90° (since the pole at the origin already contributes a fixed -90°): tan⁻¹(ωT1)+tan⁻¹(ωT2) = 90°, which occurs at ω = 1/√(T1T2).

Substituting this crossing frequency back into the magnitude expression and simplifying (using the fact that at this frequency, ωT1·ωT2 = T1T2×(1/(T1T2)) = 1, giving a helpful algebraic simplification), the magnitude at the real-axis crossing works out to:

Step 4 — Applying the Nyquist stability criterion: since P=0 (no open-loop poles in the right-half plane), the Nyquist criterion requires that the number of closed-loop poles in the right-half plane, Z, equals the number of clockwise encirclements N of the critical point (-1,0) by the Nyquist plot, plus P: Z=N+P=N. For Z=0 (closed-loop stability), the Nyquist plot must make zero net clockwise encirclements of the (-1,0) point — for this particular system's polar plot shape (spiraling inward from -90° at ω→0⁺ toward -270° at ω→∞, crossing the negative real axis exactly once at the frequency found above), this requires that the real-axis crossing point lies to the right of (i.e., has smaller magnitude than) the critical point (-1,0):

Conclusion: the closed-loop system is stable if and only if the gain K satisfies K < (T1+T2)/(T1T2); if K exceeds this critical value, the Nyquist plot would encircle the (-1,0) point once in the clockwise direction (N=1), giving Z=1 closed-loop pole in the right-half plane, and the system would become unstable. This result is entirely consistent with (and can be independently verified using) the Routh-Hurwitz criterion applied to the corresponding closed-loop characteristic equation s(1+sT1)(1+sT2)+K=0, which would yield the identical maximum stable gain condition K < (T1+T2)/(T1T2) through the algebraic Routh array method, confirming the two stability-analysis techniques (Nyquist and Routh-Hurwitz) agree, as they must, since both determine the identical underlying stability property of the same system.

Verification via the Routh Array

Expanding the closed-loop characteristic equation s(1+sT1)(1+sT2)+K=0 explicitly: s(1+sT1+sT2+s²T1T2)+K = T1T2s³+(T1+T2)s²+s+K=0. Constructing the Routh array from coefficients (T1T2, T1+T2, 1, K):

For stability, all first-column entries must be positive (assuming T1,T2>0, so T1T2>0 and T1+T2>0 automatically): the s¹ row requires (T1+T2)-KT1T2>0, i.e., K<(T1+T2)/(T1T2), and the s⁰ row requires K>0 — exactly reproducing the identical stability condition derived from the Nyquist criterion above, confirming both methods agree perfectly, as they must for any linear system, since both ultimately test for the same underlying condition (absence of right-half-plane characteristic roots).

Practical Significance of the Two Approaches

This agreement is more than a coincidental check — it highlights the complementary practical roles of the two techniques. The Routh-Hurwitz method used here required first deriving the exact closed-loop characteristic polynomial algebraically, which becomes increasingly cumbersome for higher-order or more complex open-loop transfer functions, especially those containing transport delay (dead-time) terms e^(-sTd) that cannot be expressed as a finite polynomial at all. The Nyquist criterion, by contrast, works directly from the open-loop frequency response G(jω)H(jω) — which can be obtained either analytically or purely experimentally by direct frequency-response measurement on real hardware — without ever needing to form the closed-loop characteristic equation explicitly, and it extends naturally to systems with transport delay or other non-rational transfer functions simply by including the appropriate additional phase contribution (-ωTd radians) in the polar plot construction. This is precisely why the Nyquist criterion, despite being graphically/geometrically based rather than purely algebraic, remains an indispensable tool in classical control theory alongside Routh-Hurwitz, particularly for systems where an explicit closed-form characteristic polynomial is unavailable or impractical to derive.