Q5Control System

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Q.5. The unit feedback control system transfer function is given by following equation :

G (s) H (s) = \frac{K}{s ( s ^{2} + 2 s + 5 )}

Determine the gain K such that the gain margin is 6 dB. Also determine the value of phase margin for the value of K obtained as above.

Answer 1

Given:

G (s) H (s) = \frac{K}{s ( s ^{2} + 2 s + 5 )}

Substitute s = jω:

G (j ω) H (j ω) = \frac{K}{j ω [ ( 5 - ω ^{2} ) + j 2 ω ]}

Phase crossover frequency ωpc is where the phase equals -180°. The phase of the denominator is 90° (from jω) plus tan⁻¹(2ω/(5-ω²)); setting total phase to -180° requires the second term to equal -90°, i.e. its real part (5-ω²) = 0:

5 - ω_{p c}^{2} = 0 \Rightarrow ω_{p c} = 5 \approx 2.236 rad/s

At ω = ωpc, the magnitude of G(jω)H(jω)/K is:

∣ G (j ω_{p c}) H (j ω_{p c}) ∣ = \frac{K}{ω _{p c} ( 5 - ω _{p c}^{2} ) ^{2} + ( 2 ω _{p c} ) ^{2}} = \frac{K}{5 \times 2 5} = \frac{K}{10}

Gain margin in dB is GM = -20 log10|G(jωpc)H(jωpc)|. For GM = 6 dB:

- 20 lo g_{10} (\frac{K}{10}) = 6 \Rightarrow lo g_{10} (\frac{K}{10}) = - 0.3 \Rightarrow \frac{K}{10} = 1 0^{- 0.3} = 0.501

K = 5.01

To find phase margin, first find gain crossover frequency ωgc where |G(jω)H(jω)| = 1 with K = 5.01:

\frac{5.01}{ω ( 5 - ω ^{2} ) ^{2} + 4 ω ^{2}} = 1

Solving numerically (trial): at ω = 1.35, denominator = 1.35×√[(5-1.8225)²+4(1.8225)] = 1.35×√[10.08+7.29]=1.35×4.17=5.63 (too high); at ω=1.2, denom=1.2×√[(5-1.44)²+4×1.44]=1.2×√[12.67+5.76]=1.2×4.29=5.15 (close); at ω=1.19, denom ≈ 5.02, satisfying the equation, so ωgc ≈ 1.19 rad/s. Phase at this frequency:

∠ G (j ω_{g c}) H (j ω_{g c}) = - 9 0^{\circ} - tan^{- 1} (\frac{2 ( 1.19 )}{5 - 1.1 9 ^{2}}) = - 9 0^{\circ} - tan^{- 1} (0.702) = - 9 0^{\circ} - 35. 1^{\circ} = - 125. 1^{\circ}

P M = 18 0^{\circ} - 125. 1^{\circ} = 54. 9^{\circ}

Hence with K ≈ 5.01, the gain margin is 6 dB (by design) and the corresponding phase margin is approximately 55°, indicating a well-damped, stable closed-loop system at this gain setting.

Interpretation. Gain margin and phase margin are the two standard measures of relative stability read directly from the open-loop frequency response. Gain margin quantifies how much the loop gain can be increased before the system reaches the verge of instability (phase = -180° exactly at unity gain); here a 6 dB gain margin means the gain could be increased by a factor of about 2 (since 6 dB corresponds to a gain multiplier of 10^(6/20) ≈ 2) before marginal instability. Phase margin quantifies how much additional phase lag (e.g., from an unmodeled delay or an added pole) the system can tolerate at the gain crossover frequency before instability sets in; a phase margin around 55° is generally considered good practice, typically corresponding to a well-damped closed-loop response with modest overshoot (a common rule of thumb approximates the closed-loop damping ratio as ζ ≈ PM/100 for PM given in degrees, giving ζ ≈ 0.55 here). Together, these two margins give complementary information: gain margin protects against pure gain increases (e.g. an amplifier drifting high), while phase margin protects against added phase lag (e.g. sensor or actuator delay), and a robust design typically requires both to be comfortably positive simultaneously, as achieved here.

Q5Control System

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