Q2Control System

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Q.2. The open loop transfer function of a unity feedback control system is :

G (s) = \frac{K}{s ( 1 + s T )}

By what factor should the gain K be multiplied so that damping ratio increases from 0.3 to 0.69?

Answer 1

The closed-loop transfer function for unity feedback with G(s) = K/[s(1+sT)] is:

\frac{C ( s )}{R ( s )} = \frac{G ( s )}{1 + G ( s )} = \frac{K / T}{s ^{2} + \frac{1}{T} s + \frac{K}{T}}

Comparing with the standard second-order form s² + 2ζωn s + ωn²:

ω_{n}^{2} = \frac{K}{T}, 2 ζ ω_{n} = \frac{1}{T}

From these, eliminating ωn:

ω_{n} = \frac{K}{T} \Rightarrow 2 ζ \frac{K}{T} = \frac{1}{T} \Rightarrow ζ = \frac{1}{2 K T}

This shows ζ ∝ 1/√K for fixed T, i.e. K ∝ 1/ζ². If K1 corresponds to ζ1 = 0.3 and K2 corresponds to ζ2 = 0.69, then:

\frac{K _{2}}{K _{1}} = (\frac{ζ _{1}}{ζ _{2}})^{2} = (\frac{0.3}{0.69})^{2} = (0.4348)^{2} = 0.189

Hence the gain K must be multiplied by a factor of approximately 0.189 (i.e., reduced to about 18.9% of its original value) to raise the damping ratio from 0.3 (lightly damped, large overshoot) to 0.69 (close to optimal damping with minimal overshoot). This inverse-square relationship between K and ζ is a standard and important result for Type-1 second-order servo systems: increasing gain always reduces damping (and vice versa), which is why gain and transient response cannot be tuned independently without additional compensation.

Effect on transient performance. To appreciate why this gain reduction is desirable, consider the percentage peak overshoot formula for a unit-step input to a standard underdamped second-order system:

% O S = e^{- π ζ / 1 - ζ^{2}} \times 100

At the original damping ratio ζ1 = 0.3, the exponent is -π(0.3)/√(1-0.09) = -0.943/0.954 = -0.988, giving %OS = e^(-0.988)×100 ≈ 37.2% — a large, often unacceptable overshoot for a precision positioning servo. At the improved damping ratio ζ2 = 0.69, the exponent is -π(0.69)/√(1-0.4761) = -2.168/0.7238 = -2.995, giving %OS = e^(-2.995)×100 ≈ 5.0% — a much more acceptable, lightly-oscillatory response close to the commonly cited 'optimal' engineering damping range of ζ ≈ 0.6–0.7, which balances a reasonably fast response against minimal overshoot. This concretely demonstrates why a designer would deliberately choose to reduce the loop gain K to about 18.9% of its original value: the resulting seven-fold reduction in peak overshoot (from ≈37% down to ≈5%) is a dramatic improvement in transient quality, illustrating the fundamental gain-versus-damping trade-off inherent to simple proportional-gain (uncompensated) Type-1 servo systems, and motivating the use of dedicated compensators (as discussed elsewhere in this paper) when both high steady-state accuracy (which favors high K) and low overshoot (which favors low K) are simultaneously required.

Settling time consideration. It is also worth checking the settling time impact of this gain change, since reducing K reduces ωn (as ωn = √(K/T)) even as ζ increases. Using the 2% settling time approximation ts ≈ 4/(ζωn), and noting that the product ζωn = 1/(2T) is actually independent of K (from the earlier relation 2ζωn = 1/T, which holds regardless of the specific K chosen), the settling time ts ≈ 8T remains unchanged by this gain adjustment. This is a subtle but important point: for this particular Type-1 configuration, reducing the gain to improve damping and overshoot does not come at the cost of a slower settling time, because the real part of the closed-loop poles (which governs the settling time via its reciprocal) is fixed by T alone and does not depend on K. What does change is the damped natural frequency ωd = ωn√(1-ζ²) and hence the peak time tp = π/ωd, which increases somewhat as ζ increases (fewer, slower oscillations before settling), but the overall envelope of decay to within 2% of the final value is unaffected — making this an unusually 'free' improvement in transient quality achievable simply by re-tuning the proportional gain.

Q2Control System

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