How Are Overshoot, Decay Ratio, And Oscillation Frequency Evaluated? Master Underdamped Pilot-Plant Analysis

For an underdamped unit-operation response, evaluate overshoot from the first peak, decay ratio from successive peak amplitudes, and oscillation frequency from the time spacing between cycles. For a damping factor (0<\zeta<1), the normalized overshoot is (\exp[-\pi\zeta/\sqrt{1-\zeta^2}]), while the decay ratio is its square. The damped radian frequency is (\omega_d=\sqrt{1-\zeta^2}/\tau), where (\tau) is the system’s natural time constant or period parameter.

The practical method is to read the transient response relative to its ultimate steady-state value, measure peak heights and peak spacing, and then compare the results with the second-order underdamped equations.

Establishing the Reference Response

Identify the Ultimate Value

First determine the initial value (y_0) and the final steady-state value (y_\infty) from the pilot-plant response curve. The total steady-state change is

[ B = y_\infty-y_0 ]

For a step test, the response should be allowed to continue long enough that the final value is reliable rather than estimated from an early portion of the curve.

Measure Deviations from Steady State

Peak heights must be measured relative to (y_\infty), not relative to the initial value. This distinction matters because the oscillation amplitudes decay around the final value.

For a response that overshoots above the final value, let (A) be the first peak’s height above (y_\infty). Let (C) be the height above (y_\infty) of the next corresponding peak.

Evaluating Overshoot

Calculate the Experimental Overshoot

The fractional overshoot is

[ M_p=\frac{A}{B} ]

where:

(A) is the first peak excursion above the ultimate value.
(B) is the total steady-state change caused by the input step.

As a percentage,

[ %OS=100\frac{A}{B} ]

For example, if the first peak is 20% above the final value relative to the total step change, then (M_p=0.20) and the overshoot is (20%).

Relate Overshoot to Damping

For an underdamped second-order response,

[ M_p=\exp\left(\frac{-\pi\zeta}{\sqrt{1-\zeta^2}}\right) ]

A smaller (\zeta) produces a larger overshoot because the system dissipates less energy during each cycle.

If the measured overshoot is used to estimate the damping factor, the relationship can be rearranged as

[ \zeta= \frac{-\ln(M_p)} {\sqrt{\pi^2+\left[\ln(M_p)\right]^2}} ]

This estimate is meaningful only when (0<M_p<1), corresponding to a clearly underdamped response.

Evaluating the Decay Ratio

Compare Successive Peak Amplitudes

The decay ratio is obtained by dividing the amplitude of one oscillation peak by the amplitude of the next corresponding peak:

[ DR=\frac{C}{A} ]

Both (C) and (A) are measured from the ultimate value. The initial-to-next peak ratio is therefore less than one for a stable decaying oscillation.

Use the Overshoot Relationship

For a standard underdamped second-order system,

[ DR= \exp\left(\frac{-2\pi\zeta}{\sqrt{1-\zeta^2}}\right) ]

Therefore,

[ DR=M_p^2 ]

This provides a useful consistency check: the decay ratio calculated from measured successive peaks should approximately equal the square of the measured fractional overshoot.

Clarify Ratio Conventions

Process-control texts sometimes report decay as a first-to-second amplitude ratio, such as 4:1 or 10:1. That convention is the reciprocal of (C/A).

Thus:

(C/A=0.25) corresponds to a first-to-second ratio of (4:1).
(C/A=0.10) corresponds to a first-to-second ratio of (10:1).

The numerical convention must be stated explicitly when reporting pilot-plant results.

Evaluating Oscillation Frequency

Measure the Period from the Response

Measure the time between two successive peaks of the same type. If the peak times are (t_1) and (t_2), the damped oscillation period is

[ T_d=t_2-t_1 ]

The corresponding damped angular frequency is

[ \omega_d=\frac{2\pi}{T_d} ]

This is usually the most direct experimental evaluation because it uses observable features of the process response.

Use the Second-Order Model

For a second-order system represented with damping factor (\zeta) and time parameter (\tau), the damped radian frequency is

[ \omega_d=\frac{\sqrt{1-\zeta^2}}{\tau} ]

The associated period is

[ T_d=\frac{2\pi\tau}{\sqrt{1-\zeta^2}} ]

Here, (\omega_d) describes the actual frequency of the decaying oscillations. It should not be confused with the undamped natural frequency, which is commonly written as (\omega_n=1/\tau) under this parameterization.

Compare Measured and Predicted Values

Students can calculate (\zeta) from overshoot or decay ratio, estimate (\tau) from the process model, and then calculate the predicted (\omega_d). Comparing it with (2\pi/T_d) helps identify measurement errors, model mismatch, sensor delays, or nonlinear pilot-plant behavior.

Understanding the Trade-offs

Faster Response Can Increase Oscillation

A lightly damped loop may reach the desired region quickly but produce substantial overshoot and repeated oscillations. Increasing damping generally reduces overshoot and decay persistence, although the response may become slower.

This is the central controller-tuning trade-off: speed must be balanced against stability, actuator movement, and deviation of the controlled variable from its target.

Decay Ratio Is a Stability Indicator

A decay ratio close to one indicates that successive oscillations are decreasing slowly. A much smaller ratio indicates that the transient dies out more rapidly.

Educational experiments may describe a first-to-second amplitude ratio between approximately 4:1 and 10:1 as a practical balanced range. In the (C/A) convention, this corresponds approximately to (0.25) to (0.10).

Avoid Measuring Noise as Oscillation

Noise, actuator resolution, process disturbances, and sensor filtering can distort peak amplitudes. Peak measurements should be taken from clearly identifiable cycles, and several successive peak pairs should be compared when the signal quality allows it.

Account for Digital Sampling

A digital control system must sample an oscillatory process variable at a rate greater than twice its oscillation frequency to avoid aliasing. In practice, four or more samples per oscillation period provide a more reliable representation for monitoring and tuning.

If the sampling rate is too low, the interface may display a false lower-frequency oscillation or an apparently constant signal.

How to Apply This to Your Experiment

Use the response plot and model equations together rather than relying on a single measurement.

If your primary focus is overshoot: Measure the first peak above the ultimate value, divide by the total steady-state change, and report both (M_p) and (100M_p%).
If your primary focus is damping or stability: Measure two successive peak amplitudes relative to steady state, report (C/A), and state clearly whether the reciprocal first-to-second convention is being used.
If your primary focus is oscillation speed: Measure the time between successive corresponding peaks and calculate (\omega_d=2\pi/T_d); compare it with (\sqrt{1-\zeta^2}/\tau).
If your primary focus is digital control quality: Verify that the sampling rate is well above two samples per oscillation cycle, with approximately four or more samples per period preferred.

These measurements turn the pilot-plant waveform into a quantitative description of damping, transient persistence, and oscillation speed.

Summary Table:

Parameter	How to Measure	Key Equation	What It Indicates
Overshoot	First peak excursion above the final steady-state value divided by the total step change	`M_p = A/B`; `%OS = 100M_p`	Peak deviation and damping level
Damping factor	Calculate from the measured fractional overshoot	`zeta = -ln(M_p) / sqrt(pi^2 + [ln(M_p)]^2)`	How quickly oscillations dissipate
Decay ratio	Next corresponding peak amplitude divided by the first peak amplitude, both measured from steady state	`DR = C/A = M_p^2`	Persistence of the transient response
Oscillation period	Time between two successive corresponding peaks	`T_d = t_2 - t_1`	Duration of one damped cycle
Damped frequency	Convert the measured period to angular frequency	`omega_d = 2pi/T_d`	Speed of the decaying oscillation
Sampling quality	Compare the sampling rate with the oscillation frequency	At least 2 samples per cycle; 4 or more preferred	Reliability of digital response data

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