Step disturbances are modeled as sustained changes, while impulse disturbances are modeled as concentrated, short-duration inputs. A step of magnitude (A) is written (u_d(t)=A,\mathcal{H}(t)), where (\mathcal{H}(t)) is the unit-step function, and has Laplace transform (U_d(s)=A/s). An impulse is written (u_d(t)=B,\delta(t)), where (B) is its area or total injected quantity, and has Laplace transform (U_d(s)=B); a unit impulse has (B=1).

A step tests how a pilot plant moves from one operating condition to another, while an impulse reveals the plant’s dynamic distribution of response over time. With a process model (G(s)), the corresponding output is found from (Y(s)=G(s)U_d(s)).

Modeling Disturbances in Pilot Plants

Step Inputs Represent Sustained Operating Changes

A step disturbance with magnitude (A) is modeled as

[ d(t)=A,\mathcal{H}(t) ]

where

[ \mathcal{H}(t)= \begin{cases} 0, & t<0\ 1, & t\geq 0 \end{cases} ]

Its Laplace transform is

[ D(s)=\mathcal{L}{A,\mathcal{H}(t)}=\frac{A}{s}. ]

This represents a sudden change that remains in effect, such as a new valve position, feed flow rate, heating duty, or controller setpoint.

Impulse Inputs Represent Concentrated Events

An impulse disturbance is modeled as

[ d(t)=B,\delta(t), ]

where (\delta(t)) is the Dirac delta function and (B) is the impulse area:

[ \int_{-\infty}^{\infty} B,\delta(t),dt=B. ]

Its Laplace transform is

[ D(s)=\mathcal{L}{B,\delta(t)}=B. ]

For a unit impulse, (B=1), so the transform is simply

[ D(s)=1. ]

In a pilot plant, an ideal impulse approximates a very rapid tracer, catalyst, or reactant injection whose duration is negligible relative to the process response time.

Calculating the Transient Response

Process Models Link Inputs to Outputs

For a linearized process model with transfer function

[ G(s)=\frac{Y(s)}{D(s)}, ]

the output transform is

[ Y(s)=G(s)D(s). ]

The time-domain transient response is obtained by taking the inverse Laplace transform of (Y(s)).

This framework allows the same plant model to be tested against different disturbance types without changing the underlying process dynamics.

Response to a Step Disturbance

For a step of magnitude (A),

[ Y_{\text{step}}(s)=G(s)\frac{A}{s}. ]

The factor (1/s) accounts for the fact that the disturbance remains at its new value after the change.

For a first-order process,

[ G(s)=\frac{K}{\tau s+1}, ]

where (K) is the steady-state gain and (\tau) is the time constant. The step response becomes

[ Y_{\text{step}}(s)

\frac{KA}{s(\tau s+1)}. ]

Its time-domain form is

[ y_{\text{step}}(t)=KA\left(1-e^{-t/\tau}\right), \qquad t\geq 0. ]

The final change in output is (KA), and the time constant indicates how quickly the plant approaches that new steady state.

Response to an Impulse Disturbance

For an impulse of area (B),

[ Y_{\text{impulse}}(s)=G(s)B. ]

For a unit impulse, this reduces to

[ Y_{\text{impulse}}(s)=G(s). ]

Therefore, the impulse response is the inverse Laplace transform of the process transfer function itself.

For the same first-order process,

[ Y_{\text{impulse}}(s)=\frac{KB}{\tau s+1}, ]

[ y_{\text{impulse}}(t)=\frac{KB}{\tau}e^{-t/\tau}, \qquad t\geq 0. ]

Unlike the step response, the impulse response generally returns toward zero after the disturbance has passed.

What Each Test Reveals

Step Tests Estimate Gain and Time Constants

A step test is particularly useful for identifying:

Steady-state gain: the final output change divided by the input change.
Time constant: the characteristic speed of the transient response.
Process order: whether the response behaves like a first-order, second-order, or more complex system.
Stability: whether the output settles, oscillates, or diverges.

For a first-order response, the output reaches approximately (63.2%) of its final change after one time constant.

Impulse Tests Reveal Residence-Time Behavior

In flow systems, an impulse tracer test produces a concentration response at a downstream location. The normalized response is commonly interpreted as a residence-time distribution, or RTD.

If (C(t)) is the measured tracer concentration, a normalized RTD is often written as

[ E(t)=\frac{C(t)}{\int_0^\infty C(t),dt}. ]

The function (E(t)) describes the fraction of material leaving the system at each residence time. Its integral is one:

[ \int_0^\infty E(t),dt=1. ]

The corresponding cumulative RTD is

[ F(t)=\int_0^t E(\lambda),d\lambda. ]

Thus, an impulse experiment provides the distribution (E(t)), while a step tracer experiment provides its cumulative form (F(t)).

Step and Impulse Responses Are Mathematically Related

For a causal linear system, the unit-step response is the integral of the impulse response:

[ g_{\text{step}}(t)=\int_0^t g_{\text{impulse}}(\lambda),d\lambda. ]

Conversely, the impulse response is the time derivative of the step response:

[ g_{\text{impulse}}(t)=\frac{d}{dt}g_{\text{step}}(t). ]

This relationship explains why step tests identify how the process accumulates toward a new condition, whereas impulse tests expose the timing and shape of the process’s internal dynamic response.

Representing Dead Time and Delays

Transportation Lags Shift the Response

Pilot plants commonly include transport delays caused by piping, fluid residence time, sensor placement, and analyzer response. A pure delay of (\theta) units is represented in the Laplace domain by

[ e^{-\theta s}. ]

A first-order-plus-dead-time model can therefore be written as

[ G(s)=\frac{K e^{-\theta s}}{\tau s+1}. ]

The delayed step input has the transform

[ D_{\text{step}}(s)=\frac{A e^{-\theta s}}{s}, ]

and the delayed impulse response has the transform

[ D_{\text{impulse}}(s)=B e^{-\theta s}. ]

In the time domain, the response is zero before (t=\theta), after which the process transient begins.

Delays Affect Control Evaluation

Dead time reduces the controller’s ability to respond before the process output reflects a change. Ignoring it can produce overly aggressive PID settings and can make a simulated control loop appear more stable than the physical pilot plant.

Including (e^{-\theta s}) gives the mathematical model a closer representation of real measurement and transportation behavior.

Understanding the Trade-offs

Ideal Inputs Are Approximations

A physical valve cannot change position infinitely quickly, and a tracer injection always has finite duration. Real tests therefore approximate ideal steps and impulses using rapid valve movements or short injection pulses.

The approximation is useful only when the input duration is small compared with the dominant process time scale.

Step Tests Can Disturb the Operating Point

A large step may drive temperature, pressure, composition, or flow outside the safe or linear operating range. It can also change the process gain or time constant during the experiment.

The test magnitude should be large enough to rise above measurement noise but small enough to preserve approximately linear behavior.

Impulse Tests Are Sensitive to Measurement Quality

An impulse response can be difficult to interpret when the injected quantity is too small, the sensor is noisy, or the injection is not well characterized. A finite-width pulse may need to be treated as the actual input rather than as an ideal delta function.

For RTD work, the measured concentration must also be normalized consistently, because the integral of the response represents the total detected tracer quantity.

Linear Models Have Limited Scope

The relation

[ Y(s)=G(s)D(s) ]

assumes a linear, time-invariant model, usually obtained by linearizing the plant around a selected operating point. Strong nonlinearities, phase changes, saturation, fouling, and composition-dependent properties may require nonlinear dynamic simulation or tests at multiple operating conditions.

How to Apply This to Your Project

Use the disturbance model that matches the physical question being asked:

If your primary focus is steady-state gain and process speed: Apply a small step disturbance, fit the transition curve, and estimate the gain, time constant, process order, and possible dead time.
If your primary focus is residence-time distribution: Apply a short tracer pulse, measure the downstream concentration history, and normalize it to obtain (E(t)).
If your primary focus is controller tuning: Use a step response model that includes process gain, time constants, and transportation delay before selecting PID parameters.
If your primary focus is rapid injection behavior: Model the disturbance as (B\delta(t)), where (B) is the measured injected amount or pulse area rather than automatically assuming a unit impulse.
If your primary focus is model validation: Compare predicted and measured transient curves while checking whether the input duration, sensor delay, and operating range justify the linear approximation.

Step inputs characterize the plant’s transition between operating states, while impulse inputs reveal how the plant distributes and dissipates a disturbance over time.

Summary Table:

Disturbance	Mathematical model	Best used to evaluate	Key result
Step	`d(t) = A H(t)`; `D(s) = A/s`	Gain, time constant, process order, stability, and controller tuning	Transition to a new operating condition
Impulse	`d(t) = B delta(t)`; `D(s) = B`	Residence-time distribution and dynamic response shape	Timing and dissipation of a short event
First-order step response	`y(t) = KA(1 - e^(-t/tau))`	Steady-state change and process speed	Final change is `KA`; 63.2% is reached at `t = tau`
First-order impulse response	`y(t) = (KB/tau)e^(-t/tau)`	Short-duration injection behavior	Response decays toward zero
Dead-time model	`G(s) = K e^(-theta s)/(tau s + 1)`	Transport delay and realistic control evaluation	Response begins after delay `theta`
Tracer impulse test	`E(t) = C(t)/integral C(t)dt`	Residence-time distribution in flow systems	Material residence-time profile

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