Time-Domain and Frequency-Domain Responses

Scope: first-order and second-order transients; dead time estimation; disturbances and periodic forcing. Cross-cutting case: an instrumented heat-exchanger outlet temperature step test.

What this unit builds

A time response must be read together with the input that created it. A step response without the actuator's recorded movement can conceal valve saturation. A smooth trace without sensor location can conceal delay. When two mechanisms both explain the nominal trend, they must be tested using a disturbance or experiment that separates their effects, not selected on visual appearance alone.

Lesson sequence

1. [First-order step and impulse responses](../lessons/unit-02-lesson-01.html) — A first-order stable system reaches 63.2% of its final step change at one time constant. Its impulse response decays exponentially; a measured step trace can suggest K and tau but not automatically establish linearity. Governing relation: `y(t)=K Delta u [1-exp(-t/tau)]`.

2. [Second-order damping and overshoot](../lessons/unit-02-lesson-02.html) — A pair of dynamic states can create underdamped oscillation. Damping ratio controls overshoot; natural frequency sets the basic clock. A negative derivative signal or actuator limitation can change an apparently textbook response. Governing relation: `G(s)=omega_n²/(s²+2 zeta omega_n s+omega_n²)`.

3. [FOPDT identification and uncertainty](../lessons/unit-02-lesson-03.html) — A first-order-plus-dead-time model compresses process behavior into gain, lag and effective delay. Parameter fitting is a modeling convenience; different triples can fit a noisy short experiment, especially if the final value is unknown. Governing relation: `G(s)=K exp(-theta s)/(tau s+1)`.

4. [Disturbance and sinusoidal frequency response](../lessons/unit-02-lesson-04.html) — Transient step tests emphasize low-frequency gain; sinusoidal forcing probes attenuation and phase shift. A system that passes slowly varying disturbances may still suppress faster oscillations or suffer added delay. Governing relation: `|G(jw)|=K/sqrt(1+(w tau)^2)`.

Model connections and physical interpretation

First-order step and impulse responses

A first-order stable system reaches 63.2% of its final step change at one time constant. Its impulse response decays exponentially; a measured step trace can suggest K and tau but not automatically establish linearity. A real actuator step may be ramp-limited, and sampled measurements introduce time resolution and uncertainty. The calculation that anchors this lesson is deliberately synthetic: If K=2 K/% and input changes 3%, the final temperature gain is 6 K. At tau=40 s, the step response is 6(1−exp(−1))=3.79 K above baseline. The reasoning should be checked in the next topic, because a dynamic approximation useful for one type of experiment may be inappropriate when the controlled input, disturbance or measurement location changes.

Second-order damping and overshoot

A pair of dynamic states can create underdamped oscillation. Damping ratio controls overshoot; natural frequency sets the basic clock. A negative derivative signal or actuator limitation can change an apparently textbook response. The stated overshoot relation requires a linear, normalized, no-zero, underdamped second-order step response. The calculation that anchors this lesson is deliberately synthetic: For zeta=0.40, percent overshoot is exp(-pi zeta/sqrt(1−zeta²))*100=25.4%. With omega_n=0.20 rad/s, the ideal first peak is t_p=pi/(omega_n sqrt(1−zeta²))=17.1 s. The reasoning should be checked in the next topic, because a dynamic approximation useful for one type of experiment may be inappropriate when the controlled input, disturbance or measurement location changes.

FOPDT identification and uncertainty

A first-order-plus-dead-time model compresses process behavior into gain, lag and effective delay. Parameter fitting is a modeling convenience; different triples can fit a noisy short experiment, especially if the final value is unknown. The tangent method and direct nonlinear least squares may produce different estimates. Input saturation should be excluded. The calculation that anchors this lesson is deliberately synthetic: A 4% input step yields a 6 °C steady output change: K=1.5 °C/%. A discernible first response occurs after theta=12 s, and 63.2% of the remaining change is observed at t≈62 s, suggesting tau≈50 s. The reasoning should be checked in the next topic, because a dynamic approximation useful for one type of experiment may be inappropriate when the controlled input, disturbance or measurement location changes.

Disturbance and sinusoidal frequency response

Transient step tests emphasize low-frequency gain; sinusoidal forcing probes attenuation and phase shift. A system that passes slowly varying disturbances may still suppress faster oscillations or suffer added delay. Steady harmonic formulas omit startup transients and rely on linearity at the forced amplitude. The calculation that anchors this lesson is deliberately synthetic: For K=3, tau=20 s and angular frequency 0.10 rad/s, amplitude ratio is 3/sqrt(5)=1.342 and phase shift is −atan(2)=−63.4°. The reasoning should be checked in the next topic, because a dynamic approximation useful for one type of experiment may be inappropriate when the controlled input, disturbance or measurement location changes.

Unit synthesis: a reproducible experiment

Start from an instrumented heat-exchanger outlet temperature step test. Draw the boundary and place every relevant measurement on it. Give the operating point, consistent units and any assumptions about constant density, heat capacity or sensor behavior. Construct a hypothesis that could be tested with a small safe, synthetic input perturbation. Predict the output sign and a first approximation for the relevant time scale before computing anything.

Next compare a nominal model with one deliberately misspecified version. Change a gain, a lag, a delay or a sensor bias rather than modifying all uncertain features at once. Quantify the difference in output peak, settling time, residual steady error and manipulated-variable excursion. Record when the model ceases to be a good representation of the exercise. A successful classroom exercise is a transparent chain of assumptions and checks, not merely a curve.

Finally design a holdout test. In a process setting this might be a second flow rate, a different operating temperature or a load disturbance that was not included in model fitting. Compare model and measurement on the same time grid, keep a reproducible data table, and explain why any fitted coefficient cannot be assumed constant outside its observed range. Treat alarm and protective requirements as a separate design decision.

Unit knowledge check

1. For First-order step and impulse responses, state the terms of y(t)=K Delta u [1-exp(-t/tau)], choose a limiting or equilibrium case, and explain when it would be invalid.
2. For Second-order damping and overshoot, state the terms of G(s)=omega_n²/(s²+2 zeta omega_n s+omega_n²), choose a limiting or equilibrium case, and explain when it would be invalid.
3. For FOPDT identification and uncertainty, state the terms of G(s)=K exp(-theta s)/(tau s+1), choose a limiting or equilibrium case, and explain when it would be invalid.
4. For Disturbance and sinusoidal frequency response, state the terms of |G(jw)|=K/sqrt(1+(w tau)^2), choose a limiting or equilibrium case, and explain when it would be invalid.

Continue to the [Unit 2 assessment](../assessments/unit-02.html) or [Unit 2 project](../projects/unit-02.html).

Unit 2, Lesson 1: First-order step and impulse responses

Unit: Time-Domain and Frequency-Domain Responses. Physical setting: an instrumented heat-exchanger outlet temperature step test. Reference model: y(t)=K Delta u [1-exp(-t/tau)].

Learning outcomes

By the end of this lesson you should be able to (i) explain first-order step and impulse responses in physical terms, (ii) identify the variables and assumptions of y(t)=K Delta u [1-exp(-t/tau)], (iii) reproduce and challenge a numerical benchmark, (iv) recognize how measurement, delay and actuator limits modify the model, and (v) design a specific experiment that can distinguish the current model from an alternative interpretation.

Physical reasoning, not just a formula

A first-order stable system reaches 63.2% of its final step change at one time constant. Its impulse response decays exponentially; a measured step trace can suggest K and tau but not automatically establish linearity.

Consider the unit-wide equipment case: an instrumented heat-exchanger outlet temperature step test. In this setting the same numerical trend may have more than one physical cause. A change observed at the analyzer may reflect a real process response, a changed flow residence time or the instrument dynamics. The analyst must first specify which quantity is manipulated, which is observed and which variables are treated as external disturbances. This is particularly important for first-order step and impulse responses, where confusing a state with a measured output can give a mathematically solvable but physically incorrect model.

The chosen governing expression is y(t)=K Delta u [1-exp(-t/tau)]. It is a compact model, not a complete plant specification. Terms on opposite sides of an equality must have compatible units and consistent time bases. When a parameter is estimated from one transient, its result belongs to that operating range unless new data demonstrate transferability. The task is to explain the meaning of the relation before substituting numbers and then to compare its predictions with a checkable boundary condition or steady state.

Working the numerical reference case

Synthetic worked example. If K=2 K/% and input changes 3%, the final temperature gain is 6 K. At tau=40 s, the step response is 6(1−exp(−1))=3.79 K above baseline.

First transcribe every numerical input into a short table with its physical unit. Second, compute the baseline result without a software solver. Third, identify which output should change and which should remain unchanged when the manipulated input is zero. In this first-order step and impulse responses exercise, the reported figures are not fitted plant data: they are an internally stated teaching benchmark. Any extension to different equipment requires new coefficients and new validation.

A useful sensitivity check is to change the selected coefficient or input by +10% while retaining all other conditions. This is not equivalent to assuming the output must also rise by 10%; exponential responses, ratios, saturation and feedback can change the mapping. Write the original expression, substitute the perturbed value and compare the sign of the result against physical intuition. Next repeat with a −10% change. An asymmetric response is evidence of nonlinearity in the selected mapping, not necessarily a numerical error.

Derivation and model restrictions

A real actuator step may be ramp-limited, and sampled measurements introduce time resolution and uncertainty. To turn a physical inventory into a differential state equation, write accumulation as incoming rate minus outgoing rate plus net production, if relevant. To turn an instrument or actuator into a dynamical element, specify its output, time constant and initial condition. To build a transfer function, take deviations about a steady operating point and verify that the assumed initial deviations are zero. Do not apply frequency-domain algebra blindly to a strongly nonlinear or switching regime.

For first-order step and impulse responses, the limiting checks should include a zero-change input, a long-time equilibrium where one exists and at least one infeasible input that the physical equipment cannot realize. If a parameter has dimensions of reciprocal time, use it only with a time expressed in matching units. If a parameter multiplies a temperature difference, its unit must convert the difference into a heat rate before the quantity is added to the energy balance. These simple checks detect mistakes that a graph renderer will not.

How to identify the model experimentally

Use the previously studied topic First-order step and impulse responses as a source of prior knowledge and the next topic Second-order damping and overshoot as an independent challenge. Define the reference operating point for first-order step and impulse responses in the process case and record a small safe input sequence, the actual actuator trajectory, the measurement times and instrument range. A step, a pulse and a slow ramp probe different properties, so do not collapse them into one claimed validation. If the fitted output differs from the recorded trace, inspect the timestamp alignment and sensor dynamics before increasing model order.

Where possible, obtain at least two input amplitudes. Normalize the measured output by the change in input and test whether the curves approximately collapse. Failure to collapse can indicate nonlinearity, saturation, temperature-dependent coefficients or changed initial conditions. A second holdout operating point is especially important when the main formula was derived by linearization. Report the residual error and uncertainty rather than relabeling an imperfect approximation as a universal physical law.

Uncertainty, troubleshooting, and design decisions

Failure mode to investigate: Forcing a non-exponential response into a one-lag model can hide parallel heat-transfer paths or a poorly mixed vessel. The first diagnostic question is which conservation law, measurement chain or actuator path the proposed explanation implicates. Reconstruct the actual manipulated-variable history and compare it with what the controller requested; a large difference is a plant-side constraint, not a parameter-estimation error that can be fixed by fitting a more elaborate process transfer function.

Decision from the evidence: Calculate gain from the final difference, estimate tau from 63.2%, and reserve later test data for validation. State what the experiment supports and what remains unidentifiable. The decision should include whether the next useful action is to add a sensor, run a second input condition, change an assumed model structure or restrict the admissible operating range. When model outputs could guide actual equipment, independent engineering review and safety analysis are necessary before any implementation.

A challenge to the simplest explanation

Compare the present explanation with FOPDT identification and uncertainty, which appears elsewhere in this unit. Imagine obtaining a smooth measured response that both models could plausibly produce. Choose one additional measurement that would change differently under the two explanations. Define the expected sign and time scale under each. For first-order step and impulse responses, record not just peak error but also the total actuator travel, disturbances, measurement noise and how the reference signal was generated. This exercise forces you to separate apparent goodness of fit from the ability to predict new conditions.

Numerical implementation and independent checks

The companion [interactive model, Laboratory 01](../labs/lab-01.html) is First-order thermal step response. Change one slider at a time, export the series and independently reproduce at least one value from a governing equation. The browser tool is deterministic and offline. It represents a defined educational model, not an approved process simulator. Its chart scale may rescale as parameters change, so compare numerical coordinates rather than visual slope alone.

For a numerical transient, verify step-size sensitivity by recomputing with half the time increment. If a predicted instability disappears merely when the integration time step shrinks, it was likely numerical rather than physical. On the other hand, an unstable physical model may diverge more quickly when more accurately integrated. Preserve initial conditions, parameter units and exact software version in the report so another student can repeat the test.

Practice and oral-examination questions

1. Starting from y(t)=K Delta u [1-exp(-t/tau)], identify its state(s), manipulated input(s) and measured output(s). State how you would verify all dimensions.
2. Recalculate the example after a 10% increase of the principal process parameter. What changes in the predicted steady value and in the transient, and why are these not necessarily the same?
3. Propose a sensor or actuator failure that would make an otherwise correct first-order step and impulse responses model appear wrong. What extra logged variable would distinguish that failure?
4. Explain how FOPDT identification and uncertainty might confound the identification of this lesson’s mechanism and propose a discriminating test.
5. State one safety or operating limit that the simplified y(t)=K Delta u [1-exp(-t/tau)] relation does not capture. How should an engineer document that omission?

Answer guide and reflection

Question 1 is assessed by correct units and by separating the actual physical inventory from its measured signal; a memorized symbolic expression without those definitions is insufficient. For Question 2, the correct method uses the original equation twice and recomputes the equilibrium or time constant where applicable; simple proportional scaling is not accepted without showing that the equation is linear in the varied coefficient. For Question 3, the measurement or actuator diagnosis must lead to a different predicted trace. For Question 4, a proposed experiment is credible only if it can distinguish two candidate mechanisms. For Question 5, a constraint must be named and its omission explicitly stated.

Takeaway for Time-Domain and Frequency-Domain Responses. The lesson is complete only after the model can be challenged with a changed initial condition, a disturbance and a non-ideal measurement. A plotted agreement under one nominal set of synthetic conditions is the start of investigation, not its end. The next step is to test whether the assumptions retain their meaning when second-order damping and overshoot is included in the model.

Unit 2, Lesson 2: Second-order damping and overshoot

Unit: Time-Domain and Frequency-Domain Responses. Physical setting: an instrumented heat-exchanger outlet temperature step test. Reference model: G(s)=omega_n²/(s²+2 zeta omega_n s+omega_n²).

Learning outcomes

By the end of this lesson you should be able to (i) explain second-order damping and overshoot in physical terms, (ii) identify the variables and assumptions of G(s)=omega_n²/(s²+2 zeta omega_n s+omega_n²), (iii) reproduce and challenge a numerical benchmark, (iv) recognize how measurement, delay and actuator limits modify the model, and (v) design a specific experiment that can distinguish the current model from an alternative interpretation.

Physical reasoning, not just a formula

A pair of dynamic states can create underdamped oscillation. Damping ratio controls overshoot; natural frequency sets the basic clock. A negative derivative signal or actuator limitation can change an apparently textbook response.

Consider the unit-wide equipment case: an instrumented heat-exchanger outlet temperature step test. In this setting the same numerical trend may have more than one physical cause. A change observed at the analyzer may reflect a real process response, a changed flow residence time or the instrument dynamics. The analyst must first specify which quantity is manipulated, which is observed and which variables are treated as external disturbances. This is particularly important for second-order damping and overshoot, where confusing a state with a measured output can give a mathematically solvable but physically incorrect model.

The chosen governing expression is G(s)=omega_n²/(s²+2 zeta omega_n s+omega_n²). It is a compact model, not a complete plant specification. Terms on opposite sides of an equality must have compatible units and consistent time bases. When a parameter is estimated from one transient, its result belongs to that operating range unless new data demonstrate transferability. The task is to explain the meaning of the relation before substituting numbers and then to compare its predictions with a checkable boundary condition or steady state.

Working the numerical reference case

Synthetic worked example. For zeta=0.40, percent overshoot is exp(-pi zeta/sqrt(1−zeta²))*100=25.4%. With omega_n=0.20 rad/s, the ideal first peak is t_p=pi/(omega_n sqrt(1−zeta²))=17.1 s.

First transcribe every numerical input into a short table with its physical unit. Second, compute the baseline result without a software solver. Third, identify which output should change and which should remain unchanged when the manipulated input is zero. In this second-order damping and overshoot exercise, the reported figures are not fitted plant data: they are an internally stated teaching benchmark. Any extension to different equipment requires new coefficients and new validation.

A useful sensitivity check is to change the selected coefficient or input by +10% while retaining all other conditions. This is not equivalent to assuming the output must also rise by 10%; exponential responses, ratios, saturation and feedback can change the mapping. Write the original expression, substitute the perturbed value and compare the sign of the result against physical intuition. Next repeat with a −10% change. An asymmetric response is evidence of nonlinearity in the selected mapping, not necessarily a numerical error.

Derivation and model restrictions

The stated overshoot relation requires a linear, normalized, no-zero, underdamped second-order step response. To turn a physical inventory into a differential state equation, write accumulation as incoming rate minus outgoing rate plus net production, if relevant. To turn an instrument or actuator into a dynamical element, specify its output, time constant and initial condition. To build a transfer function, take deviations about a steady operating point and verify that the assumed initial deviations are zero. Do not apply frequency-domain algebra blindly to a strongly nonlinear or switching regime.

For second-order damping and overshoot, the limiting checks should include a zero-change input, a long-time equilibrium where one exists and at least one infeasible input that the physical equipment cannot realize. If a parameter has dimensions of reciprocal time, use it only with a time expressed in matching units. If a parameter multiplies a temperature difference, its unit must convert the difference into a heat rate before the quantity is added to the energy balance. These simple checks detect mistakes that a graph renderer will not.

How to identify the model experimentally

Use the previously studied topic First-order step and impulse responses as a source of prior knowledge and the next topic FOPDT identification and uncertainty as an independent challenge. Define the reference operating point for second-order damping and overshoot in the process case and record a small safe input sequence, the actual actuator trajectory, the measurement times and instrument range. A step, a pulse and a slow ramp probe different properties, so do not collapse them into one claimed validation. If the fitted output differs from the recorded trace, inspect the timestamp alignment and sensor dynamics before increasing model order.

Where possible, obtain at least two input amplitudes. Normalize the measured output by the change in input and test whether the curves approximately collapse. Failure to collapse can indicate nonlinearity, saturation, temperature-dependent coefficients or changed initial conditions. A second holdout operating point is especially important when the main formula was derived by linearization. Report the residual error and uncertainty rather than relabeling an imperfect approximation as a universal physical law.

Uncertainty, troubleshooting, and design decisions

Failure mode to investigate: A measured ringing trace need not mean the process itself has two storage elements; controller tuning can create closed-loop oscillation. The first diagnostic question is which conservation law, measurement chain or actuator path the proposed explanation implicates. Reconstruct the actual manipulated-variable history and compare it with what the controller requested; a large difference is a plant-side constraint, not a parameter-estimation error that can be fixed by fitting a more elaborate process transfer function.

Decision from the evidence: Extract the first peak and oscillation period, then cross-check damping from multiple successive peaks. State what the experiment supports and what remains unidentifiable. The decision should include whether the next useful action is to add a sensor, run a second input condition, change an assumed model structure or restrict the admissible operating range. When model outputs could guide actual equipment, independent engineering review and safety analysis are necessary before any implementation.

A challenge to the simplest explanation

Compare the present explanation with Disturbance and sinusoidal frequency response, which appears elsewhere in this unit. Imagine obtaining a smooth measured response that both models could plausibly produce. Choose one additional measurement that would change differently under the two explanations. Define the expected sign and time scale under each. For second-order damping and overshoot, record not just peak error but also the total actuator travel, disturbances, measurement noise and how the reference signal was generated. This exercise forces you to separate apparent goodness of fit from the ability to predict new conditions.

Numerical implementation and independent checks

The companion [interactive model, Laboratory 04](../labs/lab-04.html) is Second-order damping and overshoot. Change one slider at a time, export the series and independently reproduce at least one value from a governing equation. The browser tool is deterministic and offline. It represents a defined educational model, not an approved process simulator. Its chart scale may rescale as parameters change, so compare numerical coordinates rather than visual slope alone.

For a numerical transient, verify step-size sensitivity by recomputing with half the time increment. If a predicted instability disappears merely when the integration time step shrinks, it was likely numerical rather than physical. On the other hand, an unstable physical model may diverge more quickly when more accurately integrated. Preserve initial conditions, parameter units and exact software version in the report so another student can repeat the test.

Practice and oral-examination questions

1. Starting from G(s)=omega_n²/(s²+2 zeta omega_n s+omega_n²), identify its state(s), manipulated input(s) and measured output(s). State how you would verify all dimensions.
2. Recalculate the example after a 10% increase of the principal process parameter. What changes in the predicted steady value and in the transient, and why are these not necessarily the same?
3. Propose a sensor or actuator failure that would make an otherwise correct second-order damping and overshoot model appear wrong. What extra logged variable would distinguish that failure?
4. Explain how Disturbance and sinusoidal frequency response might confound the identification of this lesson’s mechanism and propose a discriminating test.
5. State one safety or operating limit that the simplified G(s)=omega_n²/(s²+2 zeta omega_n s+omega_n²) relation does not capture. How should an engineer document that omission?

Answer guide and reflection

Question 1 is assessed by correct units and by separating the actual physical inventory from its measured signal; a memorized symbolic expression without those definitions is insufficient. For Question 2, the correct method uses the original equation twice and recomputes the equilibrium or time constant where applicable; simple proportional scaling is not accepted without showing that the equation is linear in the varied coefficient. For Question 3, the measurement or actuator diagnosis must lead to a different predicted trace. For Question 4, a proposed experiment is credible only if it can distinguish two candidate mechanisms. For Question 5, a constraint must be named and its omission explicitly stated.

Takeaway for Time-Domain and Frequency-Domain Responses. The lesson is complete only after the model can be challenged with a changed initial condition, a disturbance and a non-ideal measurement. A plotted agreement under one nominal set of synthetic conditions is the start of investigation, not its end. The next step is to test whether the assumptions retain their meaning when fopdt identification and uncertainty is included in the model.

Unit 2, Lesson 3: FOPDT identification and uncertainty

Unit: Time-Domain and Frequency-Domain Responses. Physical setting: an instrumented heat-exchanger outlet temperature step test. Reference model: G(s)=K exp(-theta s)/(tau s+1).

Learning outcomes

By the end of this lesson you should be able to (i) explain fopdt identification and uncertainty in physical terms, (ii) identify the variables and assumptions of G(s)=K exp(-theta s)/(tau s+1), (iii) reproduce and challenge a numerical benchmark, (iv) recognize how measurement, delay and actuator limits modify the model, and (v) design a specific experiment that can distinguish the current model from an alternative interpretation.

Physical reasoning, not just a formula

A first-order-plus-dead-time model compresses process behavior into gain, lag and effective delay. Parameter fitting is a modeling convenience; different triples can fit a noisy short experiment, especially if the final value is unknown.

Consider the unit-wide equipment case: an instrumented heat-exchanger outlet temperature step test. In this setting the same numerical trend may have more than one physical cause. A change observed at the analyzer may reflect a real process response, a changed flow residence time or the instrument dynamics. The analyst must first specify which quantity is manipulated, which is observed and which variables are treated as external disturbances. This is particularly important for fopdt identification and uncertainty, where confusing a state with a measured output can give a mathematically solvable but physically incorrect model.

The chosen governing expression is G(s)=K exp(-theta s)/(tau s+1). It is a compact model, not a complete plant specification. Terms on opposite sides of an equality must have compatible units and consistent time bases. When a parameter is estimated from one transient, its result belongs to that operating range unless new data demonstrate transferability. The task is to explain the meaning of the relation before substituting numbers and then to compare its predictions with a checkable boundary condition or steady state.

Working the numerical reference case

Synthetic worked example. A 4% input step yields a 6 °C steady output change: K=1.5 °C/%. A discernible first response occurs after theta=12 s, and 63.2% of the remaining change is observed at t≈62 s, suggesting tau≈50 s.

First transcribe every numerical input into a short table with its physical unit. Second, compute the baseline result without a software solver. Third, identify which output should change and which should remain unchanged when the manipulated input is zero. In this fopdt identification and uncertainty exercise, the reported figures are not fitted plant data: they are an internally stated teaching benchmark. Any extension to different equipment requires new coefficients and new validation.

A useful sensitivity check is to change the selected coefficient or input by +10% while retaining all other conditions. This is not equivalent to assuming the output must also rise by 10%; exponential responses, ratios, saturation and feedback can change the mapping. Write the original expression, substitute the perturbed value and compare the sign of the result against physical intuition. Next repeat with a −10% change. An asymmetric response is evidence of nonlinearity in the selected mapping, not necessarily a numerical error.

Derivation and model restrictions

The tangent method and direct nonlinear least squares may produce different estimates. Input saturation should be excluded. To turn a physical inventory into a differential state equation, write accumulation as incoming rate minus outgoing rate plus net production, if relevant. To turn an instrument or actuator into a dynamical element, specify its output, time constant and initial condition. To build a transfer function, take deviations about a steady operating point and verify that the assumed initial deviations are zero. Do not apply frequency-domain algebra blindly to a strongly nonlinear or switching regime.

For fopdt identification and uncertainty, the limiting checks should include a zero-change input, a long-time equilibrium where one exists and at least one infeasible input that the physical equipment cannot realize. If a parameter has dimensions of reciprocal time, use it only with a time expressed in matching units. If a parameter multiplies a temperature difference, its unit must convert the difference into a heat rate before the quantity is added to the energy balance. These simple checks detect mistakes that a graph renderer will not.

How to identify the model experimentally

Use the previously studied topic Second-order damping and overshoot as a source of prior knowledge and the next topic Disturbance and sinusoidal frequency response as an independent challenge. Define the reference operating point for fopdt identification and uncertainty in the process case and record a small safe input sequence, the actual actuator trajectory, the measurement times and instrument range. A step, a pulse and a slow ramp probe different properties, so do not collapse them into one claimed validation. If the fitted output differs from the recorded trace, inspect the timestamp alignment and sensor dynamics before increasing model order.

Where possible, obtain at least two input amplitudes. Normalize the measured output by the change in input and test whether the curves approximately collapse. Failure to collapse can indicate nonlinearity, saturation, temperature-dependent coefficients or changed initial conditions. A second holdout operating point is especially important when the main formula was derived by linearization. Report the residual error and uncertainty rather than relabeling an imperfect approximation as a universal physical law.

Uncertainty, troubleshooting, and design decisions

Failure mode to investigate: A gain sign error in process identification can turn intended negative feedback into positive feedback. The first diagnostic question is which conservation law, measurement chain or actuator path the proposed explanation implicates. Reconstruct the actual manipulated-variable history and compare it with what the controller requested; a large difference is a plant-side constraint, not a parameter-estimation error that can be fixed by fitting a more elaborate process transfer function.

Decision from the evidence: Report units, residuals, a holdout test, and confidence bounds rather than one over-precise fitted constant. State what the experiment supports and what remains unidentifiable. The decision should include whether the next useful action is to add a sensor, run a second input condition, change an assumed model structure or restrict the admissible operating range. When model outputs could guide actual equipment, independent engineering review and safety analysis are necessary before any implementation.

A challenge to the simplest explanation

Compare the present explanation with First-order step and impulse responses, which appears elsewhere in this unit. Imagine obtaining a smooth measured response that both models could plausibly produce. Choose one additional measurement that would change differently under the two explanations. Define the expected sign and time scale under each. For fopdt identification and uncertainty, record not just peak error but also the total actuator travel, disturbances, measurement noise and how the reference signal was generated. This exercise forces you to separate apparent goodness of fit from the ability to predict new conditions.

Numerical implementation and independent checks

The companion [interactive model, Laboratory 03](../labs/lab-03.html) is First-order plus dead time. Change one slider at a time, export the series and independently reproduce at least one value from a governing equation. The browser tool is deterministic and offline. It represents a defined educational model, not an approved process simulator. Its chart scale may rescale as parameters change, so compare numerical coordinates rather than visual slope alone.

For a numerical transient, verify step-size sensitivity by recomputing with half the time increment. If a predicted instability disappears merely when the integration time step shrinks, it was likely numerical rather than physical. On the other hand, an unstable physical model may diverge more quickly when more accurately integrated. Preserve initial conditions, parameter units and exact software version in the report so another student can repeat the test.

Practice and oral-examination questions

1. Starting from G(s)=K exp(-theta s)/(tau s+1), identify its state(s), manipulated input(s) and measured output(s). State how you would verify all dimensions.
2. Recalculate the example after a 10% increase of the principal process parameter. What changes in the predicted steady value and in the transient, and why are these not necessarily the same?
3. Propose a sensor or actuator failure that would make an otherwise correct fopdt identification and uncertainty model appear wrong. What extra logged variable would distinguish that failure?
4. Explain how First-order step and impulse responses might confound the identification of this lesson’s mechanism and propose a discriminating test.
5. State one safety or operating limit that the simplified G(s)=K exp(-theta s)/(tau s+1) relation does not capture. How should an engineer document that omission?

Answer guide and reflection

Question 1 is assessed by correct units and by separating the actual physical inventory from its measured signal; a memorized symbolic expression without those definitions is insufficient. For Question 2, the correct method uses the original equation twice and recomputes the equilibrium or time constant where applicable; simple proportional scaling is not accepted without showing that the equation is linear in the varied coefficient. For Question 3, the measurement or actuator diagnosis must lead to a different predicted trace. For Question 4, a proposed experiment is credible only if it can distinguish two candidate mechanisms. For Question 5, a constraint must be named and its omission explicitly stated.

Takeaway for Time-Domain and Frequency-Domain Responses. The lesson is complete only after the model can be challenged with a changed initial condition, a disturbance and a non-ideal measurement. A plotted agreement under one nominal set of synthetic conditions is the start of investigation, not its end. The next step is to test whether the assumptions retain their meaning when disturbance and sinusoidal frequency response is included in the model.

Unit 2, Lesson 4: Disturbance and sinusoidal frequency response

Unit: Time-Domain and Frequency-Domain Responses. Physical setting: an instrumented heat-exchanger outlet temperature step test. Reference model: |G(jw)|=K/sqrt(1+(w tau)^2).

Learning outcomes

By the end of this lesson you should be able to (i) explain disturbance and sinusoidal frequency response in physical terms, (ii) identify the variables and assumptions of |G(jw)|=K/sqrt(1+(w tau)^2), (iii) reproduce and challenge a numerical benchmark, (iv) recognize how measurement, delay and actuator limits modify the model, and (v) design a specific experiment that can distinguish the current model from an alternative interpretation.

Physical reasoning, not just a formula

Transient step tests emphasize low-frequency gain; sinusoidal forcing probes attenuation and phase shift. A system that passes slowly varying disturbances may still suppress faster oscillations or suffer added delay.

Consider the unit-wide equipment case: an instrumented heat-exchanger outlet temperature step test. In this setting the same numerical trend may have more than one physical cause. A change observed at the analyzer may reflect a real process response, a changed flow residence time or the instrument dynamics. The analyst must first specify which quantity is manipulated, which is observed and which variables are treated as external disturbances. This is particularly important for disturbance and sinusoidal frequency response, where confusing a state with a measured output can give a mathematically solvable but physically incorrect model.

The chosen governing expression is |G(jw)|=K/sqrt(1+(w tau)^2). It is a compact model, not a complete plant specification. Terms on opposite sides of an equality must have compatible units and consistent time bases. When a parameter is estimated from one transient, its result belongs to that operating range unless new data demonstrate transferability. The task is to explain the meaning of the relation before substituting numbers and then to compare its predictions with a checkable boundary condition or steady state.

Working the numerical reference case

Synthetic worked example. For K=3, tau=20 s and angular frequency 0.10 rad/s, amplitude ratio is 3/sqrt(5)=1.342 and phase shift is −atan(2)=−63.4°.

First transcribe every numerical input into a short table with its physical unit. Second, compute the baseline result without a software solver. Third, identify which output should change and which should remain unchanged when the manipulated input is zero. In this disturbance and sinusoidal frequency response exercise, the reported figures are not fitted plant data: they are an internally stated teaching benchmark. Any extension to different equipment requires new coefficients and new validation.

A useful sensitivity check is to change the selected coefficient or input by +10% while retaining all other conditions. This is not equivalent to assuming the output must also rise by 10%; exponential responses, ratios, saturation and feedback can change the mapping. Write the original expression, substitute the perturbed value and compare the sign of the result against physical intuition. Next repeat with a −10% change. An asymmetric response is evidence of nonlinearity in the selected mapping, not necessarily a numerical error.

Derivation and model restrictions

Steady harmonic formulas omit startup transients and rely on linearity at the forced amplitude. To turn a physical inventory into a differential state equation, write accumulation as incoming rate minus outgoing rate plus net production, if relevant. To turn an instrument or actuator into a dynamical element, specify its output, time constant and initial condition. To build a transfer function, take deviations about a steady operating point and verify that the assumed initial deviations are zero. Do not apply frequency-domain algebra blindly to a strongly nonlinear or switching regime.

For disturbance and sinusoidal frequency response, the limiting checks should include a zero-change input, a long-time equilibrium where one exists and at least one infeasible input that the physical equipment cannot realize. If a parameter has dimensions of reciprocal time, use it only with a time expressed in matching units. If a parameter multiplies a temperature difference, its unit must convert the difference into a heat rate before the quantity is added to the energy balance. These simple checks detect mistakes that a graph renderer will not.

How to identify the model experimentally

Use the previously studied topic FOPDT identification and uncertainty as a source of prior knowledge and the next topic Disturbance and sinusoidal frequency response as an independent challenge. Define the reference operating point for disturbance and sinusoidal frequency response in the process case and record a small safe input sequence, the actual actuator trajectory, the measurement times and instrument range. A step, a pulse and a slow ramp probe different properties, so do not collapse them into one claimed validation. If the fitted output differs from the recorded trace, inspect the timestamp alignment and sensor dynamics before increasing model order.

Where possible, obtain at least two input amplitudes. Normalize the measured output by the change in input and test whether the curves approximately collapse. Failure to collapse can indicate nonlinearity, saturation, temperature-dependent coefficients or changed initial conditions. A second holdout operating point is especially important when the main formula was derived by linearization. Report the residual error and uncertainty rather than relabeling an imperfect approximation as a universal physical law.

Uncertainty, troubleshooting, and design decisions

Failure mode to investigate: Fitting a curve at one frequency cannot identify both lag and dead time unambiguously. The first diagnostic question is which conservation law, measurement chain or actuator path the proposed explanation implicates. Reconstruct the actual manipulated-variable history and compare it with what the controller requested; a large difference is a plant-side constraint, not a parameter-estimation error that can be fixed by fitting a more elaborate process transfer function.

Decision from the evidence: Use at least three forcing frequencies with a stable operating point, allowing transients to decay before estimating gain. State what the experiment supports and what remains unidentifiable. The decision should include whether the next useful action is to add a sensor, run a second input condition, change an assumed model structure or restrict the admissible operating range. When model outputs could guide actual equipment, independent engineering review and safety analysis are necessary before any implementation.

A challenge to the simplest explanation

Compare the present explanation with Second-order damping and overshoot, which appears elsewhere in this unit. Imagine obtaining a smooth measured response that both models could plausibly produce. Choose one additional measurement that would change differently under the two explanations. Define the expected sign and time scale under each. For disturbance and sinusoidal frequency response, record not just peak error but also the total actuator travel, disturbances, measurement noise and how the reference signal was generated. This exercise forces you to separate apparent goodness of fit from the ability to predict new conditions.

Numerical implementation and independent checks

The companion [interactive model, Laboratory 05](../labs/lab-05.html) is Sinusoidal forcing and phase lag. Change one slider at a time, export the series and independently reproduce at least one value from a governing equation. The browser tool is deterministic and offline. It represents a defined educational model, not an approved process simulator. Its chart scale may rescale as parameters change, so compare numerical coordinates rather than visual slope alone.

For a numerical transient, verify step-size sensitivity by recomputing with half the time increment. If a predicted instability disappears merely when the integration time step shrinks, it was likely numerical rather than physical. On the other hand, an unstable physical model may diverge more quickly when more accurately integrated. Preserve initial conditions, parameter units and exact software version in the report so another student can repeat the test.

Practice and oral-examination questions

1. Starting from |G(jw)|=K/sqrt(1+(w tau)^2), identify its state(s), manipulated input(s) and measured output(s). State how you would verify all dimensions.
2. Recalculate the example after a 10% increase of the principal process parameter. What changes in the predicted steady value and in the transient, and why are these not necessarily the same?
3. Propose a sensor or actuator failure that would make an otherwise correct disturbance and sinusoidal frequency response model appear wrong. What extra logged variable would distinguish that failure?
4. Explain how Second-order damping and overshoot might confound the identification of this lesson’s mechanism and propose a discriminating test.
5. State one safety or operating limit that the simplified |G(jw)|=K/sqrt(1+(w tau)^2) relation does not capture. How should an engineer document that omission?

Answer guide and reflection

Question 1 is assessed by correct units and by separating the actual physical inventory from its measured signal; a memorized symbolic expression without those definitions is insufficient. For Question 2, the correct method uses the original equation twice and recomputes the equilibrium or time constant where applicable; simple proportional scaling is not accepted without showing that the equation is linear in the varied coefficient. For Question 3, the measurement or actuator diagnosis must lead to a different predicted trace. For Question 4, a proposed experiment is credible only if it can distinguish two candidate mechanisms. For Question 5, a constraint must be named and its omission explicitly stated.

Takeaway for Time-Domain and Frequency-Domain Responses. The lesson is complete only after the model can be challenged with a changed initial condition, a disturbance and a non-ideal measurement. A plotted agreement under one nominal set of synthetic conditions is the start of investigation, not its end. The next step is to test whether the assumptions retain their meaning when disturbance and sinusoidal frequency response is included in the model.

Unit 2 applied project: Time-Domain and Frequency-Domain Responses

Decision problem

You are an analyst assigned to an instrumented heat-exchanger outlet temperature step test. A proposed controller or model change claims better performance. Your job is not to accept the claim but to define a reproducible counterfactual using the physical assumptions of this unit, quantify expected benefits, identify failure modes and explain what evidence would change the decision. Use deliberately synthesized teaching data unless real data are supplied with permission.

Required technical workflow

Stage 1 — First-order step and impulse responses. Use y(t)=K Delta u [1-exp(-t/tau)] to predict a measurable effect. Calculate gain from the final difference, estimate tau from 63.2%, and reserve later test data for validation. Explain how forcing a non-exponential response into a one-lag model can hide parallel heat-transfer paths or a poorly mixed vessel. Use the example conditions in the lesson only as a preliminary reference case; the final project should vary at least two different operating conditions so that conclusions do not depend on one curve.

Stage 2 — Second-order damping and overshoot. Use G(s)=omega_n²/(s²+2 zeta omega_n s+omega_n²) to predict a measurable effect. Extract the first peak and oscillation period, then cross-check damping from multiple successive peaks. Explain how a measured ringing trace need not mean the process itself has two storage elements; controller tuning can create closed-loop oscillation. Use the example conditions in the lesson only as a preliminary reference case; the final project should vary at least two different operating conditions so that conclusions do not depend on one curve.

Stage 3 — FOPDT identification and uncertainty. Use G(s)=K exp(-theta s)/(tau s+1) to predict a measurable effect. Report units, residuals, a holdout test, and confidence bounds rather than one over-precise fitted constant. Explain how a gain sign error in process identification can turn intended negative feedback into positive feedback. Use the example conditions in the lesson only as a preliminary reference case; the final project should vary at least two different operating conditions so that conclusions do not depend on one curve.

Stage 4 — Disturbance and sinusoidal frequency response. Use |G(jw)|=K/sqrt(1+(w tau)^2) to predict a measurable effect. Use at least three forcing frequencies with a stable operating point, allowing transients to decay before estimating gain. Explain how fitting a curve at one frequency cannot identify both lag and dead time unambiguously. Use the example conditions in the lesson only as a preliminary reference case; the final project should vary at least two different operating conditions so that conclusions do not depend on one curve.

Evidence package

Provide a diagram with input, state, output and disturbance labels; a parameter dictionary with units and evidence status; the precise model equations; a csv or spreadsheet of numerical results; two labelled charts; and a structured comparison of nominal and perturbed cases. The input changes must be safe in the synthetic scenario and no physical equipment actuation is authorized by this assignment.

Model comparison and failure analysis

Compare a simple low-order model with a second model that adds one relevant limitation from first-order and second-order transients; dead time estimation; disturbances and periodic forcing. Keep the initial conditions and disturbance realization consistent. Calculate peak deviation, time to regain a chosen tolerance, integrated absolute error and the total excursion of the manipulated input when a controller is present. Explain any model disagreement using equation terms, not subjective plot descriptions.

Deliverables and evaluation

Submit a 3–5 page technical memo, runnable offline files or documented equations, a small data table and an audit statement. The recommended change must be limited to the modeled conditions. State at least two uncertainties, one additional validation experiment, and any safety question outside the authority of a classroom model. The grading priority is transparent reasoning and reproducibility, not a claimed industrial optimum.

Unit 2 assessment: Time-Domain and Frequency-Domain Responses

Assessment type: open-book problem set. Total: 16 tasks plus one integrative evaluation. Show reasoning, units, model assumptions, and at least one verification check for each numerical answer. Data are illustrative rather than plant measurements.

Part 1: First-order step and impulse responses

Question 1. Derive or interpret y(t)=K Delta u [1-exp(-t/tau)]. Define its coefficient units, the state variable, its initial condition and the operating point if linearized.

Question 2. Using the synthetic benchmark described in the corresponding lesson (If K=2 K/% and input changes 3%, the final temperature gain is 6 K. At tau=40 s, the step response is 6(1−exp(−1))=3.79 K above baseline.), repeat the arithmetic, then alter one explicitly stated input by +15% and explain the sign of the difference.

Question 3. Challenge the approximation: A real actuator step may be ramp-limited, and sampled measurements introduce time resolution and uncertainty. Describe an experiment, instrument placement and observation that would invalidate this assumption.

Question 4. Troubleshoot the known risk: Forcing a non-exponential response into a one-lag model can hide parallel heat-transfer paths or a poorly mixed vessel. Propose a controlled test and a quantitative metric by which two possible causes can be separated.

Part 2: Second-order damping and overshoot

Question 5. Derive or interpret G(s)=omega_n²/(s²+2 zeta omega_n s+omega_n²). Define its coefficient units, the state variable, its initial condition and the operating point if linearized.

Question 6. Using the synthetic benchmark described in the corresponding lesson (For zeta=0.40, percent overshoot is exp(-pi zeta/sqrt(1−zeta²))*100=25.4%. With omega_n=0.20 rad/s, the ideal first peak is t_p=pi/(omega_n sqrt(1−zeta²))=17.1 s.), repeat the arithmetic, then alter one explicitly stated input by +15% and explain the sign of the difference.

Question 7. Challenge the approximation: The stated overshoot relation requires a linear, normalized, no-zero, underdamped second-order step response. Describe an experiment, instrument placement and observation that would invalidate this assumption.

Question 8. Troubleshoot the known risk: A measured ringing trace need not mean the process itself has two storage elements; controller tuning can create closed-loop oscillation. Propose a controlled test and a quantitative metric by which two possible causes can be separated.

Part 3: FOPDT identification and uncertainty

Question 9. Derive or interpret G(s)=K exp(-theta s)/(tau s+1). Define its coefficient units, the state variable, its initial condition and the operating point if linearized.

Question 10. Using the synthetic benchmark described in the corresponding lesson (A 4% input step yields a 6 °C steady output change: K=1.5 °C/%. A discernible first response occurs after theta=12 s, and 63.2% of the remaining change is observed at t≈62 s, suggesting tau≈50 s.), repeat the arithmetic, then alter one explicitly stated input by +15% and explain the sign of the difference.

Question 11. Challenge the approximation: The tangent method and direct nonlinear least squares may produce different estimates. Input saturation should be excluded. Describe an experiment, instrument placement and observation that would invalidate this assumption.

Question 12. Troubleshoot the known risk: A gain sign error in process identification can turn intended negative feedback into positive feedback. Propose a controlled test and a quantitative metric by which two possible causes can be separated.

Part 4: Disturbance and sinusoidal frequency response

Question 13. Derive or interpret |G(jw)|=K/sqrt(1+(w tau)^2). Define its coefficient units, the state variable, its initial condition and the operating point if linearized.

Question 14. Using the synthetic benchmark described in the corresponding lesson (For K=3, tau=20 s and angular frequency 0.10 rad/s, amplitude ratio is 3/sqrt(5)=1.342 and phase shift is −atan(2)=−63.4°.), repeat the arithmetic, then alter one explicitly stated input by +15% and explain the sign of the difference.

Question 15. Challenge the approximation: Steady harmonic formulas omit startup transients and rely on linearity at the forced amplitude. Describe an experiment, instrument placement and observation that would invalidate this assumption.

Question 16. Troubleshoot the known risk: Fitting a curve at one frequency cannot identify both lag and dead time unambiguously. Propose a controlled test and a quantitative metric by which two possible causes can be separated.

Integrative design problem

For the case of an instrumented heat-exchanger outlet temperature step test, select two lessons from this unit and build a consistent process-and-measurement model. Include one realistic actuator bound, one sensor imperfection and one disturbance. Explain the trade-off between fast response and robust operation. Present at least one table of simulated values and one limiting-value check. Label synthetic data distinctly from measured data.

Marking rubric

• 30% governing equations and physical interpretation; 20% internally consistent numerical reasoning; 20% model/assumption checks; 20% alternative hypothesis and experiment design; 10% clear figures, units, and communication.

Examiner notes

Answers are not unique. A valid derivation states a control volume, assumptions, and dimensions before substituting numbers. Parameter changes should be recomputed using the actual model rather than asserted by proportion. If the student finds that the available inputs do not identify a required quantity, a defensible bound and an appropriate additional measurement earn more credit than an invented precise number.