Transfer Functions, Block Diagram Algebra, and Signal Flow Graphs

Lesson 2.1 — First-order transients and time constants

Unit: Time responses and second-order dynamics. Prerequisites: differential equations, complex numbers, and the earlier lessons in the unit. Study: derive the relationship, reproduce the numerical case, investigate one changed assumption and complete the four problems. All numbers are synthetic and the models are not validated for hardware use.

Physical meaning and governing model

The step response of K/(tau s+1) from rest is y(t)=KA[1−exp(−t/tau)] for a step amplitude A. The time constant marks 63.2% of the total change; it is not the settling time. For a monotone exponential, 2% settling occurs around 3.91 tau.

This directly links a pole at −1/0.75=−1.333 rad/s to a measured response. Two step amplitudes can test linearity: if their normalized curves differ, the first-order LTI model is doubtful.

Governing relationship: y(t)=KA(1−exp(−t/tau))

The formula should not be used until its input, output, initial state, sign convention and units have been declared. Distinguish a plant characteristic from the characteristic of a feedback interconnection. An algebraically valid transfer function is not an empirical model until its parameters and operating regime have supporting evidence. If an idealized element has no physical saturation or bandwidth, identify the missing implementation constraint before attempting design.

Worked example and interpretation

For K=2, tau=0.75 s and A=1.5, the final output is 3. At t=1.5 s the response is 3(1−e^(−2))≈2.594. The nominal 2% settling time is 2.93 s.

Check the magnitude against a relevant limit. In a continuous model, a pole measured in reciprocal seconds gives a time constant through its reciprocal; in a digital model, a pole must be inside the unit circle. A calculation of an output amplitude does not identify a suitable control input unless the actuator relationship is also known. For this particular example, copy the expression into a fresh worksheet, write the intermediate substitution with units and test whether the result changes in the expected direction when one parameter changes.

Limits, counterexamples and model evidence

Dead time, sensor smoothing and actuator slew rate can mimic a larger time constant. Fit several data points rather than relying on one 63% crossing.

A useful counterexample changes one important feature at a time: initial state, operating point, saturation level, sensor filtering, delay or an unmodeled fast mode. The fact that a trace is well-fitted is not proof of predictive validity. A good validation experiment uses data withheld during fitting; it should also record the actual input, since the requested command may not equal the delivered signal. If evidence is missing, write a conditional conclusion instead of inventing a parameter or tolerance.

Application perspective — temperature control experiment

The actuator might be a resistive heater with finite maximum power, and the temperature sensor may have its own lag. Comparing the heater input to the measured chamber temperature without separating these dynamics can produce a false plant time constant. Thermal capacitance also depends on which masses reach approximately the same temperature during a short pulse.

The experiment should define the performance metric before looking at a plot. Appropriate measures differ by task: a transient model can be checked by peak, delay and settling behavior; a frequency response by gain and phase; a state model by observable signals and sensitivity to starting conditions. A controller that improves one metric but violates current, force, thermal, bandwidth or latency limits is not an acceptable solution. Explain which additional measurement best distinguishes the model from its main competing explanation.

Problems for independent work

1. Computation and sensitivity. A recorded output reaches 95% in 2.5 s. Estimate tau and compare it with the estimate from a noisy 63% crossing. Show how the relevant parameter enters the governing equation. If the data do not determine a single number, list the missing information and work symbolically.

2. Measurement planning. Identify the dimensional parameters and state what changes if all time units are reported in milliseconds rather than seconds. Explain how the proposed measurement separates a parameter estimate from an unobserved initial condition.

3. Failure analysis. Describe an independent way to check the worked calculation without copying the exact procedure in the text. Trace the proposed change through input, state and measured output rather than saying only that “the system may become unstable.”

4. Comparison. Relate this model to the following lesson, Second-order damping and natural frequency. Name one assumption shared by the two topics and one assumption requiring a different experiment or mathematical analysis.

Answer guidance and reproducibility checklist

A convincing solution starts with the original relation, defines the physical symbols, substitutes the scenario values with consistent units and checks the limit or sign of the result. For the sensitivity problem, use at least two plausible values and state which conclusion stays invariant. For the measurement problem, specify a real sensor signal and a time or frequency range. For the failure problem, identify the omitted dynamic or nonlinearity and predict the direction of at least one observable discrepancy. A comparison should not assume the next lesson solves the same question. Preserve input and output data, disclose parameter fitting, and independently recompute one representative point. Remember that numerical plausibility does not validate equipment tuning or safe deployment.

Lesson 2.2 — Second-order damping and natural frequency

Unit: Time responses and second-order dynamics. Prerequisites: differential equations, complex numbers, and the earlier lessons in the unit. Study: derive the relationship, reproduce the numerical case, investigate one changed assumption and complete the four problems. All numbers are synthetic and the models are not validated for hardware use.

Physical meaning and governing model

The canonical second-order denominator s²+2ζωn s+ωn² separates natural frequency from damping ratio. Under 0<ζ<1 the response oscillates with damped frequency ωd=ωn sqrt(1−ζ²). For ζ≥1 the real modes are nonoscillatory.

Overshoot depends on damping much more directly than on the natural frequency; increasing ωn mainly changes the clock speed for this canonical normalized model.

Governing relationship: Mp=exp(−πζ/sqrt(1−ζ²)); omega_d=omega_nsqrt(1−ζ²)

The formula should not be used until its input, output, initial state, sign convention and units have been declared. Distinguish a plant characteristic from the characteristic of a feedback interconnection. An algebraically valid transfer function is not an empirical model until its parameters and operating regime have supporting evidence. If an idealized element has no physical saturation or bandwidth, identify the missing implementation constraint before attempting design.

Worked example and interpretation

At ζ=0.4 and ωn=5 rad/s, the overshoot fraction is exp(−1.371)≈0.254, or 25.4%. The oscillation frequency is 4.583 rad/s; nominal 2% settling time 4/(ζωn)=2 s.

Check the magnitude against a relevant limit. In a continuous model, a pole measured in reciprocal seconds gives a time constant through its reciprocal; in a digital model, a pole must be inside the unit circle. A calculation of an output amplitude does not identify a suitable control input unless the actuator relationship is also known. For this particular example, copy the expression into a fresh worksheet, write the intermediate substitution with units and test whether the result changes in the expected direction when one parameter changes.

Limits, counterexamples and model evidence

A zero in the numerator, input smoothing, saturation or a third pole can invalidate canonical overshoot formulas. A reduced-order fit is a hypothesis to be checked.

A useful counterexample changes one important feature at a time: initial state, operating point, saturation level, sensor filtering, delay or an unmodeled fast mode. The fact that a trace is well-fitted is not proof of predictive validity. A good validation experiment uses data withheld during fitting; it should also record the actual input, since the requested command may not equal the delivered signal. If evidence is missing, write a conditional conclusion instead of inventing a parameter or tolerance.

Application perspective — temperature control experiment

The actuator might be a resistive heater with finite maximum power, and the temperature sensor may have its own lag. Comparing the heater input to the measured chamber temperature without separating these dynamics can produce a false plant time constant. Thermal capacitance also depends on which masses reach approximately the same temperature during a short pulse.

The experiment should define the performance metric before looking at a plot. Appropriate measures differ by task: a transient model can be checked by peak, delay and settling behavior; a frequency response by gain and phase; a state model by observable signals and sensitivity to starting conditions. A controller that improves one metric but violates current, force, thermal, bandwidth or latency limits is not an acceptable solution. Explain which additional measurement best distinguishes the model from its main competing explanation.

Problems for independent work

1. Computation and sensitivity. Solve for ζ when overshoot must not exceed 10%, and discuss whether raising ωn alone can meet that bound. Show how the relevant parameter enters the governing equation. If the data do not determine a single number, list the missing information and work symbolically.

2. Measurement planning. Construct a plausible situation in which the model prediction fails, naming the omitted state, nonlinearity or timing element. Explain how the proposed measurement separates a parameter estimate from an unobserved initial condition.

3. Failure analysis. Name a second performance variable that could become worse while the headline performance metric improves. Trace the proposed change through input, state and measured output rather than saying only that “the system may become unstable.”

4. Comparison. Relate this model to the following lesson, Higher-order modes and approximations. Name one assumption shared by the two topics and one assumption requiring a different experiment or mathematical analysis.

Answer guidance and reproducibility checklist

A convincing solution starts with the original relation, defines the physical symbols, substitutes the scenario values with consistent units and checks the limit or sign of the result. For the sensitivity problem, use at least two plausible values and state which conclusion stays invariant. For the measurement problem, specify a real sensor signal and a time or frequency range. For the failure problem, identify the omitted dynamic or nonlinearity and predict the direction of at least one observable discrepancy. A comparison should not assume the next lesson solves the same question. Preserve input and output data, disclose parameter fitting, and independently recompute one representative point. Remember that numerical plausibility does not validate equipment tuning or safe deployment.

Lesson 2.3 — Higher-order modes and approximations

Unit: Time responses and second-order dynamics. Prerequisites: differential equations, complex numbers, and the earlier lessons in the unit. Study: derive the relationship, reproduce the numerical case, investigate one changed assumption and complete the four problems. All numbers are synthetic and the models are not validated for hardware use.

Physical meaning and governing model

Real plants often have many poles. A dominant-pole approximation is warranted only when faster modes contribute little at the frequencies, initial states and output of interest. The residue attached to a pole matters as much as its location; a fast pole with large residue can affect the initial slope.

At low frequencies the product has nearly the same gain as the slow stage, but near the fast pole there is extra phase lag. That lag may matter in feedback even if the plot of an open-loop step seems close.

Governing relationship: G(s)=K/[(tau1s+1)(tau2s+1)]

The formula should not be used until its input, output, initial state, sign convention and units have been declared. Distinguish a plant characteristic from the characteristic of a feedback interconnection. An algebraically valid transfer function is not an empirical model until its parameters and operating regime have supporting evidence. If an idealized element has no physical saturation or bandwidth, identify the missing implementation constraint before attempting design.

Worked example and interpretation

For tau1=1.0 s and tau2=0.1 s, the slow pole is −1 rad/s and the fast pole −10 rad/s. Replacing the fast stage by unity may give acceptable late-time behavior while mispredicting the first few tenths of a second.

Check the magnitude against a relevant limit. In a continuous model, a pole measured in reciprocal seconds gives a time constant through its reciprocal; in a digital model, a pole must be inside the unit circle. A calculation of an output amplitude does not identify a suitable control input unless the actuator relationship is also known. For this particular example, copy the expression into a fresh worksheet, write the intermediate substitution with units and test whether the result changes in the expected direction when one parameter changes.

Limits, counterexamples and model evidence

Do not eliminate an unstable pole or a nonminimum-phase zero merely because it appears fast. Model reduction should preserve the unstable and critical dynamics explicitly.

A useful counterexample changes one important feature at a time: initial state, operating point, saturation level, sensor filtering, delay or an unmodeled fast mode. The fact that a trace is well-fitted is not proof of predictive validity. A good validation experiment uses data withheld during fitting; it should also record the actual input, since the requested command may not equal the delivered signal. If evidence is missing, write a conditional conclusion instead of inventing a parameter or tolerance.

Application perspective — temperature control experiment

The actuator might be a resistive heater with finite maximum power, and the temperature sensor may have its own lag. Comparing the heater input to the measured chamber temperature without separating these dynamics can produce a false plant time constant. Thermal capacitance also depends on which masses reach approximately the same temperature during a short pulse.

The experiment should define the performance metric before looking at a plot. Appropriate measures differ by task: a transient model can be checked by peak, delay and settling behavior; a frequency response by gain and phase; a state model by observable signals and sensitivity to starting conditions. A controller that improves one metric but violates current, force, thermal, bandwidth or latency limits is not an acceptable solution. Explain which additional measurement best distinguishes the model from its main competing explanation.

Problems for independent work

1. Computation and sensitivity. Bound the extra phase lag of the 0.1 s pole at ω=1, 5 and 10 rad/s and decide at what crossover it becomes significant. Show how the relevant parameter enters the governing equation. If the data do not determine a single number, list the missing information and work symbolically.

2. Measurement planning. Describe an independent way to check the worked calculation without copying the exact procedure in the text. Explain how the proposed measurement separates a parameter estimate from an unobserved initial condition.

3. Failure analysis. Explain how an engineer could communicate the numerical result without presenting it as a certified equipment setting. Trace the proposed change through input, state and measured output rather than saying only that “the system may become unstable.”

4. Comparison. Relate this model to the following lesson, Interpreting response measurements. Name one assumption shared by the two topics and one assumption requiring a different experiment or mathematical analysis.

Answer guidance and reproducibility checklist

A convincing solution starts with the original relation, defines the physical symbols, substitutes the scenario values with consistent units and checks the limit or sign of the result. For the sensitivity problem, use at least two plausible values and state which conclusion stays invariant. For the measurement problem, specify a real sensor signal and a time or frequency range. For the failure problem, identify the omitted dynamic or nonlinearity and predict the direction of at least one observable discrepancy. A comparison should not assume the next lesson solves the same question. Preserve input and output data, disclose parameter fitting, and independently recompute one representative point. Remember that numerical plausibility does not validate equipment tuning or safe deployment.

Lesson 2.4 — Interpreting response measurements

Unit: Time responses and second-order dynamics. Prerequisites: differential equations, complex numbers, and the earlier lessons in the unit. Study: derive the relationship, reproduce the numerical case, investigate one changed assumption and complete the four problems. All numbers are synthetic and the models are not validated for hardware use.

Physical meaning and governing model

A measured rise-time or settling-time number is not self-defining. State the command shape, initial condition, final-value estimator, tolerance band, sample interval, filtering, and whether the signal is noisy. Fit a physically based model and report uncertainty instead of declaring a single beautiful trace to be proof.

The RMSE reports overall fit but not systematic lag. Plot residuals against time: an alternating sign pattern may reveal a missed oscillatory mode and a long same-sign segment can reveal sensor bias.

Governing relationship: e(t)=y_measured(t)−y_model(t); RMSE=sqrt(mean(e(t)^2))

The formula should not be used until its input, output, initial state, sign convention and units have been declared. Distinguish a plant characteristic from the characteristic of a feedback interconnection. An algebraically valid transfer function is not an empirical model until its parameters and operating regime have supporting evidence. If an idealized element has no physical saturation or bandwidth, identify the missing implementation constraint before attempting design.

Worked example and interpretation

Suppose five residuals in volts are [0.04, −0.02, 0.05, −0.03, 0.01]. Their squared sum is 0.0055 V²; RMSE=sqrt(0.0055/5)≈0.0332 V.

Check the magnitude against a relevant limit. In a continuous model, a pole measured in reciprocal seconds gives a time constant through its reciprocal; in a digital model, a pole must be inside the unit circle. A calculation of an output amplitude does not identify a suitable control input unless the actuator relationship is also known. For this particular example, copy the expression into a fresh worksheet, write the intermediate substitution with units and test whether the result changes in the expected direction when one parameter changes.

Limits, counterexamples and model evidence

Training and checking a model on the same test trace overstates predictive confidence. Repeat at new command levels and operating temperatures if safe.

A useful counterexample changes one important feature at a time: initial state, operating point, saturation level, sensor filtering, delay or an unmodeled fast mode. The fact that a trace is well-fitted is not proof of predictive validity. A good validation experiment uses data withheld during fitting; it should also record the actual input, since the requested command may not equal the delivered signal. If evidence is missing, write a conditional conclusion instead of inventing a parameter or tolerance.

Application perspective — temperature control experiment

The actuator might be a resistive heater with finite maximum power, and the temperature sensor may have its own lag. Comparing the heater input to the measured chamber temperature without separating these dynamics can produce a false plant time constant. Thermal capacitance also depends on which masses reach approximately the same temperature during a short pulse.

The experiment should define the performance metric before looking at a plot. Appropriate measures differ by task: a transient model can be checked by peak, delay and settling behavior; a frequency response by gain and phase; a state model by observable signals and sensitivity to starting conditions. A controller that improves one metric but violates current, force, thermal, bandwidth or latency limits is not an acceptable solution. Explain which additional measurement best distinguishes the model from its main competing explanation.

Problems for independent work

1. Computation and sensitivity. Fit one time constant using the first half of a hypothetical trace; state which second-half observations would falsify your model. Show how the relevant parameter enters the governing equation. If the data do not determine a single number, list the missing information and work symbolically.

2. Measurement planning. Name a second performance variable that could become worse while the headline performance metric improves. Explain how the proposed measurement separates a parameter estimate from an unobserved initial condition.

3. Failure analysis. Which measured quantity would most directly separate an initial-condition error from a plant-parameter error? Give the signal units and observation window. Trace the proposed change through input, state and measured output rather than saying only that “the system may become unstable.”

4. Comparison. Relate this model to the following lesson, First-order transients and time constants. Name one assumption shared by the two topics and one assumption requiring a different experiment or mathematical analysis.

Answer guidance and reproducibility checklist

A convincing solution starts with the original relation, defines the physical symbols, substitutes the scenario values with consistent units and checks the limit or sign of the result. For the sensitivity problem, use at least two plausible values and state which conclusion stays invariant. For the measurement problem, specify a real sensor signal and a time or frequency range. For the failure problem, identify the omitted dynamic or nonlinearity and predict the direction of at least one observable discrepancy. A comparison should not assume the next lesson solves the same question. Preserve input and output data, disclose parameter fitting, and independently recompute one representative point. Remember that numerical plausibility does not validate equipment tuning or safe deployment.

Unit 2 engineering project — Second-order design envelope

Research question. Can a compact model predict a second operating condition reliably, and what do its residuals mean? This is a synthetic computation-only assignment. No energized or safety-critical tests are required.

Case data and required work

Analyze canonical step responses for ζ in {0.2,0.4,0.7} and ωn in {2,5,8} rad/s. For each combination estimate overshoot, nominal settling time and peak time. Compare analytic formulas against numerical response points, note any mismatch, and identify one design satisfying overshoot below 15% and nominal settling faster than 2 s, if feasible.

Reproducible workflow

1. Prepare a model card containing the independent variable, measured output, units, initial conditions, parameter values and restrictions. Distinguish fixed known parameters from values estimated using the training observations. Set a reproducible numerical tolerance and record tool versions if code is used.
2. Write the parent differential equation or characteristic relationship. Derive one check by hand and one by independent numerical evaluation. Use enough precision internally to avoid arithmetic drift, but round reported results to the underlying input precision.
3. Build a graph that includes a physically meaningful axis range, a legend, a command trace, model output and any withheld validation output. If a logarithmic frequency axis is used, never include zero frequency as a point on that axis.
4. Investigate at least two plausible parameter alternatives. Report which output feature changes most and whether that sensitivity points to an important new measurement. If a parameter cannot be estimated uniquely, say why instead of choosing a convenient number.
5. Document the principal discrepancy between model and validation trace. State whether the discrepancy is likely random noise, a coherent time delay, a calibration offset, unmodeled dynamics or a regime change. Propose a specific follow-up experiment capable of distinguishing at least two possibilities.
6. Finish with a decision memo stating what the experiment establishes, what it cannot establish, and whether further evidence is needed before any real use.

Measurement and interpretation lens

Response metrics and parameter uncertainty: Measure the final value using a plateau only when there is evidence that the system really reached equilibrium. Record peak timing, peak amplitude, ringing, 10–90% rise time and a precisely defined settling band. Perturb estimated damping and natural frequency separately to see which measurement responds most.

The baseline computation should be viewed as a model prediction, not a design certificate. The next experiment should be chosen for information gain: an extra trace is helpful only if it can distinguish competing explanations. A practical model is sometimes deliberately simpler than the physical plant; defend simplification on the basis of bandwidth, residuals, risk, and data rather than convenience. Be explicit about initial states, since two identical inputs can yield different measured trajectories when starting conditions differ.

Required submitted artifacts

• A short derivation with dimensions, sign convention and limits.
• A machine-readable table of inputs and outputs, with units in headers and an explanation of any synthetic perturbation.
• Two labeled plots: an explanatory baseline plot and a separate uncertainty or hold-out prediction plot.
• An audit table separating assumptions, directly measured quantities, fitted parameters and conditions outside the model.
• One paragraph on model usefulness; one paragraph on the strongest counterexample or failure scenario.

Evaluation rubric and common mistakes

25% theory: valid differential equation, transfer function, characteristic equation or discrete update with correct signals. 25% arithmetic: independently checked numerical values and declared units. 20% evaluation: withheld-case prediction, uncertainty and an honestly labeled residual plot. 15% communication: reproducible data, readable graphs and explicit limitations. 15% safe experimental reasoning: no substitution of classroom simulation for real commissioning or protective design.

A common failure is tuning a parameter on every available point and then citing the same points as validation. Another is reporting an impressive percentage error despite a missing or nearly zero denominator. Do not hide cases with poor fit. If the model predicts instability or impossible inputs, document the finding and stop at simulation; do not seek to reproduce an unsafe condition on a real instrument.

Unit 2 assessment — Time responses and second-order dynamics

Requirements: Independently show calculations and state assumptions; answers should have units. These numbers are invented teaching examples, not equipment recommendations. Suggested duration: 2–3 hours plus an optional computational verification.

Part A — short quantitative checks

A1. Analyze the case τ=0.5 s. State the relevant relationship, what physical variable is being calculated, and the result with correct dimensions. Explain one assumption required for the answer and one reason the same expression would be inappropriate for a more complicated plant.

A2. Analyze the case ζ=0.5, ωn=4. State the relevant relationship, what physical variable is being calculated, and the result with correct dimensions. Explain one assumption required for the answer and one reason the same expression would be inappropriate for a more complicated plant.

A3. Analyze the case ζ=0.6. State the relevant relationship, what physical variable is being calculated, and the result with correct dimensions. Explain one assumption required for the answer and one reason the same expression would be inappropriate for a more complicated plant.

A4. Analyze the case pole −2 rad/s. State the relevant relationship, what physical variable is being calculated, and the result with correct dimensions. Explain one assumption required for the answer and one reason the same expression would be inappropriate for a more complicated plant.

Part B — synthesis problems

B1. First-order transients and time constants. A recorded output reaches 95% in 2.5 s. Estimate tau and compare it with the estimate from a noisy 63% crossing. Derive or select the governing equation; include at least one dimensional or limiting-case check. Contrast the method with the previous lesson in the unit. What extra measured variable would most reduce uncertainty? State when the calculation is valid and when it is not.

B2. Second-order damping and natural frequency. Solve for ζ when overshoot must not exceed 10%, and discuss whether raising ωn alone can meet that bound. Derive or select the governing equation; include at least one dimensional or limiting-case check. Contrast the method with the previous lesson in the unit. What extra measured variable would most reduce uncertainty? State when the calculation is valid and when it is not.

B3. Higher-order modes and approximations. Bound the extra phase lag of the 0.1 s pole at ω=1, 5 and 10 rad/s and decide at what crossover it becomes significant. Derive or select the governing equation; include at least one dimensional or limiting-case check. Contrast the method with the previous lesson in the unit. What extra measured variable would most reduce uncertainty? State when the calculation is valid and when it is not.

B4. Interpreting response measurements. Fit one time constant using the first half of a hypothetical trace; state which second-half observations would falsify your model. Derive or select the governing equation; include at least one dimensional or limiting-case check. Contrast the method with the previous lesson in the unit. What extra measured variable would most reduce uncertainty? State when the calculation is valid and when it is not.

Part C — model criticism and experimental planning

C1. A student produces a smooth simulated plot with an error below 1% on synthetic training data and claims the controller can be installed. Identify at least four missing kinds of evidence, including one physical limit and one timing or sensing issue. Explain how these could change the outcome even without changing the algebra.

C2. Design a safe, offline-only demonstration for one of the four lessons. The demonstration should include at least three inputs, a falsifiable hypothesis, a baseline outcome and a deliberately perturbed parameter. State how you would decide which hypothesis survived and why the experiment does not authorize real-machine deployment.

C3. Compare two candidate models that fit a single step response equally well but have different unobserved dynamics. Propose a measurement or excitation that could distinguish them without relying on an unsafe extreme command. Discuss whether the problem is identification, sensing, control design or all three.

Instructor reference answers and marking guidance

A1 key. For τ=0.5 s, the compact result is 2% settling approximately 1.96 seconds. Require the student's derivation and units, not just this answer. The explanation should note the idealized assumption—linear time-invariant dynamics, zero initial state where applicable, or the explicitly chosen sign and sampling convention. An unmodeled delay, actuation limit or sensor error can invalidate a direct application.

A2 key. For ζ=0.5, ωn=4, the compact result is damped frequency approximately 3.464 rad/s. Require the student's derivation and units, not just this answer. The explanation should note the idealized assumption—linear time-invariant dynamics, zero initial state where applicable, or the explicitly chosen sign and sampling convention. An unmodeled delay, actuation limit or sensor error can invalidate a direct application.

A3 key. For ζ=0.6, the compact result is canonical overshoot about 9.48%. Require the student's derivation and units, not just this answer. The explanation should note the idealized assumption—linear time-invariant dynamics, zero initial state where applicable, or the explicitly chosen sign and sampling convention. An unmodeled delay, actuation limit or sensor error can invalidate a direct application.

A4 key. For pole −2 rad/s, the compact result is e-fold decay time 0.5 seconds. Require the student's derivation and units, not just this answer. The explanation should note the idealized assumption—linear time-invariant dynamics, zero initial state where applicable, or the explicitly chosen sign and sampling convention. An unmodeled delay, actuation limit or sensor error can invalidate a direct application.

B series guidance. Mark four components independently: (i) a correct relationship and explanation of variables; (ii) a quantitative or logically determined result where data suffice; (iii) a nontrivial limiting-case or alternative-parameter test; and (iv) a credible discrimination experiment. A general statement such as “the model may be inaccurate” earns little without identifying an omitted mechanism. The expected examples include the unique cases from Lessons 2.1–2.4.

C series guidance. Real evidence would require verified sensor scale, actual input authority, operating envelopes, repeat tests, missing-mode investigation, documented failure handling and independent oversight. The strongest student answers name the input and output at which a discrepancy would appear, not only a generic safety list. Encourage responses that reject an overly precise conclusion when the model does not identify the needed variable.

Short self-evaluation

After submission, label each conclusion as an identity derived from the mathematical model, a numerical result for synthetic assumptions, an empirical observation, or an engineering judgment. Confusing these categories is more serious than a minor arithmetic slip. Check every angular-frequency value against its rad/s unit and every sampled quantity against the actual sample period. Where two tests support different conclusions, do not suppress the disagreement; state the evidence gap.