Classical Lead, Lag, and PID Controller Design in Frequency Domain

Lesson 8.1 — Sampling and aliasing

Unit: Sampling, digital loops and practical verification. 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

Sampling converts a continuous signal to discrete observations. If frequency components above half the sample rate are present, they can alias into misleading lower frequencies. The ideal Nyquist sampling condition requires bandlimiting; physical sensors need an anti-alias analog filter with finite transition bandwidth.

The sample frequency and sensor filtering must be selected together. Increasing the controller update frequency does not fix aliasing already introduced by a slower upstream sensor.

Governing relationship: f_alias=|f_signal−k*f_s| for a suitable integer k

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

A 70 Hz sine sampled at 100 samples/s appears as a 30 Hz alias because |70−100|=30 Hz. A digital filter applied after sampling cannot reliably distinguish the original 70 Hz component from genuine 30 Hz content.

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 reconstructed signal can be corrupted by clock jitter, quantization or multi-rate timing. Do not interpret the alias formula as permission to omit physical filtering.

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 — networked laboratory rig

A PC, USB sensor and embedded actuator can each operate on different clocks; timestamps, buffering and missed packets create delays. The ideal sampled recurrence is a reference case rather than a complete architecture. A controlled test should log actual acquisition, computation and actuation times, not merely the nominal timer period.

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. For a 350 Hz unwanted component and a 200 Hz sampling rate, compute the apparent frequency and propose an upstream mitigation. 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, Zero-order hold and discrete poles. 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 8.2 — Zero-order hold and discrete poles

Unit: Sampling, digital loops and practical verification. 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 zero-order hold keeps each computed actuator value constant between updates. For y_dot=(u−y)/τ, exact integration over one constant-input interval yields y[k+1]=a y[k]+(1−a)u[k] with a=e^(−Ts/τ). This is not Euler integration; it exactly samples this ideal first-order plant.

The discrete pole is inside the unit circle because the continuous pole is stable. Feedback can alter that multiplier enough to create oscillation or divergence.

Governing relationship: a=exp(−Ts/tau); y[k+1]=ay[k]+(1−a)u[k]

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

With τ=0.5 s and Ts=0.1 s, a=e^(−0.2)=0.8187. Starting at y0=0 and holding u=1, y1=0.1813, y2=0.3297, matching the continuous response at the sample instants.

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 communication delay or actuator that does not hold constant invalidates this simple recurrence. Sample-to-sample agreement does not reveal intersample spikes from omitted fast dynamics.

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 — networked laboratory rig

A PC, USB sensor and embedded actuator can each operate on different clocks; timestamps, buffering and missed packets create delays. The ideal sampled recurrence is a reference case rather than a complete architecture. A controlled test should log actual acquisition, computation and actuation times, not merely the nominal timer period.

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. Compute the multiplier for Ts=0.25 s and predict how many sample intervals are needed to reach 63% of the final value. 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, Discrete controller implementation. 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 8.3 — Discrete controller implementation

Unit: Sampling, digital loops and practical verification. 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 sampled proportional law u[k]=Kp(r[k]−y[k]) combined with a zero-order-held plant produces a recursion that can be analyzed directly. Derive the coefficient rather than transplanting a continuous-time pole formula into a discrete controller. Stability requires discrete characteristic roots strictly inside the unit circle.

Fast sampling improves correspondence with continuous control only if latency and computation scale appropriately. A stable continuous controller does not automatically remain stable under an arbitrary digital update interval.

Governing relationship: For plant a=e^(−Ts/tau): y[k+1]=[a−(1−a)Kp]y[k]+(1−a)Kp*r[k]

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

Take a=0.8 and Kp=5. The closed-loop multiplier is 0.8−0.2×5=−0.2: stable but alternating. For Kp=10 the multiplier is −1.2, so errors grow in this ideal sampled model.

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

Direct-feedthrough plants and simultaneous samples can introduce algebraic-loop conventions. Real-time deadlines and jitter must be verified, not assumed.

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 — networked laboratory rig

A PC, USB sensor and embedded actuator can each operate on different clocks; timestamps, buffering and missed packets create delays. The ideal sampled recurrence is a reference case rather than a complete architecture. A controlled test should log actual acquisition, computation and actuation times, not merely the nominal timer period.

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 the exact stable gain interval for the coefficient a=0.8 under nonnegative proportional gain. 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, Validation, uncertainty and failure testing. 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 8.4 — Validation, uncertainty and failure testing

Unit: Sampling, digital loops and practical verification. 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 practical control workflow runs from model assumptions to simulation, offline tests, supervised low-energy experiments and documented acceptance criteria. Change one variable at a time when identifying a cause. Reserve separate data for validation and check behavior under reference changes, disturbances, sensor faults, actuator limits and communication dropouts.

Model fit and control safety are different kinds of evidence. A correct mathematical trace can be unusable when energy, thermal, mechanical or timing limits are violated.

Governing relationship: Performance acceptance example: max|e|, peak|u|, settling band, and overshoot specified before test

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 a specification requires |e|<0.05 after 3 s, |u|≤5, and no response excursion above 1.2 for a unit reference. A controller that passes settling time but demands u=7 fails the acceptance criterion.

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

No single test proves safety. Hardware safety analysis, emergency stop pathways, independent protection and relevant standards are outside the scope of this teaching simulator.

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 — networked laboratory rig

A PC, USB sensor and embedded actuator can each operate on different clocks; timestamps, buffering and missed packets create delays. The ideal sampled recurrence is a reference case rather than a complete architecture. A controlled test should log actual acquisition, computation and actuation times, not merely the nominal timer period.

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. Build a test matrix with at least six cases: nominal command, supply variation, sensor drift, loss of feedback, saturation, and sampling delay. State pass/fail signals for each. 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, Sampling and aliasing. 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 8 engineering project — Sampled proportional loop and unit-circle criterion

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

Use tau=0.5 s, Ts in {0.02,0.10,0.25} s and Kp in {0.5,2,5,10}. Compute a=exp(−Ts/tau), closed-loop multiplier a−(1−a)Kp, steady-state response and overshoot from the recurrence. Classify stable/unstable or marginal cases and give a timing-related reason a physical controller might disagree.

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

Sampling, intersample behavior and missed deadlines: Repeat a sampled-data experiment at several sample intervals and compare multipliers of the closed-loop discrete recurrence. Include an imposed one-update delay in a separate model to expose why timing assumptions matter. Document digital sensor scaling and the time at which a sample actually becomes available.

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 8 assessment — Sampling, digital loops and practical verification

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, Ts=0.1. 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 a=0.8, Kp=5. 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 a=0.8, Kp=10. 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 f=70 Hz, fs=100 Hz. 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. Sampling and aliasing. For a 350 Hz unwanted component and a 200 Hz sampling rate, compute the apparent frequency and propose an upstream mitigation. 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. Zero-order hold and discrete poles. Compute the multiplier for Ts=0.25 s and predict how many sample intervals are needed to reach 63% of the final value. 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. Discrete controller implementation. Solve the exact stable gain interval for the coefficient a=0.8 under nonnegative proportional gain. 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. Validation, uncertainty and failure testing. Build a test matrix with at least six cases: nominal command, supply variation, sensor drift, loss of feedback, saturation, and sampling delay. State pass/fail signals for each. 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, Ts=0.1, the compact result is a≈0.81873. 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 a=0.8, Kp=5, the compact result is closed-loop multiplier −0.2, stable. 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 a=0.8, Kp=10, the compact result is multiplier −1.2, unstable. 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 f=70 Hz, fs=100 Hz, the compact result is alias 30 Hz. 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 8.1–8.4.

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.