Time-Domain Performance of First- and Second-Order Systems

Lesson 3.1 — Closed-loop transfer equations

Unit: Feedback, sensitivity and tracking. 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

Write the summing junction before manipulating blocks. For negative feedback with a unity sensor, E=R−Y and U=CE; the plant output Y=P(U+D_in)+D_out. The disturbance location decides whether it is multiplied by P. Distinguish reference tracking from disturbances rather than calling all inputs commands.

The three transfer functions share a closed-loop denominator but carry different numerators. This distinction predicts which signals are removed well at low frequency and which are not.

Governing relationship: Y=TR+PSD_in+SD_out; S=1/(1+PC)

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

Use P=2/(s+2), C=4. Then T=8/(s+10), S=(s+2)/(s+10) and P S=2/(s+10). At DC, the reference gain is 0.8 and an output disturbance is attenuated to 0.2 of its open-loop effect.

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

This result uses linear superposition and ignores actuator limits. A measurement sensor H adds another dynamic block and complicates the algebra.

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 — servo-position response

A small motor driving a geared carriage may exhibit backlash, Coulomb friction and torque limits. A linear oscillator with fitted poles can predict one operating regime well yet fail for a reversal. Distinguish a command that tests tracking from an external load that tests disturbance rejection. A fast step could be physically inadmissible when it exceeds travel or force limits.

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. Insert a measurement-noise term n into e=r−(y+n); derive the noise-to-output transfer and explain its sign. 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, Sensitivity and disturbance rejection. 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 3.2 — Sensitivity and disturbance rejection

Unit: Feedback, sensitivity and tracking. 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

Sensitivity S=1/(1+L) measures how the closed-loop response changes with certain normalized plant perturbations and output disturbances. Complementary sensitivity T=L/(1+L) describes reference tracking and the transmission of measurement noise in a common architecture. Both are frequency-dependent complex functions.

High loop gain by magnitude alone does not ensure low sensitivity; phase may place the loop close to its critical point. Disturbance rejection is a vector relationship in the complex plane.

Governing relationship: S(s)+T(s)=1; L(s)=P(s)C(s)

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 a frequency where L(jω)=9 is purely real, S=0.1 and T=0.9. If L(jω)=−0.9 instead, the sensitivity magnitude is 10 despite the small loop magnitude change: proximity to −1 matters.

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

Large sensitivity peaks amplify modeling errors and may signal poor robustness. Loop-shaping objectives at low and high frequency conflict if sensor noise and delay are significant.

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 — servo-position response

A small motor driving a geared carriage may exhibit backlash, Coulomb friction and torque limits. A linear oscillator with fitted poles can predict one operating regime well yet fail for a reversal. Distinguish a command that tests tracking from an external load that tests disturbance rejection. A fast step could be physically inadmissible when it exceeds travel or force limits.

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. Draw S and T magnitude across a chosen crossover, and identify why both cannot be arbitrarily small at the same frequency. 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, Steady-state error and system type. 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 3.3 — Steady-state error and system type

Unit: Feedback, sensitivity and tracking. 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

Final error can often be derived with the final-value theorem using E(s)=R(s)/(1+L(s)) for unity feedback. A type-0 loop has finite position constant Kp; a type-1 loop contains one integrator and can ideally eliminate constant step error when stable. Ramp tracking involves the velocity error constant.

Adding an integrator improves certain low-frequency errors but changes loop order, phase and recovery from saturation. Tracking claims require a stable closed loop and feasible control action.

Governing relationship: e_ss(step)=1/(1+lim[s→0]L(s)); e_ss(ramp)=1/lim[s→0]s*L(s)

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 L(s)=10/(s+2), Kp=5 and e_ss for a unit step is 1/6≈0.1667. For L(s)=6/[s(s+3)], Kv=2 s⁻¹ and the unit-ramp error is 0.5.

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

The final-value theorem is invalid if poles of sE(s) violate its stability conditions. An integrator does not remove sensor bias or force an impossible constant output.

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 — servo-position response

A small motor driving a geared carriage may exhibit backlash, Coulomb friction and torque limits. A linear oscillator with fitted poles can predict one operating regime well yet fail for a reversal. Distinguish a command that tests tracking from an external load that tests disturbance rejection. A fast step could be physically inadmissible when it exceeds travel or force limits.

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. Compare the step and ramp errors of the two given loops; identify which design objective actually motivated the integrator. 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, Actuator constraints and sensor noise. 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 3.4 — Actuator constraints and sensor noise

Unit: Feedback, sensitivity and tracking. 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

An actuator converts the computed command into a limited physical action. A digital-to-analog command can clip, slew, quantize or be delayed; a sensor can have gain bias and noise. A linear transfer diagram without these elements may predict accurate tracking while the real signal stalls.

A clipped controller is nonlinear: doubling a reference does not necessarily double a response. Noise may drive high-frequency control effort even when output noise looks small due to plant filtering.

Governing relationship: u_actual=clip(u_cmd,−Umax,+Umax)

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 temperature control plant requires steady input u=8 units for a target but the actuator maximum is 6. No integral gain can establish the requested equilibrium under that operating condition. The feasibility error precedes controller tuning.

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

Noise spectra matter more than a single root-mean-square value. Test practical input ranges, sensor disconnects, bias and stuck-actuator faults before any deployment.

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 — servo-position response

A small motor driving a geared carriage may exhibit backlash, Coulomb friction and torque limits. A linear oscillator with fitted poles can predict one operating regime well yet fail for a reversal. Distinguish a command that tests tracking from an external load that tests disturbance rejection. A fast step could be physically inadmissible when it exceeds travel or force limits.

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. List three experiments that distinguish a tracking error from an actuator limit, a sensor offset and a genuine unstable mode. 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. Explain how an engineer could communicate the numerical result without presenting it as a certified equipment setting. Explain how the proposed measurement separates a parameter estimate from an unobserved initial condition.

3. Failure analysis. Identify the dimensional parameters and state what changes if all time units are reported in milliseconds rather than seconds. 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, Closed-loop transfer equations. 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 3 engineering project — Reference tracking versus disturbance rejection

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

For plant P(s)=2/(s+2) and controller C=K with K in {1,2,5,10}, derive transfer paths for a unit reference, plant-input disturbance and measurement noise. Compare DC gains, time constants and initial actuator demands. Add a saturation at ±5 units and explain where a linear conclusion becomes unreliable.

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

Disturbance path and control authority: Introduce a plant-input load before the controlled process, then repeat with an additive output disturbance; the two outputs need not have the same transfer path. Measure control effort and report the effect of a nonunity sensor gain. Use a consistent sign convention so “positive disturbance” has an unambiguous meaning.

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 3 assessment — Feedback, sensitivity and tracking

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 L(0)=9. 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 L(0)=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 P=2/(s+2), C=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.

A4. Analyze the case maximum command 3, required equilibrium command 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.

Part B — synthesis problems

B1. Closed-loop transfer equations. Insert a measurement-noise term n into e=r−(y+n); derive the noise-to-output transfer and explain its sign. 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. Sensitivity and disturbance rejection. Draw S and T magnitude across a chosen crossover, and identify why both cannot be arbitrarily small at the same frequency. 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. Steady-state error and system type. Compare the step and ramp errors of the two given loops; identify which design objective actually motivated the integrator. 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. Actuator constraints and sensor noise. List three experiments that distinguish a tracking error from an actuator limit, a sensor offset and a genuine unstable mode. 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 L(0)=9, the compact result is S(0)=0.1, T(0)=0.9. 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 L(0)=4, the compact result is unity-step tracking error 0.2. 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 P=2/(s+2), C=4, the compact result is closed-loop reference DC gain 0.8. 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 maximum command 3, required equilibrium command 5, the compact result is steady target infeasible. 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 3.1–3.4.

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.