Root Locus Analysis, Sketching Rules, and Parameter Sensitivity

Lesson 5.1 — Sinusoidal steady-state response

Unit: Frequency response and robustness margins. 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

In a stable LTI system, evaluating G(jω) gives the complex steady-state gain at angular frequency ω. Its magnitude scales sinusoidal amplitude and its argument supplies phase shift, after natural transients decay. This frequency response is not meaningful as a guaranteed bounded steady state if unstable modes dominate.

The frequency is 5 rad/s, not 5 Hz; confusing them shifts design interpretations by a factor of 2π. Frequency response is a useful experimental identification tool if excitation remains safe and within the linear region.

Governing relationship: For G(s)=1/(taus+1): |G(jω)|=1/sqrt(1+(omegatau)^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

With tau=0.2 s and ω=5 rad/s, magnitude is 1/√2≈0.707 and phase is −45°. A 2 V amplitude input yields approximately 1.414 V amplitude output at that frequency.

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 sinusoidal response can be obscured by drift or harmonic distortion. Measure phase against a common time reference and allow sufficient cycles after transients.

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 — audio amplifier feedback

The feedback loop can improve small-signal fidelity and yet be sensitive to capacitive loads or additional high-frequency phase lag. A narrowband test at a comfortable audio frequency is insufficient evidence of wideband stability. A laboratory plan must identify which physical measurements are safe and what separate protective circuitry limits energy.

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. Repeat the calculation at 5 Hz; demonstrate how dramatically the response differs from the 5 rad/s result. 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, Constructing Bode plots. 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 5.2 — Constructing Bode plots

Unit: Frequency response and robustness margins. 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 Bode plot separates magnitude in decibels and phase in degrees as functions of logarithmic angular frequency. Poles introduce slopes near their break frequencies and zeros introduce compensating slope changes. The straight-line approximation is a planning device; exact evaluation is needed near crossovers.

At each break the exact magnitude departs from the naive corner-intersection sketch. Phase lags accumulate smoothly, not as an instantaneous 90° jump at a break.

Governing relationship: G(s)=10/[(1+s/2)(1+s/20)]

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

DC magnitude is 20 dB. The first pole at 2 rad/s introduces an approximate −20 dB/decade trend, and after 20 rad/s a second pole steepens the asymptote toward −40 dB/decade.

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 hidden resonance or significant delay can invalidate a clean minimum-phase sketch. Always label frequency units and account for sign changes in gain.

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 — audio amplifier feedback

The feedback loop can improve small-signal fidelity and yet be sensitive to capacitive loads or additional high-frequency phase lag. A narrowband test at a comfortable audio frequency is insufficient evidence of wideband stability. A laboratory plan must identify which physical measurements are safe and what separate protective circuitry limits energy.

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. Calculate exact magnitude and phase at ω=2, 10 and 20 rad/s and compare with the asymptotic sketch. 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, Gain and phase margin interpretation. 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 5.3 — Gain and phase margin interpretation

Unit: Frequency response and robustness margins. 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 classical phase margin is measured at a frequency where the loop magnitude is 1, comparing phase to −180° for the standard negative-feedback convention. Gain margin concerns loop magnitude where phase reaches −180°. The definitions need attention for multiple crossings, unstable open loops and delays.

Delay has unit magnitude yet adds frequency-dependent phase. It can destabilize a loop that appears attractive from its gain plot alone.

Governing relationship: PM=180°+arg[L(jω_gc)] where |L(jω_gc)|=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

If |L| crosses unity at 4 rad/s and the unwrapped phase is −135°, phase margin is 45°. If an extra 0.05 s pure delay is inserted, it adds about −11.46° at 4 rad/s and reduces local phase margin to roughly 33.5°.

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

A single quoted phase-margin number does not capture closed-loop peak sensitivity, multiple unity-gain crossings or nonlinearity. Avoid claiming guaranteed overshoot from an approximate rule of thumb.

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 — audio amplifier feedback

The feedback loop can improve small-signal fidelity and yet be sensitive to capacitive loads or additional high-frequency phase lag. A narrowband test at a comfortable audio frequency is insufficient evidence of wideband stability. A laboratory plan must identify which physical measurements are safe and what separate protective circuitry limits energy.

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. Calculate the largest additional delay that would consume 20° of phase margin at a crossover of 4 rad/s. 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. Which measured quantity would most directly separate an initial-condition error from a plant-parameter error? Give the signal units and observation window. Explain how the proposed measurement separates a parameter estimate from an unobserved initial condition.

3. Failure analysis. Construct a plausible situation in which the model prediction fails, naming the omitted state, nonlinearity or timing element. 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, Nyquist reasoning and loop shaping. 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 5.4 — Nyquist reasoning and loop shaping

Unit: Frequency response and robustness margins. 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 Nyquist criterion relates encirclements of the point −1 to open-loop right-half-plane poles and resulting closed-loop right-half-plane zeros of 1+L. Its power is handling systems whose open loop is unstable, where simple margin intuition can mislead. The contour and sign conventions must be stated explicitly.

The geometry emphasizes why phase and gain jointly matter. A point at magnitude 0.9 and angle −180° is only 0.1 from the critical point and may have very high sensitivity.

Governing relationship: Closed-loop characteristic: 1+L(s)=0; critical point L=−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 a stable open loop with a polar curve that stays away from −1 and makes zero net encirclements, the closed loop has no right-half-plane poles under the standard criterion assumptions. By contrast, an open-loop right-half-plane pole requires a compensating encirclement to obtain stable feedback.

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

Time delays require frequency-dependent phase; samples, right-half-plane zeros and hidden modes limit how aggressively a loop can be shaped. Numerical Nyquist plots require sufficient frequency coverage.

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 — audio amplifier feedback

The feedback loop can improve small-signal fidelity and yet be sensitive to capacitive loads or additional high-frequency phase lag. A narrowband test at a comfortable audio frequency is insufficient evidence of wideband stability. A laboratory plan must identify which physical measurements are safe and what separate protective circuitry limits energy.

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. Sketch two polar trajectories with the same closest magnitude to unity but different phase and compare their distance from −1. 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, Sinusoidal steady-state response. 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 5 engineering project — Bode loop-shaping investigation

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 G(s)=10/((1+s/2)(1+s/20)). Evaluate logarithmic points from 0.1 to 200 rad/s, extract exact magnitude and phase, and overlay asymptotic straight-line segments. Examine a pure delay of 0.02 and 0.08 s. Explain how crossover phase changes, and never treat a gain-only plot as sufficient proof of robust stability.

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

Frequency sweep and delay margin: Perform a logarithmic sweep, collecting output amplitude and phase relative to the input at each frequency. At crossover interpolate magnitude and phase rather than choosing the nearest sample. Test the effect of adding a small explicit delay and distinguish modeled delay from sampling latency.

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 5 assessment — Frequency response and robustness margins

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=0.2 s, ω=5 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.

A2. Analyze the case gain ratio=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.

A3. Analyze the case phase at crossover=−145°. 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 delay 0.05 s at 4 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. Sinusoidal steady-state response. Repeat the calculation at 5 Hz; demonstrate how dramatically the response differs from the 5 rad/s result. 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. Constructing Bode plots. Calculate exact magnitude and phase at ω=2, 10 and 20 rad/s and compare with the asymptotic sketch. 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. Gain and phase margin interpretation. Calculate the largest additional delay that would consume 20° of phase margin at a crossover of 4 rad/s. 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. Nyquist reasoning and loop shaping. Sketch two polar trajectories with the same closest magnitude to unity but different phase and compare their distance from −1. 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=0.2 s, ω=5 rad/s, the compact result is magnitude 0.7071 and phase −45°. 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 gain ratio=10, the compact result is 20 dB. 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 phase at crossover=−145°, the compact result is phase margin 35°. 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 delay 0.05 s at 4 rad/s, the compact result is additional phase lag ≈11.46°. 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 5.1–5.4.

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.