Frequency-Response Techniques: Bode Plots and Relative Stability Margins

Lesson 6.1 — Proportional control and offset

Unit: P, PI, PD and PID control. 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 proportional controller sets u=Kp(r−y) and requires nonzero error to sustain nonzero output in most type-0 plants. Increasing Kp generally reduces some low-frequency offsets but also increases control effort and can expose delay or neglected poles. A gain that is appropriate for a constant reference may be poor for fast commands.

A response can be fast and still fail accuracy or actuator requirements. Decide which metric matters before choosing gain.

Governing relationship: For P(s)=1/(s+1), T(s)=Kp/(s+1+Kp)

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 Kp=4, a unit-step command yields final y=4/5=0.8, offset 0.2 and closed-loop time constant 0.2 s. Raising Kp to 9 reduces offset to 0.1 but makes initial controller demand 9 units.

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

With a finite voltage supply a proportional step may immediately saturate. An anti-noise or filtering requirement can also limit usable 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 — robot wheel velocity control

Encoder quantization and friction near zero speed can dominate the low-speed error, while motor current limits dominate accelerations. A gain selected from one operating point should be tested at multiple commanded speeds and load torques. The model must be rewritten in consistent angular units if rpm is used for telemetry but rad/s is used in derivations.

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 hypothetical actuator ceiling of 5 units, determine whether Kp=9 can act linearly at the instant of a unit step. 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, Integral action and PI control. 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 6.2 — Integral action and PI control

Unit: P, PI, PD and PID control. 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

Integral action adds a state proportional to accumulated tracking error, helping remove a constant error when the closed loop stabilizes and the steady command is feasible. In parallel notation C(s)=Kp+Ki/s, Ki has different units from Kp. The integrator must be initialized and limited in a realizable implementation.

The double pole at −2 makes a critically damped characteristic, but numerator zeros mean the output step is not automatically identical to a canonical zero-free second-order response.

Governing relationship: For P(s)=1/(s+1), C(s)=Kp+Ki/s; denominator s²+(1+Kp)s+Ki

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

Choose Kp=3 and Ki=4: the characteristic is s²+4s+4=(s+2)². The loop is asymptotically stable and, with no saturation or bias, has zero error for a constant step.

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

Integral windup is possible when the target is unreachable. Measurement offsets can produce perfect measured tracking of the wrong physical quantity.

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 — robot wheel velocity control

Encoder quantization and friction near zero speed can dominate the low-speed error, while motor current limits dominate accelerations. A gain selected from one operating point should be tested at multiple commanded speeds and load torques. The model must be rewritten in consistent angular units if rpm is used for telemetry but rad/s is used in derivations.

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 for Ki=16, identify the damping ratio, and explain the expected transient tradeoff. 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, Derivative action and filtered PID. 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 6.3 — Derivative action and filtered PID

Unit: P, PI, PD and PID control. 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

Derivative action reacts to how the measured output changes and can add phase lead, but ideal differentiation amplifies unbounded high-frequency measurement noise. An implementable derivative branch uses a filter, often Kd s/(1+τf s), and derivative-on-measurement avoids a setpoint kick.

A small-amplitude, high-frequency noise source can overwhelm an actuator when differentiated. Filtering limits this at the cost of some phase lead and an additional dynamic state.

Governing relationship: u=Kpe+Kiintegral(e)−Kd*d(y_filtered)/dt

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 measured noise is n(t)=0.01 sin(200t) V, its ideal derivative amplitude is 0.01×200=2 V/s. Multiplying by Kd=0.5 produces a control ripple amplitude of 1 in the chosen consistent units.

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 filter cutoff must not be chosen solely to make a plotted signal smooth. Its phase shift influences the actual closed-loop crossover and stability margins.

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 — robot wheel velocity control

Encoder quantization and friction near zero speed can dominate the low-speed error, while motor current limits dominate accelerations. A gain selected from one operating point should be tested at multiple commanded speeds and load torques. The model must be rewritten in consistent angular units if rpm is used for telemetry but rad/s is used in derivations.

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 noise amplification at 200 rad/s for τf=0.01 s versus an ideal differentiator, keeping Kd fixed. 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, Saturation, windup and bumpless transfer. 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 6.4 — Saturation, windup and bumpless transfer

Unit: P, PI, PD and PID control. 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

When u_cmd exceeds available actuator authority, the difference between command and actual actuation can drive an integrator in the wrong direction. Conditional integration or back-calculation can mitigate windup, but the antiwindup logic and state transitions must be tested. Bumpless transfer aligns integrator states when switching between manual and automatic modes.

During saturation, the plant behavior is dominated by physical limits rather than by the linear closed-loop poles. Avoid drawing conclusions from a purely linear pole table.

Governing relationship: u_actual=clip(u_unsat,umin,umax); I_dot=e+Kaw*(u_actual−u_unsat)

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 PI output requests 12 V while the supply permits 5 V. With Kaw=0.5 s⁻¹, the back-calculation term contributes −3.5 V-equivalent per second in an appropriately scaled integrator equation, opposing excessive accumulation.

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

An antiwindup method can itself cause undesirable transitions if measurement noise, quantization or switching logic is poorly handled. Its implementation requires defined units.

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 — robot wheel velocity control

Encoder quantization and friction near zero speed can dominate the low-speed error, while motor current limits dominate accelerations. A gain selected from one operating point should be tested at multiple commanded speeds and load torques. The model must be rewritten in consistent angular units if rpm is used for telemetry but rad/s is used in derivations.

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. Design a manual-to-auto transfer test that checks whether the actuator command jumps. State the acceptable jump threshold and how it would be measured. 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, Proportional control and offset. 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 6 engineering project — PI control with actuator saturation

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 plant y_dot=−y+u and parallel PI gains Kp=2, Ki=1, then Kp=4, Ki=4. Start with a unit step, then a reference of 10 while |u|≤2, and return to zero. Compare integrator state with and without conditional antiwindup. Report tracking, command clipping, recovery and whether any demanded reference is impossible.

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

Actuator saturation and integrator recovery: Apply reference steps that are both achievable and unachievable, and compare PI loops with and without antiwindup. Record the integrator state, actual saturation-limited command and recovery time after reducing the reference. The control signal is as important as the plotted output.

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 6 assessment — P, PI, PD and PID control

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 P=1/(s+1), Kp=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.

A2. Analyze the case Kp=3, Ki=4 on P=1/(s+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.

A3. Analyze the case Kp=3, Ki=16 on P=1/(s+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.

A4. Analyze the case required actuator 8, maximum 6. State the relevant relationship, what physical variable is being calculated, and the result with correct dimensions. Explain one assumption required for the answer and one reason the same expression would be inappropriate for a more complicated plant.

Part B — synthesis problems

B1. Proportional control and offset. For a hypothetical actuator ceiling of 5 units, determine whether Kp=9 can act linearly at the instant of a unit step. 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. Integral action and PI control. Repeat for Ki=16, identify the damping ratio, and explain the expected transient tradeoff. 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. Derivative action and filtered PID. Compare noise amplification at 200 rad/s for τf=0.01 s versus an ideal differentiator, keeping Kd fixed. 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. Saturation, windup and bumpless transfer. Design a manual-to-auto transfer test that checks whether the actuator command jumps. State the acceptable jump threshold and how it would be measured. 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 P=1/(s+1), Kp=4, the compact result is steady error 0.2 for a unit step. 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 Kp=3, Ki=4 on P=1/(s+1), the compact result is characteristic (s+2)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 Kp=3, Ki=16 on P=1/(s+1), the compact result is damping ratio 0.5. 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 required actuator 8, maximum 6, the compact result is infeasible steady target. 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 6.1–6.4.

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.