Nyquist Stability Criterion, Polar Plots, and Cauchy Principle

Lesson 7.1 — Building state equations

Unit: State-space representations and state feedback. 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

State variables give a first-order representation of system memory. A mass–spring–damper has position x1 and velocity x2; its differential equation yields a two-state linear model. A and B depend on physical parameters and unit choices. A state transform can alter their entries while preserving system behavior.

The measured position is not enough to specify the full initial state: initial velocity matters. A state-space model makes that dependency explicit for simulation and estimation.

Governing relationship: x_dot=[[0,1],[-k/m,−c/m]]x+[0,1/m]F; y=[1,0]x

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 m=2 kg, c=4 N·s/m and k=18 N/m, A=[[0,1],[-9,−2]] in consistent SI units and B=[0,0.5]ᵀ. The characteristic poles satisfy s²+2s+9=0, giving −1±j√8 rad/s.

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

Limits, counterexamples and model evidence

Friction, travel stops, backlash and force saturation can invalidate a linear oscillator. State units should be recorded so gain dimensions are not silently lost.

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 — vibration suppression platform

An accelerometer observes acceleration rather than displacement directly. Estimating position requires integration and bias handling, so a clean state-feedback design may fail without an estimator. A model-based vibration controller must avoid commanding forces beyond actuator stroke and thermal ratings. Experimental tests should use a low-risk fixture and follow institutional lab controls.

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. Assume initial displacement 0.1 m and zero velocity; calculate initial acceleration at zero applied force. 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, Controllability and reachability. 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 7.2 — Controllability and reachability

Unit: State-space representations and state feedback. 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 controllability matrix [B AB ... A^(n−1)B] reveals whether a finite-dimensional linear state can be influenced throughout its state space by unconstrained inputs. Full rank is a structural criterion, not a practical guarantee that the required inputs are safe or inexpensive.

The force input affects velocity directly and position indirectly through integration. The matrix calculation formalizes that physical intuition.

Governing relationship: For n=2, Wc=[B, A*B]; full rank=2 needed

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=[[0,1],[-9,−2]] and B=[0,0.5]ᵀ, AB=[0.5,−1]ᵀ. The determinant of Wc=[ [0,0.5],[0.5,−1] ] is −0.25, so the state pair is controllable.

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 nearly singular controllability matrix signals numerical difficulty even if its rank is mathematically full; the required effort may exceed the actuator range.

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 — vibration suppression platform

An accelerometer observes acceleration rather than displacement directly. Estimating position requires integration and bias handling, so a clean state-feedback design may fail without an estimator. A model-based vibration controller must avoid commanding forces beyond actuator stroke and thermal ratings. Experimental tests should use a low-risk fixture and follow institutional lab controls.

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. Set the actuator to influence only a decoupled second state in a diagonal A; construct an uncontrollable example and identify the inaccessible 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. 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, Observability and estimation. 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 7.3 — Observability and estimation

Unit: State-space representations and state feedback. 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 output sensor may measure only a combination of states. The observability matrix [C; CA; ...] tests whether ideal, noise-free output history identifies the initial state in the LTI model. Estimators combine models and measurements; their design is a second feedback problem with its own error dynamics.

Velocity can be inferred indirectly from changing position. Numerically differentiating noisy position is not equivalent to a well-designed observer with explicit bandwidth and noise assumptions.

Governing relationship: For n=2, Wo=[C; C*A]; full rank=2 needed

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=[[0,1],[-9,−2]] and C=[1,0], CA=[0,1] and Wo is the identity, hence the states are observable from position history in the ideal model.

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

Limits, counterexamples and model evidence

Sampling, sensor bias, limited resolution and delays reduce practical estimation quality; mathematically full rank alone says nothing about measurement signal-to-noise ratio.

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 — vibration suppression platform

An accelerometer observes acceleration rather than displacement directly. Estimating position requires integration and bias handling, so a clean state-feedback design may fail without an estimator. A model-based vibration controller must avoid commanding forces beyond actuator stroke and thermal ratings. Experimental tests should use a low-risk fixture and follow institutional lab controls.

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. Change C to [0,1], compute Wo and identify whether this oscillator remains observable. Explain your result physically. 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, State feedback and pole placement. 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 7.4 — State feedback and pole placement

Unit: State-space representations and state feedback. 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

State feedback u=−Kx+Nr modifies the matrix to A−BK. If (A,B) is controllable, desired poles can often be assigned algebraically, but extreme pole locations can demand unrealistic control effort. Reference prefilters affect command tracking without changing the closed-loop eigenvalues.

Pole assignment can be checked by direct matrix multiplication and eigenvalue evaluation. A design requiring large acceleration commands may still be unacceptable.

Governing relationship: A_cl=A−B*K; det(sI−A_cl)=desired polynomial

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

Using A=[[0,1],[-9,−2]], B=[0,0.5]ᵀ and K=[k1,k2], the characteristic is s²+(2+0.5k2)s+(9+0.5k1). To request (s+3)(s+4)=s²+7s+12, choose k2=10 and k1=6.

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

Full-state feedback presumes accurate state availability or an observer. Unmodeled poles, quantization and saturation threaten aggressive assigned dynamics.

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

Application perspective — vibration suppression platform

An accelerometer observes acceleration rather than displacement directly. Estimating position requires integration and bias handling, so a clean state-feedback design may fail without an estimator. A model-based vibration controller must avoid commanding forces beyond actuator stroke and thermal ratings. Experimental tests should use a low-risk fixture and follow institutional lab controls.

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. Check the state feedback gains by recomputing trace and determinant of A−BK, then compare initial force for x=[0.1,0]ᵀ. 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, Building state 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 7 engineering project — Mass–spring model, observer and state-feedback

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 m=2 kg, c=4 N·s/m, k=18 N/m and position sensing. Build A, B, C, calculate controllability and observability ranks, and find K placing poles at −3 and −4. Compare maximum demanded force for initial displacement 0.1 m and 0.5 m, and explain how an omitted force limit changes outcomes.

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

State measurement and feedback effort: Construct A, B and C from parameter units, compute the reachability and observability matrix ranks, then request poles and recover the actual gain matrix. Record maximum force or voltage and determine whether the requested poles remain feasible after imposing physical limits.

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 7 assessment — State-space representations and state feedback

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 m=2,c=4,k=18. 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 B=[0,0.5]ᵀ and A=[[0,1],[-9,-2]]. 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 C=[1,0]. 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 poles −3,−4 with given A and B. 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. Building state equations. Assume initial displacement 0.1 m and zero velocity; calculate initial acceleration at zero applied force. 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. Controllability and reachability. Set the actuator to influence only a decoupled second state in a diagonal A; construct an uncontrollable example and identify the inaccessible 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.

B3. Observability and estimation. Change C to [0,1], compute Wo and identify whether this oscillator remains observable. Explain your result physically. 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. State feedback and pole placement. Check the state feedback gains by recomputing trace and determinant of A−BK, then compare initial force for x=[0.1,0]ᵀ. 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 m=2,c=4,k=18, the compact result is A lower row [−9,−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.

A2 key. For B=[0,0.5]ᵀ and A=[[0,1],[-9,-2]], the compact result is controllability determinant −0.25. 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 C=[1,0], the compact result is observability matrix identity. 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 poles −3,−4 with given A and B, the compact result is K=[6,10]. 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 7.1–7.4.

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.