Controllability, Observability, and Kalman Canonical Decompositions

Lesson 3.1 — Observability and reconstructible state

Unit: State estimation and Kalman filtering. Prerequisites: Book 11 feedback and state-space fundamentals; preceding lessons in this unit. Learning objectives: explain the model, reproduce the worked case, identify an important failure condition, and independently solve the problems.

Concept and mechanism

A linear system may have internal modes not visible from one sensor. The observability matrix stacks C, CA and higher powers; full column rank allows state recovery for a finite-dimensional LTI system in principle. An observable pair may still be poorly conditioned, making numerical estimates sensitive to measurement noise. The test describes a model, not whether a cheap sensor can achieve a required physical accuracy.

Governing relation: O=[C; C A; ...; C A^(n-1)]; rank(O)=n for observable n-state model

To interpret this expression, mark each symbol as a state, command, observation, uncertainty, noise statistic or design weight. Determine whether the result is continuous-time, sampled-time, a static mapping, or a bound. The statement may require nonsingular matrices, bounded variation, a linear model or known timing. List the specific hypotheses before drawing an engineering conclusion.

Worked example

For A=[[0,1],[-4,-3]] and C=[1,0], the two-state observability matrix is [[1,0],[0,1]], so rank=2. A position signal contains information about velocity through time evolution. If C=[0,0], the matrix is zero and no state reconstruction can follow from output observations.

Independent check: Write the same example as an equation whose two sides must agree after substitution. Check the sign and limiting cases; then change only one number and predict whether the outcome increases, decreases, or requires a full recalculation. Do not mistake numerical agreement under synthetic data for physical validation. A result describing a covariance is conditional on the noise model; one describing stability is conditional on the model and its defined perturbations; one describing an optimum is conditional on both the objective and the feasible set.

Interpretation and decision

Observability is an information question: it is neither a demand to measure all states directly nor a proof that the estimate remains accurate during a sensor fault. Modeling assumptions about A and timing matter greatly.

The design choice should identify the goal being improved—tracking, robustness, estimation, control effort or timing—and a quantity that may become worse. Distinguish a sufficient theorem condition from a necessary condition. For optimization explicitly report actuator limits; for estimation report which measurements support the inference. A calculated number is useful only when its assumptions and units are available to the reader.

When the argument fails

Two eigenmodes with nearly identical sensor signatures can make a model mathematically observable but practically difficult. Singular values and noise sensitivity give information beyond a binary rank calculation.

Describe a falsifiable counterexample by naming an omitted dynamic, a violated constraint, an unobservable mode or a mischaracterized uncertainty set. Indicate which measured variable or computed check would reveal the discrepancy. Repeating the original calculation at the same nominal point cannot test a modeling assumption that never appeared in the equation.

Engineering case — Two-sensor vehicle motion estimate

A notional low-energy carriage provides position from an encoder and acceleration from an inertial sensor. Encoder quantization and accelerometer bias are physically distinct noise sources. The estimate can track position while velocity remains uncertain during a prolonged data gap. A validation plan compares predicted and measured sensor outputs on withheld trajectories and checks for innovation autocorrelation. Explicitly decide which state components are estimated, measured or fundamentally ambiguous.

Lab protocol: Information audit. Draw the causal arrows between state, input and measured output. Compare a measured trace with the estimator prediction without using that same trace to retune every unknown coefficient. Record where unobservable modes, unmeasured bias or timing uncertainty can enter.

Create an assumption ledger listing sensor availability, actuator limits, initial state, operating point and the variable intentionally changed. For the main claim, separate what follows algebraically from what would require a second independent experiment. If the proposed experiment is unable to distinguish competing hypotheses, redesign it before collecting more repeated data.

Exercises

1. Calculation: For A=diag(−1,−2) and C=[1,1], compute O and its determinant. Repeat for C=[1,0] and interpret the missing mode. Show the intermediate arithmetic, a dimensions or sign check, and the condition under which the result is valid.

2. Alternative model: Construct an alternative interpretation of the same nominal observation, then name one additional signal that would separate the alternatives.

3. Sensitivity: Change one uncertain parameter by 20%. State which original claim remains logically valid, which result needs recalculation, and what new numerical input is required.

4. Connection: Compare the present method to Luenberger observer and error dynamics. Name an assumption they share and one kind of answer each method cannot produce alone.

5. Verification: Draft a five-line test specification listing input, expected outcome, numeric tolerance, potential failure and the reason a passing simulation would not establish hardware safety.

Reference guidance

A complete answer to the calculation question begins from the displayed governing relation and shows substitutions. The alternative-model answer must predict a different outcome in a specified observation. The sensitivity answer must distinguish parameter error from measurement noise. The connection question should identify a difference in mathematical object or guarantee, not just a difference in terminology. A suitable test reports reproducible settings and a clear fail condition. Use the accompanying unit assessment for explicit numerical checks and worked answer guidance.

Lesson 3.2 — Luenberger observer and error dynamics

Unit: State estimation and Kalman filtering. Prerequisites: Book 11 feedback and state-space fundamentals; preceding lessons in this unit. Learning objectives: explain the model, reproduce the worked case, identify an important failure condition, and independently solve the problems.

Concept and mechanism

A deterministic observer evolves a model state and corrects its trajectory when a measurement differs from prediction. The correction term L(y−C xhat) drives estimation error according to A−LC in the ideal model. Observer gain is chosen for adequate convergence without excessive sensitivity to noise. Controller and observer poles serve different purposes and need not coincide.

Governing relation: xhat_dot=A xhat+B u+L(y-C xhat); e_dot=(A-LC)e

To interpret this expression, mark each symbol as a state, command, observation, uncertainty, noise statistic or design weight. Determine whether the result is continuous-time, sampled-time, a static mapping, or a bound. The statement may require nonsingular matrices, bounded variation, a linear model or known timing. List the specific hypotheses before drawing an engineering conclusion.

Worked example

Consider a scalar plant x_dot=−x+u with measurement y=x and observer gain L=4. Estimation error follows e_dot=−5e, so a starting error e(0)=2 becomes 2e^(−5t). At t=0.2 s it is approximately 0.7358. If sensor noise v enters y, the observer correction includes Lv, which can amplify high-frequency noise when L is increased.

Independent check: Write the same example as an equation whose two sides must agree after substitution. Check the sign and limiting cases; then change only one number and predict whether the outcome increases, decreases, or requires a full recalculation. Do not mistake numerical agreement under synthetic data for physical validation. A result describing a covariance is conditional on the noise model; one describing stability is conditional on the model and its defined perturbations; one describing an optimum is conditional on both the objective and the feasible set.

Interpretation and decision

Faster observer poles are attractive in an ideal noiseless model; they are not free. A modest observer may provide a better noise-versus-delay tradeoff for a quantized instrument. The appropriate bandwidth depends on the controller dynamics and sample rate.

The design choice should identify the goal being improved—tracking, robustness, estimation, control effort or timing—and a quantity that may become worse. Distinguish a sufficient theorem condition from a necessary condition. For optimization explicitly report actuator limits; for estimation report which measurements support the inference. A calculated number is useful only when its assumptions and units are available to the reader.

When the argument fails

If the actual plant has bias or differs from A, the clean homogeneous error equation no longer holds. An observer can converge to a biased estimate rather than the true state.

Describe a falsifiable counterexample by naming an omitted dynamic, a violated constraint, an unobservable mode or a mischaracterized uncertainty set. Indicate which measured variable or computed check would reveal the discrepancy. Repeating the original calculation at the same nominal point cannot test a modeling assumption that never appeared in the equation.

Engineering case — Two-sensor vehicle motion estimate

A notional low-energy carriage provides position from an encoder and acceleration from an inertial sensor. Encoder quantization and accelerometer bias are physically distinct noise sources. The estimate can track position while velocity remains uncertain during a prolonged data gap. A validation plan compares predicted and measured sensor outputs on withheld trajectories and checks for innovation autocorrelation. Explicitly decide which state components are estimated, measured or fundamentally ambiguous.

Lab protocol: Information audit. Draw the causal arrows between state, input and measured output. Compare a measured trace with the estimator prediction without using that same trace to retune every unknown coefficient. Record where unobservable modes, unmeasured bias or timing uncertainty can enter.

Create an assumption ledger listing sensor availability, actuator limits, initial state, operating point and the variable intentionally changed. For the main claim, separate what follows algebraically from what would require a second independent experiment. If the proposed experiment is unable to distinguish competing hypotheses, redesign it before collecting more repeated data.

Exercises

1. Calculation: For plant pole −2 and desired observer error pole −8 with C=1, find L. Compute remaining error after 0.5 s from e0=1. Show the intermediate arithmetic, a dimensions or sign check, and the condition under which the result is valid.

2. Alternative model: Construct an alternative interpretation of the same nominal observation, then name one additional signal that would separate the alternatives.

3. Sensitivity: Change one uncertain parameter by 20%. State which original claim remains logically valid, which result needs recalculation, and what new numerical input is required.

4. Connection: Compare the present method to Kalman prediction and measurement update. Name an assumption they share and one kind of answer each method cannot produce alone.

5. Verification: Draft a five-line test specification listing input, expected outcome, numeric tolerance, potential failure and the reason a passing simulation would not establish hardware safety.

Reference guidance

A complete answer to the calculation question begins from the displayed governing relation and shows substitutions. The alternative-model answer must predict a different outcome in a specified observation. The sensitivity answer must distinguish parameter error from measurement noise. The connection question should identify a difference in mathematical object or guarantee, not just a difference in terminology. A suitable test reports reproducible settings and a clear fail condition. Use the accompanying unit assessment for explicit numerical checks and worked answer guidance.

Lesson 3.3 — Kalman prediction and measurement update

Unit: State estimation and Kalman filtering. Prerequisites: Book 11 feedback and state-space fundamentals; preceding lessons in this unit. Learning objectives: explain the model, reproduce the worked case, identify an important failure condition, and independently solve the problems.

Concept and mechanism

The Kalman filter forms a minimum-variance linear estimator under specified linear-Gaussian assumptions; even without Gaussian noise, the covariance recursion still follows for the linear minimum mean-square formulation under appropriate second moments. Process covariance Q describes uncertainty in model evolution and measurement covariance R describes sensor uncertainty. A larger R reduces trust in an individual noisy measurement, all else equal.

Governing relation: Pminus=A P A^T+Q; K=Pminus H^T(H Pminus H^T+R)^(-1); Pplus=(I-KH)Pminus

To interpret this expression, mark each symbol as a state, command, observation, uncertainty, noise statistic or design weight. Determine whether the result is continuous-time, sampled-time, a static mapping, or a bound. The statement may require nonsingular matrices, bounded variation, a linear model or known timing. List the specific hypotheses before drawing an engineering conclusion.

Worked example

For a scalar random-walk model A=H=1, prior covariance P=1, process covariance Q=0.25, measurement covariance R=0.75, the prediction covariance is Pminus=1.25. Kalman gain is 1.25/(1.25+0.75)=0.625 and posterior covariance is (1−0.625)×1.25=0.46875. If predicted state is 4 and measured value is 6, the corrected state is 4+0.625(6−4)=5.25.

Independent check: Write the same example as an equation whose two sides must agree after substitution. Check the sign and limiting cases; then change only one number and predict whether the outcome increases, decreases, or requires a full recalculation. Do not mistake numerical agreement under synthetic data for physical validation. A result describing a covariance is conditional on the noise model; one describing stability is conditional on the model and its defined perturbations; one describing an optimum is conditional on both the objective and the feasible set.

Interpretation and decision

The covariance reduction measures uncertainty under the model assumptions, not guaranteed physical error. For numerical implementations the Joseph covariance update form can better preserve symmetry and positive semidefiniteness when rounding or matrix conditioning matters.

The design choice should identify the goal being improved—tracking, robustness, estimation, control effort or timing—and a quantity that may become worse. Distinguish a sufficient theorem condition from a necessary condition. For optimization explicitly report actuator limits; for estimation report which measurements support the inference. A calculated number is useful only when its assumptions and units are available to the reader.

When the argument fails

If sensor noise has an unknown constant offset, treating it as independent zero-mean Gaussian noise is mis-specified. The filter may show low internal covariance while producing a biased physical estimate.

Describe a falsifiable counterexample by naming an omitted dynamic, a violated constraint, an unobservable mode or a mischaracterized uncertainty set. Indicate which measured variable or computed check would reveal the discrepancy. Repeating the original calculation at the same nominal point cannot test a modeling assumption that never appeared in the equation.

Engineering case — Two-sensor vehicle motion estimate

A notional low-energy carriage provides position from an encoder and acceleration from an inertial sensor. Encoder quantization and accelerometer bias are physically distinct noise sources. The estimate can track position while velocity remains uncertain during a prolonged data gap. A validation plan compares predicted and measured sensor outputs on withheld trajectories and checks for innovation autocorrelation. Explicitly decide which state components are estimated, measured or fundamentally ambiguous.

Lab protocol: Information audit. Draw the causal arrows between state, input and measured output. Compare a measured trace with the estimator prediction without using that same trace to retune every unknown coefficient. Record where unobservable modes, unmeasured bias or timing uncertainty can enter.

Create an assumption ledger listing sensor availability, actuator limits, initial state, operating point and the variable intentionally changed. For the main claim, separate what follows algebraically from what would require a second independent experiment. If the proposed experiment is unable to distinguish competing hypotheses, redesign it before collecting more repeated data.

Exercises

1. Calculation: Set R=3 while holding the other scalar values fixed. Recompute gain and posterior covariance, and explain the direction of the measurement weighting change. Show the intermediate arithmetic, a dimensions or sign check, and the condition under which the result is valid.

2. Alternative model: Construct an alternative interpretation of the same nominal observation, then name one additional signal that would separate the alternatives.

3. Sensitivity: Change one uncertain parameter by 20%. State which original claim remains logically valid, which result needs recalculation, and what new numerical input is required.

4. Connection: Compare the present method to Innovations, noise tuning and estimator validation. Name an assumption they share and one kind of answer each method cannot produce alone.

5. Verification: Draft a five-line test specification listing input, expected outcome, numeric tolerance, potential failure and the reason a passing simulation would not establish hardware safety.

Reference guidance

A complete answer to the calculation question begins from the displayed governing relation and shows substitutions. The alternative-model answer must predict a different outcome in a specified observation. The sensitivity answer must distinguish parameter error from measurement noise. The connection question should identify a difference in mathematical object or guarantee, not just a difference in terminology. A suitable test reports reproducible settings and a clear fail condition. Use the accompanying unit assessment for explicit numerical checks and worked answer guidance.

Lesson 3.4 — Innovations, noise tuning and estimator validation

Unit: State estimation and Kalman filtering. Prerequisites: Book 11 feedback and state-space fundamentals; preceding lessons in this unit. Learning objectives: explain the model, reproduce the worked case, identify an important failure condition, and independently solve the problems.

Concept and mechanism

The innovation is the difference between observed and predicted measurement. Its expected covariance can be derived from the prediction covariance and sensor model. Innovations should be examined over time for autocorrelation, changing variance, sign bias and outliers; a single small innovation says little. Estimator validation also compares reconstructed states against an independent reference when possible.

Governing relation: nu_k=y_k-H xhatminus_k; S_k=H Pminus_k H^T+R; NIS=nu^T S^(-1)nu

To interpret this expression, mark each symbol as a state, command, observation, uncertainty, noise statistic or design weight. Determine whether the result is continuous-time, sampled-time, a static mapping, or a bound. The statement may require nonsingular matrices, bounded variation, a linear model or known timing. List the specific hypotheses before drawing an engineering conclusion.

Worked example

Suppose a scalar innovation is ν=2 with predicted variance S=4. Its normalized innovation squared is 2²/4=1. For another innovation ν=6 with the same variance, NIS=9, signaling a much larger discrepancy relative to modeled noise. This statistic should be interpreted using a specified distribution and repeated data, not by applying an arbitrary alarm threshold to one sample.

Independent check: Write the same example as an equation whose two sides must agree after substitution. Check the sign and limiting cases; then change only one number and predict whether the outcome increases, decreases, or requires a full recalculation. Do not mistake numerical agreement under synthetic data for physical validation. A result describing a covariance is conditional on the noise model; one describing stability is conditional on the model and its defined perturbations; one describing an optimum is conditional on both the objective and the feasible set.

Interpretation and decision

An innovation can be small because the filter follows a faulty sensor too closely, especially under a mis-tuned R. Independent physical checks are essential for critical signals. Changing Q and R simply until a plot looks smooth risks hiding a model defect.

The design choice should identify the goal being improved—tracking, robustness, estimation, control effort or timing—and a quantity that may become worse. Distinguish a sufficient theorem condition from a necessary condition. For optimization explicitly report actuator limits; for estimation report which measurements support the inference. A calculated number is useful only when its assumptions and units are available to the reader.

When the argument fails

An intermittent delay can make innovations temporally correlated even when the logged covariance looks reasonable. Stale timestamps need to be represented in the model or rejected by a verified data-handling rule.

Describe a falsifiable counterexample by naming an omitted dynamic, a violated constraint, an unobservable mode or a mischaracterized uncertainty set. Indicate which measured variable or computed check would reveal the discrepancy. Repeating the original calculation at the same nominal point cannot test a modeling assumption that never appeared in the equation.

Engineering case — Two-sensor vehicle motion estimate

A notional low-energy carriage provides position from an encoder and acceleration from an inertial sensor. Encoder quantization and accelerometer bias are physically distinct noise sources. The estimate can track position while velocity remains uncertain during a prolonged data gap. A validation plan compares predicted and measured sensor outputs on withheld trajectories and checks for innovation autocorrelation. Explicitly decide which state components are estimated, measured or fundamentally ambiguous.

Lab protocol: Information audit. Draw the causal arrows between state, input and measured output. Compare a measured trace with the estimator prediction without using that same trace to retune every unknown coefficient. Record where unobservable modes, unmeasured bias or timing uncertainty can enter.

Create an assumption ledger listing sensor availability, actuator limits, initial state, operating point and the variable intentionally changed. For the main claim, separate what follows algebraically from what would require a second independent experiment. If the proposed experiment is unable to distinguish competing hypotheses, redesign it before collecting more repeated data.

Exercises

1. Calculation: With a two-dimensional innovation vector [1,2] and diagonal S=diag(1,4), compute NIS. Identify one assumption needed to interpret that number statistically. Show the intermediate arithmetic, a dimensions or sign check, and the condition under which the result is valid.

2. Alternative model: Construct an alternative interpretation of the same nominal observation, then name one additional signal that would separate the alternatives.

3. Sensitivity: Change one uncertain parameter by 20%. State which original claim remains logically valid, which result needs recalculation, and what new numerical input is required.

4. Connection: Compare the present method to Observability and reconstructible state. Name an assumption they share and one kind of answer each method cannot produce alone.

5. Verification: Draft a five-line test specification listing input, expected outcome, numeric tolerance, potential failure and the reason a passing simulation would not establish hardware safety.

Reference guidance

A complete answer to the calculation question begins from the displayed governing relation and shows substitutions. The alternative-model answer must predict a different outcome in a specified observation. The sensitivity answer must distinguish parameter error from measurement noise. The connection question should identify a difference in mathematical object or guarantee, not just a difference in terminology. A suitable test reports reproducible settings and a clear fail condition. Use the accompanying unit assessment for explicit numerical checks and worked answer guidance.

Unit 3 applied project — Estimator consistency and bias detection

Engineering question

Simulate a scalar random walk with Q=0.1, R=0.4, a deterministic measurement perturbation, and a late injected sensor bias of +1. Compare innovation, estimated state and covariance before and after bias. Test a second filter with R=4 and describe how innovation and responsiveness differ. The validation report must distinguish assumed covariance from observed model mismatch.

This is an offline design and evidence exercise. Its output is a reproducible mathematical study and a bounded conclusion, not permission to commission a physical controller. Write the study question in a sentence that could be falsified by a measured trajectory or a computed counterexample. Explain why the chosen input variation contains more information than an arbitrary random curve.

Setup, model and data protocol

Specify each variable, unit and initial state. The baseline model must be documented independently of the controller. List any linearization point, noise covariance, uncertainty interval, constraints or bounded disturbance. If the project contains a numerical experiment, provide either a short script or a CSV data table, and log the equation and numerical integration interval that produced it. Avoid a test in which the same output is used both to choose every model parameter and to certify accuracy. If this project cannot be evaluated without a missing sensor or unsupported assumption, make the gap an explicit result.

Required analytical work

1. Derive the governing state equation, transfer equation, cost, covariance update or stability inequality appropriate to this unit. Where possible, independently derive a special case without a library solver.
2. Calculate a baseline numerical value and check it using a distinct method: direct substitution, a matrix identity, a simulated recurrence or an analytically known limit. Include the expected dimensions.
3. Declare a testable range for one key parameter. Sample several values and describe what is and is not proven by that sample. For a mathematical worst-case guarantee, state why the chosen analytic bound applies to the full nominated uncertainty set.
4. Check performance and feasibility separately. A low cost does not imply constraints are met; a stable trajectory does not prove that an actuator command is allowed. Where the optimizer or model predicts an impossible demand, report infeasibility plainly.
5. Record at least one deliberately adverse scenario: sensor bias, delayed packet, perturbation outside the fit range, insufficient excitation, a saturated actuator, or a change in the model’s region of validity.

Measurement and critical interpretation

Information audit: Draw the causal arrows between state, input and measured output. Compare a measured trace with the estimator prediction without using that same trace to retune every unknown coefficient. Record where unobservable modes, unmeasured bias or timing uncertainty can enter.

A technically useful report separates mathematical truths under explicit assumptions, simulations of those assumptions, and empirical claims that require physical evidence. Explain which additional variable would most improve confidence. A compelling graph should show labeled axes and a model-versus-reference comparison, not just a beautiful output trace. If a proposed controller improves response time but worsens effort, the tradeoff belongs in the conclusion. Do not discard a failed scenario merely because it disrupts the nominal result.

Required report package

• Model card: signals and dimensions, operating range, nominal and uncertain parameters, initial state and constraints.
• One derivation checked by a second independent calculation, with all rounding disclosed.
• A baseline numeric table and an adverse-case numeric table, each with reproducible values.
• At least two plots with captions explaining what the curves demonstrate and what they do not.
• A decision paragraph about the next cheapest test that could reject an important assumption.
• A separate limitations note identifying implementation, sensor, timing, actuator and safety evidence not supplied by this exercise.

Assessment rubric

25% analytical correctness means the mathematical object and its conditions are identified precisely. 20% numerical traceability means the results can be recomputed from the stated data. 20% comparison requires a meaningful alternative, sensitivity or failure scenario. 20% engineering interpretation requires an explicit distinction between a passed model test and an unverified deployment assumption. 15% communication covers units, graph labels, repeatability and clear identification of the limits of conclusions. Strong work may conclude that a proposed design is not justified by the evidence; that is not a failure of the assignment.

Unit 3 assessment — State estimation and Kalman filtering

Instructions: Show the derivation and dimensions, not only the final value. Work independently before reading the reference key. Every number refers to a synthetic model and should not be interpreted as a device setting. The assessment combines computation, experimental design, and evaluation of whether a claim follows from the evidence.

Part A — quantitative short questions

A1. Given A=[[0,1],[-4,-3]],C=[1,0], calculate the named mathematical quantity. State the model assumptions that make the requested result well defined. Explain whether the answer represents a transfer characteristic, a state estimate, an optimal action, a bounded inequality or a verification verdict. Show the intermediate arithmetic and one reason an actual data set might disagree.

A2. Given A=−1,C=1,L=4, calculate the named mathematical quantity. State the model assumptions that make the requested result well defined. Explain whether the answer represents a transfer characteristic, a state estimate, an optimal action, a bounded inequality or a verification verdict. Show the intermediate arithmetic and one reason an actual data set might disagree.

A3. Given Pminus=1.25,R=0.75, calculate the named mathematical quantity. State the model assumptions that make the requested result well defined. Explain whether the answer represents a transfer characteristic, a state estimate, an optimal action, a bounded inequality or a verification verdict. Show the intermediate arithmetic and one reason an actual data set might disagree.

A4. Given innovation 2, covariance 4, calculate the named mathematical quantity. State the model assumptions that make the requested result well defined. Explain whether the answer represents a transfer characteristic, a state estimate, an optimal action, a bounded inequality or a verification verdict. Show the intermediate arithmetic and one reason an actual data set might disagree.

Part B — distinct lesson problems

B1. Observability and reconstructible state. For A=diag(−1,−2) and C=[1,1], compute O and its determinant. Repeat for C=[1,0] and interpret the missing mode. Include the model, numerical or logical solution, and a limiting-case check. Then design one follow-up measurement or computed counterexample that can challenge the result under an untested assumption.

B2. Luenberger observer and error dynamics. For plant pole −2 and desired observer error pole −8 with C=1, find L. Compute remaining error after 0.5 s from e0=1. Include the model, numerical or logical solution, and a limiting-case check. Then design one follow-up measurement or computed counterexample that can challenge the result under an untested assumption.

B3. Kalman prediction and measurement update. Set R=3 while holding the other scalar values fixed. Recompute gain and posterior covariance, and explain the direction of the measurement weighting change. Include the model, numerical or logical solution, and a limiting-case check. Then design one follow-up measurement or computed counterexample that can challenge the result under an untested assumption.

B4. Innovations, noise tuning and estimator validation. With a two-dimensional innovation vector [1,2] and diagonal S=diag(1,4), compute NIS. Identify one assumption needed to interpret that number statistically. Include the model, numerical or logical solution, and a limiting-case check. Then design one follow-up measurement or computed counterexample that can challenge the result under an untested assumption.

Part C — synthesis and model criticism

C1. Data and claims. A team plots attractive curves on its calibration data and calls the controller robust and safe. Identify the exact robust or feasible property that would need to be established, a missing uncertainty or fault class, and a test that could falsify the unsupported conclusion. Avoid generic assurances.

C2. Evidence hierarchy. Separate a theoretical identity, a result from a discrete simulation, a laboratory observation and a physical safety approval. For the current unit, provide one concrete example of each, then explain why one cannot silently replace another.

C3. Design comparison. Select two approaches studied in this unit. Compare their required model detail, computational or measurement burden, failure sensitivity and interpretability. Name one operating condition in which the less elaborate approach could be preferable.

C4. Integration with Book 11. Start from a textbook PID or linear state-space example and show which assumption must be changed to require the advanced tool from the present unit. Identify an actuator constraint, sensor characteristic or nonlinear effect the simple representation had omitted.

Reference answers and grading notes

A1 reference. A=[[0,1],[-4,-3]],C=[1,0] leads to observability rank 2 in the stated ideal model. The answer is incomplete without the numerical relationship, relevant units, and an explicit condition such as positivity, stability, nonsingularity, observability or a known sample interval. Where an inequality appears, the conclusion is restricted to that bound and the defined uncertainty class; where an estimated value appears, the inference is conditioned on its assumed noise model.

A2 reference. A=−1,C=1,L=4 leads to error pole=−5 in the stated ideal model. The answer is incomplete without the numerical relationship, relevant units, and an explicit condition such as positivity, stability, nonsingularity, observability or a known sample interval. Where an inequality appears, the conclusion is restricted to that bound and the defined uncertainty class; where an estimated value appears, the inference is conditioned on its assumed noise model.

A3 reference. Pminus=1.25,R=0.75 leads to Kalman gain 0.625 in the stated ideal model. The answer is incomplete without the numerical relationship, relevant units, and an explicit condition such as positivity, stability, nonsingularity, observability or a known sample interval. Where an inequality appears, the conclusion is restricted to that bound and the defined uncertainty class; where an estimated value appears, the inference is conditioned on its assumed noise model.

A4 reference. innovation 2, covariance 4 leads to NIS=1 in the stated ideal model. The answer is incomplete without the numerical relationship, relevant units, and an explicit condition such as positivity, stability, nonsingularity, observability or a known sample interval. Where an inequality appears, the conclusion is restricted to that bound and the defined uncertainty class; where an estimated value appears, the inference is conditioned on its assumed noise model.

B series marking. Award credit for four independent elements: correct mathematical relationship, transparent substitution, meaningful limiting-case or sensitivity check, and a proposed measurement that would discriminate among possible models. The four named lesson scenarios demand different reasoning; a generic statement about noise or risk is not sufficient. Problems involving constraints must state whether a proposed solution is feasible rather than only desirable.

C series marking. Accept conclusions that identify an information gap rather than inventing missing values. A strong answer describes the exact signal at which the model would first fail, a safe way to expose that failure in software, and which human or independent engineering review would be needed for any later physical test. For stochastic examples insist on a distinction between model variance and guaranteed error. For optimization distinguish an unconstrained minimum from a feasible command. For robust control do not confuse sufficient tests with necessary ones.

Post-assessment research note

Write a final one-page technical summary with the columns: claim, equation, required assumptions, numerical evidence, strongest counterexample and next verification step. The purpose is to practice deciding what is established, what is conditional and what remains unknown. A model may be educationally useful even when publication-quality validation is absent. Never treat an optimized cost, a stable nominal trace, or an intact software archive as an independent functional-safety assessment.