Linear Quadratic Regulator (LQR) Optimal Control and Algebraic Riccati Equations

Lesson 6.1 — Persistent excitation and identifiability

Unit: Adaptive control, identification and learning. 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

Estimating unknown plant coefficients requires informative data. If all observed inputs are constant and the measured output changes little, several models may explain the same trace. Persistent excitation formalizes a richness property in the regression signal that permits parameter convergence under appropriate assumptions. A good input experiment excites relevant dynamics while respecting an allowable operating envelope.

Governing relation: y_k=phi_k^T theta+epsilon_k; information matrix M=sum_k phi_k phi_k^T

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 theta has two entries and every regression vector is φk=[1,1]ᵀ for ten samples. Then M=10[[1,1],[1,1]], which has rank one. Only θ1+θ2 is identified. If five rows use [1,0]ᵀ and five rows [0,1]ᵀ, M=diag(5,5), which has full rank and separates both parameters in this ideal regression.

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

A large dataset is not automatically rich data. Ten thousand nearly repeated measurements can reveal less about an independent parameter than a short, carefully varied experiment.

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

A plant operated inside a tight safe zone might not allow the excitation needed for every parameter. In that case the uncertainty should be reported rather than forcing an unreliable point 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 — Adaptive gain in a changing load

A synthetic motor model changes effective inertia mid-experiment. An estimator that has only seen one constant-speed trajectory has little information about inertia, even though thousands of samples exist. Apply an admissible sequence of small command changes and evaluate held-out prediction error. Contrast a slow, conservative fixed controller with an adaptive design under noisy measurements, and log when the adaptive gain reaches a projection boundary.

Lab protocol: Excitation audit. Check the rank and conditioning of the regressor information matrix before evaluating estimated coefficients. Add a held-out dataset; never label the same dataset both training and validation. Track whether the adaptation law saturates at a parameter boundary.

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 regression vectors [1,0], [1,1] and [1,−1], form M, find its eigenvalues and decide whether both parameter directions are distinguishable. 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 Recursive least squares with forgetting. 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 6.2 — Recursive least squares with forgetting

Unit: Adaptive control, identification and learning. 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

Recursive least squares updates a linear parameter estimate as new measurements arrive and uses a covariance-like matrix to describe which regression directions remain uncertain. A forgetting factor below one reduces the weight of old data, improving response to drift but making estimates more sensitive to noise. The update requires careful numerical conditioning and bounded covariance.

Governing relation: K=P phi/(lambda+phi^T P phi); theta_new=theta+K(y-phi^T theta); P_new=(P-K phi^T P)/lambda

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

Take scalar θ0=0, P0=4, λ=1, regressor φ=1 and measurement y=2. The gain is K=4/(1+4)=0.8, updated θ1=0+0.8×2=1.6, and P1=(4−0.8×4)=0.8. A second identical, noiseless observation gives gain 0.8/(1+0.8)=0.4444 and θ2≈1.7778. The estimator moves toward 2 as evidence accumulates.

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

RLS is an identification algorithm, not a universal adaptive controller. If the regressor is a function of biased estimated states or the noise is correlated with input, a fitted parameter may be biased.

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

Choosing a very small forgetting factor can cause old information to disappear rapidly. The estimate may then move wildly in a weakly excited region or attempt to track measurement noise.

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 — Adaptive gain in a changing load

A synthetic motor model changes effective inertia mid-experiment. An estimator that has only seen one constant-speed trajectory has little information about inertia, even though thousands of samples exist. Apply an admissible sequence of small command changes and evaluate held-out prediction error. Contrast a slow, conservative fixed controller with an adaptive design under noisy measurements, and log when the adaptive gain reaches a projection boundary.

Lab protocol: Excitation audit. Check the rank and conditioning of the regressor information matrix before evaluating estimated coefficients. Add a held-out dataset; never label the same dataset both training and validation. Track whether the adaptation law saturates at a parameter boundary.

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: Repeat the first scalar update with λ=0.5, P0=4, θ0=0, φ=1, y=2. Compare the gain and interpretation with λ=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 Gradient adaptation and bounded parameters. 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 6.3 — Gradient adaptation and bounded parameters

Unit: Adaptive control, identification and learning. 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

Gradient updates change parameters in the direction that reduces a chosen instantaneous prediction error. The adaptation step depends on a learning rate, regressor scale and loss definition. Projection operators can constrain estimated parameters to a physically plausible range, but projection alone does not make an unverified adaptive closed loop stable. Separating the estimator’s goal from the controller’s stability argument is essential.

Governing relation: theta_next=Proj_[thetamin,thetamax](theta+etaphi(y-phi*theta))

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 scalar θ0=0.5, φ=2, y=3, learning rate η=0.1, the prediction is 1, error is 2 and the unprojected update is θ1=0.5+0.1×2×2=0.9. If the admissible interval is [0.2,0.8], projection sets θ1=0.8. The residual does not disappear merely because the estimate is kept inside a declared physical range.

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

Parameter clipping can protect the admissible set and make logs interpretable, but the adaptation law may still be too fast for the sample rate. The data can also be insufficient to distinguish true drift from a sensor-calibration change.

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 η is too large, the scalar least-squares update can oscillate or diverge, even for fixed clean data. Normalized gradients and stability analysis can avoid some of these problems but still depend on correct assumptions.

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 — Adaptive gain in a changing load

A synthetic motor model changes effective inertia mid-experiment. An estimator that has only seen one constant-speed trajectory has little information about inertia, even though thousands of samples exist. Apply an admissible sequence of small command changes and evaluate held-out prediction error. Contrast a slow, conservative fixed controller with an adaptive design under noisy measurements, and log when the adaptive gain reaches a projection boundary.

Lab protocol: Excitation audit. Check the rank and conditioning of the regressor information matrix before evaluating estimated coefficients. Add a held-out dataset; never label the same dataset both training and validation. Track whether the adaptation law saturates at a parameter boundary.

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 noiseless y=θtrueφ with φ=3, derive the interval of positive η for stable scalar error recursion under the displayed unprojected update. 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 Adaptive control versus robust fixed design. 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 6.4 — Adaptive control versus robust fixed design

Unit: Adaptive control, identification and learning. 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 fixed robust controller trades optimal nominal performance for tolerance to an explicit range of plant uncertainty. An adaptive controller attempts to update its model or gain as the plant changes. Neither dominates automatically. Adaptive schemes require excitation, bounded update behavior, and independent limits on what changing parameters may command; robust designs can be overly conservative when uncertainty is wide.

Governing relation: P(s,k)=k/(s+1); with fixed C=K>0, pole=-(1+Kk); changing K_hat(t) changes the system class

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 k between 0.5 and 1.5 and fixed K=2, the simple loop pole lies between −2 and −4 s^-1, stable over the stated positive interval. An adaptive Khat that is allowed to become −2 would instead yield pole −(1−2k), which is positive for k>0.5; an uncontrolled update can create instability even though every evaluated fixed positive gain was stable.

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 decision should include operational cost: required sensors, computational cycles, validation effort and the consequences of a faulty adaptation transient. A stable modest controller may be preferable when performance requirements are achievable without learning online.

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

Demonstrations that update only on synthetic noiseless data do not prove performance under real drift, saturation or lost measurements. A good trial includes a deliberately unidentifiable case and reports the failed parameter convergence.

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 — Adaptive gain in a changing load

A synthetic motor model changes effective inertia mid-experiment. An estimator that has only seen one constant-speed trajectory has little information about inertia, even though thousands of samples exist. Apply an admissible sequence of small command changes and evaluate held-out prediction error. Contrast a slow, conservative fixed controller with an adaptive design under noisy measurements, and log when the adaptive gain reaches a projection boundary.

Lab protocol: Excitation audit. Check the rank and conditioning of the regressor information matrix before evaluating estimated coefficients. Add a held-out dataset; never label the same dataset both training and validation. Track whether the adaptation law saturates at a parameter boundary.

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 k∈[0.5,1.5], find the fixed K interval that makes the scalar pole strictly negative for all k. Distinguish that interval from a physical actuator limit. 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 Persistent excitation and identifiability. 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 6 applied project — Excitation and estimator convergence experiment

Engineering question

Estimate y=θ1φ1+θ2φ2 using one rank-deficient dataset with φ=[1,1] repeated, then one full-rank dataset with φ drawn from [1,0],[0,1] and [1,1]. Generate deterministic noise. Compare ordinary least squares and a simple RLS update; report information matrix rank, recovered parameters and validation error. Do not claim identifiable individual parameters from the rank-one case.

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

Excitation audit: Check the rank and conditioning of the regressor information matrix before evaluating estimated coefficients. Add a held-out dataset; never label the same dataset both training and validation. Track whether the adaptation law saturates at a parameter boundary.

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 6 assessment — Adaptive control, identification and learning

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 ten repeats of phi=[1,1], 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 RLS theta=0,P=4,lambda=1,phi=1,y=2, 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 eta=0.1,phi=2,y=3,theta=0.5, 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 fixed scalar plant k in [0.5,1.5],K=2, 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. Persistent excitation and identifiability. For regression vectors [1,0], [1,1] and [1,−1], form M, find its eigenvalues and decide whether both parameter directions are distinguishable. 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. Recursive least squares with forgetting. Repeat the first scalar update with λ=0.5, P0=4, θ0=0, φ=1, y=2. Compare the gain and interpretation with λ=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. Gradient adaptation and bounded parameters. For noiseless y=θtrueφ with φ=3, derive the interval of positive η for stable scalar error recursion under the displayed unprojected update. 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. Adaptive control versus robust fixed design. For k∈[0.5,1.5], find the fixed K interval that makes the scalar pole strictly negative for all k. Distinguish that interval from a physical actuator limit. 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. ten repeats of phi=[1,1] leads to information matrix rank 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.

A2 reference. RLS theta=0,P=4,lambda=1,phi=1,y=2 leads to K=0.8, theta_new=1.6 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. eta=0.1,phi=2,y=3,theta=0.5 leads to theta_new=0.9 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. fixed scalar plant k in [0.5,1.5],K=2 leads to pole between −2 and −4 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.