Full-Order and Reduced-Order State Observer Synthesis

Lesson 5.1 — Local linearization and equilibrium analysis

Unit: Nonlinear systems, Lyapunov reasoning and feedback linearization. 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

Nonlinear dynamics can be approximated near a named equilibrium by their Jacobian matrices. This local model predicts sufficiently small perturbations if the system is smooth and neglected higher-order terms are small. It does not guarantee behavior over a distant state region. Operating-point selection matters because the derivative changes with state.

Governing relation: x_dot=f(x,u); A=(partial f/partial x)|_(x,u), B=(partial f/partial u)|_(x,u)

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 x_dot=−x³+u. At equilibrium x=1, u=1, the local derivative with respect to x is A=−3 and B=1, so perturbations approximately satisfy δx_dot=−3δx+δu. At equilibrium x=0, u=0, the Jacobian A=0 and linearization alone gives no asymptotic-stability conclusion, even though the full system x_dot=−x³ decays toward zero for nonzero initial states.

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 zero linearized eigenvalue is inconclusive about nonlinear stability; it is not proof of instability. Strongly nonlinear actuation or state limits can make the local model inaccurate even after a short excursion.

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

For x_dot=x−x³ with zero input, equilibria occur at −1, 0 and 1. The Jacobian has different signs at these points. One local controller is not guaranteed to work identically at all three.

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 — Speed-dependent friction model

A demonstration stage exhibits nearly linear viscous friction at moderate velocity and a deadband near zero. A controller derived by canceling the measured friction curve can fail where the curve is discontinuous or the sensor velocity estimate is noisy. Use phase portraits to compare local convergence and overshoot while keeping the simulated state inside a declared envelope. State precisely whether the proof shows local stability, global stability or only a numerical observation.

Lab protocol: Region-of-validity audit. Write the exact nonlinear model, the equilibrium, the proposed Lyapunov function or transformed coordinates and the domain in which all manipulations hold. Compare a small initial perturbation with a much larger one and note whether a local proof still applies.

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: Find and classify the equilibria of x_dot=2x−x³ using the scalar derivative test where it applies. State what happens if the derivative equals zero. 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 Lyapunov functions and regions of attraction. 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 5.2 — Lyapunov functions and regions of attraction

Unit: Nonlinear systems, Lyapunov reasoning and feedback linearization. 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 Lyapunov function is a positive measure of state displacement whose time derivative along the system trajectories reveals whether the measure decreases. For stability about the origin, V should be positive definite near the origin; asymptotic stability follows under appropriate conditions if its derivative is negative definite. Global claims need global hypotheses, not merely a local graph.

Governing relation: V(x)=0.5x²; dV/dt=xf(x)

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 x_dot=−x−x³, choose V=0.5x². Then Vdot=x(−x−x³)=−x²−x⁴<0 for x≠0. V is positive definite and radially unbounded; for this smooth scalar system this establishes global asymptotic stability of the origin. Compare x_dot=x−x³: the same V gives Vdot=x²−x⁴, which is positive for sufficiently small nonzero x, so the origin is unstable.

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 derivative’s sign must be calculated for the actual closed-loop dynamics. Finding a function that decreases along one simulated path is not equivalent to proving a decrease for all states in a region.

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 actuator saturation may modify the vector field outside a narrow band. A Lyapunov certificate derived without saturation does not automatically extend to the clipped system.

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 — Speed-dependent friction model

A demonstration stage exhibits nearly linear viscous friction at moderate velocity and a deadband near zero. A controller derived by canceling the measured friction curve can fail where the curve is discontinuous or the sensor velocity estimate is noisy. Use phase portraits to compare local convergence and overshoot while keeping the simulated state inside a declared envelope. State precisely whether the proof shows local stability, global stability or only a numerical observation.

Lab protocol: Region-of-validity audit. Write the exact nonlinear model, the equilibrium, the proposed Lyapunov function or transformed coordinates and the domain in which all manipulations hold. Compare a small initial perturbation with a much larger one and note whether a local proof still applies.

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 x_dot=−2x+x³ and V=x²/2, find where Vdot is negative. Can the argument establish global asymptotic stability? 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 Feedback linearization and singularities. 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 5.3 — Feedback linearization and singularities

Unit: Nonlinear systems, Lyapunov reasoning and feedback linearization. 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

Input-output feedback linearization cancels known nonlinearities to produce a simpler desired response when the input acts with adequate relative degree and the cancellation is defined. Such a law is fragile to model error and can fail at points where the input effectiveness vanishes. Internal dynamics can remain after output linearization and must be checked separately.

Governing relation: For x_dot=−x³+u, choose u=x³−k(x-r), giving (x-r)_dot=−k(x-r) for constant r

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

Let k=2 s^-1 and constant r=1. Under the exact scalar model and unbounded actuation, the proposed law yields e_dot=−2e and e(t)=e0e^(−2t). If x0=0, error begins at −1 and at t=1 s is approximately −0.1353. Although the nonlinear term cancels perfectly in equations, a plant input limit may prevent u=x³−2e when x is large.

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

Exact cancellation presumes an accurate model. If the real cubic coefficient is 1.2 but the control cancels 1.0, a residual −0.2x³ changes the trajectory and could dominate far from the chosen operating point.

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

For dynamics x_dot=−x+u x, solving for u to impose a desired x_dot divides by x. At x=0 the input has no instantaneous effect, so a naive inverse law is singular.

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 — Speed-dependent friction model

A demonstration stage exhibits nearly linear viscous friction at moderate velocity and a deadband near zero. A controller derived by canceling the measured friction curve can fail where the curve is discontinuous or the sensor velocity estimate is noisy. Use phase portraits to compare local convergence and overshoot while keeping the simulated state inside a declared envelope. State precisely whether the proof shows local stability, global stability or only a numerical observation.

Lab protocol: Region-of-validity audit. Write the exact nonlinear model, the equilibrium, the proposed Lyapunov function or transformed coordinates and the domain in which all manipulations hold. Compare a small initial perturbation with a much larger one and note whether a local proof still applies.

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 x_dot=−0.8x³+u with controller u=x³−3x and r=0, calculate the actual nonlinear closed-loop dynamics. Determine whether V=x²/2 decreases for all x. 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 Sliding-mode intuition and chattering limits. 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 5.4 — Sliding-mode intuition and chattering limits

Unit: Nonlinear systems, Lyapunov reasoning and feedback linearization. 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

Switching feedback can drive a chosen error variable toward a surface despite certain bounded disturbances. A textbook discontinuous sign law is useful for a proof but not directly implementable by finite-bandwidth actuators. Boundary layers soften the switch and reduce chattering at the cost of finite steady residual error. Disturbance bounds must be genuine rather than assumed without evidence.

Governing relation: s_dot=d-ksign(s), |d|<=D, k>D; V=0.5s² gives Vdot<=-(k-D)|s|

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=3 and |d|≤1, the derivative of V satisfies Vdot≤−2|s| for nonzero s. The ideal reaching-time bound from |s0|=4 is at most 4/(3−1)=2 s. The statement depends on the disturbance being matched to this scalar channel and the actuator being able to realize the discontinuous command.

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 analysis distinguishes a mathematically finite reaching time from a physical switching transient. Sampling, sensor noise, unmodeled high-frequency modes and input-rate limits may invalidate the ideal bound.

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 small boundary-layer function sat(s/φ) typically gives smoother commands. It no longer produces the same exact sign-law inequalities for arbitrarily small s, so claims of exact finite-time arrival should be revised.

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 — Speed-dependent friction model

A demonstration stage exhibits nearly linear viscous friction at moderate velocity and a deadband near zero. A controller derived by canceling the measured friction curve can fail where the curve is discontinuous or the sensor velocity estimate is noisy. Use phase portraits to compare local convergence and overshoot while keeping the simulated state inside a declared envelope. State precisely whether the proof shows local stability, global stability or only a numerical observation.

Lab protocol: Region-of-validity audit. Write the exact nonlinear model, the equilibrium, the proposed Lyapunov function or transformed coordinates and the domain in which all manipulations hold. Compare a small initial perturbation with a much larger one and note whether a local proof still applies.

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=5, disturbance bound D=2 and |s0|=1.5, compute the ideal reaching-time bound. What changes if D≥k? 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 Local linearization and equilibrium analysis. 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 5 applied project — Nonlinear energy proof and bounded actuation

Engineering question

For xdot=−x−αx³+u with α={0.5,1,2}, derive Vdot for u=0 and compare trajectories from x0={0.2,1,3}. Test u=clip(−kx, ±1) in software and explain how the clipped vector field differs from the ideal law. Distinguish an exact proof for the unforced mathematical model from numerical evidence under bounded control.

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

Region-of-validity audit: Write the exact nonlinear model, the equilibrium, the proposed Lyapunov function or transformed coordinates and the domain in which all manipulations hold. Compare a small initial perturbation with a much larger one and note whether a local proof still applies.

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 5 assessment — Nonlinear systems, Lyapunov reasoning and feedback linearization

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 f(x)=−x³+u at x=1,u=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 x_dot=−x−x³,V=x²/2, 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 x0=0,r=1,k=2 in e_dot=−2e, 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 k=3,D=1,s0=4 ideal sliding, 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. Local linearization and equilibrium analysis. Find and classify the equilibria of x_dot=2x−x³ using the scalar derivative test where it applies. State what happens if the derivative equals zero. 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. Lyapunov functions and regions of attraction. For x_dot=−2x+x³ and V=x²/2, find where Vdot is negative. Can the argument establish global asymptotic stability? 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. Feedback linearization and singularities. For x_dot=−0.8x³+u with controller u=x³−3x and r=0, calculate the actual nonlinear closed-loop dynamics. Determine whether V=x²/2 decreases for all x. 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. Sliding-mode intuition and chattering limits. For k=5, disturbance bound D=2 and |s0|=1.5, compute the ideal reaching-time bound. What changes if D≥k? 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. f(x)=−x³+u at x=1,u=1 leads to A=−3,B=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. x_dot=−x−x³,V=x²/2 leads to Vdot=−x²−x⁴ 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. x0=0,r=1,k=2 in e_dot=−2e leads to e(1)=−e^(−2)≈−0.1353 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. k=3,D=1,s0=4 ideal sliding leads to reaching bound≤2 s 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.