Full-State Feedback Controller Design and Pole Placement via Ackermann

Lesson 4.1 — Prediction models and lifted dynamics

Unit: Model predictive control and constraints. 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 sampled linear model forecasts future states from a current estimate and candidate input sequence. Repeated substitution produces a lifted, or stacked, prediction relation with matrices that reveal how each future input influences every predicted state. This is the foundation of quadratic model predictive control. A reliable implementation must clarify whether commands are absolute inputs or changes relative to an operating point.

Governing relation: x_{k+1}=A x_k+B u_k; x_{k+2}=A² x_k+A B u_k+B u_{k+1}

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 A=0.8, B=0.2, current x0=1, u0=2 and u1=0, x1=0.8×1+0.2×2=1.2 and x2=0.8×1.2+0.2×0=0.96. A controller that incorrectly treats u1 as acting on x1 at the previous time will produce a different forecast and may violate constraints.

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 stacked forecast is causal: only inputs earlier than a state appear in that state expression. Disturbance predictions can be appended, but their uncertainty should not be disguised as known deterministic inputs.

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

Model mismatch in A compounds across a horizon. An inaccurate long-horizon forecast can be less useful than a shorter well-calibrated one, particularly around saturation or rapidly changing operating points.

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 — Microgrid energy scheduling in simulation

Consider a purely synthetic battery scheduling model where state of charge changes with commanded power, subject to energy and power limits. A short-horizon optimizer can shift power to reduce a cost, but a physically infeasible target must be reported rather than hidden in the objective weights. Charge efficiency and battery limits are operating assumptions, not optional plotting details. Compare a one-step controller with a longer prediction that anticipates an upcoming demand increase.

Lab protocol: Feasibility audit. Write all dynamic equations and hard limits before minimizing the cost. Solve one tiny case by hand. Then create an impossible target on purpose and insist that the optimizer reports infeasible instead of pretending the output was achieved.

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=0.7, B=0.3, x0=−1, u0=1, u1=2, compute x1 and x2 independently and via the lifted formula. 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 Quadratic MPC optimization with hard bounds. 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 4.2 — Quadratic MPC optimization with hard bounds

Unit: Model predictive control and constraints. 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

MPC usually minimizes a predicted tracking error plus an input or input-change cost subject to explicit dynamics and constraints. A quadratic objective with linear constraints can be a convex quadratic program when its Hessian is positive semidefinite. In a one-step scalar case the constrained solution is often the clipped unconstrained optimum; in multi-step coupled cases independent clipping is generally incorrect.

Governing relation: min_u (x1-r)^2+rhou², x1=ax0+b*u, umin<=u<=umax

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 a=1, b=1, x0=2, reference r=0, ρ=1 and bound −1≤u≤1. Without the bound the minimizer is u=−1, which exactly reaches the lower boundary. With tighter −0.5≤u≤0.5, the constrained minimum is u=−0.5, x1=1.5 and J=1.5²+0.5²=2.5. Setting u=−2 would give zero state error but an input penalty 4 and violates both bounds.

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

Hard bounds turn a preference into a feasible set. An optimizer returning the nominal unconstrained gain when it violates an explicit bound is not solving the stated problem. Even feasible numerical outputs must be checked against solver tolerances.

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 with actuator slew constraints |u_k−u_{k−1}|≤δ couples decisions across the horizon, so simply clipping each unconstrained input may not yield a feasible sequence.

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 — Microgrid energy scheduling in simulation

Consider a purely synthetic battery scheduling model where state of charge changes with commanded power, subject to energy and power limits. A short-horizon optimizer can shift power to reduce a cost, but a physically infeasible target must be reported rather than hidden in the objective weights. Charge efficiency and battery limits are operating assumptions, not optional plotting details. Compare a one-step controller with a longer prediction that anticipates an upcoming demand increase.

Lab protocol: Feasibility audit. Write all dynamic equations and hard limits before minimizing the cost. Solve one tiny case by hand. Then create an impossible target on purpose and insist that the optimizer reports infeasible instead of pretending the output was achieved.

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 x0=3, a=b=1, ρ=2, r=0 and bound |u|≤1, calculate the unconstrained and constrained input and their objective values. 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 Receding horizon and disturbance rejection. 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 4.3 — Receding horizon and disturbance rejection

Unit: Model predictive control and constraints. 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

MPC applies only the first decision of each optimized sequence, measures or estimates the new state, and solves again. Re-optimization can compensate for moderate forecast errors, but not for uncontrolled model uncertainty, lost sensors or impossible targets. Offset-free tracking often adds a disturbance model or integral action. Horizon length changes anticipation, numerical workload and terminal behavior.

Governing relation: At each k: estimate x_k -> solve {u_{k|k},...,u_{k+N-1|k}} -> apply u_{k|k} -> measure -> repeat

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

Imagine a one-state plant x_{k+1}=0.9x_k+0.1u_k+d with constant disturbance d=0.1. For a nominal model that omits d, applying u=0 at x=0 yields observed x1=0.1. At the following update the controller can react to this unexpected change, but if the disturbance persists and the design lacks an integral or disturbance-estimation mechanism, a nonzero equilibrium error can remain.

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

Re-optimization is feedback and makes MPC different from a long open-loop command script. Nevertheless, a large enough disturbance can violate constraints between sample times before new optimization is possible.

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 inaccurate state estimate can turn optimization into confident action based on the wrong initial state. Solver feedback should record feasibility, status, iteration count and whether a fallback input was used.

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 — Microgrid energy scheduling in simulation

Consider a purely synthetic battery scheduling model where state of charge changes with commanded power, subject to energy and power limits. A short-horizon optimizer can shift power to reduce a cost, but a physically infeasible target must be reported rather than hidden in the objective weights. Charge efficiency and battery limits are operating assumptions, not optional plotting details. Compare a one-step controller with a longer prediction that anticipates an upcoming demand increase.

Lab protocol: Feasibility audit. Write all dynamic equations and hard limits before minimizing the cost. Solve one tiny case by hand. Then create an impossible target on purpose and insist that the optimizer reports infeasible instead of pretending the output was achieved.

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: Derive the steady state x for x_{k+1}=0.9x_k+0.1u_k+0.1 with constant u=0. What actuator input would hold x=0 in this model? 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 Feasibility, terminal design and computation deadlines. 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 4.4 — Feasibility, terminal design and computation deadlines

Unit: Model predictive control and constraints. 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

An optimization problem is infeasible when no candidate sequence satisfies every modeled constraint. Soft constraints can assign a penalty to violation of some performance limits, but safety-critical constraints must not be softened merely to obtain a solver result. Terminal sets and terminal costs help analyze recursive feasibility and stability under specified assumptions. The optimizer must also finish before the control update deadline.

Governing relation: For scalar plant x_{k+1}=x_k+u_k, |u_k|<=1; one-step target constraint x_{k+1}=0

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

If x0=3, the one-step exact-target constraint requires u0=−3, which violates |u0|≤1. The problem is infeasible. A three-step horizon could reach zero using u0=u1=u2=−1 in this ideal integrator, provided no intermediate state constraints prohibit the path. This example shows why horizon and terminal requirements affect feasibility.

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 fallback action cannot be chosen arbitrarily. It should have its own modeled admissible region and independent safety review; otherwise a software error can convert infeasibility into an unsafe command.

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

Warm-starting can reduce solver time, but it does not prove the optimizer converges within every deadline. Compute-time distributions, missed updates and the chosen behavior on failure belong in the test plan.

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 — Microgrid energy scheduling in simulation

Consider a purely synthetic battery scheduling model where state of charge changes with commanded power, subject to energy and power limits. A short-horizon optimizer can shift power to reduce a cost, but a physically infeasible target must be reported rather than hidden in the objective weights. Charge efficiency and battery limits are operating assumptions, not optional plotting details. Compare a one-step controller with a longer prediction that anticipates an upcoming demand increase.

Lab protocol: Feasibility audit. Write all dynamic equations and hard limits before minimizing the cost. Solve one tiny case by hand. Then create an impossible target on purpose and insist that the optimizer reports infeasible instead of pretending the output was achieved.

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 x0=2.5 and |u|≤0.5, find the minimum horizon N for the ideal integrator to reach xN=0. State any implicit assumptions about fractional command values. 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 Prediction models and lifted 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.

Unit 4 applied project — MPC feasibility and receding horizon comparison

Engineering question

Use x_{k+1}=0.9x_k+0.1u_k, x0=2, |u|≤1 and reference zero. Derive the one-step quadratic optimum for two penalties and demonstrate a target or terminal constraint that is infeasible. Compare a full open-loop two-step command with first-step-only receding-horizon behavior after an injected disturbance. Record objective, bound satisfaction and fallback rule.

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

Feasibility audit: Write all dynamic equations and hard limits before minimizing the cost. Solve one tiny case by hand. Then create an impossible target on purpose and insist that the optimizer reports infeasible instead of pretending the output was achieved.

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 4 assessment — Model predictive control and constraints

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=0.8,B=0.2,x0=1,u0=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.

A2. Given x0=2,r=0,rho=1,|u|≤0.5, integrator, 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 x_next=0.9x+0.1u+0.1, u=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.

A4. Given x0=3, |u|≤1 with exact target 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.

Part B — distinct lesson problems

B1. Prediction models and lifted dynamics. For A=0.7, B=0.3, x0=−1, u0=1, u1=2, compute x1 and x2 independently and via the lifted formula. 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. Quadratic MPC optimization with hard bounds. For x0=3, a=b=1, ρ=2, r=0 and bound |u|≤1, calculate the unconstrained and constrained input and their objective values. 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. Receding horizon and disturbance rejection. Derive the steady state x for x_{k+1}=0.9x_k+0.1u_k+0.1 with constant u=0. What actuator input would hold x=0 in this model? 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. Feasibility, terminal design and computation deadlines. For x0=2.5 and |u|≤0.5, find the minimum horizon N for the ideal integrator to reach xN=0. State any implicit assumptions about fractional command values. 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=0.8,B=0.2,x0=1,u0=2 leads to x1=1.2 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. x0=2,r=0,rho=1,|u|≤0.5, integrator leads to u=−0.5, x1=1.5 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. x_next=0.9x+0.1u+0.1, u=0 leads to steady x=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.

A4 reference. x0=3, |u|≤1 with exact target 0 leads to minimum horizon 3 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.