Discrete-Time State-Space Systems, Z-Transform State Equations, and Deadbeat Control

Lesson 7.1 — Transfer matrices and channel interaction

Unit: Multivariable systems, coupling and decoupling. 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 multivariable plant has several inputs and outputs. Its transfer matrix G(s) describes how each input affects each output; off-diagonal terms represent coupling. Independent single-loop tuning may work when those terms are weak at relevant frequencies, but steady-state decoupling alone may fail near dynamic resonance. Signal scaling must be fixed before comparing numerical gains.

Governing relation: y(s)=G(s)u(s), G=[g11 g12;g21 g22]

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 static G=[[2,1],[0.5,1.5]]. Applying u=[1,0] produces y=[2,0.5], so the first actuator moves both measured outputs. Applying u=[0,1] produces y=[1,1.5]. For target y=[1,1], solve det(G)=3−0.5=2.5, yielding inverse (1/2.5)[[1.5,−1],[−0.5,2]], and u=[0.2,0.6].

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 inverse makes sense only for a nonsingular static gain matrix and actuator commands within limits. If one output represents a temperature and another a voltage, the numerical matrix depends on the measurement units.

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 dynamic G(s) can have phase lag and nonminimum-phase zeros that a static inverse ignores. Attempting to cancel those dynamics can produce unstable or noncausal control laws.

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

Engineering case — Two-channel heating and airflow

A test process has two actuators, heater power and fan duty, and two outputs, temperature and humidity proxy. Increasing airflow changes both measurements, so independent loops can oppose one another. First inspect a 2×2 steady-state gain matrix, then compare the effect of a small step in each actuator. Before trusting inverse-based decoupling, check conditioning, actuator bounds and whether dynamics or delays change the input-output relationships.

Lab protocol: Actuator-direction audit. Examine the input direction that produces the smallest measured change. Ask whether that direction is affordable in actuator effort. Report how a small gain perturbation changes an inverse-based command and whether the proposed pairing remains sensible.

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 G=[[1,0.8],[0.2,1]], compute its determinant, G^{-1} and the input vector required to produce y=[1,0]. 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 Singular values and poor conditioning. 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 7.2 — Singular values and poor conditioning

Unit: Multivariable systems, coupling and decoupling. 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

Singular values describe how strongly a matrix maps input directions into output directions after scaling. A small minimum singular value means some desired output direction requires large input, making control sensitive to noise and mismatch. Condition number is the ratio of maximum to minimum singular value, provided the minimum is nonzero. A full-rank matrix can still be practically difficult to invert.

Governing relation: kappa_2(G)=sigma_max(G)/sigma_min(G)

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 diagonal G=diag(10,0.1), singular values are 10 and 0.1, so condition number is 100. To obtain y=[1,1] requires u=[0.1,10], with a huge second input relative to the first. Rescaling the second output units can change the raw condition number, so an engineering interpretation must begin with normalized variables.

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 low singular value need not imply instability; it implies low authority in an input-output direction. Controller optimization can penalize that costly direction instead of demanding exact inversion.

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

Measuring frequency-dependent singular values can reveal interaction peaks that are invisible in a DC matrix. A matrix well-conditioned at zero frequency may be poorly conditioned near a fast actuator resonance.

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

Engineering case — Two-channel heating and airflow

A test process has two actuators, heater power and fan duty, and two outputs, temperature and humidity proxy. Increasing airflow changes both measurements, so independent loops can oppose one another. First inspect a 2×2 steady-state gain matrix, then compare the effect of a small step in each actuator. Before trusting inverse-based decoupling, check conditioning, actuator bounds and whether dynamics or delays change the input-output relationships.

Lab protocol: Actuator-direction audit. Examine the input direction that produces the smallest measured change. Ask whether that direction is affordable in actuator effort. Report how a small gain perturbation changes an inverse-based command and whether the proposed pairing remains sensible.

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 G=diag(4,0.5), calculate the condition number and input required for y=[2,1]. What happens if the second actuator is limited to |u2|≤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 Relative gain array and input-output pairing. 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 7.3 — Relative gain array and input-output pairing

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

Concept and mechanism

The relative gain array compares the open-loop effect of an actuator with the effect remaining when other channels are controlled. For a nonsingular steady-state gain matrix G, RGA=G elementwise-multiplied by (G^{-1})^T. Values close to one on proposed pairings can support decentralized pairing under stated assumptions; negative or very large elements warn of interactions. RGA is not a complete dynamic stability analysis.

Governing relation: Lambda=G .* (G^(-1))^T, elementwise product

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 G=[[2,1],[0.5,1.5]], the inverse is [[0.6,−0.4],[−0.2,0.8]]. RGA=[[2×0.6,1×(−0.2)],[0.5×(−0.4),1.5×0.8]]=[[1.2,−0.2],[−0.2,1.2]]. Row and column sums each equal one. Diagonal pairings appear preferable to off-diagonal ones for this static model.

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

This analysis assumes the matrix represents the relevant process gain near the operating point. Transport delay, changing process direction and actuator saturation can reverse an apparently satisfactory pairing decision.

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 G is singular or near singular, its inverse does not exist or is numerically fragile; an RGA computed with an unstable inverse is not meaningful evidence.

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

Engineering case — Two-channel heating and airflow

A test process has two actuators, heater power and fan duty, and two outputs, temperature and humidity proxy. Increasing airflow changes both measurements, so independent loops can oppose one another. First inspect a 2×2 steady-state gain matrix, then compare the effect of a small step in each actuator. Before trusting inverse-based decoupling, check conditioning, actuator bounds and whether dynamics or delays change the input-output relationships.

Lab protocol: Actuator-direction audit. Examine the input direction that produces the smallest measured change. Ask whether that direction is affordable in actuator effort. Report how a small gain perturbation changes an inverse-based command and whether the proposed pairing remains sensible.

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 the RGA for diagonal G=diag(2,3). Explain why it matches the intuitive independent pairing. 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 State feedback and decentralized versus centralized 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 7.4 — State feedback and decentralized versus centralized design

Unit: Multivariable systems, coupling and decoupling. 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

In state-space form, multiple inputs can collectively control modes that a single actuator cannot influence. The controllability matrix tests mathematical reachability, while centralized feedback can coordinate channels to avoid conflicting commands. Decentralized control is simpler to deploy but may ignore strong cross-coupling. Physical authority, communication and available measurement determine whether a centralized policy is practical.

Governing relation: x_dot=A x+B u; controllability matrix C=[B AB ... A^(n-1)B]

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=diag(−1,−2), B=I2. Both states can be driven independently and the pair is controllable because B itself has rank two. If instead B=[[1],[0]] with only one scalar input, the second state is uncontrollable; it remains stable in this example, but its trajectory cannot be assigned arbitrarily. Controllability alone does not say whether the commanded controls are within rated voltage or force.

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 centralized state-feedback law can take advantage of cross-sensor information, but it may depend on data that arrive late. A decentralized law may keep essential local loops operating when a network fails, at the cost of weaker coordination.

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 pole-placement calculation that asks for unachievable poles on an uncontrollable unstable mode is not a solver failure; it is a limitation of the selected actuator arrangement.

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

Engineering case — Two-channel heating and airflow

A test process has two actuators, heater power and fan duty, and two outputs, temperature and humidity proxy. Increasing airflow changes both measurements, so independent loops can oppose one another. First inspect a 2×2 steady-state gain matrix, then compare the effect of a small step in each actuator. Before trusting inverse-based decoupling, check conditioning, actuator bounds and whether dynamics or delays change the input-output relationships.

Lab protocol: Actuator-direction audit. Examine the input direction that produces the smallest measured change. Ask whether that direction is affordable in actuator effort. Report how a small gain perturbation changes an inverse-based command and whether the proposed pairing remains sensible.

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

Exercises

1. Calculation: For A=diag(1,−2) and B=[[0],[1]], identify the uncontrollable mode and explain why the system cannot be stabilized using only this input. 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 Transfer matrices and channel interaction. 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 7 applied project — Two-input two-output process pairing case

Engineering question

Use steady gain matrix G=[[2,1],[0.5,1.5]] and desired outputs [1,1]. Compute inverse input, RGA and condition number. Perturb g12 by +20% and recalculate required inputs. Introduce a ±0.5 actuator ceiling and explain the resulting compromise when exact outputs cannot be achieved. Compare diagonal pairing with an attempted cross-pairing.

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

Actuator-direction audit: Examine the input direction that produces the smallest measured change. Ask whether that direction is affordable in actuator effort. Report how a small gain perturbation changes an inverse-based command and whether the proposed pairing remains sensible.

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 7 assessment — Multivariable systems, coupling and decoupling

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 G=[[2,1],[0.5,1.5]], 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.

A2. Given G=diag(10,0.1), 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.

A3. Given RGA for G=[[2,1],[0.5,1.5]], 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 A=diag(1,−2),B=[[0],[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.

Part B — distinct lesson problems

B1. Transfer matrices and channel interaction. For G=[[1,0.8],[0.2,1]], compute its determinant, G^{-1} and the input vector required to produce y=[1,0]. 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. Singular values and poor conditioning. For G=diag(4,0.5), calculate the condition number and input required for y=[2,1]. What happens if the second actuator is limited to |u2|≤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. Relative gain array and input-output pairing. Find the RGA for diagonal G=diag(2,3). Explain why it matches the intuitive independent pairing. 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. State feedback and decentralized versus centralized design. For A=diag(1,−2) and B=[[0],[1]], identify the uncontrollable mode and explain why the system cannot be stabilized using only this input. 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. G=[[2,1],[0.5,1.5]] leads to det G=2.5 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.

A2 reference. G=diag(10,0.1) leads to condition number 100 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. RGA for G=[[2,1],[0.5,1.5]] leads to RGA=[[1.2,−0.2],[−0.2,1.2]] 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.

A4 reference. A=diag(1,−2),B=[[0],[1]] leads to unstable first mode uncontrollable 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.