Routh-Hurwitz Stability Criterion and Steady-State Error Analysis

Lesson 4.1 — Poles and internal versus external stability

Unit: Stability, characteristic equations and root locus. Prerequisites: differential equations, complex numbers, and the earlier lessons in the unit. Study: derive the relationship, reproduce the numerical case, investigate one changed assumption and complete the four problems. All numbers are synthetic and the models are not validated for hardware use.

Physical meaning and governing model

For a minimal finite-dimensional continuous-time LTI model, asymptotic stability requires poles strictly in the left half-plane. An imaginary-axis simple pole creates nondecaying oscillation in some contexts; a right-half-plane pole causes an exponential instability. Internal state modes matter even if a transfer-function cancellation makes them invisible in one input-output path.

A stability statement must specify the equilibrium, model, inputs and outputs. Bounded-input bounded-output stability of a transfer function need not establish internal stability under pole-zero cancellation.

Governing relationship: x_dot=Ax; mode exp(lambdat); stable if Re(lambda)<0

The formula should not be used until its input, output, initial state, sign convention and units have been declared. Distinguish a plant characteristic from the characteristic of a feedback interconnection. An algebraically valid transfer function is not an empirical model until its parameters and operating regime have supporting evidence. If an idealized element has no physical saturation or bandwidth, identify the missing implementation constraint before attempting design.

Worked example and interpretation

For A=diag(−2,+1), one state decays as e^(−2t) and the other grows as e^t. If the sensor only reads the first state, the measured transfer may appear perfectly stable while a hidden state diverges.

Check the magnitude against a relevant limit. In a continuous model, a pole measured in reciprocal seconds gives a time constant through its reciprocal; in a digital model, a pole must be inside the unit circle. A calculation of an output amplitude does not identify a suitable control input unless the actuator relationship is also known. For this particular example, copy the expression into a fresh worksheet, write the intermediate substitution with units and test whether the result changes in the expected direction when one parameter changes.

Limits, counterexamples and model evidence

Nonlinear or switched systems need different analysis. A stable local Jacobian does not certify large-disturbance behavior or a fault-safe controller.

A useful counterexample changes one important feature at a time: initial state, operating point, saturation level, sensor filtering, delay or an unmodeled fast mode. The fact that a trace is well-fitted is not proof of predictive validity. A good validation experiment uses data withheld during fitting; it should also record the actual input, since the requested command may not equal the delivered signal. If evidence is missing, write a conditional conclusion instead of inventing a parameter or tolerance.

Application perspective — power-converter current loop

Inductor current dynamics and computational delay are often decisive, but the switching process, output filter and sensed current scale set additional limits. Frequency-response estimates must be confined to an appropriate small-signal region. Avoid interpreting a pedagogical model as guidance for connecting live electronics or tuning protection circuitry.

The experiment should define the performance metric before looking at a plot. Appropriate measures differ by task: a transient model can be checked by peak, delay and settling behavior; a frequency response by gain and phase; a state model by observable signals and sensitivity to starting conditions. A controller that improves one metric but violates current, force, thermal, bandwidth or latency limits is not an acceptable solution. Explain which additional measurement best distinguishes the model from its main competing explanation.

Problems for independent work

1. Computation and sensitivity. Construct C=[1,0] for the sample A; explain why the second state is invisible and how a new sensor might help. Show how the relevant parameter enters the governing equation. If the data do not determine a single number, list the missing information and work symbolically.

2. Measurement planning. Describe an independent way to check the worked calculation without copying the exact procedure in the text. Explain how the proposed measurement separates a parameter estimate from an unobserved initial condition.

3. Failure analysis. Explain how an engineer could communicate the numerical result without presenting it as a certified equipment setting. Trace the proposed change through input, state and measured output rather than saying only that “the system may become unstable.”

4. Comparison. Relate this model to the following lesson, Routh–Hurwitz tests. Name one assumption shared by the two topics and one assumption requiring a different experiment or mathematical analysis.

Answer guidance and reproducibility checklist

A convincing solution starts with the original relation, defines the physical symbols, substitutes the scenario values with consistent units and checks the limit or sign of the result. For the sensitivity problem, use at least two plausible values and state which conclusion stays invariant. For the measurement problem, specify a real sensor signal and a time or frequency range. For the failure problem, identify the omitted dynamic or nonlinearity and predict the direction of at least one observable discrepancy. A comparison should not assume the next lesson solves the same question. Preserve input and output data, disclose parameter fitting, and independently recompute one representative point. Remember that numerical plausibility does not validate equipment tuning or safe deployment.

Lesson 4.2 — Routh–Hurwitz tests

Unit: Stability, characteristic equations and root locus. Prerequisites: differential equations, complex numbers, and the earlier lessons in the unit. Study: derive the relationship, reproduce the numerical case, investigate one changed assumption and complete the four problems. All numbers are synthetic and the models are not validated for hardware use.

Physical meaning and governing model

The Routh array converts a polynomial characteristic equation into sign information about root locations. For s³+a2 s²+a1 s+a0, with positive coefficients, the first column includes 1, a2, (a2 a1−a0)/a2 and a0. All entries must be positive for stability in this uncomplicated cubic case.

A sign change in the first column indicates right-half-plane roots, not just vague poor damping. Report the gain interval and boundary cases separately.

Governing relationship: For s³+3s²+4s+K, first column: 1,3,(12−K)/3,K

The formula should not be used until its input, output, initial state, sign convention and units have been declared. Distinguish a plant characteristic from the characteristic of a feedback interconnection. An algebraically valid transfer function is not an empirical model until its parameters and operating regime have supporting evidence. If an idealized element has no physical saturation or bandwidth, identify the missing implementation constraint before attempting design.

Worked example and interpretation

The conditions are K>0 and K<12, so 0

Check the magnitude against a relevant limit. In a continuous model, a pole measured in reciprocal seconds gives a time constant through its reciprocal; in a digital model, a pole must be inside the unit circle. A calculation of an output amplitude does not identify a suitable control input unless the actuator relationship is also known. For this particular example, copy the expression into a fresh worksheet, write the intermediate substitution with units and test whether the result changes in the expected direction when one parameter changes.

Limits, counterexamples and model evidence

Zero first elements and entire zero rows need special construction; naive division by zero silently invalidates a table. Coefficient rounding can alter marginal conclusions.

A useful counterexample changes one important feature at a time: initial state, operating point, saturation level, sensor filtering, delay or an unmodeled fast mode. The fact that a trace is well-fitted is not proof of predictive validity. A good validation experiment uses data withheld during fitting; it should also record the actual input, since the requested command may not equal the delivered signal. If evidence is missing, write a conditional conclusion instead of inventing a parameter or tolerance.

Application perspective — power-converter current loop

Inductor current dynamics and computational delay are often decisive, but the switching process, output filter and sensed current scale set additional limits. Frequency-response estimates must be confined to an appropriate small-signal region. Avoid interpreting a pedagogical model as guidance for connecting live electronics or tuning protection circuitry.

The experiment should define the performance metric before looking at a plot. Appropriate measures differ by task: a transient model can be checked by peak, delay and settling behavior; a frequency response by gain and phase; a state model by observable signals and sensitivity to starting conditions. A controller that improves one metric but violates current, force, thermal, bandwidth or latency limits is not an acceptable solution. Explain which additional measurement best distinguishes the model from its main competing explanation.

Problems for independent work

1. Computation and sensitivity. For K=15 construct the first column, count sign changes and compare against numerical roots as a separate verification. Show how the relevant parameter enters the governing equation. If the data do not determine a single number, list the missing information and work symbolically.

2. Measurement planning. Name a second performance variable that could become worse while the headline performance metric improves. Explain how the proposed measurement separates a parameter estimate from an unobserved initial condition.

3. Failure analysis. Which measured quantity would most directly separate an initial-condition error from a plant-parameter error? Give the signal units and observation window. Trace the proposed change through input, state and measured output rather than saying only that “the system may become unstable.”

4. Comparison. Relate this model to the following lesson, Root-locus geometry and gain. Name one assumption shared by the two topics and one assumption requiring a different experiment or mathematical analysis.

Answer guidance and reproducibility checklist

A convincing solution starts with the original relation, defines the physical symbols, substitutes the scenario values with consistent units and checks the limit or sign of the result. For the sensitivity problem, use at least two plausible values and state which conclusion stays invariant. For the measurement problem, specify a real sensor signal and a time or frequency range. For the failure problem, identify the omitted dynamic or nonlinearity and predict the direction of at least one observable discrepancy. A comparison should not assume the next lesson solves the same question. Preserve input and output data, disclose parameter fitting, and independently recompute one representative point. Remember that numerical plausibility does not validate equipment tuning or safe deployment.

Lesson 4.3 — Root-locus geometry and gain

Unit: Stability, characteristic equations and root locus. Prerequisites: differential equations, complex numbers, and the earlier lessons in the unit. Study: derive the relationship, reproduce the numerical case, investigate one changed assumption and complete the four problems. All numbers are synthetic and the models are not validated for hardware use.

Physical meaning and governing model

The root locus traces solutions of 1+K G(s)=0 while K varies over a specified nonnegative range. Branches originate at open-loop poles and end at zeros or infinity. The real-axis sign rule and asymptote angles are consequences of the argument condition, not decoration.

A gain increase shifts pole locations, but there is no blanket rule that more gain is safer or faster. The closed-loop natural frequency is √K and damping ratio 2/√K in this simple example.

Governing relationship: For G(s)=1/[s(s+4)], characteristic s²+4s+K=0

The formula should not be used until its input, output, initial state, sign convention and units have been declared. Distinguish a plant characteristic from the characteristic of a feedback interconnection. An algebraically valid transfer function is not an empirical model until its parameters and operating regime have supporting evidence. If an idealized element has no physical saturation or bandwidth, identify the missing implementation constraint before attempting design.

Worked example and interpretation

At K=4, roots are −2±0? The discriminant is 16−16=0, giving a repeated root −2. At K=9 the roots are −2±j√5 and the loop becomes more oscillatory without losing stability.

Check the magnitude against a relevant limit. In a continuous model, a pole measured in reciprocal seconds gives a time constant through its reciprocal; in a digital model, a pole must be inside the unit circle. A calculation of an output amplitude does not identify a suitable control input unless the actuator relationship is also known. For this particular example, copy the expression into a fresh worksheet, write the intermediate substitution with units and test whether the result changes in the expected direction when one parameter changes.

Limits, counterexamples and model evidence

Open-loop dynamics change with actuator bandwidth, load or operating point. Once a delay or additional pole is present, the root-locus of the simplified model can be misleading.

A useful counterexample changes one important feature at a time: initial state, operating point, saturation level, sensor filtering, delay or an unmodeled fast mode. The fact that a trace is well-fitted is not proof of predictive validity. A good validation experiment uses data withheld during fitting; it should also record the actual input, since the requested command may not equal the delivered signal. If evidence is missing, write a conditional conclusion instead of inventing a parameter or tolerance.

Application perspective — power-converter current loop

Inductor current dynamics and computational delay are often decisive, but the switching process, output filter and sensed current scale set additional limits. Frequency-response estimates must be confined to an appropriate small-signal region. Avoid interpreting a pedagogical model as guidance for connecting live electronics or tuning protection circuitry.

The experiment should define the performance metric before looking at a plot. Appropriate measures differ by task: a transient model can be checked by peak, delay and settling behavior; a frequency response by gain and phase; a state model by observable signals and sensitivity to starting conditions. A controller that improves one metric but violates current, force, thermal, bandwidth or latency limits is not an acceptable solution. Explain which additional measurement best distinguishes the model from its main competing explanation.

Problems for independent work

1. Computation and sensitivity. Find K for a damping ratio of 0.5 and compute the corresponding closed-loop pole pair. Show how the relevant parameter enters the governing equation. If the data do not determine a single number, list the missing information and work symbolically.

2. Measurement planning. Explain how an engineer could communicate the numerical result without presenting it as a certified equipment setting. Explain how the proposed measurement separates a parameter estimate from an unobserved initial condition.

3. Failure analysis. Identify the dimensional parameters and state what changes if all time units are reported in milliseconds rather than seconds. Trace the proposed change through input, state and measured output rather than saying only that “the system may become unstable.”

4. Comparison. Relate this model to the following lesson, Robustness under parameter variation. Name one assumption shared by the two topics and one assumption requiring a different experiment or mathematical analysis.

Answer guidance and reproducibility checklist

A convincing solution starts with the original relation, defines the physical symbols, substitutes the scenario values with consistent units and checks the limit or sign of the result. For the sensitivity problem, use at least two plausible values and state which conclusion stays invariant. For the measurement problem, specify a real sensor signal and a time or frequency range. For the failure problem, identify the omitted dynamic or nonlinearity and predict the direction of at least one observable discrepancy. A comparison should not assume the next lesson solves the same question. Preserve input and output data, disclose parameter fitting, and independently recompute one representative point. Remember that numerical plausibility does not validate equipment tuning or safe deployment.

Lesson 4.4 — Robustness under parameter variation

Unit: Stability, characteristic equations and root locus. Prerequisites: differential equations, complex numbers, and the earlier lessons in the unit. Study: derive the relationship, reproduce the numerical case, investigate one changed assumption and complete the four problems. All numbers are synthetic and the models are not validated for hardware use.

Physical meaning and governing model

Nominal stability is not robust stability. Model coefficients vary with temperature, aging, operating point and manufacturing tolerances. Even a small change can matter when poles sit near the imaginary axis. Explore an interval of plausible parameters and compare analytical inequalities with numerical eigenvalue checks.

A robust stability statement is not a robust performance statement. Slow decay and excessive actuator demand are failures even if all tested roots are left of zero.

Governing relationship: For s²+a*s+b, asymptotic stability requires a>0 and b>0

The formula should not be used until its input, output, initial state, sign convention and units have been declared. Distinguish a plant characteristic from the characteristic of a feedback interconnection. An algebraically valid transfer function is not an empirical model until its parameters and operating regime have supporting evidence. If an idealized element has no physical saturation or bandwidth, identify the missing implementation constraint before attempting design.

Worked example and interpretation

If a can fall between 0.2 and 0.6 and b between 1 and 4, all frozen-coefficient cases pass the second-order Hurwitz test, but their damping ratios vary from 0.05 to 0.3. Their transient overshoots could be very different.

Check the magnitude against a relevant limit. In a continuous model, a pole measured in reciprocal seconds gives a time constant through its reciprocal; in a digital model, a pole must be inside the unit circle. A calculation of an output amplitude does not identify a suitable control input unless the actuator relationship is also known. For this particular example, copy the expression into a fresh worksheet, write the intermediate substitution with units and test whether the result changes in the expected direction when one parameter changes.

Limits, counterexamples and model evidence

Time variation, correlated uncertainty and nonlinear friction are not fully captured by checking arbitrary frozen polynomials. Practical margins need validated uncertainty bounds.

A useful counterexample changes one important feature at a time: initial state, operating point, saturation level, sensor filtering, delay or an unmodeled fast mode. The fact that a trace is well-fitted is not proof of predictive validity. A good validation experiment uses data withheld during fitting; it should also record the actual input, since the requested command may not equal the delivered signal. If evidence is missing, write a conditional conclusion instead of inventing a parameter or tolerance.

Application perspective — power-converter current loop

Inductor current dynamics and computational delay are often decisive, but the switching process, output filter and sensed current scale set additional limits. Frequency-response estimates must be confined to an appropriate small-signal region. Avoid interpreting a pedagogical model as guidance for connecting live electronics or tuning protection circuitry.

The experiment should define the performance metric before looking at a plot. Appropriate measures differ by task: a transient model can be checked by peak, delay and settling behavior; a frequency response by gain and phase; a state model by observable signals and sensitivity to starting conditions. A controller that improves one metric but violates current, force, thermal, bandwidth or latency limits is not an acceptable solution. Explain which additional measurement best distinguishes the model from its main competing explanation.

Problems for independent work

1. Computation and sensitivity. Create a 4×4 parameter grid for a and b; report the least damping ratio and discuss why the grid is not a mathematical proof of all intermediate cases. Show how the relevant parameter enters the governing equation. If the data do not determine a single number, list the missing information and work symbolically.

2. Measurement planning. Which measured quantity would most directly separate an initial-condition error from a plant-parameter error? Give the signal units and observation window. Explain how the proposed measurement separates a parameter estimate from an unobserved initial condition.

3. Failure analysis. Construct a plausible situation in which the model prediction fails, naming the omitted state, nonlinearity or timing element. Trace the proposed change through input, state and measured output rather than saying only that “the system may become unstable.”

4. Comparison. Relate this model to the following lesson, Poles and internal versus external stability. Name one assumption shared by the two topics and one assumption requiring a different experiment or mathematical analysis.

Answer guidance and reproducibility checklist

A convincing solution starts with the original relation, defines the physical symbols, substitutes the scenario values with consistent units and checks the limit or sign of the result. For the sensitivity problem, use at least two plausible values and state which conclusion stays invariant. For the measurement problem, specify a real sensor signal and a time or frequency range. For the failure problem, identify the omitted dynamic or nonlinearity and predict the direction of at least one observable discrepancy. A comparison should not assume the next lesson solves the same question. Preserve input and output data, disclose parameter fitting, and independently recompute one representative point. Remember that numerical plausibility does not validate equipment tuning or safe deployment.

Unit 4 engineering project — Routh and root-locus gain interval

Research question. Can a compact model predict a second operating condition reliably, and what do its residuals mean? This is a synthetic computation-only assignment. No energized or safety-critical tests are required.

Case data and required work

Investigate characteristic s³+3s²+4s+K for K in {1,6,11,12,13,18}. Calculate a Routh first column, count sign changes and independently obtain numerical roots with Python or another transparent method. Separate a stable interval, marginal boundary and unstable interval. Discuss sensitivity near K=12.

Reproducible workflow

1. Prepare a model card containing the independent variable, measured output, units, initial conditions, parameter values and restrictions. Distinguish fixed known parameters from values estimated using the training observations. Set a reproducible numerical tolerance and record tool versions if code is used.
2. Write the parent differential equation or characteristic relationship. Derive one check by hand and one by independent numerical evaluation. Use enough precision internally to avoid arithmetic drift, but round reported results to the underlying input precision.
3. Build a graph that includes a physically meaningful axis range, a legend, a command trace, model output and any withheld validation output. If a logarithmic frequency axis is used, never include zero frequency as a point on that axis.
4. Investigate at least two plausible parameter alternatives. Report which output feature changes most and whether that sensitivity points to an important new measurement. If a parameter cannot be estimated uniquely, say why instead of choosing a convenient number.
5. Document the principal discrepancy between model and validation trace. State whether the discrepancy is likely random noise, a coherent time delay, a calibration offset, unmodeled dynamics or a regime change. Propose a specific follow-up experiment capable of distinguishing at least two possibilities.
6. Finish with a decision memo stating what the experiment establishes, what it cannot establish, and whether further evidence is needed before any real use.

Measurement and interpretation lens

Pole migration and stability boundaries: Vary gain over a declared bounded interval, calculate the characteristic polynomial at each tested point and compare roots with the predicted stability interval. Near a boundary, reduce the parameter step. Do not infer that no instability exists between two coarse samples.

The baseline computation should be viewed as a model prediction, not a design certificate. The next experiment should be chosen for information gain: an extra trace is helpful only if it can distinguish competing explanations. A practical model is sometimes deliberately simpler than the physical plant; defend simplification on the basis of bandwidth, residuals, risk, and data rather than convenience. Be explicit about initial states, since two identical inputs can yield different measured trajectories when starting conditions differ.

Required submitted artifacts

• A short derivation with dimensions, sign convention and limits.
• A machine-readable table of inputs and outputs, with units in headers and an explanation of any synthetic perturbation.
• Two labeled plots: an explanatory baseline plot and a separate uncertainty or hold-out prediction plot.
• An audit table separating assumptions, directly measured quantities, fitted parameters and conditions outside the model.
• One paragraph on model usefulness; one paragraph on the strongest counterexample or failure scenario.

Evaluation rubric and common mistakes

25% theory: valid differential equation, transfer function, characteristic equation or discrete update with correct signals. 25% arithmetic: independently checked numerical values and declared units. 20% evaluation: withheld-case prediction, uncertainty and an honestly labeled residual plot. 15% communication: reproducible data, readable graphs and explicit limitations. 15% safe experimental reasoning: no substitution of classroom simulation for real commissioning or protective design.

A common failure is tuning a parameter on every available point and then citing the same points as validation. Another is reporting an impressive percentage error despite a missing or nearly zero denominator. Do not hide cases with poor fit. If the model predicts instability or impossible inputs, document the finding and stop at simulation; do not seek to reproduce an unsafe condition on a real instrument.

Unit 4 assessment — Stability, characteristic equations and root locus

Requirements: Independently show calculations and state assumptions; answers should have units. These numbers are invented teaching examples, not equipment recommendations. Suggested duration: 2–3 hours plus an optional computational verification.

Part A — short quantitative checks

A1. Analyze the case s²+3s+2. State the relevant relationship, what physical variable is being calculated, and the result with correct dimensions. Explain one assumption required for the answer and one reason the same expression would be inappropriate for a more complicated plant.

A2. Analyze the case s³+3s²+4s+K. State the relevant relationship, what physical variable is being calculated, and the result with correct dimensions. Explain one assumption required for the answer and one reason the same expression would be inappropriate for a more complicated plant.

A3. Analyze the case s²+4s+9. State the relevant relationship, what physical variable is being calculated, and the result with correct dimensions. Explain one assumption required for the answer and one reason the same expression would be inappropriate for a more complicated plant.

A4. Analyze the case A=diag(−1,+2). State the relevant relationship, what physical variable is being calculated, and the result with correct dimensions. Explain one assumption required for the answer and one reason the same expression would be inappropriate for a more complicated plant.

Part B — synthesis problems

B1. Poles and internal versus external stability. Construct C=[1,0] for the sample A; explain why the second state is invisible and how a new sensor might help. Derive or select the governing equation; include at least one dimensional or limiting-case check. Contrast the method with the previous lesson in the unit. What extra measured variable would most reduce uncertainty? State when the calculation is valid and when it is not.

B2. Routh–Hurwitz tests. For K=15 construct the first column, count sign changes and compare against numerical roots as a separate verification. Derive or select the governing equation; include at least one dimensional or limiting-case check. Contrast the method with the previous lesson in the unit. What extra measured variable would most reduce uncertainty? State when the calculation is valid and when it is not.

B3. Root-locus geometry and gain. Find K for a damping ratio of 0.5 and compute the corresponding closed-loop pole pair. Derive or select the governing equation; include at least one dimensional or limiting-case check. Contrast the method with the previous lesson in the unit. What extra measured variable would most reduce uncertainty? State when the calculation is valid and when it is not.

B4. Robustness under parameter variation. Create a 4×4 parameter grid for a and b; report the least damping ratio and discuss why the grid is not a mathematical proof of all intermediate cases. Derive or select the governing equation; include at least one dimensional or limiting-case check. Contrast the method with the previous lesson in the unit. What extra measured variable would most reduce uncertainty? State when the calculation is valid and when it is not.

Part C — model criticism and experimental planning

C1. A student produces a smooth simulated plot with an error below 1% on synthetic training data and claims the controller can be installed. Identify at least four missing kinds of evidence, including one physical limit and one timing or sensing issue. Explain how these could change the outcome even without changing the algebra.

C2. Design a safe, offline-only demonstration for one of the four lessons. The demonstration should include at least three inputs, a falsifiable hypothesis, a baseline outcome and a deliberately perturbed parameter. State how you would decide which hypothesis survived and why the experiment does not authorize real-machine deployment.

C3. Compare two candidate models that fit a single step response equally well but have different unobserved dynamics. Propose a measurement or excitation that could distinguish them without relying on an unsafe extreme command. Discuss whether the problem is identification, sensing, control design or all three.

Instructor reference answers and marking guidance

A1 key. For s²+3s+2, the compact result is poles −1 and −2. Require the student's derivation and units, not just this answer. The explanation should note the idealized assumption—linear time-invariant dynamics, zero initial state where applicable, or the explicitly chosen sign and sampling convention. An unmodeled delay, actuation limit or sensor error can invalidate a direct application.

A2 key. For s³+3s²+4s+K, the compact result is stable for 0. Require the student's derivation and units, not just this answer. The explanation should note the idealized assumption—linear time-invariant dynamics, zero initial state where applicable, or the explicitly chosen sign and sampling convention. An unmodeled delay, actuation limit or sensor error can invalidate a direct application.